Modeling and Dynamic Stability Analysis of the Grid-Following Inverter Integrated With Photovoltaics

The power transfer capacity of transmission lines is limited by the stability of the power system. In addition, the dynamics of photovoltaic (PV) integration through the grid-following inverter (GFI) affects the stability limits, which are not well studied in the literature. This article, therefore, focuses on the stability-constrained power transfer capabilities of transmission lines with single-phase PV-GFI integration. An adaptive tuning method is proposed based on the Lyapunov function (LF) analysis that increases the power transfer capability limit. Outer and inner control loop parameters are considered as candidates for the proposed method. The proposed tuning law is not affected by the active and reactive power requirements, low-voltage or overvoltage ride-through, network impedance, and load changes. In addition, an active damping method using a capacitor voltage feedback loop is proposed to eliminate the effect of resonance with improved power quality. The proposed method is validated through simulation and experimental verification for various test scenarios, such as sudden changes in solar irradiance, dynamic load changes, network parameter variation, and grid voltage sag–swell conditions. The proposed tuning method achieves an enhanced power transfer capacity limit compared with the existing method.

Abstract-The power transfer capacity of transmission lines is limited by the stability of the power system. In addition, the dynamics of photovoltaic (PV) integration through the gridfollowing inverter (GFI) affects the stability limits, which are not well studied in the literature. This article, therefore, focuses on the stability-constrained power transfer capabilities of transmission lines with single-phase PV-GFI integration. An adaptive tuning method is proposed based on the Lyapunov function (LF) analysis that increases the power transfer capability limit. Outer and inner control loop parameters are considered as candidates for the proposed method. The proposed tuning law is not affected by the active and reactive power requirements, low-voltage or overvoltage ride-through, network impedance, and load changes. In addition, an active damping method using a capacitor voltage feedback loop is proposed to eliminate the effect of resonance with improved power quality. The proposed method is validated through simulation and experimental verification for various test scenarios, such as sudden changes in solar irradiance, dynamic load changes, network parameter variation, and grid voltage sag-swell conditions. The proposed tuning method achieves an enhanced power transfer capacity limit compared with the existing method.

NOMENCLATURE L i , r i
Inverter-side inductor with parasitic resistance. L g , r g Grid-side inductor with parasitic resistance. C f , C dc Filter capacitance and dc-link capacitance.
Effective resistance and inductance of BC. v g , i g Grid voltage and grid current. v cf , v dc Voltage across C f and dc-link voltage. i L , i dcl Load current and loss component of i g . v gm Peak magnitude of the grid voltage. θ g , ω g Grid phase angle and angular frequency. f swi , f g Inverter switching frequency and grid frequency. f r Resonance frequency of LC L filter. i pcc Estimated PCC current. v pv , i pv PV voltage and current. p pv Solar PV power being supplied.
Inverter-side current and BC inductor current. s b , s i Switching function for BC and GFI. p g Average power injected into the grid.
Steady-state and perturbed value of x i .

