Dynamic Voltage Stiffness Control Technique for a Virtual Oscillator based Grid-forming Controller

Virtual oscillator control is the latest control technique for grid-forming inverters. Virtual Oscillator based Controllers (VOCs) provide all the steady-state droop functionalities of conventional droop controllers and, in addition, the time-domain synchronization with a connected electrical network. However, existing literature does not consider the aspect of dynamic control over the voltage stiffness of a VOC. Voltage stiffness is a vital parameter for a grid-forming inverter. If the voltage stiffness is too high, the inverter picks up all the reactive power demand of the PCC. In contrast, if the stiffness is too low, the inverter does not participate in voltage regulation at all. Limiting the reactive power output during a higher voltage sag, especially when connected to a weak grid, is challenging for a VOC. Entering into the current control mode is the existing solution, but it severely affects the effective synchronization between the VOC and the voltages of the PCC. As a result, the grid-forming mode of operation becomes inefficient. This article has introduced a Virtual Impedance (VIm) based dynamic voltage stiffness control technique for VOCs. The systematic design procedure for the proposed voltage stiffness controller is presented. In addition, a rigorous approach for stability analysis is presented.


I. INTRODUCTION
on-linear limit cycle oscillator (Virtual Oscillator) based control is the latest control strategy for the grid-forming inverters [1]- [3].Virtual Oscillator (VO) control meets all the steady-state functionalities of conventional droop control and virtual synchronous generator control but with improved dynamic performance [4], [5].Among different models of oscillators, the Andronov-Hopf oscillator and Dispatchable Virtual Oscillator (d-VOC) based controllers are proved to be the best fits for grid-forming controllers [6], [7].The elemental form of an Andronov-Hopf oscillator and a d-VOC is similar [6].A grid-forming controller should meet the system-level requirements such as fault ride-through capability, MPPT capability, and capability to perform under unbalanced grid conditions [8].The recent research works [9]- [11] have provided a very general VO-based control architecture where all the above-mentioned system-level functionalities are included.
However, the existing literature on VOCs does not consider the dynamic control over the voltage stiffness.Commonly a VOC is designed for less than a 10% tolerable voltage sag range [6].The reason behind a shorter low voltage bound is that the control over the reactive and active power droop characteristics of VOCs are not decoupled.The reactive power droop cannot be changed independently without changing the active power droop.It is important to mention that voltage stiffness is an important parameter for any grid-forming inverter, especially when connected to a weak grid [8].During the addition of a large load in an electrical network, the gridforming inverter, which is electrically nearest, is affected first [12].If the connected grid is weak, the voltage drop becomes large.Hence, the reactive power demand on the nearest gridforming inverter becomes significantly high.At this point, if the voltage stiffness of the grid-forming inverter is too low, the inverter does not participate in voltage regulation at all [8].As the grid is weak, the voltage amplitude drops further.On the other hand, if the voltage stiffness of the inverter is too high, the inverter picks up all the reactive power demand on itself [8], and the output current may tend to exceed the maximum rating.When the output current tends to the limit, the existing VOCs have two options.The first option is to be disconnected from the electrical network.However, the new grid codes prevent grid-forming converters from being disconnected easily from the network in such a condition as mentioned above [13].The second option is to enter into current control mode by activating the fault ride-through controller [9], [14].However, the effective synchronization of a grid-forming controller with the connected network gets severely affected in the current control mode [8].The exact reason behind the loss of synchronization of a VOC in the current control mode is investigated using a simulation study in Section IV.It is not a very good choice to enter into the current control mode during every voltage sag caused by a load transient at the PCC.Also, a discrete and higher jump in the value of the virtual resistor, as presented in [14] reduces the effective X/R ratio in the resultant source impedance of the grid-forming inverter.A lower X/R ratio cannot ensure superior decoupling between the active and reactive power output of the inverter [12].
