**1. Introduction**

The smart grid is considered to be the future of electricity network and the microgrid is an important part of it [1,2,3]. Microgrid enables the integration of distributed generation units (DG units) at the distribution level [4,5]. The use of DG units at the distribution level provides many advantages like less loss in power transmission from generation to consumption. The microgrid also allows the islanded operation, in case of power failure from the main grid. There are many challenges involved in the integration of DG units at the distribution level that need to be solved [6,7,8]. The voltage source inverter (VSI) is the key element for integrating DG units in microgrid [9,10,11]. Proper control mechanisms on VSI are required for voltage and frequency stability and for proper power sharing among multiple DG units in a microgrid [12]. A microgrid can be operated in either grid-connected mode or in islanded mode [13,14,15], and VSI can be controlled in either voltage control mode (VCM) or in power control mode (PCM) [16]. In the islanded mode of operation, either all VSIs operate in VCM with power sharing achieved by solving optimal power flow equations [17,18] or by using the droop control method [19,20,21] or by operating in a master-slave configuration. In the master-slave configuration, a master VSI operates in VCM, and other VSIs operate in PCM [22,23]. In grid-connected mode, the grid works as “swing” for optimal power flow equations with all DG units operating in VCM; or in a master-slave configuration, the grid works as a master unit with all DG units operating in PCM. The control systems for VCM or PCM can be designed in a natural ($abc$) reference frame, stationary ($a\xdf$) reference frame, or synchronous ($dq$) reference frame [24]. Signals in these reference frames can be converted using Clarke and Park transformations [25]. DG units and loads frequently join and leave the microgrid resulting in a change in topology, so the control system must work on locally available signals [26].

This paper focuses on the control system design of a voltage source inverter (VSI) in power control mode (PCM). Many control techniques have been presented in the literature on this problem. In [27], PI controller-based active and reactive power control is shown in the $dq$ reference frame. This controller uses two control loops for both active and reactive power control. This makes the tuning of the controller difficult. In [28], the power control strategy is presented for photovoltaic and battery storage-based inverters in the AC microgrid. It utilizes the multiple loops of PI control to achieve active and reactive power control in the $dq$ reference frame. In [29], a power controller is presented to improve the power quality of the grid-connected inverter. The power controller is designed in $a\xdf$ reference frame, and it generates the reference values for the current controller. The current controller is a hysteresis-based PWM controller designed in $abc$ reference frame. A direct active and reactive power control of grid-tied inverter is presented in [30]. The fuzzy logic-based controller is designed to minimize the errors between active and reactive powers and their reference values. In [31], an artificial neural network (ANN) based power control is presented to enhance the power quality of a photovoltaic-based inverter connected to a grid-tied AC microgrid. An optimal controller is designed for power control of VSI connected in a grid-tied microgrid [32]. The controller parameters are obtained by using particle swarm optimization (PSO) to minimize a cost function. In [33], a power control strategy based on cascaded voltage-current control is presented. In this voltage-current control, the disturbance signal containing grid and load current is also estimated and compensated. A sliding mode controller-based power control is presented in [34]. The controller is designed in a $dq$ reference frame for a grid-connected inverter. In [35], the power control of the inverter connected to islanded microgrid is presented. Sliding mode control is presented to provide robustness against external disturbances, including communication failures. In [36], a voltage-oriented power coordination control is presented for a grid-tied inverter in an AC microgrid. An active and reactive power control strategy are shown in [37]. The P-V and Q-f droop curves provided the reference values for cascaded voltage-current controllers. The control is implemented in the $dq$ reference frame. In [38], a power control strategy is presented for VSI in a grid-tied microgrid. In [39], a model predictive control incorporated with the droop method is shown for load sharing in an AC microgrid. A particle swarm optimization (PSO) is also implemented to find the optimal required active and reactive power reference values of inverters to minimize the operational cost of the microgrid. Active and reactive power control is presented in [40] for grid-connected microinverter. The control design is implemented in the $dq$ reference frame. In [41], voltage support and harmonic compensation are provided in a grid-connected microgrid by active and reactive power control. Using the PI control, a reference for the current controller is generated in the $a\xdf$ reference frame, which is controlled by a PR controller. An adaptive fractional fuzzy sliding mode control (AFFSMC) is presented to regulate active and reactive power injected by DG unit into a grid-connected microgrid in [42]. In a microgrid, the controllers of all VSIs need a synchronization signal, which is provided by a global positioning system (GPS) [43].