I. INTRODUCTION
P HOTOVOLTAIC (PV) systems with integration of the grid-following inverter (GFI) have gained popularity among distributed energy resources in recent decades due to their ease of extension and use in rural areas [1]. Concerns about pollution and noise levels are also driving the government policy to increase PV generation in the electrical grid. However, the intermittent nature of PV systems has an impact on the stability limits of power system transmission networks, as well as power quality [2], further degrading the reliability of the two-stage PV-GFI system. As a result, a strong and decisive controller is required to maintain the stability of the PV-GFI system in the face of changing solar irradiation, load demand, and external disturbances [3].
A two-stage PV power generation system cascades dc-dc boost converter (BC) to the dc-link of GFI. Some literature has investigated the limits of power transfer capability of the GFI-based power systems; however, the stability limits for transmission lines originating from the solar PV-GFI systems have not been well explored. In renewable-source- under distortion, transients, fluctuations, harmonics, and dc offsets is more challenging. Therefore, the control design for such PV-GFI systems must meet the criteria of maximum power extraction from input sources, constant dc-link voltage level during linear/nonlinear load operation, and adjustment of the power factor while providing robust and stable dynamic behavior. The control of a PV-GFI system can be divided into two parts: the dc-side control and the ac-side control [4]. The dc-side control implies to the BC, which uses perturb and observe (P&O) [5]-based maximum power point tracking (MPPT) algorithm to extract maximum power from the PV array despite changes in solar irradiation and temperature. To stabilize the maximum power point (MPP) operating point, the proportional-integral (PI) controller is commonly used, but the PI controller often struggles to maintain stability when the operating point drifts away from the MPP [6]. In spite of this, it has various instability issues due to the sampling of the controller and parameter dependency. The fundamental disadvantage of these closed-loop systems is that they do not guarantee global stability when substantial disturbances are introduced. The Lyapunov-function-based control (LFBC) technique improves dynamic performance and ensures global stability during transients. The LFBC technique for BC is used under uncertainty and external disturbances to achieve global stability in a standalone PV system [7]. However, such an LFBC approach can be used for BC systems in grid-connected applications.
Again, the ac-side control performs an inverter switching action to achieve grid synchronization and dc-link voltage control, in addition to satisfying the grid power quality requirements. Two approaches in [8] and [9] are concerned with adjusting the dc-link voltage in response to the changes in grid voltage and minimizing the losses in the GFI. Current harmonics created by ripples in the dc-link and high-frequency switching of the GFI can be decreased in a PV-GFI system using filters L and LC L [10]. Compared with filters L and LC, the LC L filter delivers greater switching harmonic attenuation, less power loss, smaller component size, and lower inverter current ripple. The PI-based current controllers are most effective in the case of three-phase GFI due to their simplicity and the ability to achieve zero steady-state tracking error with a specific operating point [11]. However, complexity in controller tuning gains along with incapacity to deal with periodic disturbances restrict the PI controller to being used in the single-phase GFI system [12], which brings effective usage of the proportional resonant (PR) controller in the single-phase GFI systems. Appropriate damping is essential to eliminate intrinsic resonance in the LC L filter to solve the instability of the power system [13]. A passive damping strategy adds power loss due to additional resistance in the system. When the PR controller approach is used for a GFI system without active damping, the poles are positioned on the imaginary axis of the s-plane. As a result, the grid current suffers frequency oscillation, which could lead to instability in the GFI system. The PR controller uses active damping (a virtual resistor concept) to minimize frequency fluctuation in the grid current, which moves the pole to the left of the s-plane [14]. The performance of the PV-GFI system with a hybrid damping approach is explored in [15], which