Economic challenges, technological advancements and environmental impacts are now demanding distributed generation in place of the conventional centralized generation [1]. Power operation companies are now confronted with unprecedented difficulties in terms of meeting the load requirements, consumer satisfaction and environmental considerations. Thus, distributed generation has received good attention because of its potential to alleviate pressure from the main transmission system by supplying a few local loads [2]. The waste heat generated from the fuel to electricity conversion is exploited by the distributed generation system with the help of microturbines, reciprocating engines and fuel cells to provide heat and power to the customers. Adding to the system distributed energy sources (DES) like photovoltaic (PV) panels, wind turbines, energy storage devices such as batteries and capacitors, generators and controllable loads. The distributed generators can extract energy from other renewable nearby systems and provide momentous contributions to future energy generation and distribution. Another noteworthy feature is that the carbon emission is reduced to a large extent satisfying the commitment of many nations concerning decrease of carbon footprints [3]. However, the distributed generation faces technical issues regarding its connection to the intermittent renewable generation and feeble areas of the distribution network. Further, owing to the distinct behavior of the distributed generation unlike the conventional load, alteration in power flow results in problems. To counter the irregular behavior and increasing penetration of the distributed generation, the microgrid was introduced.
The microgrid has paved its way into distributed generation and looks promising for future prospects. It has the ability to respond to changes in the load, while decreasing feeder losses and improving local reliability. Basically designed to cater the heat and power requirements of local customers, it can serve as uninterruptible power supply for critical loads. Several control strategies for the microgrid have been proposed in the literature including PI controllers in [4][12]. Robust
One efficient way to resolve the foregoing microgrid problem is to cast the integrated dynamic model into the SoS framework for better operation and organized control. The concept of SoS has opened up a new school of thought in systems engineering. SoS has emerged as a hot topic for research over the past few years. Although still in the infant stages, the concept of SoS has managed to achieve widespread acclaim. Being restricted to defense and IT, at one point of time, SoS has now entered a plethora of domains [16], [17]. It is worth noting that a microgrid is a complex system comprising of a variety of systems which are nonlinear in nature and possess strong crosscoupling between them. Hence viewing the microgrid from an intelligent system of systems perspective is need of the hour. Moreover an efficient control methodology based on SoS has to be established in order to overcome the challenges posed by the microgrid. The concept of SoS is now widespread and has entered several domains including defense, IT, health care, manufacturing, energy and space stations and exploration to name a few.
Recent progress in analysis and design issues of networked control systems is reported in [18][22]. Additional related work are published in [23]. Networked control of system of systems has been introduced in [24]. A control system consisting of a real time network in its feedback can be termed as a networked control system (NCS) [25]. As mentioned in [24] that the need to design a SoS control system which can tolerate packet loss and delays is one of the prime challenges in SoS networked control, we considered a network which is subjected to both delays and nonstationary packet dropouts and the controller stabilizes the system in the presence of these communication issues. The controller design for such a networked control is presented in [26]. A further extension is made in the controller design by introducing a distinct
The main objective of this paper is to formulate a SoS framework including a communication network for the distributed generation units of a microgrid and consequently design a feedback controller to guarantee closedloop stabilization in the presence of communication constraints, network delays and packet dropouts. The considered microgrid system comprised two sets of microalternator and PV panels as distributed generation units, which are eventually connected to a load and the main grid. To achieve our objective, the work reported here in combines two parts: a detailed modeling and analysis part leading to establish a stabilizing controller and realtime implementation part to illustrate the performance. This latter part is achieved at the SoS research lab at KFUPM using pilotscale lab equipment to build an experimental microgrid including up to 12 DG units. The simulation tools are MatlabSimulink environment, particularly the products Simscape Power Systems Examples and RAPSim an open source simulation software for microgrids.
The main contributions of this work are:
1) A generalized approach to modeling microgrids with multiple distribution generation units is developed based on SoS framework. This is a novel approach in the context of renewable energy systems.
2) The microgrid system incorporated communication network to enable processing control signals and power measurements.
3) An improved output feedback networked controller is constructed to stabilize the microgrid system in presence of delays and non stationary packet dropouts.
The paper is organized as follows. A brief introduction of microgrids and SoS is presented in Section Ⅰ. Detailed modeling of both the microalternator and PV system is presented in Section Ⅱ. A combined framework of microalternator and PV system connected to a load and the main grid is proposed. A state space representation of the combined system is presented. Further the detailed networked control system, its stability analysis and controller design is explained in Sections Ⅲ and Ⅳ with relevant theorems. The controller design is implemented on the proposed framework and simulation results are shown for two sets of PV and microalternator systems. Finally conclusions are drawn and the effectiveness of the proposed framework and control strategy is discussed.
Ⅱ. MODELING OF THE MICRO ALTERNATORPV SYSTEMTo model the microgrid system consisting of microalternator and PV system, we consider the separate modeling of microalternator and PV system initially. After modeling both these individual systems, we integrate them into a microgrid system which is connected to a load and the main grid. Two sets of such systems are considered eventually which form a SoS structure for the microgrid system.
A. Micro AlternatorThe swing equation of the alternator can be written as two first order differential equations [27]:
$ \begin{eqnarray} \frac{d\delta}{dt}& =& \omega_{0}(\omega 1), \;\;\;\frac{d\omega}{dt}=\frac{1}{2H}(P_{m}P_{e}) \label{meq1} \end{eqnarray} $  (1) 
where
$ \begin{eqnarray} \frac{de_{q}'}{dt}& =& \frac{1}{T_{do}'}[E_{fd}e_{q}'(x_{d}x_{d}')i_{td}] \label{meq2} \end{eqnarray} $  (2) 
where
$ \begin{eqnarray} \frac{dE_{fd}}{dt}=\frac{1}{T_{A}}[K_{A}(V_{tref}V_{t})(E_{fd}E_{fdo})] \label{meq3} \end{eqnarray} $  (3) 
where
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Fig. 1 Microalternator connected to grid. 
The terminal voltage of the alternator is given as
$ \begin{eqnarray*} V_{t}& =& V_{s}+(r_{t}+jx_{t})i_{t} \end{eqnarray*} $ 
In
$ \begin{eqnarray*} V_{d}+jV_{q}& =& V_{sd}+jV_{sq} \nonumber\\ &&+(r_{t}+jx_{t})(i_{td}+ji_{tq}) \nonumber\\ x_{q}i_{tq}+j(e_{q}'x_{d}'i_{td})& =& (V_{sd}+r_{t}i_{td}x_{t}i_{tq}) \nonumber\\ &&+ j(V_{sq}+r_{t}i_{tq}+x_{t}i_{td}). \end{eqnarray*} $ 
The real part can be written as
$ \begin{eqnarray} V_{sd}=(x_{q}+x_{t})i_{tq}r_{t}i_{td}\label{meq4} \end{eqnarray} $  (4) 
and the imaginary part can be written as
$ \begin{eqnarray} e_{q}'V_{sq}=r_{t}i_{tq}+(x_{d}'+x_{t})i_{td}.\label{meq5} \end{eqnarray} $  (5) 
Substituting
$ \begin{eqnarray} i_{td} & =& \frac{r_{t}V_{sd}+(e_{q}'V_{sq})(x_{q}+x_{t})}{r_{t}^{2}+(x_{d}'+x_{t})(x_{q}+x_{t})} \nonumber\\ \bar{V}_{sd}& =& V_{sd}[(x_{d}'+x_{t})(x_{q}+x_{t})] \nonumber\\ &&+ r_{t}(e_{q}'V_{sq})(x_{q}+x_{t})\nonumber\\ i_{tq}& =& \frac{\bar{V}_{sd}}{(x_{q}+x_{t})[r_{t}^{2}+(x_{d}'+x_{t})(x_{q}+x_{t})]}. \label{meq6} \end{eqnarray} $  (6) 
Hence, the terminal voltage and power output of the alternator are given as
$ \begin{eqnarray} V_{t}& =& (V_{d}^{2}+V_{q}^{2})^{\frac{1}{2}} \;=\; ((x_{q}i_{tq})^{2}+(e_{q}'x_{d}'i_{td})^{2})^{\frac{1}{2}} \nonumber\\ P_{e}& =& V_{d}i_{td}+V_{q}i_{tq} \;=\; (e_{q}'i_{tq})+(x_{q}x_{d}')i_{td}i_{tq}. \label{meq7} \end{eqnarray} $  (7) 
By perturbing the above nonlinear equations around a normal operating point and letting
$ \begin{eqnarray} \Delta\dot{\delta}& =& \omega_{0}\Delta \omega \nonumber\\ \Delta\dot{\omega}& =& \frac{1}{2H}[\Delta P_{e}] \nonumber\\ \Delta\dot{e_{q}'}& =& \frac{1}{T_{do}'}[\Delta E_{fd}\Delta e_{q}'(x_{d}x_{d}')\Delta i_{td}] \nonumber\\ \Delta\dot{E_{fd}}& =& \frac{K_{A}}{T_{A}}\Delta V_{t}\frac{1}{T_{E}}\Delta E_{fd} \label{meq8} \end{eqnarray} $  (8) 
along with the generator output current
$ \begin{eqnarray} \Delta i_{t}=\Delta i_{td}+j\Delta i_{tq}. \label{meq9} \end{eqnarray} $  (9) 
The firstorder difference of
$ \begin{eqnarray} \Delta i_{td}=\frac{r_{t}}{r_{t}^{2}+x_{1}x_{2}}\Delta V_{sd}+\frac{x_{2}}{r_{t}^{2}+x_{1}x_{2}}(\Delta e_{q}'\Delta V_{sq}). \label{eq9} \end{eqnarray} $  (10) 
Similarly obtaining the change in the
$ \begin{eqnarray} \Delta i_{tq} =\frac{\Delta V_{sd}}{x_{1}} + \frac{r_{t}}{x_{d}'+x_{t}}(\Delta e_{q}'\Delta V_{sq}). \label{meq10} \end{eqnarray} $  (11) 
Further algebraic manipulation yields
$ \begin{eqnarray} \Delta V_{t}& =& \frac{V_{do}}{V_{to}}\Delta V_{d}+\frac{V_{qo}}{V_{to}}\Delta V_{q} \nonumber\\ & =& \frac{V_{do}}{V_{to}}(x_{q}\Delta i_{tq})+\frac{V_{qo}}{V_{to}}(\Delta e_{q}'x_{d}'\Delta i_{td}) \label{meq11} \end{eqnarray} $  (12) 
$ \begin{eqnarray} \Delta P_{e}& =& e_{qo}'\Delta i_{tq}+i_{tqo}\Delta e_{q}' \nonumber\\ &&+ (x_{q}x_{d}')[i_{tdo}\Delta i_{tq}+i_{tqo}\Delta i_{td}]. \end{eqnarray} $  (13) 
A mathematical expression describing the ⅠⅤ characteristics of a solar cell has been studied extensively. An equivalent model of a solar cell shown in Fig. 2 includes a photo diode, a shunt resistor depicting leakage current and a series resistor representing an internal resistance to current flow. The difference between photovoltaic current
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Fig. 2 Model of PV cell. 
