,

(College of Engineering,Temple University,Philadelphia 19122,USA)

Abstract: As a growing number of microgrids (MGs)has been integrated into the modern power grids,the interconnection and applicable cooperation among multiple MGs motivate the development of networked MGs.Dynamic MGs,as an advanced networked MGs structure,can not only integrate multiple MGs into the distribution system but also fulfill the requested system network reconfiguration with improved flexibility.A general distributed control approach for networked MGs is reviewed.A distributed control framework for dynamic MGs operation is developed,along with an extensible architecture with considerations of large-scale distributed energy resources (DERs)integration.A scalable small-signal stability analysis is conducted per the proposed distributed control strategies and the conditions under which the system is exponentially stable are derived.At last,the effectiveness of the proposed control framework and stability analysis are verified using a 6-bus test feeder.

Keywords:Dynamic microgrids,distributed control,networked microgrids,small-signal stability analysis

1 Introduction1

The increasing integration of inverter-interfaced distributed generators (DGs)has motivated the emerging technology of microgrids (MGs).A MG is comprised of a cluster of DGs and loads and has explicitly identified electrical boundaries.It is capable of operating in both islanded and grid-connected modes.In other words,each MG along the distribution feeder can not only operate as a dispatchable source at the point of common coupling (PCC)but also operate autonomously without the energization from the main power grid[1-2].

Additional system operating challenges are presented when multiple adjacent MGs are interconnected,which motivates the development of networked MGs.The operating characteristics of each individual MG within the network could be various in load profile,power generation,and dynamic response,while the system operational resiliency could be enhanced through proper coordination among the MGs[3-4].For instance,in networked MGs,a MG can support its neighboring MG to restore loads if there is generation deficiency caused by extreme events (e.g.,power outages induced by natural disasters).However,there exist many technical challenges in networked MGs management.For example,the system’s operating characteristics would become increasingly complex due to the heterogeneous dynamics of each MG and the interactions among the MGs through both physic and cyber networks[5].As the framework of networked MGs becomes complicated,additional control efforts are required to ensure proper system operation[6-7].

An advanced structure of networked MGs,known as dynamic MGs,has recently been favored[8].Compared with the conventional networked MGs that operate under static topologies and interact through static PCCs,dynamic MGs have flexible electric boundaries and interact with their adjacent MGs at multiple points of interconnection (POIs).Furthermore,dynamic MGs could enable optimal network reconfiguration plans for the system operators with increased flexibility.Specifically,if there is a contingency in the power system and a network reconfiguration is requested for system restoration,the topology of the distribution feeder could be seamlessly varied by changing the on-off status of target switches in the context of dynamic MGs[9].A schematic structure of conventional MGs,networked MGs,and dynamic MGs is shown in Fig.1,and the comparison among them is summarized in Tab.1.

Fig.1 Comparison among the structure

Tab.1 Comparison among the conventional,networked,and dynamic MGs

The three-level hierarchical control framework including primary control,secondary control,and tertiary control has been adopted in networked and dynamic MGs[10-11].Droop control has been widely implemented at the primary level due to its simplicity and effectiveness in automatic system stabilization[12-13].However,since there are no intercommunications between local primary controllers and other DG units,the secondary control is used to realize global regulations in MGs.Recent research interests have mainly been focused on decentralizing the control strategies in the secondary control level when coordinating multiple DGs.Compared to the conventional centralized secondary control strategy where a central controller is used to regulate all the DGs,the distributed secondary control strategy implements a local secondary controller to each DG.Each DG controller exchanges its local measurements (e.g.,frequency and voltage amplitude)with its neighboring DGs using peer-to-peer communications links[14].In Ref]15],a distributed secondary control scheme is presented for the system network reconfiguration in an unbalanced system.In Ref]16],a set of distributed averaging proportional integral (DPAI)controllers are proposed for islanding operation,which achieve frequency regulation and proportional active power sharing and allow for a tunable compromise between the voltage regulation and the reactive power sharing.In Ref]17],a wireless-based robust communication algorithm is developed to improve the reliability of distributed secondary control.The proposed algorithm combines communications and control functionalities to absorb the transmission errors by performing an averaging operation in every controller.As discussed,the distributed secondary control strategy could deliver the same control efforts with improved system operation reliability and resilience[18].

