Angular Droop Control

• Angular droop control is a family of strategies that regulate active-power mismatch via phase-angle deviations, improving grid synchronization.
• It employs distinct formulations, including direct voltage-angle, power-factor-angle, and complex droop approaches, for precise control.
• Experimental and theoretical studies indicate that angular droop control offers scalable, inverse-optimal performance with enhanced transient response compared to frequency droop.

Searching arXiv for papers on angular droop control and closely related formulations. Angular droop control is a class of droop-based control strategies in which active-power imbalance is mapped primarily to a phase-angle deviation, or more broadly to an angle-related state, rather than only to a steady-state frequency deviation. In converter-based power systems, this idea appears in several distinct but related forms: direct voltage-angle droop for interconnected microgrids, distributed angular droop for phase-angle stabilization of voltage-source converters, power-factor-angle droop for cascaded inverters, and generalized complex-droop formulations in which angle and voltage-magnitude dynamics are coupled in a common complex state (Sivaranjani et al., 2018, Sivaranjani et al., 2018, Jouini et al., 2021, Sun et al., 2019, He et al., 2022). Across these formulations, the common objective is to exploit the strong dependence of active power on electrical angle while improving synchronization, transient response, power sharing, or nominal-frequency restoration relative to conventional frequency droop.

1. Conceptual scope and defining features

Angular droop control is not a single controller but a family of controllers centered on an angle variable. In direct voltage-angle droop for microgrid interconnections, the regulated quantity is the PCC phase angle, and the controller uses angle measurements from distribution PMUs to realize direct real-power control (Sivaranjani et al., 2018, Sivaranjani et al., 2018). In distributed converter angle stabilization, the control law acts on converter phase-angle dynamics and yields an active-power-to-angle droop relation at steady state (Jouini et al., 2021). In cascaded inverters, the drooped quantity can be the power factor angle, giving a decentralized angle-based law for both grid-connected and islanded operation (Sun et al., 2019). In complex droop control, the control variable is a “complex angle” combining $\ln v_k$ and $\theta_k$, so phase and amplitude are regulated jointly (He et al., 2022).

A central distinction from conventional frequency droop is the steady-state channel through which active-power mismatch is absorbed. Standard frequency droop leads to a stationary frequency deviation unless a secondary layer restores frequency. Angular droop instead aims to absorb active-power mismatch in a steady-state angle displacement while recovering exact nominal frequency, or equivalently to regulate the phase-angle trajectory directly rather than indirectly through $\dot\theta=\omega$ alone (Jouini et al., 2021, Jouini et al., 14 Jul 2025). This is why recent hardware-oriented work characterizes angular droop as a grid-forming strategy with “exact frequency synchronization” and “no stringent separation between primary and secondary frequency control” (Jouini et al., 14 Jul 2025).

The literature also uses adjacent formulations that are not direct $P$-$\theta$ droop in the narrow sense but remain dynamically relevant. Frequency-based droop always shapes angle because $\omega=\dot\theta$, and several papers explicitly interpret droop control as a way of shaping angular dynamics without introducing explicit synchronous-machine emulation (Liu et al., 2022, Mallada, 2016, Jiang et al., 2019). This suggests that the boundary between “angular droop” and “frequency droop with angular interpretation” is methodological rather than absolute.

2. Core mathematical formulations

In interconnected microgrids, the starting point is the nonlinear AC power-flow coupling among neighboring PCCs. For microgrid $j$, with neighbors $\mathcal N_j$, PCC voltage magnitude $V_j(t)$, phase angle $\delta_j(t)$, and $\theta_k$0, the injected real and reactive powers are written as (Sivaranjani et al., 2018)

$\theta_k$1

with $\theta_k$2 the line admittance. Direct angle droop then regulates real power through angle deviation. In angle-droop mode, the primary dynamics are (Sivaranjani et al., 2018)

$\theta_k$3

$\theta_k$4

where $\theta_k$5 and $\theta_k$6 represent local generation-load mismatch.

The converter-based distributed angular droop formulation of Schiffer and coauthors is explicit about the steady-state active-power-to-angle law. For converter $\theta_k$7, the optimal feedback law is (Jouini et al., 2021)

$\theta_k$8

and the induced steady state $\theta_k$9 satisfies

$\dot\theta=\omega$0

This is the precise $\dot\theta=\omega$1-$\dot\theta=\omega$2 droop relation: power mismatch is balanced by angle displacement, while $\dot\theta=\omega$3 yields zero steady-state frequency error (Jouini et al., 2021).

