Accurate Active and Reactive Power Sharing Based on a Modified Droop Control Method for Islanded Microgrids

When multiple paralleled distributed generation (DG) units operate in an islanded microgrid, accurate power sharing of each DG unit cannot be achieved with a conventional droop control strategy due to mismatched feeder impedance. In this paper, a small AC signal (SACS)-injection-based modified droop control method is presented for accurate active and reactive power sharing among DG units. The proposed control method adjusts the voltage amplitude of each DG unit by injecting small AC signals to form a reactive power control loop. This strategy does not need communication links or to specifically obtain the physical parameter of the feeder impedance and only requires the local information. Moreover, the parameter design procedure and stability analysis are given full consideration. Finally, simulation and experimental results verify the effectiveness of the proposed control scheme, and accurate active and reactive power sharing can be achieved.


Introduction
The microgrid, which consists of a variety of distributed generation (DG) units, such as photovoltaic systems, wind power systems, full cells and other energy storage systems, will become an effective supplement to the main power grid and the potential energy structure [1][2][3]. These renewable DG units have the advantages of reductions in the pollution caused by fossil fuels, decreased power transmission losses, ease of installation, and so on. However, the high level of penetration of renewable DG units will introduce serious challenges to the main power grid or microgrid, such as frequency and voltage deviations and power sharing and fluctuation [4,5]. In an islanded microgrid in particular, multiple voltage source converter (VSC)-based DG units paralleled together should not only provide voltage and frequency support for the loads but should also achieve accurate power sharing according to their power rating [6][7][8].
The voltage and frequency droop control method is widely adopted for the parallel connection of multiple inverter-based DG units, and accurate active power sharing can be achieved in a steady state because the feeder impedance is usually mostly inductive [1]. However, due to the coupling of active and reactive power and mismatched feeder impedance, high performance cannot be guaranteed by the conventional droop control method with respect to reactive power sharing [9][10][11]. Various improved droop control strategies have been proposed, and they are mainly divided into two categories: communication-based improved droop control techniques [12][13][14][15][16][17][18] and communication-less improved droop control techniques [19][20][21][22][23][24][25][26][27].
High voltage and frequency regulation performance and acceptable power sharing can be obtained via communication-based improved droop control techniques in islanded microgrids, and these improved methods can be divided into several families: hierarchical con- In addition, some other methods without communication lines have been proposed [28,29]. A droop control strategy was proposed in [28] to improve reactive power sharing in which an integral term is used in the Q-V droop controller to restore the voltage amplitude. However, the sharing accuracy cannot be guaranteed during the restoration process [16]. A distributed optimal control strategy based on a Kalman filter state estimator was proposed for performing reactive power sharing and system voltage restoration via local measurement [29], but it requires a solution for the optimal regulators to be achieved by computing an optimization cost function. The high computational burden is not suitable for industrial controllers, and it is not feasible for practical microgrid application scenarios [4].
In [30], a small AC signal (SACS)-injection-based control method for achieving frequency restoration and accurate active power sharing was proposed; however, reactive power sharing was not considered. Therefore, a SACS-injection-based modified droop control method is proposed in this paper for accurate active and reactive power sharing. The main contributions of the proposed control strategy can be summarized as follows: (1) since no communications lines are needed, a low-cost and highly adaptable microgrid configuration can be achieved; (2) the specific parameter of the feeder impedance does not need to be known; and (3) accurate fundamental active and reactive power sharing can be achieved with only local information. Table 1 shows a comparison of various control methods. This paper is organized as follows. In Section 2, a brief introduction of the conventional droop control method is provided, and a SACS-injection-based modified droop control strategy for accurate power sharing is proposed in Section 3. In Section 4, the parameter design procedure and stability analysis are described in detail. Simulation and experimental results are provided in Section 5 to verify the correctness and effectiveness of the proposed droop control method. Finally, the conclusions of this paper are presented in Section 6.  Figure 1 shows the simplified structure of the islanded microgrid with multiple DG units in parallel operation, and the block diagram with the conventional droop control method for power sharing is illustrated in Figure 2.  Figure 1 shows the simplified structure of the islanded microgrid with multiple DG units in parallel operation, and the block diagram with the conventional droop control method for power sharing is illustrated in Figure 2.

Outer Droop Control Loop
In islanded mode, the inverter-based DG units work as voltage source converters (VSCs) in parallel to provide voltage and frequency support for the microgrid, and all DG units with different power capacities must achieve power sharing according to their own rated power for the economical and reliable operation of the microgrid. As shown in Figure 2, each DG unit calculates its own active and reactive power by measuring the local output voltage Uc and current Io. The conventional P-ω and Q-E control algorithm, as described in Equations (1) and (2), is also implemented for load power sharing and as the synthetic voltage reference. The mathematical expression of the droop control method can be shown as: where P and Q are the output active and reactive power of the inverter, respectively. P0 and Q0 are the reference active and reactive power, respectively, and they are usually set to zero in islanded microgrids. ωref and Eref are the synthetic reference frequency and voltage amplitude, respectively. ω0 and E0 are the nominal frequency and voltage amplitude, respectively. kp and kq are the droop coefficients for the inverter-based DGs operating in islanded mode, and the values are chosen according to the allowable maximum frequency and voltage amplitude deviation of the inverter. When multiple DGs with different power   Figure 1 shows the simplified structure of the islanded microgrid with multiple DG units in parallel operation, and the block diagram with the conventional droop control method for power sharing is illustrated in Figure 2.

