Accurate Reactive Power Sharing Strategy for Droop-based Islanded AC Microgrids

—In islanded AC microgrids, the mismatched impedances of the interfacing feeders between the inverters and the load bus cause poor reactive power sharing when the conventional frequency and voltage droop con-trol technique is employed. Such operation endangers the whole microgrid reliability as it may lead to overloading certain inverters and, consequently, triggering protection relays and causing cascaded failure. Thus, this paper proposes an accurate reactive power sharing strategy that considers the mismatched feeder impedances in islanded AC microgrids. It is based on the optimal tuning of the virtual complex impedance for each inverter. The proposed strategy has several advantages. First, it has a physical meaning as it establishes an explicit relationship between the mismatched values of the actual resistive-inductive feeders and the assigned values for the proposed virtual complex impedance to each inverter. Hence, further degradations in the microgrid voltages are prevented. Second, there is no need for prior knowledge about the actual feeder impedances in the design stage as they are estimated online from the available measurements. Lastly, the proposed control is reliable and fault-tolerant as it copes with unexpected failures such as failure of (or sudden switching off)some inverters and communication disruptions/delays. It ensures accurate power sharing even under the primary controllers after losing communication links with the secondary controller. The simulation and experimental veriﬁ-cation results are presented to validate the performance of the proposed technique.


I. INTRODUCTION
T HE fast growth in the global electricity demand and the need to utilize environment-friendly energy sources accelerate the advancements in the power electronics systems.As a sequence, there is a substantial annual increase in the number of distributed power generation systems (DPGSs) installed close to the load centers [1].Additionally, clustering of small-scale generation units forms microgrids to provide reliable and optimal integration of inverter-based DPGSs, as they can be operated in the grid-connected or islanded mode [2].
The conventional frequency and voltage droop control technique is widely used to control the islanded microgrids.This control approach of the DPGSs mimics the operation of synchronous generators.The droop characteristics are utilized to self-regulate the DPGSs output powers [3].The conventional droop control approach approximates the line impedance to be purely inductive.However, this is not the case in resistive low voltage microgrids.[4].
While the conventional droop control can achieve proper active power sharing, it has severe limitations related to the reactive power sharing [5] in low voltage islanded microgrids.This is mainly due to the mismatched feeder impedances and the differences in the design parameters of the inverters.In microgrids, the detrimental effects of the mismatched feeder impedances on the reactive power sharing remain a major problem.For example, the inaccurate power sharing between parallel-connected inverters poses instability problems due to circulating current within the microgrid's inverters [6].Such operation endangers reliability as it can overload certain inverters, consequently triggering overloading protection that may cause cascaded failure in the microgrid.
Among many suggested methods which aim to enhance the reactive power sharing in islanded microgrids, the virtual output impedance approach has gained significant popularity due to its simple implementation [7].This technique relies on reshaping the characteristics of the inverter output impedance by establishing an additional control loop.Therefore, the coupling between the active and reactive power flow is minimized.Consequently, the reactive power sharing accuracy is improved even under mismatched feeder impedances [8].
Various studies in the literature have proposed different virtual impedance approaches.This includes designing and implementing virtual inductive [8], virtual resistive [9], and virtual complex impedance.Moreover, the virtual complex impedance is divided further into virtual resistive-inductive impedance [7], [10] and virtual resistive-capacitive impedance [11].Even though there is substantial work available in the literature to address the reactive power sharing using the virtual impedance approach, the procedure of assigning optimal values to the virtual impedance for each inverter has not been thoroughly investigated.On the one hand, some studies recommend setting fixed values of the virtual impedances for all inverters much larger than the maximum value of line impedance expected [9].However, this approach is impractical as good power sharing is accomplished at the expense of voltage degradation in the microgrid.Another approach is to assign fixed values for the virtual impedances based on the summation approach [12].It relies on setting the value of the virtual impedance of one inverter to zero while a specific value is assigned to the virtual impedance of the second inverter [12].While the summation approach improves the reactive power sharing, it fails to regulate the errors in power sharing to zero.Furthermore, the information of the feeder impedances is assumed to be known [12].
The idea of adaptive virtual impedance has gained more attention recently [8], [13], [14].In this approach, the value assigned to the virtual impedance is adaptively varied to improve the reactive power sharing.In [8], the virtual output impedance is implemented using an adaptive high pass filter.In [13], the tuning of the adaptive virtual impedance for each unit is performed based on the received power share reference.The adaptive virtual impedance control scheme based on small-AC-signal injection is proposed [14].Despite the available improvements in the reactive power sharing, the major shortcomings of the existing adaptive virtual impedance methods include 1) the complex implementation, 2) the good reactive power sharing is obtained at the cost of voltage regulation in the microgrid, 3) the continuous dependence on the reliability of the communication network, and 4) most importantly, the works mentioned above have not established an explicit relation between the assigned values to the adaptive virtual impedances and the mismatched values in the physical feeder impedances, which are the cause of the inaccurate reactive power sharing phenomena in the microgrids.
This paper aims to extend the previous work [15], where further analysis, a generalized methodology, simulation results, and HIL experimental validation are considered.The proposed strategy ensures accurate reactive load power sharing between the inverters operating in islanded AC microgrids.It relies on the optimal tuning of the virtual impedances for all inverters based on the estimated feeder impedances.Hence, the main merits of this methodology include the following aspects: 1) An optimal tuning approach of virtual complex impedance for each inverter is proposed.It has a physical meaning as it considers the exact mismatched values in the physical feeder impedances.2) An implementation of an online feeder impedance (both resistance and inductance) estimation algorithm in the microgrid central controller (MGCC) is presented.The estimation technique is simple; it uses the available measurements and ensures high accuracy and low computational burden.3) Unnecessary voltage degradation in the microgrid is avoided.The voltage drops in the microgrid are reduced due to the use of the proposed optimal values of the virtual impedances.4) The proposed method is reliable as it ensures accurate power sharing amongst inverters even in extreme operating conditions such as communication-link failure and sudden losing/turning off/on some inverters units.So, it supports plug and play feature.
The rest of this paper is structured as follows: Section II reviews the conventional droop control, the reactive power sharing, and the virtual impedance for islanded AC microgrids.
Section III illustrates the proposed virtual impedance control scheme.Section IV and Section V present the simulation and experimental verification results.Finally, conclusions are given in Section VI.

