The power coefficient cp and the thrust coefficient ct are modeled as functions of tip speed ratio λ and blade pitch angle β by the corresponding LUTs. The fore–aft deflection of the WT tower, excited by the thrust force Ft, is modeled as a mass–spring–damper oscillator with mass mt, damping coefficient dt, and stiffness coefficient kt. The aeroelastic model outputs the WT mechanical torque Mm (in p.u. of mR) for given inputs (vw,β,Ω); i.e., 4aPw=12ρπr2vw3/pR,4bMm=PmΩ=Pwcp(λ,β)Ω,λ=ΩωRrṽw,ṽw=vw-s˙t,4cs¨t=1mtFt-dts˙t-ktst,Ft=12ρπr2vw2ct(λ,β),st,0=Ftkt,s˙t,0=0, with wind power Pw (in p.u. of pR); WT aerodynamic or mechanical power Pm (in p.u. of pR); air density ρ; rotor radius r; relative wind speed ṽw; WT fore–aft tower displacement st; and initial steady-state values st,0,s˙t,0,λ0, and β0. Using HR (in s) to indicate the WT total physical drivetrain inertia constant, the WT mechanical dynamics are approximated by a one-mass model ; i.e., 5Ω˙=Mm-Me2HR,Ω(0)=Ω0.

The electrical power for MPPT is given by 6Pmppt:=ΩMmppt, where the MPPT torque is the function of rotor speed Mmppt:=Mmppt(Ω), shown in Fig. .

Below rated wind speed in region II, Mmppt increases proportionally to Ω2 for optimal operation at the MPP. Above rated wind speed in region III, the pitch control limits the WT rotor speed to Ω=1 such that Pmppt=Mmppt=1. Below cut-in wind speed vw,cut-in in region I, the WT does not generate power; i.e., Mmppt=0 for Ω<Ωmin⁡. For a smooth transition to region II, a non-optimal operation is accepted in the small transition region I–II, defined by Ωmin⁡≤Ω≤Ωmin⁡+ΔΩI-II; i.e., Mmppt is obtained by multiplying the optimal torque at the MPP by a factor that is linearly interpolated between 0 at Ωmin⁡ and 1 at Ωmin⁡+ΔΩI-II. For a more complete description of the MPPT curve, see Sect. III.B.1.

The MRS-based derating maximizes the WT kinetic energy reserve for inertia provision. The derating power setpoint Pset is defined relative to the MPP; i.e., Pset=1 corresponds to MPPT, and Pset<1 corresponds to derating. Increasing derating (decreasing Pset) reduces the electrical power setpoint Pset⋆:=P‾⋆Pset with available power P‾⋆∈(0,1]; i.e., the WT accelerates. As a consequence, the tip speed ratio λ increases, and the power coefficient cp decreases. The pitch controller additionally increases β if necessary to limit the rotor speed to Ω=1. In general, the MRS prioritizes increasing WT speed over pitching to provide power reserve.

With the measured or estimated rotor-effective wind speed vw , the available power in Fig.  is defined as 7P‾⋆:=Pw(vw)c‾p⋆(vw):=12ρπr2vw3c‾p⋆(vw)/pR, where the MPPT (Pset=1) power coefficient is given by 8c‾p⋆(vw)=cp(Ω‾⋆,β‾⋆,vw)ifvw,cut-in≤vw<vw,II,min⁡,cp⋆:=max⁡cpifvw,II,min⁡≤vw≤vw,R,1Pw(vw)ifvw,R<vw≤vw,cut-out, with cut-in wind speed vw,cut-in, minimum region-II wind speed vw,II, rated wind speed vw,R, cut-out wind speed vw,cut-out, and steady-state values (Ω‾⋆,β‾⋆).

Limiting Pset⋆ by Pmppt for rotor speed transients and wind measurement errors, the saturated power setpoint is given by 9P‾set⋆:=min⁡(Pset⋆,Pmppt),ifPset<1(derating),Pmppt,ifPset=1(MPPT).withPset⋆:=P‾⋆Pset. Pset⋆ is ignored for Pset=1 (MPPT) in Eq. (), since (i) no wind measurements are required, and (ii) smaller transient rotor speed overshoots occur due to higher power setpoint adaptation. For example, if Ω>1 due to a wind gust, it follows that P‾set⋆=Pmppt=ΩMmppt>1, whereas Pset⋆≤1.