$ \begin{eqnarray} I_{pv}=I_{ph}I_{D}\frac{V_{pv}+I_{pv}R_{s}}{R_{sh}}. \end{eqnarray} $  (14) 
The diode current
$ \begin{eqnarray} I_{D}=I_{s}(e^{\frac{(V_{pv}+R_{s}I_{pv})}{nV_{T}}}1) \end{eqnarray} $  (15) 
where
The solar irradiation and the working temperature of the cell determines the photo current. At given cell temperature
$ \begin{eqnarray} I_{ph}=[I_{sc}+a(TT_{ref})]G \end{eqnarray} $  (16) 
where
$ \begin{eqnarray} I_{s}=I_{sref}\left(\frac{T}{T_{ref}}\right)^{\frac{3}{n}}e^{\frac{qE_{g}}{nk}(\frac{1}{T}\frac{1}{T_{ref}})} \end{eqnarray} $  (17) 
where
$ \begin{eqnarray} I_{pv}=I_{sc}I_{s}[e^{q(\frac{V_{pv}+I_{pv}R_{s}}{nV_{T}})}1]. \end{eqnarray} $  (18) 
Because of the fact that the power generated by a solar cell is low, multiple solar cells are connected in series and parallel to generate power in the range of watts. Thus the characteristic equation of the photovoltaic array consisting of
$ \begin{eqnarray} I_{pv}=N_{p}I_{ph}N_{p}I_{s}[e^{\frac{(\frac{V_{pv}}{N_{s}}+\frac{I_{pv}R_{s}}{N_{p}})}{nV_{T}}}1] \label{eqcurr} \end{eqnarray} $  (19) 
which is a nonlinear current voltage relationship of a PV array. Alternatively, it can be rewritten as
$ \begin{eqnarray} V_{T}=N_{s}[{\text{ln}}(\frac{N_{p}I_{ph}I_{pv}}{N_{p}I_{s}}+1)nV_{T}\frac{I_{pv}R_{s}}{N_{p}}]. \label{eqvolt} \end{eqnarray} $  (20) 
This can be iteratively solved using NewtonRaphson algorithm [28] of the form
$ \begin{eqnarray} x_{n+1}=x_{n}\frac{f(x_{n})}{f'(x_{n})}\xrightarrow[{}]{\text{continues}} \frac{x_{n+1}x_{n}}{x_{n}}\leq E_{s} \end{eqnarray} $  (21) 
where
$ \begin{eqnarray*} && x_n = V_{T, n}, \\ && f(V_T)= V_TN_{s}[nV_{T}*{\text{ln}}(\frac{N_{p}I_{ph}I_{pv}N_{p}I_{s}}{N_{p}I_{s}}) \nonumber\\ &&~~~~~~~~~~~~ + \frac{I_{pv}R}{N_{p}}]. \end{eqnarray*} $ 
Since
$\begin{eqnarray} V_{n+1} & =& N_{s}[nV_{T}\times {\text {ln}}(\frac{N_{p}I_{ph}I_{pv}N_{p}I_{s}}{N_{p}I_{s}})\nonumber\\ &&+ \frac{I_{pv}R}{N_{p}}]. \label{eqvoltRec} \end{eqnarray} $  (22) 
The power conditioning unit (PCU) consists of devices needed to connect the PV array to the microgrid [29]. The significant components of the PCU are
1) DC/DC Converter Model
The primary function of the DC/DC converter is to either increase or decrease the DC output voltage. Out of the various topologies of DC/DC converters, buck and boost converters are the more fundamental ones. Because the PV output voltage has be to stepped up, a boost converter is used in this case. A typical converter configuration is shown in Fig. 3. The boost converter steps up the DC voltage level. It consists of an inductor, a diode and a power electronic switch.
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Fig. 3 DC/DC converter configuration. 
The dynamics of the converter can be expressed as
$ \begin{eqnarray} && V_{pv}={L}_{dc}\dot{I}_{pv}+(1d_{c})V_{dcp}\nonumber\\ &&~~~~~~~ \longrightarrow \;\; \dot{I}_{pv}=\frac{1}{L_{dc}}(V_{pv}(1d_{c})V_{dcp}) \label{eqcurrpv} \end{eqnarray} $  (23) 
where
2) DC Link Capacitor Model
The DC link capacitor functions as an energy storage and filter for the DC voltage.
By applying Kirchhoff current law (KCL) at the DClink node, the dynamics of the DC link capacitor can be obtained as
$ \begin{eqnarray} \frac{dV_{dcp}}{dt}=\frac{1}{C_{dc}}(I_{dc1}I_{dc2}) \label{eqvoltRate} \end{eqnarray} $  (24) 
where
3) Inverter Model
The inverter is responsible for the conversion of the PV array DC output and giving it to the grid at an appropriate frequency. A voltage gain model of a voltage source inverter (VSI) operating in PWM mode is considered as shown in Fig. 4.
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Fig. 4 Inverter model. 
