Frequency security certification based energy shaping control for wind power participation in primary frequency regulation

1 Introduction

Frequency security in a power system depends not only on the amount of active power reserve but also on how quickly that reserve is delivered after a disturbance. In conventional grids, synchronous-machine inertia and turbine-governor response provide a natural first line of defense. Wind turbines connected through converters do not contribute in the same way because the rotor speed is decoupled from the grid frequency and maximum power point tracking (MPPT) normally keeps the turbine close to its aerodynamic optimum. As the synchronous share decreases, this decoupling reduces the apparent inertia and primary frequency response available to the system (Wu et al., 2018; Bao and Zhang, 2022; Ramesh et al., 2021).

A large body of work has, therefore, investigated frequency support from wind turbines. Virtual-inertia and rate of change of frequency (RoCoF) loops extract rotor kinetic energy during the first few seconds, whereas droop loops increase active power in proportion to frequency deviation (Shin et al., 2024; Morren et al., 2006; Wang et al., 2013). Power reserve may also be created through rotor overspeed or pitch deloading, and grid-forming control can provide converter-level inertia and damping for fully rated or direct-drive turbines (Ghosh et al., 2016; Miao et al., 2015; Abouzeid et al., 2019). These controls improve the first frequency nadir, but the kinetic-energy source is finite. If rotor speed is allowed to fall too far, the turbine may enter a low-speed operating region in which stall risk, prolonged mechanical loading, and protection-triggered termination become important (Bao et al., 2020; Wang and Wu, 2018; Zhao et al., 2025; Han et al., 2026). The energy released during support must also be restored. When many turbines recover over a similar interval, their combined active-power reduction may cause a secondary frequency dip (SFD), even though the first nadir has been improved (Huang et al., 2025; Ren et al., 2025; Zhao et al., 2025; Han et al., 2026).

Wind–storage coordination is a practical way to relieve this recovery burden. Battery energy-storage systems (BESSs), supercapacitors, and hybrid storage units can share the fast active-power response, smooth the transition from support to recovery, and improve the post-disturbance trajectory (Zhao et al., 2026; Zhang et al., 2024; Li et al., 2022; Yu et al., 2024). Recent studies also account for fatigue loading, state of charge (SOC), turbine health, and the allocation of regulation duties among multiple wind farms (Sun et al., 2026; Shu et al., 2024; Liu et al., 2026; Ding et al., 2022). Probabilistic and data-driven methods have been used to assess frequency-regulation capability under uncertain wind conditions (Wu et al., 2026; Prakash et al., 2022). In parallel, grid-forming wind turbines and offshore wind farms connected through HVDC or MTDC grids have broadened the design space for wind-based frequency support (Hu et al., 2024; Zhang et al., 2025a; b; Kou et al., 2018; Bianchi and Domínguez-García, 2016). Despite their different turbine technologies and grid interfaces, all these studies have to reconcile fast support with the recovery burden created by kinetic-energy extraction.

Existing studies address the main elements of the problem, but the coupling between them is often weak. A wind turbine that injects extra power immediately after a disturbance creates an energy debt that reappears during rotor speed recovery. In many existing schemes, the initial support command, the storage sharing rule, and the recovery trajectory are tuned as separate modules. Such a structure is convenient for implementation, but it does not explicitly ask whether the system has enough frequency margin to tolerate the recovery absorption that follows a large early injection. In addition, adaptive droop or virtual inertia gains are often updated from local turbine states, whereas the admissible support level should also depend on the system security margin. This motivates a supervisory formulation in which fast support and recovery are treated as a constrained energy shaping process rather than two loosely connected stages.

Table 1 places the proposed formulation in relation to several representative research streams. The comparison is intentionally organized by the role assigned to rotor energy recovery. In the proposed frequency security certification-based energy shaping control (FS-ESC) framework, recovery is not appended as a task scheduled after the event; it is modeled as an absorption process seen by the grid, and its admissible magnitude is checked online against the available frequency margin.

TABLE 1. Position of the proposed method relative to representative studies.

