Adoptive Inverter Controller for Microgrid in Islanded Mode

Abstract

The utilization of distributed generation (DG) in Microgrids has posed challenges in modeling and operation and has been resolved with power electronic-based interfacing inverters and associated controllers. The inverter controller in both transient and steady states is of paramount importance, as the stability of Microgrid in grid-connected or islanded mode is dependent on inverter control. Again, the type of the network whether low-voltage or high-voltage distribution sector is of prime importance. In high-voltage network, the reactance/resistance ratio is high and hence no issue of decoupling. But in low-voltage networks as this ratio is low and decoupling is an issue. Normally, the Microgrid is located nearer to load centers. Microgrids are active networks with bi-directional current flow. This research work consists of modeling the Microgrid in a transient state, with a special focus on the study of dynamic and static loads pattern in islanded mode. This chapter presents adoptive control of inverter in an islanded Microgrid. A special focus is given to the state space modeling of all the components of the network and then developing linearized models. The sinusoidal time-varying quantities have been transformed into park transformation, to mitigate decoupling.

1. Introduction

The latest civilized nation needs reliable and quality power for day-to-day domestic, commercial, transport, entertainment, etc. The utilization of Renewable Energy Sources (RESs) in Microgrids is not only reducing the Green House Gases (GHGs) but also adhering to the latest regulations of IEEE-1547-2018. Then, the passive distribution networks are becoming active networks with the bidirectional inverters. The distribution generators are small powered renewable energies from solar, wind, micro-turbines, fuel cells, battery systems, compressed air, super capacitors, which are connected in the low-voltage network, nearer to the loads. As the number of small resources is of smaller capacity, the dynamic modeling is a bit complex due to more variable parameters. To attain optimal power flow, proportional load sharing of inverter-based micro-sources and to take care of the transient states for voltage and frequency regulation, new control strategy controllers are to be designed, for Microgrid in islanded mode. The chapter comprises of six sections. The literature review is done in Section 2 with a comparison to the proposed methodology in this chapter. The mathematical equations and dynamic modeling are discussed in Section 3. Section 4 depicts network and MATLAB design parameters. Section 5 gives out the MATLAB results and discussion. Section 6 finally gives out conclusions and future work proposals.

2. Literature review

The references of about 10 mentioned in the references section have been reviewed in this section.

Juan C. Vasquez et al. [1]: proposed an adoptive droop controller for voltage source inverter, which can cater to the grid and islanded mode. The controller monitors the parameters of voltage, frequency, and virtual impedance amplitude and angle. The controller dynamics are controlled by decoupling to control the power flow in both grid and islanded modes. The system stability is also assessed by plotting the poles and zeros of the characteristic equation in real and imaginary axes with root locus method. However, the system dynamics are better controlled by the proposed method in this chapter, which is validated in MATLAB/Simulink and discussed.

Liang Zhang et al. [2]: the authors proposed a virtual synchronous generator (VSG) technology to address the issues of inertial support in AC/DC Microgrids. But while improving the stability of the system, fast performance is neglected. To accelerate the frequency differential and to reduce the fluctuation in DC voltage, the virtual inertia is dynamically adjusted. The proposed method in this chapter improves the stability of the controller better than the method proposed by authors Liang Zhang et al., which is proved through MATLAB simulations.

Zhiyong Chen et al. [3]: the authors proposed an adoptive sliding mode controller (ASMC), to obviate the issues of performance of PID controllers in Microgrid islanded mode. This method also enhances the system performance by rejection of disturbance. This method depends upon the nonlinear model of the controller for the performance of interfacing inverters of islanded Microgrid. However, the proposed method in this chapter is capable of managing the dynamics of the system, which is shown in the simulation results.

Qusay Salem et al. [4]: the authors presented a control method with a synchronous reference frame (SRF) with PI controllers for voltage source inverters of DGs. This modified droop control has enhanced Microgrid performance in a grid as well as islanded mode. The proposed method in this chapter can reduce the dynamics and improve the stability, which is shown in MATLAB simulations for different types of loads.

Masoud Dashtdar et al. [5]: the authors presented a self-tuning PI controller with a combination of Genetic Algorithm (GA) and Artificial Neural Network (ANN) to control the frequency of Microgrid in islanded mode. ANN parameters like frequency, voltage, etc., are optimized and adjusted by training with the help of GA, to optimize and adjust the coefficients of the PI controller. The frequency control is located in the secondary control loop of Microgrid. However, the methodology in this chapter is a more robust controller, which will manage the dynamics and help in enhancing the stability of Microgrid. The discussions of simulations in MATLAB platform confirm the proposed methodology in this chapter.

Bahador Fani et al. [6]: the authors have discussed different control topologies of interfaced inverters in islanded mode of Microgrid like droop control techniques, control based on various philosophies like distributed, decentralized and centralized controls, virtual impedance-based control, small signal stability control, etc. The authors suggested future work on the management of inverter fault currents, controller design techniques to control frequency, and voltage while regulating power flow. The proposed method in this chapter is more efficient in getting the reliability of the Microgrid during islanded mode. The method proved its efficiency through the MATLAB simulations.

Bernhard Hammer et al. [7]: the authors developed a method of reducing the oscillations in the Microgrid by tuning controller parameters. The authors have not considered the system speed restriction with the cut-off frequency of the low-pass filter. The proportional load sharing is achieved by tuning the system parameters and balancing the system’s natural damping. For this purpose, the authors considered three cascaded controllers. They are to regulate active and reactive powers, to control voltage and the last one is to control current. In this method, the pole assignment method is used, to tune the parameters of the controllers, which may be a bit difficult. However, the method proposed in this chapter is superior to this method, as it can reduce the dynamics and enhance the stability of the system.

Seyed Mohammad Sadegh Hosseinimoghadam et al. [8]: the authors proposed a method to overcome the droop control in the islanded Microgrid. The droop control depends on the proportionality of active power to voltage and reactive power to frequency in islanded Microgrid. The proportional load sharing is attained by incorporating a virtual impedance between the inverter and PCC, to overcome the resistive behavior of low-voltage Microgrids. But this method is not suitable in islanded mode of Microgrid. However, the method proposed in this chapter has more flexibility in managing the system dynamics during transient states.

