Primary Frequency Response in Power Systems

• Primary Frequency Response is the fast-acting process that adjusts real power through governors, inverters, and responsive loads to stabilize grid frequency after disturbances.
• It is mathematically formulated via augmented swing equations and lag-based models to precisely capture metrics like RoCoF, frequency nadir, and steady-state deviation.
• Its implementation in modern, low-inertia grids involves adaptive reserve sizing, market integration, and coordinated distributed energy resources for enhanced reliability.

Primary Frequency Response (PFR) is the vital, fast-acting process by which an electric power system stabilizes frequency immediately following a significant disturbance, such as generator or load outages. It operates via automatic real-power adjustments from synchronous machine governors, inverter-based resources, responsive loads, and increasingly, distributed energy resources aggregated by virtual power plants. PFR is essential for maintaining frequency security—limiting both rate-of-change-of-frequency (RoCoF) and the frequency nadir—buying crucial time for slower secondary controls and preventing involuntary load shedding or system collapse. Its implementation, mathematical characterization, control, and system integration span classical swing-equation dynamics, optimization-based market designs, explicit modeling of resource-limited assets (e.g., batteries, EVs), and modern frequency-stability constraints for operation in high-renewable, low-inertia contexts.

1. Mathematical Formulation of Primary Frequency Response

PFR is governed by the classic swing equation, augmented to include aggregate inertia ($H$), damping ($D$), net contingency power imbalance ($P_{\rm cont}$), and the trajectory of real-time primary response $p(t)$: $$\frac{d\Delta f(t)}{dt} + \frac{D'}{2H} \Delta f(t) = \frac{1}{2H} [p(t) - P_{\rm cont}]$$ where $\Delta f(t)$ is the grid frequency deviation, $H$ is system inertia, $D'$ the load relief (often $D' = D \cdot P_{\text{load}}$), and $p(t)$ the composite primary response including both conventional governor actions and inverter-based resource contributions (Susanto et al., 2020).

Standard closed-form solutions allow explicit calculation of:

• RoCoF (initial): $-P_{\rm cont}/(2H)$,
• Frequency nadir: analtytically via transient response formulas,
• Steady-state deviation: $(PFR - P_{\rm cont})/D'$.

The model generalizes to multi-speed PFR by decomposing $p(t)$ as the sum of fast (e.g., batteries, inverter-based systems) and slow (conventional turbine governor) providers, each modeled as a first-order lag with distinct time constants ($\tau_1$, $\tau_2$). The combined frequency response can then be approximated by a single-lag with fitted composite parameters for tractable, accurate assessment of RoCoF, nadir, and steady-state, enabling direct use in frequency-constrained optimization (Susanto et al., 2020).

2. Classical and Advanced Control Mechanisms

Governor and Droop Mechanisms

The core generator PFR mechanism is the proportional droop: $\Delta P_{\mathrm{gov}} = -\frac{1}{R} \Delta f$ with $R$ the droop slope, subject to deadband ($f_{dz}$) and ramping limitations: $$\Delta P_g(\Delta f) = \begin{cases} 0,&|\Delta f|<f_{dz}\ -\frac{1}{R}(\Delta f - \operatorname{sgn}(\Delta f) f_{dz}),&|\Delta f|\ge f_{dz} \end{cases}$$ Explicit modeling of dead-zone effects (nonlinearity) in system optimization is critical for avoiding over- or under-procurement of reserve and is addressed in convex chance-constrained OPF frameworks (Chertkov et al., 2017).

Load-side and Demand Response Participation

Load-side primary response employs frequency-responsive demand (fast controllable loads): $d_j = K_p \omega_j$ (Guo et al., 2018). Distributed and decentralized implementations modeled via primal–dual algorithms solve optimal load control problems, guaranteeing robust and fair allocation (Zhao et al., 2013, Kasis et al., 2016). Adaptive demand response, leveraging local measurements and real-time estimation of inertia (e.g., via solar irradiance sensors), supports dynamic allocation of frequency support in high-renewables scenarios (You, 2020).

Converter-based and Fast Frequency Response

Grid-forming and grid-following inverters are equipped with programmable droop (and optional virtual inertia) controllers, supporting PFR with tunable time constants and response profiles (Collados-Rodriguez et al., 12 Nov 2024). Their participation is modeled as first-order (or higher) lags, and appropriate parameter selection can transition system response from classical under-damped second-order (synchronous) to nearly first-order, offering design freedom in shaping frequency trajectories and improving nadir/RoCoF trade-offs (Kenyon et al., 2021, Collados-Rodriguez et al., 12 Nov 2024).

Hybrid schemes, including mode-switching wind turbine controls with formal safety guarantees (region-of-safety/barrier certificates), further extend PFR provision under operational constraints (Zhang et al., 2018).

3. Primary Frequency Response in Modern, Low-Inertia Grids

High penetration of non-synchronous renewables dramatically reduces system inertia, increasing both RoCoF and depth of frequency nadir. Analytical and simulation studies consistently conclude:

• Fast-acting PFR via batteries, supercapacitors, responsive loads, and aggregated distributed resources is essential for mitigating degradation (You, 2021, You, 2021).
• Multi-speed PFR composition is necessary: batteries (sub-second response) are coupled with conventional governors (seconds) and demand response (hundreds of ms to seconds).
• Explicit design targets include: RoCoF limits (often $<0.5$ Hz/s), minimum nadir (e.g., $>59.0$ Hz for North America), steady-state error suppression.
• Proper aggregation and optimization of distributed resource parameters (inertia, droop, delay) are addressed through virtual power plant (VPP) frameworks, which also support market-based procurement and co-optimization of energy and ancillary services (Feng et al., 6 Mar 2025).