A. Conventional Frequency Droop Control
Fig. 1 depicts the equivalent model of an islanded microgrid that consists of two single-phase inverters and a resistiveinductive load.E j δ j (j=1,2) represents the output voltage of DG j.V pcc θ pcc is the microgrid voltage.Z f1 and Z f2 are the impedance of feeder 1 and feeder 2, respectively.R f1 , R f2 , X f1 and X f2 represent the resistances and reactances of the feeders.The output active power and reactive power of the j th inverter is expressed as [4] Corresponding to the small power angle δ j : it valid to assume sinδ j ≈ δ j and cosδ j ≈ 1.Furthermore, when the conventional droop assumes the feeder impedance is purely inductive, 90 • , (1a) and (1b) can be reduced to It is worth mentioning that assuming feeder impedances are purely inductive is unrealistic as microgrids feeders will have non-negligible resistance [16].Hence, the inaccurate reactive power sharing is considered an inherent drawback of the conventional droop control technique, and vice versa with active power sharing if the microgrids feeders are assumed purely resistive.
As can be seen from the assumption made in (2a) and (2b), the output active P j and reactive Q j power of the j th inverter can be controlled by δ j and E j , respectively.This goal is achieved through adjusting the frequency and the amplitude of the voltage reference provided to the outer voltage control loop by the conventional droop control as follows [3], [6]: where ω * and E * are the rated values of angular frequency and voltage amplitude, respectively.D p and D q are the droop coefficients considering the inverter capability of supplying and absorbing reactive power.These droop coefficients are calculated from the acceptable deviation limits of the frequency (f max , f min ) and voltage (V max , V min ) in the microgrids [6], [17].