Michele Tucci et al. [9]: the authors suggested a synthetic control method for stabilizing frequency and voltage control of the Microgrid in the islanded mode with DGs including power lines and loads. This method is suitable to use with plug and play resources also the modeling of power lines and DGs are not required. The regulator tuning is not required when new DGs are plugged into islanded Microgrid. The model cannot regulate power flow, as it cannot work in parallel with local regulators. At this juncture, the method proposed in this chapter can work efficiently as it is having cascaded controller, which is proved with simulations in MATLAB.

Lasantha Meegahapola et al. [10]: the authors proposed a strategy for Microgrid operation and control for seamless synchronous islanding from grid mode by keeping the dynamics of frequency and phase angle, without interruption to sensitive loads. The method is so efficient while resynchronizing to the grid without disrupting the system dynamics. The synchro-phasor problems and telecom delays are hurdles in maintaining the system stability. But this chapter’s methodology is more superior in managing the system and the low inertia effect. The MATLAB simulations proved the effectiveness of this method over the referred author’s strategy.

3. Modeling of islanded microgrid with loads

The Microgrid enters an islanded mode in case of non-availability of the grid or when grid parameters are beyond tolerance limits. The important factors to be considered in modeling islanded Microgrid are the optimum values for sizing, placing, accuracy, and validity in different modules [11]. These can be like generation module, network module, and load module. The model may be linear dynamic, utilizing a synchronously rotating reference frame or stationary frame [12].

The load module can be sub-divided into active, passive, and dynamic load modules. The dynamic modules may be rectifier interfaced load (RIAL) or induction motor load (IM). The modeling of different modules is done in the following sections.

3.1 Generation controller

The controller of the distributed generation inverter comprises of three loops. The first loop is the outer power loop which adjusts the voltage amplitude of the inverter output according to the droop control philosophy. The second and third loops are the voltage loop and current loop, which regulate the frequency variations and create damping for the output filter. Figure 1 shows the details of the generation controller.

Figure 1.

Power and control diagram of inverter interfaced DG.

Transforming the mathematical model into state variables, the virtual control outputs are taken to power, voltage and current controllers, to get the references [6]. The low-pass filter filters the instantaneous power components passing through it. The selection of low-pass filter bandwidth is so selected that the coefficients of droop control will have the control on damping and oscillations of the inverter, so that the control and stability are achieved [13]. This phenomenon will reduce the circulating currents, and thus, the damage due to the overloading of renewable sources is mitigated [14]. The islanded inverter interfaced DG reference voltage and frequency are given by,

$${\omega }_{\mathit{dr}}^{\ast }={\omega }_n-m_p\mathrm{P}.$$E1

$$v_{\mathit{od}-\mathit{dr}}^{\ast }=V_n-n_Q\mathrm{Q}.$$E2

where ${\omega }_n$ and $V_n$ are nominal set values of d-axis frequency and voltage as, $v_{\mathit{oq}-\mathit{dr}}^{\ast }$is set to zero, to align the q-axis voltage to x-axis on cartesian co-ordinates and hence the reactive power and active power droops for the standard range are given by,

$$m_p={\omega }_{\mathit{max}}-{\omega }_{\mathit{min}}/P_{\mathit{max}}$$E3

$$n_Q=v_{{\mathit{od}}_{\mathit{max}}}-v_{{\mathit{od}}_{\mathit{min}}}/Q_{\mathit{max}}$$E4

The average values of reactive and active powers through the LCL filter in Figure 2 are given by,

Figure 2.

Outer loop power controller.

$$\mathrm{P}=\frac{{\omega }_c}{s+{\omega }_c}\times 1.5\left(v_{\mathit{od}}{i}_{\mathit{od}}+v_{\mathit{oq}}{i}_{\mathit{oq}}\right)$$E5

$$\mathrm{Q}=\frac{{\omega }_c}{s+{\omega }_c}\times 1.5\left(v_{\mathit{oq}}{i}_{\mathit{od}}-v_{\mathit{od}}{i}_{\mathit{oq}}\right).$$E6

where 1.5 ($v_{\mathit{od}}{i}_{\mathit{od}}+v_{\mathit{oq}}{i}_{\mathit{oq}}$) is active power and 1.5 ($v_{\mathit{od}}{i}_{\mathit{od}}+v_{\mathit{oq}}{i}_{\mathit{oq}})$ is reactive power. The parameters $v_{\mathit{od}}{i}_{\mathit{od}}$ and $v_{\mathit{oq}}{i}_{\mathit{oq}}$ are output voltage and current of “dq” reference frame and the LCL filter cutoff frequency is ${\mathrm{\omega }}_c$.

3.1.1 Controllers of inner loop voltage and current control

The aim of the current and voltage regulators is to mitigate the high-frequency disturbances and to provide natural damping to the filter circuit [15]. The reference current to the filter inductor $i_{\mathit{ldq}}$ is given by voltage controller and $v_{\mathit{idq}}$ is given by current controller are to be given to PWM module as shown in Figure 3. The voltage controller regulates the voltage and the current controller modifies the voltage wave of the filter inductor [16]. PI controllers are used in voltage and current controllers with feed forward and backward terms. Both these controllers control the output voltage and filter inductor current [17]. The proportional and integral gains of these PI controllers along with forward gain are given by,

Figure 3.

Inverter controllers of DG for voltage and current.