Most literature on power control is on grid-connected microgrids, where the voltage at the point of common coupling (PCC) is maintained by the main grid. The voltage provided by the main grid does not have disturbances related to the PWM inverter, whereas in the islanded microgrid the voltage is maintained by the master VSI. Due to the voltage at PCC maintained by a PWM inverter in islanded mode, the decoupling of active and reactive power is difficult. This paper presents a state feedback controller with disturbance cancellation for VSI control in power control mode (PCM), connected to an islanded AC microgrid. This control strategy utilizes only a three-phase current and a three-phase voltage sensor. The disturbance is also estimated using an extended high gain observer (EHGO). This EHGO-based control utilizes only a current sensor for active and reactive power control of the DG unit. Stability analyses and simulation results have shown the effectiveness of the proposed control scheme. The contribution of the presented work lies in the direct active and reactive power control in the presence of non-constant disturbance due to PCC voltage maintained by another PWM inverter working as master VSI. Another contribution is the estimation of this disturbance by EHGO; this EHGO-based control uses only a three-phase current sensor and saves the requirement of a voltage sensor.

A mathematical model of the microgrid and control problem is presented in Section 2. The control scheme is presented in Section 3. Section 4 presents the stability analyses. The simulation results are presented in Section 5. The paper is concluded in Section 6.

**2. Problem Formulation**

The voltage source inverter (VSI) is shown in Figure 1. The VSI in this figure is six switches PWM inverter with an output $RLC$ filter. The $RLC$ filter reduces the harmonics from the VSI’s output PWM voltages. In this figure, all signals are represented in the $abc$ reference frame. Here, ${\mathbf{V}}_{\mathbf{t},\mathbf{abc}}$ is the VSI input signal, to be compared with a triangular carrier wave to produce the gating signals. The gating signals apply on PWM inverter switches to produce the PWM ${\mathbf{V}}_{\mathbf{t},\mathbf{abc}}$ signals. ${\mathbf{I}}_{\mathbf{t},\mathbf{abc}}$ is filter input current, ${\mathbf{I}}_{\mathbf{L},\mathbf{abc}}$ is filter output current and ${\mathbf{V}}_{\mathbf{abc}}$ is filter output voltage. ${\mathbf{I}}_{\mathbf{L},\mathbf{abc}}$ is also the current injected into the microgrid from VSI. ${R}_{t}$, ${C}_{t}$, and ${L}_{t}$ are the resistance, capacitance, and inductance of the VSI output filter, respectively. The VSI is connected to the microgrid at the point of common coupling (PCC).

The signals shown in Figure 1 are three-phase signals, represented in $abc$ reference frame, and can be converted into $dq$ reference frame using the park’s transformation. The signals in the $abc$ reference frame and $dq$ reference frame are linked as follows:

${\mathbf{V}}_{\mathbf{a}\mathbf{b}\mathbf{c}}?{V}_{d},{V}_{q}$

${\mathbf{V}}_{\mathbf{t},\mathbf{a}\mathbf{b}\mathbf{c}}?{V}_{td},{V}_{tq}$

${\mathbf{I}}_{\mathbf{t},\mathbf{a}\mathbf{b}\mathbf{c}}?{I}_{td},{I}_{tq}$

${\mathbf{I}}_{\mathbf{L},\mathbf{a}\mathbf{b}\mathbf{c}}?{I}_{Ld},{I}_{Lq}$

The system shown in Figure 1 can be mathematically represented in the $dq$ reference frame by the following dynamical equations [44]. Here, ${?}_{\xb0}$ is the angular frequency of the system.