The power on DC side of the inverter is given as
$ \begin{eqnarray} P_{dc}=V_{dcp}I_{dc2}. \end{eqnarray} $  (25) 
The instantaneous active power on the AC side of the inverter is expressed as
$ \begin{eqnarray} P_{ac}={\text{Re}}[V_{p}I_{pf}^{*}] \end{eqnarray} $  (26) 
where
$ \begin{eqnarray*} V_{p}& =& V_{pd}+jV_{pq}, \;\;\;\;I_{pf}=I_{pfd}+jI_{pfq}. \label{eqIVdq} \end{eqnarray*} $ 
Little algebra yields the power relations
$ \begin{eqnarray} P_{ac}& =& V_{pd}I_{pfd}+V_{pq}I_{pfq}\nonumber\\ P_{dc}& =& V_{pd}I_{pfd}+V_{pq}I_{pfq}. \label{eqPdcPac} \end{eqnarray} $  (27) 
During its operation in PWM mode and referring to Fig. 4, the output voltage of the inverter can be written as
$ \begin{eqnarray} V_{p}=m_{p}+V_{dcp}\psi_{p} \end{eqnarray} $  (28) 
where
$ \begin{eqnarray} V_{pd}& =& m_{p}\times V_{dcp}\times {\text{cos}}(\psi_{p}+\theta)\nonumber\\ V_{pq}& =& m_{p}\times V_{dcp}\times {\text{sin}}(\psi_{p}+\theta). \end{eqnarray} $  (29) 
Finally, we obtain the expression for
$ \begin{eqnarray} I_{dc2}& =& (I_{pfd}m_{p}{\text{cos}}(\psi_{p}+\theta) \nonumber\\ &&+ I_{pfq}m_{p}{\text{sin}}(\psi_{p}+\theta)). \end{eqnarray} $  (30) 
4) LC Filter and Coupling Inductance Model
The purpose of using a low pass filter is attenuation of switching frequency ripple of the output voltage of an inverter. The filter is a T section of an RL circuit shunted by a capacitor. While the inductor blocks high frequency harmonics, the capacitor stops low frequency harmonics. Collectively, they block most of the harmonics, thereby reducing ripples from going through the system [30]. By applying KCL around the PV inverter and filter capacitor, we obtain a nonlinear relation as
$ \begin{eqnarray} V_{p}& =& I_{pf}R_{pf}+L_{pf}\frac{dI_{pf}}{dt}+V_{cp}\nonumber\\ &&+ (I_{pf}I_{p})R_{pdr} \label{eqVp} \end{eqnarray} $  (31) 
where
$ \begin{eqnarray} \frac{dI_{pfd}}{dt}& =& \frac{\omega_{0}R_{pf}}{L_{pf}}I_{pfd}+\omega_{0}\omega I_{pfq}\nonumber\\ &&+ \frac{\omega_{0}m_{p}V_{dcp}{\text{cos}}(\psi_{p}+\theta)}{L_{pf}} \nonumber\\ && \frac{\omega_{0}V_{cpd}}{L_{pf}}\omega_{0}R_{pdr}I_{pcd} \end{eqnarray} $  (32) 
$ \begin{eqnarray} \frac{dI_{pfq}}{dt}& =& \frac{\omega_{0}R_{pf}}{L_{pf}}I_{pfq}\omega_{0}\omega I_{pfd}\nonumber\\ &&+ \frac{\omega_{0}m_{p}V_{dcp}{\text{sin}}(\psi_{p}+\theta)}{L_{pf}} \frac{\omega_{0}V_{cpq}}{L_{pf}}\nonumber\\ && \omega_{0}R_{pdr}I_{pcq}. \end{eqnarray} $  (33) 
By coupling the transmission line between microgrid and PV filter capacitor, we obtain a nonlinear equation as
$ \begin{eqnarray} V_{cp}& =& I_{p}R_{p}+L_{p}\frac{dI_{p}}{dt}+V_{s}\nonumber\\ && (I_{pf}I_{p})R_{pdr} \end{eqnarray} $  (34) 
Rewriting in
$ \begin{eqnarray} \frac{dI_{pd}}{dt}& =& \frac{\omega_{0}R_{p}}{L_{p}}I_{pd}+\omega_{0}\omega I_{pq} \nonumber\\ &&+ \frac{\omega_{0}}{L_{p}}(V_{cpd}V_{sd})+\omega_{0}R_{pdr}I_{pcd} \end{eqnarray} $  (35) 
$ \begin{eqnarray} \frac{dI_{pq}}{dt}& =& \frac{\omega_{0}R_{p}}{L_{p}}I_{pq}+\omega_{0}\omega I_{pd} \nonumber\\ &&+ \frac{\omega_{0}}{L_{p}}(V_{cpq}V_{sq})+\omega_{0}R_{pdr}I_{pcq} \end{eqnarray} $  (36) 
where
$ \begin{eqnarray} C_{pf}\frac{dV_{cp}}{dt}=(I_{pf}I_{p}) \end{eqnarray} $  (37) 
where
$ \begin{eqnarray} \frac{dV_{cpd}}{dt}& =& \frac{1}{C_{pf}}(I_{pfd}I_{pd})+\omega_{0}\omega V_{cpq}\nonumber\\ \frac{dV_{cpq}}{dt}& =& \frac{1}{C_{pf}}(I_{pfq}I_{pq})+\omega_{0}\omega V_{cpd}. \end{eqnarray} $  (38) 
5) Linearized Model of the PV System
A linearized model of the PV system includes a small signal model of characteristic equation of the PV array and power conditioning unit. From
$ \begin{eqnarray*} V_{pv} & =& N_{s}[{\text{ln}}(\frac{N_{p}I_{ph}I_{pv}}{N_{p}I_{s}}+1)nV_{T}\nonumber\\ && \frac{I_{pv}R_{s}}{N_{p}}] \end{eqnarray*} $ 
Making firstorder changes of (20) with
$ \begin{eqnarray} \Delta V_{pv}& =& K_{pv}\Delta I_{pv}\nonumber\\ K_{pv}& =& N_{s}[\frac{nV_{T}}{N_{p}I_{ph}I_{pvo}+N_{p}I_{s}}+ \frac{R_{s}}{N_{p}}]. \end{eqnarray} $  (39) 
By linearization each of the components of the power conditioning unit, a small signal model can be readily obtained. The linearized state equations are given as [29]
$ \begin{eqnarray} \Delta \dot{V}_{dcp}& =& \frac{1}{C_{dcp}}[I_{pfd0}m_{p}{\text{sin}}(\psi_{p}+\theta)\Delta \psi_{p} \nonumber\\[1mm] &&+ m_{p}{\text{cos}}(\psi_{p}+\theta)\Delta I_{pfd}\nonumber\\[1mm] &&+ I_{pfd0}{\text{cos}}(\psi_{p}+\theta)\Delta m_{p} \nonumber\\[1mm] &&+ I_{pfq0}m_{p}{\text{cos}}(\psi_{p}+\theta)\Delta \psi_{p}+m_{p}{\text{sin}}(\psi_{p}\nonumber\\[1mm] &&+ \theta)\Delta I_{pfq} + I_{pfq0}{\text{sin}}(\psi_{p}+\theta)\Delta m_{p}\nonumber\\[1mm] && (1d_{c})\Delta I_{pv}]\\[2mm] \end{eqnarray} $  (40) 
$ \begin{eqnarray} \Delta \dot{I}_{pfd}& =& \frac{\omega_{0}R_{pf}}{L_{pf}}\Delta I_{pfd}\nonumber\\[1mm] &&+ \omega_{0}(\Delta I_{pfq}+I_{pfq0}\Delta \omega) \nonumber\\[1mm] &&+ \frac{\omega_{0}}{L_{pf}}[m_{p}{\text{cos}}(\psi_{p}+\theta)\Delta V_{dcp}\nonumber \\[1mm] && m_{p}V_{dcp0}{\text{sin}}(\psi_{p}+\theta)\Delta \psi_{p} \nonumber\\[1mm] &&+ V_{dcp0}{\text{cos}}(\psi_{p}+\theta)\Delta m_{p}] \frac{\omega_{0}\Delta V_{cpd}}{L_{pf}} \nonumber \\[1mm] &&\frac{\omega_{0}R_{pdr}}{L_{pf}}(\Delta I_{pfd}\Delta I_{pq})\\[2mm] \end{eqnarray} $  (41) 
$ \begin{eqnarray} \Delta \dot{I}_{pfq}& =& \frac{\omega_{0}R_{pf}}{L_{pf}}\Delta I_{pfq}\nonumber\\[1mm] && \omega_{0}(\Delta I_{pfd}+I_{pfd0}\Delta \omega) \nonumber\\[1mm] &&+ \frac{\omega_{0}}{L_{pf}}[m_{p}{\text{sin}}(\psi_{p}+\theta)\Delta V_{dcp} \nonumber \\[1mm] &&+ m_{p}V_{dcp0}{\text{cos}}(\psi_{p}+\theta)\Delta \psi_{p} \nonumber\\[1mm] &&+ V_{dcp0}{\text{sin}}(\psi_{p}+\theta)\Delta m_{p}] \frac{\omega_{0}\Delta V_{cpq}}{L_{pf}}\nonumber \\[1mm] && \frac{\omega_{0}R_{pdr}}{L_{pf}}(\Delta I_{pfq}\Delta I_{pq}) \end{eqnarray} $  (42) 
$ \begin{eqnarray} \Delta\dot {I}_{pd}& =& \frac{R_{p}}{L_{p}}\Delta I_{pd}+\omega_{0}(\Delta I_{pq}+ I_{pq0}\Delta \omega)\nonumber\\ &&+\frac{1}{L_{p}}(\Delta V_{cpd}\Delta V_{sd}) \nonumber\\ &&+ \frac{\omega_{0}R_{pdr}}{L_{p}}(\Delta I_{pfd}\Delta I_{pd}) \label{eqdIpd} \end{eqnarray} $  (43) 
$ \begin{eqnarray} \Delta\dot {I}_{pq}& =& \frac{R_{p}}{L_{p}}\Delta I_{pq}+ \omega_{0}(\Delta I_{pd}+I_{pd0}\Delta \omega)\nonumber\\ &&+\frac{1}{L_{p}}(\Delta V_{cpq}\Delta V_{sd}) \nonumber\\ &&+ \frac{\omega_{0}R_{pdr}}{L_{p}}(\Delta I_{pfq}\Delta I_{pq}). \label{eqdIpq} \end{eqnarray} $  (44) 
The voltage across the filter capacitor
$ \Delta \dot{V}_{cpd} = \omega_{0}(\Delta V_{cpq} + V_{cpq0}\Delta \omega) + \frac{1}{C_{pf}}(\Delta I_{pfd}  \Delta I_{pd}) \label{eqVd} $  (45) 
$ \begin{eqnarray} \Delta \dot{V}_{cpq} = \omega_{0}(\Delta V_{cpd} + V_{cpd0}\Delta \omega)+ \frac{1}{C_{pf}}(\Delta I_{pfq}  \Delta I_{pq}). \label{eqVq} \end{eqnarray} $  (46) 
In the sequel, we consider a system where the microalternator and the PV system are the supply sources for a combined system. This eventually means that we have a microgrid with two distributed generation units. The modeling proceeds done in two parts. Initially both subsystems are modeled individually as presented in the earlier sections. An integrated modeling approach is presented here in this section from a novel view.
The microalternatorPV system is modeled to represent a system of systems, where two different subsystems are operating independently to achieve a common goal. There exists an integration between the two subsystems which is clearly evident from 5. It shows that both systems are connected through a common bus having voltage
In the end, the combined model consisting of both systems is considered for the networked control system (NCS) design implementation. For this reason, the state matrices shown are for the overall system. However, the
Fig. 5 shows a microgrid system with a microalternator and PV generator along with a load connected to the main grid. All of these are connected through a common bus having voltage
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Fig. 5 Combined system. 