$K_{\mathit{pv}}$—voltage controller proportional gain, $K_{\mathit{iv}}$—voltage controller integral gain, $K_{\mathit{pi}}$—current controller proportional gain, $K_{\mathit{ii}}$—current controller integral gain, and F—feed-forward gain. The dynamical equations of voltage and current controllers are given by,

$$i_{\mathit{ld}}^{\ast }=K_{\mathit{pv}}\left(v_{\mathit{od}}^{\ast }-v_{\mathit{od}}\right)+K_{\mathit{iv}}{\mathrm{\varphi }}_d-{\omega }_n{C}_f{v}_{\mathit{oq}}+\mathrm{F}{i}_{\mathit{od}}$$E7

$$i_{\mathit{lq}}^{\ast }=K_{\mathit{pv}}\left(v_{\mathit{oq}}^{\ast }-v_{\mathit{oq}}\right)+K_{\mathit{iv}}{\mathrm{\varphi }}_q+{\omega }_n{C}_f{v}_{\mathit{od}}+\mathrm{F}{i}_{\mathit{oq}}$$E8

$$v_{\mathit{ld}}^{\ast }=K_{\mathit{pi}}\left(i_{\mathit{ld}}^{\ast }-i_{\mathit{ld}}\right)+K_{\mathit{ii}}{\lambda }_d-{\omega }_n{L}_f{i}_{\mathit{lq}}$$E9

$$v_{\mathit{lq}}^{\ast }=K_{\mathit{pi}}\left(i_{\mathit{lq}}^{\ast }-i_{\mathit{lq}}\right)+K_{\mathit{ii}}{\lambda }_q+{\omega }_n{L}_f{i}_{\mathit{ld}}$$E10

In which voltage controller reference currents are $i_{\mathit{ld}}^{\ast }$ and $i_{\mathit{lq}}^{\ast }$, and used as set values to the current controller. The current controller reference voltage magnitude and angles are given by $v_{\mathit{ld}}^{\ast }$, $v_{\mathit{lq}}^{\ast }$ and ${\lambda }_d$, ${\lambda }_q$ respectively, and are used as reference set values to PWM—Pulse Width Modulated inverter, which is depicted in Figure 1.

3.1.2 Module of LCL filter output and inductance

The LCL filter output along with the inductance of coupling are considered as reference values. Coupling inductance adjusts the inverter output impedance and so decouples the active and reactive powers [18]. The LCL filter (low-pass filter consisting of inductance, capacitance, and inductance) output dynamical voltage and current are given in the following equations [19]. The inter-facing inverter generates voltage which is equal to the reference voltage, $v_i$ = $v_i^{\ast }$, as depicted in Figure 3. The d- and q-axes reference currents and voltages are given by,

$$i_{\mathit{od}}^{\ast }=\frac{1}{L_c}\left(v_{\mathit{od}}-v_{\mathit{md}}-r_c{i}_{\mathit{od}}\right)+\mathrm{\omega }{i}_{\mathit{oq}}$$E11

$$i_{\mathit{oq}}^{\ast }=\frac{1}{L_c}\left(v_{\mathit{oq}}-v_{\mathit{mq}}-r_c{i}_{\mathit{oq}}\right)-\mathrm{\omega }{i}_{\mathit{od}}$$E12

$$i_{\mathit{ld}}^{\ast }=\frac{1}{L_f}\left(v_{\mathit{ld}}-v_{\mathit{od}}-r_f{i}_{\mathit{ld}}\right)+\mathrm{\omega }{i}_{\mathit{lq}}$$E13

$$i_{\mathit{lq}}^{\ast }=\frac{1}{L_{\mathit{ff}}}\left(v_{\mathit{lq}}-v_{\mathit{oq}}-r_f{i}_{\mathit{lq}}\right)+\mathrm{\omega }{i}_{\mathit{ld}}$$E14

$$v_{\mathit{od}}^{\ast }=\frac{1}{C_f}\left(i_{\mathit{ld}}-i_{\mathit{od}}\right)+\mathrm{\omega }{v}_{\mathit{oq}}+R_d\left(i_{\mathit{ld}}^{\ast }-i_{\mathit{od}}^{\ast }\right).$$E15

$$v_{\mathit{oq}}^{\ast }=\frac{1}{C_f}\left(i_{\mathit{lq}}-i_{\mathit{oq}}\right)-\mathrm{\omega }{v}_{\mathit{od}}+R_d\left(i_{\mathit{lq}}^{\ast }-i_{\mathit{oq}}^{\ast }\right).$$E16

In which $v_{\mathit{md}}$ = Microgrid d-axis voltage, $v_{\mathit{mq}}$ = Microgrid q-axis voltage,

$r_f$, $r_c$ = parasitic resistances, $L_f$, $L_c$ = parasitic inductances,

$R_d$ = damping resistor connected in series with $C_f$.

3.2 Network control module

In inverter interfaced DGs, the VSIs (Voltage source inverters), the time constant is small and hence the network dynamics have an impact on stability [20]. The network RL form is shown in Figure 4. The dynamical equations of the line connected between Microgrid bus and main grid in DQ reference frame (the capital letters DQ shows for network) are given by,

Figure 4.

Line configuration with RL.

$$i_{\mathit{\text{lineD}}}^{\ast }=\frac{-R_{\mathit{\text{line}}}}{L_{\mathit{\text{line}}}}{i}_{\mathit{\text{lineD}}}+\frac{1}{L_{\mathit{\text{line}}}}\left(v_{\mathit{mD}}-v_{\mathit{gD}}\right)+\mathrm{\omega }{i}_{\mathit{\text{lineQ}}}$$E17

$$i_{\mathit{\text{lineQ}}}^{\ast }=\frac{-R_{\mathit{\text{line}}}}{L_{\mathit{\text{line}}}}{i}_{\mathit{\text{lineQ}}}+\frac{1}{L_{\mathit{\text{line}}}}\left(v_{\mathit{mQ}}-v_{\mathit{gQ}}\right)+\mathrm{\omega }{i}_{\mathit{\text{lineD}}}$$E18

Eqs. (17) and (18) can be written in the small scale linearized state space,

$$\text{∆}{i}_{\mathit{\text{lineD}}}^{\ast }=\frac{-R_{\mathit{\text{line}}}}{L_{\mathit{\text{line}}}}∆i_{\mathit{\text{lineD}}}+\frac{1}{L_{\mathit{\text{line}}}}\left(∆v_{\mathit{mD}}-∆v_{\mathit{gD}}\right)+\text{∆}{\omega }_{\mathit{com}}{I}_{\mathit{\text{lineQ}}}$$E19

$$\text{∆}{i}_{\mathit{\text{lineQ}}}^{\ast }=\frac{-R_{\mathit{\text{line}}}}{L_{\mathit{\text{line}}}}∆i_{\mathit{\text{lineQ}}}+\frac{1}{L_{\mathit{\text{line}}}}\left(∆v_{\mathit{mQ}}-∆v_{\mathit{gQ}}\right)+\text{∆}{\omega }_{\mathit{com}}{I}_{\mathit{\text{lineD}}}$$E20

where $v_{\mathit{mD}},v_{\mathit{gD}}$, $v_{\mathit{mQ}}$, $v_{\mathit{gQ}}$ are the voltages of Microgrid bus and main grid or distribution grid bus in DQ axes, respectively. The ∆ $i_{\mathit{\text{lineD}}}^{\ast }$ and ∆ $i_{\mathit{\text{lineQ}}}^{\ast }$ are the incremental reference current values in DQ axes.

