Stability and Robustness of a Coupled Microgrid Cluster Formed by Various Coupling Structures

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(Discipline of Engineering and Energy,Murdoch University,Perth 6150,Australia)

Abstract:A standalone microgrid (MG) may frequently experience overloading owing to insufficient power generation or excessive renewable-based generation,which can cause unacceptable voltage and frequency deviations.Such problems are conventionally alleviated by load-shedding or renewable curtailment.Alternatively,autonomously operating MGs can be provisionally connected to facilitate temporary power exchange.The power-exchange link among the MGs can be of different types,e.g.,three-phase ac,single-phase ac,or dc-link and power electronic converter-interfaced.All these topologies can facilitate power exchange,but they differ with regard to stability and robustness.In the present study,the stability and robustness of such structures are investigated,and the effects of factors such as the length of the interconnecting line among the MGs,the amount of power supplied to the troubled MGs,and the number of coupled MGs are compared.The stability and robustness of the structures are evaluated in Matlab.

Keywords:Microgrid,coupled microgrids,stability,robustness

1 Introduction

A microgrid (MG) is usually referred to as a small-scale interconnected network of predominantly renewable energy-based and power electronic converter-interfaced distributed energy resources(DERs),distributed loads,and energy storage[1].Such a system can operate under a centralized,semi-centralized,or decentralized control scheme[2],which may have two modes of operation:grid-connected and autonomous.When a utility grid is unavailable because of economic constraints,the MGs can operate under a decentralized control scheme and remain permanently autonomous[3].For electrifying remote and regional areas,autonomous MGs are cost-effective solutions and are preferred over long transmission and distribution lines.DERs in such a scheme typically use droop control to realize the desired power sharing while also regulating the desired frequency and voltage[4].Thus,the active power and reactive power are controlled at the output of each DER,through droop control,to regulate the voltage and frequency within acceptable limits[5].

Alternatively,a suitable distributed control method,such as that proposed in Ref]6],can be employed.

MGs can experience over-generation and overloading despite the design considerations owing to the high intermittency and uncertainty of renewable energy sources[5,7].Load-shedding is the simplest method to alleviate this problem by returning the frequency and voltage to acceptable limits.Conversely,over-generation can be addressed by curtailing renewable energy sources.However,both solutions are uneconomic and undesirable.Careful planning considerations are required for autonomous MGs to account for the seasonal dependency,unpredictability,and intermittency associated with renewable energy sources[3],which can reduce the probability of facing such problems.The problems of over-generation and overloading can be alleviated using oversized energy-storage systems,but this increases the installation and operational costs of the MGs[7-8].

A third operation mode can be implemented to address these problems,in which the neighboring autonomous MGs interconnect temporarily to exchange power in such situations[9].This is highly preferred—particularly when a locality consists of several neighboring MGs.In this case,the overloading or over-generation management strategy can be realized by properly interconnecting two or more neighboring MGs.The model of a system of coupled MGs (CMGs) and its operation based on the availability of communication infrastructure and the concept of the power market were discussed in Refs]9-10].According to this concept,a common power-exchange link (i.e.,distribution lines) that enables a physical connection among the MGs is introduced to connect the neighboring MGs.Such an interconnecting link can be in either AC or DC[10-12].A direct AC-AC connection between an MG and the power-exchange link can be easily realized through a conventional circuit breaker or an interconnecting static switch,which is highly economical[12];however,a direct AC-AC connection reduces the operational autonomy of the MGs.To add proper isolation among the MGs for retaining operational autonomy,a power electronic converter-based interlinking structure was proposed[8,13]for the MGs to function independently,as well as to facilitate power exchange with neighboring MGs.Power sharing between two islanded MGs during mutual contingency was studied in Ref]8],and an autonomous control approach for sharing power with neighboring MGs through a back-to-back converter was presented.Additionally,neighboring MGs can interconnect to support each other during the occurrence of a fault[14]or during normal conditions to minimize the levelized cost of electricity.If one of the MGs temporarily observes a high-power generation by its DERs,the neighboring MG(s) can be connected to one another to import power and thus offer electricity at a lower price.A transformative architecture was proposed for coupling the nearby MGs to improve the system resiliency during faults[15].In Ref]16],the reliability aspects of a CMG were analyzed,and in Ref]17],the voltage and current controllability in CMGs was analyzed.In Ref]18],the dynamic security of CMGs was examined.Several structures to form such CMGs,as well as their control mechanisms,have been proposed.Fig.1a schematically shows this concept.Such a provisional interconnection of neighboring MGs to form a cluster that facilitates power exchange enhances the system reliability and resiliency[16-19].