Due to the increasing number of DGs,it is impractical to implement a fully connected communication network among all DGs to make the system states globally accessible for regulations.However,consensus-based algorithms can calculate the average of locally measured parameters in a distributed way using a sparse communication network[19].The conventional consensus-based algorithm,called average consensus,is only applicable to a group of static inputs.However,considering the requirements of appropriate coordination among DGs in distributed control,it make senses to develop the tracking of dynamical linear consensus on time- varying inputs[20].

The small-signal stability analysis is critical to the operation of the dynamic MGs due to its complicated structure and operational dynamics[21].Dynamic MGs have complicated control structures and operating dynamics.In the context of dynamic MGs,it is challenging in deriving and expanding small-signal stability analysis accordingly to the general dynamic MGs system.In Ref]9],the sufficient conditions for dynamic MGs exponential stability are derived; In Ref]16],A small-signal stability analysis has been presented for the proposed voltage controller along with a performance study in networked MGs.

In this work,a distributed control framework for dynamic MGs operation is presented,along with a scalable architecture with considerations of large-scale distributed energy resources (DERs)integration.Furthermore,a scalable small-signal stability analysis is derived based on the proposed distributed control strategies and the conditions under which the system is exponentially stable are derived.The remainder of this paper is structured as follows.Section 2 presents a review of control strategies for networked MGs.Section 3 proposes a distributed secondary control strategy for dynamic MGs operation,and introduce a scalable small- signal stability analysis.The proposed work is validated using the case studies in Section 4.The conclusion is drawn in Section 5.

2 Review of hierarchical control framework and consensus-based distributed control strategies

2.1 Primary control and secondary control

The conventional primary and secondary control framework is depicted in Fig.2.It is worth mentioning that since the tertiary control relies on the present electricity market and price,it is not the key point of this study[10].

Fig.2 Conventional primary and secondary control framework

2.1.1 Primary control

The primary control coordinates multiple inverters to automatically stabilize the system’s operating frequency and voltage.When the system operates under the islanded mode,each dispatchable inverter acts as a grid-forming voltage source inverter (VSI)[10].Hence,to guarantee an ideal output impedance at the fundamental frequency,the voltage,active power,and reactive power are regulated using the primary control framework consisting of inner voltage and current control loops,droop control,voltage reference generators,and optional virtual impedance loops.Among them,droop control,as an adaptive control,is used to adjust frequency and voltage magnitude based on the changing loading conditions.When integrating multiple VSIs in parallel,the widely adoptedP-fandQ-Vdroop characteristics for the inductive system are represented as follows

whereωiandEiare the operating frequency and output voltage amplitude;ω*andE*are the nominal frequency and voltage amplitude;miandniare droop coefficients;PiandQiare the active and reactive power flows,respectively.

2.1.2 Secondary control

However,DGs that are only implemented with droop control will be regulated with steady-state deviations.Referring to the droop characteristics in Eqs.(1a)and (1b),PiandQimay not be guaranteed to remain at zero,thus the operating frequencyωiand output voltage amplitudeEiwould deviate from their nominal values (ω*andE*),respectively[16].

The secondary level control is implemented to eliminate the steady-state error generated by the droop control.For the centralized secondary control,the system operating frequency (ωi) and voltage magnitude (Ei)at PCC are measured and compared to their reference values (ω*andE*),respectively.And then the errors processed by the compensators (ωseci and Eseci)are sent to the primary control level of the inverter to restore the frequency and output voltage magnitude[10].In other words,Eqs.(1a)and (1b)becomeωi=ω*-miPi+ωseciwhileEi=E-niQi+Eseci.A set of PI controllers could be used to realize the frequency and voltage amplitude restorations[14]

wherekω_P,kω_I,kE_P,andkE_Iare the control gains of the secondary control.