3.3 Load control module

Various types of loads like passive, active, constant power and dynamic loads are switched on and thrown off at different times at Microgrid bus to find out the stability of the islanded Microgrid [21]. MATLAB/Simulink software package is used for simulating the model and the Microgrid interfaced inverter controller of DG found to be stable with proper power regulation, voltage, and frequency deviations with acceptable deviations as prescribed by IEEE-1847-2018 standards [22]. The network circuit, for the total simulations, is shown in Figure 5.

Figure 5.

Total network circuit for testing inverter interfaced DG.

3.3.1 Passive load module

The tested passive loads considered are RL load and constant power load CPL [23]. The resistive loads, which can be considered are incandescent lamps, heaters, etc. The electric motors, solenoids, etc., can be categorized as RL loads [24]. The passive load of constant power load (CPL) considered is instantaneous impedance, which is of positive value [25]. The passive load modeling is done in the following sections [26]. The network model for passive loads is shown in Figure 6. The passive load dynamical equations, which are connected at PCC, are given by,

Figure 6.

Passive network model.

$$i_{\mathit{PLD}}^{\ast }==\frac{-R_{\mathit{PL}}}{L_{\mathit{PL}}}{i}_{\mathit{PLD}}+\frac{1}{L_{\mathit{PL}}}\left(v_{\mathit{mD}}\right)+\mathrm{\omega }{i}_{\mathit{PLQ}}$$E21

$$i_{\mathit{PLQ}}^{\ast }=\frac{-R_{\mathit{PL}}}{L_{\mathit{PL}}}{i}_{\mathit{PLQ}}+\frac{1}{L_{\mathit{PL}}}\left(v_{\mathit{gD}}\right)-\mathrm{\omega }{i}_{\mathit{PLD}}$$E22

Eqs. (21) and (22) are common for passive loads and PL, has to be replaced by $R_{\mathit{\text{LOAD}}}$ for resistive, ${\mathit{RL}}_{\mathit{\text{LOAD}}}$ for inductive load and $R_{\mathit{CPL}}$ for constant power load. But constant power load has to be expressed in real and imaginary parts ($R_{\mathit{CPL}}+\mathrm{j}{X}_{\mathit{CPL}}$),

For which the real part is $-r_{\mathit{CPL}}cos\alpha$ and imaginary part is $-r_{\mathit{CPL}}sin\alpha$, where $\alpha$ is angle of delay of rectifier controller.

3.3.2 Module of active load

The UPSs—un-interrupted power systems are nowadays being used in industries and residential buildings. The three-phase inverters are being used extensively in lifts and cranes [27]. The single-line power and control circuit is depicted in Figure 7. The control circuit is having of DC voltage control and AC current control part of the Rectifier Interfaced Active Load [28]. These consist of PI controllers. The DC voltage controller controls the voltage of the capacitor $C_{\mathit{dc}}$ and the inner current of $L_f$ is controlled by AC controller [29]. The reference input to AC controllers is given by the DC voltage controller and this in turn controls the current through $L_{\mathit{fAL}}$. The feed forward terms being given by AC controllers, which helps in decoupling the d and q. The d-q axes inductor currents are given by $i_{\mathit{ldq}\mathit{AL}}$, the frequency of nominal system voltage is ${\omega }_n$ and inductance $L_{\mathit{fAL}}$. The control circuit is shown in Figure 8.

Figure 7.

Control circuit of RIAL – Rectified interfaced active load.

Figure 8.

AC d-q current and DC voltage controllers of RIAL.

3.3.3 Dynamic load module

The dynamic load considered is an induction motor (IM) and the dynamic equations in the D-Q reference frame, considering the resistance ($r_S,)$, inductance ($L_{\mathit{SS},})$, mutual inductance ($L_m)$ of motor, rotor slip (s), and stator angular frequency (ω) are given by,

$$V_{\mathit{Qs}}=r_s{i}_{\mathit{Qs}}+L_{\mathit{ss}}{i}_{\mathit{Qs}}^{\ast }+L_m{i}_{\mathit{Qr}}^{\ast }+\mathrm{\omega }{L}_{\mathit{ss}}{i}_{\mathit{Ds}}+\mathrm{\omega }{L}_m{i}_{\mathit{Dr}}$$E23

$$V_{\mathit{Ds}}=r_s{i}_{\mathit{Ds}}+L_{\mathit{ss}}{i}_{\mathit{Ds}}^{\ast }+L_m{i}_{\mathit{Dr}}^{\ast }-\mathrm{\omega }{L}_{\mathit{ss}}{i}_{\mathit{Qs}}-\mathrm{\omega }{L}_m{i}_{\mathit{Qr}}$$E24

$$V_{\mathit{Qr}}=r_S{i}_{\mathit{Qr}}+L_{\mathit{rr}}{i}_{\mathit{Qr}}^{\ast }+L_m{i}_{\mathit{Qs}}^{\ast }+\mathrm{s\omega }{L}_{\mathit{rr}}{i}_{\mathit{Dr}}+\mathrm{s\omega }{L}_m{i}_{\mathit{Ds}}$$E25

