A Model Predictive Approach for Enhancing Transient Stability of Grid-Forming Converters

A model-based approach for controlling post-fault transient stability of grid-forming (GFM) converter energy resources is designed and analyzed. This proposed controller is activated when the converter enters into the saturated current operation mode. It aims at mitigating the issues arising from insufficient post-fault deceleration due to current saturation and thus improving the transient stability of the GFM Inverter Based Resources (IBRs). The considered approach conveniently modifies the post-fault trajectory of GFM IBRs by introducing appropriate corrective phase angle jumps and power references. These corrections are optimised following a model predictive approach (the model referring to post-fault dynamics of GFM IBRs in both saturated and normal operation modes). While constructing the proposed controller, the situation for GFM IBRs to enter into the saturated operation mode are identified. The effectiveness of this transient stability enhancement approach by means of dynamic simulations under various grid conditions is tested and discussed. The results demonstrate much better transient stability performance.


I. INTRODUCTION
In response to climate change concerns, many countries have outlined national strategies to achieve net-zero emissions in power systems [1], [2].This largely relies on decarbonizing energy supply by developing renewables and phasing out fossil fuels.Unlike traditional fossil-fuel synchronous generators (SGs) interfaced resources, most renewables (such as wind and solar) are interfaced with power grids through power electronics, termed inverter-based resources (IBRs).Nearly all the IBRs in service today are 'grid-following' (GFL), which inherently do not independently contribute to power system inertia and strength [3].This poses major challenges for system stability as the generation mix continues to evolve towards higher shares of renewables, in the sense that power disturbances cause faster frequency variation due to the low system inertia, and voltage disturbances are propagated much farther due to the low grid strength.To maintain the stability of the electrical system, one may then use 'grid-forming' (GFM) technologies, which allow IBRs to contribute to system stabilisation like SGs by providing virtual inertia, voltage, and frequency support [4]- [6].
Operating such GFM IBRs is simultaneously challenging and providing opportunities for enhancing the system transient stability.A major challenge arises from very limited overcurrent capabilities of inverters [7], which inhibits their ability to provide sufficient active and reactive power during and after large disturbances (e.g., grid faults), thereby affecting the transient stability.At the same time, oversizing the converter to achieve higher fault currents is rather costly.Given that the transient behaviors of GFM IBRs are mainly dictated by their control strategies, it is more promising and cost-effective to enhance transient stability by refining the control system design, which admits a larger degree of freedom and flexibility in comparison to SGs of which the behaviors are tightly linked to the rotating physical components.
A substantial body of literature has been dedicated to the transient stability enhancement of GFM IBRs.References [8]- [10] proposed modifying the GFM IBR's control loops for transient stability enhancement.However, these approaches do not consider the impact of current saturation on transient stability.In practical implementation of GFM IBRs, current saturation is an essence to protect power electronic switches from overcurrent.There are two typical categories of current saturation [11]: (a) virtual impedance-based approaches, which emulate a virtual impedance after the reference terminal voltage to reduce the over-current, and (b) current reference saturation (CRS), which puts a hard limit on the reference current generated by voltage controller of the GFM IBR.The authors of [12]- [15] proposed transient stability enhancement methods that consider the virtual impedance based current saturation.While the virtual impedance-based methods might exhibit a higher margin of stability compared to current reference saturation (CRS)-based approaches, they may fail to promptly constrain the inverter's output current within the limit.As a result, the inverter may be overloaded until virtual impedance limits the current [11] Reference [16] shows if the angle of current saturation in CRS approach is not set correctly, GFM IBRs may be stuck in the saturated operation mode.For this purpose, [16] proposes to optimise the current angle to prevent the GFM IBRs from being locked in the saturated current operation mode.The authors in [17] proposed to change the reference power of GFM IBRs again to enhance transient stability.Data-driven predictive control was proposed in [18], where a corrective power reference change was introduced to achieve enhanced transient stability.The possible benefits of direct corrective change of the angle alongside power reference change was not considered in [16]- [18].Therefore, the whole potential of modifying the synchronization control for transient stability enhancement was not investigated in these studies.
In this paper, the goal is to untap this opportunity by modifying the post-fault trajectory of GFM IBRs via introduc-ing appropriate corrective phase jumps and power reference change into the Active Power Control (APC) loop of the VSG.To explore this approach, following are checked: 1) Optimizing these two control signals through Model Predictive Control (MPC) approaches.More specifically, this is achieved by formulating an MPC with an objective function that is designed to minimize deviations of APC angle ( i.e., the angle produced by the APC loop).This, in turn, follows the intuition that reducing this deviation decreases the risk of approaching the instability angle threshold.2) How well this approach performs, and thus get intuitions and results relatively to its limitations and capabilities in rejecting network disturbances.The proposed MPC strategy is designed essentially to help the GFM IBR return to steady-state post-fault equilibria as fast as possible.Since the complete post-fault transient behavior of GFM IBRs is affected by both saturated current and normal operation modes, this approach requires modeling the system in both these cases, plus assess the conditions of the transitions during and between these modes.Indeed, as analysed in Appendix B, switching into the saturation modes happens when APC angle exceeds a certain threshold.This exceeding angle causes the reference current produced by the voltage controller to exceed the maximum allowable current.Added to this is the analysis of the dynamics of the GFM IBR's APC angle.Although the corrective signals are only applied when the GFM IBR is in the saturated mode, the MPC optimization's horizon is the whole transient until reaching the steady-state.The corrective signals are set to zero when the GFM IBR exits saturated operation mode.
The reminder of this paper is organized as follows.Section II presents a model for the operation of a GFM IBR in both normal and saturated modes, and proposes a solution for removing high frequency oscillations.Section III analyses the transient stability of GFM IBRs with a focus on the impact of the current saturation.This section also briefly reviews two benchmarks when analysing transient stability enhancements.Section IV presents and discusses the implemented MPC approach for achieving such transient stability enhancement.Section V analyses the performance of such a controller by means of a set of selected dynamic simulations.This is then followed by a set of conclusions about the suitability of the proposed approach, and its capabilities in reducing the risk of transient instability.