Grid-forming control is required to limit the initial ROCOF . This paper simplifies the grid-forming VSM control proposed in by neglecting fast electromagnetic transients and low-level current control loops. However, the grid synchronization dynamics of grid-forming control define the inertial response and must therefore be taken into account.

For VSM control, the grid synchronization dynamics are similar to the dynamics of a real (grid-connected) SM, as illustrated in Fig. .

The VSM acceleration Ω˙v is proportional to the sum of virtual torques; i.e., the VSM mechanical model is based on a one-mass model with virtual inertia Hv instead of physical inertia HR in Eq. (). At steady state, the difference between VSM mechanical and electromagnetic torque is zero (i.e., Mm,v-Me,v=0), and the VSM damping torque is zero (i.e., Mdp,v=0). Since only the inertial response to ROCOFs or electromagnetic changes is of interest, the VSM mechanical torque is set to zero (i.e., Mm,v=0), resulting in the freely spinning VSM in Fig. . Denormalization of the power system frequency, i.e., ωs=Ωsωs,R with ωs,R:=2πfs,R, and subsequent integration of ωs yield the grid or system angle ϕs. Similarly, denormalization of the VSM rotor speed, i.e., ωv=Ωvωs,R, and subsequent integration of ωv yield the VSM rotor angle ϕv. The VSM electromagnetic torque Me,v depends on the (real) load angle δ:=ϕv-ϕs multiplied by the electromagnetic feedback gain ke. Due to unknown ϕs or ke, the VSM controller calculates the torque or power feedback directly based on current and grid voltage measurements. The VSM damping torque Mdp,v is proportional to the VSM slip Ωv-Ωs, which emulates the effect of damper windings in SMs. However, unlike real SMs, the VSM enables flexible tuning of the VSM damping Dv. Grid voltage measurements are required to determine Ωs for VSM damping torque calculation.

The VSM power for inertia provision, added to the power setpoint P‾set⋆ in Fig. , is defined as 10Pv=ΩvMe,v, where Ωv and Me,v are given by the grid synchronization loop in Fig.  with input Ω˙s.

The inertial response in the Laplace domain of the VSM is given by 11Me,vΩ˙s=-keωs,Rs2+2ζvωn,vs+ωn,v2,ωn,v2:=keωs,R2Hv,ζv:=Dv8Hvkeωs,R=1⇒Dv:=Dv(Hv):=8Hvkeωs,R, with natural angular velocity ωn,v and damping ratio chosen as ζv=1 to avoid overshooting. With grid-synchronized VSM speed Ωv=Ωs≈1 in Eq. () and setting s=0 in Eq. (), the steady-state VSM power for a constant ROCOF simplifies to 12Pv=ΩvMe,v≈Me,v≈-2HvΩ˙s.

The power system dynamics in Fig.  are defined by a reference frequency event. The electromagnetic feedback in Fig.  depends on the load angle given by the angle difference between the VSM and the grid; i.e., Me,v=keδ=keϕv-ϕs. For simplicity, this paper assumes a constant electromagnetic feedback gain ke. Actually, ke depends on the WT operating point and the WT grid connection; i.e., ke is a nonlinear function of the load angle delta δ, the grid voltage, and the grid impedance . Type 3 WTs use doubly fed induction machines (DFIMs), where ke also depends on the DFIM rotor current or excitation level Sect. III.E. If not negligible, the dependency of ke on δ and the excitation level should be taken into account based on the WT operating point. With admissible limits for grid voltage and impedance defined by grid codes , ke should be chosen based on the WT grid connection, as explained in Appendix .

This paper assumes internal damping of the VSM ; i.e., the damping torque Mdp,v in Fig.  is solely virtual and is not converted into real electrical output power. In contrast, for external damping of a real SM, the damping power is part of the electrical output power . In this regard, Hv of a VSM and H of a (real) SM differ; i.e., assuming Hv=H and equal damping gains, the actually extracted kinetic energy during the inertial response is smaller for a VSM-controlled WT than for a (real) SM.

Figure  shows an exemplary inertial power response to a ROCOF with quasi-steady-state amplitude ΔP approximated by Eq. ().