$$V_{\mathit{Dr}}=r_r{i}_{\mathit{Dr}}+L_{\mathit{rr}}{i}_{\mathit{Dr}}^{\ast }+L_m{i}_{\mathit{Ds}}^{\ast }-\mathrm{s\omega }{L}_{\mathit{rr}}{i}_{\mathit{Qr}}-\mathrm{s\omega }{L}_m{i}_{\mathit{Qs}}$$E26

$$T_e=\frac{3}{2}\frac{P}{2}{L}_m\left(i_{\mathit{Qs}}{i}_{\mathit{Dr}}-i_{\mathit{Ds}}{i}_{\mathit{Qr}}\right)$$E27

$$T_e-T_L=\mathrm{J}\frac{d}{\mathit{dt}}\left\{\left(1-s\right)\mathrm{\omega }\right)\}$$E28

in which the terms in Eqs. (27) and (28)$T_e$ is motor torque, $T_L$ is load torque, p is the number of poles of the motor and J is the inertia of the combined motor and load. It is assumed that the stator supply frequency is $\mathrm{\omega }$ having an impact on inverter droop equations [30]. The linearized equations of induction motor considering the small signal deviations are given by,

$$\frac{{∆}_{\mathit{uIM}}}{\left[\begin{array}{c}{∆}_{v_{\mathit{Qs}}}\\ {∆}_{v_{\mathit{Ds}}}\\ {∆}_{v_{\mathit{Qr}}}\\ {∆}_{v_{\mathit{Dr}}}\\ {∆}_{T_L}\end{array}\right]}=F_{\mathit{IM}}\frac{{∆}_{\mathit{XIM}}}{\left[\begin{array}{c}{∆}_{i_{\mathit{Qs}}}\\ {∆}_{i_{\mathit{Ds}}}\\ {∆}_{i_{\mathit{Qr}}}\\ {∆}_{i_{\mathit{Dr}}}\\ {∆}_s\end{array}\right]}+E_{\mathit{IM}}\frac{{∆}_{\mathit{XIM}}}{{\left[\begin{array}{c}{∆}_{i_{\mathit{Qs}}}\\ {∆}_{i_{\mathit{Ds}}}\\ {∆}_{i_{\mathit{Qr}}}\\ {∆}_{i_{\mathit{Dr}}}\\ {∆}_s\end{array}\right]}^{\ast }}+D_{1\mathit{IM}}∆\omega +D_{2\mathit{IM}}∆{\omega }^{\ast }$$E29

The linearized small signal in state-space model of the induction motor load is given by,

$$\begin{array}{ll}{\left[∆x_{\mathit{IM}}^{\ast }\right]}_{5\times 1}=& {\left[∆\begin{array}{c}{A}_{\mathit{IM}}\end{array}\right]}_{5\times 5}{\left[∆\begin{array}{c}{x}_{\mathit{IM}}\end{array}\right]}_{5\times 1}+{\left[\begin{array}{c}{B}_{1\mathit{IM}}\end{array}\right]}_{5\times 5}{\left[∆\begin{array}{c}{u}_{\mathit{IM}}\end{array}\right]}_{5\times 1}\\ & +{\left[\begin{array}{c}{B}_{2\mathit{IM}}\end{array}\right]}_{5\times 1}{\left[∆\begin{array}{c}\omega \end{array}\right]}_{1\times 1}+{\left[\begin{array}{c}{B}_{3\mathit{IM}}\end{array}\right]}_{5\times 1}{\left[∆\begin{array}{c}{\omega }^{\ast }\end{array}\right]}_{1\times 1}\end{array}$$E30

$${\left[∆\begin{array}{c}{y}_{\mathit{IM}}\end{array}\right]}_{5\times 1}={\left[∆\begin{array}{c}{C}_{\mathit{IM}}\end{array}\right]}_{5\times 5}{\left[∆\begin{array}{c}{x}_{\mathit{IM}}\end{array}\right]}_{5\times 1}$$E31

In which $A_{\mathit{IM}}$ = state variable matrix of induction motor load

$B_{1\mathit{IM}}$ = input matrix of voltage

$B_{2\mathit{IM}}$ = input matrix of load torque

$B_{3\mathit{IM}}$ = input matrix of load frequency and derivatives

$C_{\mathit{IM}}$ = output matrix of dynamic load of induction motor

From Eq. (30), $∆x_{\mathit{IM}}$ and $∆u_{\mathit{IM}}$ can be written as,

$$\left[∆x_{\mathit{IM}}\right]={\left[\begin{array}{ccccc}∆i_{\mathit{Qs}}& ∆i_{\mathit{Ds}}& ∆i_{\mathit{Qr}}& ∆i_{\mathit{Dr}}& ∆s\end{array}\right]}^T$$E32

$$\left[∆u_{\mathit{IM}}\right]={\left[\begin{array}{ccccc}∆v_{\mathit{Qs}}& ∆v_{\mathit{Ds}}& ∆v_{\mathit{Qr}}& ∆v_{\mathit{Dr}}& ∆T_L\end{array}\right]}^T$$E33

The small signal linearized equations of induction motor dynamic load are given by,

$$\begin{array}{ll}{\left[∆\begin{array}{c}{x}_{\mathit{IM}}^{\ast }\end{array}\right]}_{5n_{\mathit{IM}\times 1}}=& {\left[∆\begin{array}{c}{A}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }5n_{\mathit{IM}}}{\left[∆\begin{array}{c}{x}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }1}+{\left[\begin{array}{c}{B}_{1\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }5n_{\mathit{IM}}}{\left[∆\begin{array}{c}{u}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }1}\\ & +{\left[\begin{array}{c}{B}_{2\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times 1}}{\left[∆\begin{array}{c}{\omega }_{\mathit{\text{IcomM}}}\end{array}\right]}_{1_{\times }1}+{\left[\begin{array}{c}{B}_{3\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }1}{\left[∆\begin{array}{c}{\omega }_{\mathit{com}}^{\ast }\end{array}\right]}_{1\times 1}\end{array}$$E34

$${\left[∆\begin{array}{c}{y}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }1}={\left[\begin{array}{c}{C}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }5n_{\mathit{IM}}}{\left[∆\begin{array}{c}{x}_{\mathit{IM}}\end{array}\right]}_{5n_{\mathit{IM}\times }1}$$E35