2.2 Consensus-based distributed secondary control

In Eqs.(2a) and (2b),the conventional centralized secondary controller is used to regulate the frequency and voltage amplitude of all the DGs.However,a significant drawback of the centralized secondary controller is that the system could cease the services if any failures occur in the controller.In contrast,the distributed secondary control strategy uses a local secondary controller in each DG to communicate with its neighboring DGs,which enhances the reliability of the system.

Furthermore,as the system configuration becomes complex and the flexible network reconfiguration is requested in the system (e.g.,dynamic MGs),it is not applicable to build a fully connected communication network among all DGs for control implementation.However,the limitation can be released using consensus algorithms over a sparse communication network.Two types of consensus-based algorithms that can be applied in the distributed secondary control of dynamic MGs are also introduced in Section 2.2.

2.2.1 Average consensus algorithm

The communication network of the multi-agent system can be illustrated as a graphG=(V,ε,A),where V={V1,V2,...,Vn} represents the mark of each agent,ε⊆V×V refers to effective communications among agents,and A is the adjacency matrix with entriesaij=aji,whereaij=1 if there is communication between nodesiandj(i.e.,(i,j)∈ε),or elseaij=0.Fig.3 presents a sample topology of the communication network among the DGs in the networked MGs and its corresponding adjacency matrix[7].Based on the communication network topology,the commonly used continuous-time average consensus algorithm can be written as

Fig.3 Communication network and adjacency matrix

wherexi(t)refers to the static state of agentiat the timet(i=1,…,n);xi(0)is the initial value att=0;k′is a measurement of the interaction strength that modifies the system states among agents[22].Eq.(3b)indicates all the states in Eq.(3a)asymptotically achieve the average consensus[22].

2.2.2 Dynamic consensus algorithm

Compared to the average consensus that tracks the average of static states,the average of dynamic input signals could be tracked using the dynamic consensus.The continuous-time dynamic consensus algorithm is expressed as

wherezi(t)refers to the dynamic state of agentiat the timet(i=1,…,n);zi(0)represents the initial value att=0 andxi(0)=zi(0); Eq.(4b)indicates all the system states in Eq.(4a)reach the dynamic consensus without steady-state errors[20].

3 Distributed control strategies for dynamic MGs and scalable small-signal stability analysis

3.1 Dynamic MGs distributed control strategies

Compared to the networked MGs,the dynamic MGs with flexible electric boundaries demonstrate higher scalability and resilience for stabilizing the system operation,but it requires a more complex system architecture and control framework.The requested network reconfiguration is realized in the context of dynamic MGs by changing the status of smart switches (SSWs),which are used to achieve the boundary variation of dynamic MGs[23].Moreover,a concept of minimum MG (min-MG)that is the smallest set of DGs and SSWs to support local loads is presented as the basic building block to construct dynamic MGs.As shown in Fig.1c,the dynamic MGs are assumed as the min-MGs,and the SSW located at the position of dynamic POI can identify the electric boundary of min-MGs[9].

The regulations developed for the system seamless network reconfiguration are fulfilled at the secondary control level.Based on the different requirements of the system reconfiguration,the various operation modes (i.e.,Type I,II,and III modes)included in the distributed secondary control level are developed below.Note that each DG can only activate one operation mode at a time to avoid introducing contradictory regulation efforts.For theithDG,the distributed secondary frequency and voltage controllers are written as