II. OPERATIONAL CHARACTERISTICS OF GRID-FORMING CONVERTER
This section provides an overview of GFM IBRs (virtual synchronous machines in particular).It also discusses the situation when a GFM IBR enters into the saturated current mode or oscillates between saturated and normal modes.
As modeled in Fig. 1 [8], [9], [19], [20], GFM IBRs are typically connected to the grid through LC filters with inductance L f and capacitance C, and a transformer with impedance Z t .Modeling the grid via a Thevenin equivalent, i.e., a voltage source V g behind an impedance Z g leads to an equivalent impedance of the grid plus transformer as Z := R+jX = Z g +Z t .GFM IBRs typically consist of three layers of controllers [23].The outer layer includes voltagereactive power droop control and APC.The inner layer in its turn is typically a voltage controller, and current controller is the innermost layer.All of these controllers operate within the synchronous dq-framework.A GFM IBR participates in reactive power sharing through a voltage-reactive power droop rule.This controller determines a reference V ref d for the inner control layer (i.e., the voltage controller), and a reference V ref q -that is set to zero in normal modes.In cascade, setting these references makes the voltage controller generates current references that drive the innermost controller (i.e., the current controller) in the normal operation mode.The GFM IBR synchronizes itself to the grid through APC, obtained by means of active power droop and virtual inertia control [9].This also means that the arc of the AC variable in the dqcoordinates (the argument of the cosine form the AC variable) is set by such APC controller.To avoid confusion, recall that the arc of the AC variable is not the relative difference between the arcs of the actual signal and reference signals, i.e. the angle (details in Appendix A).
In practice, the dynamics of the voltage and current control loops are much faster than APC control loops.This leads to the first approximation they can be neglected in the transient stability analysis.We thus share the same assumptions as in [9], [10], [15], [16].
To wrap up, Fig. 1 shows the control diagram of a GFM IBR.In this plot, the presence of the typical VSM control with frequency bound and internal controllers, i.e. voltage controller and current controller (in blue) are observed [8], [9], [19], [20].The logical position of the controller proposed in this paper is highlighted in purple.The voltage and current phasor diagram associated with the GFM IBRs' normal operation is shown in Fig 2. Note that the Thevenin grid voltage and arc of its DQ-coordinate are virtual variables.The total output current of the inverter I s is the summation of the current flowing the filter capacitor (jωCV ) and the actual current fed into the grid (I = (V − V g )/Z), which should not exceed the inverter current limit I max s .