In which $A_{\mathit{IM}}$ = state matrix of induction motor dynamic load

$B_{1\mathit{IM}}$ = input matrix of voltage

$B_{2\mathit{IM}}$ = input matrix of load torque

$B_{3\mathit{IM}}$ = input matrix of load frequency and derivatives

$C_{\mathit{IM}}$ = output matrix of dynamic load of induction motor

From Eq. (34), $∆x_{\mathit{IM}}$ and $∆u_{\mathit{IM}}$ of linearized transpose matrix form, can be written as,

$$\left[∆x_{\mathit{IM}}\right]={\left[\begin{array}{ccccc}∆x_{\mathit{IM}1}& ∆x_{\mathit{IM}2}& \dots & \dots & ∆x_{{\mathit{IMn}}_{\mathit{IM}}}\end{array}\right]}^T$$E36

$$\left[∆u_{\mathit{IM}}\right]={\left[\begin{array}{ccccc}∆u_{\mathit{IM}1}& ∆u_{\mathit{IM}2}& \dots & \dots & ∆u_{{\mathit{IMn}}_{\mathit{IM}}}\end{array}\right]}^T$$E37

3.4 Total microgrid model

The total system of the entire state space model in the linearized notation of Microgrid interfaced inverter can be derived from mapping all models of matrixes, which are connected to PCC [31]. The Microgrid may either export power or import from the grid (−ve sign) or import to the grid (+ve sign) depending upon the D, Q matrixes of the network. The generation, net, and load modules are given mapping matrixes [32]. ${\mathrm{M}}_{\mathrm{GEN}}\left(\mathrm{m},\mathrm{g}\right)$ is the DG module with Microgrid bus with “m” and grid bus with “g” of the order $2_{\mathrm{nN}}\times 2_{\mathrm{nG}}$ with line elements and is given by mapping matrix of generation is given by,

$${\mathrm{M}}_{\mathrm{GEN}}\left(\mathrm{m},\mathrm{g}\right)=\left\{\begin{array}{c}1\text{if inverter}\mathrm{DG}\text{is connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\\ 0\text{if inverter}\mathrm{DG}{\mathrm{isn}}^{\prime }\mathrm{t}\text{connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\end{array}\right.$$E38

The network module mapping matrix with all the connecting lines and nodes is expressed by ${\mathrm{M}}_{\mathrm{NET}}$ and is given by,

$${\mathrm{M}}_{\mathrm{NET}}\left(\mathrm{m},\mathrm{g}\right)=\left\{\begin{array}{c}-1\text{if inverter}\mathrm{DG}\text{is exporting to grid}\mathrm{at}\mathrm{PCC}\text{node}\\ +1\text{if inverter}\mathrm{DG}\text{is importing from grid}\mathrm{at}\mathrm{PCC}\text{node}\end{array}\right.$$E39

The mapping matrix of “n” passive and “n” active loads of the order of ${\mathrm{n}}_{\mathrm{PL}}$ and ${\mathrm{n}}_{\mathrm{AL}}$ is given by,

$${\mathrm{M}}_{\mathrm{PL}}/{\mathrm{M}}_{\mathrm{AL}}\left(\mathrm{m},\mathrm{g}\right)=\left\{\begin{array}{c}-1\text{if active or passive load is connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\\ 0\text{if active or passive load is not connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\end{array}\right.$$E40

The dynamic load mapping matrix of induction motor of the order of ${\mathrm{n}}_{\mathrm{nN}}\times {\mathrm{n}}_{\mathrm{nIM}}$ is given by,

$${\mathrm{M}}_{\mathrm{IM}}\left(\mathrm{m},\mathrm{g}\right)=\left\{\begin{array}{c}-1\text{if}\mathrm{IM}\left(\text{induction motor}\right)\text{stator is connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\\ 0\text{if}\mathrm{IM}\text{of stator is not connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\end{array}\right.$$E41

The assumption is that all nodal voltages are inputs to every module and currents are outputs [33]. To reduce the deviations in dynamic stability inverter interfaced DG, a large value virtual resistor ${\mathrm{r}}_n$is introduced in the circuit between every node and ground, which is shown in Figure 9(a). The purpose of inserting this resistance is to reduce the dynamics of the inverter and to improve the stability of the system. The Microgrid bus voltage is given by,

Figure 9.

(a). Virtual resistor ${\mathrm{r}}_N$ between microgrid bus and ground. (b). Disturbance at grid bus created from microgrid bus.

$${\mathrm{v}}_{\mathit{gDQ}}={\mathrm{r}}_n\left({\mathrm{i}}_{\mathit{gDQ}}+{\mathrm{i}}_{\mathit{\text{lineDQ}}}-{\mathrm{i}}_{\mathit{\text{lineDQ}}}\right)$$E42

To analyze the small signal stability, a disturbance is created at the Microgrid bus by switching on sudden dynamic load and throwing off, and the response is recorded [34]. There are no remarkable deviations in frequency and voltage, which are as per IEEE-1547-2018, it should be in the limits for voltage 0.88 $\le$ v $\le$ 1.11p.u., and for frequency 49.5 $\le$ Hz $\le$ 50.5. Figure 9(b) depicts the scenario of load current [35]. The corresponding nodal currents and voltages in terms of mapping matrixes are given by,