whereηiandλiare a pair of binary variables to indicate the operation mode of theithDG;μsswrepresents the status of SSW (μssw=1 as closed andμssw=0 as open);kf,kp1,2,kv,kq1,2andkθare the designed control gains;Pi′andQi′are the active and reactive power output in per unit,respectively; ˉandˉare the average active and reactive power generation of DGs in per unit,respectively;P′SSWandQ′SSWare the per-unit active/reactive power flow at the target SSW;θSSWandESSWare the voltage phasor mismatch and amplitude at the target SSW;Eoi=Ei-∫aij(Eoi-Eoj)dtis the observed average DG operating voltage that can be regulated as rated (i.e.,Eoi=Eoj=E*) using the dynamic consensus-based average observer[24]; as shown in Fig.4,ωM0andEM0are the designed control variables for active and reactive power sharing within the min-MG;ωM1,ωM2,ωM3andEM1,EM2,EM3are the designed control variables for the operation of Type I,II and III modes,respectively.They are detailed as follows.

Fig.4 Proposed distributed secondary control framework

Type I mode (ηi=0 andλi=0)is designed for dynamic MG operating under static topology without reconfiguration requests.Under Type I mode,ωi=ω*,as the controller converges,meaning that the operating frequency and voltage are regulated as rated,and proportional active and reactive power sharing is achieved among the DGs between the neighboring min-MGs.

Type II mode (ηi=0 andλi=1)is designed for networked MG network reconfiguration when a closed SSW is requested to open.To guarantee seamless system topology transition,power flows through the target SSW are eliminated before it re-opens.Under Type II mode,besidesωi=ω*and=E*,it is also observed thatP'SSW=0 andQ'SSW=0 as the controller converges,meaning that the active and reactive power flow at the target SSW is minimized for seamless re-opening.

Type III mode (ηi=1 andλi=0)is designed for networked MG operating when requesting an open SSW to close.To achieve seamless system topology transit,voltage phasors on both sides of the targeted SSW are synchronized before the target SSW re-closes.Under Type III mode,besidesωi=ω*,=,it is also observed thatθSSW=0 andESSW=E*as the controller converges,meaning that the phase and voltage mismatches at the target SSW are minimized for seamless re-closure.

3.2 Scalable small-signal stability analysis

In this paper,the time domain small-signal method is used to approximate the nonlinear function at the equilibrium point by its first-order partial derivative.Since the obtained linear model parameters are closely related to the selection of the equilibrium point,the large-signal model is performed to find out the equilibrium point before the small-signal stability analysis[25].

The system under study is assumed to be inductive,and the active and reactive power flowsPiandQiat busiare given by[26]

whereEiis the output voltage magnitude;EPCCis the voltage magnitude at PCC;Xiandδiare the equivalent reactance and phase angle mismatch between busithDG and PCC,respectively.

Since the frequency is a global signal that converges fast enough in the system,the delay in adjusting the output frequency can be neglected,while any delay in adjusting the output voltage magnitude is modeled as a first-order low-pass filter[16]

whereωcis the cut-off frequency.

Based on Eqs.(1a),(1b),(7a),(7b)and (8),the system can be written as

where it is assumed that sinδi≈δiandEPCC≈E*are constants and so thate=(niE*/Xi+1)E*is constant and it is not considered in the following stability analysis.

Then the linearized approximation method is used to convert the nonlinear system into a linear system.It is assumed thatEi≈E'and cosδi≈cosδ'are constants in Eqs.(9a)and (9b),respectively.The linearized system modeling with the proposed frequency and voltage regulations is thereby modeled as follows

whereMi=E′E*/XiandNi=(E*/Xi)cosδ′ are constants,respectively.

For presentation purposes,it is denoted that M=diag(Mi),N=diag(Ni),m=diag(mi)and n=diag(ni).The state-space representations of Eqs.(10a)and (10b)are expressed as

where Xδ=[δ1,…,δn,ωsec1,…,ωsecn]T; XE=[E1,…,En,Esec1,…,Esecn]T; The corresponding system matrices Wδand WEare given by

where Hhand Kh(h=1,2,3)are the coefficient matrices that correspond to different operation modes,respectively; It is assumed that the variablesnLandnRare the numbers of DGs on the left and right sides of the target SSW so that T=[[1]nL/nL,[1]nL,nR/nR; [1]nR,nL/nR,[1]nR/nL]describes the average voltageϕiobserved at theithDG.