Based on this idea, a FS-ESC method is developed for primary frequency regulation by wind farms. The technical contributions are as follows:

• An aggregate supervisory model is formulated in which system frequency, wind release of kinetic energy, rotor speed recovery, and BESS power are represented on a common system power base.
• A kinetic energy admissibility law is introduced for the virtual inertia and droop request, so that the wind farm support command is reduced before the aggregate rotor approaches its lower speed limit.
• A recovery shaping rule is derived from a control-barrier condition. When the certified absorption margin is non-negative, the rule bounds the net recovery absorption; when it is negative, the rule prevents rotor recovery from adding a positive net load. BESS power is used to offset the recovery demand within its available headroom.
• Deterministic and Monte Carlo simulations are used to examine both the first nadir response and the delayed recovery stage, showing that the proposed controller reduces SFD without sacrificing the initial frequency support effect.

The remainder of this article is organized as follows. Section 2 defines the supervisory frequency, wind energy, and storage models. Section 3 presents the FS-ESC controller and the recovery certificate. Section 4 evaluates the method in deterministic and uncertain operating conditions. Section 5 discusses the engineering implications and model limitations, and Section 6 concludes the study.

2 Materials and methods

2.1 System frequency response model

The controller is designed at the supervisory level. The center-of-inertia frequency is, therefore, represented by a low-order system frequency response (SFR) model, while fast converter and network transients are left to the underlying plant controllers. Let Δ f denote the normalized frequency deviation in p.u. with respect to the nominal frequency f 0; the corresponding physical frequency deviation is f 0 Δ f in Hz. All active power variables are expressed on the same system base. Here, H s is the equivalent synchronous inertia constant, D is the load damping coefficient, Δ P m is the mechanical power increment from conventional governors, Δ P w is the net wind-farm active power increment, Δ P b is the BESS active power increment, and Δ P L is the load disturbance. The aggregate frequency dynamics areThe conventional primary response is modeled aswhere R g and T g are the governor droop and time constant, respectively. Equation 1 and Equation 2 are not intended to replace detailed electromagnetic, shaft, or turbine control models. Their role is to provide a tractable layer in which the energy released by the wind farm and the energy absorbed during recovery can be related directly to the system frequency margin. In practice, the parameters can be identified from simulations with higher fidelity or measured records of frequency response.

2 H s Δ f ̇ = Δ P m + Δ P w + Δ P b − Δ P L − D Δ f . (1)

T g Δ P ̇ m = − Δ P m − 1 R g Δ f , (2)

2.2 Aggregate wind kinetic energy state

For the supervisory layer, the wind farm is reduced to an aggregate rotor speed state ω r and the corresponding kinetic energy statewhere H w is the turbine inertia constant on the wind-farm base and S w is the installed wind-farm capacity share on the system base. Thus, S w = 0.34 represents a 34-MW wind-farm rating on the 100-MW system base, not the instantaneous pre-disturbance active-power output. Equation 3 defines the aggregate kinetic-energy state used by the supervisory layer. This aggregation does not describe the speed dispersion among individual turbines; it provides a supervisory estimate of the kinetic-energy headroom available to the farm-level controller. Let ω min and ω 0 denote the minimum admissible rotor speed and the pre-disturbance rotor speed, respectively. The lower bound is a supervisory proxy for the point at which continued kinetic-energy extraction should be terminated to avoid low-speed operation and the associated stall or protection risk. It is not a substitute for detailed aerodynamic, converter-current, or drive-train constraints. The base case represents MPPT operation through the initial rotor-energy state. Because the model does not include an explicit C p ( λ , β ) relation, it does not reproduce the non-linear change in aerodynamic power as the rotor speed departs from the MPPT trajectory. Deloaded operation can be incorporated only after its pre-disturbance curtailment and aerodynamic reserve are specified in a more detailed turbine model. The extractable kinetic-energy margin and the reference energy levels are given by Equations 4–6: The commanded wind-power increment is decomposed aswhere P rel ≥ 0 represents the release of kinetic energy used for frequency support and P rec ≥ 0 represents the active power absorption used for rotor speed recovery. The aggregate wind active-power response used in the simulations isand the rotor-energy state is updated from the actual aggregate wind power: Equation 9 makes the main tradeoff explicit. A large P rel helps arrest the initial frequency decline, but it also increases the energy that must be returned to the rotor. If P rec is then scheduled only from the local turbine speed error, the recovery process may appear to the grid as an additional load and produce an SFD.