Outer Droop Control Loop
In islanded mode, the inverter-based DG units work as voltage source converters (VSCs) in parallel to provide voltage and frequency support for the microgrid, and all DG units with different power capacities must achieve power sharing according to their own rated power for the economical and reliable operation of the microgrid. As shown in Figure 2, each DG unit calculates its own active and reactive power by measuring the local output voltage Uc and current Io. The conventional P-ω and Q-E control algorithm, as described in Equations (1) and (2), is also implemented for load power sharing and as the synthetic voltage reference. The mathematical expression of the droop control method can be shown as: where P and Q are the output active and reactive power of the inverter, respectively. P0 and Q0 are the reference active and reactive power, respectively, and they are usually set to zero in islanded microgrids. ωref and Eref are the synthetic reference frequency and voltage amplitude, respectively. ω0 and E0 are the nominal frequency and voltage amplitude, respectively. kp and kq are the droop coefficients for the inverter-based DGs operating in islanded mode, and the values are chosen according to the allowable maximum frequency and voltage amplitude deviation of the inverter. When multiple DGs with different power

Outer Droop Control Loop
In islanded mode, the inverter-based DG units work as voltage source converters (VSCs) in parallel to provide voltage and frequency support for the microgrid, and all DG units with different power capacities must achieve power sharing according to their own rated power for the economical and reliable operation of the microgrid. As shown in Figure 2, each DG unit calculates its own active and reactive power by measuring the local output voltage U c and current I o . The conventional P-ω and Q-E control algorithm, as described in Equations (1) and (2), is also implemented for load power sharing and as the synthetic voltage reference. The mathematical expression of the droop control method can be shown as: where P and Q are the output active and reactive power of the inverter, respectively. P 0 and Q 0 are the reference active and reactive power, respectively, and they are usually set to zero in islanded microgrids. ratings are connected in parallel, the droop coefficients should be designed to share the load power in proportion to each rated power [19].
where k pn and k qn (n = 1, 2, ..., N) are the droop coefficients corresponding to the nth DG unit, and P n and Q n are the nominal active and reactive power of each DG, respectively. Notably, Equation (1) holds only if the line impedance between the PCC and the DG unit is mainly inductive, as shown in Figure 3, and the active and reactive power flow from the DG unit to the PCC can be expressed as follows [31]: where E and U p are the output voltages of the DG and PCC voltage amplitudes, respectively; X and R represent the feeder reactance and resistance, respectively; and δ is the power angle difference between the DG and PCC voltages. When X >> R, Equation (4) can be rewritten as: Usually, the power angle δ is small, so it satisfies sinδ ≈ δ and co (5) can be simplified as: Moreover, when the reactance is much less than the resistance o X << R, the delivered active and reactive power can be approximated  Usually, the power angle δ is small, so it satisfies sinδ ≈ δ and cosδ ≈ 1, and Equation (5) can be simplified as: Moreover, when the reactance is much less than the resistance of the line impedance X << R, the delivered active and reactive power can be approximated as:

Inner Voltage Regulation Loop
The reference voltage is obtained through the above droop controller, and then the output voltage of each DG unit is generated based on the voltage regulation loop. The inner voltage regulation loop also consists of an outer voltage loop and an inner current loop based on the αβ stationary frame, as shown in Figure 4. G V (s) and G I (s) are the transfer functions of the voltage-loop controller and the current-loop controller, respectively, and they can be expressed as [14]: 2k ivh s s 2 +2ω cvh s+ω 2 h G I (s) = k pi (10) where G V (s) adopts a proportional resonant (PR) controller, G I (s) uses a proportional controller, k pv and k pi are the proportional gains, k ivh is the fundamental or harmonic resonant gain term, ω cvh is the cutoff frequency of the resonant controllers and ω h is the resonant frequency.

Inner Voltage Regulation Loop
The reference voltage is obtained through the above droop controller, and then the output voltage of each DG unit is generated based on the voltage regulation loop. The inner voltage regulation loop also consists of an outer voltage loop and an inner current loop based on the αβ stationary frame, as shown in Figure 4. GV(s) and GI(s) are the transfer functions of the voltage-loop controller and the current-loop controller, respectively, and they can be expressed as [14]: where GV(s) adopts a proportional resonant (PR) controller, GI(s) uses a proportional controller, kpv and kpi are the proportional gains, kivh is the fundamental or harmonic resonant gain term, ωcvh is the cutoff frequency of the resonant controllers and ωh is the resonant frequency. When the feeder impedance is mainly an inductor, the conventional droop technique can achieve accurate active power sharing; however, it has poor reactive power sharing performance due to the mismatch of the feeder impedance [1,4]. In the following section, a modified droop control strategy based on SACS injection is proposed for accurate active and reactive power sharing without communication links.