The CMG can be realized in three forms,which were described in Refs]20-22]:a three-phase line connected to each MG via a three-phase back-to-back power electronic converter,a single-phase line connected to each MG via a three-phase/single-phase back-to-back power electronic converter,and a dc line connected to each MG via a three-phase converter.The corresponding power electronic-based interfaces are shown in Fig.1b.

Fig.1 MG interconnection topologies

Properly coordinating and efficiently managing the power transaction from one MG to another through voltage source converters (VSCs) and lines is the key operational challenge that must be addressed.Although appropriate strategies for power transaction management among MGs have been proposed[20-22],for these strategies,the stability and robustness of the CMG should be investigated.The main objective of the present study was to investigate and demonstrate the stability and robustness of these coupling strategies.Factors such as the distances between the MGs and the amount of power to be exchanged were considered,and the CMG stability and robustness were evaluated against them.The findings of this study can help system designers to select a suitable coupling structure depending on the design constraints,such as the expected amount of power support at each interconnection and the distances between the MGs.

2 Operation of autonomous MG

In the autonomous mode,MGs typically operate under droop control to realize the desired power sharing among the DERs while regulating the voltage and frequency at their output[2-3].Consider the network shown in Fig.1a,where each MG consists of several DERs,loads,and an interlinking converter coupled to a power-exchange link.The interlinking converter can have any of the three topologies shown in Fig.1b.The DERs are assumed to operate in theP-fQ-Vdroop control mode,where the voltage and frequency at the output of each DER are determined in accordance with Ref]5].

Here,imandniare the droop coefficients of theDER of the MG and can be obtained as follows

where the subscripts max,min,and rated represent the maximum,minimum,and rated limits for the frequencies and voltages of the MGs,respectively,andiPandiQrepresent the active power and reactive power,respectively,injected by each DER to the MG.All the DERs within an MG are assumed to have the same;Vratedrepresents the rated voltage of the DER’s point of common coupling,that is,one per unit.However,these parameters can differ among neighboring MGs.Each DER is provided with local energy storage,thus,it can be considered as a dispatchable DER.The structure and operation of the DERs were discussed in Ref]7].Finally,each MG is connected to a common power-exchange link via its interlinking converter to exchange power with the other MGs in the event of power deficiency or over-generation.Each MG can operate independently at any predefined frequency and voltage level without affecting the operation of neighboring MGs.

3 CMGs

Detailed analyses of different topologies and architectures of CMGs were presented in Refs]10-11].The optimal control of a utility grid-connected CMG was investigated in Ref]23],and an interactive control method for sharing the load in a CMG ensuring a wide range of system stability was presented in Ref]15].Optimization-based techniques were developed in Refs]23-24] for ensuring optimal power exchange between MGs.A decision-making-based approach was proposed for determining the most suitable MG(s) to connect to an overloaded MG[25].The ultimate objective is that an MG can be coupled to any other MG (and not necessarily an adjacent MG) if a general link is available to function as a power-exchange highway[21-22],which can be either DC or AC.Back-to-back power electronic converter-based interfaces were proposed in Refs]8,26] that can be used between each MG and the interconnecting three-phase AC line to ensure operational autonomy.This scheme is relatively expensive,as two VSCs are employed for each MG.However,the main objective of such a connection is to provide isolation between the MGs;thus,MGs with different operational regulations can exchange power[17,26].Furthermore,such an interconnecting link can be realized through a single-phase AC power line instead of a three-phase AC power line[12]to reduce the cost while fulfilling the requirements of the power transfer during overloading and/or over-generation,as the numbers of lines and power electronic filter components will be reduced.Simultaneously,the stability of the CMG network must be ensured.

