Dynamic Behavior Improvement of Control System in Inverter-Based Island Microgrid by Adding a Mixed Virtual Impedance Loop to Voltage Control Loop
Subject Areas : International Journal of Smart Electrical Engineering
Saeid Farhang
1
,
Ghazanfar Shahgholian
2
,
Bahador Fani
3
1 - Najafabad Branch, Islamic Azad University
2 - Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran
3 - Najafabad Branch, Islamic Azad University, Najafabad, Iran
Keywords: Stability, small signal analysis, Island microgrid, distributed generation sources, frequency and voltage droop strategy, power distribution, transient mode response,
Abstract :
A common method for controlling a group of parallel converters in decentralized control strategy structure in an island microgrid, the use is the droop-down characteristics of frequency ω-P and voltage E-Q. However, the problem with using this method is that the reactive power is not properly distributed (in proportion to the capacity of the micronutrients) between the micronutrients, which may lead to overload in the converters. Microgrids may also suffer from dynamic stability problems such as power fluctuations, which can be increased by switching between active and reactive power control. To avoid this problem, the X / R ratio of transmission lines is an important parameter that should be carefully considered in the design of micronutrient controllers. By linearizing and simplifying conditions, the control system conversion function model becomes a single input-single output system, which is efficient enough to show the relationship between control parameters such as slope of droop characteristics and derivative sentences, virtual impedance, and voltage controllers. Using this model, stability conditions for different parameters are analyzed. Also, to improve power distribution stability, common droop strategies are modified by adjusting the slope as well as adding nonlinear sentence sections. This approach reduces the coupling between active and reactive power control and reduces the dependence of power distribution on grid parameters such as the X / R ratio. To evaluate the reliability of the proposed model, the simulation results in a sample island microgrid in MATLAB software are presented.
8 International Journal of Smart Electrical Engineering, Vol.10, No.1, Winter 2021 ISSN: 2251-9246
EISSN: 2345-6221
pp. 1:6 |
Dynamic Behavior Improvement of Control System in Inverter-Based Island Microgrid by Adding a Mixed Virtual Impedance Loop to Voltage Control Loop
Saeid Farhang1,4, Ghazanfar Shahgholian*2,4, Bahador Fani3,4
1M.Sc., Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran,
2Associate Professor, Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran, shahgholian@iaun.ac.ir
3Associate Professor, Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran, b.fani@pel.iaun.ac.ir
4Smart Microgrid Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Abstract
A common method for controlling a group of parallel converters in decentralized control strategy structure in an island microgrid, the use is the droop-down characteristics of frequency ω-P and voltage E-Q. However, the problem with using this method is that the reactive power is not properly distributed (in proportion to the capacity of the micronutrients) between the micronutrients, which may lead to overload in the converters. Microgrids may also suffer from dynamic stability problems such as power fluctuations, which can be increased by switching between active and reactive power control. To avoid this problem, the X / R ratio of transmission lines is an important parameter that should be carefully considered in the design of micronutrient controllers. By linearizing and simplifying conditions, the control system conversion function model becomes a single input-single output system, which is efficient enough to show the relationship between control parameters such as slope of droop characteristics and derivative sentences, virtual impedance, and voltage controllers. Using this model, stability conditions for different parameters are analyzed. Also, to improve power distribution stability, common droop strategies are modified by adjusting the slope as well as adding nonlinear sentence sections. This approach reduces the coupling between active and reactive power control and reduces the dependence of power distribution on grid parameters such as the X / R ratio. To evaluate the reliability of the proposed model, the simulation results in a sample island microgrid in MATLAB software are presented.
Keywords: distributed generation sources, island microgrid, frequency and voltage droop strategy, stability, small signal analysis, power distribution, transient mode response
Article history: Received 20-Apr-2021; Revised 01-May-2021; Accepted 15-May-2021.
