A Nonlinear Integral Backstepping Controller to Regulate the Voltage and Frequency of an Islanded Microgrid Inverter

: The islanded operation mode of a microgrid system is usually affected by the system uncertainties, such as the load, source, and parameter variations. In such systems, the voltage and frequency must be regulated to maintain the power quality during islanded operation. As an approach to control the voltage and frequency, in this study, a decentralized nonlinear integral backstepping controller for the voltage source inverter used in an islanded microgrid is developed. First, the dynamical model of the inverter-based distribution generations (DGs) in microgrid system is developed. Subsequently, the model-based controller for the microgrid is built using dynamics of inverter-based DGs and Lyapunov theory, which could eliminate the voltage and frequency deviations in the system under different uncertainties. To ensure the system stability, a control Lyapunov function is adopted. Considering the inﬂuence of irradiations and other meteorological variables ﬂuctuations a battery energy storage (BESS) is applied on the DC side to suppress the ﬂuctuations of output power of DGs. Furthermore, the efﬁciency of the designed controller was validated through simulations in the MATLAB/Simulink environment under different scenarios and effectiveness of the proposed framework is further validated by real-time hardware in loop (HIL) experiments. In addition, the performance of the proposed controller is compared with a conventional backstepping (BS) controller. The comparison results demonstrate that the efﬁciency of the designed controller in terms of obtaining steady-state operating conditions is better than that of the BS controller.


Introduction
A microgrid (MG) is an advanced small-level power network, which consists of distribution generation (DG) local loads and a storage system. MGs can improve the power quality of a system, reduce CO 2 emissions and energy losses, decrease the load of the transmission lines and offer an efficient approach to integrate renewable energy systems (RESs) [1]. MGs usually operate in the grid-connected mode to supply power to the grid. In this mode, the main grid controls the voltage and frequency of the system [2]. Nevertheless, the MG exhibits unique advantages when operating in the islanded mode. In this mode, the MG is disconnected from the main grid and operates autonomously. Furthermore, in the absence of the utility grid support in this mode, the MG must be suitably controlled to regulate the associated voltage and frequency to be within their normal ranges [3]. In addition, in the islanded mode, the DG sources must also regulate the frequency and voltage levels to ensure matching between the generated power to the load demand to maintain the power quality for consumers [4]. Nevertheless, in the islanded operation mode, because of the low inertia of the RESs, the variation in the load demand and uncertainty in the output of the DGs renders the voltage and frequency control of the MG highly challenging [5][6][7].
In an MG framework, the energy sources are usually RESs connected to the MG through DC/AC or AC/AC inverters. These inverters usually operate as voltage source inverters (VSI) in the islanded operation mode of the MG. To achieve the desired performance level and maintain the system voltage and frequency in the required ranges, the VSI must be suitably controlled [8,9]. The inverter efficiency primarily depends on the utilized control method. In particular, the applied control technique should be able to address the system's non-linearity and load variations. Moreover, the transient response of the control technique should be prompt under all the operating conditions [10].
Several control techniques have been implemented to realize inverter control in the microgrid operation. These methods include linear techniques such as proportionalintegral (PI) control with pulse width modulation (PWM) and space vector modulation (SVM) [11,12], dead-beat control [1,[3][4][5][6][7][8][9][10][11][12][13][14][15], proportional-resonant (PR) [16,17] control, hysteresis control [18], H-infinity control [3,19], repetitive control [20,21], and linear quadratic control (LQR) [22]. However, these control methods involve certain disadvantages. In the presence of distortion in the electrical parameters, the performance of the PI controller is not satisfactory. Although the dead-beat controller can realize a fast-transient response, its efficiency degrades under system uncertainties and parameter disturbances. The main limitations of the PR controller are its sensitivity to the variation in frequency and the necessity of accurate tuning. Furthermore, the operation of the hysteresis controller mandates perfect mathematical understanding of the system, and its performance is satisfactory only under slow system dynamics. All the aforementioned control techniques exhibit low accuracy in RES-based islanded MG systems [10,23]. Droop control is the most common approach to realize voltage control in MGs under islanded operation [24]. However, the traditional droop control involves several limitations such as the strong dependency on the filter impedance and slow transient response [5]. Furthermore, this method is not robust in the presence of external disturbances and the disruption of MG parameters [25]. Some researchers proposed an adaptive droop controller to realize the voltage control in MGs; however, the voltage was unstable under load changes [26].
To overcome this issue, a centralized fuzzy based gain-scheduling control strategy was proposed [27] to realize voltage regulation and ensure the energy balance in MGs. The centralized controller uses the information of different parts of the MG and maintains the system stability. However, the major limitation of this controller is its dependence on the training process; specifically, the controller cannot attain the desired objective without proper fuzzy model training. Various centralized control techniques [28][29][30][31] have been developed to realize voltage and frequency control of the MG system. However, these centralized strategies have several disadvantages. In particular, the stability of the microgrid is degraded in the event of a communication failure, and the control system and necessary devices are often prohibitively expensive. Therefore, such controllers are not reliable.
In this context, a decentralized controller can overcome the limitations of centralized controllers. These controllers use only the local information of various MG elements, thereby avoiding the problems of communication delays [32]. To realize voltage control, a generalized decentralized control-based method was developed [33]; however, this method cannot satisfy the voltage stability criteria. Moreover, several decentralized control techniques have been implemented [34][35][36] to realize voltage regulation in MG applications. However, the operation of these decentralized controllers is limited to specific operating points and is sensitive to the parameters of different components in the MGs.
Nonlinear controllers, which can realize the operation in a wide operating range can overcome the limitations of linear controllers. Recently, many nonlinear sliding mode controller-based approaches were proposed [37][38][39][40][41] realize the voltage and frequency regulation in both grid-connected and islanded mode operations, and these approaches exhibited a reasonable efficiency. Nevertheless, such controllers often exhibit the chattering phenomenon, which severely reduces the life of power electronic-based devices in the MG system. Even though the SMC is less sensitive to parameter variations and external disturbances, the selection of the time-varying sliding surface is complex, owing to the intermittency of the RESs within the MG.
The abovementioned problems can be solved by using a backstepping controller as it can operate in a wide range and stabilize the overall system [42]. A sliding modebased backstepping controller was proposed in [43] for an islanded MG but the controller has chattering problems and it does not guarantee global robustness against external disturbance. In [44] an adaptive backstepping controller was used in a control loop along with a PI controller to control voltage of the MG by considering the resistance load variation only; nevertheless, the stability against the source uncertainties and parameter variations in the system was not considered. An adaptive backstepping controller was developed in [45] taking into account the parametric uncertainties; but the external disturbances were neglected. However, the existing backstepping techniques in the literature does not exhibit stability and cannot satisfy the system requirements in the face of the intermittency of RESs, external parameter variations, and load variations [46]. Furthermore, steady state errors often occur in the output response of such frameworks. In this regard, to improve the robustness and steady state performance of the MGs, the backstepping technique must be improved to ensure stable MG operation To this end, this paper proposes an integral-based backstepping controller scheme for inverter-based DG sources in an MG system. The integral action is applied during the backstepping design processes to improve convergence to a steady-state for tracking error vectors under model uncertainties and external disturbance. The objective of the controller is to regulate the voltage and frequency of the system under load variations, source uncertainty, and parameter variations. The stability analysis of the overall system is based on the Lyapunov function. In particular, the contributions of this work are as follows.

