Stability analysis and decentralized control of inverterbased ac microgrid
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Abstract
This work considers the problem of decentralized control of inverterbased ac microgrid in different operation modes. The main objectives are to (i) design decentralized frequency and voltage controllers, to gather with power sharing, without information exchange between microsources (ii) design passive dynamic controllers which ensure stability of the entire microgrid system (iii) capture nonlinear, interconnected and largescale dynamic of the microgrid system with meshed topology as a portHamiltonian formulation (iv) expand the property of shiftedenergy function in the context of decentralized control of ac microgrid (v) analysis of system stability in large signal point of view. More precisely, to deal with nonlinear, interconnected and largescale structure of microgrid systems, the portHamiltonian formulation is used to capture the dynamic of microgrid components including microsource, distribution line and load dynamics as well as interconnection controllers. Furthermore, to deal with large signal stability problem of the microgrid system in the gridconnected and islanded conditions, the shiftedHamiltonian energy function is served as a storage function to ensure incremental passivity and stability of the microgrid system. Moreover, it is shown that the aggregating of the microgrid dynamic and the decentralized controller dynamics satisfies the incremental passivity. Finally, the effectiveness of the proposed controllers is evaluated through simulation studies. The different scenarios including gridconnected and islanded modes as well as transition between both modes are simulated. The simulation conforms that the decentralized control dynamics are suited to achieve the desired objective of frequency synchronization, voltage control and power sharing in the gridconnected and islanded modes. The simulation results demonstrate the effectiveness of the proposed control strategy.
Keywords
Decentralized control Inverterbased ac microgrid Frequency and voltage control Active and reactive power sharing Incremental passivity PortHamiltonian framework Shiftedenergy function1 Introduction
In recent years, renewable energy systems have been increasingly employed to mainly reduce the cost of energy prices and solve the environmental issues. A cluster of elements such as loads, distributed generators (DG), distributed storage units and controllable loads, connected through a medium or low voltage interconnecting power grid is defined as a Microgrid system [1, 2]. When the number of elements in each microgrid is increased, the problem of controlling them can be very challenging in a centralized way. Using communication links, especially in largescale system, might not be economical and practical. Moreover, due to the communication link delay or failure, the reliability of the system may deteriorate. Thus, each microgrid has the responsibility of controlling their own units locally. From the decentralized point of view, the controller of microsource units only needs the local variables [3].
Different decentralized control strategies including droopbased and nondroopbased controllers have been proposed for ac microgrid systems with different operation modes. Early work on decentralized parallel inverter control concepts suitable for microgrid operation was reported in [4]. Subsequent work [5, 6] extended the droop concept to ensure sharing of harmonic currents of nonlinear loads. In further investigation of the droop concept, some researchers [7, 8, 9] have proposed powerangle droop control, in which the phase angle of the distributed source voltage, relative to a systemwide common timing reference, is set according to a droop law. A static droop compensator is utilized for power sharing in [10]. An enhanced droop control featuring a transient droop performance is proposed [11]. To improve the active and reactive power decoupling performance, improved droop controllers with virtual output impedance are reported [11, 12]. To account for nonlinear loads, harmonicbased droop controllers are investigated [13, 14]. To realize realtime decentralized control strategies a decentralized control strategy extracted from offline optimization results is designed [15], but this control design requires prior knowledge of the grid structure and extensive computations are needed if the grid structure changes. Furthermore, the methods proposed in [16, 17] allow for the seamless pluggingin, unplugging and replacement of microsource units without spoiling microgrid stability. Control design procedures with these features have been termed plugandplay [16, 18, 19, 20, 21]. However, existing droop controllers are synthesized in the sense of the smallsignal model of the power transfer mechanisms. To improve the performance of parallel operation, a stationaryreferenceframe droopcontrolled model of parallelconnected inverters is introduced and inner voltage and current control loops are proposed [22]. Decoupled droop control techniques are proposed and analyzed in [23] to obtain independent association of frequency with active power and voltage with reactive power. To exploit the flexibility and fast dynamics of the inverterbased