© 2021 IAUCTB-IJSEE Science. All rights reserved
1. Introduction
In recent years, distributed generation (DG) resources have been widely used due to their numerous economic and environmental benefits [1,2]. In addition to providing consumer demand, these resources increase reliability, reduce system losses, and simultaneously generate electricity and heat while reducing costs [3,4]. Distributed generation sources use renewable energy to generate electricity. Inverter-based distributed generation sources, such as photovoltaic systems, are more widely used than other distributed generation sources [5,6]. Ease of installation and operation, utilization of renewable energy and high efficiency of production capacity, are the reasons for more use of these resources [7,8]. In recent years, the construction of solar power plants in various dimensions and production capacities is increasing, so that it is predicted that in the near future, more than ten percent of the total electricity produced by these sources will be provided [9,10].
A set of distributed energy sources, local loads, information sending and receiving platforms, monitoring control systems to manage the performance of resources, networks and shared loads form a microgrid formed in the context of a distribution system and acts as a controllable set by the main network [11,12].
The microgrid can be operated in two modes of network connection or autonomous (island) [13,14]. The ability of the microgrid to become an island and its autonomous function is very important. This capability provides energy for sensitive loads in the microgrid in the event of network interruptions, which improves the reliability of consumers within the microgrid [15,16].
A. Reviews of relevant literature:
Microgrids are part of the distribution network, and there is usually a combination of active and reactive power that is not properly controlled independently [17,18]. In other words, the effect of active and reactive powers on each other reduces the margin of stability of resources [19,20]. The droop-down characteristics ω-P (voltage frequency-active power) and E-Q (voltage amplitude-reactive power) are commonly used to control real and reactive capacities in island microgrids [21,22].
A particle optimization algorithm is proposed to adjust the controller parameters in [23] in which the usual droop characteristics are used to improve the transient stability of the system. Also, to investigate the effect of load dynamics on low frequency fluctuations of the system, different types of loads have been studied.
An islanded microgrid as a linear multivariable dynamic system is modelled in [24], and the multivariable analysis tools are employed. Droop control dependency on the X/R ratio of the microgrid distributed energy sources is recognized and its type is presented using the increase ratio concept array. Also, three different controllers including H-infinity, H-2 and sequential PID controls are designed and compared.
The virtual and reactive power virtual frame method is proposed for the desired X/R ratio of the lines in [25], in which the two parameters of virtual active power and virtual reactive power are defined as a linear combination of real and real reactive powers. For the desired X/R ratio of the lines, there is an angle between the independent variables of P and Q with the independent variables E and ω equal to the impedance angle. By rotating the real power frame to obtain the virtual power frame along the frame (E, ω), the virtual real power can be controlled directly by the frequency and the virtual reactive power directly by the voltage range. Therefore, the use of virtual values in droop characteristics improves the system stability margin and dynamic response. In this method, power sharing is done on virtual values.
In [26] by means of voltage recovery mechanism, the amount of output voltage of the sources is also kept within the allowable range. In this method, using the parametric consensus control method as the reactive power distribution error, based on the information about each source and its neighbors in the directional graph, the equivalent of the system is calculated. The calculated reactive power distribution error is due to the difference between the resistance term and the self-terminal impedance of the communication lines of the adjacent sources. To calculate the amount of virtual impedance, the difference between the reactive power distribution error is calculated and the reactive power generated by each source is entered into a PI controller and in the permanent term is used to correct the adaptive virtual impedance. In other words, the term determines how much of the voltage drop across the impedance of the lines is related to the resistance part and how much is related to its reactance part. Then, based on that resistance and selfie term, the virtual selfie impedance is determined.
Another way to reduce the coupling of real and reactive power and improve system stability is to use virtual output impedance [27,28].
A method based on adaptive mixed virtual impedance has been proposed to ideally distribute reactive power and improve the stability of island microgrids in [29]. The proposed virtual impedance consists of two terms of resistance and selfie. The virtual impedance resistance term is obtained to adjust the voltage droop across the output of the E-Q characteristic by adjusting the characteristic slope ΔRv-P, which uses an integral control loop to adjust the characteristic slope. The virtual impedance self-semester adjusts the output current angle of the sources to increase the selfie property of the system and reduce the combination of active and reactive power control. Also, a low-bandwidth telecommunication link is used to send the reference bus information to the local controller of each micro-source to adjust the virtual impedance in this method.