B. Reactive Power Sharing Considering Mismatched Feeder Impedances
Even though the conventional droop ensures accurate active power sharing in the microgrids under inductive feeder impedance assumption, circulating and inaccurate reactive power between the inverters in case of unequal feeder impedances are the main challenges in such a system as will be shown below [3], [6], [8].
To simplify the analysis, the equivalent model considering two identical inverters depicted in Fig. 1 is investigated.The assumption of identical inverters is justified as identical gridforming inverters are usually found in industrial applications to interface batteries to the islanded microgrids to maintain, for example, reliable voltage and frequency regulation [18].
The voltage drop across the impedance of feeder j is approximated by [19] Considering the voltage drop in (4), the terminal voltages of inverters 1 and 2 are obtained by (5a) and (5b), respectively.
Now, if Z f1 is larger than Z f2 , the mismatched impedance components between the two feeders are calculated as follows: By solving the obtained formulas, the reactive power sharing error (∆Q) due to the mismatched components of feeder 1 and feeder 2 is derived as [20], [21] where P tot = P 1 +P 2 and Q tot = Q 1 +Q 2 are the total active and reactive power demand.Equation (7) indicates that the reactive power sharing error is a function of the mismatched impedance value between the two feeders.As a result, for identical grid-forming inverters, the inverter with the higher feeder impedance will supply less reactive power.In conclusion, the impact of the mismatched feeder impedances on the reactive power sharing between inverters j and j + 1 at the PCC (also at the terminal) is summarized in (8).

C. Virtual Impedance for Droop-based Controlled Microgrids
To rectify the inevitable shortcomings of the conventional droop control under mismatched feeder impedances, the virtual impedance approach (Z v = R v + jX v ) is suggested to ensure inductive behavior at the line frequency [6], [8].Its working principle relies on dropping of the reference voltage of the j th inverter (v refj ), generated by both (3a) and (3b) and by the added errors from the secondary controller, in proportion to its output current (i j ) as summarized in (9).
where v * j is the modified voltage reference for the outer voltage control loop.Adopting Z v not only improves the accuracy of the reactive power sharing [5], [13] but also enhances the stability of the system as it effectively suppresses the circulating currents among the inverters [6].
The remaining challenge is determining the optimal value for Z v to be assigned to each inverter.Unlike such approaches, this paper proposes an optimal approach to determine the virtual impedance for each inverter based on online estimation of the feeder impedances, as illustrated next.

III. PROPOSED REACTIVE POWER SHARING STRATEGY
Fig. 2 to Fig. 4 depict the structure of the microgrid under investigation and the detailed implementation of the proposed reactive power sharing strategy that consists of two stages.