E w = H w S w ω r 2 , (3)

E av = E w − E min , (4)

E 0 = H w S w ω 0 2 , (5)

E min = H w S w ω min 2 . (6)

Δ P w ⋆ = P rel − P rec , (7)

T w Δ P ̇ w = − Δ P w + Δ P w ⋆ , (8)

E ̇ w = − Δ P w . (9)

2.3 Battery energy storage model

The BESS is modeled as a fast active-power actuator with a first-order power response and SOC constraints: Here, T b is the BESS response time, E b is its energy capacity in p.u. h, S O C b is the state of charge, and S O C b min and S O C b max are the lower and upper SOC limits, respectively. The factor 3,600 in Equation 11 converts hours to seconds because the dynamic simulation uses seconds in Equation 12. A positive Δ P b denotes discharge to the grid. The storage unit is represented as a lithium-ion BESS because the target service includes primary-frequency support and recovery buffering over tens of seconds. A supercapacitor is better suited to a sub-second inertial burst and can be represented by the same actuator form with a smaller energy capacity and a faster time constant. The selected T b represents the converter-limited active-power response rather than the electrochemical discharge duration. For compactness, P b max in the control equations denotes the discharge power currently available after the local converter and SOC limits have been applied; it equals the nameplate converter limit when those local limits are inactive. The underfrequency command is, therefore, limited by 0 ≤ Δ P b ⋆ ≤ P b max. Overfrequency events can be represented with the opposite sign convention and by allowing BESS charging or wind-power curtailment, but they are not included in the numerical study.

T b Δ P ̇ b = − Δ P b + Δ P b ⋆ , (10)

S O C ̇ b = − Δ P b 3600 E b , (11)

S O C b min ≤ S O C b ≤ S O C b max . (12)

2.4 Control objective

The controller has two linked objectives: it should improve the first frequency nadir and limit the frequency risk introduced by rotor recovery. Let Δ f min < 0 denote the prescribed normalized frequency floor in p.u. The desired supervisory operating conditions are while reducing the integral absolute frequency deviationIn Equation 16, t 0 is the disturbance time and t f is the end of the evaluation horizon. Equations 13–15 state the intended operating bounds for the supervisory references. Because the actual wind and BESS powers follow first-order dynamics, hard invariance of the rotor-energy and SOC bounds also depends on the local actuator and protection layers. In the present work, J f is an evaluation metric rather than the objective of an online optimization problem; the online law uses explicit saturation and recovery certification instead of solving an optimization problem at each step.

Δ f t ≥ Δ f min , (13)

E min ≤ E w t ≤ E 0 , (14)

S O C b min ≤ S O C b t ≤ S O C b max , (15)

J f = ∫ t 0 t f f 0 Δ f t d t , (16)

3 Proposed FS-ESC methods

3.1 Frequency support request

At each sampling instant, the wind-farm supervisory controller first forms a fast support request using a combined droop and virtual inertia signal:where K d is the wind droop gain, K i is the virtual inertia gain, Δ f ̇ ℓ is a filtered RoCoF signal, P w max is the largest active power support request allowed by the wind farm controller, and [ ⋅ ] 0 P w max denotes saturation between zero and P w max. The RoCoF term is filtered to reduce amplification of measurement noise. Equation 17 has the same form as a conventional wind virtual inertia and droop request; the difference is that this request is treated as a demand to be allocated, not as the final turbine command.

P req = − K d Δ f − K i Δ f ̇ ℓ 0 P w max , (17)

3.2 Energy admissible wind support

The normalized kinetic energy margin is defined asThe wind kinetic energy release isBecause P req is already saturated by Equation 17 and η w ∈ [ 0,1 ], Equation 19 gradually reduces the wind contribution as the aggregate rotor-energy margin is consumed. When E w is close to E 0, the wind farm can supply the full request. As the rotor approaches the lower speed bound, the command is reduced before the bound is reached, avoiding the abrupt withdrawal of support associated with a fixed droop gain.

η w = s a t 0,1 E w − E min E 0 − E min . (18)

P rel = η w P req . (19)

The remaining part of the fast support is assigned to the BESS:where K b is the storage droop gain. The first two terms in the bracket assign to the BESS the part of the support request that cannot be delivered by the rotor energy channel. The last term adds a direct storage response when the frequency deviation remains large.