A different angle-based variant appears in power factor angle droop control for cascaded inverters. There the controller is (Sun et al., 2019)

$\dot\theta=\omega$4

$\dot\theta=\omega$5

with $\dot\theta=\omega$6. Here the drooped quantity is the angle of complex power rather than the bus voltage phase angle. The method is therefore angle-based, but the relevant angle is $\dot\theta=\omega$7, not $\dot\theta=\omega$8 itself (Sun et al., 2019).

In complex droop control, the state is

$\dot\theta=\omega$9

and the control law becomes (He et al., 2022)

$P$0

Its imaginary part gives the angle/frequency dynamics, while the real part regulates voltage amplitude. This formulation is directly connected to dispatchable virtual oscillator control and generalizes conventional $P$1-$P$2 and $P$3-$P$4 droop in a coupled nonlinear form (He et al., 2022).

3. Dynamic rationale and comparison with frequency droop

The principal rationale for angular droop is that real power is strongly coupled to electrical angle. In inverter-dominated microgrids, the direct use of angle can improve dynamic behavior relative to conventional frequency droop, which regulates real power only indirectly through $P$5 (Sivaranjani et al., 2018, Sivaranjani et al., 2018). The mixed angle/frequency droop papers state that angle droop gives increased stability, smaller frequency deviations, and faster dynamic response, whereas conventional frequency droop exhibits undesirable frequency deviations, chattering, synchronization loss, and degraded active-power sharing accuracy in inverter-rich microgrids (Sivaranjani et al., 2018).

The inverse-optimal angle-stabilization work makes the same contrast in steady-state terms. Standard first-order frequency droop,

$P$6

leads to stationary frequency error and therefore requires a separate secondary layer for exact restoration. Angular droop augments the power mismatch term with explicit angle feedback and drives the system to a synchronized steady state with $P$7 (Jouini et al., 2021).

A related but broader line of work studies dynamic droop as a way to shape angle/frequency behavior without direct angle feedback. The iDroop literature emphasizes that power flow is angle-based and that frequency is the derivative of angle, so any inverter law acting on $P$8 also acts through the angle-synchronization channel (Mallada, 2016, Jiang et al., 2019). In these models,

$P$9

and dynamic droop is used to improve transient synchronization, frequency nadir, or noise rejection while preserving the steady-state sharing of conventional droop (Mallada, 2016).

Another adjacent contribution is the “Equivalent Inertia Provided by Droop Control of Fast Frequency Regulation Resources,” which is not a direct angle-droop paper but explicitly interprets droop control as shaping angular behavior through frequency. With

$\theta$0

the method affects $\theta$1, and the paper shows that constant droop yields time-varying equivalent inertia, while a time-varying droop coefficient

$\theta$2

can emulate a constant inertia increment without introducing generator-like dynamics (Liu et al., 2022). This suggests a broader interpretation in which angular droop and frequency-based dynamic shaping lie on a continuum of angle-centered control design.

4. Distributed, switched, and complex generalizations

One major generalization of angular droop addresses measurement reliability. Direct angle droop depends on synchronized D-PMU angle measurements, which may be lost because GPS synchronization is unavailable. The mixed angle-and-frequency droop framework therefore defines a local switching variable

$\theta$3

with $\theta$4 for angle-droop mode and $\theta$5 for frequency-droop mode (Sivaranjani et al., 2018, Sivaranjani et al., 2018). When angle measurements are lost, the controller temporarily switches to frequency droop: $\theta$6

$\theta$7

The network is modeled as a nonlinear switched system with $\theta$8 global modes, and a distributed secondary controller is synthesized so that the nonlinear interconnection is locally $\theta$9-dissipative and therefore locally $\omega=\dot\theta$0-stable under arbitrary switching (Sivaranjani et al., 2018).

The state structure used for synthesis is

$\omega=\dot\theta$1

and the mode-dependent secondary law is

$\omega=\dot\theta$2

Because $\omega=\dot\theta$3 inherits the sparsity of the power-flow Jacobian $\omega=\dot\theta$4, each microgrid uses only measurements from immediate neighbors (Sivaranjani et al., 2018).