The use of a dynamic slope characteristic (ArctanP-ω) instead of the ω-P characteristic is suggested in [30]. This characteristic has a high slope in loading around the origin and therefore has a suitable response and sharing speed, and in other parts of the work, especially in high loading, it has a low slope, which will bring better stability and dynamic response. Due to the limited amplitude of changes in the Arctang function, the frequency remains guaranteed. In this method, the dynamic response may not be acceptable at operating points where the characteristic slope is high, and at points where the characteristic slope is low, a division error may occur even for real power. Given that these points usually correspond to the nominal limits of the resources, there is a possibility of overload for the resources due to the sharing error.
B. Main study and novelty:
The objectives of this article can be stated as follows:
- Investigation and analysis of changes in the parameters of common droop strategies on the margin of system stability and stability of active and reactive power distribution.
- Investigation of the effect of resource operating point changes on the transient response of power control system and voltage control of inverter sources.
- Modify the control structure of common droop strategies according to their disadvantages so that the stability of reactive power distribution is improved.
C. Main structure of the article:
After stating the importance of the issue in the introduction, in the second part, the use of the virtual impedance loop in the voltage control loop is stated. In the third part, the system under study is mentioned. In the fourth part, the simulation results using MATLAB software are given for two modes. Also, the simulation results of the system in terms of small signal stability are shown using the Nyquist diagram. Finally, in the fifth part, conclusions and suggestions for further research are given.
2. Application of virtual impedance loop in voltage control loop
The use of conventional droop methods in all cases does not cause the ideal distribution of active and reactive power in the microgrid [31,32]. And various factors such as differences in the size of the impedance of the lines connected to the sources, the X/R ratio of the system and the load level, the accuracy of power distribution and the margin of stability of the system are affected [33,34].
Due to the high dependence of common droop methods on the impedance of lines connected to sources, by changing the impedance values of these lines and creating a dominant resistance or inductive property can improve the distribution of active and reactive power, which reduces the coupling between active and reactive power control [35,36]. Increases the margin of stability in the system. Therefore, a large amount of inductor can be used to connect the voltage source converter to the ac bus, which causes the line impedance to become the dominant inductor and improves reactive power distribution and reduces the combination of active and reactive power sources in the system [37,38].
Although the use of this method can help reduce the error of reactive power distribution in the microgrid, but it increases the size of the inductor used and also increases the cost of installation and commissioning of the system [39,40]. Also, using this method causes high losses and significant voltage droop in the dc bus of the source. Therefore, using the above method is not a suitable solution for the system. Self-generating in-line induction using virtual impedance is used to improve reactive power distribution and improve the stability margin. Virtual impedance is the addition of a section to the controller that causes the controller to see a virtual voltage droop across the network that does not actually exist [41,42].
The virtual impedance parameter is defined as complex. RV is the value of the real part of the virtual impedance or in other words the virtual resistance and LV is the value of the virtual inductor. In this case, by selecting the appropriate values for virtual resistance and virtual inductor, the X/R ratio of the system in inductive networks can be increased and reduced in resistance networks to reduce the combination of active and reactive power, thus increasing the stability margin of the system.
3. System under study
The microgrid structure of the studied sample is shown in Fig. 1. This microgrid consists of 3 DG units that are connected in parallel to the point of common coupling (PCC) and feed the local and common loads. Loads L1 to L3 are local loads connected to distributed generation sources and loads L4 and L5 are loads connected to a common bus. Three DG units have the same capacity and each includes a DC voltage source and an inverter to convert DC voltage to AC voltage. The inverter is connected to the microgrid via an LC filter.
Z1, Z2 and Z3 are the impedance of the lines between the DGs and the common bus.
Fig. 1. System under study
4. Simulation results
In this part, the studied system is simulated to investigate the performance of derivative sentences in frequency and voltage droop characteristics, as well as the effect of the virtual impedance loop on improving the microgrid dynamic response. In these simulations, in order to better determine the effect of the corrections made on the droop characteristics and voltage control loop, the value of frequency droop slope characteristic (mp=0.0084) has been selected to increase the frequency of oscillating modes of the system and reduce the stability margin of the system. In this section, two different simulation scenarios are investigated with the aim of modeling small and large changes in the working point of design resources.