Proposed SACS-Injection-Based Modified Droop Method for Power Sharing
In this section, a SACS-injection-based modified droop control method without communication links is proposed which is motivated by the aforementioned secondary control strategy [14,30,32], and a detailed introduction is illustrated in the following section.

Motivated by the Secondary Control Strategy through PI Controller for Power Sharing
The authors of [14,32] proposed a secondary control strategy for accurate reactive power sharing. First, the secondary controller obtains the output reactive power Q of each DG unit via communication links, and then the reference reactive power Q* is calculated and sent back to the primary control. Finally, accurate reactive power sharing can be achieved by adding the compensator ∆E, calculated through the PI controller to the nominal voltage amplitude E0, which is provided in the following equations: where ∆E = kps (Q* −Q) + kis/s (Q* − Q), and kps and kis are the proportion and integration gains, respectively. Equation (11) can also be rewritten as where E = E0 + ∆E. As shown in Figure 5, the output reactive power Q of each DG unit can be adjusted due to the voltage compensator ∆E, and it can also achieve the reference value Q* due to When the feeder impedance is mainly an inductor, the conventional droop technique can achieve accurate active power sharing; however, it has poor reactive power sharing performance due to the mismatch of the feeder impedance [1,4]. In the following section, a modified droop control strategy based on SACS injection is proposed for accurate active and reactive power sharing without communication links.

Proposed SACS-Injection-Based Modified Droop Method for Power Sharing
In this section, a SACS-injection-based modified droop control method without communication links is proposed which is motivated by the aforementioned secondary control strategy [14,30,32], and a detailed introduction is illustrated in the following section.

Motivated by the Secondary Control Strategy through PI Controller for Power Sharing
The authors of [14,32] proposed a secondary control strategy for accurate reactive power sharing. First, the secondary controller obtains the output reactive power Q of each DG unit via communication links, and then the reference reactive power Q* is calculated and sent back to the primary control. Finally, accurate reactive power sharing can be achieved by adding the compensator ∆E, calculated through the PI controller to the nominal voltage amplitude E 0 , which is provided in the following equations: where ∆E = k ps (Q* − Q) + k is /s (Q* − Q), and k ps and k is are the proportion and integration gains, respectively. Equation (11) can also be rewritten as where E = E 0 + ∆E. As shown in Figure 5, the output reactive power Q of each DG unit can be adjusted due to the voltage compensator ∆E, and it can also achieve the reference value Q* due to the existence of the integral term k is /s. However, communication line failure and communication delay may result in poor reactive power sharing.

SACS-Injection-Based Modified Droop Control Method for Power Sharing
The above-mentioned method can achieve active reactive power sharing by adding the voltage compensator ∆E to the Q-E droop expression; however, communication lines are needed, which increase the complexity and high cost of the system. In this section, a SACS-injection-based modified droop control method, which is similar to Equation (11), is proposed. The difference between the method and equation is that the compensator ∆E is composed of the reactive power of the injected SACS, and it can be expressed as follows: where E′0 = E0 + Gq Qss, and Qss is the reactive power of the injected SACS, which will be described in detail in the following section. Gq is the amplifier gain, which is used for amplifying the reactive power of the SACS. To achieve accurate reactive power sharing for each DG unit, the following droop control strategy for the injected SACS is also used: where ωss and ωss0 are the reference frequency and the nominal frequency of the injected SACS, respectively, and ksq is the SACS droop coefficient for reactive power sharing. If two DG units are connected in parallel, according to droop Equation (14), the frequency difference ∆ωss in the injected SACSs for the two DG units can be expressed as follows: where ωss1 and ωss2 represent the injected SACS frequency of each DG unit. By integrating the frequency difference ∆ωss, the phase difference δss of the injected SACSs of each DG can be deduced as

SACS-Injection-Based Modified Droop Control Method for Power Sharing
The above-mentioned method can achieve active reactive power sharing by adding the voltage compensator ∆E to the Q-E droop expression; however, communication lines are needed, which increase the complexity and high cost of the system. In this section, a SACS-injection-based modified droop control method, which is similar to Equation (11), is proposed. The difference between the method and equation is that the compensator ∆E is composed of the reactive power of the injected SACS, and it can be expressed as follows: where E 0 = E 0 + G q Q ss , and Q ss is the reactive power of the injected SACS, which will be described in detail in the following section. G q is the amplifier gain, which is used for amplifying the reactive power of the SACS. To achieve accurate reactive power sharing for each DG unit, the following droop control strategy for the injected SACS is also used: where ω ss and ω ss0 are the reference frequency and the nominal frequency of the injected SACS, respectively, and k sq is the SACS droop coefficient for reactive power sharing. If two DG units are connected in parallel, according to droop Equation (14), the frequency difference ∆ω ss in the injected SACSs for the two DG units can be expressed as follows: where ω ss1 and ω ss2 represent the injected SACS frequency of each DG unit. By integrating the frequency difference ∆ω ss , the phase difference δ ss of the injected SACSs of each DG can be deduced as

