Dynamic Complex-Frequency Control

• Dynamic complex-frequency control is a unified framework that represents amplitude and phase as a complex frequency, extending traditional frequency-based models.
• It enhances system performance by enabling ultra-fast transient damping, precise synchronization, and robust control in power converters, photonic devices, and oscillator networks.
• Control strategies like dVOC, η-control, and complex sliding mode leverage this framework for systematic tuning, improved inertia emulation, and tailored dynamic responses.

Dynamic complex-frequency control is an advanced paradigm in the analysis and regulation of dynamical systems—particularly electric power converters, photonic structures, and oscillator networks—where both the amplitude and phase evolution of system variables are treated jointly via a complex-valued frequency quantity. By extending conventional frequency-centric control to incorporate the amplitude rate of change as a fundamental variable, dynamic complex-frequency control provides a unified and multivariable framework for robust synchronization, improved transient response, and more flexible controller synthesis in a variety of domains, notably low-inertia power systems, resonant photonics, and coupled oscillator networks (Moutevelis et al., 2022, Domingo-Enrich et al., 2024, Bernal et al., 2024, Giordano et al., 8 Apr 2026, Krasnok, 22 May 2025).

1. Mathematical Foundation: Complex Frequency

Dynamic complex-frequency control is grounded in the precise analytic treatment of physical variables as complex phasors in a rotating (typically dq) reference frame. For a generic three-phase quantity $\tilde{u}(t) = u_d(t) + j u_q(t)$, the polar representation is $\tilde{u}(t) = u(t) e^{j\varphi(t)}$. Differentiation yields:

$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$

Here,

• $\tilde{s} = \sigma + j\omega$ is the complex frequency,
• $\sigma = \dot{u}/u$ is the per-unit amplitude sweep rate,
• $\omega = \dot{\varphi}$ is the instantaneous angular frequency.

This formalism generalizes to bus voltages ($\tilde{v}$), injected currents ($\tilde{i}$), and other phasor state variables, making it possible to capture not just frequency deviations but also amplitude dynamics—explicitly linking local power perturbations to both frequency and voltage trajectories (Moutevelis et al., 2022, Domingo-Enrich et al., 2024).

2. Unified Power–Frequency Relationships and Dynamics

By expressing instantaneous complex power as $S_h = v_h\,i_h^*$ and differentiating in the complex domain, the time derivative of power becomes:

$\dot{S}_h = (s_v + s_i)\,S_h$

When re-referenced to a local frame, the dynamics incorporate the action of controllers—such as phase-locked loops (PLLs), grid-forming or grid-following loops—directly into the evolution of $\tilde{u}(t) = u(t) e^{j\varphi(t)}$0. Every control architecture that manipulates amplitude ($\tilde{u}(t) = u(t) e^{j\varphi(t)}$1) and phase ($\tilde{u}(t) = u(t) e^{j\varphi(t)}$2) manifests as a real or imaginary translation in $\tilde{u}(t) = u(t) e^{j\varphi(t)}$3, leading to a taxonomy in which control actions are precisely mapped to their effect on amplitude and phase dynamics (Moutevelis et al., 2022, Domingo-Enrich et al., 2024, Bernal et al., 2024).

Complex frequency formulations support a two-time-scale separation: fast dynamics govern synchronization (complex frequency locking), while slow dynamics govern voltage magnitude stabilization. This decomposition enables tractable stability analysis: fast modes can be treated via linear spectral techniques, with slow modes addressed through reduced-order Lyapunov or frequency-domain passivity arguments (He et al., 2022, Domingo-Enrich et al., 2024).

3. Frameworks for Dynamic Complex-Frequency Control

Several control methodologies harness the complex-frequency structure. Representative frameworks include:

• Dynamic Complex Droop / dVOC: Generalizes scalar frequency-power and voltage-reactive droop to a vectorial (complex) error law. Static gains in conventional droop can be replaced with dynamic (frequency-dependent) transfer functions, e.g., $\tilde{u}(t) = u(t) e^{j\varphi(t)}$4 (synthetic inertia plus droop) and $\tilde{u}(t) = u(t) e^{j\varphi(t)}$5 for voltage, supporting sophisticated dynamic shaping and precise inertia emulation (Domingo-Enrich et al., 2024, He et al., 2022).
• η-Control for IBRs: Seeks to constrain both $\tilde{u}(t) = u(t) e^{j\varphi(t)}$6 and $\tilde{u}(t) = u(t) e^{j\varphi(t)}$7 simultaneously, driving the terminal CF to $\tilde{u}(t) = u(t) e^{j\varphi(t)}$8. This approach replaces separate frequency and voltage regulation loops with a single complex-valued integrator acting on both error components, tightly coordinating active and reactive power feedbacks (Bernal et al., 2024).
• Complex-Frequency Sliding/Feedforward Control in Oscillator Networks: Extends Kuramoto-like phase networks to complex-valued systems, allowing for exact phase, unit-modulus, and prescribed-frequency locking using switched feedforward and sliding mode strategies, with explicit finite-time convergence guarantees even under network heterogeneity (Giordano et al., 8 Apr 2026).

These frameworks treat amplitude and phase as coupled but distinctly controllable dynamical quantities, often leading to improved transient damping, enhanced synchronization robustness, and systematically tunable trade-offs between inertia, damping, and voltage/frequency response.