**(1a)**$\begin{array}{cc}\hfill \frac{d{V}_{d}}{dt}& \hfill ={?}_{\xb0}{V}_{q}+\frac{1}{{C}_{t}}{I}_{td}-\frac{1}{{C}_{t}}{I}_{Ld}\hfill \end{array}$

**(1b)**$\begin{array}{cc}\hfill \frac{d{I}_{td}}{dt}& \hfill =\frac{1}{{L}_{t}}{V}_{td}-\frac{{R}_{t}}{{L}_{t}}{I}_{td}-\frac{1}{{L}_{t}}{V}_{d}+{?}_{\xb0}{I}_{tq}\hfill \end{array}$

**(1c)**$\begin{array}{cc}\hfill \frac{d{V}_{q}}{dt}& \hfill =-{?}_{\xb0}{V}_{d}+\frac{1}{{C}_{t}}{I}_{tq}-\frac{1}{{C}_{t}}{I}_{Lq}\hfill \end{array}$

**(1d)**$\begin{array}{cc}\hfill \frac{d{I}_{tq}}{dt}& \hfill =\frac{1}{{L}_{t}}{V}_{tq}-\frac{{R}_{t}}{{L}_{t}}{I}_{tq}-\frac{1}{{L}_{t}}{V}_{q}-{?}_{\xb0}{I}_{td}\hfill \end{array}$

The active power (${P}_{\xb0}$) and reactive power (${Q}_{\xb0}$) delivered into the microgrid by the VSI are calculated as follows:

**(2)**$\begin{array}{cc}\hfill {P}_{\xb0}& \hfill =\frac{3}{2}\left({V}_{d}{I}_{Ld}+{V}_{q}{I}_{Lq}\right)\hfill \\ \hfill {Q}_{\xb0}& \hfill =\frac{3}{2}\left({V}_{q}{I}_{Ld}-{V}_{d}{I}_{Lq}\right)\hfill \end{array}$

The problem considered in this paper is to control the active power (${P}_{\xb0}$) and reactive power (${Q}_{\xb0}$) delivered into the microgrid by VSI in the $dq$ reference frame and minimize the following tracking error signals to an ultimate bound in finite time. The control signals generated by the controller will be the variables ${V}_{td}$ and ${V}_{tq}$. The reference signals for active power and reactive power are ${P}_{\xb0}^{?}$ and ${Q}_{\xb0}^{?}$, respectively.

**(3)**$\begin{array}{cc}\hfill {e}_{P}& \hfill ={P}_{\xb0}-{P}_{\xb0}^{?}\hfill \\ \hfill {e}_{Q}& \hfill ={Q}_{\xb0}-{Q}_{\xb0}^{?}\hfill \end{array}$

**3. Control Scheme**

System shown in Equations (1a)–(1d) and (2) is a multiple-input multiple-output system, with input variables $\left\{{V}_{td},{V}_{tq}\right\}$ and output variables $\left\{{P}_{\xb0},{Q}_{\xb0}\right\}$. As shown in Figure 2, a microgrid of three parallel connected VSI in master/slave configuration is presented here. Master VSI works in VCM and controls the voltages, whereas slave VSIs work in PCM and control its output active and reactive power $\left\{{P}_{\xb0},{Q}_{\xb0}\right\}$.

Considering the power control problem for a slave VSI, the voltages considered at PCC are as follows:

**(4)**$\begin{array}{cc}\hfill {V}_{dss}& \hfill \u02dc\sqrt{2}\times 220\hfill \\ \hfill {V}_{qss}& \u02dc0\hfill \end{array}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}\begin{array}{cc}\hfill \frac{d{V}_{dss}}{dt}& \hfill \u02dc0\hfill \\ \hfill \frac{d{V}_{qss}}{dt}& \hfill \u02dc0\hfill \end{array}$

The voltage values given in Equation (4) are steady-state values, but using these values within transient period reduces the requirement of a current sensor.

Using Equation (4) values with Equations (1a) and (1c) provide the following relations between $RLC$ filter’s input current $\left\{{I}_{td},{I}_{tq}\right\}$ and current injected into the microgrid $\left\{{I}_{Ld},{I}_{Lq}\right\}$.

**(5)**$\begin{array}{cc}\hfill {I}_{Ld}& ={I}_{td}\hfill \\ \hfill {I}_{Lq}& ={I}_{tq}-{?}_{\xb0}{C}_{t}{V}_{dss}\hfill \end{array}$

Using Equations (4) and (5) with Equation (2) gives following equations to calculate the output active (${P}_{\xb0}^{\text{'}}$) and reactive powers (${Q}_{\xb0}^{\text{'}}$).

