Virtual Power-Based Technique for Enhancing the Large Voltage Disturbance Stability of HV Grid-Forming Converters

Grid-forming control plays an essential role in the modernization of HV electrical transmission grids, particularly in mitigating challenges imposed by large voltage disturbances. In this context, the concept of virtual power has gained prominence as a potential approach that improves stability following such disturbances. This paper illustrates the use of virtual power method in enhancing the large disturbance stability given by the grid-forming control. An adaptive virtual impedance is also proposed to further improve system dynamics. The methods are evaluated through large disturbance stability analysis and time-domain simulations.


I. INTRODUCTION
The power grid is undergoing significant transformations such as the increasing penetration of Renewable Energy Systems (RESs).These RESs are connected to the grid through Power Electronic (PE) converters.Nowadays, most PE converters are connected to the grid with grid-following control.This type of control allows the PE converter to adjust the output current according to the power reference while being synchronized with the voltage measured at the Point of Common Coupling (PCC) [1].The PE converter is therefore referred to as This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version may no longer be accessible a controlled current source.However, as the number of RESs increases, new grid codes have specified that PE converters must be able to regulate voltage and frequency [2].This has led to the development of grid-forming as a potential control to form the sinusoidal voltage waveform by operating as a controlled voltage source [3]- [6].The implementation of grid-forming presents several challenges due to the constraints set by power generation.In particular, voltage sags, which are a common occurrence in power grids, can cause voltage drops below the minimum operating level of PE converters, resulting in an overcurrent phenomenon [7].In contrast to the behavior of synchronous generators, PE converters do not have an embedded overcurrent capability.Thus, it is crucial to ensure that the converter operates within its rated capacity.As such, the grid-forming control must include a current saturation strategy to limit its output current.Two types of current limitation techniques are predominant in the literature: the first one is based on a virtual impedance whose value is adjusted to decrease the current below a specified threshold [8], [9].The latter -the main focus of this paper -consists of integrating current control in the grid-forming system, and activating the current saturation mechanism whenever an overcurrent is detected.Subsequently, the system will provide saturated current references for the current control to effectively handle the overcurrent situation.[10], [11].
Even though implementing current saturation is an effective technique for protecting the power system from significant disturbances, it can cause system instability if not carefully implemented.[12]- [15] have shown that current saturation reduces the critical clearing time and limits the maximum allowable power angle.One solution is to use a Phased Locked Loop (PLL) when the current is saturated and to adjust the synchronization angle control accordingly [16], [17].However, it requires a complex control architecture along with the study of the additional dynamics of the PLL.Another method consists of deploying the virtual impedance-based current limitation [8], [18], [19].However, further research is needed to determine the optimal control since knowledge of the output grid impedance is required.In [13], a virtual power-based method is proposed as a potential approach to improve system synchronization following fault clearance.This method aims to widen the stability margin of the system by increasing the maximum value that the virtual power can reach.This paper builds upon this approach and features updates that further improve the system's dynamics.
This paper has two main objectives.The first one is to conduct a comprehensive analysis of the system behavior upon the application of virtual power in response to diverse grid fault scenarios.The second one is to propose an adaptive virtual impedance to enhance the large disturbance stability behavior of a grid-forming converter subject to any kind of voltage sags while ensuring adherence to grid code specifications.
This paper is structured as follows: Section II defines the topology and the control of the PE converter.Furthermore, it describes the application of the virtual power method.Section III presents a study of the system behavior following the application of the virtual power under two fault scenarios.Section IV illustrates the mechanism of a new adaptive virtual impedance.The methods are then evaluated for several types of grid faults.

II. PRESENTATION OF THE SYSTEM: STRUCTURE AND PURPOSE
A. Physical aspect of the structural design The hardware configuration illustrated in Fig. 1 serves as the main reference system.It is an HV system consisting of a Voltage Source Converter (VSC) connected to an equivalent Thevenin grid model (L g ) via a transformer (R c , L c ).A per-unit (pu) model is considered: the transformer is replaced by its inductance (the resistance is considered negligible) and the grid impedance ratio is R g /X g ≈ 0. Table I presents the hardware characteristics of the system.