$$\begin{array}{ll}{\left[\begin{array}{c}{v}_{\mathit{gDQ}}\end{array}\right]}_{2n_{N\times }1}=& {\left[\begin{array}{c}{M}_{\mathit{GEN}}\end{array}\right]}_{2n_{N\times }2n_G}{\left[\begin{array}{c}∆i_{\mathit{ODQ}}\end{array}\right]}_{2n_{G\times }1}+{\left[\begin{array}{c}{M}_{\mathit{NET}}\end{array}\right]}_{2n_{N\times }2n_L}{\left[\begin{array}{c}∆i_{\mathit{\text{LINEDQ}}}\end{array}\right]}_{L1}\\ & +{\left[\begin{array}{c}{M}_{\mathit{PL}}\end{array}\right]}_{2n_{N\times }2n_L}{\left[\begin{array}{c}∆i_{\mathit{\text{PLDQ}}}\end{array}\right]}_{2\mathit{nPL}\times 1}+{\left[\begin{array}{c}{M}_{\mathit{AL}}\end{array}\right]}_{2n_{N\times }2n_{\mathit{AL}}}{\left[\begin{array}{c}∆i_{\mathit{\text{ALDQ}}}\end{array}\right]}_{2\mathit{nAL}\times 1}\\ & +{\left[\begin{array}{c}{M}_{\mathit{IM}}\end{array}\right]}_{2n_{N\times }2n_{\mathit{IM}}}{\left[\begin{array}{c}∆i_{\mathit{DQ}}\end{array}\right]}_{2\mathit{nIM}\times 1}+{\left[\begin{array}{c}{R}_N\end{array}\right]}_{2n_{N\times }2n_N}{\left[\begin{array}{c}{M}_{\mathit{\text{dist}}}\end{array}\right]}_{2n_{N\times }2}{\left[∆\begin{array}{c}{i}_{\mathit{\text{distDQ}}}\end{array}\right]}_{2\times 1}\end{array}$$E43

In which $M_{\mathit{\text{dist}}}$ is the disturbance mapping matrix of order $2n_{N\times }2$ and maps the load disturbance node into network node. Hence the $M_{\mathit{\text{dist}}}$ elements are given by,

$${\mathrm{M}}_{\text{dist}}\left(\mathrm{m},\mathrm{g}\right)=\left\{\begin{array}{c}-1\text{if load disturbance is connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\\ 0\text{if load disturbance is not connected to grid}\mathrm{at}\mathrm{PCC}\text{node}\end{array}\right.$$E44

4. Design parameters

The design values calculated and used in this work are given in the Tables 1–3.

Line	Value and units
1	(0.4 + j038) Ω
2	(0.5 + j0.25) Ω
3	(0.23 + j0.11) Ω
4	(0.35 + j0.58) Ω
5	(0.3 + j0.47) Ω
6	(0.25 + j0.15) Ω

Table 1.

Parameters of line.

Component	Value and units
DG and Inverter output	100kVA
Filter Resistance, $R_f$	0.22 Ω
Filter Inductance, $L_f$	1.45 mH
Filter Resistance, $C_f$	5.5 $\mathit{\mu F}$
Cut off frequency, ${\omega }_c$	31.41 rad/sec
proportional gain constant of Voltage controller, ${\mathrm{K}}_{\mathrm{pv}}$	0.08
integral gain constant of Voltage controller, ${\mathrm{K}}_{\mathrm{iv}}$	400
proportional gain constant of Current controller, ${\mathrm{K}}_{\mathrm{pi}}$	11.5
proportional gain constant of Current controller, ${\mathrm{K}}_{\mathrm{ii}}$	14,000
Feed forward gain, F	0.75
Coupling Resistor, $R_c$	0.02 Ω
Coupling Inductor, $L_c$	0.4 mH

Table 2.

Inverter interfaced DG parameters.

Load	Parameter	Value and unit
Resistive Load (RL)	Active Power Load (AL)	18 kW
	Reactive power load (Inductive)	10 kVA
Constant Power Load (CPL)	Active Power	20 kW
	Reactive power load (Inductive + Capacitive)	10 kVA + 10 kVAr
	$cos\alpha$ ($\alpha$- rectifier delay angle)	0.86
Rectifier Interfaced Active Load (RIAL) with DC Motor	Armature Resistance $R_a$	0.6 Ω
	Armature Inductance $L_a$	0.12 H
	Field Resistance $R_f$	240 Ω
	Field inductance, $L_f$	120 H
	Field-armature mutual inductance, $L_{\mathit{af}}$	1.8 H
	Armature voltage, $V_a$	300 V
	Field voltage, $V_f$	240 V
	Motor rating	5 HP
	Motor speed	1750 rpm
Induction Motor Load; (IM)	Motor Capacity	7.5HP/10 kVA
	Frequency	50 Hz
	Speed	1500 rpm
	Volts	415 V
	Number of poles, P	4
	Stator resistance/Leakage inductance	1.77 Ω/0.0139H; $R_{s/}{Z}_s$
	Rotor resistance/Leakage inductance	1.34 Ω/0.0121H; $R_{r/}{Z}_r$
	Magnetizing inductance	$\left(L_m\right)$0.3687H
	Torque, $T_L$	48.83 Nw.m

Table 3.

Load parameters.

5. Simulation results and discussion

The model which is developed, is validated through simulations in MATLAB/Simulink platform. There are four types of loads considered here for analysis purposes, which are passive load RL, constant power load CPL, RIAL-rectifier interfaced active load and induction motor dynamic load. The switching time for all loads is 0.4 secs and the switching effect is analyzed at PCC.

5.1 Passive load $\left(R_L\right)$

The Microgrid model developed is tested for analyzing the effect of sudden transients like loads switching on and throwing off with the passive load, which consists of resistance and inductance with the values of $R_L$ 18 kW and $L_L$ 10 kVA, respectively. The load is switched on at PCC through the circuit breaker at 0.4 secs and thrown off at 0.8 secs, over a sampling time of 1 sec in MATLAB/Simulink. The parameters such as voltage and frequency are monitored for deviations and found to be within limits. The controller is stable without any issue of dynamics and the profiles of frequency, voltage and active and reactive powers are investigated at PCC, which are shown in Figures 10–13.

Figure 10.

Frequency profile with $R_L$ load.

Figure 11.

Voltage profile with $R_L$ load.

Figure 12.

Active power profile with $R_L$ load.

Figure 13.

Reactive power profile with $R_L$ load (inductive reactive power).