The damping energy corresponds to the area between the two curves in Fig. . Strictly speaking, the VSM concept violates the law of conservation of energy since the VSM braking energy is not fully converted into electrical energy. However, high internal damping avoids power overshoots and is beneficial for grid frequency stability . Also, grid codes require sufficient damping and consider (internal) damping power separately from electrical output power Kap. 5.1.1.11, Anmerkung 1. Finally, Hv is comparable to H of a real SM when neglecting the transient damping, i.e., when considering the quasi-steady-state power change ΔP=-2HvΩ˙s, as shown in Fig. . Thus, Hv is a suitable measure for the inertial power response and the grid frequency support by inertia provision.

The recent draft for certification of grid-forming IBRs quantifies inertia provision by the mean power change over a time window starting 0.5s after a ROCOF change and ending at the beginning of the next ROCOF change during the reference frequency event; i.e., TA:=mean|ΔP(t)/Ω˙s|≈2Hv. For example, due to a constant initial ROCOF for 1s during the reference frequency event starting at t=0, the first time window for quantifying inertia provision is 0.5s≤t≤1s. For simplicity, this paper quantifies inertia provision by the control parameter Hv (see also Fig. ), which results in a (slight) overestimation of the inertia provision compared to .

This paper assumes an ideal inertial power response; i.e., the VSM power Pv is added to the original machine power setpoint P‾set⋆ in Fig. . Moreover, the final electromagnetic torque equals the electromagnetic torque reference, i.e., Me=Me,ref in Fig. , neglecting low-level current controls with closed-loop time constants that are significantly smaller than the ones of high-level WT or VSM control. Clearly, this is a simplified representation of the actual VSM control, which adjusts the voltage or current phase angle based on the VSM angle ϕv to achieve the grid-forming capability . Although the implementation details are beyond the scope of this paper, the simplified representation should take into account the general differences between existing VSM control strategies, as discussed in Appendices –.

For a negative ROCOF, the WT output power increases for inertia provision, which decelerates the WT. According to Eq. (), the MPPT would counteract the deceleration by reducing Pmppt when Ω decreases. To avoid this, the so-called MPPT compensation manipulates the MPPT input Ω . This paper simplifies the MPPT compensation proposed in Sect. III.B.2. The speed change due to inertia provision is estimated by replacing the numerator of the one-mass model in Eq. () by the inertial torque change Pv/Ω. Thus, the manipulated MPPT input, equal to the theoretical WT rotor speed for zero inertia provision, is given by 13ΩH0:=Ω+min⁡a,12HR∫t0tPvΩdt,if|Ω˙v|>ϵ(active MPPT comp.),0,if|Ω˙v|≤ϵ(inactive MPPT comp.),,a:=max⁡1-Ω,0, with the threshold ϵ used for detecting active inertia provision based on the VSM acceleration Ω˙v≈Ω˙s. The integral in Eq. () is reset to zero for inactive MPPT compensation. The actual implementation of Eq. () includes an additional rate limiter, which ensures |Ω˙H0|≤Ω˙H0,max⁡ with maximum acceleration Ω˙H0,max⁡ for a smooth transition between active and inactive MPPT compensation Sect. III.B.2.

Assuming active MPPT compensation, a prolonged MPP deviation during a long time period with a small negative ROCOF would lead to excessive WT rotor deceleration. Thus, the threshold ϵ in Eq. () should not be chosen too small. This also implies less inertia provision for small negative ROCOFs |Ω˙s|≈|Ω˙v|≤ϵ than expected by Hv due to the MPPT compensation being inactive. Also, for ϵ<|Ω˙s|<Ω˙s,max⁡ with normalized worst-case ROCOF magnitude Ω˙s,max⁡, there may be cases when the inertia provision is (slightly) smaller than expected by Hv, if output power saturation is required to protect the rotor speed due to prolonged MPP deviations. However, the proposed approach ensures unsaturated or full inertia provision when reaching the worst-case or reference ROCOF |Ω˙s|=Ω˙s,max⁡.