A. The Causes and Effects of Entering to the Current Saturation mode
During the normal operation, the current from the inverter is usually below the maximum allowable one.However, when a disturbance causes either changes in the angle of the dqcoordinate, i.e.APC angle, (as in Fig. 3) or in the Thevenin voltage magnitude (as schematized in Fig. 4), the inverter output current may exceed the maximum allowable one.In that case, the native current saturation block ("Saturation" in Fig. 1) is activated.This generates a new reference current with magnitude I max s and angle of β from d-axis reference frame.The result of generating this new reference current is that of deactivating the "Voltage Control" block in Fig. 1).As proven in Appendix B, a GFM IBR enters into the the Fig. 1: A block schematic viewpoint of the typical control structure of a Grid-Forming Converter.The additional control structure that is proposed and analysed in this paper is highlighted in purple.saturated current operation mode if the APC angle θ exceeds the threshold θ sat defined as Note that, neglecting the effect of the filter capacitors (since these capacitors are small and have little impact on the actual voltage and currents in nominal frequency), such threshold actually depends on the grid strength condition and grid's Thevenin voltage (as summarized in Fig. 7).More verbosely, for a strong grid, a deep voltage sag will cause saturation no matter what the APC angle θ is.However, as the grid becomes weaker, the saturation threshold θ sat becomes larger, and such increase is more significant when the grid voltage is relatively lower.If the grid is very weak, the GFM IBR will never fall into the saturated current operation mode during severe voltage dips.This is because of the fact that a smaller current (I) in Fig. 4 can provide the same ZI if the grid is weaker (bigger Z).
As mentioned earlier in this section, current saturation causes deactivation of the current reference generated by the voltage controller.Hence, the GFM IBR's voltage V in this case does not follow its reference , and it is determined by V g , I max s , β, θ, and Z as depicted in Fig. 6  and 5. Neglecting again the impact of the filter capacitors, one may perform a vector analysis as in Appendix C, and obtain that the voltage under saturated current operation modes evolves as

B. Causes and Mitigation of Oscillations During Post-fault Recovery
While in a saturated current operation mode, the voltage controller is deactivated, but still generates a current reference.If this reference is bigger than the maximum allowed current, then the GFM IBR can not switch back to the normal mode [21].The magnitude of this reference in its turn depends The diagram of how voltages and currents may actually be if, due to a deep voltage sag, the system enters in a current saturated mode.This diagram also demonstrates why, while in saturation mode, Again, the difference of this case and Fig. 5 is the cause of saturation.on the GFM IBR's APC angle, the parameters of voltage controller, the anti wind-up strategy, as well as the post-fault grid conditions.It may thus happen that the GFM IBR returns to a normal mode, but the conditions are such that it soon reenters in saturation.Intuitively, there exist conditions for which the GFM IBR switches back-and-forth between saturated and normal modes, inducing oscillations in the system [21].In other words, there is a range APC angles in which if the GFM IBR operates in the normal operation mode, its output current is more than the maximum allowed value, and if it operates in the saturated mode, its voltage controller provides a current reference less than the maximum allowed value.In this range of APC angle, it oscillates between these two operating modes.
The condition for oscillations is implicitly defined by intersection of two set: a) the set of APC angles at which i ref,sat dq , the output of the voltage controller during saturated mode, is lower than the maximum allowed current, and b) the set of APC angles in which θ sat , the GFM IBR's APC angle, exceeds the saturation threshold.More formally, the quantity (a) is formulated as Summarizing, if θ ∈ R and θ sat ≤ θ then the GFM IBR switches back-and-forth between saturated and normal operation modes, inducing oscillations in the system.R is called as the "set of return angle" in the remainder of the paper.
The presence of these oscillations introduces a high frequency disturbance for a couple of hundreds of milliseconds.To mitigate this phenomenon, This paper proposes to hold the GFM IBR in a saturated current operation mode if the APC angle θ is bigger than the saturation threshold θ sat according to Appendix B regardless of the magnitude of the reference current produced by the voltage controller.Indeed, if the estimator of the APC angle returns perfect estimates, then such oscillations can be fully mitigated.However, imperfect estimates may cause the presence of minor oscillations.