adds to the complexity of the control algorithm and the power loss. Although the PR used in [16] for a single-phase PV-GFI system has around zero steady-state error, it cannot eliminate the current harmonics that feed into the grid. To avoid control saturation in the existing PR controller, a novel approach is developed in [17], which also achieves accurate current control with low total harmonic distortion (THD) values. An adaptive quasi-PR controller is proposed in [18] for the control of single-phase grid-connected converters operating under multiple perturbations. The grid-integrated PV systems under distorted grid voltage need enhanced grid current harmonic compensation and power ripple mitigation. The control strategy described in [19] includes an advanced proportional multiresonant controller and a compensation for grid voltage sag.
The GFI system, which has its pole on the left side of the s-plane, can be controlled by the LBFC technique without active damping. The LFBC controller provides global stability without active damping. The LBFC with a capacitor voltage feedback approach can enhance the damping ratio by shifting the conjugate poles far away from the imaginary axis. The LFBC can outperform the PR controller due to a larger damping ratio and a quicker transient response. Hence, [20] integrates the LFBC with the PR controller to show good performance for the single-phase grid-connected GFI with fixed dc source. However, the impacts of nonlinear demand and external disturbances caused by PV intermittency are not considered in [20]. Literatures in [21], [22], and [23] assessed the stability analysis of LFBC techniques for the PV-GFI system with L and LC filters. The stability of the PV-GFI system is established in [21] using the Lyapunov function (LF) to achieve robustness against nonideal grid conditions. Since all the phase-locked loops (PLLs) and multiloop controllers have been removed, high dynamic and tracking performance can be further achieved with LFBC. In [24], grid-forming inverters are discussed with transient stability analysis, considering system damping as part of a large-signal analysis. With the aid of an energy-function-based stability assessment, [24] discusses how GFI can cause instability due to loss of synchronism. The stability of PV power generation based on dc-link voltage and reactive power regulation is evaluated in [23], while [25] discusses the stability of PV-GFI with partial shading conditions. It is obvious from [22] and [26] that the PV control parameters have an impact on the transient stability. Increased power transmission capability can result in improved stability by modifying the PV control parameters.
The analysis presented in [27] was insufficient to represent the two-stage control action of the PV-GFI system. Under significant perturbations away from the operating point induced by environmental and load circumstances, this article proposes a controller that can provide a global asymptotically stable performance. The proposed controller can inject the grid current and maintain the global stability of the system without affecting the system dynamics. The proposed control mechanism for the PV-GFI system compensates for nonlinear load requirements while delivering active power from the PV array to the grid. In contrast to the approaches in [28], there is no time delay in acquiring the compensating current reference during a linear/nonlinear load scenario. The current article incorporates the following original contributions.
1) This article discusses the performance analysis of a single-phase PV-GFI system and develops an LFBC strategy while considering the effect of the LC L filter and intermittency of the PV system. 2) The dc-link voltage is a time-varying function that injects dc offset and second-order harmonic ripple content into the grid. The dc-link capacitor C dc is included for analysis to show the intermittency of the PV system in relation to the grid voltage.
3) The proposed LFBC uses a root locus approach to examine closed-loop poles for global stability and active damping to reduce resonance without affecting stability, allowing adaptive tuning of parameters and PLL block using LFs with PV signal measurements to improve power transfer efficiency and performance under solar irradiation and filter parameter variations. Section II presents the PV-GFI system and its modeling, while Section III explains the LFBC control for GFI and BC with parameter tuning methods for GFI and BC control. Various test situations are discussed in Section IV. The conclusion is drawn in Section V.