A. Online Estimation of Feeder Impedances
The first step to implement the proposed power sharing strategy is to know the actual values of the feeder impedances.To do so, various impedance estimation methods (intrusive and non-intrusive) can be utilized.The intrusive methods require the injection of additional disturbances to the network such as pulse injection [22], the PRBS [23], or the PQ variations [24].On the other hand, the non-intrusive techniques estimate the grid impedance using the existing measurements without injecting additional disturbance to the network.The nonintrusive techniques include, for example, the recursive least squares (RLS) algorithm [25] and the Kalman filter [26].
This paper uses the RLS algorithm to estimate the feeder resistance and inductance.Compared to its nonintrusive counterparts, the RLS algorithm has a faster estimation time, low computational complexity, and it is suitable for impedancebased applications.Inverter The RLS algorithm is implemented in the secondary control loop.To estimate the feeder impedance of a particular feeder, the RLS algorithm requires the current and voltage measurements at the two terminals of this feeder.For example, the impedance of feeder 1 is estimated from the i 1 , v 1 and v pcc measurements and so on.
To further elaborate, the rest of this section presents the online impedance estimation of feeder j by the j th inverter, where j= 1,2, 3 .... Similar analysis can be followed for the online impedance estimation for other feeders.
First, the steady state operation of the j th inverter is expressed as: where v j and i j are the output voltage and current measurements of the j th inverter, respectively.v pcc is the common ac bus voltage.If v j and v pcc are considered to be the system input variables, and i j is the state variable, then, the continuoustime state space model of ( 10) is expressed as follows: Eq. ( 11) is written in discrete-time form as shown in (12).
By considering the sampling interval (T s ) of the measurements, the solution of the state equations shown in (12) is obtained by mapping the input and output as Eq. ( 13) can be written in a linear regression form as: where y(k) is the output vector that contains i(k).ϕ T (k) is the input vector that consists of two variables, the delayed voltage (v j (k − 1)) and current (i j (k − 1)) vectors by one sample.θ is the regression vector of the unknown parameters (R fj , L fj ) The RLS algorithm can be applied now to estimate the unknown components based on the three measurement quantities (v j , i j and v pcc ).
The equations of the RLS algorithm used in the estimation are expressed in (15) at the sampling instance (k).
where, ρ is the forgetting factor with values between 0 and 1.It compromises the convergence rate and the steady-state error [27], [28].Initialization Fig. 3: Proposed algorithm to determine the optimal values of the virtual impedances for the inverters operating in the islanded AC microgrid.
Fig. 4: Implementation of the proposed reactive power sharing strategy using the optimal virtual complex impedance, which is based on the online estimation of the feeder impedances for the j th inverter (j= 1, 2, 3, ....).
Unlike other algorithms that intend to reduce the mean square error (MSE), the RLS algorithm recursively estimates the unknown parameters that minimize the weighted linear least-squares cost function relating to the measured signals; see (15b).After a few iterations, the best fits for the unknown parameters are selected upon reaching the convergence.Finally, the resistance and inductance components of the j th feeder are calculated directly from the estimated regression vector θ in (15) as shown below: It is worth mentioning that the estimated feeder impedances are performed before enabling the proposed control, where the RLS algorithm is executed in the secondary control loop only for a short time (a few ten milliseconds).As the RLS algorithm utilizes remote measurements (the inverter output current and voltage) for the impedance measurement based on ( 15) and ( 16), the estimation accuracy could be impacted by possible communication delays.To minimize such delays, the sampling rate of the RLS algorithm can be set to a slow sampling rate than the sampling rate of the inverter control algorithm.Finally, the estimated impedances are saved to calculate the proposed optimal virtual impedance for each inverter, as will be explained next.

B. Implementation of the Proposed Optimal Virtual Impedance
Fig. 3. shows the methodology followed to optimally tune the virtual impedances of all inverters.The calculation of these virtual impedances is performed in the secondary control loop.The estimated feeder impedances based on ( 16) of all feeders (e.g., Ẑf1 = Rf1 + j Xf1 , Ẑf2 = Rf2 + j Xf2 , and Ẑfj = Rfj + j Xfj ) are compared to find the highest value to be taken as the base virtual impedance, Z v−base = R v−base + jX v−base , as summarized in (17a).The control laws of the proposed optimal virtual impedances Z v1 , Z v2 , ...,Z vj for inverters 1, 2, ..., j in the microgrid shown Fig. 2 are obtained based on the difference (mismatched) between Z v−base and the estimated impedances of their own feeders, as given in (17b)-(17e): where R v1 , R v2 , R vj , X v1 , X v2 , X vj are the real and imaginary components of the proposed optimal virtual impedances for inverters 1, 2, and j, respectively.
Next, the assigned values for the virtual impedances by (17) in the secondary controller are sent to the inverters to be used in their primary control loops (9).For the j th inverter, implementing the term Z vj × i j requires the time derivative of the inverter output current, making the system highly sensitive to the output current noise.To solve this issue, the secondorder generalized integrator (SOGI) approach is utilized [7].Hence, (9) can be rewritten for the j th inverter under the proposed control as follows: where i αj and i βj are the two components of the inverter output current in the αβ0 stationary reference frame.
The assigned optimal virtual impedances to all inverters equalize the voltage drop across these feeders of the microgrid.As a result, the negative impacts of the mismatched feeder impedances on the reactive power sharing are fully mitigated.In other words, the performance of the reactive power sharing after enabling the proposed control strategy can be summarized as follows: where Q 1 , Q 2 and Q j are the injected reactive power by inverter 1, inverter 2 and inverter j, respectively.
It is worth mentioning that while other design approaches in ( 17) could be adopted, only the proposed approach of fixing the base virtual impedance to the originally maximum feeder impedance ensures the optimal solution.Simultaneously with achieving 100% accurate reactive power sharing among the inverters, it prevents unnecessary voltage drops in the microgrid even after communication link failure as the virtual impedance of each inverter is required to be initialized only for one time regardless of the load changes.Hence, after losing the communication between the secondary control and primary control layers, the proposed control strategy continues to ensure the accurate reactive power sharing by using the virtual impedance sent earlier from the secondary control loop.