III. TRANSIENT STABILITY OF GRID-FORMING CONVERTER
The concept of transient stability of a GFM IBR relates to its ability to retain synchronism after a big disturbance [8].Synchronism is mainly associated with the APC angle and frequency.Based on the control design shown in Fig. 1, the APC control loop follows the swing equations [22], i.e., Neglecting the effect of the damping term D p enables performing a transient angle stability analysis of GFM IBRs by means of the traditional equal area criterion [22].However, because of the current saturation, the relation between output power and APC angle changes.Indeed, the power-angle re-lations during normal and saturated modes can be expressed, respectively, as [16] P unsat = V g V X sin θ (7) These equations are decent approximations only if power losses (i.e., the resistive component of Z) are insignificant.
According to these equations, the saturated power output is considerably less dependent on X.Indeed, if filter capacitor is very small, P sat is fully independent of X.Therefore, while a GFM IBR is in the saturated mode, the traditional action of reducing X for enhancing its transient stability is no longer beneficial.
To graphically compare effect of normal and saturated output powers on transient stability, Fig. 8 shows how the deceleration area for saturated power output (area A 1 ) is considerably smaller than the unsaturated deceleration one (area A 1 plus area A 2 ).Here θ 0 is the initial APC angle, θ af is the APC angle immediately after the fault is cleared, θ sat is the saturation threshold, while θ sat U E and θ sat ZC are the APC angles corresponding to the saturated unstable equilibrium angle and zero power output angle, respectively.These two angles are in their turn smaller than those of the unsaturated case (i.e., θ unsat U E and θ unsat ZC = π).To summarize, entering the saturated current operation mode reduces the deceleration area, the unstable equilibrium angle θ U E and the zero power angle θ ZC .
Once the APC angle exceeds the unstable equilibrium angle θ sat U E , the GFM IBR will be unable to return to the stable equilibrium without corrective control.Adopting an effective corrective control means then -given the intuitions abovesteering the system so that even if the APC angle exceeds θ sat U E the system can be brought back to stable operation.However, after passing θ sat ZC from which the GFM IBR may absorb power reversely, the system is being operated in an unsafe condition and is considered unstable.Operating GFM IBR in APC angles less than θ sat ZC can be considered safe because there is a chance for the inverter to return to the normal operation mode.