A. System Description
The two-stage configuration of a single-phase PV-GFI system with the LC L filter is illustrated in Fig. 1. The PV-GFI system is made up of a PV array, a dc-dc BC, and a GFI that is also linked to an LC L filter. The LC L filter is used to reduce the switching harmonics at the ac terminals of the GFI. BC is used to extract power from the PV array at MPP as solar irradiation changes, which involves the first stage of the conversion process. In the second stage of the conversion process, the GFI feeds power into the single-phase grid. The GFI dc-link voltage is adaptive and adjusts the reference voltage based on the voltage variation at the PCC. In addition, the PCC also feeds a mix of linear and nonlinear local loads. By connecting an R L load bank through a diode bridge rectifier, a nonlinear load is emulated.

B. Small-Signal Modeling of GFI
As mentioned, GFI feeds power to the grid through the LC L filter consisting of the capacitor (C f ), inverter-side inductor (L i ), and grid-side inductor (L g ) with internal resistances r i and r g , respectively. To design an improved model of a GFI system with PV, the nonlinearity and intermittency of the PV system are taken into account by including the effect of the dc-link capacitor (C dc ) voltage. The dynamics involved in both the ac and dc sides of GFI can be evaluated using the following equation: where v g , v cf , and v dc are the measured grid voltage, the voltage across C f , and dc-link voltage, respectively, through three voltage sensors. i i and i L are the inverter-side current and the load current, respectively, and are measured through two current sensors. Let x i ∈ {v dc , v cf , i i , i g , s i } be the state variables for the GFI small-signal modeling, that is, x i = x i +x i , wherex i andx i denote its steady-state and perturbed value, respectively. Here, s i is the GFI switching function. The steady-state value ofî i ,v cf , andŝ i can be represented as follows:î Using an appropriate PLL, the peak magnitude of the grid voltage (v gm ) at PCC can be obtained in the PCC with the phase angle (θ g ). For appropriate GFI control, the reference/steadystate value of dc-link voltage (v dc ) must be maintained at µv gm , i.e.,v dc = µv gm . The variable µ is considered as 1.1-1.2 to compensate for the various losses that occur in the operation of the GFI. Moreover, the difference between v dc andv dc is fed to a PI regulator with an appropriate limiter to estimate the loss component of the current (î dcl ). i dcl further combining with the estimated PCC current (î pcc ) and the load current (i L ) provides the estimated value of the peak grid currentî gm . Multiplying sinθ g byî gm provides the reference/steady-state value of the grid currentî g , i.e., i g =î gm sinθ g . The value of dc-link capacitor (C dc limits the GFI lifetime. Hence, a small value of C dc is desired to regulate the ripples in v dc within its reference value ofv dc and can be expressed as follows: where ω g (= 2π f g ) is the grid angular frequency considering f g as the grid frequency, and P g is the average power injected into the grid. Fig. 2(a) indicates the ripple voltage at dc-link for different values of C dc . It can be seen that the ripple voltage decreases with increasing dc-link capacitor. However, the large value of C dc leads to an increase in the size, cost, and weight of the GFI with a slower dynamic response.
Taking into account the small-signal model in (1), the averaged linearized state-space model of the GFI system can be further simplified as follows: where τ g (=L g /r g ) and τ i (=L i /r i ) represent the grid-side and GFI-side time constant of the LC L filter, respectively. In the PV-GFI system, the design of the LC L filter is essential to attenuate the harmonics and dampen out the filter resonance. The design of L i is based on the current ripple in i i at the inverter switching frequency f swi , while the design of L g depends on the attenuation of current harmonics injected into the grid. From Fig. 1, the transfer function G i (s) can be obtained as (5), where the effect of parasitic r g and r i is not considered in the calculation The filter is selected based on the methods given in [29] and [30]. Considering the design parameters of Table II, the Bode plot is obtained in Fig. 2(b). The resonance frequency can be selected as 1044.28 Hz according to the calculation given as follows: where f swi is considered as the switching frequency of the GFI. The design of the LC L filter is based on the selection of resonance frequency ( f r ) and will be selected within the frequency band of the closed-loop system given by (6). Furthermore, Fig. 2(c) depicts the effect of changes in the LC L filter parameters during resonance damping. Increasing LC L filter parameters by 15%, the resonance frequency decreases and causes undesirable oscillations in i g . However, at 15% lower value of LC L filter, the resonance frequency increases by introducing harmonics into the grid. Hence, it is possible to filter out the high switching frequency component of i i , while damping out the resonance in grid current i g using an effective active damping scheme through linearization process mentioned in Fig. 3. Fig. 1(left) depicts the circuit diagram of a PV array that supplies solar power ( p pv ) to the dc-link of GFI through a BC. The generated p pv (= v pv i pv ) needs to be injected into PCC after differencing conversion losses (dc-link loss, switching loss, parasitic resistance losses) and the local load power. The estimated PCC current (î pcc ) injected into the infinite grid can be deduced in terms of p pv andv gm , i.e.,î pcc = 2 p pv /v gm . The further utilization ofî pcc is indicated in Section II-B and Fig. 4(a). Furthermore, the purpose of the MPPT control loop for BC is to regulate the inductor current to match the reference PV current generated by the current-oriented MPPT algorithm, as discussed in [31].

C. Small-Signal Modeling of PV With BC
The nonlinearity and intermittency of the PV system are considered when designing reformable modeling of the boost system with PV, as follows: where r b and L b are the resistance and inductance of BC, respectively. The perturbation signals (x b ) about the nominal steady-state values (x b ) are introduced to obtain the BC smallsignal model with the variable x b ∈ {v pv : PV voltage, i pv : PV current, i b : BC inductor current, and s b : switching function for BC}, i.e., x b =x b +x b . By linearizing (7), the average state-space model can be written as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. where is the time constant of the BC. From (7), the steady-state value of the switching function can be obtained as follows: III. PROPOSED LF-BASED GFI CONTROL The behavior of the PV-GFI system at its equilibrium point using Lyapunov's stability criteria is studied in this section. This can be accomplished by considering the fact that energy from the PV-GFI system dissipates continuously along its trajectories. Thus, the energy dissipation gradually converges to zero as the system state trajectories approach the equilibrium point. A generalized LFBC methodology is proposed in Section III-A, which can be extended to the GFI and BC, as discussed later in this section.