In addition to inertia provision, which supports grid frequency by injecting inertial VSM power Pv proportional to the ROCOF, active power droop control supports grid frequency by injecting (saturated) droop power P‾d proportional to the frequency deviation ΔΩs:=1-Ωs. Thus, the final power reference in Fig.  is given by Pref=P‾set⋆+Pv+P‾d. For WTs, droop control is inactive during normal operation within a tolerance band of |ΔΩs|≤0.4% ; i.e., the (unsaturated) droop power is Pd=P‾d=0 in Fig. . During a critical system state outside of the tolerance band, the WTs have to support grid frequency by a proportional power adaptation when possible. This means that P‾d=Pd>0 is only required if wind power reserves are available due to previous derating . For the present control strategy, this implies that the inertia provision based on additional kinetic energy extraction is prioritized over droop control; see Appendix  for details.

Ignoring the two max⁡ blocks in Fig. , the maximum droop power Pd,max⁡ is given by the total currently available power min⁡(P‾⋆,Pmppt) minus the sum of the power setpoint P‾set⋆ and the VSM power Pv. The saturation P‾d:=min⁡(Pd,Pd,max⁡) prevents excessive WT overloading since, without it, the droop power Pd would add to the inertial power even if the output or reference power Pref already exceeds the available one. In other words, the WT control prioritizes inertia provision over droop control. Similarly, for real (grid-connected) SMs, the droop control or speed governor response time is significantly slower than the SM inertial response; i.e., only the SM inertial power limits the initial ROCOF.

The additional saturations by the two max⁡ blocks in Fig.  ensure that the droop control power Pd does not counteract the VSM power Pv for inertia provision. Without the upper max⁡ block, Pd=Pd,max⁡<0 could counteract Pv>0 for a high negative initial ROCOF; without the lower max⁡ block, Pd=Pd,max⁡>0 could counteract Pv<0 more than expected for a subsequent positive ROCOF during frequency recovery. The presented control is a simplified version of the actual control with dynamic droop saturation of Sect. III.F.

The maximum feasible VSM inertia constant Hv,max⁡ is obtained by solving the optimization problem 14∀t≥t0:Hv,max⁡:=arg maxHvHv,s.t.Ω(t)≥Ωmin⁡Me(t)≤Me,max⁡M˙e(t)≤M˙e,max⁡Pe(t)≤Pe,max⁡. Here, the WT electromagnetic torque rate M˙e:=ddtMe is in s-1; the WT electrical power Pe is in p.u. of rated power pR=ωRmR (in W); and Ωmin⁡, Me,max⁡, M˙e,max⁡, and Pe,max⁡ denote the admissible operating limits. Note that, although the objective function and optimization argument in Eq. () are the same, the solution of this problem is not trivial due to the presence of nonlinear optimization constraints. Depending on the grid codes and WT design, additional constraints may have to be considered, e.g., to account for aerodynamic stall limits (here implicitly taken into account by Ωmin⁡ in Eq. ). Appendix  discusses the case of recovery power limits, with reference to the examples shown later in Sect. .

evaluate the capability of grid-following WTs to emulate inertia by considering WT rotor speed or available kinetic energy reserve. Here, we depart from that approach by deriving a simplified solution of Eq. () for grid-forming WTs that takes all operating constraints into account.

Neglecting any changes in aerodynamic conditions during the inertial response, i.e., assuming constant values for wind speed, pitch angle, and tip speed ratio, it follows that the mechanical power is constant; i.e., 15∀t0≤t≤t0+Δt:vw(t)=vw(t0)=:vw,0β(t)=β(t0)=:β0λ(t)=λ(t0)=:λ0⇒Pm(t)=Pm(t0)=:Pm,0.