P b,f = P req − P rel − K b Δ f 0 P b max , (20)

3.3 Recovery shaping certificate

After the first frequency arrest, the rotor speed controller tends to restore E w toward E 0. This restoration must be moderated when the system frequency is close to the security floor. The safety function is defined asFor an initial state satisfying h ( t 0 ) ≥ 0, the differential inequality,is sufficient to retain h ( t ) ≥ 0 over any interval on which it is enforced, whereSubstituting Equation 1 into Equation 22 yields an admissible upper bound for the net recovery absorption P abs = P rec − P b,r. Here, P b,r denotes the BESS power specifically assigned to compensate rotor recovery: The unconstrained recovery demand is first computed from the rotor energy deficit aswhere T rec is the nominal rotor energy recovery time constant and P rec max is the turbine-side limit for recovery absorption. The BESS is then assigned to cover as much of this recovery demand as allowed by its remaining power headroom:The final recovery command is limited by the component covered by the BESS and by the positive certified absorption margin:The resulting BESS command is

h Δ f = Δ f − Δ f min . (21)

h ̇ + α h ≥ 0 , (22)

α > 0 . (23)

P abs ≤ P abs max , (24)

P abs max = Δ P m + P rel + P b,f − Δ P L − D Δ f + 2 H s α Δ f − Δ f min . (25)

P rec 0 = E 0 − E w T rec 0 P rec max , (26)

P b,r = min max 0 , P b max − P b,f , P rec 0 . (27)

P rec = min P rec 0 , P b,r + max 0 , P abs max . (28)

Δ P b ⋆ = P b,f + P b,r . (29)

Proposition 1 Consider the SFR dynamics in Equation 1 over an interval [ t 0 , t 1 ] . Assume that the states and disturbance estimate used in Equation 25 are exact, the allocated wind and BESS powers track their references without error, and Δ f ( t 0 ) ≥ Δ f min . If P abs max ( t ) ≥ 0 throughout the interval and the recovery command is selected by Equation 28 , then, for all t ∈ [ t 0 , t 1 ]. Thus, under these assumptions, rotor recovery does not drive the SFR model below the prescribed frequency floor (Equation 30) holds.

Proof: Equation 27 gives 0 ≤ P b,r ≤ P rec 0. Therefore, when P abs max ≥ 0, Equation 28 yieldsWith ideal reference tracking, the combined wind and BESS contribution during recovery is P rel + P b,f − P abs. Equation 31 establishes the required upper bound on the net recovery absorption. Substitution into Equation 1, followed by use of Equation 25, gives h ̇ + α h ≥ 0. Integration of this scalar differential inequality gives h ( t ) ≥ h ( t 0 ) e − α ( t − t 0 ) ≥ 0, which proves the result.

P abs = P rec − P b,r = min P rec 0 − P b,r , P abs max ≤ P abs max . (31)

The non-negative-margin condition is essential. If P abs max < 0, Equation 28 sets P rec ≤ P b,r and, therefore, prevents rotor recovery from creating a positive net absorption. It does not, however, enforce the negative bound P abs ≤ P abs max; doing so would require additional fast injection from storage or another reserve source, subject to available headroom. The proposition is also stated for ideal power tracking. Because the simulated wind and BESS powers follow the first-order actuators in Equation 8 and Equation 10, a practical certificate should use measured actuator power or subtract a bound on tracking error from P abs max.

The proposition gives a supervisory guarantee for the SFR layer on intervals that satisfy its stated assumptions. It should not be read as a claim that the simplified model captures converter current limits, shaft torsion, pitch dynamics, wake interaction, or network constraints. In an implementation, the certificate would sit above the detailed turbine and converter controllers. If Δ P L or D is uncertain, a conservative robustness margin can be subtracted from P abs max before applying Equation 28.

3.4 Implementation procedure

Figure 1 summarizes the implementation sequence used in the simulations. At each sampling instant, the controller receives the measured frequency, the filtered RoCoF estimate, the aggregate rotor energy estimate, the BESS SOC, and the estimated load disturbance. It then forms the raw support request in Equation 17, applies the rotor energy admissibility law in Equation 19, allocates the remaining fast component to the BESS through Equation 20, and finally limits rotor recovery using Equations 24–28. This sequence preserves compatibility with conventional turbine-level active power controllers because FS-ESC provides only the farm-level active power references.