A. Benchmarks in transient stability Enhancement
Among different GFM control architectures from the literature that can provide transient stability enhancement, two can be considered as benchmarks: (1) Bounding Frequency [8] and (2) Compensating Saturation [7].
Bounding Frequency essentially corresponds to limit the acceleration area in Fig. 8 by limiting frequency generated by APC.This limitation contributes to transient stability enhancement in two ways.First of all, the post-fault frequency is smaller, meaning that less deceleration is needed to bring the frequency back to normal operating frequency.Secondly, the post-fault APC angle (θ af in Fig. 8) becomes considerably smaller than when operating in the unbounded cases.Note that the boundary (maximum) frequency should be sufficiently large so that the GFM IBR can participate in active power sharing during normal operation.Compensating Saturation aims at mitigating the adverse effect of saturation on transient stability by means of subtracting a virtual power from the reference power in the APC control loop.As suggested in [7], to compute such a virtual power one may use the unsaturated current reference, a method though only applicable for some specific voltage controllers (e.g., a virtual impedance voltage controller) considered in this strategy.For PI voltage controller (which is used in our paper), current saturation makes the voltage controller deactivated as explained in subsection II-B.Therefore, the current reference generated by the voltage controller is not a meaningful signal in the control schema of Fig. 1.In this paper, in order to adopt this approach as a benchmark (not as the main proposed methodology) for assessing our main methodology, the difference between unsaturated and saturated power is first computed based on the relation of active power and the estimated APC angle, and then subtracted from the reference power.This approach provides extra post-fault deceleration compared to uncompensated control.
This extra deceleration might, then, be close to the required one.However, if it is smaller than the required amount, the GFM IBR might become unstable, while if it is bigger, this might lead to additional frequency and angle oscillations (vanishing in time as the system settles to its steady-state equilibrium).This suggests that this virtual power that should be removed from the reference power in the APC control loop should be optimized.This challenge is though met by an opportunity: since the synchronization mechanism of the GFM IBR is dependent on a re-designable control system instead of on less controllable rotating elements, there is the opportunity to directly change the GFM IBR's APC angle during a transient event, and in this way enhance the controllability over the trajectory of the GFM IBR.In other words there exists a degree of freedom to improve transient stability while the system oscillates.How to exploit this possibility will be discussed in more details in Section IV.

IV. MPC FOR TRANSIENT STABILITY ENHANCEMENT
In this section, an MPC approach to appropriately modify post-fault system trajectories is proposed.As shown in Fig. 1, the added logic (in violet) contribute two control signals, ∆P ref and ∆θ c , each providing extra controllability to the post-fault trajectory of the GFM IBR.The proposed scheme applies a corrective action only when the GFM IBR transitions into a post-fault saturated current operation mode, so that there is no risk of worsening the stability properties of the system if the disturbances are not large enough to bring the GFM IBR into a saturated mode.
Notation-wise, k ∈ 0, 1, . . ., T T d − 1 is used as the time index within the MPC horizon (T thus being the absolute length of the rolling horizon, and T d being the time step of such a discretized horizon).κ is used as the absolute time index, even if we often keep this index as tacit (in other words, every instant k within the MPC rolling time window refers to the absolute time κ + k).
The MPC controller will thus provide for every κ two reference vectors ∆P ref (k) and ∆θ c (k), k ∈ 0, 1, . . ., T T d − 1 , built to minimize the sum of the squares of the APC angle deviation from the post-fault equilibrium point, i.e., min subject to a set of constraints C.Here the set of decision variable D comprises the corrective signals ∆P ref and ∆θ c in each time step and variables that model dynamics of the system.As for the set of constraints C, they either describe the dynamical behaviour of the GFM IBR in its transient (i.e., subset C 1 ) or impose physical limits (i.e., subset C 2 ).