For the simplified solution, we assume that the ROCOF f˙s is constant and equal to the worst-case initial ROCOF, until reaching the minimum frequency nadir fs,min⁡ at time t=ts,min⁡; i.e., the considered time duration is given by 16Δt:=ts,min⁡-t0=fs,min⁡-fs,Rf˙s. Assuming that the initial electrical power equals the mechanical power in Eq. (), i.e., Pe,0=Pm,0, and approximating the electrical power change during Δt by an ideal power pulse ΔPe according to the simplified inertial response in Eq. (), the electrical power constraint in Eq. () simplifies to 17Pe:=Pe,0+ΔPe:=Pm,0+2HvΩ˙s,max⁡≤Pe,max⁡, where Ω˙s,max⁡:=|f˙s,0|/fs,R>0. It follows that additional output power ΔPe:=2HvΩ˙s,max⁡ is extracted from the WT kinetic energy reserve according to 18∫t0ts,min⁡Pm-Pedt=Ekin(ts,min⁡)-Ekin(t0), from which we get ΔPeΔt=HRΩ02-Ω(ts,min⁡)2. Finally, the minimum rotor speed constraint in Eq. () simplifies to 19Ω(ts,min⁡)=Ω02-2HvHR1-Ωs,min⁡≥Ωmin⁡, where Ω0:=Ω(t0), and HR includes the total inertia of the WT drivetrain. Note that Eq. () depends on the normalized frequency nadir Ωs,min⁡:=fs,min⁡/fs,R and not explicitly on the ROCOF, which justifies the aforementioned assumption of a constant ROCOF in Eq. (). Based on Eqs. () and (), the torque constraint in Eq. () simplifies to 20max⁡Me=PeΩ(ts,min⁡)≤Me,max⁡. Finally, with Eqs. ()–() and the simplified torque rate constraint derived in Appendix Eq. (), a nonlinear optimization algorithm solves Eq. () for given initial values Ω0 and Pm,0.

The optimization problem expressed by Eq. () can be solved in a more general way, where dynamic simulations of the WT inertial response to a worst-case or reference frequency event replace the aforementioned simplified expressions. More precisely, an optimization algorithm iterates the simulations with varying Hv to find the maximum inertia constant Hv,max⁡ that does not violate any operating limits (see Fig. ). Clearly, this approach is generic due to its applicability to different WT models and their controllers. Moreover, this approach allows for more accurate solutions. For instance, derating strategies can provide additional wind power reserves , which are only taken into account by the complete numerical solution but not by the simplified one. For the iterative simulations during optimization, we rely on appropriate WT modeling, with steady-state initializations derived in Appendix .

Inertia provision requires power headroom. Accordingly, all saturations or manipulations of the WT power reference Pref for protection are not just removed from the WT control model, but are converted into corresponding inequality constraints; i.e., 21∀t≥t0:c:=c1c2c3c4:=Ωmin⁡-min⁡Ω(t),max⁡Me(t)-Me,max⁡,max⁡M˙e(t)-M˙e,max⁡,max⁡Pe(t)-Pe,max⁡,where∀i∈{1,2,3,4}:ci≤0. Based on Eq. (), a nonlinear optimization algorithm solves the optimization problem in Eq. (). The ith constraint is considered active if ci=0 or inactive if ci<0.

Although grid codes can vary among countries and system operators, the core requirements for inertia provision are similar . This paper focuses on the German code for grid-forming control and its requirements for inertia provision . This grid code defines two reference frequency events with maximum initial ROCOF magnitudes of |f˙s|=2Hzs-1: one for negative inertia provision due to a high positive initial ROCOF and another for positive inertia provision due to a high negative initial ROCOF. The latter is considered the worst-case reference frequency event for WFs, since the output power has to increase for inertia provision, which decelerates the WTs. Emulating this reference frequency event and evaluating the WF power response are required to verify inertia provision.

The considered grid code defines various tests based on the reference frequency events to verify inertia provision, including operation in (i) fictive or simulated island mode with changing power imbalance ΔP due to varying electrical load; (ii) grid-emulator-connected mode with changing ROCOF Ω˙s; and (iii) real grid-connected mode with changing controller-internal ROCOF, corresponding to the VSM acceleration Ω˙v (see also Fig. ). In the latter case (iii), the frequency signal defined by the reference frequency events is added as a disturbance to the controller-internal frequency, corresponding to the VSM speed Ωv. It should be noted that some tests consider deactivated droop control. However, the verification principle is always the same; see also . In all tests, a ROCOF changes Ω˙, and the inertial power response ΔP is measured or vice versa. The actual inertia provision is quantified by the measured inertia constant Hmeas=|ΔP/(2Ω˙)| at quasi-steady state, as obtained from Eq. (). Note that ΔP corresponds to the measured power change ΔPmeas only if droop control is deactivated. Otherwise, for the correct calculation of Hmeas, the droop power change ΔP‾d (depending on the frequency deviation) should be added to the inertial power (depending on the ROCOF) during the frequency event; i.e., ΔP=ΔPmeas-ΔP‾d. Clearly, Hmeas must match the expected VSM inertia constant; i.e., Hmeas≈Hv.