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A decentralized control approach is designed for voltage regulation of an MG despite source uncertainty, parameter variations and load changes to reduce the overall computational complexity of the system.

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To improve the robustness and steady state performance of the controller, an integral action is added to the nonlinear backstepping controller.

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The proposed controller is compared with simple backstepping controller to control the MG voltage and frequency. The rest of the paper is as follows: The description and mathematical model of the microgrid are explained in Section 2. Section 3 discusses the controller design process. Simulation and HIL experiment results of the NLIB controller are discussed in Section 4. Comparison of the proposed controller is given in Section 5. In the end, the conclusion is given in Section 6.

System Description
The schematic, single-line diagram of a DGs network that is interfaced via VSI and series filter represented by R f and L f to the utility grid is illustrated in Figure 1. At the point of common coupling (PCC), a balanced three-phase load is connected. The parameters of the system are provided in Tables 1 and 2. Based on the dynamic model of the islanded MG network, a new controller is designed which is discussed in the next part. An internal oscillator is usually responsible for the control of the islanded system's frequency which is performed in an open-loop fashion. As clear from Figure 1 the frequency of the internal oscillator is the same as the nominal frequency of the system. Therefore is the pre-specified frequency of MG while operating in an islanded mode both for voltage and current signals.

Islanded Microgrid Methametical Model
In islanded operation mode of MG, the schematic diagram of an inverter-based DGs is presented in Figure 1. The state-space mathematical model is described in this section  The MG network must work in both grid-connected and islanded modes. The DGs converter operates as a current-controlled VSC in grid-connected mode. In this operation mode the main grid regulates the frequency and voltage magnitude at PCC. But during islanded operation. Due to the variations in load demands there is a power mismatch between load and DGs. Therefore, the system frequency and voltage magnitude at PCC fluctuate and cannot hold constant value as their reference value. Therefore, in this operation if the DGs units will not offer the frequency and voltage regulation the load voltage and frequency can fluctuate greatly. Thus for the uninterruptible operation of islanded MG system frequency and voltage should be controlled. In order to regulate frequency and voltage of load a new controller is necessary.
Based on the dynamic model of the islanded MG network, a new controller is designed which is discussed in the next part. An internal oscillator is usually responsible for the control of the islanded system's frequency which is performed in an open-loop fashion. As clear from Figure 1 the frequency of the internal oscillator is the same as the nominal frequency of the system. Therefore w 0 is the pre-specified frequency of MG while operating in an islanded mode both for voltage and current signals.