C. Stability Analysis
This paper analyzes only the stability of Fig. 5, as it is the main part of the proposed scheme.As stated earlier, the SOGI is used to implement the proposed virtual impedance for each inverter, as shown in Fig. 4. It is used to obtain both the alpha and beta components of the output current of the j th inverter.The SOGI closed-loop transfer functions for alpha and beta channels are given by ( 20) and (21), respectively [29].
where K is the gain of the SOGI closed-loop transfer functions.It is set to 0.35 in this paper to compromise between the time response and the bandwidth of the filter [7], [29].ω refj is the angular frequency reference of the j th inverter.The closed-loop transfer function of the proposed control (the SOGI and the virtual impedance) for the j th inverter shown in Fig. 4 is given as follows: Considering the transfer function G v (s) in ( 22) and the parameters listed in Table I, the Bode plot and root locus of the system are plotted in Fig. 6(a) and Fig. 6(b), respectively.It can be noticed from Fig. 6(a) that the proposed combination of the SOGI and the optimal virtual impedance approach operates G s Fig. 5: Simplified control block diagram of Fig. 4 that is being used for implementing the proposed virtual impedance for the j th inverter.mainly at the fundamental frequency to effectively modify the output impedance of the inverter in order to ensure the accurate reactive power sharing.Z v behaves as a low pass filter (LPF) after the system frequency 50 Hz with a magnitude slope of -20 dB/dec.The root locus of the transfer function G v (s) shown in Fig. 6(b) indicates that increasing Z v leads to moving the zeros closer towards the origin while the system two poles are kept constant and far from the imaginary axis.

IV. SIMULATION RESULTS
To demonstrate the efficacy of the proposed control strategy, the islanded AC microgrid shown in Fig. 2 is considered.The system is simulated in MATLAB/Simulink software.Table.I lists the system parameters used in the simulation.

A. Power Sharing Under Identical Power Ratings
To obtain the key simulation results, the performance of the system is tested under different conditions as follows: 1) Test 1 (t < 3 sec): the system operation under the conventional control is examined for a microgrid load equal to P load = 3 kW and Q load = 3 kVar.
2) Test 2 (t = 3 sec): the proposed control is enabled, where the virtual impedances of inverters 1, 2, and 3 are set according to (17).3) Test 3 (t = 4 sec): a step change in the microgrid load to P load = 6 kW and to Q load = 6 kVar is applied under the proposed control.4) Test 4 (t = 5 sec): a failure in the communication links between the secondary controller and the inverters is tested.
Fig. 7(a) and Fig. 7(b) show the active and reactive power of the microgrids.It can be seen that the delivered powers to the load track perfectly the references as long as the secondary controller is enabled.Furthermore, regardless of the mismatch impedances of the feeders, the active load is shared equally between the three inverters, 1 kW each for t < 4 sec and 2 kW each for t ≥ 4 sec.However, the reactive load power is not shared accurately by the three inverters under the conventional droop control.For example, the injected reactive power by the 1 st inverter is equal to Q 1 = 0.75 kW while Q 2 = 1.25 kW, where Z f1 > Z f2 ; see (8).Fig. 7(b) shows that the errors in the reactive power sharing are remarkably reduced, and the three inverters share the same amount of reactive power under the proposed control scheme for t ≥ 4 sec.For instance, the delivered power to the PCC by each inverter is equal to 1 kVar for 3-4 sec and 2 kVar for 4-5 sec.Fig. 7 shows the system response remains stable after enabling the virtual impedances at t = 3 sec.This finding of stable operation is predicted by the stability analysis shown in Fig. 6(b) in which the closed-loop transfer function of the added virtual impedance loop has two poles (P 1 and P 2 ) located in the left hand side of the s-plane.Hence,  it can be seen that the reactive power sharing manifests an excellent dynamic performance and no instability phenomenon has occurred.Fig. 8 depicts the current waveforms in the microgrid.The magnitudes and phase of current waveforms are not identical under the traditional droop control.In contrast, the current waveforms of the three inverters have identical magnitude and phase after enabling the proposed control for t ≥ 3 sec.Fig. 9 presents the online estimation results of the feeder resistance and inductance by inverter 1.It can be observed that the RLS algorithm provides very accurate estimation results after it is enabled for t >= 3 sec.Note that the accuracy of impedance estimation mainly relies on the forgotten factor parameter of the RLS algorithm.In this paper, this parameter is set to 0.995 to provide the accurate and stable estimation.Fig. 10 shows the amplitudes of voltage references provided by droop control of each inverter.Under the traditional control, the highest voltage reference value is for inverter 1 as the  impedance of its feeder has the largest value.The differences in these voltages are compensated completely after enabling the proposed control.Fig. 10 also shows the voltage amplitude of the microgrid load and its reference.It can be seen that, before the communication fault, the secondary control loop regulates the microgrid voltage to its nominal value of 230 √ 2 V.