For the considered VSM control, Hmeas≈Hv holds if no power saturation is active (see also Fig. ). In other words, it is assumed that the actual VSM control implementation would pass all tests with Hmeas≈Hv for an arbitrarily chosen Hv if no power saturation exists. It follows that Hmeas≈Hv holds if no operating limits are violated in the simulations of the WT model. Otherwise, in reality, the protection methods would saturate the output power as the desired inertia provision is not feasible; i.e., Hmeas<Hv due to Hv>Hv,max⁡. Thus, assuming proper VSM control allows this study to focus on the relevance of interactions with WT control, WT characteristics, and operating limits.

Running and passing all tests defined in the grid code verifies proper grid-forming control implementation and inertia provision for a given Hv at a given operating point. However, this approach is not suitable for evaluating the maximum feasible inertia provision of WFs over a wide operating range. Thus, this paper considers a single worst-case test scenario to simulate WTs for varying Hv and varying operating point (vw,Pset). The reference frequency event for positive inertia provision with an initial ROCOF of f˙s=-2Hzs-1 defines the input Ω˙s in Fig. . This can be interpreted as ideal ROCOF emulation at the point of common coupling. As in real grid-connected operation mode, droop control is activated. Finally, a WT survives the worst-case test scenario if no operating limit is violated, i.e., for Hv≤Hv,max⁡ at a given operating point (vw,Pset), according to Eq. ().

This section considers simulation results of the WT inertial response to the reference frequency event with optimized Hv=Hv,max⁡ at vw=9ms-1<vw,R for two different power setpoints: (i) Pset=100% for MPPT in Fig.  and (ii) Pset=95% for MRS-based derating in Fig. . The grid frequency is identical in both cases (panel a); i.e., Ωs decreases with the initial ROCOF Ω˙s=-4%s-1 for t0=0s≤t<1s and further decreases with Ω˙s=-2/3%s-1 afterwards, until reaching the nadir Ωs,min⁡=95% at ts,min⁡=2.5s. Then, Ωs increases with Ω˙s=2%s-1, before staying constant at Ωs=98% for t≥4s. In addition, panel (a) shows the VSM speed Ωv, the WT speed Ω, and the adjusted MPPT input ΩH0. Panel (b) shows the mechanical power Pm, the electrical power Pe, the electrical torque Me, and its limit Me,max⁡. Panels (d), (e), and (f) show the tip speed ratio λ, the power coefficient cp, and the blade pitch angle β, respectively. Finally, panel (c) shows the resulting aerodynamic trajectory (red line).

In Fig. , the power coefficient starts at its maximum value due to MPPT, i.e., (β,λ)=(β⋆,λ⋆), which yields cp=cp⋆. The VSM inertial response increases the electrical power Pe during the negative ROCOF, whereas the aerodynamic or mechanical power Pm remains almost constant. Thus, the WT rotor decelerates, and the tip speed ratio λ decreases. After the ROCOF changes from negative to positive at ts,min⁡=2.5s, Pe rapidly decreases below Pm, and the rotor accelerates. Accordingly, considering the trajectory (cp,λ) in Fig. c, the WT leaves its MPP (λ⋆,cp⋆) during the negative ROCOF, with significantly decreasing λ but almost constant cp. Due to the grid synchronization delay of the VSM, the WT reaches its rotor speed nadir or (λmin⁡,cp,min⁡) at t=2.593s, i.e., shortly after the frequency nadir at ts,min⁡=2.5s. Afterwards, with increasing Ω, the trajectory converges to (λ⋆,cp⋆) for t→∞.

At t≈4.65s in Fig. a, the MPPT compensation resets ΩH0 to Ω such that Pe in panel (b) rapidly decreases to re-accelerate the WT to its MPP, called WT rotor speed recovery. Clearly, decreasing the rate limit Ω˙H0,max⁡ for the transition between active and inactive MPPT compensation in Eq. () leads to a smoother change in Pe, which is expected to cause less severe secondary frequency disturbances . However, this would slow down the WT rotor speed recovery. Finding a reasonable compromise is beyond the scope of this paper, but see Appendix  for further discussion on this point.