$ \begin{eqnarray*} I_{t}+I_{p}=I_{b}+I_{l} \end{eqnarray*} $ 
where
$ \begin{eqnarray} I_{td}+I_{pd} & =& I_{bd}+I_{ld} \end{eqnarray} $  (47) 
$ \begin{eqnarray} I_{tq}+I_{pq} & =& I_{bq}+I_{lq}. \label{eqIq} \end{eqnarray} $  (48) 
Further, we express the nonstate currents
Load Current:
At the microgrid, the load is modeled as admittance
$ \begin{eqnarray*} I_{ld}+jI_{lq}=(V_{sd}+jV_{sq})(gjb). \end{eqnarray*} $ 
Equating real and imaginary parts, we get
$ \begin{eqnarray} I_{ld}=gV_{sd}+bV_{sq}, \;\; I_{lq}=gV_{sq}bV_{sd}. \end{eqnarray} $  (49) 
Grid Current:
The main grid current
$ \begin{eqnarray*} I_{b}& =& \frac{V_{s}V_{b}}{r_{b}+jx_{b}} \\ I_{bd}+jI_{bq}& =& \frac{V_{sd}+jV_{sq}(V_{b}{\text{sin}}\delta + jV_{b}{\text{cos}}\delta)}{r_{b}+jx_{b}}. \end{eqnarray*} $ 
Equating real and imaginary parts
$ \begin{eqnarray} I_{bd}& =& \frac{(V_{sq}V_{b}{\text{sin}}\delta)r_{b} + (V_{sq}V_{b}{\text{cos}}\delta)x_{b}}{r_{b}^{2}+x_{b}^{2}} \end{eqnarray} $  (50) 
$ \begin{eqnarray} I_{bq}& =& \frac{(V_{sq}V_{b}{\text{cos}}\delta)r_{b}+ (V_{sq}V_{b}{\text{sin}}\delta)x_{b}}{r_{b}^{2}+x_{b}^{2}}. \end{eqnarray} $  (51) 
Standard manipulations yield the bus voltage components
$ \begin{eqnarray} V_{sd}& =& \frac{1}{\Theta_{1}}[z_{b}z_{1}I_{pd}+z_{b}x_{2}e_{q}' \nonumber\\[3mm] &&+ V_{b}z_{1}(r_{b}{\text{sin}}\delta+x_{b}{\text{cos}}\delta)\Theta_{2}V_{sq}] \label{eqVsd} \end{eqnarray} $  (52) 
$ \begin{eqnarray} V_{sq}& =& \Theta_{4}I_{pd}+\Theta_{5}e_{q}'+ \Theta_{6}I_{pq}+\Theta_{7}V_{b} \end{eqnarray} $  (53) 
where
$ \begin{eqnarray*} ~~~\Theta_{1}& =& [gz_{b}z_{1}+z_{1}r_{b}+r_{t}z_{b}]\nonumber\\ ~~~\Theta_{2}& =& [bz_{b}z_{1}+x_{b}z_{1}+x_{2}z_{b}] \\ ~~~\Theta_{3}& =& [x_{b}x_{2}z_{1}+bz_{b}x_{2}z_{1}+z_{1}z_{b}r_{t}^{2}z_{b}]\nonumber\\ ~~~\Theta_n & =& \Theta_{2}\Theta_{3}+\Theta_{1}^{2}x_{2} \\ ~~~\Theta_{4}& =& \frac{1}{\Theta_n}\Theta_{3}z_{b}z_{1}, \nonumber\\ ~~~\Theta_{5}& =& \frac{1}{\Theta_n}(\Theta_{3}z_{b}x_{2}+z_{b}x_{2}r_{t}\Theta_{1}) \\ ~~~\Theta_{6}& =& \frac{1}{\Theta_n}(\Theta_{1}x_{2}z_{1}z_{b})\nonumber\\ ~~~\Theta_{7}& =& \frac{1}{\Theta_n}[z_{1}\Theta_{3}(r_{b}{\text{sin}}\delta +x_{b}{\text{cos}}\delta) \nonumber\\ &&+\Theta_{1}x_{2}z_{1}(r_{b}{\text{cos}}\delta x_{b}{\text{sin}}\delta)]. \end{eqnarray*} $ 
Linearized Model of the Combined System:
A linearized model of the combined system is obtained by expressing the linearized microgrid voltage components
$ \begin{eqnarray} \Delta V_{sq} = \Theta_{4} \Delta I_{pd}+\Theta_{5} \Delta e_{q}'+ \Theta_{6} \Delta I_{pq}+\Theta_{0}\Delta \delta \end{eqnarray} $  (54) 
where
$ \begin{eqnarray*} \Theta_{0}& =& V_{b}[\frac{1}{Den}(z_{1}\Theta_{3}(r_{b}{\text{cos}}\delta_{0}x_{b}{\text{sin}}\delta_{0}) \nonumber\\ && \Theta_{1}x_{2}z_{1}(r_{b}{\text{sin}}\delta_{0}+x_{b}{\text{cos}}\delta_{0}))]. \end{eqnarray*} $ 
Similarly, we also obtain
$ \begin{eqnarray} \Delta V_{sd}& =& \frac{1}{\Theta_{1}}[z_{b}z_{1}\Delta I_{pd}+z_{b}x_{2}\Delta e_{q}' \nonumber\\ && + V_{b} z_{1}(r_{b}{\text{cos}}\delta_{0}x_{b}{\text{sin}}\delta_{0})\Delta \delta \nonumber\\ && \Theta_{2}(\Theta_{4} \Delta I_{pd} + \Theta_{5} \Delta e_{q}' + \Theta_{6} \Delta I_{pq} + \Theta_{0} \Delta \delta)]. \end{eqnarray} $  (55) 
Finally, the values of
$ \begin{eqnarray*} A_{3, 1}& =& \frac{(x_{d}x_{d}')i_{td1}}{T_{do}'}, \;\; A_{3, 3}=\frac{(x_{d}x_{d}')i_{td3}}{T_{do}'} \\ A_{9, 9}& =& k_{p}(R_{eq1}+\Theta_{11})\\ A_{9, 10}& =& \omega_{0}k_{p}(\frac{\Theta_{2}\Theta_{6}}{\Theta_{1}}) \\ A_{10, 9}& =& \omega_{0}k_{p}\Theta_{4}, \;\;A_{10, 10}=k_{p}(R_{eq1}+\Theta_{6})\\ \Theta_{11}& =& (\frac{z_{b}z_{1}}{\Theta_{1}}\frac{\Theta_{2}\Theta_{4}}{\Theta_{1}}). \end{eqnarray*} $ 
Consider the microgrid with two sets of microalternator and PV system representing a SoSNCS with random communication delays, where the sensor is clock driven and the controller and the actuator are event driven. The discretetime linear timeinvariant plant model is as follows:
$ \begin{eqnarray} x_p(k+1) = A x_p + B u_p, \;\;y_p = C x_p \label{neq1} \end{eqnarray} $  (56) 
$ A=\\ \left[ \begin{array}{cccccccccccc} 0 & \omega_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{P_{e1}}{2H} & 0 & \frac{P_{e3}}{2H} & 0 & 0 & 0 & P_{e1} & P_{e2} & 0 & 0 & 0 & 0 \\ A_{3,1} & 0 & A_{3,3} & \frac{1}{T_{do}'}& 0 & 0 & e_{q1} & e_{q2} & 0 & 0 & 0 & 0 \\ \frac{K_{A}}{T_{A}}V_{t1} & 0 & \frac{K_{A}}{T_{A}}V_{t3} & \frac{1}{T_{A}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \omega_{0}I_{pfq0} & 0 & 0 & \frac{k_{pv}}{L_{dc}} & \frac{(d_{c}1)}{L_{dc}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \omega_{0}I_{pfd0} & 0 & 0 & \frac{(1d_{c})}{C_{dcp}} & 0 & m_{p}k_{pd1} & m_{p}k_{pd2} & 0 & 0 & 0 & 0 \\ k_{p}D_{1} & \omega_{0}I_{pd0} & k_{p}C_{1} & 0 & k_{pf} & 0 & k_{pf}R_{eq} & \omega_{0} & k_{pf}R_{d} & 0 & k_{pf} & 0\\ k_{p}G & \omega_{0}I_{pq0}& k_{p}C_{1} & 0 & 0 & 0 & \omega_{0} & k_{pf}R_{eq} & 0 & k_{pf}R_{d} & 0 & k_{pf}\\ 0 & \omega_{0}V_{cpq0} & 0 & 0 & 0 & 0 & k_{p}R_{d} & 0 & A_{9,9} & A_{9,10} & k_{p} & 0\\ 0 & \omega_{0}V_{cpq0} & 0 & 0 & 0 & 0 & 0 & k_{p}R_{d} & A_{10,9} & A_{10,10} & 0 & k_{p}\\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{\omega_{0}}{C_{pf}} & 0 & \frac{\omega_{0}}{C_{pf}} & 0 & 0 & \omega_{0}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\omega_{0}}{C_{pf}} & 0 & \frac{\omega_{0}}{C_{pf}} & \omega_{0} & 0\\ \end{array} \right]$ 
$ B=\left[ \begin{array}{cc} 0&0\\ 0&0\\ 0&0\\ 0&0\\ k_{pf}V_{dcp0}{\text{cos}}(\psi_{p})&k_{pf}V_{dcp0}m_{p}{\text{sin}}(\psi_{p})\\ k_{pf}V_{dcp0}{\text{sin}}(\psi_{p})&k_{pf}V_{dcp0}m_{p}{\text{cos}}(\psi_{p})\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ \frac{(I_{pfd0}{\text{cos}}(\psi_{P})+I_{pfq0}{\text{sin}}(\psi_{P}))}{C_{dcp}}&\frac{m_{p}(I_{pfd0}{\text{sin}}(\psi_{P})+I_{pfq0}{\text{cos}}(\psi_{P}))}{C_{dcp}}\\ \end{array} \right] $ 
where
Remark 1: We note that the developed renewable energy model is formulated in linearized continuoustime form as it emerges from physical considerations. For all practical purposes, we seek a discretetime system of the type of (56). As will be mentioned in the simulation results section, the microalternatorPV system containing
For a more general case, we assume that the measurement after passing through the network exhibits a randomly varying communication delay and is described by [26]
$ \begin{eqnarray} y_c(k) = \left \{ \begin{array}{lll} & y_p(k  \tau_k^m), &\delta(k) = 1 \\ & y_p(k), &\delta(k) = 0 \end{array} \right. \label{neq2} \end{eqnarray} $  (57) 