Virtual impedance (VIm) based control technique has been a popular choice for voltage-source and current-source converters in recent years [15].VIm-based fault ride-through technique is presented in [16], [17] for grid-forming converters.The converter enters into the grid-following mode when the current limiting controller is activated.However, a fault ride-through technique is not suitable for dealing with typical voltage sags (10-20%) frequently occurring in power systems [18].A VIm-based control technique is used for improving the reactive power sharing accuracy among parallelly connected grid-forming inverters in AC islanded microgrids in [19], [20].However, the requirement of communication links makes the control strategies proposed in [19], [20] less reliable.VIm-based decentralized control technique for accurate reactive power sharing among parallelly connected grid-forming inverters in AC islanded microgrids is presented in [12], [21 N are used in [22], [23] to mitigate voltage unbalance in the islanded microgrid.The proposed method presented in [22] requires a secondary centralized controller, which reduces the reliability of the system.The literature [23] considers a maximum voltage unbalance of 8.2%, whereas a voltage unbalances of 2-15% for a duration of 3 cycles to 1 minute is very typical in electrical power systems [18].This article introduces a VIm-based dynamic voltage stiffness control method for VO-based grid forming controllers.The discrete research contributions of this article are summarized as follows.1.This article has presented a simulation study and analysis to point out the reason behind the loss of effective synchronization of a VOC in the current control mode.2. The proposed dynamic voltage stiffness controller provides an extra degree of freedom to a VO-based grid-forming controller to modify the reactive power droop characteristic without changing the active power droop characteristic.3. The proposed controller dynamically vary the virtual impedance to ensure a lower voltage stiffness under normal condition and a higher voltage stiffness under voltage sags while maintaining a higher X/R ratio throughout the operating range.4. The proposed controller has decoupled control over individual phases.It helps the VOC to withstand significantly larger unbalance voltage sag (more than 15%). 5. Using the proposed controller, an existing VOC can maintain effective synchronization with a connected electrical network during balanced and unbalanced sags and remain in the grid-forming mode under significantly higher voltage sags (more than 20%).6.The proposed controller can be integrated into an existing VO-based system-level grid-forming controller without the requirement of any change in the existing controller or any extra sensor, as shown in Fig. 1. 7. It is challenging to design the parameters of the proposed adaptive VIm-based voltage stiffness controller.A large and fixed value of virtual impedance can increase the range of voltage sag that a grid-forming inverter can withstand.However, it decreases the participation towards voltage regulation from the inverter.This is why instead of putting a fixed virtual impedance, the proposed technique adopts variable virtual impedance.This article has presented a detailed design procedure that can achieve a suitable trade-off between the increase in tolerable voltage sag limit and a decrease in the reactive power output.8.This article has introduced a rigorous approach of small signal stability analysis for VO-based grid-forming controller where the detailed model of the power system, voltage dynamic of the VOC, nested voltage and current loops, and the proposed controller are taken into account.9.A brief overview is presented in the Conclusion on how the present research work will be extended in the future to provide newer functionalities such as improvement in reactive power sharing among parallelly connected converters.
The rest of the paper is organized as follows.Section II introduces the proposed adaptive VIm-based dynamic voltage stiffness controller.Section II also presents the detailed design procedure for the key parameters of the proposed controller.In Section III, a small-signal analysis is presented to find the range of the key parameters of the proposed controller.Simulation and experimental studies are presented in Section IV and Section V to validate the functionalities offered by the proposed VIm method when integrated into a VO-based grid forming controller.Finally, the paper is concluded in Section VI with a brief overview of how the present research work will be extended in the future.