B. Comparison With Conventional Methods
As the available literature does not utilize the online estimation of the feeder impedances to tune the virtual impedances, the performance compared in paper against ventional methods, where there is no virtual impedance and with fixed virtual impedance scheme.two methods are considered base case studies in the literature.Similar to the proposed method, these two methods are implemented in the centralized microgrid.
Fig. 11(a) and Fig. 11(b) provide the reactive load power sharing results obtained with no virtual impedance and a fixed virtual impedance scheme, respectively.Both control schemes fail to achieve zero steady-state errors in the reactive power sharing between the identical inverters due to the mismatched feeder impedances.Hence, the proposed control shown earlier in Fig. 7(b) outperforms these two power sharing techniques.

C. Power Sharing Under Different Power Ratings
Unlike the other cases presented in this paper, this section investigates a special case that considers the reactive power sharing between non-identical inverters with different power ratings.Two inverters are considered with 5 kW and 2.5 kW rating power, respectively.The impedance components of feeders 1 and 2 are identical to those listed in Table .I. The only difference in this case study is the tuning procedure of the virtual impedances.Considering the difference in the power ratings, the injected power by inverter 1 in the ideal operation should be double the injected power by inverter 2. Hence, to achieve accurate reactive power sharing through equal voltage droop across the feeders, Z v of inverter 2 is taken double to its value calculated by (17).Fig. 12 shows the reactive load power sharing among inverters 1 and 2. The conventional control (t <3 sec) not only fails to ensure the proportional reactive power sharing, but also the inverter with the lower power rating (inverter 2) supplies reactive power more than the inverter with the higher power rating (inverter 1).For t >3 sec, fair sharing of the load reactive power among the two inverters is achieved concerning their power rating differences.

D. Performance During Faults in the System
Fig. 13 presents the reactive power sharing for four different scenarios: normal operation before t= 5 sec, communication fault at t=5 sec, switching off inverter 1 (DG1) at t=6 sec, and load change at t ≥ 7 sec.In these scenarios, the proposed control validates to be reliable as it ensures the accurate active and reactive power sharing among the inverters.For instance, after losing inverter 1 at 6 sec, the load power is distributed equally between inverters 2 and 3. Furthermore, the control strategy copes with the sudden load change at 7 sec while communication is unavailable.This validates the robustness of  the proposed reactive power sharing against various abnormal operating conditions.

V. HARDWARE-IN-THE-LOOP EXPERIMENTAL RESULTS
To further verify the power sharing performance of the proposed strategy, hardware-in-the-loop (HIL) experiments are conducted on a single-phase ac microgrid with two DG units, as depicted in Fig. 14.The power stage (electric part of the circuit) is emulated using PLECS RT Box.The control algorithm is realized through a TMS320F28069M LaunchPad from Texas Instruments.The parameters chosen for the HIL test bench are identical to those used for the simulation, as listed in Table .I.This section has tested three different cases under the mismatch in the impedances of feeders 1 and 2. In case 1, the power sharing between the two inverters is investigated when supplying a resistive-inductive microgrid load.Then, case 2 considers only a resistive load to verify the performance improvement by eliminating the circulating reactive power between the two inverters.In case 3, the performance of the proposed control is tested for the loss of communication links between the secondary controller and the two inverters.