The impedance lines for system simulation are: Z1=1.6+j2.45 Ω, Z2=1.1+j1.51 Ω and Z3=0.8+ji.13 Ω. The nominal power of the source and the nominal frequency are 1 KVA and 50 Hz, respectively. DC link voltage is 208 V. The local loads are the same and equal to: 1215 W and 1030 Var.
A. Locating local and shared loads:
In this case, which models small changes in the working point of microgrid resources, the switching process of common and local loads is considered. The frequency droop characteristic slope is set to 0.0084.
In this mode of simulation, the resources in the microgrid experience a significant increase in load in 0.5 sec seconds and a sudden decrease in load in moment 1 sec.
Fig. 2 shows the output of active and reactive power sources in a situation where the usual droop characteristics are used. As can be seen, a significant increase in power fluctuation has occurred in both active and reactive power due to a significant increase in the mp parameter.
In other words, with the change of the working point of the resources due to the switching process of common and local loads, the frequency of oscillation modes of the system has increased and the response of the sources has been associated with the occurrence of these oscillating disturbances with low damping speed.
Figs. 3 and 4 show the simulation results for the use of the modified droop characteristics and the use of the modified droop characteristics with the virtual impedance loop, respectively.
By comparing the simulation results qualitatively, we can see the improvement of the dynamic and transient behavior of the microgrid in the case that derivative sentences are used in the resource power control characteristics.
(a) Active power
(b) Reactive power
Fig. 2. Using common droop characteristics
With the addition of a virtual impedance loop to the voltage control loop, the transient response sources are further improved and the power fluctuations in the output of the active and reactive power generation sources are completely damped.
As the slope of the frequency and voltage droop characteristics increases, the coupling between active and reactive power control increases. Therefore, by selecting the appropriate virtual impedance value, the factors of increasing the X/R ratio of the system and subsequently reducing the coupling between the active and reactive power controllers are provided and the transient response and stability of the system are improved. The value of virtual impedance has been selected to significantly increase the X / R ratio of the system as ZV = RV + JXV = 0.2 + j1.75 Ω.
A. Installation properties and setting up:
In this case, the ability to connect new resources without the need to lose the microgrid or part of it has been examined. Comparing Figs. 5 and 6, it can be seen that without using derivative sentences that modify the dynamic behavior of the droop characteristics, the transient due to the arrival of a new source in 0.5 seconds has a large amplitude range in active and reactive powers and also, the oscillating response is at a low damping rate.
However, the use of derivative sentences has reduced the fluctuations of active and reactive powers and improved transient response.
However, the amplitude is still high at the time of the major malfunction, which may cause the protection systems to malfunction.
(a) Active power
(b) Reactive power
Fig. 3. Using modified droop characteristics with derivative sentences
(a) Active power
(b) Reactive power
Fig. 4. Using modified droop characteristics with derivative sentences with virtual impedance loop
On the other hand, it can be seen in Fig. 7 that, using the virtual impedance loop, the amplitude of the uplift in the transient response is acceptable.
B. Small signal analysis of voltage control loop with virtual impedance loop:
Fig. 8 shows the Nyquist diagram of the voltage control loop without the use of virtual impedance and Fig. 9 shows the Nyquist diagram with the presence of the virtual impedance loop.
(b) Active power
(b) Reactive power
Fig. 5. Using common droop characteristics
(b) Active power
(b) Reactive power
Fig. 6. Using the droop characteristics modified with derivative sentences
As can be seen, the control system function is stable in both cases. However, the use of virtual impedance has led to an increase in the stability margin of the control system. The phase limit of the system has increased from 109o to 123o and the maximum amount of overheating has decreased from 2.78 dB to 1.15 dB. Therefore, the use of virtual impedance improves the frequency response of the voltage control loop, which will improve the stability margin.