distributed energy resources, a piecewise linear voltagecurrent droop controller is introduced [24]. To improve the reactive power sharing performance of droop control, a virtual impedance optimization method for reactive power sharing in networked microgrid is proposed [25]. To control parallelconnected Inverters, a synchronous reference frame virtual impedance loop is proposed [26]. To soft synchronization of microgrid, an approach using robust control theory is reported in [27]. In [28], a reactivepower sharing scheme for two inverterbased distributed generations with unequal line impedances in islanded microgrids is reported. An adaptative droop control for balancing the state of charge of multiple energy storage systems in decentralized microgrids is studied in [29]. Moreover, numerous control methodologies have been proposed to improve the ac microgrid performance [30, 31, 32, 33, 34, 35, 36, 37]. A fully decentralized control and a f P/Q droop control scheme of gridconnected cascaded inverters are reported [30, 31]. A costfunctionbased decentralized power quality compensation method realizing simultaneously bus voltage compensation and inverter current sharing is proposed [32]. A dual control strategy for power sharing improvement in islanded mode of AC microgrid is reported [33]. A decentralized optimal servo control system for implementing instantaneous reactive power sharing in microgrids is proposed [34]. In [38], a passivitybased decentralized control strategy for currentcontrolled inverters in ac microgrids is reported. an adaptive droop control is proposed including a fuzzybased droop coefficient adjustment in [35]. Decentralized parallel operation of single phase inverters in islanded mode is reported in [36]. In the study by [37], twostage adaptive virtual resistor (AVR) control scheme for lowvoltage parallel inverters is proposed. A decentralized control with unique equilibrium point for cascadedtype microgrid is proposed in [39]. By far the largest body of research work done in decentralized microgrid control has been for radial architectures (see also [3]). Notably, the droop control has several limitations and challenges including: inability to guarantee large signal stability, weak transient performance, weak performance for mixedline microgrids with resistive–inductive line conditions, and coupled dynamics between active and reactive power.
Early work towards stability analysis of decentralized controlled intverters based on droopcontrol was reported in [40]. A generalized approach for analyzing the smallsignal stability of interconnected inverter systems was proposed. This result was extended in [41] with the inclusion of reactive powervoltage magnitude droops for the interface inverters. In [42], the authors propose a method, based on the operating point, to set droop gains adaptively. A computational approach to determining radial microgrid stability, scalable to large systems, is presented in [43]. Moreover, numerous methodologies have been proposed to enhance the microgrid stability performance via modifying the inverter control loops [44, 45, 46, 47, 48]. These techniques and methods include a combination of repetitive and deadbeat control with feedforward compensation for disturbance rejection [46], secondary control based on potentialfunction [47], realtime small signal stability analysis for droop gain schedule [45] and droop gain variation for increasing virtual inertia [48]. The various analytical stability tools for microgrid controllers design are reported in [49, 50, 51, 52, 53, 54, 55, 56]. The small signal models of the inverter based microgrid is established in [50, 51, 56]. The active load impacts on system smallsignal stability are analyzed in [52], it shows the active loads with a large dc voltage controller gain may lead to instability. A combination of smallsignal and timedomain simulation for a comprehensive stability analysis are used in [49, 53, 57, 58, 59]. Another impedancebased stability criterion for gridconnected inverters is reported in [55]. The frequency regulation of microgrid by utilizing the kinetic energy from the induction motor loads is studied in [60]. An adaptive droop adjustment based on the microgrid small signal model is introduced in [42]. The authors in [61] use the online grid parameter estimation combined with small signal stability analysis to detect the islanding and adjust the inverters droop control parameters. The droop scheduling scheme based on bifurcation theory is presented in [62] to find the worst primary reserve share that is closest to instability. A modeadaptive droop control method is proposed in [63]. Sufficient conditions for voltage stability in a droopcontrolled lossy microgrid is reported in [64]. Furthermore, sufficient conditions for stability of lossless microgrids based on energy function are established in [65]. Two nonlinear droop controls are proposed in [66, 67] to enhance microgrid frequency regulation. Nonetheless, it may either require multiple small signal stability analyzes to determine the acceptable droop gains [66] or lead to instability due to a high droop gain when the frequency is close to the nominal value [67, 68]. Moreover, the complete reviews and classifications of stability in microgrids are presented [69, 70, 71].
