Primary Frequency Response (PFR) in Modern Grids

• Primary Frequency Response (PFR) is the autonomous corrective action that immediately stabilizes grid frequency using inertia from rotating machines and fast droop control.
• PFR combines traditional synchronous generator responses with modern converter-based interventions from resources like PEVs to manage rapid load changes.
• MILP-based scheduling co-optimizes energy dispatch and reserve provisioning, reducing frequency deviations and operating costs in low-inertia, renewable-rich systems.

Primary Frequency Response (PFR) is the immediate, autonomous corrective action taken by power system resources to counteract sudden generation-load imbalances and stabilize grid frequency. PFR is essential for arresting the initial rate-of-change-of-frequency (RoCoF) and limiting the depth of the frequency nadir after a disturbance such as a generator trip or a rapid load change. This function historically has been provided by synchronous generators via their natural inertia and governor-controlled droop response, but with growing integration of fast-acting power electronic devices and distributed energy resources—including plug-in electric vehicles (PEVs)—contemporary power systems employ an increasingly heterogeneous portfolio of PFR resources, whose coordinated scheduling and market remuneration are active research topics (Fernandes et al., 2022).

1. Principles of Primary Frequency Response in Heterogeneous Systems

PFR operates on post-contingency timescales of sub-seconds to a few seconds. It incorporates two key physical mechanisms:

• (i) Inertia response: Kinetic energy from spinning synchronous machines is transferred to the grid in the first fractions of a second, instantaneously counteracting frequency decline.
• (ii) Governor-droop response: Synchronous machines and inverter-based resources (like PEVs) alter their active power output proportionally to frequency deviation, ramping up/down mechanical torque or injecting/extracting power to arrest and reverse frequency excursions.

In mixed-resource grids with high converter penetration and reduced synchronous inertia, fast-acting energy storage—especially PEV batteries—can supply PFR with rapid active power adjustments, effectively emulating governor droop through converter control (Fernandes et al., 2022). For PEVs, PFR provision is formulated as an additional control layer that dynamically modulates charging/discharging in response to frequency deviations, subject to the physical and market constraints imposed by battery state-of-charge and reservation bids.

2. Mathematical Formulation and Day-Ahead Scheduling

The co-optimization of commitment, dispatch, and PFR reserve scheduling is implemented via a mixed-integer linear program (MILP) (Fernandes et al., 2022). The objective function aggregates system operating costs, unserved demand penalties, spillage costs, frequency deviation penalties, and explicit compensation to PEVs for reserved PFR capacity and real-time deployment:

$$\min \Bigg\{ \sum_{t,g}(C_gp_{gt} + c_{gt}^{SU} + c_{gt}^{SD}) + \sum_{t,d}C^{UD}p_{dt}^{UD} + \sum_{t,g \in G^R}C^P s_{gt} + \sum_{t,d,k} (C^{UD}p_{dtk}^{UD,PR} - C^{\Delta F}\Delta f_{tk}) + \sum_{t,n,v} \Big[ Cc_{vnt}^{V,PR}c_{vnt}^{V,PR} + \sum_{k} Cp_{vnt}^{V,PR}p_{vntk}^{V,PR} \Big] \Bigg\}$$

Decision variables encode unit commitments, energy production, renewable spillage, load interruptions, frequency deviations, and for each PEV group, reserved and delivered PFR at every bus and time.

Key PEV-related constraints:

• Droop-control law: PEV PFR contribution is modeled as a virtual machine:

$p_{vntk}^{V,PR} = -\frac{1}{DR_v} \Delta f_{tk}$

subject to the maximum reserved capacity $c_{vnt}^{V,PR}$.

• Energy-power coupling: PEV response over a reserve duration $D^{PR}$:

$$e_{vntk}^{C,PR} = D^{PR} p_{vntk}^{V,PRC},\quad e_{vntk}^{D,PR} = D^{PR} p_{vntk}^{V,PRD}$$

• State-of-charge (SoC) dynamics: PEVs’ participation in PFR affects their SoC trajectory, requiring the model to track both routine charging/discharging and contingency-driven interventions.