To list the elements of C 1 , we note that first and foremost the key variable in the transient stability analysis is the APC angle θ.Accounting for ∆θ c , discretizing equation ( 6) leads to Similarly, C 1 includes the swing equation adjusted to account for the active power reference change ∆P ref , and thus Note how equations ( 10) and (11) suggest that ∆P ref (k) influences frequency, while ∆θ c (k) directly affects the APC angle.Consequently, these two variables offer control over θ and ω.
The power output of the GFM IBR depends on the APC angle θ and and its operational mode, i.e. whether in saturated current operation or not.To capture this state, a binary variable n(k) is introduced, where n(k) = 0 and n(k) = 1 represent respectively saturated and normal mode at time k.As discussed in subsection II-A, the GFM IBR is in the saturation mode if θ(k) is more than θ sat .To capture the fact that n would be zero or one depending on this binary comparison, thus the ancillary parameter M is added, to be chosen sufficiently big, and with it the relations is introduced as so that n = 1 only if θ(k) is less than θ sat .
According to subsection III, the active power output of the GFM IBR depends on whether it is in the saturated or normal modes.Being in one of these two modes is modeled with n(k) as introduced in ( 12) and (13).Therefore, this variable enables also representing the active power in saturated and normal operation modes for the GFM IBR as: Aforementioned equations form C 1 .In addition to these constraints, another subset of constraints C 2 is needed to ensure variables of the system behave within desired range.C 2 should ensure that the APC angle estimation module can accurately track its variations.Since fast changes of APC angle makes it challenging to have an accurate estimation of the APC angle, the changes of θ should be constrained, i.e., Similar considerations on the fact that the upper and lower values of frequency should be constrained to avoid large frequency deviations lead to include in C 2 also the bounding box Moreover, to ensure that the GFM IBR does not absorb active power from the grid, θ should be maintained within the range of greater than zero and less than the saturated zero crossing APC angle, i.e., When the GFM IBR is in normal operation mode, ∆P ref should be set to zero.Otherwise, APC control loop will receive corrective signal from the proposed MPC.This is not desired because the corrective signal should be applied only in saturated mode, so C 2 should include According to (10), the APC angle changes is summation of ∆θ c (k + 1) − ∆θ c (k) and integral of frequency.When the GFM IBR returns to the normal operation mode, ∆θ c is maintained at a constant value, with no corrective phase jump.Otherwise, the APC control loop will be exposed to corrective signals from the proposed MPC during the normal operation mode.This is not desired as GFM IBRs' APC does not need any corrective signal for the transient stability during the normal operation mode.On the other hand, during the saturated current operation mode, ∆θ c decreases so to restore the APC angle to a stable equilibrium.This requires adding the following constraint to C 2 : Before introducing the following constraint, we note that it has then been added after realizing that the search space, without this constraint, was sufficiently big to significantly slow down the numerical computation of the optimum.The constraint imposes that once the GFM IBR transitions from saturated to normal operation mode, it should remain in that state.This means imposing One may apprehend that this type of constraint may cause the optimization problem to potentially become infeasible.In our simulations, this had never happened, even if we do not have mathematical proofs guaranteeing this particular fact.Finally, the proposed optimization problem is characterized by a set of initial conditions (e.g., the APC angle and corrective control signals, θ(0), ∆θ c (0) and ∆P ref (0)).They are initialized according to their current point values or estimates.
Compliant with the MPC framework, once the objectiveoptimal solution of problem ( 9) is computed to a certain predefined numerical precision, the proposed control approach uses ∆θ c (1) and ∆P ref (1) as the corrective control signals.After this, the optimization problem is reinitialized and resolved until coming back to normal operation mode, so that the solutions are actuated in a rolling horizon manner.