4 Results

4.1 Simulation setup

The base case uses a single-area incremental SFR equivalent on a 100-MW power base, as shown in Figure 2. The model is linearized about a balanced operating point: 100 MW is the adopted power base rather than a specified pre-disturbance load, and no transmission-network topology is represented. This level of detail is consistent with the supervisory certificate, which acts on center-of-inertia frequency, aggregate wind kinetic energy, and storage power references rather than on line flows or electromagnetic transients. The pre-disturbance conventional dispatch is 0.66 p.u., and the installed wind-farm capacity share is 0.34 p.u. on the system base. The 0.66 p.u. value is an operating dispatch in the equivalent model, not a conventional-unit nameplate rating; the governor saturation and unit commitment are not represented. Table 2 lists the absent absolute-load, network-topology, and conventional-nameplate quantities explicitly rather than inferring them from the 100-MW base. The value S w = 0.34 denotes the rated wind-farm capacity used to scale kinetic energy and farm-side power limits. A 0.15 p.u. load increase, equivalent to 15 MW, is applied at t = 5 s. The frequency floor is set to 49.58 Hz, or Δ f min = − 0.0084 p.u., for f 0 = 50 Hz. This is a case-study design threshold used to test the recovery certificate, not a limit quoted from a particular national grid code. With the parameters in Table 2, the threshold lies 0.1784 Hz above the no-wind-support nadir and 0.0243 Hz below the fixed-droop first nadir, so the base case leaves little frequency margin during recovery. To avoid tying the interpretation to one numerical floor, the Monte Carlo samples are also post-processed at nearby floors in Section 4. Unless otherwise stated, all powers are expressed on the system base. Table 2 lists the principal parameters reported in the model.

TABLE 2. Main simulation parameters.

Four control strategies are compared under the same disturbance and parameter set:

• No wind support: only the conventional governor responds to the load increase.
• Fixed wind droop: the wind farm follows the virtual inertia and droop request directly, without energy admissibility or certified recovery limiting.
• Conventional wind and storage coordination: wind support is weighted by the remaining rotor-energy margin, and the BESS covers a fixed fraction of the recovery demand.
• Proposed FS-ESC: Equations 17–29 are applied.

The comparison strategies use the same plant parameters and raw support request. In the supervisory model, the wind release and recovery references form the command in Equation 7; the actual aggregate wind power and BESS power then follow the first-order responses in Equation 8 and Equation 10. The state equations are integrated by the explicit Euler method with Δ t = 0.02 s. The RoCoF signal is filtered by a first-order low-pass filter with T r = 0.14 s, and IAFD is evaluated by trapezoidal integration over the post-disturbance interval. In the fixed wind droop case, recovery starts after 16 s when rotor energy has been depleted and uses a 12 s recovery time constant. In the conventional wind–storage case, the wind support weight is 0.35 + 0.65 η w, recovery starts after 15 s once the physical frequency deviation f 0 Δ f is above −0.24 Hz, the recovery time constant is 16 s, and the BESS covers up to 62% of the recovery absorption within the remaining 0.08 p.u. converter headroom. The FS-ESC case uses the 22-s recovery time constant and the certificate in Equations 24–28.

4.2 Deterministic disturbance response

Figure 3 compares the frequency, wind active power increment, BESS dispatched power, and aggregate rotor speed trajectories. Without wind support, the frequency nadir reaches 49.4016 Hz, which is below the 49.58 Hz floor. Fixed wind droop improves the first nadir to 49.6043 Hz, but the support is sustained without considering the subsequent energy debt; during recovery, this produces an SFD of 0.1028 Hz. The conventional wind and storage strategy reaches a similar level of the first nadir, 49.6582 Hz, but the fixed recovery sharing rule still leaves an SFD of 0.0848 Hz and drives the rotor close to its lower margin. The BESS trace shows why the proposed method behaves differently: after the initial frequency support period, FS-ESC keeps the storage unit active during rotor recovery so that less of the recovery absorption is passed through to the grid. FS-ESC gives a first nadir of 49.6590 Hz and reduces the dip caused by recovery to 0.0062 Hz. The aggregate rotor speed remains above 0.93 p.u., indicating that the improvement is achieved without deep depletion of kinetic energy.