In addition to the droopbased control strategies, nondroopbased approaches for voltage and frequency control of the islanded microgrids have also been developed, e.g. [72, 73, 74, 75, 76, 77, 77, 78, 79, 80]. A robust servomechanism controller for a singleDG/ MultiDG islanded system, which uses the internal oscillator to control the frequency in an openloop manner, has been proposed in [73, 74, 78, 79]. In addition, the following methods have been proposed for singleDER microgrids: a voltage controller, designed using an H_{∞} approach and repetitive control technique, to mitigate voltage harmonics of the point of common coupling (PCC) [81] and a robust control scheme for a microgrid designed based on an H_{∞} approach to provide a robust performance [82]; and a robust servomechanism approach for PCC voltage control [73]. However, these methods are only applicable to singleDER microgrids. In the study by [83], a decentralized servomechanism controller based on robust approach for the islanded operation of radial connection of two distributed generation (DG) units is proposed. In [84] state feedback is combined with a decentralized LMI strategy to ensure stabilization and frequency regulation, while [85] studies the performance of decentralized frequencycontrol algorithm based on integral action. To improve currentsharing, a robust controller for parallelconnected inverterbased DGs in lossy microgrid networks is reported in [86]. Although extensive research has been carried out on the development of nondroopbased control of microgrids, they suffer from one or more of the following drawbacks: in applicability to multiplemicrosources, inability to guarantee stability and/or performance with respect to several microsources, and highorder controller structures.
To overcome the disadvantages of the existing microgrid control approaches, various challenges associated with large signal stability, seamless transition between different operation modes and lowcomplexity of the local controllers must be addressed.
In this study, to deal with nonlinear, interconnected and largescale structures of microgrid systems, the portHamiltonian formulation is used to capture the dynamic of microgrid components and interconnection controllers. In addition, the decentralized controllers are also formulated as portHamiltonian systems that are connected to the microsources with interaction ports. This framework describes the dynamics in terms of the system stored energy, interconnection and dissipation structures [87, 88, 89]. Furthermore, we focus on a more accurate and higher order model for the inverterbased microsources than conventionally used in the literature [65, 80, 90].
In this work, to deal with large signal stability problem of microgrid system, the shiftedHamiltonian energy function is served as a storage function to ensure incrementalpassivity of system. Notably, the large signal stability analysis is necessary to guarantee seamless transition between islanded/gridconnected modes and the convergence of equilibrium states of nonlinear microgrid dynamic. Although historically energy functions have played a crucial role to cope with accurate models of power systems (see also [80, 90, 91, 92, 93, 94, 95, 96, 97]), our approach based on the incremental passivity allows us to cover an accurate dynamic of microgrids and paves the way for the design of decentralized controllers.
In the reported research works, the microgrids have been almost studied in smallsignal point of view while no solution has been proposed regarding the controller to face with largesignal stability. In this paper, authors aim to model the microgrid components and propose new decentralized passivitybased control strategy for an microgrid consisting microsource units and local loads based on portHamiltonian framework, shiftedenergy function, and incremental passivity to ensure stability of the entire system and enhance its performance for transition between gridconnected to islanded modes and guarantee the desired frequency synchronization, voltage tracking as well as power sharing. This paper presents theoretical concepts, requirements, and necessary conditions for stability and performance of microgrid with meshed topology. Finally, in addition to stability and performance analysis, the effectiveness of the proposed controller is evaluated by set of simulation studies.

Design decentralized frequency and voltage controllers, to gather with power sharing, without information exchange between microsources and their local controllers,

Design passive dynamic controllers which ensure stability of the entire microgrid system,

Capture nonlinear, interconnected and largescale dynamic of the microgrid system with meshed topology as a portHamiltonian formulation,

Expand the property of shiftedenergy function in the context of decentralized control of microgrid,

Serve the shiftedHamiltonian energy function as a storage function to ensure incremental passivity,

Analysis of system stability in large signal point of view.