B. Grid-forming control
As introduced earlier, the grid-forming control is considered for the low-level control of the converter.In this particular scheme, the converter controls the instantaneous three-phase modulated voltage v * mabc in both magnitude V m and synchronization angle θ * m .1) Active Power Control: In this section, the fundamental concepts of active power control for HV applications are recalled.Indeed, the angle θ m is controlled by the active power control.Since R g /X g ≈ 0, the active and reactive power at PCC can be expressed as follows: (1) Where δ is the angle between the phasors associated with the instantaneous variables v m and e g , respectively.V m and E g are the RMS values of these vectors.
Referring to the work demonstrated in [6], the active power control used is an IP controller which consists of embedding an inertial effect.The active power control architecture depicted in Fig. 2 comprises two primary control parameters: the inertial constant H that determines the responsiveness of the control system, and the proportional gain K p that dampens frequency oscillations.The closed-loop system can be assimilated to a second-order system with and H = 3 s and a damping factor (ζ SY S ) of 0.7 for the computation of K p .V m and E g are approximated to 1 pu.A strong grid of SCR = 20 is considered.The initial state of the active power is 1 pu and the voltage is 1 pu.
2) Voltage references generation: There are various methods to generate voltage references, and the choice of the appropriate method depends on the specific requirements and characteristics of the grid-forming system.In this paper, the final modulated voltage references are generated by a current control, which receives its current references from the electromagnetic model between v m and v g .[20] revealed that a quasi-static model leads to an overall more stable transient behavior than a dynamic electromagnetic model.The quasi-static model is illustrated in (2).A PI controller is implemented in the current loop with a 2 ms time response.The corresponding control structure is shown in Fig. 2.

C. Current limitation process
Grid-forming converters are designed to limit the current injected into the grid.This is achieved by introducing a current saturation block between the quasistatic model of (2) and the current controllers, as shown in Fig. 2. Its mechanism relies on saturating the current references when their amplitude exceeds I max Besides, priority management is integrated within the current saturation block to manage axis priority.In the approach adopted in this paper, the current is limited in polar coordinates without assigning any priority to a specific axis, as demonstrated in (3).
Thus, the current saturation block of Fig. 2 can be represented by ( 4) and the measured current by (5).
Where K is the ratio between the RMS values of i * g and i * g sat .

D. Virtual Power Method
As demonstrated in [13], the current saturation can restrict the synchronization control by making the active power insensitive to changes in phase.[13] suggested considering the unsaturated current references i * gdq as a basis for PCC active power measurements, i.e. the active power feedback is chosen as illustrated in (6).
Where i * gdq are the unsaturated current references at the output of the quasi-static model.
Likewise, when designing a control system for a gridforming converter, the cascaded loop structure requires that the inner loops must reach their setpoints first.This ensures that the upper loop can be modified without causing interactions that may result in system instability.Consequently, when the current saturation is activated, the actual output current is the saturated one, while the unsaturated current references are deployed to provide a more stable active power control.Hence, when the current is saturated, the use of virtual power allows an equivalent of the unsaturated virtual current to flow through the converter.However, the current flowing in the grid is the real physical current in order to limit the current injection into the grid.This is depicted in Fig. 3a.
In order to showcase the new representation of the system, the representative equations are recalled.Indeed, (2) represents the quasi-static model, (4) represents the current saturation bloc, and (5) represents the approximate current measured following the current control.Merging all of these equations, it results in the virtual system of (7).The equivalent virtual circuit of the system is drawn in 3b where the current is the unsaturated current i * gdq and the impedance at the grid side is divided by K.
v gdq = e gdq + jX g i gdq (7) Thus, the two active powers P and P virt can be written in the following form:

III. APPLICATION OF VIRTUAL POWER-BASED METHOD UNDER BALANCED GRID FAULTS
A. Study of the system behavior under a solid fault According to the equations of (8), Fig. 4 provides a graphical representation of P (δ) and P virt (δ) character- Fig. 4: Impact of virtual power method on system synchronization during a solid fault.
istics.Besides, the virtual system evolution during a solid fault is illustrated.Under normal operating conditions, P virt (δ) intersects with the maximum power transfer curve, denoted by P * , at point 1.When a solid fault is applied, v g drops to zero, and subsequently, P virt (δ) decreases to zero.This phase spans from point 1 to point 2.During the fault, δ starts to increase, and P virt (δ) remains at 0, resulting in the movement from point 2 to point 3. Once the fault is cleared, the voltage increases and the operating point moves to reach point 4 of P virt (δ) characteristic at E g = 1 pu.If the virtual power method had not been applied, the system would have followed the black curve of Fig. 4, which is below the level of P * , leading to instability.In the last phase (point 4 to point 1), the operating point returns to its initial position, and the system resumes its normal operation.With that being said, when a solid fault occurs, the virtual power improves the large disturbance stability of the system.In order to test the robustness of the virtual power method in transient scenarios, a short circuit is applied at t = 0.1 s.The length of the solid fault is considered to be 100 ms.Fig. 5 illustrates the evolution of P , P virt , and I g with respect to time, while Fig. 6 represents P virt evolution with respect to δ. Accordingly, the synchronization process goes through the following steps: • The system is first desynchronized during the fault.
• Then, the system re-synchronizes after the fault clearance.
It is noteworthy that the current is effectively limited to its maximum value I max = 1.2 pu during the fault occurrence and returns to its initial value after the fault clearance.
To better evaluate the effectiveness of the virtual power method in longer solid faults, the length of the short circuit is increased to 200 ms.The system behavior after the fault occurrence is shown in Fig. 7 and Fig. 8.According to the results, the system is always able to resynchronize, even for a longer fault duration.However, different phenomena appear during the synchronization process.When the fault is cleared, P virt (δ) is below the maximum power transfer curve P * .Therefore, the stability point cannot be reached.Consequently, P virt (δ) completes an additional complementary round searching for its stability point.During this round, the active power reaches negative values, as shown in Fig. 8.This phenomenon occurs because the system requires time to establish a new stability point following fault clearance.After the complementary round, the system re-synchronizes.

B. Study of the system response to grid faults with residual voltage levels
As mentioned previously, the P virt characteristic of ( 8) is valid as soon as |i g | > I max .Hence, it can also be represented during the fault when the voltage is low.The example of E g = 0.x pu is taken in Fig. 9.In case the characteristic P virt (δ) of E g = 0.x pu crosses P * = 1 pu, it is then possible to re-synchronize the virtual system and the power angle during the fault.
To illustrate such a theoretical explanation, a 200 ms grid fault with a residual voltage E g = 0.2 pu is considered.Fig. 10 and Fig. 11 show that P virt is still able to re-synchronize during the fault, as it moves to point 2, the intersection between the maximum power transfer P * = 1 pu and P virt (δ) for E g = 0.2 pu.When the fault is cleared, the operating point reaches P virt (δ) for E g = 1 pu, which allows the same re-synchronization phenomenon to occur as in the solid fault case.However, as illustrated in ( 8), the maximum reachable value of P virt depends on E g and X c .The expression for the maximum achievable virtual power is given by: Where V m is assumed to be 1 pu for simplicity.As a high SCR is used, X g /K can be approximated to 0. Thus, the threshold at which the virtual power intersects with P * = 1 pu can be determined as: Consequently, when E g falls below 0.15 pu, the virtual power is unable to re-synchronize during faults, resulting in angle divergence.Nevertheless, after the fault is cleared, the system is capable of re-synchronizing, as depicted in Figure 7.