A. Proposed Control Methodology
A step-by-step description for designing the LFBC methodology using the linearized average state-space model of any switching converter is provided below, and its flowchart is depicted in Fig. 3.
Step 1: Derive the state-space averaged model for the switching converter.
Step 2: Linearize the average state-space model around the steady-state point corresponding to the control target s j . The small-signal model for the switching converter can be defined as follows:ẋ where j ∈ {b : for BC, i : for GFI}; A j and B j are the system and input matrices, respectively.x j (= x j −x j ) and s j (= s j −ŝ j ) represent the state variable and input switching function of the jth converter, respectively.
Step 3: Develop Lyapunov's energy function E j (x j ) using Here, E j (x j ) is considered positive definite, that is, E j (x) > 0, as the matrix Q j in the following equation is symmetrical and positive definite Step 4: Define s j such that E j (x j ) must decay over time and can be achieved by taking the time derivative of the energy function (Ė j (x j )) as follows: Step 5: Obtain the global stability of the converter system by solving the inequalityĖ j (x j ) < 0, i.e., term1 and term2 contribute to a negative value as follows: where P j is a positive semidefinite matrix and can be evaluated using (14). However, the optimal value of λ j should be selected such that the value λ j should be a tradeoff between the overshoot and the settling time of the converter system.
Step 6: Evaluate the stabilizing control input variations j from the second part of (14) as follows: Step 7: Compute the linearized closed-loop system matrix (A cj ) as (16) and find the eigenvalue of A cj for different values of λ j Step 8: Compute the total stabilizing control input s j and simulate the nonlinear closed-loop system numerically.

B. Control for GFI With Stability Analysis
The primary goal of controlling the GFI with the LC L filter is to feed clean PV power into the grid. This can be accomplished through an outer voltage loop control and an inner current loop control. The outer loop control generates the reference current to the inner loop controller by regulating the dc-link voltage v dc . The inner loop control injects current into the grid so that the grid current i g should follow the reference currentî g . In this section, the LFBC method for GFI is used to determine the control input s i so that the tracking error asymptotically converges to zero. The Lyapunov energy function for the GFI is defined as E i (x) = (1/2)x T i Q ixi according to (11). Here, Q i can be constituted as Q i = diag{L i , L g , C f , C dc } based on (12).
To achieve the global stability of the GFI, the differentiation of E i (x i ) should be negative, i.e.,Ė i (x i ) < 0. This leads to term1 ofĖ i (x i ) [as in (13)] being negative and yields A T i Q i + Q i A i = −P i . The matrix P i can be obtained as P i = diag{2r i , 2r g , 0, 0} by substituting the matrix of Q i and A i from (4) and (12). The matrix P i is found to be positive semidefinite. Therefore, the stabilizing control inputs i can be effectively calculated ass i = −λ i B i T Q ixi considering that term2 ofĖ i (x i ) [as in (13)] is negative. Here, the gain λ i is a positive scalar, i.e., λ i > 0. By substituting the matrices Q i , A i , and B i from (4),s i can be further evaluated as follows: Equation (17) does not eliminate the damping effect of the resonance. Therefore, (17) provides a globally stable system with a smaller damping factor, as detailed in Section III-D.
Equation (17) (17) to decrease the grid current oscillation, which is expressed as The switching function of GFI s i can be obtained by adding (18) and (2), which are represented by Equation (19) can lower the resonance of a closed-loop single-phase PV-GFI system without compromising its overall stability. The closed-loop state matrix A cgi (=A i − B i λ i B T i Q i ) can be obtained as (20) using (16) A cgi The stability analysis of the GFI system matrix A cgi may be produced by forming a characteristic equation as where D g0 , D g1 , D g2 , D g3 , and D g4 are the coefficients of the characteristic equation (21) and can be represented by Fig. 5(a) shows a damping ratio performance curve with different values of λ i and λ g by analyzing (21). It can be seen that the value of ζ decreases as λ g increases. Therefore, the proposed LFBC system with grid current feedback is unable to enhance damping. This implies that the grid current feedback damping does not suppress the fluctuation of the grid current, as mentioned in Fig. 6(a).   2) Case-2 (Capacitor Voltage Damping): In the proposed LFBC, an extra capacitor feedback loop is used in (17) to actively dampen the oscillations of the grid current. Now, (17) can be updated to (23) by including the capacitor voltage errorṽ cfs With this modification, the proposed control strategy for the PV-GFI inverter is shown in Fig. 4(a). The switching function of GFI s i can be expressed using (23) and (2) as follows: The instantaneous value of s i is then used for the pulse (S 1 , S 2 , S 3 , S 4 ) generation through simple sinusoidal pulse width modulation (SPWM) technique, which uses the career frequency of f swi . Equation (24) can dampen the resonance without destroying the global stability of the closed-loop single-phase PV-GFI system. The closed-loop state matrix A cvi (=A i − B i λ i B T i Q i ) can be obtained as (25), as shown at the bottom of the next page, using (16).
Stability analysis of the GFI system matrix A cvi can be obtained through the formation of characteristics as follows: where s is an operator in the Laplace domain, and I is the 4 × 4 identity matrix. The coefficients D v0 , D v1 , D v2 , D v3 , and D v4 are represented as follows: dc + L i r g + L g r i + C dc Cr g λ iv 2 dc + L i + L g + C f r i r g +v dc λ v L g D v3 = C f L i L g λ iî 2 i + C dc L g λ iv 2 dc + C dc L i r g + C dc L g r i D v4 = C dc C f L i L g .
From the known values of the system parameters, the stability of the system can be obtained at λ i > 0 and λ v > 0.