The following lemmas are used for the subsequent stability analysis.

Lemma 1[27]: if M is positive definite and N is a positive-definite scalar,then MN=NM and M+N are positive-definite.

Lemma 2[16]: if both A1and A2are positive definite,the characteristics polynomial det(s2I+sA1+A2)=0 has all its roots satisfyRe(s)＜0,i.e.,all characteristic roots are in the left-half complex plane.

Lemma 3[9]: if M=diag(Mi)is a positive-definite scalar,L is a Laplacian matrix of a connected graph[28],andris a positive constant,L+r[1]nM is positive definite.

In Type I mode,the system matrix of the frequency regulation (h=1)is

Referring to Schur complement,the characteristic polynomial of Wδis

where α1=sI+kfI and det(α1)=0 satisfiesRe(s)＜0 for all its roots;β1=s2I+s(mM)+kp1H1M and det(β1)=0 satisfiesRe(s)＜0 for all the roots by referring to Lemma 2.Therefore,in this scenario,the system is exponentially stable ifMi≈Mj.

The system matrix of the voltage regulation (h=1)for Type I mode is

In Eqs.(13)and (15),both H1and K1describe the communication among DGs within the individual min-MG and the communication among DGs between the min-MGs at each side of the target SSW.

By using the same method in Eq.(14),whenkv=0,the characteristic polynomial of WEis expressed as

where α2=sI+I+nN and det(α2)=0 satisfiedRe(s)＜0 for all its roots; similarly,β2=s2I+s(I+nN)+(kq1K1N) and det(β2)=0 satisfyRe(s)＜0 for all its roots according to Lemma 2.Since eigenvalues are a continuous function of matrix parameters,the system is exponentially stable ifkv＞0 is sufficiently small.

In Type II mode,based on the power balance between generation (Pi,Pj)and consumption (PL)when the system operates autonomously,the active power flow in per-unitP'SSWat the target SSW can be expressed as

Therefore,the system matrix of the frequency regulation (h=2)is

The matrix P=2kp2diag([1]nL,nL,[1]nR,nR)introduces the through active power at the target SSW with proper direction.Referring to Lemma 3,H2is positive-definite so that H2M is positive-definite based on Lemma 1.Finally,Similar to the derivations in Type I mode,the system is exponentially stable ifMi≈Mj.

The reactive power flow in per-unitQ'SSWat the target SSW can be derived similarly in Eq.(17)so that the system matrix of the voltage regulation (h=2)for Type II mode is

The matrix Q=2kq2diag([1]nL,nL,[1]nR,nR)introduces the through reactive power at the target SSW with proper direction.Referring to Lemma 3,K2is positive definite so that K2N is positive definite as per Lemma 1.Similar to Type I mode,the system is exponentially stable ifkv＞0 is sufficiently small andNi≈Nj.

In Type III mode,since SSW is requested to open,the system stability analysis is applied for each individual min-MG.according to Millman’s theorem[29],the measured voltage phasor at the target SSW can be written as

Thus,the system matrix of the frequency regulation (h=3)is

The matrix θ represents the phase mismatch at the target SSW compared to the common reference.WhenMi≈Mj,M-1is positive definite scalar so that H3is positive-definite referring to Lemma 3.Similar to Type I and II modes,the system is exponentially stable ifMi≈Mj.

The system matrix of the voltage regulation (h=3)for Type III mode is

The matrix V represents the voltage magnitude mismatch at the target SSW.WhenNi≈Nj,N-1is a diagonal positive definite scalar so that K3is positive definite based on Lemma 3.Similar to Type I and II modes,the system is exponentially stable ifNi≈Nj.The value ofkvdoes not affect the system stability in this mode since the average DG voltage regulation T is disabled.