The remainder of this paper is organized as follows. In Section 2, the portHamiltonian formulation and shiftedHamiltonian energy function as well as incremental passivity are defined as a background of this study. The microgrid topology and structure are presented in Section 3. In Section 4, the microgrid component dynamics are also formulated. In addition, the microgrid units (e.g. microsource, load and distribution line) are modeled based on port Hamiltonian formulation. The openloop analysis of overall microgrid is established in Section 5. In Sections 6 and 7, the different passive controllers for frequency and voltage control are also proposed and analyzed in different operation modes. Furthermore, to deal with large signal stability problem, the shiftedHamiltonian energy function is served as a storage function to ensure incremental passivity of entire system. Finally, the paper is closed with simulation results in Section 8 and conclusions remarks as well as discussion of future work in Section 9.
2 Background: portHamiltonian and incremental passivity formulation
As mentioned, to deal with nonlinear, interconnected and largescale structures of microgrid systems, the portHamiltonian formulation is used to capture the dynamic of microgrid components and interconnection controllers. In addition, the decentralized controllers are also formulated as portHamiltonian systems that are connected to the microsources with interaction ports. This framework describes the dynamics in terms of the system stored energy, interconnection and dissipation structures [87, 88, 89]. Moreover, the portHamiltonain formulation is described as follows:
Definition 1
With state \(x\in {{\mathbb {R}}^{n}}\), input \(u\in {{\mathbb {R}}^{m}}\) and output \(y\in {{\mathbb {R}}^{m}}\) [87, 88]. In addition, \({\mathcal {J}}^{\top }(x)={\mathcal {J}}(x)\) is interconnection matrix, \({\mathcal {R}}^{\top }(x)={\mathcal {R}}(x)\ge 0\) is dissipation matrix, and \({\nabla }{\mathcal {H}}(x)\) is also the vector of partial derivatives of the Hamiltonian energy function \({\mathcal {H}}(x)\) with respect to the state x.
It is shown that both the microgrid and the controller dynamics admit a portHamiltonian representation which are then interconnected to obtain a closedloop portHamiltonian system. This allows to easily identify the Hamiltonian energy function and give conditions for stability of the equilibrium state.
In addition, to deal with large signal stability problem of microgrid system, the shiftedHamiltonian energy function is served as a storage function to ensure incrementalpassivity of system (see [88]). Notably, the incrementalpassivity of the shifted portHamiltonian system is defined as follows:
Definition 2
3 Microgrid structure and topology
The microgrid is an emerging concept for an efficient integration of renewable microsource units (see [1, 3, 100, 101, 102, 103] and references herein). An inverterbased ac microgrid consists of microsources (e.g. wind turbine and solar panel equipped with inverters), distribution lines and loads that are connected to maingrid via static switch (common coupling point).
Definition 3
The inverterbased ac microgrid is composed to several units, i.e. microgrid units, including microsource, distribution line and load units (see [3, 90, 100] and references herein). The microsource unit is referred to unit injecting an amount of power into the microgrid and load unit is also correspond to unit absorbing an amount power from the microgrid. In addition, distribution unit is referred to unit transferring power within microgrid. Moreover, the maingrid unit is also corresponded to common coupling port injecting (absorbing) an amount of power into (from) the microgrid (maingrid).