**(6)**$\begin{array}{cc}\hfill {P}_{\xb0}^{\text{'}}& =\frac{3}{2}{V}_{dss}{I}_{td}\hfill \\ \hfill {Q}_{\xb0}^{\text{'}}& =-\frac{3}{2}{V}_{dss}\left({I}_{tq}-{?}_{\xb0}{C}_{t}{V}_{dss}\right)\hfill \end{array}$

Using Equation (9), $\left\{{P}_{\xb0}^{\text{'}},{Q}_{\xb0}^{\text{'}}\right\}$ can be calculated using only $\left\{{I}_{td},{I}_{tq}\right\}$. From Equation (6), the relative degree of system is 1. Only Equation (1b) is required for controller design of active power (${P}_{\xb0}^{\text{'}}$) and Equation (1d) is required for controller design of reactive power (${Q}_{\xb0}^{\text{'}}$).

**3.1. Controller Design for Active Power (${P}_{\xb0}^{\text{'}}$)**

The controller for active power (${P}_{\xb0}^{\text{'}}$) can be designed in error coordinates by assuming the following variables:

${e}_{P}^{\text{'}}={P}_{\xb0}^{\text{'}}-{P}_{\xb0}^{?}$, $\frac{d{P}_{\xb0}^{?}}{dt}=0$, ${d}_{d}=-{V}_{d}$, ${u}_{d}={V}_{td}$, ${a}_{d}=\frac{3{V}_{dss}}{2{L}_{t}}$

The system in error coordinates is as follows:

**(7)**$\frac{d{e}_{P}^{\text{'}}}{{d}_{t}}=-\frac{{R}_{t}}{{L}_{t}}\left({e}_{P}^{\text{'}}+{P}_{\xb0}^{?}\right)+{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+{a}_{d}{d}_{d}+{a}_{d}{u}_{d}$

The system shown in Equation (7) can be stabilized by the control equation given below for the properly chosen values of ${k}_{1d}$ and ${k}_{2d}$.

**(8)**$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \hfill {u}_{d}={M}_{d}sat\left[\frac{1}{{M}_{d}{a}_{d}}\left(-{a}_{d}{d}_{d}-{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+\frac{{R}_{t}}{{L}_{t}}{P}_{\xb0}^{?}-{k}_{1d}{e}_{P}^{\text{'}}-{k}_{2d}{e}_{P\xb0}^{\text{'}}\right)\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{d{e}_{P\xb0}^{\text{'}}}{{d}_{t}}={e}_{P}^{\text{'}}\hfill \end{array}$

The controller shown in Equation (8) requires current signals $\left\{{I}_{td},{I}_{tq}\right\}$ and a voltage signal ${V}_{d}$. To save the requirement of a voltage sensor for ${V}_{d}$, ${d}_{d}$ is estimated using extended high gain observer (EHGO) [45] as follows. Let

${\widehat{a}}_{d}{s}_{d}={a}_{d}{d}_{d}+\left[{a}_{d}-{\widehat{a}}_{d}\right]{u}_{d}$; ${\widehat{a}}_{d}$ is the nominal value of ${a}_{d}$

The system shown in Equation (7) can be converted into an augmented system as follows:

**(9)**$\begin{array}{cc}\hfill \frac{d{e}_{P}^{\text{'}}}{{d}_{t}}& =-\frac{{R}_{t}}{{L}_{t}}\left({e}_{P}^{\text{'}}+{P}_{\xb0}^{?}\right)+{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+{\widehat{a}}_{d}{s}_{d}+{\widehat{a}}_{d}{u}_{d}\hfill \\ \hfill \frac{d{s}_{d}}{{d}_{t}}& ={f}_{d}\hfill \end{array}$

The EHGO for the augmented system in Equation (9) is as follows:

**(10)**$\begin{array}{cc}\hfill \frac{d{\widehat{e}}_{P}^{\text{'}}}{{d}_{t}}& =-\frac{{R}_{t}}{{L}_{t}}\left({e}_{P}^{\text{'}}+{P}_{\xb0}^{?}\right)+{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+{\widehat{a}}_{d}{\widehat{s}}_{d}+{\widehat{a}}_{d}{u}_{d}+({a}_{1d}/{?}_{d})({e}_{P}^{\text{'}}-{\widehat{e}}_{P}^{\text{'}})\hfill \\ \hfill \frac{d{\widehat{s}}_{d}}{{d}_{t}}& =({a}_{2d}/{?}_{d}^{2})({e}_{P}^{\text{'}}-{\widehat{e}}_{P}^{\text{'}})\hfill \end{array}$