Islanded Microgrid Methametical Model
In islanded operation mode of MG, the schematic diagram of an inverter-based DGs is presented in Figure 1. The state-space mathematical model is described in this section for an islanded microgrid [36]. A balanced three-phase RCL load is connected to the system. The state-space model of the islanded MG system is given as Equation (1) in the ABC frame: where V t,abc , i L abc , i abc, v abc are 3 × 1 vectors consisting of each phase quantities. Clark and Park's transformation shows that each phase quantity can be indicated in a rotating (d-q) coordinate system. The dynamics of islanded microgrid has been modelled mathematically using park transformation are given below: . . . . .
where V d , V q , I d , I q represent the output voltages and current of VSI in d-q axis; I ld , I lq are load inductor currents in d-q axis. The variables C f , L f , R f are equivalence capacitance inductance and resistance of the LC filter. Whereas L l , R l express load inductance and its equivalence resistance. The output voltage angular frequency w o is given as: where w * is reference set-point, m represents droop gain, s indicates operator of Laplace transform and w c is the cut-off frequency of the filter.

Energy Management Scheme of BESS.
The PV power fluctuates due to intermittent nature of solar irradiations and other meteorological variables. Therefore, a rechargeable battery module is incorporated in MG to assure the power stability of PV source. A simple energy management scheme of PV and BESS is provided in Figure 2.
In Figure 2 P d power represents the require for the system, P bt indicates the power of battery module and P pv is the power supplied by the PV panels.  In Figure 2 power represents the require for the system, indicates the power of battery module and is the power supplied by the PV panels.

Integral Backstepping Controller Design
This section describes the design process of the proposed NLIBC. The controller's main purpose is to enforce the output voltage to track the reference voltage which assures the system stability. The backstepping controller formulation with an integral action in dq frame is given in this section. The steps to obtain the control laws are as follows: Step 1: Defining the voltage tracking error for VSI: where * is the reference voltage signal for the system. The error must converge to zero for achieving the required objective. Taking the derivative of the : putting the values of in Equation (10) from Equation (2): introducing an integral element for robustness: The Lyapunov function can be selected such that it not only ensures stability, but also improves performance of the controller. Different types of Lyapunov function i.e., nonquadratic Lyapunov functions are used in literature to ensure stability [47]. In this study first Lyapunov candidate is defined as following in Equation (13) for analysing the system stability:

Integral Backstepping Controller Design
This section describes the design process of the proposed NLIBC. The controller's main purpose is to enforce the output voltage to track the reference voltage which assures the system stability. The backstepping controller formulation with an integral action in d-q frame is given in this section. The steps to obtain the control laws are as follows: Step 1: Defining the voltage tracking error for VSI: where v * d is the reference voltage signal for the system. The error z 1 must converge to zero for achieving the required objective. Taking the derivative of the z 1 : .
putting the values of . v d in Equation (10) from Equation (2): .
introducing an integral element for robustness: The Lyapunov function can be selected such that it not only ensures stability, but also improves performance of the controller. Different types of Lyapunov function i.e., non-quadratic Lyapunov functions are used in literature to ensure stability [47]. In this study first Lyapunov candidate is defined as following in Equation (13) for analysing the system stability: where v 1 is first Lyapunov candidate and γ 1 is the design parameter and it must be positive. The system will be asymptotically stable if where . v 1 is the derivative of Lyapunov canidate and ζ 1 shows the integral element. Putting values in Equation (14) from Equation (11): introducing virtual control input as a stabilization function α 1 : to enforce z 1 to vanish we can choose: c 1 is the controller gain. From Equations (16) and (17), Equation (15): introducing the 2 nd error: . Substituting it in Equation (10) and using Equations (16) and (17) we get: Step 2: For z 1 to vanish, z 2 should vanish first, taking the derivative of z 2 : .
defining the second Lyapunov function v 2 for islanded MG system stability assurance: taking the derivative of v 2 : .
putting values of . z 1 from Equation (20): where c 2 is the design parameter of the controller and it should be c 2 > 0. Comparing Equations (27) and (31) and solving for v d we can get the control law as following: by using the same steps for v q with following errors: The stabilization function is given as: defining the Lyapunov function as: .
where c 3 > 0, c 4 > 0. Subsequently, we can obtain control law for v q as follows: Electronics 2021, 10, 660 9 of 24 where K 5 , K 6 , K 7 , K 8 are given as: The control laws derived in Equations (33) and (40) are used to track the reference voltages and converge the error dynamic to zero and it also ensures the asymptotic stability of the system.