The numerical metrics in Table 3 show the same tradeoff. Relative to fixed wind droop, FS-ESC lowers SFD by 93.9% and reduces the integral absolute frequency deviation (IAFD) by 32.4%. Relative to conventional coordination between wind and storage, it lowers IAFD by 16.9%. The improvement is accompanied by higher BESS energy use. This is expected, because the controller deliberately transfers part of the recovery burden from the rotor speed channel to the storage channel whenever the frequency margin is tight.

TABLE 3. Deterministic simulation metrics.

4.3 Additional operating scenarios

Additional cases are used to examine the effects of load increase, wind-farm share, online inertia, and initial rotor speed. The base deterministic case starts from the MPPT-equivalent state ω 0 = 1.08 p.u. A separate case starts from ω 0 = 1.15 p.u. to test the controller with a larger initial kinetic-energy margin. Only the initial energy state is changed in this aggregate model; the case does not represent a complete deloaded operating point because pre-disturbance curtailment and the aerodynamic reserve are not modeled. In the wind-share cases, the wind active-power and recovery limits are scaled with S w, while the BESS rating is unchanged. The combined severe case also reduces H s to 2.70 s. Table 4 summarizes the results.

TABLE 4. Additional scenario validation under different load steps, wind-farm shares, synchronous inertia levels, and initial rotor speeds.

FS-ESC gives the smallest SFD in each reported case. When the load increase rises to 0.20 p.u., for example, the SFD decreases from 0.1688 Hz with fixed wind droop and 0.1088 Hz with conventional wind–storage coordination to 0.0080 Hz with FS-ESC. The combined severe case also has a lower nadir of 49.5236 Hz, showing that the recovery certificate does not replace the need for adequate primary reserve. Its role is narrower: it prevents rotor recovery from adding a large secondary dip. The higher-initial-speed case shows that the same certificate can accommodate a larger kinetic-energy state. It should not be interpreted as validation of a complete deloaded aerodynamic control scheme.

4.4 Power allocation and recovery certification

Figure 4 shows the internal power allocation of FS-ESC. Immediately after the disturbance, the wind farm supplies most of the requested support because the rotor energy margin is still sufficient. As the margin decreases, the wind contribution is reduced and the BESS takes a larger share of the fast response. During recovery, the desired rotor speed restoration is admitted only to the extent allowed by the certified absorption margin. When the margin is small, the BESS offsets more of the recovery demand before it appears to the grid; when the margin is negative, the rule blocks positive net recovery absorption. This produces a gradual active-power withdrawal and limits the SFD in the reported cases.

4.5 Uncertainty assessment

The deterministic case does not reflect the variability of wind speed, disturbance size, and online synchronous inertia. A Monte Carlo study with 120 samples is, therefore, performed using random seed 23. Wind speed v is expressed in m/s and sampled as v = 8.9 z, where z follows a unit-scale Weibull distribution with shape factor 2.25; thus, 8.9 m/s is the Weibull scale parameter. The initial rotor speed is mapped by ω 0 = c l i p ( 0.72 + 0.044 v , 0.82 , 1.18 ) p.u. The load step is sampled from a normal distribution with mean 0.150 p.u. and standard deviation 0.022 p.u. and then truncated to 0.095 p.u.–0.205 p.u. The synchronous inertia H s is sampled from a normal distribution with the mean 3.2 s and standard deviation 0.35 s and then truncated to 2.45 s–4.1 s.

Figure 5 reports the nadir distribution and the violation rate. Fixed wind droop violates the 49.58-Hz floor in 40.0% of the samples, mainly because the same gain is applied regardless of the available rotor energy and the subsequent recovery burden. The conventional wind and storage scheme gives a violation rate of 7.5%, and FS-ESC gives 5.0%. These rates correspond to 48, nine, and six violations, respectively, for fixed wind droop, conventional coordination, and FS-ESC. With only 120 samples, the 2.5 percentage-point difference between the two wind–storage methods should be interpreted descriptively rather than as a statistically conclusive separation. The same Monte Carlo samples were also post-processed with nearby frequency floors of 49.55 and 49.61 Hz. The violation counts for fixed wind droop, conventional coordination, and FS-ESC are 33/3/3 at 49.55 Hz and 69/27/26 at 49.61 Hz. This sensitivity check shows that the main comparison with fixed wind droop is not caused by a single threshold choice, while the smaller difference between the two wind–storage methods remains descriptive. The remaining FS-ESC violations occur in samples with a large disturbance and synchronous inertia near the lower end of the tested range. The controller also suppresses recovery-related net absorption when the certified margin is insufficient; it does not remove an initial reserve deficit.