4 Case studies

To validate the performance of the proposed dynamic MGs framework and the controllers developed for each operation mode,a 6-bus test system with five DGs is established in Fig.5,where the system consists of two min-MGs,one SSW,and five DGs with local loads,respectively.The requested system topology variation (Topologies #1-#4) is implemented in the context of dynamic MGs.Initially,both the main breaker and the SSW are closed,so that system operates under grid-connected mode (Topology #1).It is then assumed that the distribution feeder is disconnected from the main power grid due to a grid contingency.The main breaker is open,and the system operates autonomously (Topology #2).The networks are possessively reconfigured as per the system operator requests (Topology #3 and Topology #4).The detailed system parameters are shown in Tab.2.

Tab.2 System setting and control parameters

Fig.5 Dynamic MGs reconfiguration schemes

4.1 Case I:System dynamic response with load change

Topology #1 in Fig.5 indicates that the system operates in the grid-connected mode with the main breaker and the target SSW closed.The corresponding time-domain waveforms are shown in Fig.6,whent＜t1=4 s,the system frequency,and voltage are regulated by the upstream network.Further,Topology #2 in Fig.5 illustrates the proposed controller performance for system operation without the request of reconfiguration.Whent≥t1=4 s in Fig.6,the system starts operating in islanded mode with Type I mode being activated.Whent=t2=8 s,the load profile is changed on each bus (e.g.,parallel a large impedance load).As seen,the system frequency and voltage are regulated as reference values,and accurate proportional active and reactive power sharing is achieved within each MG.

Topologies #3 in Fig.5 explains the proposed controller performance for system reconfiguration with SSW opening.The requested reconfiguration is executed with the proposed controller under Type II mode being activated.First of all,SSW receives the reconfiguration request att=t3=10 s as shown in Fig.6,where the power flowing through SSW is mitigated before reconfiguration.Second,the load profile is varied on each bus by paralleling a large load att=t4=15 s.Finally,the SSW is opened att=t5=20 s.Topology #4 in Fig.5 shows the proposed controller performance for system reconfiguration with SSW closure.The requested reconfiguration is executed with the Type III mode being activated.Specifically,Fig.6 shows that the SSW receives request att=t6=22 s,changes the load profile att=t7=24 s,and turns off the SSW att=t8=30 s.Aftert＞t8,the proposed controller switches back to Type I mode and proportional power sharing among DGs 1-5 is continuously achieved.

Fig.6 System dynamic response

4.2 Case II:System stability with control parameter change

In this case,the effectiveness of the presented small-signal stability analysis is verified using various control parameters.As mentioned,for the voltage/reactive controller of both Type I and Type II modes,the system is exponentially stable if the control gainkv＞0 is sufficiently small.As a consequence,the voltage/reactive controller of Type II mode is taken as an example to explain the consistency between the theoretical analysis and simulation results.Whenkvis positive but not sufficiently small (e.g.,kv=300),the system is unstable,as shown in Fig.7a.On the other hand,the corresponding eigenvalue distributions derived from the theoretical stability analysis is depicted in Fig.7b,where the eigenvalueλ1is located at the right-half plane and this result agrees with the simulation results in Fig.7a.

Fig.7 System operation status with control parameter change (kv=300)

5 Conclusions

In this paper,based on the hierarchical control framework and the consensus-based averaging approach,a distributed control strategy is proposed for dynamic MGs operation.The developed operation modes regulate the system operating frequency and voltage and guarantee proportional power sharing among the DGs.Moreover,the corresponding small-signal stability analysis is developed for the presented controllers along with various operation modes,where the exponential stability of the linearized system is illustrated leveraging specific sufficient conditions,respectively.Finally,through the system reconfiguration testing and the real-time simulation results,the developed works are validated.