(b) Active power
(b) Reactive power
Fig. 7. Using modified droop characteristics with derivative sentences with virtual impedance loop
Fig. 8. Nyquist diagram of the voltage control ring conversion function without the presence of virtual impedance
Fig. 9. Nyquist diagram of the voltage control loop conversion function using virtual impedance
5. Conclusions and suggestions
In this paper, a suitable dynamic structure (coefficients of active and reactive power derivatives of sources) is considered to improve the dynamic and transient behavior of the frequency and voltage control characteristics of an inverter voltage source. As mentioned, the ratio of derivative sentence coefficients in drop characteristics is directly related to the X/R ratio of lines connected to sources. Therefore, the use of a virtual impedance loop in the system voltage control loop to increase the X/R ratio of the system was presented virtually. The use of virtual impedance increases the selfie property of the sources and reduces the coupling between active and reactive power control. Small signal analysis was also shown to use virtual impedance loops to improve transient response and system stability margin. Using the time domain simulation results for the studied microgrid, the efficiency of the dynamic structure added to the power control characteristics and the virtual impedance loop added to the voltage controller were shown in terms of improving the dynamic and transient behavior of the microgrid.
According to the contents and the characteristics of the methods discussed, the following suggestions are provided to continue the work:
• Provide an algorithm for the optimal selection of derivative sentence coefficients in a way that both improves dynamic behavior and increases the accuracy of reactive power distribution.
• Investigation of the effect of different loads such as motor loads and nonlinear loads for microgrid on dynamic performance of microgrid.
• Practical implementation of proposed methods for controlling inverter resources in a laboratory complex.
References
[1]. M. Salari, F. Haghighatdar-Fesharaki, “Optimal placement and sizing of distributed generations and capacitors for reliability improvement and power loss minimization in distribution networks”, Journal of Intelligent Procedures in Electrical Technology, vol. 11, no. 43, pp. 83-94, Dec. 2020.
[2]. J. Peng, B. Fan, W. Liu, "Voltage-based distributed optimal control for generation cost minimization and bounded bus voltage regulation in dc microgrids", IEEE Trans. on Smart Grid, vol. 12, no. 1, pp. 106-116, Jan. 2021.
[3]. S. F. Zarei, M. Parniani, "A comprehensive digital protection scheme for low-voltage microgrids with inverter-based and conventional distributed generations", IEEE Trans. on Power Delivery, vol. 32, no. 1, pp. 441-452, Feb. 2017.
[4]. M. Alilou, S. Sadi, S. Zamanian, J. Gholami, S. Moshari, "Improving the efficiency of actual distribution system by allocating multi-DG and DSTATCOM", Journal of Intelligent Procedures in Electrical Technology, vol. 12, no. 45, pp. 1-15, June 2021 (in Persian).
[5]. A.B. Nassif, "A protection and grounding strategy for integrating inverter-based distributed energy resources in an isolated microgrid", CPSS Trans. on Power Electronics and Applications, vol. 5, no. 3, pp. 242-250, Sept. 2020
[6]. Q.Yang, A. Ehsan, L. Jiang, X. Fang, “19 - Optimal energy dispatch in residential community with renewable DGs and storage in the presence of real-time pricing”, Smart Power Distribution Systems, pp. 447-465, 2019
[7]. A.Z.A. Shaqsi, K. Sopian, A. Al-Hinai, “Review of energy storage services, applications, limitations, and benefits”, Energy Reports, vol. 6, pp. 288-306, Dec. 2020.
[8]. A. Zerrahn, W.P. Schill, C. Kemfert, “On the economics of electrical storage for variable renewable energy sources”, European Economic Review, vol. 108, pp. 259-279, Sept. 2018.
[9]. O. Sharifiyana, M. Dehghani, G. Shahgholian, S. Mirtalaee, M. Jabbari, “Overview of dc-dc non-insulated boost converters (Structure and improvement of main parameters)”, Journal of Intelligent Procedures in Electrical Technology, vol. 12, no. 48, pp. 1-29, March 2022 (in Persian).
[10]. F. Katiraei, J.R. Aguero, "Solar PV integration challenges", IEEE Power and Energy Magazine, vol. 9, no. 3, pp. 62-71, 2011.
[11]. D. E. Olivares, A. Mehrizi-Sani, A. H. Etemadi, C. A. Cañizares, R. Iravani, M. Kazerani, A.H. Hajimiragha, O. Gomis-Bellmunt, M. Saeedifard, R. Palma-Behnke, G. A. Jiménez-Estévez, N. D. Hatziargyriou, "Trends in microgrid control", IEEE Trans. on Smart Grid, vol. 5, no. 4, pp. 1905-1919, May 2014.