Remark 1
The inverterbased microgrid model used in this work consists of g=1 maingrid bus (in gridconnected mode), s microsource buses, ℓ load buses and also one reference bus. Therefore, the total number of buses (nodes or vertices) of the \(\mathcal {G}\) microgrid (graph) is \(\mathcal {N}=n+1\), with n=g+s+ℓ. Without loss of generality, it is assumed that the set of nodes (vertices) \(\mathcal {N}\) can be partitioned into four ordered subsets called \({\mathcal {N}}_{G}, {\mathcal {N}}_{S}, {\mathcal {N}}_{L}\) and \({\mathcal {N}}_{0}\) and these subsets are associated to maingrid, microsource, loads nodes and the reference node respectively. We call \(\mathcal {V}\in {\mathbb {R}}^{\mathcal {N}}\) the vector of node potentials. The microgrid graph consists of maingrid, microsource, load and distribution line edges. In other word, there are maingrid edge associated to a main grid unit, microsource edge associated to a microsource unit and also a load edge associated to a load unit and these edges define between main grid node, every microsource node and also every load node and the reference node respectively. Furthermore, distribution edges associated to distribution units that connect maingrid, microsource and load buses Therefore, there are in total g=1 maingrid edge (gridconnected mode), s microsource, ℓ load and d distribution edges. Hence, there are in total e=g+s+ℓ+d edges in microgrid. Without loss of generality, it is assumed that the set of edges \(\mathcal {E}\) can be partitioned into four ordered subsets called \({\mathcal {E}}_{G}, {\mathcal {E}}_{S}, {\mathcal {E}}_{L}\) and \({\mathcal {E}}_{D}\) associated to maingrid (gridconnected mode), microsource, load and distribution edges respectively. We call \((V_{e},I_{e})\in {\mathbb {R}}^{e\times e}\) the vectors pair associated to edge voltages and currents respectively. In addition, it is assumed that all microgrid units share a port of the same dimension p=p_{dq}=2 in dqform.
4 Inverterbased ac microgrid: portHamiltonian formulation
4.1 Maingrid Dynamic
Where \(\bar {V}\) and \(\bar {\omega }\) are desired voltage and phase frequency of microgrid. In addition, P_{e} and Q_{e} are active and reactive power of microgrid and d_{G} and a_{G} are constant coefficients. Furthermore, we have \({\omega }_{G}=\bar {\omega }\) and \(v_{G}=\bar {V}\).
Where maingrid connects to microgrid through a port of dimension p=p_{dq}=2 in dqform.
4.2 Microsource dynamics
Definition 4
Where ϕis the flux linkage across the inductors, q and q_{dc} the charge in the capacitors. In addition, L and C denote the inductance and capacitance of the microgrid acside. In addition C_{dc} denote the capacitance of microsource dcside.
Where \(\widehat {\boldsymbol {s}}={\left ({\widehat {s}}_{a},{\widehat {s}}_{b},{\widehat {s}}_{c}\right)}^{\top }=\boldsymbol {\Upsilon }\boldsymbol {s}\), \(\boldsymbol {\Upsilon }=\frac {1}{3}\left [\begin {array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end {array}\right ] \)and s=(s_{a},s_{b},s_{c}). In addition, G_{dc} is the conductance of microsource dcside and x=(ϕ,q,q_{dc})^{⊤} is state vector of microsource unit. Furthermore, the y_{A} and \(\bar {x}\) are auxiliary output and desired states respectively.
where \(\boldsymbol {\mathrm {J}}= \left [\begin {array}{cc} 0 & 1 \\ 1 & 0 \end {array}\right ]\) and also ΥT_{0dq}=T_{0dq}.
In addition, n_{i}=5,m_{i}=2 and p_{i}=2. The portHamiltonian energy function is also defined as \({\mathcal {H}}_{i}:{\mathbb {R}}^{n_{i}}\to \mathbb {R}\).
Remark 2
Where x_{S}=(ϕ_{S,dq},q_{S,dq},q_{dc,S}) is state vector of microsource units; \({{\mathcal {H}}_{S}}(x_{S})\)is Hamiltonian energy function; \({\mathcal {J}}_{S_{u}}\) and \({\mathcal {J}}_{S_{c}}\) are interconnection matrices and \({\mathcal {R_{S}}}\) is dissipation matrices; (w_{S},y_{S})=(i_{S,dq},v_{S,dq}) is conjugated interaction port variables; \(\left (w_{S_{0}},y_{S_{0}}\right)=\left ({\boldsymbol {i}}_{S_{0}},{\boldsymbol {v}}_{dc,S}\right)\) is also conjugated microsource port variables. Other matrices can be defined as follows, F_{S} interaction port matrix and \(F_{S_{0}}\) microsource port matrix. Furthermore, the \(y_{S_{A}}\) is auxiliary passive outputs.
4.3 Distribution line dynamics
Where ϕ is the magnetic flux in the inductor.