A. Active and Reactive Load Power Sharing
In this case, the power rating of each inverter is 5 kW.Hence, it is expected that the load power to be proportionally and equally shared among the two parallel inverters.The power sharing of the initial 3 kW and 3 kVar microgrid load is shown in Fig. 15(a the reactive load power sharing is poor under the conventional control strategy.As a sequence, the output current waveforms of the two inverters are significantly different from each other, as shown in Fig. 15(c).
When the proposed control strategy is enabled, the output currents of the inverters are gradually equalized.Fig. 15(a

B. Eliminating the Circulating Reactive Power Between Inverters
The microgrid load is set to 3 kW.Then, the efficacy of the proposed control strategy under different impedances for feeders 1 and 2 is tested.Fig. 16(a) and Fig. 16(b) show the obtained results.While the inverters supply the same amount of active power to the load under the conventional control, the circulating reactive power between the inverters caused by the mismatched impedances is relatively large; see Fig. 16(a).After enabling the proposed control strategy that compensates adaptively for the mismatched voltage drops, the output currents of both inverters become almost identical.sharing between the two inverters is maintained accurately after t 1 .

C. Loss of Communication Links
The superiority of the proposed control accurate reactive power sharing over three existing techniques is summarized in Tab.II.It can be observed that the proposed method has several advantages that include achieving very accurate reactive power sharing under abnormal conditions such as communication faults and load, whereas other methods fail in such operational scenarios.

VI. CONCLUSIONS
In this paper, a power sharing strategy scheme is proposed to enhance the accuracy of reactive power sharing in islanded AC microgrids.The control strategy deploys the online estimation of the feeder impedances for optimal tuning of the complex virtual impedance assigned to each inverter.The followed control approach provides an effective mechanism for real and reactive power decoupling.Hence, despite mismatched feeder impedances, proportional load sharing among parallelconnected inverters is achieved.Additionally, the voltage degradation in the microgrid load is avoided due to: 1) optimizing the value of the proposed virtual impedances; and 2) deploying the secondary control level.Finally, the simulation and HIL experiment results validate the effectiveness of the proposed scheme.
Several important aspects can be considered in future studies.First, microgrids with multiple PCCs and different typologies (e.g., meshed microgrids, interconnected DC and AC microgrid clusters) can be investigated.Second, future studies can also consider non-identical inverters, including the aging factor that could cause asymmetrically degradation of the components among different inverters.

Fig. 2 :
Fig. 2: A schematic diagram of the investigated islanded AC microgrid with the proposed accurate reactive power sharing strategy.

Fig. 7 :
Fig. 7: The simulation results of the power sharing in the microgrid: (a) active power, and (b) reactive power.

Fig. 10 :
Fig. 10: The voltages magnitudes of the inverters and the microgrid load.

Fig. 13 :
Fig. 13: Reactive load power sharing in the microgrid under abnormal operation: communication failures at 5 sec, switching off DG1 at 6 sec, and load change while communication is unavailable at 7 sec.

Fig. 14 :
Fig. 14: Hardware-in-the-loop setup: (a) Block diagram of the setup, (b) Screenshot of the setup.
) and Fig.15(b) also show that the proposed control strategy ensures the proportional and equal reactive power sharing after the load change to 6 kW and 6 kVar at t 1 .

Fig. 17
Fig.17(a) and Fig. 17(b) verify the reliability of the proposed power sharing strategy against a failure in the communication links in the microgrid at t 1 .The reactive power Fig.17(a) and Fig. 17(b) verify the reliability of the proposed power sharing strategy against a failure in the communication links in the microgrid at t 1 .The reactive power

1 t
Fig.17(a) and Fig. 17(b) verify the reliability of the proposed power sharing strategy against a failure in the communication links in the microgrid at t 1 .The reactive power

Fig. 15 :
Fig. 15: The experimental results of the power sharing under the conventional and the proposed control: (a) Active power, (b) Reactive power, (c) current waveforms.

Fig. 16 :
Fig. 16: Experimental results of under the conventional and the proposed control: (a) Circulating reactive power, (b) Current waveforms.

TABLE I :
The parameters of the islanded AC microgrid.