A different generalization is complex droop control, which merges angle and voltage regulation in a common complex state and is equivalent to dVOC (He et al., 2022). Here a non-nominal synchronous steady state is characterized through the dominant eigenvalue $\omega=\dot\theta$5 of the auxiliary closed-loop operator, with synchronized frequency

$\omega=\dot\theta$6

and drooped voltage magnitude

$\omega=\dot\theta$7

Under explicit parametric conditions, the system is almost globally asymptotically stable with respect to the set of non-nominal synchronous steady states (He et al., 2022). In this sense, complex droop extends angular droop from a purely phase-centered mechanism to a joint amplitude-phase synchronization law.

5. Stability, optimality, and scalability results

A notable feature of the angular droop literature is the diversity of analytical frameworks. The inverse-optimal angle-stabilization paper proves that distributed angular droop is the locally optimal controller for a nonlinear infinite-horizon cost functional (Jouini et al., 2021). The objective is

$\omega=\dot\theta$8

and the Lyapunov/value function combines a quadratic angle term with a cosine-based network potential. Under the security assumption

$\omega=\dot\theta$9

the induced steady state is locally asymptotically stable, and the controller is inverse optimal in a neighborhood of $j$0 (Jouini et al., 2021).

The same work provides a linearized coherence comparison with frequency droop. For uniform parameters, the angle-coherence metric satisfies

$j$1

for angular droop, whereas for frequency droop

$j$2

The paper concludes that angular droop admits the bound

$j$3

uniformly in network size and topology, while the frequency-droop expression deteriorates with sparse-graph scaling (Jouini et al., 2021). This supports the claim that angular droop has better scalability and disturbance rejection in large sparse networks.

Power factor angle droop yields a different but equally explicit small-signal structure. In islanded cascaded inverters, linearization gives

$j$4

so the small-signal matrix is a Laplacian, with eigenvalues

$j$5

The paper states that the system in islanded mode is stable and that the stability does not depend on load parameters (Sun et al., 2019). In grid-connected mode, the stability condition is

$j$6

which is independent of transmission line impedance (Sun et al., 2019).

Complex droop contributes a strong nonlinear result of a different kind. Under conditions involving $j$7, setpoint consistency, and bounded angle and voltage-ratio spread, the full nonlinear system is almost globally asymptotically stable with respect to the synchronous steady-state set $j$8 (He et al., 2022). This is among the strongest large-signal stability statements in the broader angular-droop family, though it is formulated for complex droop rather than pure $j$9-$\mathcal N_j$0 droop.

6. Practical implementation, experiments, and design issues

The most direct hardware evidence for angular droop appears in “Hardware test and validation of the angular droop control: Analysis and experiments” (Jouini et al., 14 Jul 2025). The controller is implemented on 15 kW low-voltage converter systems at KIT, each including a DC/DC boost converter, real-time control unit, three-phase two-level DC/AC converter, and output filter. The nominal values are

$\mathcal N_j$1

with initial angular-droop gains

$\mathcal N_j$2

The implemented angular droop law is

$\mathcal N_j$3

and, for uniform gains,

$\mathcal N_j$4

The paper studies two realizations: direct actuation of the modulation signal and indirect actuation through cascaded $\mathcal N_j$5-frame voltage and current loops. In the direct scheme,

$\mathcal N_j$6

whereas the indirect scheme generates a voltage reference first and then closes PI loops (Jouini et al., 14 Jul 2025).

The hardware experiments verify grid-forming capability after blackout-like initialization, power-to-angle droop under load steps, and plug-and-play synchronization in a two-converter system. In black start, the DC-link voltage reaches nominal $\mathcal N_j$7 V within about $\mathcal N_j$8 s, active power converges to $\mathcal N_j$9, and angle and frequency errors converge to zero (Jouini et al., 14 Jul 2025). After a load increase at $V_j(t)$0 s, active power overshoots to approximately $V_j(t)$1, settles at about $V_j(t)$2, and the induced steady-state angle obeys

$V_j(t)$3

which is the experimentally observed power-to-angle droop law (Jouini et al., 14 Jul 2025).

In the two-converter plug-and-play experiment, both converters synchronize to nominal frequency within about $V_j(t)$4 s, and active powers settle near setpoints. For power sharing, the paper derives the practical tuning condition

$V_j(t)$5

and if

$V_j(t)$6

then approximately

$V_j(t)$7

In the laboratory setup, this becomes $V_j(t)$8, and the symmetric choice $V_j(t)$9 yields approximate $\delta_j(t)$0 active-power sharing (Jouini et al., 14 Jul 2025).