C. Control of BC With Stability Analysis
An LFBC approach is proposed for BC to track the reference PV current under meteorological conditions. The Lyapunov energy function for BC is defined as Here, the matrix Q b can be restructured as Q b = diag{L b , C pv } using (12). The time derivative of E b (x b ) should be negative to maintain the global stability of BC, as discussed in Step 4 of the proposed methodology. Now, the condition of term 1 in (14) can be arranged for BC with The matrix P b can be obtained as P b = −diag{2r b , 0} by substituting Q b and A b from (8). It is observed that P b is positive semidefinite. Using (15) as a reference, the control inputs b can be effectively derived as s b = −λ b B T b Q bxb to stabilize BC behavior. Here, the gain λ b can be considered as a positive scalar, i.e., λ b > 0. The selection of λ b should be a tradeoff between the overshoot and the settling time of the BC performances. Now,s b can be derived as (29) by replacing the quantities of (8) The desired switching function of BC (s b ) can be expressed as (30) by adding (9) and (29) Using the instantaneous value of s b , a simple PWM technique is used to generate a pulse (S b ) to drive BC using the triangular career with f swb as the switching frequency. To formulate (30) and track the maximum power from the PV array, LFBC requires one voltage sensor (for v pv ) and one current sensor (for i pv ). Instead of using an additional current sensor to detect the inductor current i b , it can be calculated using the MPPT algorithm, as shown in Fig. 4(b). As a consequence, the cost of the current sensor can be reduced. The estimated reference inductor currentî b can be derived from (7) for the kth sampling instant and is given as follows: where T s is the sampling time of LFBC. The proposed LFBC strategy for BC is shown in Fig. 4(b). The closed-loop state matrix of BC can be computed as (32) by substituting the matrices Q b , A b , and B b The stability of BC can be evaluated from the characteristic equation of the system matrix A cb , which can be represented as follows: where ζ b and ω b represent the damping factor and the natural frequency of LFBC for BC, respectively. I indicates the 2 × 2 identity matrix. Now, the eigenvalues of the matrix A cb can be represented as follows: The fact that the real part of the eigenvalues in (34) is negative for λ b > 0 indicates the stability of the BC system. The damping factor ζ should be 1/ √ 2 for a fast transient response with less overshoot and a minimum settling time. The value λ b can be calculated to be 0.000137 A −1 V −1 by substituting Table I into (33).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