Simulation and Experimental Results
To verify the aforementioned analysis, first, the computer simulations were carried out in MATLAB/Simulink and then, real-time hardware in the loop tests were conducted to validate the effectiveness of the suggested framework. The simulation aims to ensure stable microgrid operation by maintaining constant bus voltages. The parameters and configuration of the microgrid used for the simulations are listed in Tables 1 and 2. The detailed block diagram of integral backstepping control is presented in Figure 3. Different case studies including load changes, parameters uncertainties and PV source uncertainty were investigated to analyse the performance of the proposed decentralized nonlinear integral backstepping control (NLIBC) and results are compared with conventional BS controller.
The control laws derived in Equations (33) and (40) are used to track the reference voltages and converge the error dynamic to zero and it also ensures the asymptotic stability of the system

Simulation and Experimental Results
To verify the aforementioned analysis, first, the computer simulations were carried out in MATLAB/Simulink and then, real-time hardware in the loop tests were conducted to validate the effectiveness of the suggested framework. The simulation aims to ensure stable microgrid operation by maintaining constant bus voltages. The parameters and configuration of the microgrid used for the simulations are listed in Tables 1 and 2. The detailed block diagram of integral backstepping control is presented in Figure 3. Different case studies including load changes, parameters uncertainties and PV source uncertainty were investigated to analyse the performance of the proposed decentralized nonlinear integral backstepping control (NLIBC) and results are compared with conventional BS controller.

Case 1: Change of Load (Resistive)
The system is simulated with load variations to decrease voltage dips and to achieve the reference voltage. The load is varied from 50 Ω to 100 Ω at time = 0.5 s as shown in Figure 4. The load voltage in ABC frames is presented in Figures 5 and 6. Figure 6 shows a zoomed-in view of the reference voltage tracking. It can be seen that when load increase at = 0.5 s voltage disturbance occurs. The voltage magnitude is maintained by the proposed control technique and the NLIB controller takes just = 0.02 s to acquire steadystate value.
The reference voltage tracking in the d-q frame by the proposed controller is presented in Figures 7 and 8. It can be seen from Figure 8 that the proposed controller immediately maintains the voltage when a disturbance occurs in the system. The results of voltage variation during load change with the application of NLIB controller show that the system has great robustness with this controller. Because of the extra drop of voltage across filter elements and load resistor voltage falls at time instant = 0.5 s NLIB controller works to regain the rated voltage at this time instant by manipulating the voltage to track back reference voltage by using control law presented in Equations (33) and (40).

Case 1: Change of Load (Resistive)
The system is simulated with load variations to decrease voltage dips and to achieve the reference voltage. The load is varied from 100 Ω to 50 Ω at time t = 0.5 s as shown in Figure 4. The load voltage in ABC frames is presented in Figures 5 and 6. Figure 6 shows a zoomed-in view of the reference voltage tracking. It can be seen that when load increase at t = 0.5 s voltage disturbance occurs. The voltage magnitude is maintained by the proposed control technique and the NLIB controller takes just t = 0.02 s to acquire steady-state value.
age variation during load change with the application of NLIB controller show that the system has great robustness with this controller. Because of the load decrease the voltage across filter elements and load resistor voltage increase at time instant = 0.5 s NLIB controller works to regain the rated voltage at this time instant by manipulating the voltage to track back reference voltage by using control law presented in Equations (33) and (40).   across filter elements and load resistor voltage increase at time instant = 0.5 s NLIB controller works to regain the rated voltage at this time instant by manipulating the voltage to track back reference voltage by using control law presented in Equations (33) and (40).     In Figure 6 it is observed that when the load changes the NLIB controller rapidly converges the voltage to its rated value in settling time of just 0.02 s. The behaviour of The reference voltage tracking in the d-q frame by the proposed controller is presented in Figures 7 and 8. It can be seen from Figure 8 that the proposed controller immediately maintains the voltage when a disturbance occurs in the system. The results of voltage variation during load change with the application of NLIB controller show that the system has great robustness with this controller. Because of the load decrease the voltage across filter elements and load resistor voltage increases at time instant t = 0.5 s NLIB controller works to regain the rated voltage at this time instant by manipulating the voltage to track back reference voltage by using control law presented in Equations (33) and (40). load current in case of resistive load variation of the system by NLIB controller is given in the ABC frame in Figure 9. It can be seen form Figure 9 the load current decreases when the resistive load is increased in the system. A perfect tracing of output voltage with respect to the reference voltage is achieved by NLIBC. The maintenance of reference voltage, both in the d-q and ABC frames, by the NLIBC is shown in Figures 5-8. For resistive load variation scenario, the proposed controller achieves a steady-state voltage level in less time. Therefore, it concludes that the performance of the NLIB controller is good enough for resistive load variations.   load current in case of resistive load variation of the system by NLIB controller is given in the ABC frame in Figure 9. It can be seen form Figure 9 the load current decreases when the resistive load is increased in the system. A perfect tracing of output voltage with respect to the reference voltage is achieved by NLIBC. The maintenance of reference voltage, both in the d-q and ABC frames, by the NLIBC is shown in Figures 5-8. For resistive load variation scenario, the proposed controller achieves a steady-state voltage level in less time. Therefore, it concludes that the performance of the NLIB controller is good enough for resistive load variations.   In Figure 6 it is observed that when the load changes the NLIB controller rapidly converges the voltage to its rated value in settling time of just 0.02 s. The behaviour of load current in case of resistive load variation of the system by NLIB controller is given in the ABC frame in Figure 9. It can be seen form Figure 9 the load current increases when the resistive load is decreased in the system. A perfect tracing of output voltage with respect to the reference voltage is achieved by NLIBC. The maintenance of reference voltage, both in the d-q and ABC frames, by the NLIBC is shown in Figures 5-8.