4. Taxonomy, Implementation, and Parameterization

The dynamic complex-frequency concept enables a structured taxonomy for converter and oscillator control:

Control Layer	Key Effect on $\tilde{u}(t) = u(t) e^{j\varphi(t)}$9 / Dynamics	Typical Parameters
Inner Current PI	Real shift in $$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$0 (amplitude dynamics)	$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$1
PLL (Grid-Following)	Imaginary shift in $$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$2 (frequency dynamics)	$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$3
Complex Droop / dVOC (Grid-Forming)	Complex $$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$4: coupled freq–volt dynamics	$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$5
P–f Droop & Virtual Synchronous Machine	Real + imaginary dynamic shifts, inertia/damping	$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$6
Sliding / SMC (Oscillator Network)	Feedforward/Sliding law in modulus and phase	$$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$7 (sliding gains)

In practice, optimal controller synthesis and tuning require translating system-level objectives—transient damping, inertia emulation, steady-state droop—into explicit choices of $$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$8, $$\dot{\tilde{u}} = \left( \frac{\dot{u}}{u} + j\,\dot{\varphi} \right ) \tilde{u} \equiv \tilde{s}\,\tilde{u}$$9, $\tilde{s} = \sigma + j\omega$0, $\tilde{s} = \sigma + j\omega$1, $\tilde{s} = \sigma + j\omega$2, etc. Guidelines include matching controller phase lag to network impedance angle, ensuring passivity and positive-realness of $\tilde{s} = \sigma + j\omega$3, and verifying synchronization margins through spectral and Lyapunov techniques (Moutevelis et al., 2022, Domingo-Enrich et al., 2024, He et al., 2022).

5. Application Domains and Dynamic Performance

Electric Power Systems

Dynamic complex-frequency control has been validated in both IEEE 9-bus, WSCC 9-bus, and full-scale (1479-bus Irish transmission grid) simulations. Key findings:

• Superior disturbance rejection: CF-based η-control reduces cumulative voltage/frequency deviation indices ($\tilde{s} = \sigma + j\omega$4 by up to 98%) compared to conventional droop/PI/virtual inertia across a range of contingencies.
• Ultra-fast transient damping: Voltage/frequency oscillations can be damped in sub-50 ms, as opposed to typical 300 ms ranges for classic methods.
• Grid-forming invariance: Decoupled regulation of $\tilde{s} = \sigma + j\omega$5 and $\tilde{s} = \sigma + j\omega$6 is preserved even under tight $\tilde{s} = \sigma + j\omega$7–$\tilde{s} = \sigma + j\omega$8 coupling and highly variable $\tilde{s} = \sigma + j\omega$9 network conditions (Bernal et al., 2024, Domingo-Enrich et al., 2024).

Key sensitivity analyses demonstrate robust performance even under current limiting, communication delay, measurement noise, and EMT (electromagnetic transient) regimes, with appropriate retuning as needed.

Photonics and Resonant Systems

Dynamic complex-frequency control in resonant photonic devices leverages real-time engineering of pole-zero constellations in the complex frequency plane. By steering excitation or system zeros along iso-amplitude contours, it becomes possible to:

• Achieve full $\sigma = \dot{u}/u$0 phase control at strictly fixed amplitude, thus eliminating amplitude-to-phase conversion and distortion intrinsic to conventional modulators.
• Exploit topological phase accumulation (Cauchy's argument principle), thereby conferring quantized phase robustness to environmental or fabrication fluctuations (Krasnok, 22 May 2025).

Oscillator Networks and Synchronization

Complex-valued extensions of Kuramoto synchronization allow for separate and jointly controlled amplitude and phase locking. Control laws employing sliding-mode or switched feedforward schemes ensure finite-time convergence to prescribed frequency and phase with robustness to oscillator heterogeneity, without the need for spectral gain tuning (Giordano et al., 8 Apr 2026).

6. Stability, Robustness, and Practical Tuning

Theoretical guarantees for stability rely on positive-real (passivity) conditions for controller–network interconnection and Lyapunov or spectral conditions in the fast–slow timescale decomposition. Controller gains should be chosen to keep imaginary poles above system bandwidths and ensure that real translations of $\sigma = \dot{u}/u$1 do not induce amplitude instability. Tunings can be mapped directly onto frequency-domain performance (ROCOF, damping, inertia) and practical implementation constraints (delays, noise, current limits) (Moutevelis et al., 2022, Domingo-Enrich et al., 2024, Bernal et al., 2024).

The taxonomy provided by the complex-frequency framework enables systematic assignment of control roles across all loops, mirroring block-diagram approaches for classical multi-machine systems but recast in the space of complex frequency.

7. Outlook and Cross-Domain Relevance

Dynamic complex-frequency control establishes a rigorous and unifying analytical and control framework for multi-variable, multiscale, and heterogeneous networked systems, from high-voltage power networks to precision photonic and coupled quantum devices. The theoretical apparatus supports the integration of dynamic transfer functions, topological robustness, and distributed control, paving the way for systematic advances in the performance and resilience of next-generation engineered systems (Domingo-Enrich et al., 2024, Krasnok, 22 May 2025, Giordano et al., 8 Apr 2026).

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References:

• (Moutevelis et al., 2022) Taxonomy of Power Converter Control Schemes based on the Complex Frequency Concept
• (Domingo-Enrich et al., 2024) Dynamic Complex-Frequency Control of Grid-Forming Converters
• (Bernal et al., 2024) A Complex Frequency-Based Control for Inverter-Based Resources
• (Giordano et al., 8 Apr 2026) Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework
• (Krasnok, 22 May 2025) Topological Phase Control via Dynamic Complex Pole-Zero Engineering
• (He et al., 2022) Complex-Frequency Synchronization of Converter-Based Power Systems