4.6 Sensitivity to recovery gain and storage power

Two parameters are particularly important for field tuning: the recovery certificate gain α and the BESS power limit P b max. The gain α determines how rapidly the admissible absorption margin grows as the frequency moves away from the floor. Smaller values produce a more conservative recovery process, while larger values release recovery absorption more quickly when the margin is available. The BESS power limit determines how much of the recovery demand can be hidden from the grid before it becomes a net load.

The sensitivity results are shown in Figure 6. At the studied operating point, changing α from 0.35 to 1.25 has a small effect on SFD and IAFD. The reason is that the BESS has sufficient headroom to offset recovery absorption once the barrier condition becomes active. In this range, the barrier gain mainly changes the timing of recovery rather than the security outcome. The storage power limit has a much stronger effect. When P b max is below approximately 0.065 p.u., both SFD and IAFD increase rapidly; at P b max = 0.035 p.u., SFD exceeds 0.05 Hz because storage can no longer buffer the rotor speed restoration. Above 0.065 p.u., the curves approach saturation, indicating that additional storage power brings little benefit for this particular disturbance. A practical tuning sequence is, therefore, to size the storage power for the largest expected certified recovery deficit and then choose α according to the preferred recovery speed.

5 Discussion

The results highlight a point that is sometimes hidden in studies on wind frequency control: support from kinetic energy is not a service in only one direction. The same rotor energy release that improves the first nadir creates a later recovery requirement. If this requirement is not considered at the time of support allocation, the controller may simply move the frequency security problem from the first few seconds to the recovery interval. FS-ESC handles this energy causality explicitly. The admissibility layer limits the release according to the remaining aggregate rotor energy, while the recovery certificate limits the net absorption according to the instantaneous frequency margin. The resulting behavior is physically interpretable: wind supplies fast support when rotor energy is available, and storage becomes more active when either the rotor energy margin or the frequency security margin is tight.

The proposed formulation also changes the role of storage. In many wind and storage schemes, BESS power is allocated by a combination of low-pass and high-pass filters, a coefficient that depends on SOC, or a fixed sharing ratio. Such rules are useful for smoothing and are straightforward to implement, but they do not directly answer whether the recovery absorption is safe for the current frequency condition. In FS-ESC, the storage device is assigned a task oriented to frequency security: it covers the part of rotor recovery that would violate the barrier condition if passed through to the grid. This explains the deterministic results in Table 3: the proposed controller consumes more BESS energy, extracts less kinetic energy from the rotor, and achieves a much smaller SFD.

The 49.58-Hz threshold is a design parameter for the case study rather than a grid-code limit. In an application study, Δ f min must be replaced by the applicable frequency-containment or ride-through requirement and coordinated with the corresponding reserve-delivery rules. Here, keeping the threshold fixed isolates the behavior of the recovery certificate; the numerical results should, therefore, not be read as evidence of compliance with a particular jurisdiction.

From the viewpoint of a system operator, FS-ESC should be interpreted as a plant-level supervisory allocator rather than as a permission to withhold a contracted frequency service. The raw request in Equation 17 represents the required fast active-power response within the available plant headroom. FS-ESC first allocates this request between rotor kinetic energy and storage; it then shapes the recovery absorption so that the plant does not create an avoidable secondary dip after meeting the initial support obligation. If a grid-code or reserve contract requires a support level that exceeds the admissible rotor-energy margin, the deficit must be covered by pre-disturbance deloading, storage headroom, curtailed wind reserve, or another fast reserve resource. Operating the turbine into a low-speed or protection region cannot be a sustainable way to satisfy an operator requirement, because such operation may jeopardize turbine availability and subsequent frequency support. Thus, the proposed method is intended to make the compliance boundary explicit: it reports when kinetic-energy support is no longer physically admissible and uses storage to preserve the delivered plant response where possible.