where
$ Prob \{ \delta(k) = 1 \} = p_k $ 
where
Class 1:
where
1) If there is no information about the likelihood of different values, we use the uniform discrete distribution,
2) If it is suspected that
ⅰ) If
ⅱ) If
3) If it is suspected that
4) If it is suspected that
Class 2:
$ \begin{eqnarray*} Prob(p_k & =& \frac{(ax+b)}{n}) \\ && =\left ( \begin{array}{l} n \\ x \end{array} \right ) q^x (1q)^{nx}, \;\;b>0\\ & x& = 0, 1, 2, \ldots, n, \;\;an+b<n \end{eqnarray*} $ 
Remark 2: It is significant to note that the case
When the full state information is not available and the time delay occurs on the actuation side, it is desirable to design the following observerbased controller [26]:
$ \begin{eqnarray} {\text{Observer:}}&& \nonumber\\ \hat{x}(k+1) & = &A \hat{x} + B u_p(k) + L(y_c(k)  \hat{y}_c(k)) \nonumber\\ \hat{y}_c(k)& = &\left \{ \begin{array}{lll}&C \hat{x}(k), \; &\delta(k) = 0 \\&C \hat{x}(k  \tau^m_k), \; &\delta(k) = 1 \end{array} \right. \end{eqnarray} $  (58) 
$ \begin{eqnarray} {\text{Controller}}:&&\nonumber\\ u_c(k)& = &K \hat{x}(k) \nonumber\\ u_p& = &\left\{ \begin{array}{lll}&u_c(k), \; &\alpha(k) = 0 \\ &u_c(k  \tau^a_k), \; &\alpha(k) =1 \end{array} \right. \label{neq5} \end{eqnarray} $  (59) 
where
$ Prob \{ \alpha(k) = 1 \} = s_k $ 
where
In this paper, we assume that
$ \begin{eqnarray} {\tau}^{}_m \leq \tau^m_k \leq {\tau}^{+}_m, \;\; {\tau}^{}_a \leq \tau^a_k \leq {\tau}^{+}_a. \label{neq7} \end{eqnarray} $  (60) 
Define the estimation error by
$ \begin{eqnarray} &&x_p (k+1) \nonumber\\ &&= \left\{ \begin{array}{lll} & A x_p(k) + B K x_p(k  \tau^\alpha_k)  B K e(k  \tau^\alpha_k),& \alpha(k) = 1 \\ & (A + BK)x_p(k)  B K e(k), & \alpha(k) = 0 \end{array} \right. \end{eqnarray} $  (61) 
$ \begin{eqnarray} &&e(k+1) = x_p(k+1)  \hat{x}(k+1)\nonumber\\ &&= \left \{ \begin{array}{lll} & A e(k)  L C e(k\tau^m_k), &\delta(k) = 1 \\ & (A LC)e(k), &\delta(k) = 0. \end{array} \right. \label{neq10} \end{eqnarray} $  (62) 
In terms of
$ \begin{eqnarray} \xi(k+1) & =& {A}_j \xi(k) + {B}_j \xi(k  \tau^m_k) \nonumber\\ &+& {C}_j \xi(k  \tau^a_k)\label{neq11} \end{eqnarray} $  (63) 
where
$ \begin{eqnarray} {A}_1 & =& \left[ \begin{array}{cc} A&0 \\ 0&A \end{array} \right], \; {A}_2 = \left[ \begin{array}{cc} A+BK&BK \\ 0&A \end{array} \right]\nonumber\\ {A}_3 & =& \left[ \begin{array}{cc} A+BK&BK \\ 0&ALC \end{array} \right]\nonumber\\ {A}_4 & =& \left[ \begin{array}{cc} A&0 \\ 0&ALC \end{array} \right]\nonumber\\ {B}_1 & =& \left[ \begin{array}{cc} BK&BK \\ 0&0 \end{array} \right], \; {B}_2 = \left[ \begin{array}{cc} 0&0 \\ 0&0 \end{array} \right]\nonumber\\ {B}_3 & =& \left[ \begin{array}{cc} 0&0 \\ 0&0 \end{array} \right], \; {B}_4 = \left[ \begin{array}{cc} BK&BK \\ 0&0 \end{array} \right]\nonumber\\ {C}_1 & =& \left[ \begin{array}{cc} 0&0 \\ 0&LC \end{array} \right], \; {C}_2 = \left[ \begin{array}{cc} 0&0 \\ 0&LC \end{array} \right]\nonumber\\ {C}_3 & =& \left[ \begin{array}{cc} 0 & 0 \\ 0&0 \end{array} \right], \; {C}_4 = \left[ \begin{array}{cc} 0 & 0 \\ 0&0 \end{array} \right]. \label{neq11A} \end{eqnarray} $  (64) 
Remark 3: It is remarked for simulation processing that we can express (61) and (62) in the form
$ \begin{eqnarray} x_p (k+1) & =& s_k [A x_p(k) + B K x_p(k\tau^\alpha_k) \nonumber\\ && B K e(k\tau^\alpha_k)] \nonumber\\ &&+ (1s_k) [(A+BK)x_p(k) \nonumber\\ && B K e(k)] \label{neq9Z}\\ \end{eqnarray} $  (65) 
$ \begin{eqnarray} e(k+1) & =& p_k [A e(k)  L C e(k\tau^m_k)] \nonumber\\ &&+ (1p_k) [(A LC)e(k)] \label{neq10Z} \end{eqnarray} $  (66) 
where the values of the random variables
Remark 4: It is important to note from (64) that
$ \begin{eqnarray} {A}_j + {B}_j + {C}_j = \left[ \begin{array}{cc} A + BK&BK \\ 0&A  LC \end{array} \right], \quad j = 1, \ldots, 4. \label{neq11AZY} \end{eqnarray} $  (67) 
The interpretation of this result is that
The aim of this paper is to design an observer based feedback stabilizing controller in the form of (58) and (59) such that the closed loop system (63) is exponentially stable in the mean square sense. Our approach is based on the concepts of switched timedelay systems [31]. For simplicity in exposition, we introduce
$ \begin{eqnarray} \sigma_1(k)& = &Prob\{\delta(k) = 1, \;\alpha(k) =1\}, \;\;\hat{\sigma}_1 = { I E} [\sigma_1]\nonumber\\ \sigma_2(k)& = &Prob\{\delta(k) = 1, \;\alpha(k) =0\}, \;\;\hat{\sigma}_2 = { I E} [\sigma_2]\nonumber\\ \sigma_3(k)& = &Prob\{\delta(k) = 0, \alpha(k) =0\}, \;\;\;\hat{\sigma}_3 = { I E} [\sigma_3] \nonumber\\ \sigma_4(k)& = &Prob\{\delta(k) = 0, \;\alpha(k) =1\}, \;\;\hat{\sigma}_4 = { I E} [\sigma_4] \label{eq10ZXZ} \end{eqnarray} $  (68) 
where
$ \begin{eqnarray} \hat{\sigma}_1& = &{ I E} [p_k] { I E} [s_k], \;\;\hat{\sigma}_2 = { I E} [p_k] { I E} [1s_k] \nonumber\\ \hat{\sigma}_3& = &{ I E} [1 p_k] { I E} [1s_k], \hat{\sigma}_4 = { I E} [1  p_k] { I E} [s_k]. \label{eq10ZXZRR} \end{eqnarray} $  (69) 
In this section, we will thoroughly investigate the stability and controller synthesis problems for the closedloop system (63). First, let us deal with the stability analysis problem and derive a sufficient condition under which the closedloop system (63) with the given controller (58) and (59) is exponentially stable in the mean square sense. Extending on the basis of [32], the following Lyapunov function candidate is constructed to establish the main theorem
$ \begin{eqnarray} &&V(\xi(k))= \sum\limits_{i=1}^{5} V_i(\xi(k)) \label{eq12} \end{eqnarray} $  (70) 
$ \begin{eqnarray} &&V_1(\xi(k))= \sum\limits_{j=1}^{4} \xi^T(k) P \xi(k), \;P > 0 \nonumber\\ &&V_2(\xi(k))= \sum\limits_{j=1}^{4} \sum\limits_{i = k \tau_k^m}^{k1} \xi^T(i) Q_j \xi(i), \;Q_j = Q^T_j > 0\nonumber\\ &&V_3(\xi(k))= \sum\limits_{j=1}^{4} \sum\limits_{i = k \tau^a_k}^{k1} \xi^T(i) Q_j \xi(i) \label{eq12A}\nonumber\\ &&V_4(\xi(k))= \sum\limits_{j=1}^{4} \sum\limits_{\ell={\tau}^{+}_m + 2}^{{\tau}^{}_m + 1} \sum\limits_{i = k+\ell1}^{k1} \xi^T(i) Q_j \xi(i) \nonumber\\ &&V_5(\xi(k))= \sum\limits_{j=1}^{4} \sum\limits_{\ell={\tau}^{+}_a + 2}^{{\tau}^{}_a + 1} \sum\limits_{i = k+\ell1}^{k1} \xi^T(i) Q_j \xi(i) \end{eqnarray} $  (71) 
It is not difficult to show that there exist real scalars
$ \begin{eqnarray} \mu \Vert \xi \Vert^2 \leq V (\xi(k)) \leq \upsilon \Vert \xi(k) \Vert^2. \label{eq13} \end{eqnarray} $  (72) 
Remark 5: By carefully considering Remark 4 in view of model (12), it is justified to select matrix
We now present the analysis result for system (12) to be exponentially stable.