In Fig. , the initial steady-state power Pm(t0)=Pe(t0) is 5% below its initial MPP value in Fig.  due to Pset=95% or cp(t0)=95%⋅cp⋆ for t0=0. Note that the lower initial electromagnetic torque Me(t0) leads to (i) higher initial WT speed Ω(t0)=1 in Fig.  compared with Ω(t0)=91.8% in Fig.  and (ii) active pitch control; i.e., β(t0)>β⋆ in Fig. . The trajectory (cp,λ) in Fig. c starts at (λ0,cp,0) with λ0>λ⋆ and cp,0<cp⋆. The VSM inertial response increases the electromagnetic power Pe during the negative ROCOF so that the WT decelerates and λ decreases. At the same time, the pitch control decreases β due to Ω<1. The trajectory reaches the (local) WT rotor speed nadir or (λ1,cp,1) at t≈2.62s. Afterwards, during the positive ROCOF, Pe rapidly decreases such that λ increases, reaching (λ2,cp,2) at t≈4.44s. Finally, during constant but below-rated frequency Ωs<1 for t≥4, inertia provision is inactive, but the active power droop control increases Pe to the maximum available MPPT power, which is greater than the initial value Pe(t0). Thus, (cp,λ) converges to the MPP (λ⋆,cp⋆) for t→∞. Note that the trajectory in Fig.  converges to the MPP from the right side of the MPP, whereas, for Pset=100% in Fig. , the trajectory converges to the MPP from the left side of the MPP.

After the ROCOF changes from negative to positive at t=2.5s in Fig. d, the electrical power Pe≈Pref=P‾set⋆+Pv+P‾d (panel b) decreases as the VSM inertial power becomes negative; i.e., Pv<0. At the same time, the droop power increases as power reserves become available; i.e., P‾d=Pd,max⁡>0. The droop power P‾d>0 counteracts the VSM inertial power Pv<0 during the grid frequency recovery. Thus, both droop control and inertia provision are active at the minimum Pe at t=4s in Fig. b, whereas P‾d=Pd,max⁡=0 and Pv>0 holds for the maximum Pe at t=1s.

Considering Figs.  and  (panel b), the electromagnetic torque Me reaches its limit Me,max⁡ in both cases; i.e., the second constraint of the optimization problem Eq. () is active. However, the initial torque Me(t0) for 5% power derating in Fig. b is more than 12% lower than Me(t0) for MPPT in Fig. b. The torque reduction is higher than the power derating due to an 11% higher initial rotor speed Ω(t0) in Fig. a than in Fig. a. Clearly, the MRS-based derating maximizes the torque headroom Me,max⁡-Me(t0) for inertia provision by maximizing the rotor speed.

In general, the VSM inertial response to the initial negative ROCOF increases the electrical power Pe, which decelerates the rotor in both cases (Figs.  and ). However, due to a higher initial value, the rotor speed does not fall below its MPP value Ω=91.84% for MRS-based derating in Fig. ; on the other hand, Ω decreases to 88.15% for MPPT in Fig. . Moreover, the WT rotor deceleration leads to constant (or only slightly decreasing) mechanical power Pm in Fig.  but increasing Pm in Fig. . Consequently, the MRS-based derating strategy provides both additional kinetic energy and additional wind energy reserves for inertia provision.

To summarize, the optimized values for the VSM inertia constant Hv=Hv,max⁡ are 2.26 and 3.80s for Pset=100% (MPPT) in Fig.  and for Pset=95% (MRS-based derating) in Fig. , respectively. For these two exemplary operating points, the MRS-based derating of 5% increases the inertia provision capability in terms of Hv,max⁡ by ca. 3.80/2.26-1≈68% compared with MPPT.

The proposed WF inertia forecasting approach uses LUTs to map the operating points of all WTs to the WF inertia, assuming optimal tuning Hv=Hv,max⁡ for each WT in the WF. For the proposed solution of the optimization problem in Eq. (), Fig.  depicts the resulting LUTs of the form Hv,max⁡=f(vw,Pset) for all operating points (vw,Pset) within the range vw,cut-in≤vw≤15ms-1 and 90%≤Pset≤100%. For enhanced visualization and clarity, panels (a) and (b) show the same results from different points of view. In the 2-D plot (panel a), the color coding for the power setpoint Pset is the same as in Fig. ; in the 3-D plot (panel b), the datapoint colors indicate the active optimization constraints in Eq. ().