3.1 Topology comparison

Realizing power exchange among multiple MGs requires a predefined coordinated control mechanism.Multiple MGs can be coupled to form a CMG by using single-phase ac lines to reduce the numbers of converter components and transmission lines and make the system more economical.

A CMG formed using dc interconnecting lines outperforms any type of ac link with regard to technical and economic aspects.However,there are several critical issues with dc power-exchange links,such as a lack of common protection schemes,expensive protection equipment,a high fault-current level,arching during fault-current interruption (as there are no natural zero current crossings),poor fault ride-through capability,circulating current in the network due to unequal line resistances,and inaccurate and inefficient power sharing based on the droop control technique[27-28].Meanwhile,the standard and infrastructure for ac power transmission and distribution systems are well established,and most of the commercially available equipment is ac network-based;thus,most network operators prefer ac systems.A comparative review of ac and dc technologies used in MG applications was presented in Ref]28].According to Ref]27],considering the present power transmission infrastructure,commercially available protection equipment,technical constraints,network standards,and guidelines,ac systems are more economical than dc systems despite being advantageous in several features.

An aspect that was not properly considered in the aforementioned studies is the comparative stability and robustness of these topologies.This was the focus of the present study.

3.2 Stability

The stability of a hybrid MG consisting of ac and dc buses was investigated in Ref]29].The concept can be extended to a multi-MG system,and prior to forming a CMG,the stability of the newly formed network must be examined.The numbers of inertial and non-inertial DGs,their ratings,and their loads must be evaluated to ensure the stability of the CMG[30].Ref]24] suggested a decision-making function along with small-signal stability evaluation prior to any transformation.

If properly designed,the loci of the eigenvalue of the new CMG network are approximately within the same operating point eigenvalues of each MG when operating independently[31].Additionally,a sensitivity analysis was performed along with the stability analysis in Ref]24],which revealed that an MG with smaller stability margins can cause a drop in the overall system stability margins of the other MGs after they are coupled.

Ref]30] reported that the stability margins of the CMG can be significantly affected by the nominal power of the DER,whereas no significant effect was observed for the load demand or power factor.Additionally,it was observed that the length and X/R ratio of the lines of the MGs can affect the stability.However,no effect on whether the MGs had a loop or radial configuration was observed.Furthermore,developing a general stability guideline is not straightforward,as the operation of the CMG depends on the conditions of the newly created system.In Ref]31],the network characteristics and topology that can affect the small-signal stability of a converterdominated MG were examined.Simple radial topologies were found to be the most stable versus loop configurations,although they were less resilient to faults and reduced network reliability.As the level of meshing increases,the network becomes less stable.Hence,it is recommended to keep the network topology as simple as possible and operate the CMG in a ring configuration with a tie-open point.The study also indicated that two adjacent converters connected through a lower-impedance line can adversely affect the stability margin.Thus,it is recommended to connect adjacent converters with sufficient electrical decoupling or isolation from each other to avoid unstable interactions.Moreover,it was found that the lengths of the lines of the adjacent MG can affect the stability margins of the CMG[24,31].By performing stability studies,the optimal and most robust topology for forming the CMG can be selected.

3.3 MG classification

CMG formation can be allowed or denied depending on the existing loading level of MGs,which is identified by measuring their frequencies.According to its loading level,each MG is classified as a healthy,problem,or floating MG (denoted as HMG,PMG,and FMG,respectively),as described below.

(1)“HMG”refers to the status of an MG when it is operating within its nominal frequency and voltage ranges and can support the neighboring MGs by exchanging power.