Case-2: Inductive Load Variation
In this case study, the proposed control scheme is validated for inductive load variation. The inductive load is changed from 500 mH to 100 mH at time = 0.5 s. Figure 10 illustrates the inductive load variations. NLIB controller behavior for inductive load variations in the ABC frame is given in Figure 11. It can be observed that when the inductive load is varied the voltage disturbance occurs and the proposed controller achieved the reference voltage with a minimum settling time of just 0.045 s. The load voltage for this case in the d-q frame is presented in Figure 12. The output current for inductive load case is given in Figure 13. From the results it is clear that for any (resistive and inductive) load variation the proposed controller achieves the steady-state values in minimum settling time. During the load variations scenario, the frequency of the MG system is analyzed at PCC. The behavior of NLIB controller for frequency regulation of the islanded MG system during load variation is illustrated in Figure 14. For resistive load variation scenario, the proposed controller achieves a steady-state voltage level in less time. Therefore, it concludes that the performance of the NLIB controller is good enough for resistive load variations.

Case 2: Inductive Load Variation
In this case study, the proposed control scheme is validated for inductive load variation. The inductive load is changed from 500 mH to 100 mH at time t = 0.5 s. Figure 10 illustrates the inductive load variations. NLIB controller behavior for inductive load variations in the ABC frame is given in Figure 11. It can be observed that when the inductive load is varied the voltage drops and the proposed controller achieved the reference voltage with a minimum settling time of just 0.045 s. The load voltage for this case in the d-q frame is presented in Figure 12. The output current for inductive load case is given in Figure 13. From the results it is clear that for any (resistive and inductive) load variation the proposed controller achieves the steady-state values in minimum settling time. During the load variations scenario, the frequency of the MG system is analyzed at PCC. The behavior of NLIB controller for frequency regulation of the islanded MG system during load variation is illustrated in Figure 14. It is clear that the proposed controller efficiently controls the system frequency during the load change scenario. The system runs at 49.95 Hz during the load change scenario. The maximum frequency level goes to 49.98 Hz. During the load variations the system frequency has some disturbance, but it is still within the rated value with the application of proposed controller.  It is clear that the proposed controller efficiently controls the system frequency during the load change scenario. The system runs at 49.95 Hz during the load change scenario. The maximum frequency level goes to 49.98 Hz. During the load variations the system frequency has some disturbance, but it is still within the rated value with the application of proposed controller.         From the results in Figures 4-12 it can be summarized that in case of sudden variations in loads NLIBC controller shows high robustness to retrieve and stabilize MG voltage and frequency.

Case 3: Controller Performance during Fluctuation at DC Bus.
In order to validate the effectiveness of the proposed controller a fault is introduced at time t = 0.5 s and it ends at t = 0.7 s. The voltage regulation of the MG system with proposed controller is presented in Figure 15. It is clear that the proposed controller efficiently controls the system frequency during the load change scenario. The system runs at 49.95 Hz during the load change scenario. The maximum frequency level goes to 49.98 Hz. During the load variations the system frequency has some disturbance, but it is still within the rated value with the application of proposed controller.
From the results in Figures 4-12 it can be summarized that in case of sudden variations in loads NLIBC controller shows high robustness to retrieve and stabilize MG voltage and frequency.

Case 3: Controller Performance during Fluctuation at DC Bus.
In order to validate the effectiveness of the proposed controller a fault is introduced at time t = 0.5 s and it ends at t = 0.7 s. The voltage regulation of the MG system with proposed controller is presented in Figure 15. The frequency regulation during the fault is given in Figure 16. It can be seen from the results that during the fault the system voltage and frequency fluctuates and NLIBC promptly damped the effect of fault and restore the system voltage and frequency.