The base case starts from an MPPT-equivalent rotor state and does not assume an aerodynamic power reserve. The aggregate energy model accounts for the energy released from the rotor and the power absorbed during recovery, but it does not calculate the non-linear aerodynamic-power change that accompanies a departure from the MPPT tip-speed ratio. The case with ω 0 = 1.15 p.u., therefore, tests only a larger initial kinetic-energy margin. A genuine overspeed-deloaded study would also have to specify pre-disturbance curtailment, the C p ( λ , β ) characteristic, pitch or torque control, and the resulting reserve available at each wind speed.

Storage technology should be selected for the time scale of the assigned service. Supercapacitors are appealing for a very fast inertial response because of their high power density and cycling capability. The recovery compensation considered here lasts for tens of seconds and is closer to primary-frequency regulation, so the numerical case uses a lithium-ion BESS with an 8 MW converter rating. The largest reported discharge energy is 3.4714 p.u. s, which is approximately 0.096 MWh on the 100-MW base. The assumed 55-MWh energy capacity is, therefore, non-binding in these short simulations; the results primarily test power allocation rather than energy-capacity sizing. A hybrid unit could be represented by splitting Δ P b into fast supercapacitor and sustained battery components, but that hardware allocation is outside the present supervisory-control contribution.

Several limitations should be kept in view. First, the proposition is conditional on a non-negative certified absorption margin and ideal tracking of the allocated powers. When the margin is negative, the implemented rule blocks positive net recovery absorption but does not, by itself, guarantee the frequency floor. The first-order wind and BESS responses used in the simulations also mean that an implementation needs a measured-power formulation or an explicit tracking-error margin. The rotor-energy and SOC limits are likewise applied at the supervisory-reference level; hard bounds under actuator lag require tightened limits or predictive protection in the local control layer. Field-oriented validation would additionally require detailed aerodynamic and converter models, drive-train and pitch limits, wake effects, communication delay, and network constraints. These dynamics may reduce the available tracking bandwidth or activate limits that are absent from the aggregate model. Second, Δ P L is treated as known or estimated. A practical implementation would need a disturbance observer, a conservative bound, or a robustness margin in Equation 25. Third, BESS degradation and energy-management cost are not optimized. The barrier certificate could be combined with economic model predictive control to balance frequency security, battery aging, and wind-energy capture. Fourth, the numerical study is limited to underfrequency events. Overfrequency control can be included by reversing the support direction and allowing BESS charging or wind-power curtailment, but the active constraints differ and require separate validation.

6 Conclusion

In this study, we present a FS-ESC strategy for wind power participation in primary frequency regulation. The method coordinates virtual inertia and droop support, finite rotor kinetic energy, BESS dispatch, and rotor speed recovery within one supervisory model. The energy admissible release law limits kinetic-energy extraction, and the recovery rule bounds net recovery absorption when the certified margin is non-negative; when it is negative, the rule prevents rotor recovery from adding a positive net load.

Simulation results show that FS-ESC preserves the improvement in the first nadir while suppressing the delayed SFD. For the deterministic 0.15 p.u. load step, the nadir rises from 49.4016 Hz to 49.6590 Hz, the SFD is limited to 0.0062 Hz, and the IAFD is 32.4% lower than that with fixed wind droop. Cases with different load steps, wind-farm shares, and initial rotor-energy states give the same qualitative result: the certificate limits the additional dip caused by recovery. In the Monte Carlo study, the frequency-floor violation rate decreases from 40.0% with fixed wind droop to 5.0% with FS-ESC. These results support treating the fast support command and the subsequent rotor-energy recovery as one constrained process.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. The authors declare that ChatGPT (OpenAI, GPT-5.5 Thinking) was used for language editing, readability improvement, and LaTeX formatting assistance. No simulation data, mathematical derivations, or scientific conclusions were generated by the tool. The authors reviewed and edited all content assisted by AI and take full responsibility for the final manuscript.

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Footnotes

Edited by: Agnimitra Biswas, National Institute of Technology, India

Reviewed by: André Alves, Rio de Janeiro State University, Brazil

Ali Berboucha, Université de Bejaia, Algeria

Footnotes

FS-ESC, Frequency security certification based energy shaping control; SFR, System frequency response; BESS, battery energy storage system; SOC, State of charge; SFD, Secondary frequency dip; IAFD, Integral absolute frequency deviation; RoCoF, Rate of change of frequency; MPPT, Maximum power point tracking; DFIG, Doubly fed induction generator; HVDC, High-voltage direct current; MTDC, Multi-terminal direct current.