Complex Frequency

1. Problem Statement

Traditional definitions of frequency in power systems, such as the one provided in IEEE Std IEC/IEEE 60255-118-1, define frequency solely based on the time derivative of the voltage phase angle ($\dot{\theta}$). While adequate for steady-state conditions where voltage magnitude is constant, this definition becomes problematic during transient conditions where voltage magnitude ($V$) varies significantly.

• Limitation: The standard definition fails to decouple the effects of phase angle variations from voltage magnitude variations.
• Consequence: In modern power systems with low inertia and high penetration of converter-interfaced generation (CIG), voltage magnitude fluctuations are frequent. Relying solely on phase derivatives can lead to inaccurate frequency estimation, numerical spikes during faults, and a lack of insight into the coupling between active/reactive power and frequency dynamics.

2. Methodology

The paper proposes a novel theoretical framework based on Complex Frequency, treating frequency as a complex quantity $\bar{\eta} = \varrho + j\omega$.

• Definition:
  • Imaginary Part ($\omega$): Represents the standard angular frequency deviation relative to a synchronous reference frame ($\dot{\theta}$).
  • Real Part ($\varrho$): Represents the rate of change of the natural logarithm of the voltage magnitude ($\dot{u}$, where $u = \ln V$). This term captures the dynamics of voltage magnitude variations.
• Mathematical Derivation:
  • The author starts with the instantaneous complex power injection equation $\bar{s} = \bar{v} \circ \bar{i}^*$, where $\bar{v}$ and $\bar{i}$ are Park's vectors (dynamic phasors) rather than steady-state phasors.
  • By differentiating the power flow equations with respect to time and applying the chain rule, the paper derives a fundamental differential equation linking the time derivative of complex power ($\dot{\bar{s}}$) to the complex frequency vector ($\bar{\eta}$).
  • Key Equation (32):
    $$\dot{\bar{s}} - \bar{s} \circ \bar{\eta} = \bar{S} \bar{\eta}^*$$
    Where $\bar{S}$ is a matrix derived from the network admittance and voltage magnitudes. This equation establishes a direct, exact link between power injections and complex frequency without assuming constant voltage magnitude.
• Approximations: The paper also derives simplified expressions for high-voltage transmission systems (neglecting resistance) and lossy networks, showing how reactive power can influence frequency in resistive networks.

3. Key Contributions

1. Introduction of Complex Frequency: A rigorous definition of frequency as a complex number $\bar{\eta} = \varrho + j\omega$, where the real part accounts for voltage magnitude dynamics.
2. Exact Differential Formulation: Derivation of Equation (32), which describes the link between complex power variations and complex frequencies in transient conditions without simplifying assumptions (other than neglecting electromagnetic transients).
3. Generalization of Existing Models: The proposed framework generalizes the "Frequency Divider Formula" (FDF) and other existing frequency estimation methods. It shows that FDF is a specific approximation of the more general complex frequency theory.
4. Special Case Analysis: The paper analyzes specific device behaviors:
   • Constant Power: Shows how power injections constrain the complex frequency.
   • Constant Impedance: Demonstrates that these loads are passive regarding frequency; they "take" the frequency imposed by the grid rather than setting it.
   • Constant Current/Power Factor: Derives relationships showing how current dynamics relate to bus frequency.
5. Application to Control and Estimation: The framework provides a method to estimate frequency parameters (like voltage-dependent load exponents) and design controllers that utilize both active and reactive power for frequency regulation.

4. Results

The paper validates the theory using the WSCC 9-bus system and the Dome simulation software.

• Magnitude Comparison: Simulations confirm that while $|\omega| \gg |\varrho|$ (frequency deviation is usually larger than voltage magnitude rate of change), the term $\varrho$ is non-negligible and crucial for accurate modeling, especially near disturbances.
• Robustness vs. PLL:
  • Standard Phase-Locked Loops (PLLs) exhibit significant numerical spikes and delays during faults and load disconnections.
  • The Complex Frequency approach (solving Eq. 32) provides a smooth, continuous, and "exact" estimation of frequency dynamics, avoiding the artifacts seen in PLL-based estimations.
• Load Parameter Estimation: The method successfully estimates the exponents ($\gamma_p, \gamma_q$) of Voltage Dependent Loads (VDL) with higher precision than empirical formulas, even in networks with high R/X ratios (distribution-like conditions).
• Control Effectiveness: A case study on Converter-Interfaced Generators (CIG) demonstrated that a control scheme regulating frequency via both active and reactive power (exploiting the coupling revealed by the complex frequency theory) is more effective at damping frequency oscillations than schemes using only active power.

5. Significance

• Theoretical Advancement: The paper resolves the ambiguity of defining frequency in non-sinusoidal or transient conditions by unifying angle and magnitude dynamics into a single complex quantity.
• Low-Inertia Systems: As power systems transition to inverter-dominated grids with low inertia, voltage magnitude fluctuations become more pronounced. The complex frequency concept provides the necessary tools to analyze and control these systems accurately.
• Control Design: It opens new avenues for frequency control strategies. Specifically, it suggests that reactive power control can be used to regulate frequency in resistive networks, a counter-intuitive but mathematically proven insight derived from the real part of the complex frequency.
• State Estimation: The framework offers a more robust alternative to PLLs for real-time frequency monitoring, particularly during severe transients where traditional methods fail.

In conclusion, Federico Milano's paper redefines the understanding of frequency in power systems, moving from a purely angular concept to a complex one that inherently accounts for voltage magnitude dynamics, thereby offering superior analytical and control capabilities for modern, low-inertia power grids.