Case 4: Controller Performance with Changes in Solar Irradiation
The solar irradiation varies continually during the operation of PV systems. This uncertainty in the available solar radiation affects the solar PV system operation. For checking the effectiveness of designed control scheme variations in solar irradiation was considered in this case study. Due to environmental effects the irradiations can quickly change. Therefore, the designed controller must control the voltage and frequency of MG during the uncertainty in solar PV source due to changes in irradiations.
To illustrate robust performance of controller for PV source variations, at = 1.5 s the solar irradiation was changed from 800 W/m to 1000 W/m and at = 3 s the irradiations settle back to 800 W/m as shown in Figure 17. Tracking of system voltage to its reference value of 375 V is shown in Figure 18. At = 1.5 s, and = 3 s due to variations in solar irradiations transients can be observed. NLIBC recovers reference voltage in the time of 0.07 s with minimum overshoot. Figure 19 shows that the proposed control scheme has excellent ability to achieve steady state values of bus voltage in settling time of 0.07 s.The NLIBC is used to maintain the voltage at the desired value through obtained The frequency regulation during the fault is given in Figure 16. It can be seen from the results that during the fault the system voltage and frequency fluctuates and NLIBC promptly damped the effect of fault and restore the system voltage and frequency. The frequency regulation during the fault is given in Figure 16. It can be seen from the results that during the fault the system voltage and frequency fluctuates and NLIBC promptly damped the effect of fault and restore the system voltage and frequency.

Case 4: Controller Performance with Changes in Solar Irradiation
The solar irradiation varies continually during the operation of PV systems. This uncertainty in the available solar radiation affects the solar PV system operation. For checking the effectiveness of designed control scheme variations in solar irradiation was considered in this case study. Due to environmental effects the irradiations can quickly change. Therefore, the designed controller must control the voltage and frequency of MG during the uncertainty in solar PV source due to changes in irradiations.
To illustrate robust performance of controller for PV source variations, at = 1.5 s the solar irradiation was changed from 800 W/m to 1000 W/m and at = 3 s the irradiations settle back to 800 W/m as shown in Figure 17. Tracking of system voltage to its reference value of 375 V is shown in Figure 18. At = 1.5 s, and = 3 s due to variations in solar irradiations transients can be observed. NLIBC recovers reference voltage in the time of 0.07 s with minimum overshoot. Figure 19 shows that the proposed control scheme has excellent ability to achieve steady state values of bus voltage in settling time of 0.07 s.The NLIBC is used to maintain the voltage at the desired value through obtained

Case 4: Controller Performance with Changes in Solar Irradiation
The solar irradiation varies continually during the operation of PV systems. This uncertainty in the available solar radiation affects the solar PV system operation. For checking the effectiveness of designed control scheme variations in solar irradiation was considered in this case study. Due to environmental effects the irradiations can quickly change. Therefore, the designed controller must control the voltage and frequency of MG during the uncertainty in solar PV source due to changes in irradiations.
To illustrate robust performance of controller for PV source variations, at t = 1.5 s the solar irradiation was changed from 800 W/m 2 to 1000 W/m 2 and at t = 3 s the irradiations settle back to 800 W/m 2 as shown in Figure 17. Tracking of system voltage to its reference value of 375 V is shown in Figure 18. At t = 1.5 s, and t = 3 s due to variations in solar irradiations transients can be observed. NLIBC recovers reference voltage in the time of 0.07 s with minimum overshoot. Figure 19 shows that the proposed control scheme has excellent ability to achieve steady state values of bus voltage in settling time of 0.07 s. The NLIBC is used to maintain the voltage at the desired value through obtained control laws. In fact, it is noticed from results that the changing solar irradiation has no significant effect on the system voltage.
Electronics 2021, 10, x FOR PEER REVIEW 16 of 24 control laws. In fact, it is noticed from results that the changing solar irradiation has no significant effect on the system voltage. The behavior of an NLIB controller for frequency regulation of the islanded MG system during variations in solar irradiation is illustrated in Figure 20. Due to variations in solar irradiation at = 1.5 s and = 3 s the system frequency has minimum disturbance but the NLIBC efficiently controls the system frequency to its reference value within settling time of.0.023 s.
It is clear from the Figures 4-17 that during the variation in load and solar irradiations scenarios the system promptly regains the steady state values after a short transient, which means that the proposed NLIBC can satisfy load demands efficiently by maintaining the power balance within the system and assuring constant output voltage.   The behavior of an NLIB controller for frequency regulation of the islanded MG system during variations in solar irradiation is illustrated in Figure 20. Due to variations in solar irradiation at = 1.5 s and = 3 s the system frequency has minimum disturbance but the NLIBC efficiently controls the system frequency to its reference value within settling time of.0.023 s.
It is clear from the Figures 4-17 that during the variation in load and solar irradiations scenarios the system promptly regains the steady state values after a short transient, which means that the proposed NLIBC can satisfy load demands efficiently by maintaining the power balance within the system and assuring constant output voltage.   The behavior of an NLIB controller for frequency regulation of the islanded MG system during variations in solar irradiation is illustrated in Figure 20. Due to variations in solar irradiation at t = 1.5 s and t = 3 s the system frequency has minimum disturbance but the NLIBC efficiently controls the system frequency to its reference value within settling time of 0.023 s.

Case-5. Validations of Proposed Controller with Battrey Energy Storage System.
In order to overcome the influence of irradiation fluctuation, a battery energy storage system is applied in the system to suppress the fluctuation of output Power of PV source.