(2)“PMG”refers to an operating condition in which the MG experiences over-generation or overloading.Any PMG within the CMG network is expected to connect to one or more neighboring HMGs and exchange power to keep its frequency and voltage within acceptable limits.

(3)“FMG”refers to the loading condition of an MG that is close to becoming a PMG.The MG’s frequency or voltage is near the maximum or minimum acceptable limit;hence,the MG should not be allowed to participate in power exchange within the CMG.

This classification of MGs is shown in Fig.2.

Fig.2 MG classification based on the frequency level

4 CMG operational and control technique

CMG formation among neighboring MGs is achieved through power electronic converters and interconnecting power lines using three different topologies.For Topologies-1 and 2,a back-to-back bidirectional converter structure is employed,whereas a single-stage converter is sufficient for Topology-3.The converter connected to the MG is labeled as the MG-side converter (MSC),and the other converter is labeled as the line-side converter (LSC).The isolation between the MSC and LSC in Topology-1 and 2 allows the MGs to operate with full autonomy,and no synchronization between them is required (see Fig.1b).On the other hand,in Topology-3,a VSC connects the dc power line to the MG (see Fig.1b).The operation of the VSCs is introduced below,and their detailed controller designs and sample time-domain studies were presented in Ref]32].

4.1 Three-phase AC interconnection (Topology-1)

Topology-1 is a three-phase ac interconnecting line with two back-to-back connected power electronic converters.In this topology,both the MSC and LSC are assumed to be three-phase,three-leg VSCs using IGBTs or MOSFETs.Each VSC is connected to its point of common coupling through a three-phase LCL filter.However,any other three-phase topology of VSCs can be used instead of the topologies considered here.The structures of the MSCs and LSCs are shown schematically in Fig.3a.

Fig.3 Control of MSC

4.1.1 Control of MSC

The main function of the MSC is to keep the dc-link voltageVdcat the desired reference level of.This is valid for any MG status (i.e.,HMG,PMG,and FMG).The dc-link exchanges the appropriate amount of power with the MG such that its voltage remains constant.This can be realized through the outermost loop in the MSC part of Fig.3a to determine the reference active power (Pref) that should be drawn from or injected into the MG to avoid a deviation inVdc.The MSC is unlikely to exchange any reactive power with the MG;hence,the reference for reactive power (Qref) is assumed to be zero.According to these two reference parameters,a dq-axis-based PQ controller determines the three-phase reference voltage() across the capacitorCfof the LCL filter of the MSC.The inner control loop of the MSC in Fig.3a is the voltage-tracking and switching control block.To achieve robust and optimal performance,a linear quadratic regulator (LQR) is used to track the sinusoidal reference voltage.The principle of this control mechanism was introduced by the authors in Ref]21] and is not repeated here.4.1.2 Control of LSC

Two modes of operation are considered for the LSC,depending on the MG status.To form a CMG,the LSCs of all the HMGs should operate in the droop control mode to enable power sharing.The LSCs connected to the HMGs form the CMG and provide the required droop voltage and angle references,and they absorb or inject the desired amount of power from the interconnecting power lines to alleviate PMG overloading or over-generation.The LSC of an HMG operates in the angle-voltage droop control mode,in contrast to the conventional frequency-voltage droop control mode that has been employed for the DERs within the MGs.This is the outermost control loop for the LSC of an HMG.Employing the angle-voltage droop instead of the frequencyvoltage droop results in a fixed value for the frequency of the interconnecting lines (e.g.,50 Hz).The angle and voltage droop equations for the LSC of thekthMG are as Ref]22]

wheremMG-kandnMG-kare the droop coefficients of thekthMG’s LSC and are derived from

Here,the subscripts min and max represent the minimum and maximum allowable limits for the angle and voltage,respectively,andrepresent the maximum active power and reactive power,respectively,injected/absorbed by each MG to the interconnecting lines.All LSCs have similar values of.The conventional angle-voltage droop control technique cannot ensure the desired power sharing among the HMGs unless a modified angle droop mechanism is implemented using the virtual impedance method[22].