Here for ${\widehat{a}}_{d}={a}_{d}$, ${s}_{d}={d}_{d}$. In addition, ${\widehat{s}}_{d}$ is an estimate of ${s}_{d}$, which can be used in the following control equation saving the requirement of a voltage sensor. The EHGO-based control equation is as follows:

**(11)**$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {u}_{d}={M}_{d}sat\left[\frac{1}{{M}_{d}{\widehat{a}}_{d}}\left(-{\widehat{a}}_{d}{\widehat{s}}_{d}-{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+\frac{{R}_{t}}{{L}_{t}}{P}_{\xb0}^{?}-{k}_{1d}{e}_{P}^{\text{'}}-{k}_{2d}{e}_{P\xb0}^{\text{'}}\right)\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{d{e}_{P\xb0}^{\text{'}}}{{d}_{t}}={e}_{P}^{\text{'}}\hfill \end{array}$

${M}_{d}$ is the bound on the control signal, calculated as follows:

**(12)**$\begin{array}{cc}\hfill {M}_{d}& \hfill >max\left|\frac{1}{{a}_{d}}\left(-{a}_{d}{d}_{d}-{a}_{d}{L}_{t}{?}_{\xb0}{I}_{tq}+\frac{{R}_{t}}{{L}_{t}}{P}_{\xb0}^{?}-{k}_{1d}{e}_{P}^{\text{'}}-{k}_{2d}{e}_{P\xb0}^{\text{'}}\right)\right|\hfill \end{array}$

**3.2. Controller Design for Reactive Power (${Q}_{\xb0}^{\text{'}}$)**

The controller for reactive power (${Q}_{\xb0}^{\text{'}}$) can be designed in error coordinates by assuming the following variables:

${e}_{Q}^{\text{'}}={Q}_{\xb0}^{\text{'}}-{Q}_{\xb0}^{?}$, $\frac{d{Q}_{\xb0}^{?}}{dt}=0$, ${d}_{q}=-{V}_{q}$, ${u}_{q}={V}_{tq}$, ${a}_{q}=-\frac{3{V}_{dss}}{2{L}_{t}}$

The system in error coordinates is as follows:

**(13)**$\begin{array}{cc}\hfill \frac{d{e}_{Q}^{\text{'}}}{{d}_{t}}& \hfill =-\frac{{R}_{t}}{{L}_{t}}\left({e}_{Q}^{\text{'}}+{Q}_{\xb0}^{?}\right)-{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}-{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{a}_{q}{d}_{q}+{a}_{q}{u}_{q}\hfill \end{array}$

The system shown in Equation (13) can be stabilized by the control equation given below for the properly chosen values of ${k}_{1q}$ and ${k}_{2q}$.

**(14)**$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {u}_{q}={M}_{q}sat\left[\frac{1}{{M}_{q}{a}_{q}}\left(-{a}_{q}{d}_{q}+{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}+\frac{{R}_{t}}{{L}_{t}}{Q}_{\xb0}^{?}-{k}_{1q}{e}_{Q}^{\text{'}}-{k}_{2q}{e}_{Q\xb0}^{\text{'}}\right)\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{d{e}_{Q\xb0}^{\text{'}}}{{d}_{t}}={e}_{Q}^{\text{'}}\hfill \end{array}$

The controller shown in Equation (14) requires current signals $\left\{{I}_{td},{I}_{tq}\right\}$ and a voltage signal ${V}_{q}$. To save the requirement of a voltage sensor for ${V}_{q}$, ${d}_{q}$ is estimated using EHGO [45] as follows. Let

${\widehat{a}}_{q}{s}_{q}={a}_{q}{d}_{q}+\left[{a}_{q}-{\widehat{a}}_{q}\right]{u}_{q}$; ${\widehat{a}}_{q}$ is the nominal value of ${a}_{q}$

The system shown in Equation (13) can be converted into an augmented system as follows:

**(15)**$\begin{array}{cc}\hfill \frac{d{e}_{Q}^{\text{'}}}{{d}_{t}}& =-\frac{{R}_{t}}{{L}_{t}}\left({e}_{Q}^{\text{'}}+{Q}_{\xb0}^{?}\right)-{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}-{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{\widehat{a}}_{q}{s}_{q}+{\widehat{a}}_{q}{u}_{q}\hfill \\ \hfill \frac{d{s}_{q}}{{d}_{t}}& ={f}_{q}\hfill \end{array}$

The EHGO for the augmented system in Equation (15) is as follows:

**(16)**$\begin{array}{cc}\hfill \frac{d{\widehat{e}}_{Q}^{\text{'}}}{{d}_{t}}& =-\frac{{R}_{t}}{{L}_{t}}\left({e}_{Q}^{\text{'}}+{Q}_{\xb0}^{?}\right)-{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}-{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{\widehat{a}}_{q}{\widehat{s}}_{q}+{\widehat{a}}_{q}{u}_{q}+({a}_{1q}/{?}_{q})({e}_{Q}^{\text{'}}-{\widehat{e}}_{Q}^{\text{'}})\hfill \\ \hfill \frac{d{\widehat{s}}_{q}}{{d}_{t}}& =({a}_{2q}/{?}_{q}^{2})({e}_{Q}^{\text{'}}-{\widehat{e}}_{Q}^{\text{'}})\hfill \end{array}$

Here for ${\widehat{a}}_{q}={a}_{q}$, ${s}_{q}={d}_{q}$. In addition, ${\widehat{s}}_{q}$ is an estimate of ${s}_{q}$, which can be used in the following control equation saving the requirement of a voltage sensor. The EHGO-based control equation is as follows:

**(17)**$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {u}_{q}={M}_{q}sat\left[\frac{1}{{M}_{q}{\widehat{a}}_{q}}\left(-{\widehat{a}}_{q}{\widehat{s}}_{q}+{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}+\frac{{R}_{t}}{{L}_{t}}{Q}_{\xb0}^{?}-{k}_{1q}{e}_{Q}^{\text{'}}-{k}_{2q}{e}_{Q\xb0}^{\text{'}}\right)\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{d{e}_{Q\xb0}^{\text{'}}}{{d}_{t}}={e}_{Q}^{\text{'}}\hfill \end{array}$

${M}_{q}$ is the bound on the control signal, calculated as follows:

**(18)**$\begin{array}{cc}\hfill {M}_{q}& \hfill >max\left|\frac{1}{{a}_{q}}\left(-{a}_{q}{d}_{q}+{a}_{q}{L}_{t}{?}_{\xb0}{I}_{td}+{a}_{q}{?}_{\xb0}{R}_{t}{C}_{t}{V}_{dss}+\frac{{R}_{t}}{{L}_{t}}{Q}_{\xb0}^{?}-{k}_{1q}{e}_{Q}^{\text{'}}-{k}_{2q}{e}_{Q\xb0}^{\text{'}}\right)\right|\hfill \end{array}$

The complete control design for the slave VSIs working in power control mode (PCM) is also shown in Figure 3. The EHGO design is shown in Figure 4.

**4. Stability Analyses**

For the systems in Equations (7) and (13), with designed controllers in Equations (8) and (14), the closed loop error dynamics are shown below in Equation (19). Equation (19) clearly shows that error will asymptotically go to zero for properly chosen values of ${k}_{1j}$ and ${k}_{2j}$.

**(19)**$\frac{{d}^{2}{e}_{i}^{\text{'}}}{{d}_{{t}^{2}}}+\left({k}_{1j}+\frac{{R}_{t}}{{L}_{t}}\right)\frac{d{e}_{i}^{\text{'}}}{{d}_{t}}+{k}_{2j}{e}_{i}^{\text{'}}=0$

Similarly, for the systems in Equations (7) and (13), with designed controllers in Equations (11) and (17), the closed loop error dynamics are shown below in Equation (20) for ${\widehat{a}}_{d}={a}_{d}$ and ${\widehat{a}}_{q}={a}_{q}$.