3. Representation of System Inertia and Frequency Metrics

This MILP scheduling framework does not explicitly include a system-wide inertia constant $H$ or RoCoF constraints. Instead:

• Post-contingency frequency deviation $\Delta f_{tk}$ is a variable, minimized by penalty in the objective.
• The PFR delivered by both thermal and PEV resources is determined through the droop relationship with $\Delta f_{tk}$, where larger frequency deviations (i.e., lower effective inertia) necessitate higher PFR mobilization.
• There is no direct model of inertial response; to capture true RoCoF or ensure sufficient inertial margin, supplemental swing-equation-based constraints would be needed but are not present in this paper’s MILP.

4. Market Design for PEV-Provided PFR

PEV aggregators participate in the day-ahead reserve market by:

• Bidding capacity offers: Each group offers a standby PFR capacity $c_{vnt}^{V,PR}$ at price $Cc_{vnt}^{V,PR}$ (R\$/MW), receiving payment even if not deployed.
• Deployment remuneration: Deployed PFR is paid at price $p_{vntk}^{V,PR} = -\frac{1}{DR_v} \Delta f_{tk}$0 (R\$/MWh) for the delivered energy during a disturbance.

This structure allows the market-clearing process to co-optimize resource allocation, compensate PEV owners for providing flexibility, and schedule PFR reserves that reflect both system security and economic efficiency.

5. Case Study: Microgrid Demonstration and Quantitative Results

A 64-bus test microgrid featuring 5 generators, 32 loads, 2 PV sites, and 6 PEV connection points (3 groups) was simulated for 24 hourly intervals with full MILP coprocessing (Fernandes et al., 2022). Three scenarios were considered:

• No PFR: Unserved energy often reached over 165 MWh, frequency deviations exceeded 1 Hz, and total day-ahead cost was R\$179.9 million.
• Thermal-only PFR: Unserved energy dropped to 8.6 MWh, but load-shedding remained in night-time hours with frequency deviations up to 0.6 Hz.
• Thermal + PEV PFR: Unserved demand was eliminated, frequency deviations were constrained within 0.2 Hz, and system cost was halved compared to thermal-only, reaching R\$15.0 million.

These quantitative outcomes demonstrate that PEV-based PFR can provide non-trivial primary control and system value even when microgrid inertia is low or renewable variability is high.

6. Practical Considerations, Limitations, and Recommendations

PEV-based PFR provides sub-second response via converter control, effectively compensating for lost mechanical inertia as grids become more converter-dominated. However:

• Accurate forecasts of PEV SoC, plug-in times, and location are essential to schedule reliable PFR.
• Battery health and cycling costs are not incorporated in the model; actual deployment should consider wear-and-tear and degradation implications.
• Realistic field deployment would require validation of converter control response times, empirical droop gains, and SoC tracking accuracy.
• To guarantee system protection against frequency rate limitations, explicit RoCoF and frequency-nadir constraints based on inertia constants should be included in future market models.

Recommendations include aggregator-operator coordination for robust availability forecasts, explicit modeling of battery degradation costs, and adoption of field pilots to validate the modeled PEV response characteristics (Fernandes et al., 2022).

7. Significance and Integration with Modern Frequency Control

The integration of PEVs into primary frequency response scheduling frameworks expands the arsenal of fast-acting primary reserves, especially as traditional synchronous inertia declines. MILP-based co-optimization enables economically efficient and system-secure scheduling that leverages the flexibility of distributed storage resources. The demonstrated reductions in both system operating costs and frequency deviations, as well as the elimination of load shedding in high-renewable, low-inertia contexts, underscore the practical importance of incorporating PEV-provided PFR into operational and market structures for future-proofed frequency management (Fernandes et al., 2022).