Case 1: Change in PV Power Due to Solar Irradiations
In this case study the PV system is producing large power than the load demand for time instant up to 1.5 s. In this time period the PV system is working at standard solar irradiation, i.e., 1000 W/m . Thus the PV system will provide extra power in this time period which is show in Figure 21. This extra power form PV source will be stored in to battery energy storage system after satisfying the load demand and it is presented in Figure 22. The PCC bus voltage is given in Figure 23. In order to overcome the influence of irradiation fluctuation, a battery energy storage system is applied in the system to suppress the fluctuation of output Power of PV source.

Case 1: Change in PV Power Due to Solar Irradiations
In this case study the PV system is producing large power than the load demand for time instant up to 1.5 s. In this time period the PV system is working at standard solar irradiation, i.e., 1000 W/m 2 . Thus the PV system will provide extra power in this time period which is show in Figure 21. This extra power form PV source will be stored in to battery energy storage system after satisfying the load demand and it is presented in Figure 22. The PCC bus voltage is given in Figure 23.     The solar irradiation was dropped from 1000 W/m 2 to 800 W/m 2 at = 1.5 s and the PV system operates in this situation till = 5 s. In this period due to variation in solar irradiations, the PV system will produce less power than load demand. In this case, to satisfy the load requirements and preserve the desired bus voltage, the BESS will be discharged. The change of power and voltage limits are presented in Figures 21-23 for this case. It can be seen from these figures that during the variation of solar irradiation scenarios the system promptly regains the steady state values after a short transient.

Experimental Validations
For further validation of proposed controller's performance real-time hardware in loop (HIL) simulation tests were conducted. The controller HIL test setup is shown in Figure 24. Before practical implementation C-HIL experiments are economical method to examine the performance of controller. The equipment framework incorporates a host-PC, MCU F28379D Dual-Core C2000 Delfino Micro-controllers (Texas Instruments, Dallas, TX, USA) for the execution of control laws. C2000 Delfino support from the embedded coder was used to interface micro controller with MATLAB. C2000 Delfino    The solar irradiation was dropped from 1000 W/m 2 to 800 W/m 2 at = 1.5 s and the PV system operates in this situation till = 5 s. In this period due to variation in solar irradiations, the PV system will produce less power than load demand. In this case, to satisfy the load requirements and preserve the desired bus voltage, the BESS will be discharged. The change of power and voltage limits are presented in Figures 21-23 for this case. It can be seen from these figures that during the variation of solar irradiation scenarios the system promptly regains the steady state values after a short transient.

Experimental Validations
For further validation of proposed controller's performance real-time hardware in loop (HIL) simulation tests were conducted. The controller HIL test setup is shown in Figure 24. Before practical implementation C-HIL experiments are economical method to examine the performance of controller. The equipment framework incorporates a host-PC, MCU F28379D Dual-Core C2000 Delfino Micro-controllers (Texas Instruments, Dallas, TX, USA) for the execution of control laws. C2000 Delfino support from the embedded coder was used to interface micro controller with MATLAB. C2000 Delfino The solar irradiation was dropped from 1000 W/m 2 to 800 W/m 2 at t = 1.5 s and the PV system operates in this situation till t = 5 s. In this period due to variation in solar irradiations, the PV system will produce less power than load demand. In this case, to satisfy the load requirements and preserve the desired bus voltage, the BESS will be discharged. The change of power and voltage limits are presented in Figures 21-23 for this case. It can be seen from these figures that during the variation of solar irradiation scenarios the system promptly regains the steady state values after a short transient.

Experimental Validations
For further validation of proposed controller's performance real-time hardware in loop (HIL) simulation tests were conducted. The controller HIL test setup is shown in Figure 24. Before practical implementation C-HIL experiments are economical method to examine the performance of controller. The equipment framework incorporates a host-PC, MCU F28379D Dual-Core C2000 Delfino Micro-controllers (Texas Instruments, Dallas, TX, USA) for the execution of control laws. C2000 Delfino support from the embedded coder was used to interface micro controller with MATLAB. C2000 Delfino MCU F28379D Launchpad was used for generation of control signal in HIL simulation tests and microgrid system was developed in MATLAB/Simulink. The PV power was kept constant for facilitation of HIL experiments. The entire process was performed in three steps. At first MG model was converted to MATLAB/Simulink feasible codes. Secondly, PWM input signal for convertor was generated by microcontroller. The code was uploaded in the firmware of microcontroller via host PC. Finally, the parameters to be observed, such as system voltage, current and frequency were displayed on the screen. The results of these HIL experiments were compared with simulations results for validation Electronics 2021, 10, 660 19 of 24 45 s however they are still within reference value. From Figure 26 it can be observed that frequency tracks its reference value with high precision. Although, few spikes can be observed due to the load change but they are well within the rated value. The response of load output current is shown in Figure 27. It can be seen from the results that the performance of proposed controller is satisfactory in HIL experiments tests. There is little fluctuation in experimental results in comparison with simulation results, which shows that the simulations and experimental results are in consistent with each other.  The voltage regulation for islanded operation mode of MG is shown in Figure 25. Although during load variations a few spikes can be seen at t = 30 s, t = 40 s and t = 45 s however they are still within reference value. From Figure 26 it can be observed that frequency tracks its reference value with high precision. Although, few spikes can be observed due to the load change but they are well within the rated value. The response of load output current is shown in Figure 27. It can be seen from the results that the performance of proposed controller is satisfactory in HIL experiments tests. There is little fluctuation in experimental results in comparison with simulation results, which shows that the simulations and experimental results are in consistent with each other.