[12]. G. Shahgholian, “A brief review on microgrids: Operation, applications, modeling, and control”, International Transactions on Electrical Energy Systems, vol. 31, no. 6, Artiacl Number: e12885, 2021.
[13]. S. Zamanian, S. Sadi, R. Ghaffarpour, A. Mahdavian, “Inverter-based microgrid dynamic stability analysis considering inventory of dynamic and static load models”, Journal of Intelligent Procedures in Electrical Technology, vol. 11, no. 44, pp. 91-109, March 2021 (in Persian).
[14]. J.F. Patarroyo-Montenegro, F. Andrade, J.M. Guerrero, J.C. Vasquez, "A linear quadratic regulator with optimal reference tracking for three-phase inverter-based islanded microgrids", IEEE Trans. on Power Electronics, vol. 36, no. 6, pp. 7112-7122, June 2021.
[15]. F. Katiraei, R. Iravani, N. Hatziargyriou, A. Dimeas, "Microgrids management", IEEE Power and Energy Magazine, vol. 6, no. 3, pp. 54-65, May-June 2008.
[16]. M. Sadegheian, B. Fani, I. Sadeghkhani, G. Shahgholian, "A local power control scheme for electronically interfaced distributed generators in islanded microgrids", Iranian Electric Industry Journal of Quality and Productivity, Vol. 8, No. 3, pp. 47-58, 2020.
[17]. R. An, Z. Liu, J. Liu, "Successive-approximation-based virtual impedance tuning method for accurate reactive power sharing in islanded microgrids", IEEE Trans. on Power Electronics, vol. 36, no. 1, pp. 87-102, Jan. 2021.
[18]. M.E. Romero, M.M. Seron, "Ultimate boundedness of voltage droop control with distributed secondary control loops", IEEE Trans. on Smart Grid, vol. 10, no. 4, pp. 4107-4115, July 2019.
[19]. S. Gandhar, J. Ohri, M. Singh, “Improvement of voltage stability of renewable energy sources-based microgrid using ANFIS-tuned UPFC”, Advances in Energy and Built Environment, vol 36, pp. 133-143, 2020.
[20]. M.S. Misaghian, M. Saffari, M.Kia, A.Heidari, M.Shafie-khah, J.P.S.Catalão, "Tri-level optimization of industrial microgrids considering renewable energy sources, combined heat and power units, thermal and electrical storage systems", Energy, vol. 161, pp. 396-411, Oct. 2018.
[21]. M. Ahmed, L. Meegahapola, A. Vahidnia, M. Datta, "Analyzing the effect of X/R ratio on dynamic performance of microgrids", Proceeding of the IEEE/ISGT-Europe, pp. 1-5, Bucharest, Romania, Sept./Oct. 2019.
[22]. A. Micallef, M. Apap, C.S. Staines, J.M.G. Zapata, "Secondary control for reactive power sharing and voltage amplitude restoration in droop-controlled islanded microgrids", Proceeding of the IEEE/PEDG, pp. 492-498, Aalborg, Denmark, June 2012.
[23]. M.A. Hassan, M.A. Abido, "Optimal design of microgrids in autonomous and grid-connected modes using particle swarm optimization", IEEE Trans. on Power Electronics, vol. 26, no. 3, pp. 755-769, March 2011.
[24]. M. Naderi, Y. Khayat, Y. Batmani, H. Bevrani, “Robust multivariable microgrid control synthesis and analysis”, Energy Procedia, vol. 100, pp. 375-387, Nov. 2016.
[25]. Y. Li, Y.W. Li, "Power management of inverter interfaced autonomous microgrid based on virtual frequency-voltage frame", IEEE Trans. on Smart Grid, vol. 2, no. 1, pp. 30-40, March 2011
[26]. J. Schiffer, T. Seel, J. Raisch, T. Sezi, "Voltage stability and reactive power sharing in inverter-based microgrids with consensus-based distributed voltage control", IEEE Trans. on Control Systems Technology, vol. 24, no. 1, pp. 96-109, Jan. 2016.