V. CASE STUDIES
To assess the effectiveness of the proposed oscillation removal method and model predictive control (MPC) approach, simulations were conducted using Simulink/MATLAB and compared against established benchmarks.The Artelys Knitro solver was utilized to solve the mixed-integer non-linear optimization program of section IV.The simulated system models a farm of parallel, identical generating facilities (GFM IBRs) connected to a 132 kV power system via a transformer, and modeled as a equivalent GFM IBR [23].The detailed specification of the base case is summarized in Table I.

A. Analysis of the effectiveness of the MPC approach in enhancing transient stability
This subsection investigates to what level the proposed approach enhances the transient stability by comparing the time domain responses of the MPC added system versus the system with original controller (i.e., where no transient stability enhancement measure is employed).Since a GFM IBR can be connected to either a strong or weak grid, the transient stability enhancement that the proposed approach brings is studied for both scenarios.In this section, a strong grid and a weak grid are characterized via Z g = 0.3 p.u. and Z g = 0.9 p.u.; respectively.Furthermore, the proposed MPC is compared with two benchmark methods, namely, the frequency bounding approach (Strategy B) and the compensating method for the difference between saturated and unsaturated power (Strategy C), as described in Subsection III-A.The fault is simulated by decreasing the Thevenin voltage to 0.05 p.u. 1) Tests Under Strong Grid Conditions: According to Fig. 9, a fault occurrence at t 0 = 0.1 seconds and lasting 0.45 seconds results in an increase in the GFM IBRs' APC angle from 0.407 Rad to 1.569 Rad.In the absence of corrective measures, the APC angle surpasses its instability limit even before the fault clearance.In this case the compensating control strategy (i.e., C) mitigates this effect by slightly decreasing the acceleration.However, the method is not efficient in long fault scenarios where the difference between on-fault saturated and unsaturated power is not significant enough to ensure stability.In contrast, for shorter faults, a considerable discrepancy between post-fault unsaturated and post-fault saturated power would assist in maintaining system stability.
In this case, setting the frequency bound (Strategy B) to 1.0066 p.u. limits the APC angle increase rate.However, due to insufficient post-fault deceleration, the system passes the unstable equilibrium in saturated mode shortly after the fault, as depicted by the transition in Fig. 9 from the light yellow to the full yellow area.More precisely, the area colors in this figure indicate: • green: a GFM IBR operating normally (i.e., unsaturated) after the fault; • light yellow: a GFM IBR operating in a saturated postfault operation.In this case, the APC angle is smaller than the unstable saturated-mode equilibrium, there may be a possibility to maintain stability.• full yellow: a situation where the APC angle is greater than the unstable saturated-mode equilibrium.However, this situation is so that with appropriate compensating control, the system may return to a normal operation mode.
• red: a situation where the APC angle is greater than the active power crossing zero.In this area, the GFM IBR absorbs active power from the grid, which is hazardous for the DC link capacitor.Although there may be a chance to return to normal operation with corrective control, operating in this area is unstable and unsafe.• orange: a situation where the active power is negative, and the GFM IBR absorbs power from the grid.Although this is an undesired situation, it is still an acceptable occurrence when the GFM IBR briefly enters this area and returns to the green area.The maximum duration that GFM IBR is allowed to enter this area depends on the capacitance of DC link capacitor and its maximum tolerable voltage.This situation occurs during substantial deceleration with minor damping.
The implementation of Model Predictive Control (MPC) with a 20 ms time step effectively restores the GFM IBR to the green area, as depicted in Fig. 9.The corrective control law during saturated-mode operation, as presented in Fig. 10, comprises changes in reference power and corrective phase jumps.The corrective phase jump facilitates a prompt return to operating point APC angle values, while the reference power changes dampen frequency deviations as expected.The values for the corrective control law computed by the proposed scheme for the strong grid-connected GFM simulation in Fig .9.
2) Tests Under Weak Grid Conditions: Fig. 11 depicts the transient response of a GFM IBR connected to a weak grid when subjected to a 281 ms fault.In contrast to the strong grid situation, now the inverter's response to the fault displays a greater propensity for instability, something that arises primarily from the fact that the initial APC angle is higher and closer to the instability boundary.Consequently, even a brief fault event can readily cause the inverter to cross this boundary.
During the fault event in this weak grid, the GFM IBR does not enter the saturated current operation mode, as discussed in Subsection II-A.Consequently, the fault-on behavior of strategy C remains analogous to the original strategy.However, following the fault, the GFM IBR transitions into saturated current operation mode.Regrettably, this occurs though too late, and the GFM IBR has already entered the unsafe area (red).The bounding frequency approach in this case limits frequency increases, but due to its insufficient post-fault deceleration properties, the system transitions from the light yellow to the full yellow area within a few milliseconds after the fault.On the other hand, the MPC corrective control offers an objective-optimal reference power change and phase angle jump during the saturated current operation mode, enabling the GFM IBR to return to the normal operating point swiftly.
3) Sensitivity Analysis: To demonstrate the capabilities of MPC in enhancing the transient stability of a GFM IBR, its performance is against the other mentioned strategies for varying droop coefficients (damping) and Thevenin voltages (representing the proximity of fault location to the GFM IBR).Figs. 13, 15, 14, and 16 demonstrate a significant increase in CCT due to MPC corrective control.Furthermore, these results illustrate that CCTs for strategy B and MPC exhibit the least sensitivity to parameter and situational changes.This can be attributed to the limited acceleration for these two strategies, regardless of the output power or damping coefficient.oscillation removal strategy, whereas the other case does.The former case is then depicted in the left side plots of Fig. 17, while the latter is presented in the right side ones.Note also that the color coding in Fig. 17 is not related to color coding in subsection V-A.Both cases present the GFM IBR subjected to a three-phase fault that occurred from 0.1 sec to 0.2 sec, indicated by red area (A f ) in Fig. 17.As shown by the left panels, significant oscillations were observed from 0.2 sec to approximately 0.8 sec, indicated by the gray area (A o ).However, with the application of the proposed oscillation removal control approach, the GFM IBR remained in the saturated current operation mode, indicated by the yellow area (A no ), until conditions for normal (unsaturated) operation were restored, indicated by the green area (A n ).