Theorem 1: Let the controller and observer gain matrices
$ \begin{eqnarray} &&\Lambda_j = \left[ \begin{array}{cc} \Lambda_{1j} & \Lambda_{2j} \\ \cdot& \Lambda_{3j} \end{array} \right] \; < \; 0\label {LMI01} \end{eqnarray} $  (73) 
$ \begin{eqnarray} && \Lambda_{1j} = \nonumber\\ && \left[ \begin{array}{ccc} \Psi_j + \Phi_{j1}&R_1 + S^T_1&R_2 + S^T_2 \\ \cdot&S_1 S_1^T  \hat{\sigma}_jQ_j&0 \\ \cdot&\cdot&S_2 S^T_2  \hat{\sigma}_jQ_j \end{array} \right] \nonumber\\ &&\Lambda_{2j} = \left[ \begin{array}{cc} R_1 + M^T_1  \Phi_{j2}&R_2 + M^T_2  \Phi_{j3} \\ S_1 M^T_1&0 \\ 0&S_2 M^T_2 \end{array} \right] \nonumber\\ &&\Lambda_{3j} = \left[ \begin{array}{cc} M_1  M^T_1 + \Phi_{j4}&\Phi_{j5} \\ \cdot&M_2  M^T_2 + \Phi_{j6} \end{array} \right] \nonumber\\ && \label {LMI1} \end{eqnarray} $  (74) 
where
$ \begin{eqnarray*} \Psi_j & =& P + \hat{\sigma}_j ( {\tau}^{+}_m  {\tau}^{}_m + {\tau}^{+}_a  {\tau}^{}_a + 2)Q_j \\ &&+ R_1 + R_1^T + R_2 + R_2^T \\ \Phi_{j1} & =& ({A}_j + {B}_j + { C}_j)^T \hat{\sigma}_j P ({A}_j + { B}_j + { C}_j)\\ \Phi_{j2} & =& ({ A}_j + { B}_j + { C}_j)^T \hat{\sigma}_j P { B}_j \\ \Phi_{j3} & =& ({ A}_j + { B}_j + { C}_j)^T \hat{\sigma}_j P { C}_j, \;\Phi_{j5}={ B}^T_j P { C}_j \\ \Phi_{j4} & =& { B}^T_j \hat{\sigma}_j P { B}_j, \;\Phi_{j6} = { C}^T_j \hat{\sigma}_j P { C}_j. \end{eqnarray*} $ 
Proof: Defining
$ \xi(k  \tau_k^m) = \xi(k)  \sum\limits_{i = k  \tau_k^m}^{k1} y(i) $  (75) 
$ \xi(k  \tau_k^a) = \xi(k)  \sum\limits_{i = k  \tau_k^a}^{k1} y(i). $  (76) 
Then the system (12) can be transformed into
$ \begin{eqnarray} \xi(k+1) = ({ A}_j + { B}_j + { C}_j) \xi(k)  { B}_j \lambda(k)  { C}_j \gamma(k) \label{eq19} \end{eqnarray} $  (77) 
where
$ \begin{eqnarray*} \lambda(k) = \sum\limits_{i = k  \tau^m_k}^{k1} y(i), \;\; \gamma(k) = \sum\limits_{i = k  \tau^a_k}^{k1} y(i). \end{eqnarray*} $ 
Evaluating the difference of
$ \begin{eqnarray} && {I E} [\Delta V_1(\xi(k))] = {I E} [V_1(\xi(k+1))]  V_1 (\xi(k)) \nonumber \\ && ~~~=\sum\limits_{j=1}^{4} \bigg [\xi^T(k) [\Phi_{j1}  P] \xi(k)  2 \xi^T (k) \Phi_{j2} \lambda (k) \nonumber \\ && ~~~2 \xi^T (k) \Phi_{j3} \gamma(k) + \lambda^T (k) \Phi_{j4} \lambda(k) \nonumber \\ && ~~~+2 \lambda^T(k) \Phi_{j5} \gamma(k) + \gamma^T(k) \Phi_{j6} \gamma(k) \bigg ]. \label{eq20} \end{eqnarray} $  (78) 
A straightforward computation gives
$ \begin{eqnarray} && {I E} [\Delta V_2 (\xi(k)] = \sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ \sum\limits_{i = k + 1  \tau^m_{k+1}}^{k} \xi^T(i) Q_j \xi(i) \nonumber\\ && ~~~\sum\limits_{i = k\tau^m_k}^{k1} \xi^T (i) Q_j \xi(i) \nonumber\\ &&~~~ =\sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ \sum\limits_{i = k + 1  \tau^m_{k+1}}^{k} \xi^T(i) Q_j \xi(i) \nonumber\\ && ~~~\xi(k  \tau_k^m) Q_j \xi(k\tau_k^m) + \sum\limits_{i = k +1  \tau^m_{k+1}}^{k1} \xi^T(i) Q_j \xi(i) \nonumber\\ &&~~~ \sum\limits_{i=k+1\tau^m_k}^{k1} \xi(i) Q_j \xi(i) \bigg ]. \label{eq21} \end{eqnarray} $  (79) 
In view of
$ \begin{eqnarray} && \sum\limits_{i = k+1\tau_{k+1}^m}^{k1} \xi^T(i) Q_j \xi(i) = \sum\limits_{i = k+1\tau^m_{k+1}}^{k\tau^m_k} \xi^T(i) Q_j \xi(i) \nonumber\\ &&~~~+\sum\limits_{i=k+1\tau_k^m}^{k1} \xi^T(i) Q_j \xi(i) \nonumber\\ &&~~~\leq\sum\limits_{i=k+1\tau^m_k}^{k1} \xi^T(i) Q_j \xi(i) \nonumber\\ &&~~~ +\sum\limits_{i=k+1{\tau}^{+}_m}^{k \tau^{}_m} \xi^T(i) Q_j \xi(i). \label{eq22} \end{eqnarray} $  (80) 
We readily obtain
$ \begin{eqnarray} && {I E} [\Delta V_2 (\xi(k))] \leq \sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ \xi^T(k) Q_j \xi(k) \nonumber\\ &&~~~\xi^T(k\tau^m_k) Q_j \xi(k\tau_k^m) \nonumber\\ &&~~~+\sum\limits_{i=k+1{\tau}^{+}_m}^{k  {\tau}^{}_m} \xi^T(i) Q_j \xi(i) \bigg ]. \label{eq23} \end{eqnarray} $  (81) 
Following parallel procedure, we get
$ \begin{eqnarray} && {I E} [\Delta V_3 (\xi(k))] \leq \sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ \xi^T(k) Q_j \xi(k) \nonumber\\ &&~~~\xi^T(k\tau_k^a) Q_j \xi(k\tau^a_k) \nonumber\\ &&~~~+\sum\limits_{i=k+1{\tau}^{+}_a}^{k  {\tau}^{}_a} \xi^T(i) Q_j \xi(i) \bigg ]. \label{eq24} \end{eqnarray} $  (82) 
Finally,
$ \begin{eqnarray} && {I E} [\Delta V_4 (\xi(k))] \nonumber\\ &&~~~=\sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ \sum\limits_{\ell = {\tau^{+}}_m + 2}^{ {\tau^{}}_m + 1} [\xi^T(k) Q_j \xi(k) \nonumber\\ &&~~~\xi^T(k+\ell1) Q_j \xi(k+\ell1)] \bigg ] \nonumber\\ &&~~~=\sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ ({\tau^{+}}_m  {\tau^{}}_m) \xi^T (k) Q_j \xi(k) \nonumber\\ &&~~~\sum\limits_{i=k+1 {\tau^{+}}_m}^{k  {\tau^{}}_m} \xi^T(i) Q_j \xi (i) \bigg ] \label{eq25}\\ \end{eqnarray} $  (83) 
$ \begin{eqnarray} && {I E} [\Delta V_5 (\xi(k))] \nonumber\\ &&~~~=\sum\limits_{j=1}^{4} \hat{\sigma}_j \bigg [ ({\tau^{+}}_a  {\tau^{}}_a) \xi^T (k) Q_j \xi(k) \nonumber\\ &&~~~\sum\limits_{i=k+1 {\tau^{+}}_a}^{k  {\tau^{}}_a} \xi^T(i) Q_j \xi (i) \bigg ]. \label{eq26} \end{eqnarray} $  (84) 
It follows from (75) and (76) that:
$ \xi(k)  \xi(k  \tau^m_k)  \lambda(k) = 0 $  (85) 
$ \begin{eqnarray} \xi(k)  \xi(k  \tau^a_k)  \gamma(k) = 0. \end{eqnarray} $  (86) 
Therefore, for any appropriately dimensioned matrices
$ \begin{eqnarray} && 2[\xi^T(k) R_1 + \xi^T(k\tau^m_k) S_1 + \lambda^T (k) M_1] \nonumber\\ &&~~~\times [\xi(k)  \xi(k\tau^m_k)  \tau(k)] = 0 \label{eq29} \end{eqnarray} $  (87) 
$ \begin{eqnarray} && 2[\xi^T(k) R_2 + \xi^T(k\tau^a_k) S_2 + \gamma^T (k) M_2]\nonumber\\ && ~~~\times[\xi(k)  \xi(k\tau^a_k)  \gamma(k)] = 0. \label{eq30} \end{eqnarray} $  (88) 
On combining (78)(88), we reach
$ \begin{eqnarray} && {I E} [\Delta V (\xi(k))] \leq \sum\limits_{j=1}^{4} \bigg [ \xi^T(k)\Psi_j \xi(k) \nonumber\\ &&~~~~~~+\sum\limits_{j=1}^{4} \xi^T(k) (2 R_1 + 2 S^T_1) \xi(k\tau^m_k) \nonumber\\ &&~~~~~~+\xi^T(k) (2 R_2 + 2 S^T_2) \xi(k\tau^a_k) \nonumber\\ &&~~~~~~+\xi^T(k) (2 R_1 + 2 M^T_1  2 \Phi_{j2}) \lambda(k) \nonumber\\ &&~~~~~~+\xi^T(k\tau^m_k) (S_1 S^T_1  \hat{\sigma}_j Q_j) \xi(k\tau^m_k) \nonumber\\ &&~~~~~~+\xi^T(k\tau^m_k) (2 S_1  2 M^T_1) \lambda(k) \nonumber\\ &&~~~~~~+\xi^T(k\tau^a_k) (S_2 S^T_2  \hat{\sigma}_j Q_j) \xi(k\tau^a_k) \nonumber\\ &&~~~~~~+\xi^T(k\tau^a_k) (2 S_2  2 M^T_2) \gamma(k) \nonumber\\ &&~~~~~~+\lambda^T(k) (M_1  M^T_1 + \Phi_{j4}) \lambda(k) \nonumber\\ &&~~~~~~+\gamma^T(k) (M_2  M_2^T + \Phi_{j5}) \gamma(k) \nonumber\\ &&~~~~~~+\lambda^T(k) \Phi_{j6}\gamma(k) \nonumber\\ &&~~~~~~+\xi^T(k) (2 R_2 + 2 M^T_2  2 \Phi_{j3}) \gamma(k)\bigg ] \nonumber\\ &&~~~=\sum\limits_{j=1}^{4} \bigg [ \zeta^T (k) \widetilde{\Lambda}_j \zeta(k) \bigg ] \label{eq31} \end{eqnarray} $  (89) 