The LUTs were computed on a standard desktop computer (FUJITSU Workstation CELSIUS W580, Xeon E-2246G 6C 3.60 GHz 12 MB, 2 × 16 GB DDR4-2666, NVIDIA Quadro P1000 4 GB) using MATLAB's Parallel Computing Toolbox, Optimization Toolbox, and Simulink. Four CPU cores were used in parallel via a “parfor” loop to perform solver-based nonlinear optimizations , including numerous WT simulations executed in Simulink's rapid accelerator mode . The computation time depended strongly on the simulation and solver settings, in particular the optimality and constraint tolerances. Without any attempt at optimizing the software for numerical efficiency, the wall-clock time was typically of the order of 4 h for a complete LUT generation. Although execution time is a metric that quickly becomes obsolete and is influenced by many factors, the computational cost of the proposed method is acceptable for realistic applications, especially since the LUT generation is performed offline rather than during operation.

In Fig. , the rotor speed limit Ωmin⁡ is only active for low wind speeds near vw,cut-in. For most operating points characterized by vw<6.3ms-1, the torque rate limit M˙e,max⁡ constraint is active. For Pset=1, i.e., even without derating, the maximum inertia constant Hv,max⁡=4.55s at vw=6.3ms-1 exceeds the rated WT physical inertia constant HR=3.26s. However, for 6.3ms-1<vw≤vw,R, the torque limit Me,max⁡ reduces Hv,max⁡ with increasing wind speed. For above-rated wind speeds vw>vw,R and Pset=1, the virtual inertia constant Hv,max⁡ is smaller than the WT physical inertia constant H=HR due to the electrical power limit Pe,max⁡. The MRS-based derating increases Hv,max⁡ over the complete wind speed range.

Appendix  reports the Hv,max⁡ results obtained with the simplified solution derived in Sect.  and compares them with the complete numerical solution. Appendix  reports the results obtained with an additional (optional) constraint for WT rotor speed recovery.

For each operating point (vw,Pset) and corresponding optimal Hv=Hv,max⁡ tuning, the rotor speed nadir min⁡Ω and the mechanical power extrema min⁡Pm or max⁡Pm are evaluated in the inertial response time interval defined in Appendix Eq. (). Figure a shows the normalized deviation of min⁡Ω to the initial WT speed Ω0 (i.e., (min⁡Ω-Ω0)/Ω0) and Fig. b shows the normalized deviation of min⁡Ω to the MPP speed Ωmpp (i.e., (min⁡Ω-Ωmpp)/Ωmpp, where Ωmpp:=Ω0 for Pset=1). Figure a and b show the normalized deviation of min⁡Pm and max⁡Pm to the initial WT mechanical power Pm,0, i.e., (min⁡Pm-Pm,0)/Pm,0 and (max⁡Pm-Pm,0)/Pm,0, respectively.

The WT significantly decelerates during the inertial response at low wind speeds, although the rotor speed deviation does not exceed 20% of Ω0 in Fig. a. Clearly, the lower speed limit Ωmin⁡ is only relevant for a few operating points at low wind speeds and minor derating; i.e., for all other operating points the available kinetic energy reserve cannot be fully extracted for inertia provision due to active torque or power constraints. Decreasing Pset increases Ω0 such that the rotor speed does not fall below its MPP value if the derating is high enough, i.e., for min⁡Ω-Ωmpp>0 in Fig. b. In this case, no rotor speed recovery is needed after the inertial response.

For MPPT, the rotor deceleration decreases the power coefficient cp. Thus, the aerodynamic power decreases during the inertial response; i.e., min⁡Pm<Pm,0 for Pset=1 at below-rated wind speeds in Fig. a. However, minor derating significantly increases min⁡Pm. For Pset≤95%, the reduction in Pm is negligible due to min⁡Ω≈Ωmpp, or Pm even increases due to min⁡Ω>Ωmpp, as shown in Figs. b and a. In summary, the MRS-based derating leads to less severe rotor deceleration due to higher initial rotor speed and thus also higher aerodynamic power during the inertial response.