The LSCs of the FMGs and PMGs should operate in the constant-power (PQ) control mode.Thus,the frequencies of the MGs must be continuously monitored.If an MG is overloaded,a control mechanism determines the amount of power that must be absorbed from the interconnecting link to return the frequency of the MG to the acceptable limit.This concept is equally applicable when the MG has surplus power generation and its frequency has increased beyond the acceptable limit.This mode of operation of the PMG through its LSC is denoted as the frequency-control mode and corresponds to the outermost control loop for the LSC of a PMG.If an MG becomes a PMG by overloading,a frequency deviation-based control mechanism denoted as the overload frequency controller (OLFC) is activated and determines the amount of power that should be absorbed from the neighboring MGs to keep the MG frequency within the acceptable limit.Likewise,during over-generation,an over-generation frequency controller (OGFC) is activated to determine the required amount of power to be injected into the neighboring MGs.Fig.4 shows the arrangement of the control mechanism.The reference of the active power(Pref) is determined by either the OLFC or OGFC block,and the reference of the reactive power (Qref) is assumed to be zero.Once the overloading or over-generation is alleviated,another frequency deviation-based control system determines the new status of the MG as an HMG that can resume its normal operation;hence,the LSC of the MG should change its operation from the PQ mode to the droop control mode.

Fig.4 Frequency controller of the LSC for PMG operation

The desired reference active power is determined by a frequency controller that employs two separate proportional-integral (PI) controllers and the necessary control logic.Then,the desired reference power during an overload or over-generation event is obtained by activating the OLFC or OGFC,respectively,and is passed through a selector (MUX).

Proper reference-selection logic is needed to facilitate a smooth transition between the constant PQ control mode (for FMG and PMG) and the droop control mode (for HMG) without causing unwanted transients,oscillations,and instability.Such an arrangement to change the operating modes of the LSC is shown in Fig.4.The three-phase reference voltages are either selected from the droop control block or the PQ controller block through a selector.After the overloading or over-generation is alleviated,a PMG or FMG should not be allowed to switch back to the droop mode unless all the remaining MGs within the CMG network are HMGs and operate at the same phase angle.

The frequency response of the MSC is shown in Fig.5a.The frequency responses of the LSCs of two HMGs operating under modified angle droop control,while supporting a PMG,are shown in Fig.5b.The plot shows both the angle and voltage droop responses of the LSCs.Fig.5c presents the frequency response of the LSC of the PMG.The gain and phase margins are denoted as GM and PM,respectively.As shown,for all the VSCs,the outermost control loop has a sufficiently large gain and phase margin to ensure stability.

wheremMGandPMGrepresent the dc droop coefficient and the active power absorbed/injected by the VSC,respectively,andVnomrepresents the nominal voltage of the dc-link (e.g.,1 p.u.).Active power absorption by the VSC is considered negative,whereas active power injection by the VSC is considered positive.As indicated by Eq.(9),the ratio of power absorbed (or injected) by HMGsiandjis the reciprocal of the ratio of their droop coefficients;i.e.

As the key objective of the power-exchange mechanism in the CMG network is to control the active power flow,the reference of the reactive power (Qref)is set to zero.The reference for the active power (Pref)is determined by the deviation of the PMG frequency from the acceptable limit,as shown in Fig.4.

The frequency controller determines the desired power reference through two different PI-based closed-loop control systems and the necessary control logic.Then,the desired power reference is chosen by a selector that selects the proper power reference during the overload or over-generation situation by activating the OLFC or OGFC,respectively.To switch between the droop control mode (for HMG) and the constant PQ control mode (for the FMG and PMG),proper reference-selection logic is needed

The reference selector selects the proper active power reference from either the droop control-based dc-link controller block or the frequency controller block through a selector and passes it to the PQ control block.Once a PMG becomes an HMG,the reference selector enables the VSC to switch back to the droop-based dc-link voltage control mode.