Definition 5
Where n_{i}=2 and p_{i}=2. In addition, portHamiltonian energy function defines as \({\mathcal {H}}_{i}\left ({\phi }_{{dq}_{i}}\right):{\mathbb {R}}^{n_{i}}\to \mathbb {R}\).
Remark 3
4.4 Load dynamics
Where q is the charge in the capacitor.
Definition 6
In addition, n_{i}=2 and p_{i}=2. The portHamiltonian energy function is also defined as \({\mathcal {H}}_{i}\left (q_{{dq}_{i}}\right):{\mathbb {R}}^{n_{i}}\to \mathbb {R}\).
Remark 4
4.5 Microgrid overall dynamic
In addition, the submatrix of microsource incidence matrix is also decomposed to capture the information about the microgrid different operation modes (gridconnected and islanding).
Where the reference node is considered to be a ground potential.
Remark 5
Where the matrix \(F_{T}\mathcal {T}F^{\top }_{T}\) is skewsymmetry.
Remark 6

\({\dot {{\mathcal {H}}}}_{T}\) accounts for the stored power (difference between supplied and absorbed power) in microgrid;

\({\left [{\nabla }{\mathcal {H}}_{T}\left (x_{T}\right)\right ]}^{\top }{\mathcal {R}}_{T}\left [{\nabla }{\mathcal {H}}_{T}\left (x_{T}\right)\right ]\) represents the dissipated power in microgrid;

\(w^{\top }_{G}{{\bar {y}}_{G}}+y^{\top }_{T_{0}}w_{T_{0}}\) represents the supplied power in microgrid.
5 Microgrid openloop analysis
Where \({\bar {x}}_{T}\in {\mathbb {R}}^{n_{T}}\) is the desired equilibrium with corresponding control \(\bar {u}\).
Proposition 1
Proof
Where \({\tilde {y}}_{S_{A}}=A^{\top }_{S}\left (\bar {s}\right)\left ({\nabla }{\mathcal {H}}_{S}(s){\nabla }{\mathcal {H}}_{S}\left (\bar {s}\right)\right)\). Therefore, the overall portHamiltonian system Eq. (69) verifies the incremental passivity with shiftedHamiltonian storage function \({\bar {{\mathcal {H}}}}_{T}\). □
6 Microgrid closedloop stability analysis and control in gridconnected mode
Remark 7
In gridconnected mode, it is assumed that the frequency and voltage of microgrid are stabilized by main grid. Therefore, the microsources work to manage active and reactive injected into microgrid.
Where \(K_{P}=bdg\left \{K_{P_{{dq}_{i}}}\right \}, K_{I}=bdg\left \{K_{I_{{dq}_{i}}}\right \}\) and \({\mathcal {H}}_{C_{i}}=\frac {1}{2}{\left ({\eta }_{{C_{dq}}_{i}}\right)}^{\top }K_{I_{{dq}_{i}}}\left ({\eta }_{{C_{dq}}_{i}}\right)\).
Where n_{SC}=n_{S}+n_{C} with \(n_{C}=\sum ^{s}_{i=1}{p_{i}}\).
Proposition 2
Proof
Therefore, the overall portHamiltonian system (81) with the decentralized passive (proportionalintegral) control dynamic using microsource output current (78) satisfies the incremental passivity. □
7 Microgrid closedloop stability analysis and control in Islanded mode
where \(\phantom {\dot {i}\!}{\phi }_{{dq}_{i}}\) is the flux linkage across the inductors, \(\phantom {\dot {i}\!}q_{{dq}_{i}}\) is the charge in the capacitors and ω_{i} is angle frequency.
Where \(\phantom {\dot {i}\!}u_{{dq}_{i}}=y_{C_{i}}=K_{P_{{dq}_{i}}}y_{M_{i}}+K_{I_{{dq}_{i}}}{\eta }_{{dq}_{i}}\) and \(\beta ={\bar {v}}_{{dc}_{i}}\left ({\omega C_{i}\boldsymbol {\mathrm {J}}}+{\mathbb {I}}_{2}\right)\).
Proposition 3
Proof
8 Simulation
We consider the following three scenarios.