The experiments also expose implementation constraints that are central to practical angular droop. First, because the controller integrates angle, digital implementation must handle unbounded angle growth; the paper uses modulo-$\delta_j(t)$1 wrapping of $\delta_j(t)$2 to avoid numerical precision loss (Jouini et al., 14 Jul 2025). Second, local clock drift causes angle drift and undesired active-power drift, so the hardware platform uses a shared optical-fiber master clock with synchronization accuracy of $\delta_j(t)$3 (Jouini et al., 14 Jul 2025). Third, nonzero line resistance and voltage mismatch create residual reactive-power circulation even in otherwise resistive-load experiments (Jouini et al., 14 Jul 2025). These observations indicate that angular droop is implementable, but only with careful treatment of synchronization, PWM realization, and network nonidealities.

7. Variants, controversies, and current boundaries

The principal controversy in angular droop is not whether angle-based control is useful, but which angle-centered formulation is appropriate for a given setting. Direct voltage-angle droop offers fast active-power regulation but depends on synchronized PMU angle measurements, making it fragile under GPS loss or spoofing (Sivaranjani et al., 2018, Sivaranjani et al., 2018). Mixed angle-and-frequency droop addresses this by switching to frequency droop during measurement loss, but the mode-dependent design scales as $\delta_j(t)$4 in the number of microgrids, which may become burdensome for larger systems (Sivaranjani et al., 2018).

Another boundary concerns the distinction between explicit angle droop and dynamic frequency droop. iDroop and related dynamic-droop designs are not direct $\delta_j(t)$5-$\delta_j(t)$6 laws, yet they are explicitly motivated by angle-frequency coupling and synchronization performance (Mallada, 2016, Jiang et al., 2019). This suggests that in low-inertia grids, much of the practical design space may be better described as “angular dynamics shaping” than by a strict dichotomy between angle droop and frequency droop.

A further issue is the interaction between active-power angle synchronization and reactive-power/voltage implementation. The 2025 wind-turbine comparison paper studies grid-forming converters whose active-power loop is governed by

$\delta_j(t)$7

which is an angle/frequency-power synchronization mechanism, while contrasting two reactive-power loop realizations: standard droop and droop-I (Chen et al., 26 Aug 2025). Its main conclusion is that droop-I can create structural high-frequency open-loop instability in purely inductive grids because the new characteristic equation

$\delta_j(t)$8

shifts high-frequency poles into the right-half plane under the ideal inductive assumption (Chen et al., 26 Aug 2025). This is not a critique of angular droop per se; rather, it shows that well-behaved angle/frequency synchronization at low frequency does not guarantee high-frequency converter stability if the $\delta_j(t)$9-$\theta_k$00 loop is poorly structured.

Finally, there is a modeling boundary between reduced angle-integrator theory and full converter dynamics. The inverse-optimal angular-droop proof is local and derived for a reduced converter model with fixed magnitudes and purely inductive lines (Jouini et al., 2021). The KIT experiments extend the idea to higher-order boost-converter and PWM dynamics, but they remain low-voltage and small-scale (Jouini et al., 14 Jul 2025). This suggests that the current state of the field is strongest in local nonlinear theory, structured switched-system synthesis, and laboratory validation, while large-scale, heterogeneous, and interoperability-rich deployments remain comparatively less settled.

Angular droop control therefore occupies a distinctive position in converter-based power-system control. It provides a direct and often physically transparent way to regulate active power through angle displacement, can achieve exact nominal-frequency synchronization in formulations that include explicit angle feedback, admits distributed and inverse-optimal interpretations, and has now been demonstrated experimentally in low-voltage hardware (Jouini et al., 2021, Jouini et al., 14 Jul 2025). At the same time, its dependence on synchronized angle measurements, sensitivity to implementation timing, interaction with voltage/reactive loops, and the diversity of its mathematical variants mean that it is best understood not as a single controller, but as an active research area spanning direct phase-angle droop, switched angle-frequency hybrids, power-angle formulations in cascaded inverters, and complex amplitude-phase synchronization laws (Sivaranjani et al., 2018, Sun et al., 2019, He et al., 2022).