5 Stability analysis

The effects of several key design and operational factors were investigated from the perspective of stability.The parameters considered were the activeand reactive-power droop coefficients,the length (or impedance) of the interconnecting lines among the MGs,and the power support required by the PMG.The technical parameters of the study are presented in the Appendix.The results presented studies discussed below indicate that the active-power droop coefficient significantly affects the CMG stability.

5.1 Droop coefficients

The active-power droop coefficient ofmMGchanged around its nominal value of 0.002 rad/kW for both Topology-1 and 2,while the reactive-power droop coefficient is kept constant.The eigenvalue trajectories are presented in Figs.8a and 8b.As shown,the critical active-power droop-coefficient value for Topology-1 is 0.08 rad/kW,whereas this value for Topology-2 is 0.125 rad/kW.Any further increase in the active-power droop coefficient will make both topologies unstable.This was verified by a time-domain simulation in PSIM software (not presented here).At the nominal values of the droop coefficients,the response of Topology-2 was found to be more oscillatory (less damping) than that of Topology-1,which can be explained by the eigenvalue location for the nominal values of the droop coefficients.However,Topology-1 is more sensitive to variations in the droop coefficients and becomes unstable at smaller values.If the CMG network must operate with a large droop coefficient,Topology-2 is more suitable than Topology-1.The eigenvalue trajectory for Topology-3 is presented in Fig.8c.As shown,increasing the active-power droop coefficient does not make the CMG unstable,in contrast to the previous topologies.However,the response of the DC power-exchange link is affected by this coefficient (i.e.,it becomes overdamped and sluggish as the coefficient increases).Additionally,Fig.8c shows that asmMGincreases,the dominant pole moves toward the origin;hence,the system response becomes slower (longer settling time).Although a smaller value of the active-power droop coefficient makes the system faster,it should not be smaller than a certain level to realize proper power sharing among the MGs (according to their droop coefficients).

Fig.8 Eigenvalue trajectory for variation of the active power droop coefficients

For both Topology-1 and 2,the variation ofnMGdoes not significantly affect the CMG stability,because the amount of reactive power flowing through the CMG link is small,as shown in Fig.9.

Fig.9 Eigenvalue trajectory for variation of the reactive power droop coefficients

5.2 PMG power

The dynamic response of the CMG is also affected by the amount of power absorbed from(overload) or injected to (over-generation) the CMG network through the PMG.To examine this effect,the PMG power was varied from 0.02 p.u.to 0.15 p.u.for all the topologies.The eigenvalue trajectories are presented in Fig.10.As the PMG power increased,the system damping increases,and the system response becomes slightly sluggish.Thus,the CMG network performs better when a larger amount of power is exchanged among the MGs.For Topology-1 and 3,with the increase in the power of the PMG,even though the dominant eigenvalue moves toward the origin,the system remains stable.In contrast,for Topology-2,the CMG stability improves even when the PMG power increases,as shown in Fig.10b.

Fig.10 Eigenvalue trajectory for variation of PMG power

5.3 Interconnecting-line impedance

Fig.11 shows the system eigenvalue trajectories when the impedance of the interconnecting line increases for Topology-1 and 2 and when the line resistance increases for Topology-3.For Topology-1 and 2,the X/R ratio is varied from 0.5 to 10 (i.e.,the line inductance is increased from 0.16 mH to 3.2 mH while the resistance of the line is kept constant at 0.1 Ω).It can be observed that an increase in the X/R ratio reduces the system damping for Topology-1 and 2.In contrast,for Topology-3,with an increase in the line resistance,the system damping increases,and the system response becomes slower.Even though an increase in the line resistance deteriorates the property of power sharing based on the MGs’ droop coefficients,it does not lead to instability.

Fig.11 Eigenvalue trajectory for impedance variation of the interconnecting line

6 Robustness

The sensitivities of CMGs with the three aforementioned topologies to the interconnecting line’s impedance (i.e.,the distances of the MGs from each other),the power exchanged by the PMG,and the number of MGs within the CMG are investigated.The focus is on the power loss in the interconnecting lines and the error in power sharing among the MGs.The study is conducted for all three topologies for a scenario in which two HMGs support a PMG while sharing power with a droop ratio of 2:1.