5.2 Passive load (CPL)

The passive load connected at PCC, in this case, is resistance, inductive, and capacitive reactance, with the values as shown in Table 3. The load is switched on at PCC through the circuit breaker at 0.4 secs and thrown off at 0.8 secs, over a sampling time of 1 sec in MATLAB/Simulink. The frequency deviations are greater while throwing off at 0.8 secs than at switching on an instant of 0.4 secs, but well within the standards of IEEE. Similarly, the voltage deviations are more at the time of throwing off, but within limits. The controller is stable without any issue of dynamics and the profiles of frequency, voltage, active, and reactive powers are shown in Figures 14–17.

Figure 14.

Frequency profile with CPL load.

Figure 15.

Voltage profile with CPL load.

Figure 16.

Active power profile with CPL load.

Figure 17.

Reactive power profile with CPL load (resistive, inductive and capacitive).

5.3 Rectifier interfaced active power (RIAL)

The three-phase full-wave bridge rectifier is connected to a DC Motor of 5 HP, 1750 rpm with variable field excitation 240/220/200 volts. At all three volts, the speed, armature current, field current, and torque are obtained in MATLAB/Simulink software and the results are depicted in Figures 18–20. As the field voltage is varied from 200 to 240 volts, the speed and torque characteristics are shown in the Figures below. The PCC frequency and voltages are well within permissible limits as per IEEE-1547-2018, 49.5 $\le f\le 50.5\mathrm{Hz}\text{and voltage}0.88\le V\le 1.11\text{Volts}.$ This indicates that the adoptive controller performance is as per the design values in the islanded Microgrid.

Figure 18.

DC motor characteristic curves with field voltage 200 volts.

Figure 19.

DC motor characteristic curves with field voltage 220 volts.

Figure 20.

DC motor characteristic curves with field voltage 240 volts.

5.4 Dynamic load with induction motor (IM)

The induction motor dynamic load is connected at PCC with a 10 HP Motor, 1500 rpm. The switching on of the induction motor at PCC with full load, created some transients initially but the controller is stable to absorb these dynamics, which is shown by the motor currents characteristic curves in Figure 21. The controller frequency at PCC is well within the IEEE standard deviation between 49.6 and 50.1 Hz as shown in Figure 22. The voltage variations are also well within specified limits as per standards of IEEE, which are 365 and 410 volts (for nominal volts 415) as shown in Figure 23.

Figure 21.

Induction motor load with 10HP, 1500 rpm capacity.

Figure 22.

Frequency profile with induction motor load 10HP, 1500 rpm capacity.

Figure 23.

Frequency profile with induction motor load 10HP, 1500 rpm capacity.

6. Conclusions and future work

6.1 Conclusions

Adoptive inverter control for stable operation of Microgrid in islanded mode has been presented in this chapter. A state space model for inverter interfaced Microgrid in a DQ reference frame with static and dynamic loads has been derived and used in designing of controller. The modeling is done in three parts. Part one, consists of DG module, part two network module and part three load model. The load model considered in this chapter is passive load, active load consisting of rectifier interfaced active load (RIAL), and dynamic load consisting of induction motor load (IM). The complete DQ model is obtained by mapping all sub-modules of DG, network, and loads, which consists of passive, active, and dynamic loads. The total mapping model developed in DQ frame is tested for dynamics of frequency, network, and load with all control loops. The time domain simulation results validated the linearized model. The MATLAB/Simulink simulation results prove the efficiency and effectiveness of the adoptive controller designed, in respect of Microgrid in islanded mode.

6.2 Future work

The work can be extended with hybrid renewables such as wind turbines, fuel cells, and energy storage systems (ESS), to use as DGs. The proportional load sharing with DG interfaced inverter controllers can be studied. Also, the stability of the total system can be assessed and controller’s control can be modified to obtain absolute stability by plotting poles and zeros of characteristic equation in real and imaginary axes. With this, the total stability of the Microgrid can be obtained in islanded mode.

Acknowledgments

The authors acknowledge with thanks the management and faculty of Electrical and Electronics Engineering Department of Koneru Lakshmaiah Education Institute, Guntur-522302, A.P., India, Vasavi Engineering College, Hyderabad-500031, India and Datta Meghe Institute of Engineering Technology and Research, Wardha 442004, India.

Conflict of interest

The authors declare that there is no conflict of interest.

Author contributions

This research was a collaborative effort between all the participating authors with equal contributions.

Financial disclosure

The authors declare that this work has not received any financial aid from any source.

Appendices

${\mathit{v}}_{\mathit{i}}$	inverter output voltage
${\mathit{v}}_{\mathit{i}}^{\mathit{\ast }}$	inverter reference input voltage
$\mathit{∆}$	incremental value
${\mathbf{K}}_{\mathbf{p}}$	controller proportional constant
${\mathbf{K}}_{\mathbf{i}}$	controller integral constant
${\mathit{\omega }}_{\mathit{n}}$	system normal frequency
m-bus	microgrid bus
g-bus	grid bus
${\mathit{T}}_{\mathit{e}}$	is motor torque
$T_L$	is load torque
${\mathit{r}}_{\mathit{S}}$	induction motor stator resistance
${\mathit{r}}_{\mathit{n}}$	virtual resistor of large value, between each node and ground
${\mathbf{L}}_{\mathbf{m}}$	induction motor stator resistance
ω	induction motor stator angular frequency
s	induction motor rotor slip
${\mathbf{L}}_{\mathbf{ss}}$	induction motor stator inductance
$\mathit{\alpha }$	delay angle of controller rectifier

Nomenclature

DER	distributed energy resources
DG	distributed generation
RES	renewable energy sources
MG	microgrid
PCC	point of common coupling
abc	time variable sinusoidal a, b, c quantities
DQ	network direct and quadrature axes quantities
dq	load side direct and quadrature axes quantities
RL load	resistive and inductive load
CPL load	constant Power Load
RIAL load	rectifier interfaced active load
IM load	induction motor load
LPF	low pass filter
LCL filter	inductance, capacitance and line side inductance
PI	proportional integral
AC	alternating current
DC	direct current
P	active power
Q	reactive power
UPS	uninterrupted power supply
PV	photo voltaic
VSI	voltage source inverter
AL	active load
PL	passive load
GHG	green house gases
abc to dq	park transformation, transforming abc quantities to dc rotating