8.1 Gridconnected scenario
Gridconnected scenario: Microgrid parameters
Bus No.  \(\bar {S}_{i}\) [pu]  \({\bar {P}_{i}}/{\bar {S}_{i}}\) [pu/pu]  \({\bar {Q}_{i}}/{\bar {S}_{i}}\) [pu/pu]  \(\bar {V}_{dc}\) [v] 

Bus 3  0.128  0.988  0.012  376.991 
Bus 4  0.070  1.000  0.000  500.000 
Bus 5  0.105  0.890  0.011  500.000 
Bus 6  0.116  0.792  0.198  376.991 
Bus 7  0.128  0.720  0.270  377.991 
Bus 8  0.116  0.800  0.020  500.000 
Bus 9  0.105  1.000  0.000  500.000 
Bus 10  0.128  0.720  0.270  376.991 
Bus 11  0.105  0.890  0.111  500.000 
Load parameters
Bus No.  Active power [pu]  Reactive power [pu] 

Bus 1  0.0698  0.0010 
Bus 2  0.0000  0.0000 
Bus 3  0.0257  0.0019 
Bus 4  0.0954  0.0016 
Bus 5  0.1885  0.0035 
Bus 6  0.1303  0.0012 
Bus 7  0.1164  0.0019 
Bus 8  0.0722  0.0019 
Bus 9  0.0607  0.0021 
Bus 10  0.0745  0.0017 
Bus 11  0.0954  0.0021 
In addition, the decentralized power sharing (and voltage) controllers are equipped with the PI controllers (77). In this scenario, the controller parameters of the PI controllers are chosen as follows: for microsource output current (\({i_{S_{dq_{i}}}}\)) and output voltage control, \({{K_{P}}=0.9{\mathbb {I}_{2}}}\) and \({K_{I}}=20{\mathbb {I}_{2}}\) as well as for frequency control, K_{P}=10 and K_{I}=500.
The simulation results show that all trajectories converge to desired states demonstrating the stability analysis in Section 6. After a transient, the frequencies synchronize and the amplitudes of bus voltages become constant. The frequencies at the all microsource buses are converged to nominal value. The voltage amplitudes remain within 1±0.1pu in steadystate. Therefore, the results demonstrate that the proposed controllers can regulate the microsource voltages with good tracking performance. The initial conditions have been chosen arbitrarily. Hence, the simulation result shows the stability of the decentralized frequency and voltage control dynamics, as given in (77). Furthermore, the simulation confirms that the decentralized control dynamics are suited to achieve the desired objective of frequency synchronization, voltage control and power sharing.
8.2 Islaned scenario
Islanded scenario: Microgrid parameters
Bus No.  \(\bar {S}_{i}\) [pu]  \({\bar {P}_{i}}/{\bar {S}_{i}}\) [pu/pu]  \({\bar {Q}_{i}}/{\bar {S}_{i}}\) [pu/pu]  \(\bar {V}_{dc}\) [v] 

Bus 3, \(i\in {\mathcal {E}}_{S_{F}}\)  0.128  0.988  0.012  376.991 
Bus 4, \(i\in {\mathcal {E}}_{S_{V}}\)  0.070  1.000  0.000  500.000 
Bus 5, \(i\in {\mathcal {E}}_{S_{V}}\)  0.105  0.890  0.011  500.000 
Bus 6, \(i\in {\mathcal {E}}_{S_{F}}\)  0.116  0.792  0.198  376.991 
Bus 7, \(i\in {\mathcal {E}}_{S_{F}}\)  0.128  0.720  0.270  377.991 
Bus 8, \(i\in {\mathcal {E}}_{S_{V}}\)  0.116  0.800  0.020  500.000 
Bus 9, \(i\in {\mathcal {E}}_{S_{V}}\)  0.105  1.000  0.000  500.000 
Bus 10, \(i\in {\mathcal {E}}_{S_{F}}\)  0.128  0.720  0.270  376.991 
Bus 11, \(i\in {\mathcal {E}}_{S_{V}}\)  0.105  0.890  0.111  500.000 
As mention in Section 7, for the gridforming microsources (\(i\in {{\mathcal {E}}_{S_{F}}}\)), the decentralized frequency (and voltage as well as power sharing) controllers are equipped with the PI controllers (91). In this scenario, the controller parameters of the PI controllers are chosen as follows: for microsource output current (\({i_{S_{dq_{i}}}}\)) and output voltage control, \({{K_{P}}=0.9{\mathbb {I}_{2}}}\) and \({K_{I}}=20{\mathbb {I}_{2}}\) as well as for frequency control, K_{P}=10 and K_{I}=500.