where
$ \begin{eqnarray*} \zeta(k) & =& \left[ \begin{array}{cc} \zeta^T_1(k)&\zeta^T_2(k)\end{array} \right]^T \\ \zeta_1(k)& =& \left[ \begin{array}{ccc} \xi^T(k)&\xi^T(k  \tau^m_k)&\xi^T(k  \tau^a_k) \end{array} \right]^T \\ \zeta_2(k)& =& \left[ \begin{array}{cc} \lambda^T(k)&\gamma^T(k) \end{array} \right]^T \end{eqnarray*} $ 
and
$ \begin{eqnarray} && {I E} [V(\xi(k+1))  V(\xi(k))] = \sum\limits_{j=1}^{4} \bigg [ \zeta^T(k) \widetilde{\Lambda}_j \zeta(k) \bigg ] \nonumber \\ &&~~~ \leq \sum\limits_{j=1}^{4} \bigg [  \widetilde{\Lambda}_{\rm min} (\widetilde{\Lambda}_j) \zeta^T (k) \zeta(k) \bigg ] \nonumber \\ &&~~~ <  \sum\limits_{j=1}^{4} \bigg [ \beta_j \zeta^T(k) \zeta(k)\bigg ] \label{eq32} \end{eqnarray} $  (90) 
where
$ \begin{eqnarray*} 0 < \beta_j < {\rm min} \big [ \lambda_{\rm min} (\Lambda_j), {\rm max} \{ \lambda_{\rm max}(P), \; \lambda_{\rm max}(Q_j) \} \big ] \end{eqnarray*} $ 
Inequality (90) implies that
$ \begin{eqnarray*} \xi(k)^2 \leq \frac{\upsilon}{\kappa} \xi(0)^2 (1\phi)^k + \frac{\lambda}{\mu \phi}. \end{eqnarray*} $ 
Therefore, it can be verified that the closedloop system (12) is exponentially stable.
A solution to the problem of the observerbased stabilizing controller design is provided by the following theorem:
Theorem 2: Let the delay bounds
$ \begin{eqnarray} &&\left[ \begin{array}{ccc} \widehat{\Lambda}_{1j} & \widehat{\Lambda}_{2j}&\widehat{\Omega}_j\\ \cdot&\Lambda_{3j} & 0 \\ \cdot&\cdot& \hat{\sigma}_j \widehat{X} \end{array} \right] \; < \; 0 \label {LMI1XX} \end{eqnarray} $  (91) 
$ \begin{eqnarray} \widehat{X} & =& \left[ \begin{array}{cc} X_{1}&X_{2} \\ X_{2}^{T}& X_{2} \end{array} \right]\label {LMI1XZX}\\ \widehat{\Psi}_j & =& \hat{X} + \hat{\sigma}_j({\tau}^{+}_m  {\tau}^{}_m + {\tau}^{+}_a  {\tau}^{}_a + 2)\Xi_j \nonumber\\ && +{\Pi}_1 + {\Pi}^T_1 + {\Pi}_2 + {\Pi}^T_2 \nonumber\\ \widehat{\Lambda}_{2j}& =& \left[ \begin{array}{cc} {\Pi}_1 + {\Gamma}^T_1 &{\Pi}_2 + {\Gamma}^T_2 \\ {\it\Upsilon}_1 {\it\Gamma}^T_1&0 \\ 0&{\it\Upsilon}_2 {\it\Gamma}^T_2 \end{array} \right] \nonumber\\ \widehat{\Lambda}_{3j} & =& \left[ \begin{array}{cc} {\it\Gamma}_1  {\it\Gamma}^T_1 &0 \\ \cdot&{\it\Gamma}_2  {\it\Gamma}^T_2 \end{array} \right] \nonumber\\ \widehat{\Omega}_j & =& \left[ \begin{array}{ccccc} \widehat{\Omega}_{1j}&0 &0&\widehat{\Omega}_{4j}&\widehat{\Omega}_{5j} \end{array} \right] \nonumber\\ \widehat{\Omega}_{4j} & =& \left[ \begin{array}{cc} Y^{T}_1 B^{T}Y^{T}_1 B^{T}Z_{1} &0 \\ 0&0 \end{array} \right], \;\; j=1, 4 \nonumber\\ \widehat{\Omega}_{5j} & =& \left[ \begin{array}{cc} 0 & Y^T_2 \\ 0& Y^T_2 \end{array} \right], \;\; j=1, 2 \nonumber\\ \end{eqnarray} $  (92) 
$\begin{eqnarray} \widehat{\Omega}_{4j} & =& 0, \; j=2, 3, \;\;\widehat{\Omega}_{5j} = 0, \;\; j=3, 4\end{eqnarray} $  (93) 
$ \begin{eqnarray} && \widehat{\Lambda}_{1j} = \left[ \begin{array}{ccc} \widehat{\Psi}_j&{\Pi}_1 + {\Upsilon}^T_1&{\Pi}_2 + {\Upsilon}^T_2 \\ \cdot&{\Upsilon}_1 {\Upsilon}_1^T  \hat{\sigma}_j\Xi_j&0 \\ \cdot&\cdot&{\Upsilon}_2 {\Upsilon}^T_2  \hat{\sigma}_j \Xi_j \end{array} \right] \nonumber\\ && \widehat{\Omega}_{1j} = \left[ \begin{array}{cc} X_{1} A^T + Y^T_1 B^TY^T_1 B^TZ_{1}&X_{2}A^{T}Y_{2}^{T} \\ X_{2} A^T&X_{2} A^T Y^T_2 \end{array} \right] \nonumber\\ && \;\; \forall \; j \;\;\;\;\;\;\;\;\; \label {LMI1Z} \end{eqnarray} $  (94) 
where the gain matrices are given by
$ \begin{eqnarray} K = Y_1 X_{1}^{1}, \;\;L = Y_2 X_{2}^{1} C^†. \label {LMI1ZSOL} \end{eqnarray} $  (95) 
Proof: Define
$ \Omega_j= \left[ \begin{array}{cccc} ({ A}_j + { B}_j + { C}_j)&0& { B}_j& { C}_j \end{array} \right]^T $ 
then matrix inequality (73) can be expressed as
$ \begin{eqnarray} \Lambda_j & =& \widetilde{\Lambda} + \Omega_j P \Omega^T_j < 0 \label {LMI1YY}\\ \widetilde{\Lambda}_j& =& \left[ \begin{array}{cc} \widetilde{\Lambda}_{1j}&\widetilde{\Lambda}_{2j} \\ \cdot&\widetilde{\Lambda}_{3j} \end{array} \right] \; < \; 0\nonumber\\ \widetilde{\Lambda}_{1j} & =& \left[ \begin{array}{ccc} \Psi_j&R_1 + S^T_1&R_2 + S^T_2 \\ \cdot&S_1 S_1^T  Q_j&0 \\ \cdot&\cdot&S_2 S^T_2  Q_j \end{array} \right] \end{eqnarray} $  (96) 
$ \begin{eqnarray} \widetilde{\Lambda}_{2j} & =& \left[ \begin{array}{cc} R_1 + M^T_1&R_2 + M^T_2 \\ S_1 M^T_1&0 \\ 0&S_2 M^T_2 \end{array} \right] \nonumber\\ \widetilde{\Lambda}_{3j} & =& \left[ \begin{array}{cc} M_1  M^T_1&0 \\ \cdot&M_2  M^T_2 \end{array} \right]. \label {LMI1ZZ} \end{eqnarray} $  (97) 
Setting
$ \begin{eqnarray} \left[ \begin{array}{ccc} \widetilde{\Lambda}_{1j} & \widetilde{\Lambda}_{2j}&\Omega_j \\ \cdot & \widetilde{\Lambda}_{3j} &0 \\ \cdot& \cdot&  \widehat{X} \end{array} \right] \; < \; 0 \label {LMI1ZZZ} \end{eqnarray} $  (98) 
Applying the congruence transformation
$ T_j={\text{diag}}\{\widehat{X}, \;\widehat{X}, \;\widehat{X}, \;\widehat{X}, \;\widehat{X}, \;I\} $ 
to matrix inequality in (98) and manipulating using (92) and
$ \begin{eqnarray*} \Xi_j & =& \widehat{X} Q_j \widehat{X}, \;{\it\Pi}_j = \widehat{X} R_j \widehat{X}, \;{\it\Upsilon}_j = \widehat{X} S_j \widehat{X}\\ \Gamma_j & =& \widehat{X} M_j \widehat{X}, \;Z_{1}=X_{1}^{1}X_{2}. \end{eqnarray*} $ 
We readily obtain matrix inequality (91) subject to (94).
Remark 6: Expanding the microgrid to accommodate additional number of distributed generation units will eventually lead to increase the static computational burden as well. However using linear matrix inequalities as efficient numerical tool makes the job of determining the feedback gains becomes moderate. Moreover, it is executed offline and performed only once.
Remark 7: The selection of
Remark 8: It is remarked that the implementation of Theorem 2 is online in nature as it requires calling random generators to pickup numbers corresponding to the scalars
Remark 9: In this paper, the
Remark 10: The networked control system model considered in this paper is distinct from the one presented in [26]. The model represents a typical SoS networked control model consisting of a sensor suite and an adhoc network for multiple subsystems. Moreover the illustrated example for the proposed control strategy is exclusively a microgrid SoS.