[27]. H. Karmi, B. Fani, G. Shahgholian, “Coordinated protection scheme based on virtual impedance control for loop-based microgrids”, Journal of Intelligent Procedures in Electrical Technology, vol. 12, no. 46, pp. 15-32, Dummer 2021.
[28]. J.H. Kim, Y.S. Lee, H.J. Kim, B.M. Han, “A new reactive-power sharing scheme for two inverter-based distributed generations with unequal line impedances in islanded microgrids”, Energies, vol. 10, no. 11, 1800, 2017.
[29]. F. Zandi, B. Fani, I. Sadeghkhani, A. Orakzadeh, "Adaptive complex virtual impedance control scheme for accurate reactive power sharing of inverter interfaced autonomous microgrids", IET Generation, Transmission & Distribution, vol. 12, no. 22, pp. 6021-6032, 2018.
[30]. C.N. Rowe, T.J. Summers, R.E. Betz, D.J. Cornforth, T.G. Moore, "Arctan power–frequency droop for improved microgrid stability", IEEE Trans. on Power Electronics, vol. 28, no. 8, pp. 3759-3747, 2013.
[31]. S. Eberlein, K. Rudion, “Small-signal stability modelling, sensitivity analysis and optimization of droop controlled inverters in LV microgrids”, International Journal of Electrical Power and Energy Systems, vol. 125, Article Number: 106404, Feb. 2021.
[32]. Y.C.C. Wong, C.S. Lim, M. D. Rotaru, A. Cruden, X. Kong, "Consensus virtual output impedance control based on the novel droop equivalent impedance concept for a multi-bus radial microgrid", IEEE Trans. on Energy Conversion, vol. 35, no. 2, pp. 1078-1087, June 2020.
[33]. R. Agrawal, D.D. Changan, A. Bodhe, “Small signal stability analysis of stand-alone microgrid with composite load”, Journal of Electrical Systems and Information Technology, vol. 7, Article Number: 12, 2020.
[34]. S. Tabatabaee, H. R. Karshenas, A. Bakhshai, P. Jain, "Investigation of droop characteristics and X/R ratio on small-signal stability of autonomous microgrid", Proceeding of the IEEE/PEDSTC, pp. 223-228, Tehran, Feb. 2011.
[35]. , , “An enhanced decentralized reactive power sharing strategy for inverter-based microgrid”, International Journal of Electrical Power and Energy Systems, vol. 98, pp. 531-542, June 2018.
[36]. X. Wang, Y. W. Li, F. Blaabjerg and P. C. Loh, "Virtual-impedance-based control for voltage-source and current-source converters", IEEE Trans. on Power Electronics, vol. 30, no. 12, pp. 7019-7037, Dec. 2015.
[37]. R. Moslemi, J. Mohammadpour, “Accurate reactive power control of autonomous microgrids using an adaptive virtual inductance loop”, Electric Power Systems Research, vol. 129, pp. 142-149, Dec. 2015.
[38].
[39]. E. Molina, J. E. Candelo-Becerra, F. E. Hoyos, “Control strategy to regulate voltage and share reactive power using variable virtual impedance for a microgrid”, Applied Sciences, vol. 9, no. 22, Nov. 2019.
[40]. Z. Li, C. Zang, P. Zeng, H. Yu, S. Li, "Fully distributed hierarchical control of parallel grid-supporting inverters in islanded ac microgrids", IEEE Trans. on Industrial Informatics, vol. 14, no. 2, pp. 679-690, Feb. 2018
[41]. C. Dou, Z. Zhang, D. Yue, M. Song, "Improved droop control based on virtual impedance and virtual power source in low-voltage microgrid", IET Generation, Transmission and Distribution, vol. 11, no. 4, pp. 1046-1054, March 2017.
[42]. X. Zhao, Y.W. Li, H. Tian, X. Wu, "Energy management strategy of multiple supercapacitors in a DC microgrid using adaptive virtual impedance", IEEE Journal of Emerging and Selected Topics in Power Electronics, Vol. 4, No. 4, pp. 1174-1185, Dec. 2016.