In addition, for the gridforming microsources (\(i\in {{\mathcal {E}}_{S_{V}}}\)), the decentralized power sharing (and voltage) controllers are equipped with the PI controllers (95). In this scenario, the controller parameters of the PI controllers are chosen as follows: for microsource output current (\({i_{S_{dq_{i}}}}\)) and output voltage control, \({{K_{P}}=0.9{\mathbb {I}_{2}}}\) and \({K_{I}}=20{\mathbb {I}_{2}}\) as well as for frequency control, K_{P}=10 and K_{I}=500.
The simulation results show that all trajectories converge to desired states demonstrating the stability analysis in Section 7. After a transient, the frequencies synchronize and the amplitudes of bus voltages become constant. The frequencies at the all microsource buses (\({i\in {{\mathcal {E}}_{S_{F}}}}\) and \({i\in {{\mathcal {E}}_{S_{V}}}}\)) are converged to nominal value. The voltage amplitudes remain within 1±0.1pu in steadystate. Therefore, the results demonstrate that the proposed controllers can regulate the microsource voltages with good tracking performance. The initial conditions have been chosen arbitrarily. Hence, the simulation result shows the stability of the decentralized frequency and voltage control dynamics, as given in (91) and (95). Furthermore, the simulation confirms that the decentralized control dynamics are suited to achieve the desired objective of frequency synchronization, voltage control and power sharing.
8.3 Transition scenario
The simulation results demonstrate the effectiveness of the proposed control strategy. The designed controllers guarantee the power sharing and the frequency as well as voltage stability of the entire microgrid system.
9 Conclusion
The problem of decentralized control of inverterbased ac microgrid in different operation modes is addressed in this work. In addition, we focus on the problem of decentralized frequency and voltage control of the microgrid system without information exchange between the microsources. The design of passive dynamic controllers which ensure stability of the entire microgrid system is also addressed in this paper. Furthermore, to deal with nonlinear, interconnected and largescale structure of microgrid systems, the portHamiltonian formulation is used as a powerful tool to capture the dynamic of microgrid components and interconnection controllers. It is shown that both the microgrid dynamic as well as the controller designs admit a portHamiltonian representation which are then interconnected to obtain a closedloop portHamiltonian system. Moreover, to deal with large signal stability problem of microgrid system, the shiftedHamiltonian energy function is served as a storage function to ensure incremental passivity of system. In addition, the contribution also expands the knowledge on the use of shiftedenergy functions in the context of decentralized control of microgrids. Moreover, it is shown that the aggregating of the microgrid dynamic and the decentralized controller dynamics satisfies the incremental passivity in the gridconnected and islanded modes. Finally, the effectiveness of the proposed controllers is evaluated through simulation studies. The different scenarios including gridconnected and islanded modes as well as transition between both modes are simulated. The simulation conforms that the decentralized control dynamics are suited to achieve the desired objective of frequency synchronization, voltage control and power sharing in the gridconnected and islanded modes. The simulation results demonstrate the effectiveness of the proposed control strategy.
Future works include attempting to add the hybrid microgrid systems and investigating large signal stability analysis as well as examining incremental passivity of entire system.
Notes
Funding
The authors declare that no fund is received from any financial or nonfinancial organization.
Availability of data and materials
Not applicable.
Authors’ contributions
MFF contributed the concept of shiftedenergy function and incremental passivity into the context of decentralized control of ac microgrid, modeled the microgrid components in portHamiltonian framework, designed decentralized passive dynamic controllers, analyzed the system stability and drafted the manuscript. AR and MG reviewed the manuscript. MFF and AR revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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