Overall Control Block Diagram
The overall block diagram of the proposed modified droop control strategy based on the SACS injection method for accurate fundamental active and reactive power sharing is illustrated in Figure 6.

Overall Control Block Diagram
The overall block diagram of the proposed modified droop control strategy based on the SACS injection method for accurate fundamental active and reactive power sharing is illustrated in Figure 6. First, only the local signals of the output current ioαβ and filter capacitor voltage ucαβ for each DG unit are measured, and then the fundamental current ioαβf and SACS current ioαβss are separated via signal extraction [14,30]. The fundamental active power P, reactive power Q and SACS reactive power Qss can be calculated based on the αβ frame and are as follows: where ucαf, ucβf, ioαf and ioβf are the output fundamental voltage and current components based on the αβ frame, respectively. uαss, uβss, ioαss and ioβss are the voltage and current components of the injected SACSs based on the αβ frame.
Compared with the conventional droop control method, the proposed modified droop control strategy adds a compensator ∆E to the voltage droop control loop, and an extra reference voltage u*αβ_ss generated by the injected SACS is added to the fundamental reference voltage u*αβ_f to synthesize the overall reference voltage u*αβ_sum, which is shown as follows: First, only the local signals of the output current i oαβ and filter capacitor voltage u cαβ for each DG unit are measured, and then the fundamental current i oαβf and SACS current i oαβss are separated via signal extraction [14,30]. The fundamental active power P, reactive power Q and SACS reactive power Q ss can be calculated based on the αβ frame and are as follows: 2 ω cp s+ω cp (u cβss i oαss − u cαss i oβss ) (17) where u cαf , u cβf , i oαf and i oβf are the output fundamental voltage and current components based on the αβ frame, respectively. u αss , u βss , i oαss and i oβss are the voltage and current components of the injected SACSs based on the αβ frame.
Compared with the conventional droop control method, the proposed modified droop control strategy adds a compensator ∆E to the voltage droop control loop, and an extra reference voltage u* αβ_ss generated by the injected SACS is added to the fundamental reference voltage u* αβ_f to synthesize the overall reference voltage u* αβ_sum , which is shown as follows: Meanwhile, to achieve zero-steady-error tracking of the SACS reference voltage, the proportional resonance controller G V (s) of the inner voltage regulation loop should be modified to: 2k ivh s s 2 +2ω cvh s+ω 2 h + 2k ivs s s 2 +2ω cvs s+ω 2 ss (19) where k ivs and ω cvs represent the gain and cutoff frequency of the SACS resonant controller, respectively. To ensure that Equation (14) holds, a virtual resistor is also applied to the feeder impedance to make the feeder impedance mainly resistant for the implementation of SACS frequency droop control. The application of the virtual resistor in the droop control method has been mentioned in many studies [1,19], and it will not be introduced in detail here. Importantly, the overall control system may have virtual reactance and virtual resistance control loops. If the feeder impedance for the fundamental voltage is not mainly inductive, a virtual reactance control loop must be added to ensure that Equations (1) and (2) hold. In most cases, the feeder reactance for the fundamental frequency is mainly inductive, and the virtual reactance does not need to be added. However, the virtual resistor must be added to make the feeder impedance for the injected SACS frequency mainly resistive.

Parameter Design and Stability Analysis
In this section, all the parameters, such as the frequency ω ss and amplitude E ss0 of the injected SACS, the amplified gain G q , and the droop coefficients k p , k q and k ssq , are discussed in detail.

Design of the SACS Frequency and Amplitude
The selection of the SACS frequency and amplitude are critical for the accurate control of active and reactive power sharing. A SACS with a large amplitude is easy to extract, but it will result in a high harmonic component of the output voltage. In contrast, a small voltage amplitude will make an SCAS difficult to extract. Therefore, the voltage amplitude of the injected SACS should be chosen based on the tradeoff between the signal extraction and voltage quality standards [33]. Finally, the amplitude of the injected SACS is chosen to be nearly 1% of the fundamental voltage amplitude.
The frequency of the injected SACS should also be chosen carefully. On the one hand, the frequency of the injected SACS must be different from the output voltage harmonic frequency to easily extract it. It is known that the output voltage contains only odd-order components, whether linear or nonlinear loads. To facilitate signal extraction, we chose the even component as the frequency of the SACS. On the other hand, the output end of the inverter-based DG is usually connected via an LC second-order low-pass filter to alleviate the high-order harmonic voltage components; therefore, the frequency of the injected SACS must be lower than the cutoff frequency of the LC filter. Finally, the SACS frequency is set to four times the fundamental voltage frequency [30].