VI. CONCLUSIONS
This paper presented a novel model-oriented strategy to improve the GFM IBRs transient post-fault behaviour considering current saturation, and provided a model-driven analysis of its performance.It has also been shown that GFM IBR's APC angle, grid voltage and grid strength are the main factors that contribute to the situation that GFM IBR's reference current exceeds its limitation.This model has been used to draft an MPC approach which recursively generates an objective-optimal corrective phase jump and reference power change.
The performance of this novel scheme has then been analysed by means of simulations.The proposed method exhibited better performance compared with original and benchmarks under both weak and strong grid conditions.
Although we here adopt a virtual synchronous machine approach as in [9], the proposed methods can also be extended to other types of GFM control technologies by simple modifications in the formulation.

ACKNOWLEDGMENTS
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 956433.Additionally, this paper is supported by Project ROSES (EP/T021713/1) funded by UKRI for the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) license to any Accepted Manuscript version arising.
Thanks Prof. Dr. Yusheng Xue for providing useful feedback on the research direction.

APPENDIX A: ARC AND ANGLE
Voltages and currents in an AC system are sinusoidal variables, expressable as r(t) = R cos ω 0 (t − t 0 ) + α r0 (22) where R is the magnitude of the variable and ω 0 (t−t 0 )+α r0 is the time-dependent arc which changes constantly, connected with the angular frequency ω 0 .Differences between arcs of grid's AC variables affect the active power flow insides the grid.Since it is complex to work with continuously changing

Time (s)
Non-oscillatory Fig. 17: Comparison of the benefit that the modified oscillation mitigation control approach proposed in subsection II-B by this paper brings against the original CRS saturation presented [16].The panels on the left represent the dynamics of the system under the latter strategy, while the right panels represent the dynamics under the new approach.Note that this specific group of figures aims only at investigating the effect of removing the oscillations, and thus considers a system not implementing the proposed MPC for stabilizing GFM IBRs.arc in power system analysis, concept of angle, which is arc of the variable minus arc of a reference variable is usually used.The voltage of a bus is usually considered as the reference for the angles of the system.In steady-state, this difference is a constant value which can be used for both static and dynamic analyses.
In context of GFM converters, the APC angle means the difference of the arc of APC and arc of a reference variable, which is Thevenin voltage of the grid.

Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :
Fig. 2: A graphical summary of the voltage and current signals in the synchronous reference frame during a steadystate normal operation mode.

ZFig. 7 :
Fig. 7: A schematic representation of how the saturation angle threshold θ sat varies with the grid Thevenin voltage V g for different values of grid impedance Z.The figure distinguishes between different impedance levels by means of differently colored solid and dashed lines.

Fig. 8 :
Fig. 8: Power Versus APC Angle plot, demonstrating the GFM IBR's output power in normal and saturated operating modes.The highlighted acceleration and deceleration regions play pivotal roles in the analysis of transient stability under these conditions.

Fig. 9 :
Fig. 9: Simulation results of the temporal behavior of the transient APC angle and frequency changes for a GFM IBR connected to a strong grid with different corrective control strategies.

Fig. 11 :
Fig. 11: Simulation results of the temporal behavior of the transient APC angle and frequency changes for a GFM IBR connected to a weak grid with different corrective control strategies.

TABLE I :
Parameters describing the simulated GFM farm.