**(20)**$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \frac{{d}^{2}{e}_{i}^{\text{'}}}{{d}_{{t}^{2}}}+\left({k}_{1j}+\frac{{R}_{t}}{{L}_{t}}\right)\frac{d{e}_{i}^{\text{'}}}{{d}_{t}}+{k}_{2j}{e}_{i}^{\text{'}}={a}_{j}\frac{d{d}_{j}}{{d}_{t}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {d}_{j}={d}_{j}-{\widehat{s}}_{j}\hfill \end{array}$

Here in Equations (19) and (20); and in following equations, $i=P$ and $j=d$ for active power control system, $i=Q$ and $j=q$ for reactive power control system.

For the following desired closed-loop characteristic Equation (21):

**(21)**${s}^{2}+{d}_{1}s+{d}_{2}=0$

${k}_{1j}$ and ${k}_{2j}$ are calculated as follows:

**(22)**$\begin{array}{cc}\hfill {k}_{1j}& ={d}_{1}-\frac{{R}_{t}}{{L}_{t}}\hfill \\ \hfill {k}_{2j}& ={d}_{2}\hfill \end{array}$

${d}_{j}$ in Equation (20) is the estimation error of EHGO. Equation (20) clearly shows that the error will go to zero if the following condition holds.

**(23)**$\frac{d{d}_{j}}{{d}_{t}}=0$

The condition in Equation (23) does not hold due to the PWM inverter, so there will be some difference in the performance of the controller without EHGO (Equations (8) and (14)) and with EHGO (Equations (11) and (17)).

The stability analysis of EHGO in Equations (10) and (16) is as follows:

The estimation errors are

$\begin{array}{cc}\hfill {\tilde{e}}_{i}& \hfill ={e}_{i}^{\text{'}}-{\widehat{e}}_{i}^{\text{'}}\hfill \\ \hfill {\tilde{s}}_{j}& \hfill ={s}_{j}-{\widehat{s}}_{j}\hfill \end{array}$

From systems in Equations (9) and (15) with EHGO in Equations (10) and (16), the estimation error dynamics are as follows:

**(24)**$\begin{array}{cc}\hfill \frac{d{\tilde{e}}_{i}}{{d}_{t}}& \hfill =-\left({a}_{1j}/{?}_{j}\right){\tilde{e}}_{i}+{\widehat{a}}_{j}{\tilde{s}}_{j}\hfill \\ \hfill \frac{d{\tilde{s}}_{j}}{{d}_{t}}& \hfill =-\left({a}_{2j}/{?}_{j}^{2}\right){\tilde{e}}_{i}+{f}_{j}\hfill \end{array}$

The transfer function from ${f}_{j}$ to ${\left\{{\tilde{e}}_{i},{\tilde{s}}_{j}\right\}}^{T}$ is

**(25)**${G}_{\xb0j}\left(s\right)=\frac{{?}_{j}}{{\left({?}_{j}s\right)}^{2}+{a}_{1j}{?}_{j}s+{\widehat{a}}_{j}{a}_{2j}}\left[\begin{array}{c}{\widehat{a}}_{j}{?}_{j}\\ {?}_{j}s+{a}_{1j}\end{array}\right]$

The above transfer function in Equation (25) clearly shows that estimation error goes to zero, as ${?}_{j}$ goes to zero. This can also be shown in the time domain using scaled estimation errors:

$\begin{array}{cc}\hfill {?}_{1j}& \hfill =\frac{{\tilde{e}}_{i}}{{?}_{j}}\hfill \\ \hfill {?}_{2j}& \hfill ={\tilde{s}}_{j}\hfill \end{array}$

The scaled estimation error dynamics are as follows:

**(26)**$\begin{array}{cc}\hfill {?}_{j}\frac{d{?}_{1j}}{{d}_{t}}& \hfill =-{a}_{1j}{?}_{1j}+{\widehat{a}}_{j}{?}_{2j}\hfill \\ \hfill {?}_{j}\frac{d{?}_{2j}}{{d}_{t}}& \hfill =-{a}_{2j}{?}_{1j}+{?}_{j}{f}_{j}\hfill \end{array}$

Here, it is clearly shown that the estimation error goes to zero as ${?}_{j}$ goes to zero if the following Equation (27) has all roots with a negative real part.

**(27)**${s}^{2}+{a}_{1j}s+{\widehat{a}}_{j}{a}_{2j}=0$

The transient response of EHGO suffers from the peaking phenomenon [45]. The adverse effects of the