Design of the Gain G q
Gain G q is not only related to the SACS power injected into the microgrid system but also affects the stability of the whole system. On one hand, according to the expression of the voltage compensator ∆E = G q Q ss , a small gain G q leads to the provision of a large SACS reactive power to the system and causes a large harmonic output current. On the other hand, a large gain G q with small SACS reactive power will lead to the instability of the whole system. Taking two DG units in parallel as an example, the fundamental voltage amplitude difference of two DG units can be expressed as where E i , Q i and Q ssi are the output fundamental voltage amplitude, reactive power and injected SACS reactive power of each DG unit, respectively, where i = 1, 2. In a steady state, Q 1 = Q 2 , so the above equation can be simplified to: where ∆Q ss represents the difference in the reactive power of the injected SACSs. Similar to the equivalent circuit at the fundamental frequency, the two DG units with SACSs are connected in parallel, as shown in Figure 7, and the following equation holds: where Q ssL is the total load power of the reactive power of the injected SACSs. Moreover, 1/2(Q ss1 − Q ss2 ) represents the SACSs' reactive power flow from DG1 to DG2.
where QssL is the total load power of the reactive power of the injected SACSs. Moreover, 1/2(Qss1 − Qss2) represents the SACSs' reactive power flow from DG1 to DG2.
According to Equation (14) and the realization for the high performance of reactive power sharing, a virtual resistor is adopted to make the feeder impedance mainly resistive for the SACS, which is shown in Figure 8. Essα*, Essβ*, Essα0 and Essβ0 represent the reference voltages and the nominal voltage amplitudes of the injected SACSs, respectively.  Rv represents the value of the virtual resistance. Due to the addition of virtual resistance, the feeder impedance of the two SACS DG units is mainly resistive, and the SACS reactive power flows from DG1 to DG2 can also be expressed as: According to Equation (14) and the realization for the high performance of reactive power sharing, a virtual resistor is adopted to make the feeder impedance mainly resistive for the SACS, which is shown in Figure 8. E ssα *, E ss β *, E ssα0 and E ss β0 represent the reference voltages and the nominal voltage amplitudes of the injected SACSs, respectively.
Similar to the equivalent circuit at the fundamental frequency, the two DG units with SACSs are connected in parallel, as shown in Figure 7, and the following equation holds: where QssL is the total load power of the reactive power of the injected SACSs. Moreover, 1/2(Qss1 − Qss2) represents the SACSs' reactive power flow from DG1 to DG2.
According to Equation (14) and the realization for the high performance of reactive power sharing, a virtual resistor is adopted to make the feeder impedance mainly resistive for the SACS, which is shown in Figure 8. Essα*, Essβ*, Essα0 and Essβ0 represent the reference voltages and the nominal voltage amplitudes of the injected SACSs, respectively.  Rv represents the value of the virtual resistance. Due to the addition of virtual resistance, the feeder impedance of the two SACS DG units is mainly resistive, and the SACS reactive power flows from DG1 to DG2 can also be expressed as: R v represents the value of the virtual resistance. Due to the addition of virtual resistance, the feeder impedance of the two SACS DG units is mainly resistive, and the SACS reactive power flows from DG1 to DG2 can also be expressed as: where E ssi and R ssi represent the SACS voltage and line impedance, respectively, where i = 1, 2. δ ss is the phase angle difference of the injected SACSs for the two DG units. Moreover, the difference in the reactive power of the injected SACSs is provided by: By substituting Equation (24) into Equation (21), the following expression can be derived as follows: where R ss = R ss1 + R ss2 . According to the power angle stability criteria [34], the power angle δ ss should not exceed 90 • . To maintain a sufficient stability margin, the power angle δ ss should be less than 45, and sinδ ss ≈ δ ss should be true. The minimum value of the gain G q can be derived as To simplify the calculation, the amplitudes E ss1 and E ss2 of the SACSs can be approximated to E ss0 , and the voltage amplitude difference ∆E 12 is the allowable deviation of the rated voltage E 0 .

Design of the Droop Coefficienst k p , k q and k ssq
With larger droop coefficients of k p and k q , the active and reactive power sharing performance will be better, but the frequency and amplitude of the fundamental reference voltage will deviate greatly from the nominal values. The droop parameters k p and k q can be designed according to the following criteria [19]: where P max and Q max are the maximum output of the active and reactive power of each DG unit, and ∆ω and ∆E are the maximum allowable deviations in the frequency and voltage amplitude.
The dynamic performance of accurate reactive power sharing depends on the droop parameters k ssq of the injected SACSs. A large droop parameter k ssq can accelerate the process of accurate power sharing, but it may cause system instability. In contrast, with a small droop parameter k ssq , the transient process of reactive power sharing can no longer achieve power sharing. Therefore, if the system is stable, a large droop parameter k ssq should be selected to improve the dynamic performance of the system, and the effect on the stability of the system will be described in detail in the following section.