II. THE PROPOSED VOLTAGE STIFFNESS CONTROLLER
The proposed dynamic voltage stiffness controller is integrated into the existing VO-based system level gridforming controller [11] as shown in Fig. 1.The proposed controller is indicated using red lines and words.The existing grid-forming controller is presented in detail in [11].
The proposed controller has the two following objectives (i) The voltage stiffness of a grid-forming inverter should be lower when the reactive power demand on the inverter is low.Conversely, the voltage stiffness should be dynamically increased with the increase in the reactive power demand on the inverter.(ii) A high X/R ratio in the total resultant impedance should be maintained throughout the operating range The two mentioned objectives are achieved by dynamically varying the virtual impedance as where  0 and  0 are the initial value of the virtual resistance and reactance, respectively.The parameters, , and  are the adaptive virtual resistance and inductive reactance gain in Ω/A.The parameter,  __ represents the quadrature axis phase currents of the inverter.

A. Selecting the value of RV0, XV0, m, and n
The initial virtual resistance and reactance,  0 and  0 are more important for improving reactive power sharing accuracy among the inverters in islanded microgrids [12].Since this article focused on the voltage stiffness control in dispatchable mode, the initial resistance and reactance are taken as zero To maintain the same X/R ratio throughout the operating range, the values of  and  are taken as the same  =  =  (3) The active and the reactive power droop characteristic of a symmetrical component based VOC (S-VOC) is represented by [24] where,  ∈ , , .
The reactive power droop characteristic can be modified only by changing the current scaling factor,   .However, changing the value of   simultaneously affect the active power droop characteristic.As presented in (1), the virtual impedance is a function of quadrature axis phase currents,  __ only.Hence, with a proper X/R ratio the virtual impedance can modify the reactive power droop without any significant effect on the active power droop.The resultant reactive power droop is derived as where,   and  _ are the reactive power output without and with the virtual impedance, respectively.The parameter  is the total source impedance which is the result of the addition of the filter and grid impedance.As shown in Fig. 2, the reactive power droop characteristic can be modified using the virtual impedance.With a positive value of virtual impedance gain,  the droop characteristic shifts towards the left.The mentioned modification increases the tolerable voltage sag limit.The reactive power output remains nearly the same near the nominal voltage.Conversely, the reactive power output decreases with the increase in voltage sag.The value of  is selected to achieve a proper trade-off between the increase in tolerable voltage sag limit and the decrease in the reactive power output.

III. SMALL SIGNAL STABILITY ANALYSIS
A stability analysis is required to derive the limiting value of the parameter, .This article has introduced a rigorous approach for small signal stability analysis of a VO-based grid-forming controller.The detailed model of the system and the controller is derived and considered for the stability analysis.The output phase voltage (i.e. __ * ,  __ *

, and 𝑣 𝑣𝑜𝑐_𝑐_𝑑 *
) of the S-VOC is taken as the reference for the same individual phase for the instantaneous to synchronous reference frame transformation, as shown in Fig. 3.The instantaneous and synchronous frame-based parameters are denoted by small and capital letters, respectively.The detailed derivation of the reference frame transformation is presented in [11].The model of the system and the controller is derived in the s-domain as follows.
The model of the connected electrical network is derived in terms of grid impedance and grid voltages as The model of the L-C filter of the inverter is derived as _ = −   (  +   )  _ + 1 (  +   ) ( _ −  _ ) (10) _ =  _ −  _ (13)  _ =  _ −  _ (14) The model of the virtual impedance is derived as The feed-forward term is included in the model as  _ = −   __ −    __ (19)  _ = −   __ +    __ (20) The model of the voltage controller is derived as _ * = (  +    ) ( _ +  _ −  _ ) −  _ (22) The model of the current controller is derived as The complete small-signal model of the system and the controller is shown in Fig. 4. The S-VOC is intended to maintain effective synchronization.When the S-VOC is synchronized, it can be approximated that  _ ≈  _ ≈  _ ≈ 0 (25) The main interest is to find the relation between  _ and  _ i.e. ( _ / _ ) in terms of virtual impedance gain, .
The expression of ( _ / _ ) is linearized around the point ( _ ,  _ ).The other known parameters are derived as follows.
The range of the proportional and integral gains of the current and the voltage controller is given as follows [10]   =     ;   =     (26) where   ,   are the current and voltage loop bandwidth.The value of   ,   are taken 21500 rad/s and 2400 rad/s [10].
The values of the known parameters are given in Table I.The system and control parameters are taken directly from standard references, [6] and [10].
The eigenvalue loci for the virtual impedance gain,  are presented in Fig. 5.For the given system, the limiting value of  is found to be 0.22 Ω/A.The schematic diagram of the system considered for the simulation studies is presented in Fig. 1.The parameters of the system are given in Table I.The grid impedance is taken higher to mimic a weak grid situation.At t = 5 s a large load, (2.5 Ω + 2 mH)/phase is added to the PCC.Without any reactive power support from the grid-forming inverter, the voltage amplitude of the PCC drops by 24.81%.Next, the grid-forming inverter without and with the proposed voltage stiffness controller is connected to the PCC.The active power reference,   * per phase is set to 300W.The performance of the VO-based grid-forming controller without the proposed voltage stiffness controller is shown in Fig. 6.The output current of the inverter tends to reach the over-current limit without the voltage stiffness controller.As a result, the anti-windup current limiter is activated and restricts the output current under the over-current limit.However, a large oscillation occurs in active and reactive power output.The reason for the power oscillation is as follows.Unlike a droop controller or virtual synchronous machine controller, a VOC uses the instantaneous inverter currents for asymptotic synchronization with a connected network.The output current of the inverter should be the result of the interaction between the phase voltage of the VOC and the PCC for effective synchronization.Once the anti-windup function is activated, the mentioned condition for effective synchronization is no longer satisfied.The anti-windup controller entirely controls the output current of the inverter.
Next, the proposed adaptive voltage stiffness controller is used with a virtual impedance gain,  of 0.07 Ω/A.The virtual impedance modifies the quadrature axis current output of the inverter, as shown in Fig. 7.The output current no longer exceeds the over-current limit.Therefore, the inverter can stay operating in grid-forming mode, as shown in Fig. 8. and keep supporting the PCC with reactive power.The voltage profile of the PCC is improved (from 24.8 % sag to 17.19% sag) by the support of the inverter.Finally, the active power droop response of the inverter without and with the added virtual impedance gain is obtained.The frequency of the grid is changed from 60 Hz to 59 Hz in five discrete steps, and the active power output of the inverter is plotted in Fig. 9.The voltage of the PCC is kept at the nominal value.The Active power droop characteristic of the inverter without and with the VI are nearly the same and follow the desired active power droop characteristic closely.A. Dispatchable operation At first, the normal dispatchable operation is conducted.The aim is to observe if there is any adverse effect on the normal operation due to the virtual impedance.The virtual impedance gain,  is set to 0.1 Ω/A.The three-phase active power reference,  * , is initially set to 1500 W. The phase currents and the active power outputs of the inverter are shown in Fig. 12.Initially, each phase injects 500 W of active power into the grid emulator.Then the reference is increased to 2400 W. It is observed that the controller successfully tracks the active power reference.