A highly recommended design approach is to layer resources with staggered time constants, ensuring the coordinate arrest of RoCoF, nadir, and steady-state deviation (You, 2021, You, 2021).

4. Market Integration, Reserve Sizing, and Scheduling

Adaptive Reserve Requirements

In low-inertia grids, real-time inertia estimation (via PMU/FNET frequency measurements) enables time-varying and adaptive procurement of frequency response reserves (FRR) to match instantaneous system needs, dramatically improving both security and economic efficiency (You, 2021). The relationship between system inertia and required FRR is strongly non-linear: as inertia falls, required FRR rises rapidly, and the sizing of reserves can be reduced by over 40% with adaptive policies.

Co-optimization and Market Mechanisms

Joint energy-inertia-PFR market frameworks, incorporating VPP aggregation and latency/delay modeling, enable system cost minimization under explicit frequency security constraints (RoCoF, nadir, quasi-steady-state frames) (Feng et al., 6 Mar 2025). PFR, inertia, and droop factors are treated as explicit market products, with clearing via security-constrained unit commitment (SCUC) and economic dispatch (SCED), and marginal pricing for all services. This enables distributed and inverter-based assets, formerly not visible for system frequency procurement, to participate fully and be remunerated for their frequency-support capabilities.

Day-ahead and real-time scheduling models (MILP formulations) now routinely embed closed-form PFR security constraints, including those based on the dual-speed SFR model for mixed fast/slow response (Susanto et al., 2020, Fernandes et al., 2022). The inclusion of electric vehicles as frequency-responsive storage, co-optimized for energy and ancillary services, demonstrates significant reliability and cost benefits (Tao et al., 14 Dec 2025, Fernandes et al., 2022).

5. New Resource Classes and Implementation Experience

Electric Vehicles and Vehicle-to-Grid

Heavy-duty and plug-in electric vehicles, coordinated under V1G and V2G control, offer scalable, rapid, and tunable primary frequency support (Tao et al., 16 Dec 2025, Tao et al., 14 Dec 2025, Fernandes et al., 2022). Design trade-offs between response magnitude, battery wear, fleet availability, and SoC constraints are critical. Simulation studies of the California grid confirm that V2G-enabled fleets can improve frequency nadir up to 0.5 Hz and halve settling times compared to baselines. Charging strategies (e.g., constant-power vs immediate) significantly affect both grid and battery-side outcomes.

Demand-side and Distributed Resources

Load-side participation, with optimal and passivity-based control guarantees, provides a robust means of increasing effective damping and droop, lowering both transient excursions and steady-state frequency deviations (Zhao et al., 2013, Kasis et al., 2016). Optimal placement and tuning of fast frequency reserves are highly locational, with performance and stability strongly dependent on network delay and phase margins—remote fast inverters can degrade electromechanical mode damping if naively configured (Misyris et al., 2022).

Wind and Photovoltaic Generators with Inertia and Primary Response Emulation

Doubly-fed induction generator (DFIG) wind turbines and PV inverters can be equipped with fast inertial and droop emulators. ANN-based and hybrid supervisory controllers further enhance performance by coordinating dual-speed resources and preempting governor delays (Morovati et al., 2020, Zhang et al., 2018). Analytical phase-wise models quantify time-varying contributions to effective system inertia and damping, supporting controller synthesis for high-renewable contexts.

6. System Implications, Guidelines, and Limitations

System-level Metrics

Table: Key Analytical Expressions for PFR Metrics (Single-lag Dual-speed Model, (Susanto et al., 2020))

Metric	Expression
RoCoF	$-\frac{P_{\rm cont}}{2H}$
Steady-state deviation	$\frac{PFR - P_{\rm cont}}{D'}$
Nadir	Closed-form as function of $H$, $D'$, $P_{\rm cont}$, $PFR$, $\tau$

• Resource Coordination: Effective PFR in high-renewable grids mandates multi-level resource participation, precise deadband/droop setting, and delay-aware aggregation (You, 2021, Susanto et al., 2020).
• Market and Regulatory Evolution: System codes increasingly require explicit procurement of sub-second frequency response and inertia, with price signals reflecting resource speed, quality, and location (Ding et al., 2019, Feng et al., 6 Mar 2025).
• Technical Challenges: Accurate SoC management, real-time inertia diagnostics, market integration of distributed assets, and detailed consideration of communication and actuation delays are critical for maintaining frequency security margins.

7. Outlook and Research Directions

• Increasing DR, EV, and storage participation: As these resources become ubiquitous, system operators must continue to refine their integration for PFR, leveraging real-time telemetry, adaptive control, and robust optimization.
• Advanced modeling for stability and security: Explicit multi-speed, deadband, and delay representations in both real-time operation and market-surveillance systems are now essential.
• Hybrid and hierarchical control strategies: Combining centralized and decentralized, resource-class-specific controllers—validated in full-network simulations—is crucial for ensuring secure and economically optimal frequency response over a wide range of operational and contingency scenarios.

Collectively, primary frequency response is undergoing a transformation from a legacy synchronous-machine-dominated, governor-led paradigm to a diverse, multi-speed, market-integrated system with significant distributed and inverter-based resource participation. This evolution is supported by rigorous modeling, optimization, and controller synthesis frameworks, as well as the explicit codification of frequency-security constraints in both operational and market processes (Susanto et al., 2020, You, 2021, You, 2021, Tao et al., 14 Dec 2025, Tao et al., 16 Dec 2025, Fernandes et al., 2022, Feng et al., 6 Mar 2025).