Stability Analysis
In this section, the relationship between the parameter k sq and the stability of multiple parallel DG systems will be analyzed in detail. According to Equations (15), (20) and (24), the fundamental voltage amplitude difference of two DG units can be expressed as Since low-pass filters are usually used for measuring blocks, Equation (29) can be rewritten as where ω c is the cutoff frequency of the low-pass filter. It is assumed that the total fundamental reactive power of the load is Q L , and the injected SACS frequency of each DG are equal in the steady state, which satisfies ω ss1 = ω ss2 . According to Equation (14), it can be deduced that the output reactive power of the two DG units is equal in the steady state, and it satisfies that Q 1 = Q 2 = Q L /2. However, when the system is in a transient process, such as load or output fundamental voltage changes, part of the reactive power flows into the load, and another part of the reactive power flows into other DG units. They can be expressed as follows: Since the feeder impedance for the fundamental voltage is usually mainly inductive, and according to Equation (7), the reactive power flowing from DG1 to DG2 can be deduced as Since the low-pass filter is also used for the power calculation, the above equation can be rewritten as Based on the previous analysis, the control block diagram of reactive power sharing can be derived as shown in Figure 9, and the simulation parameters of the islanded microgrid are listed in Table 1. The root locus of the open-loop transfer function can be obtained as shown in Figure 10. Different color lines represent different root locus and the direction of the arrows indicate that the value of k increases in Figure 10. It can be seen that when the value of droop coefficient k sq is greater than 0.03, the root of the closed-loop system will appear in the right half plane and become unstable. Figure 11 also shows the Bode diagram of the open-loop transfer function, and with the increase in the droop coefficient k sq , the phase margin of the system decreases gradually. Therefore, the droop coefficient k sq should be chosen based on the tradeoff between the system stability and dynamic response, and a large coefficient is selected preferentially under the condition of ensuring the stability of the system. Since the feeder impedance for the fundamental voltage is usually mainly inductive, and according to Equation (7), the reactive power flowing from DG1 to DG2 can be deduced as Since the low-pass filter is also used for the power calculation, the above equation can be rewritten as Based on the previous analysis, the control block diagram of reactive power sharing can be derived as shown in Figure 9, and the simulation parameters of the islanded microgrid are listed in Table 1. The root locus of the open-loop transfer function can be obtained as shown in Figure 10. Different color lines represent different root locus and the direction of the arrows indicate that the value of k increases in Figure 10. It can be seen that when the value of droop coefficient ksq is greater than 0.03, the root of the closed-loop system will appear in the right half plane and become unstable. Figure 11 also shows the Bode diagram of the open-loop transfer function, and with the increase in the droop coefficient ksq, the phase margin of the system decreases gradually. Therefore, the droop coefficient ksq should be chosen based on the tradeoff between the system stability and dynamic response, and a large coefficient is selected preferentially under the condition of ensuring the stability of the system.

Simulation and Experimental Results
In this section, the performance of the proposed SACS-injection-based modified droop control strategy is validated based on simulation and experimental results. The de-

Simulation and Experimental Results
In this section, the performance of the proposed SACS-injection-based modified droop control strategy is validated based on simulation and experimental results. The detailed system parameters are listed in Table 2.

Simulation and Experimental Results
In this section, the performance of the proposed SACS-injection-based modified droop control strategy is validated based on simulation and experimental results. The detailed system parameters are listed in Table 2. Gain G q 12 15 Low pass fifilters ω cp 31 31 Virtual resistor R v (Ω) 8 5 Load R 1 (mH + Ω) 10 + 15 18 + 13.8 Load R 2 (mH + Ω) 0 + 15 5 + 13.8