6.1 Interconnecting-line length

Fig.12 presents the line loss and power-sharing error with the increasing length (impedance) of the interconnecting line.As shown,the line loss and power-sharing error are significantly (almost linearly) affected by the increase in the line impedance for all the topologies.Topology-2 is the most robust with regard to the power-sharing error,and Topology-1 is the most robust with regard to the power loss.Topology-3 is the least robust(largest variation) for both the power-sharing error and the power loss.Fig.13 shows the line loss and the power-sharing error percentage as the power delivered by the two HMGs increased for all the topologies.The results of the robustness comparison of the three topologies agree with the results of previous studies.

Fig.12 Effect of line impedance

Fig.13 Effect of increase in the power provided by the HMGs

6.2 PMG power

Another study was conducted to evaluate the robustness of the CMG with respect to the power demand of the PMG.Figs.14a and 14b present the droop voltage magnitude and angle for the LSC of the HMGs supporting the PMG in Topology-1 and 2,and Fig.14c shows the droop voltage magnitude for the VSC of the HMGs in Topology-3.The results indicate an almost linear drop in these quantities when the power demand of the PMG increases.Topology-1 is more robust than Topology-2 because all the PMG power flows through a single-phase line in Topology-2,whereas almost one-third of the power flows in each phase of the three-phase line of Topology-1.Topology-2 is less robust than Topology-3.

Fig.14 CMG robustness against PMG power variation

6.3 Number of MGs in CMG

Another study was conducted to evaluate the robustness of the three topologies against the variation in the number of MGs within the CMG.In this study,the CMG is assumed to be composed of two,three,or four MGs.For all three topologies,one MG is a PMG,and the remaining MGs are HMGs.

Fig.15 illustrates the outermost control loop for the LSC of the PMG.The gain and phase margins (denoted as GM and PM,respectively) are shown for each system (2,3,and 4 MGs).The results indicate that the frequency control loop remains robust with an increase in the number of HMGs.All the topologies exhibit similar gain and phase margin robustness against an increase in the number of HMGs.Additionally,for all the topologies,the CMG becomes more robust with an increase in the number of HMGs.

Fig.15 Stability-margin comparison of the frequency control loop with varying number of MGs to form CMG

7 Conclusions

This study focused on various structures for coupling neighboring MGs.The effects of the coupling structure on the stability and robustness of the MGs were examined.The results indicated that the active-power droop coefficient significantly affects the CMG stability in Topology-1 and 2 in comparison with other factors,such as the length (impedance) of the interconnecting line and the power demand of the PMG.Among all the topologies considered,Topology-1 is the most sensitive to changes in the droop coefficients and can become unstable at small droop-coefficient values.If the CMG network must operate with a large droop coefficient,Topology-2 is more suitable.The results also indicated that for Topology-1 and 3,the system remains stable with an increase in the power exchange required by the PMG.In contrast,the stability of Topology-2 is enhanced when the power exchanged by the PMG increases.Although an increase in the line resistance deteriorates the property of power sharing based on the MGs’ droop coefficients,it does not lead to instability.With increases in the length of the interconnecting line and the power exchanged by the PMG,Topology-2 is the most robust with regard to the power-sharing error,whereas Topology-1 is the most robust with regard to the power loss in the interconnecting lines.Topology-3 is the least robust(largest variation) among the three topologies for both factors.Topology-1 is more robust than Topology-2 against the PMG power variation,as all the power flows through a single-phase line in Topology-2,whereas almost one-third of the power flows in each line phase in Topology-1.Additionally,the results indicated that the CMG is more robust when the number of HMGs within the CMG is larger,for all the topologies.

Appendix

The network and controller parameters used in the simulation studies are presented in Tab.1.

Tab.1 Parameters of the system under consideration