B. Effect of virtual impedance on reactive power output
The reactive power output of the inverter is observed under two different voltage sags (5% and 25%) with two different values of virtual impedance gain (0.05 Ω/A and 0.15 Ω/A).The reactive power outputs of the inverter in the mentioned conditions are shown in Fig. 13 and Fig.

Fig. 1 .
Fig. 1.The schematic diagram of the proposed S-VOC based grid forming controller

Fig. 2 .
Fig. 2. The reactive power droop characteristic without and with virtual impedance

Fig. 3 .
Fig. 3. (a) Output voltage vectors of the S-VOC (b) Instantaneous to synchronous reference frame transformation

Fig. 4 .Fig. 5 .
Fig.4.Small signal model of the system and the controller

Fig. 6 .Fig. 7 .Fig. 8 .
Fig. 6.Performance of a VO-based grid forming controller in the presence of large voltage sag without the proposed voltage stiffness controller

Fig. 9 .Fig. 10 .Fig. 11 .
Fig. 9. Active power droop response of the VOC without and with the proposed voltage stiffness controller V. EXPERIMENTAL RESULTS AND DISCUSSIONS The following experiments describe how the proposed VImbased dynamic voltage stiffness control technique can modify the inherent reactive power droop of an S-VOC without affecting the active power droop.The schematic diagram and the picture of the experimental setup are shown in Fig.10and Fig.11, respectively.A grid emulator, as depicted in Fig.10, is used.A constant amplitude and frequency inverter, Inv2, behind an inductance, Lg1, acts as the grid.The frequency and voltage amplitudes of Inv2 are controllable.The grid emulator acts as an ideal sink or source throughout the operating range, i.e., 25 A continuous rms current.The three-phase rectifier, Rec2, which is connected to the utility grid, energizes the dclink (Vdcp2, Vdcn2) of the inverter, Inv2.The specification of the experimental setup is presented in TableII.The OPAL-RT real-time controller (OP4510) is used to deploy the control strategy.

Fig. 12 .
Fig. 12.Normal dispatchable operation of the grid-forming inverter: (a) active power outputs of the individual phases (b) phase currents of the inverter

Table I :
Specifications of the system and controller parameters for the simulation study

Table II :
Specifications of the experimental setup Nominal frequency: grid emulator 250 rad/s   Filter inductor: sources 2 mH/Phase   Filter capacitor: sources 20 µF/Phase   ,   dc-link voltages (split Capacitor) 200 V   Sampling time of the controller 50 μs