Simulation Results
The simulation modes of the DG units with the same three-phase full-bridge topology are built based in MATLAB, and three DG units are connected in parallel with different feeder impedances, as shown in Figure 1. Figure 12 demonstrates the simulation waveforms of the output active power P, reactive power Q and fundamental frequency f for each DG unit via the conventional droop control strategy. In the initial state, R 1 is connected into the islanded microgrid as the load. At 3 s, the load R 2 is also plugged into the microgrid system, and the feeder impedance is mainly inductance. It can be observed that accurate active power sharing can be achieved with the conventional droop control strategy, but the output reactive power for each DG unit is different from the mismatched feeder impedance.
nected into the islanded microgrid as the load. At 3 s, the load R2 is also plugged into th microgrid system, and the feeder impedance is mainly inductance. It can be observed tha accurate active power sharing can be achieved with the conventional droop control stra egy, but the output reactive power for each DG unit is different from the mismatche feeder impedance. The simulation waveforms of the output active power P, reactive power Q, funda mental frequency f, voltage compensator ∆E and SACS frequency fss with the propose modified droop control strategy for the paralleled three DG units are shown in Figure 1 and it can be seen that high-performance power sharing can be achieved. Since the P − droop expression remains unchanged, accurate active power can also be achieved in The simulation waveforms of the output active power P, reactive power Q, fundamental frequency f, voltage compensator ∆E and SACS frequency f ss with the proposed modified droop control strategy for the paralleled three DG units are shown in Figure 13, and it can be seen that high-performance power sharing can be achieved. Since the P − f droop expression remains unchanged, accurate active power can also be achieved in a steady state. Meanwhile, with the SACS-injection-based modified droop method, the reactive power error for each DG unit can be eliminated. When the load R 2 is also plugged into the microgrid system at 3 s, the reactive power output of these three DG units can also be divided equally quickly. Moreover, the voltage compensator ∆E, which is composed of the SACS reactive power Q ss and gain G q for each DG unit, is also shown in Figure 13. With different compensators ∆E, the conventional Q-E droop expression is modified as shown in Figure 5, and the output voltage amplitude of each DG unit is also adjusted. Accurate reactive power will be achieved when the system reaches the steady state. into the microgrid system at 3 s, the reactive power output of these three DG units can also be divided equally quickly. Moreover, the voltage compensator ∆E, which is composed of the SACS reactive power Qss and gain Gq for each DG unit, is also shown in Figure  13. With different compensators ∆E, the conventional Q-E droop expression is modified as shown in Figure 5, and the output voltage amplitude of each DG unit is also adjusted. Accurate reactive power will be achieved when the system reaches the steady state.   Figure 14 shows the experimental prototype of two paralleled DG units connected in an islanded microgrid, and a three-phase two-level full-bridge PWM inverter is chosen as the main circuit topology for each DG unit. The control algorithm for the DG unit is implemented in the dSPACE SCALEXIO platform (dSPACE, Paderborn, Germany) as shown in Figure 15. A 14-bit analog-to-digital converter (ADC DS6221) (dSPACE) is used to sample the output voltage, inductor current and load current signals of each DG unit at a 20 kHz sampling frequency. The I/O board card (DS6202) (dSPACE) is configured as a PWM module with a 20 kHz switch frequency. A power analyzer (Tek-PA3000) (Tektronix, Inc. Beaverton, OR, USA) is used to measure the output power and voltage signals for analysis, and an oscilloscope (YOKOGAWA-DLM2024) (YOKOGAWA, Tokyo, Japen) is used to collect the output voltage, frequency and other information. Figure 16 shows the experimental waveform of the output power with the conventional droop control and the proposed SACS-injection-based droop control strategy. In the initial state, the DG units adopt the conventional droop control strategy, and 10 s later, the proposed modified droop control method is applied to the two DG units. Figure 16A demonstrates that the active power can be shared well with both control methods, and Figure 16B shows that the reactive power sharing error can be eliminated with the proposed modified droop method, which is enabled at 10 s.

Experimental Verification
Sensors 2023, 23, x FOR PEER REVIEW Figure 14 shows the experimental prototype of two paralleled DG units conn an islanded microgrid, and a three-phase two-level full-bridge PWM inverter is c the main circuit topology for each DG unit. The control algorithm for the DG un plemented in the dSPACE SCALEXIO platform (dSPACE, Paderborn, Germany) a in Figure 15. A 14-bit analog-to-digital converter (ADC DS6221) (dSPACE) is used ple the output voltage, inductor current and load current signals of each DG un kHz sampling frequency. The I/O board card (DS6202) (dSPACE) is configured as module with a 20 kHz switch frequency. A power analyzer (Tek-PA3000) (Tektro Beaverton, OR, USA) is used to measure the output power and voltage signals for and an oscilloscope (YOKOGAWA-DLM2024) (YOKOGAWA, Tokyo, Japen) is collect the output voltage, frequency and other information.   Figure 16 shows the experimental waveform of the output power with the tional droop control and the proposed SACS-injection-based droop control stra the initial state, the DG units adopt the conventional droop control strategy, and 1 the proposed modified droop control method is applied to the two DG units. Fig   Figure 14. Experimental prototype for an islanded microgrid system. Figure 14 shows the experimental prototype of two paralleled DG unit an islanded microgrid, and a three-phase two-level full-bridge PWM invert the main circuit topology for each DG unit. The control algorithm for the plemented in the dSPACE SCALEXIO platform (dSPACE, Paderborn, Germ in Figure 15. A 14-bit analog-to-digital converter (ADC DS6221) (dSPACE) i ple the output voltage, inductor current and load current signals of each D kHz sampling frequency. The I/O board card (DS6202) (dSPACE) is configu module with a 20 kHz switch frequency. A power analyzer (Tek-PA3000) ( Beaverton, OR, USA) is used to measure the output power and voltage signa and an oscilloscope (YOKOGAWA-DLM2024) (YOKOGAWA, Tokyo, Jap collect the output voltage, frequency and other information.    The dynamic experimental waveform of power sharing with the SACS injection-based modified droop control strategy is shown in Figure 17. The resistive-inductive load Z 1 is connected to the islanded microgrid, and a resistive load R 2 is switched ON or OFF every 8 s. The high performance of the transient process for power sharing is exhibited, and accurate active and reactive power sharing can be achieved in a steady state.