Ⅵ. SIMULATION RESULTSIn this section, the microgrid system which is modeled as a networked control SoS is simulated in the SoS research lab at KFUPM using pilotscale lab equipment to build an experimental microgrid including up to 12 DG units. The simulation tools are MATLABSimulink environment, particularly the products Simscape Power Systems Examples and RAPSim an open source simulation software for microgrids. A singleline diagram of the microgrid under consideration is depicted in Fig. 6.
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Fig. 6 Singleline diagram of the microgrid under consideration. 
The output feedback controller design is implemented which stabilizes the system in presence of packet dropouts and delays. The parameter values given in Table Ⅰ are substituted in the state matrices
The controller and observer gains can be obtained by using the relation
The NCS is modeled in such a way that delays are accommodated in both the measurement as well as the actuation channel. The delays are generated by employing random number generators. The number obtained from the uniform random number generator is compared to a variable probability
The gain matrices obtained for both sets are shown above.
Figs. 718 shown above represent the state response of both sets of microalternatorPV system named as set 1 and set 2. Each state represents a typical dynamics of the system.
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Fig. 7 Rotor angle of the microalternator. 
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Fig. 8 Rotor speed of the microalternator. 
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Fig. 9 Internal voltage of microalternator along 
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Fig. 10 Field voltage of microalternator along 
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Fig. 11 Photovoltaic cell current. 
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Fig. 12 Voltage across DClink capacitor. 
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Fig. 13 Inverter output current along 
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Fig. 14 Inverter output current along 
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Fig. 15 Coupling line current of the filter along 
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Fig. 16 Coupling line current of the filter along 
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Fig. 17 Capacitor voltage of the filter along 
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Fig. 18 Capacitor voltage of the filter along 
The simulation results elucidate the effectiveness of the proposed control methodology. Two sets of the microalternatorPV system are considered which have distinct parameter values. As evident from the graphs, the output feedback controller stabilizes the system in presence of delays and packet dropouts. It can be observed from the statewise response that each state of both the sets possessing different values is stabilized by the controller an a reasonably less time with minimum overshoot and very less oscillations. Thus, the simulation results obtained significantly exhibit the controller responses for different parameter values. Moreover, stabilization in presence of communication issues such as packet dropouts and delays further explain the effectiveness of the controller. It is also to be noted that the state response of both sets of microalternatorPV system is presented in one graph to demonstrate the ability of the controller to stabilize the system irrespective of the parameter values.
$ \begin{eqnarray*} K& = & \left[ \begin{array}{ccc} K_1&&K_2 \end{array} \right] \\ K_1& = & \left[ \begin{array}{cccccc} 0.0003&0.0002&0.0001&0.0002&0.0000&0.0001\\ 0.0001&0.0016&0.0002&0.0004&0.0003&0.0001 \end{array} \right]\\ K_2& = & \left[ \begin{array}{cccccc} 0.0112&0.0001&0.0000&0.0006&0.0003&0.0000\\ 0.0003&0.0001&0.0004&0.0001&0.0002&0.0067 \end{array} \right] \\ L& = &\left[ \begin{array}{cc} 0.0076&0.0172\\ 0.0172&0.0013\\ 0.0041&0.0005\\ 0.0244&0.0002\\ 0.0121&0.0042\\ 0.0061&0.0004\\ 0.0069&0.0170\\ 0.0489&0.0010\\ 0.0039&0.0009\\ 0.0002&0.0005\\ 0.0156&0.0008\\ 0.0015&0.0221 \end{array} \right]\\ %\end{eqnarray*} %\end{equation*} %\begin{equation*} %\begin{eqnarray*} K& = & \left[ \begin{array}{ccc} K_1&&K_2 \end{array} \right] \\ K_1& = & \left[ \begin{array}{cccccc} 0.0007&0.0100&0.0028&0.0004&0.0003&0.0419\\ 0.0005&0.0003&0.0000&0.0020&0.0002&0.0090 \end{array} \right]\\ K_2& = & \left[ \begin{array}{cccccc} 0.0000&0.0001&0.0016&0.0021&0.0023&0.0000\\ 0.0000&0.0000&0.0012&0.0011&0.0000&0.0047 \end{array} \right]\\ L& = &\left[ \begin{array}{cc} 0.0152&0.0036\\ 0.0030&0.0003\\ 0.0016&0.0025\\ 0.0000&0.0005\\ 0.0116&0.0010\\ 0.0008&0.0002\\ 0.0003&0.0001\\ 0.0011&0.0126\\ 0.0000&0.0003\\ 0.0000&0.0124\\ 0.0166&0.0008\\ 0.0003&0.0003 \end{array} \right] \end{eqnarray*} $ 
A networked control SoS methodology for the stabilization of a microgrid system with two sets of microalternator and PV systems are presented in this paper. The microalternator and PV system are modeled in detail. The combination of both systems connected to the main grid and a load is also modeled. The networked control system based on this model is subjected to delays and non stationary packet dropouts. An improved output feedback controller is proposed, which stabilizes the system in presence of the aforementioned communication constraints. Using pilotscale lab equipment to build an experimental microgrid, simulation of both sets of microalternatorPV systems are attained using MatlabSimulink environment, particularly the products Simscape Power Systems Examples. From the presented results, the effectiveness of the controller is demonstrated in solving a major issue in microgrids arising from stability of its constituent systems.
Future work would go along different avenues. One avenue would be to expand the model by including battery storage units and examine the transition between islanded and gridconnected modes of operation. Another avenue is to cast the model into a stochastic setup by including effects of load daily as well as solar radiation variations. A final avenue is to treat the model from a multiagent perspective and investigate various protocols. Research work along these ideas is underway.
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