Comparison of NLIBC with Conventional Backstepping
This section presents a comparison of the proposed NLIB controller performance with the conventional back-stepping controller. For evaluation in terms of achievable settling time and overshoot the BS controller were applied to the islanded MG system in MATLAB/Simulink. For load variations scenario the efficiency of each controller was evaluated for the islanded network. Figure 28 shows that the comparison of NLIB with BS controller for voltage regulation in the load change scenario. It is noticed that the NLIB controller converges to reference value within 0.09 s where in case of BS large settling time and overshoot can be observed. Due to change in load at t = 2.5 s transients can be observed but dynamic performance of NLIBC is better as compared to BS controller. From Figure 28 it is clear that the overshoot and settling time with BS controller is larger as compared to NLIBC.
considering the load variations. At = 2.5 s the load is varied and transients occurs in the system frequency but the NLIBC efficiently tracks the reference frequency with high precision. The NLIBC has smoother response and minimum settling time in comparison with BS controller. It is evident that convergence performance of NLIB is better than that of BS. This proves that the proposed technique has a faster dynamic response than BS controllers.

Conclusions
An NILB controller was designed for an inverter-based DG unit for islanded operation of MG. The model based back stepping controller was build based in the dynamic model of DG in islanded mode. In order to deal with the SteadyState error in the conventional backstepping controller an integral action was introduced in the control loop. To  Figure 29 represents the system frequency comparison of NLIBC and BS controller considering the load variations. At t = 2.5 s the load is varied and transients occurs in the system frequency but the NLIBC efficiently tracks the reference frequency with high precision. The NLIBC has smoother response and minimum settling time in comparison with BS controller. It is evident that convergence performance of NLIB is better than that of BS. This proves that the proposed technique has a faster dynamic response than BS controllers. settling time and overshoot the BS controller were applied to the islanded MG system in MATLAB/Simulink. For load variations scenario the efficiency of each controller was evaluated for the islanded network. Figure 28 shows that the comparison of NLIB with BS controller for voltage regulation in the load change scenario. It is noticed that the NLIB controller converges to reference value within 0.09 s where in case of BS large settling time and overshoot can be observed. Due to change in load at = 2.5 s transients can be observed but dynamic performance of NLIBC is better as compared to BS controller. From Figure 28 it is clear that the overshoot and settling time with BS controller is larger as compared to NLIBC. Figure 29 represents the system frequency comparison of NLIBC and BS controller considering the load variations. At = 2.5 s the load is varied and transients occurs in the system frequency but the NLIBC efficiently tracks the reference frequency with high precision. The NLIBC has smoother response and minimum settling time in comparison with BS controller. It is evident that convergence performance of NLIB is better than that of BS. This proves that the proposed technique has a faster dynamic response than BS controllers.

Conclusions
An NILB controller was designed for an inverter-based DG unit for islanded operation of MG. The model based back stepping controller was build based in the dynamic model of DG in islanded mode. In order to deal with the SteadyState error in the conventional backstepping controller an integral action was introduced in the control

Conclusions
An NILB controller was designed for an inverter-based DG unit for islanded operation of MG. The model based back stepping controller was build based in the dynamic model of DG in islanded mode. In order to deal with the SteadyState error in the conventional backstepping controller an integral action was introduced in the control loop. To validate the performance of proposed control scheme, number of simulation and HIL results were presented. The simulation and experimental results indicated that the proposed controller can realize stable bus voltage regardless of the system dynamics. Specifically, the following conclusions were derived:

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The proposed strategy can robustly restore the voltages and frequency of the overall MG to their reference values in the presence of unknown dynamics including fluctuations in the PV source, load changes, and parametric uncertainties in the MG system.

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The scheme does not require a communication link, which enhances the system reliability, affordability, and simplicity.

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In cases with load variations, source variations and parameter uncertainties, the reference value of the voltage and frequency is tracked by the NLIB controller in minimum settling time.

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Compared with conventional BS method it is clear from the results that proposed scheme outperforms that conventional method in robustness and stability.
Future work will be devoted to design controllers for the hybrid DC-AC microgrid and Opal-RT 4510 (Opal-RT, Montréal, QC, Canada) will be used to simulate the microgrid model for more accurate experimental analysis.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest:
The authors declare no conflict of interest.