Experimental Verification
Moreover, the voltage waveform of the PCC is also measured, as shown in Figure 18, and the total harmonic distortion (THD) value of the voltage waveform is also analyzed. Due to the injection of a four-order SACS, a slight voltage distortion is shown in the output voltage waveform, and the THD value of the PCC voltage is 1.30%. However, it can still meet the voltage harmonic standard [29]. The dynamic experimental waveform of power sharing with the based modified droop control strategy is shown in Figure 17. The resistiv Z1 is connected to the islanded microgrid, and a resistive load R2 is swit every 8 s. The high performance of the transient process for power sha and accurate active and reactive power sharing can be achieved in a stea Moreover, the voltage waveform of the PCC is also measured, as sh

Conclusions
In this paper, a SACS-injection-based modified droop control method is proposed for accurate active and reactive power sharing when DG units operate in islanded microgrids, and the parameter design procedure and stability analysis are described in detail. The proposed modified droop method has the advantage of no communication lines and the specific parameters of feeder impedance, a low cost, the high adaptability of the microgrid configuration and plug-and-play functionality. Finally, simulation and experimental results verify the correctness and effectiveness of the proposed modified droop control method.
However, the method proposed in this paper has some disadvantages: (1) it is necessary to inject small-AC-voltage signals into the system, which will lead to an increase in the harmonic control of the grid voltage, so they must be carefully selected. (2) This method can achieve accurate active power and reactive power sharing, but it can not achieve secondary frequency recovery, which is a direction for the future research.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

U c , I o
The local output voltage and current P, Q The output active and reactive power of the inverter P 0 , Q 0 The reference active and reactive power ω re f , E re f The synthetic reference frequency and voltage amplitude w 0 , E 0 The nominal frequency and voltage amplitude respectively k p , k q The droop coefficients for the inverter-based DGs operated in the islanded mode. k pn , k qn The droop coefficients corresponding to the nth DG unit (n = 1, 2, ..., N) P n , Q n The nominal active and reactive power of each DG E, U p The output voltage of the DG and PCC voltage amplitude X, R The feeder reactance and resistance δ The power angle difference between the DG and PCC voltages G V (s), G I (s) The transfer functions of the voltage-loop controller and the current-loop controller k pv , k pi The proportional gains k ivh The fundamental or harmonic resonant gain term w cvh The cutoff frequency of the resonant controllers w h The resonant frequency Q * The reference reactive power k ps , k is The proportion and integration gains Q ss The reactive power of the injected SACS G q The amplifier gain, which is used for amplifying the reactive power of the SACS w ss , w ss0 The reference frequency and the nominal frequency of the injected SACS k sq The SACS droop coefficient for reactive power sharing ∆w ss The frequency difference of the injected SACSs for the two DG units w ss1 , w ss2 The injected SACS frequency of each DG unit δ ss The phase difference of the injected SACS of each DG i oαβ , U cαβ The local signals of output current and filter capacitor voltage for each DG unit i oαβ f , i oαβss The fundamental current and SACS current Q ss The SACS reactive power U cα f , U cβ f The output fundamental voltage and current components i oα f , i oβ f Based on the αβ frame u αss , u βss The voltage and current components of the injected SACS i oαss , i oβss Based on αβ frame u * αβ_ss Extra reference voltage generated by the injected SACSs u * αβ_ f The fundamental reference voltage u * αβ_sum The overall reference voltage k ivs , w cvs The gain and cutoff frequency of the SACS resonant controller G q The amplified gain k p , k q , k ssq The droop coefficients E i , Q i , Q ssi the output fundamental voltage amplitude, reactive power and injected SACS reactive power of each DG unit, respectively, i = 1, 2 ∆Q ss The difference in the reactive power of the injected SACSs Q ssL The total of load power of the reactive power of the injected SACSs E * ssα , E * ssβ The reference voltage and the nominal voltage amplitude of the E ssα0 and E ssβ0 injected SACSs, respectively R v The value of the virtual resistance E ssi , R ssi The SACS voltage and line impedance, respectively, i = 1, 2 δ ss The phase angle difference of the injected SACSs for the two DG units P max , Q max The maximum output active and reactive power of each DG unit ∆w, ∆E The maximum allowable deviations in the frequency and voltage amplitudes k ssq The droop parameters of the injected SACSs w c The cut-off frequency of the low-pass filter