A. Introduction
It was illustrated in the case of E g < 0.15 pu that the maximum value of P virt (δ) is not large enough to inter- sect with P * = 1 pu, which prevents re-synchronization during the fault.To address this issue, a possible solution is to add a negative virtual impedance X v that varies with respect to the PCC voltage |v gdq |.It allows the increase of the maximum value of P virt (δ), as shown in Fig. 12. Subsequently, the expression of P virt (δ) becomes: Where X v is the negative virtual impedance added when the current is saturated.
The proposed expression of X v is illustrated in (13).
Where ∆X ∆V = 15.10 −2 Ω/V is the slope of the linear characteristic and V norm = 1.A limitation is applied to the virtual impedance so that X v +X c is always positive.The mechanism is modelled in Fig. 13 B. Study of the system behavior for E g = 0.1 pu In order to test the effectiveness of the proposed adaptive virtual inductance, a 200 ms grid fault with E g = 0.1 pu is performed.In the case where this virtual inductance was not applied, P virt (δ) could not synchronize during a fault.It is shown by the results of Fig. 14 and Fig. 15 that the application of negative virtual inductance highly increases the maximum P virt (δ) for E g = 0.1 pu.Consequently, the point at which P virt (δ) crosses P * = 1 pu occurs at a low δ leading to a synchronization of P virt (δ) during and after the fault.This indicates that the proposed adaptive virtual inductance can effectively enhance the system stability during faults, regardless of the voltage level and fault duration.This could be a new way to use the principle of virtual impedance to increase the large disturbance stability.

C. Comparison between the two methods of synchronization
In order to highlight the difference between the two synchronization methods, a comparison is made for a solid fault with a duration of 200 ms.Fig. 5 and Fig. 16 refer to the system behavior without and with the adaptive virtual impedance, respectively.Both methods result in successful system re-synchronization after fault clearance.However, there is a noticeable difference in the dynamic behavior of the system during the fault.Indeed, the solution of re-synchronization during the fault with negative virtual inductance provides a better dynamic behavior.

D. Test on a standard voltage sag
Both methods are compared to a standard voltage shape as defined in [2].The waveform of the voltage source E g is illustrated in Fig. 17.The voltage at the PCC drops to 5% of its nominal value for 250 ms.Then, the voltage is linearly increased to 1 pu over a period of 3 s.Fig. 18 and Fig. 19 show the system behavior without and with the adaptive virtual inductance, respectively.The results illustrate the limitations of the virtual powerbased method.In such cases, the system faces challenges in re-synchronizing once the fault is cleared, and its dynamic behavior degrades during the gradual voltage increase.The reason for this occurrence is that in the initial synchronization methods, the virtual power experiences a sudden surge beyond the stable operating point,  resulting in a loss of synchronism during the fault.As a consequence, the angle making it unattainable to return to its stable position.However, with the use of the negative virtual inductance, the dynamic behavior is improved as there are no oscillations during the voltage pickup, and the virtual power is re-synchronized during the grid fault and after its clearance.It is an essential aspect of the system's stability as it ensures reliable operation even in challenging scenarios.Overall, the results suggest that the virtual power-based method with an adaptive virtual impedance is a promising approach to address synchronization issues and improve the overall performance and reliability of the system.

V. CONCLUSION
As the aspirations for grid-forming control have grown, the prospects of incorporating synchronization methods have increased.The aim is to obtain a control architecture that ensures large disturbance stability in different grid conditions.It was proven throughout this paper that the virtual power method is a potential solution for addressing the synchronization issues that arise during fault scenarios.However, improvements were needed to meet the specifications of the grid codes.Accordingly, it has been demonstrated that the combination of the adaptive virtual inductance and virtual power methods enhances the stability of a gridforming converter during standard voltage ride through and thus, helps the converter to comply with the grid codes requirements.The proposed solution is currently being expanded to encompass a wider range of scenarios, including those involving lower SCR, distribution systems, and unbalanced faults.This work will be extended to cover experimental implementation.

Fig. 1 :
Fig. 1: Presentation of the main reference system.

Fig. 3 : 2 g
Fig.3: Equivalent circuit after the application of the virtual power-based method.

Fig. 15 :
Fig. 15: Power angle curve response after the addition of X v following a 200 ms grid fault with E g = 0.1 pu.

Fig. 16 :Fig. 17 :
Fig. 16: Dynamic response after the addition of adaptive virtual impedance following a 200 ms solid fault.

Fig. 18 :
Fig. 18: Dynamic response without the addition of X v following the fault with a standard voltage sag.

Fig. 19 :
Fig. 19: Dynamic response after the addition of X v following the fault with a standard voltage sag.

TABLE I :
Hardware System Setup.