D. Selection of Controller Gains for GFI
The proper design of the controller gains (λ i and λ v ) is required to achieve fast dynamic response and global stability of the GFI system. The gains λ i and λ v can be selected so that the conjugate poles of the closed-loop system should provide a higher damping ratio. The global stability of the system can be achieved by selecting the optimal value of λ i and λ v , which can be obtained from the closed-loop pole at the root locus of the system matrix A cvi .
The effect of the closed-loop pole in (25) is analyzed using the root locus technique with different values of λ i and λ v . Fig. 7(a) shows the impact of λ i ∈ [0, 0.04] on the system by considering the gain λ v as zero. With an increasing value of λ i , the conjugate poles ( p 3 and p 4 ) shift away from the imaginary axis and increase until the maximum value of the damping ratio reaches 0.116. While p 3 and p 4 move further toward the imaginary axis, the damping ratio decreases. The real poles p 1 and p 2 have also moved away from the imaginary axis. As the highest damping ratio cannot be achieved with the given λ i , the effect of resonance remains in the system. Fig. 7(b) illustrates the root locus of the conjugate pole with different values of λ i and λ v , where λ i is varied with a different value of λ v . The real poles are nondominant poles; therefore, it is not considered for analysis in Fig. 7(b). In case the value of λ v increases, the conjugate pole moves away from the left half of the s-plane and the maximum damping ratio can be achieved with λ v which eliminates the resonance effect. When ζ = 0.707, the values of λ i and λ v are chosen for the optimal transient response. To analyze the global stability of the GFI system in the worst case scenario, a ±15% change in parameters must occur at the same time. The root locus of the closed-loop pole in (25) under parameter variation in L i , L g , C, and C dc is shown in Fig. 7(c). The closed-loop poles of A cvi lie on the left side of the s-plane. Thus, the proposed LFBC for the GFI system ensures global stability in the worst case scenario. Fig. 7(d) shows the effect of the damping ratio in (26) under variations in the parameters of the GFI system at constant values of λ i and λ v . At ±15% parameter fluctuation in the GFI system, the closed-loop pole of A cvi is directed diagonally upward or downward. Moreover, Fig. 8 demonstrates the relationship between the damping ratio and λ i under parameter changes in ±15% at constant λ v . The value of ζ remains nearly constant when the system parameters are changed, and the influence of the transient response is unchanged.
The root locus approach is used to investigate the influence of global stability under the parameter variation in C PV and L B for the BC system, as shown in Fig. 7(e). For this investigation, the characteristic equation in (33) ] is considered with a parameter variation in 0 → ±15%. It is observed that the system is stable with the placement of poles on the left of the s-plane. Furthermore, Fig. 7(f) illustrates the impact of changing a parameter by a constant value of λ b on the damping ratios with loci of the poles pointing upward or downward. Consequently, the parameter variations in BC have no significant influence on the damping performance, indicating that the transient performance is unaffected.

IV. RESULTS AND ANALYSIS A. Prototype Description
The effectiveness and feasibility of the proposed LFBC technique for a single-phase two-stage PV-GFI system are initially verified by the Simulink simulation software. Furthermore, an experimental prototype is developed for such a PV-GFI system, as shown in Fig. 9. The PV-GFI prototype consists of one single-phase GFI, one BC, a PV emulator, a sensor module, a protection circuit, and a dSPACE MicroLabBoxbased DSP processor. The single-phase GFI and BC consist of SEMIKRON-made SKM100GB063D IGBT modules, driver boards, and the desired dc-link capacitance. The sensor module has three LEM LA55-P current sensors (to measure i pv , i i , i L ) and four LEM LV25-P voltage sensors (to measure v pv , v dc , v cf , v g ). A protection circuit consisting of a singlephase contactor is connected at the PCC terminal of the PV-GFI system. The trigger signal for the contactor is generated during undesirable circumstances by the MicroLabBox. As the PV emulator has a rating of 3.5 kW, the components of the entire system are designed accordingly. The parameters used for the respective system are listed in Tables I and II. The sampling frequency of 10 µs is used in MicroLabBox for seamless control with the LFBC implementation.

B. Test Scenarios
The robustness of the proposed LFBC is verified under the four different test scenarios listed below. 1) Scenario 1: Sudden change in solar irradiation.
2) Scenario 2: 15% variation in LC L filter parameters. 3) Scenario 3: Grid voltage sag/swell conditions. 4) Scenario 4: System performance under dynamic loads.   These four scenarios appropriately define the various disturbances caused by both PV and the single-phase grid. During these tests, dynamic performances such as v g , i g , and p g are evaluated along with the % THD of i g to investigate the grid conditions. In addition, the dynamic behavior of v pv , i pv , p pv , and v dc is also evaluated to analyze the PV performance during various test scenarios. Scenario-1: Fig. 10 illustrates the dynamic performance of a single-phase two-stage PV-GFI system under a sudden change in solar irradiation. It is observed from Fig. 10(a) that the increase or decrease in the PV current i pv depends on the sudden change in solar irradiation. To achieve maximum power, a small change in PV voltage v pv occurs during this scenario. The dc-link voltage v dc adaptively maintains a stable value, considering the effect of BC on the MPPT control. The maximum power of p pv = 3.32, 2.315, and 1.62 kW can be drawn from the PV panel during solar irradiation of 1000, 700, and 500 W/m 2 , respectively. Subsequently, the PV feedforward termî pcc also decreases with a decrease in solar irradiation. As mentioned earlier, the grid current decreases in accordance   Fig. 10(b). In addition, the injected grid current is also observed to be in phase with the grid voltage. The FFT analysis of i g for Scenario 1 is shown in Fig. 10(b) (bottom) with a frequency per division of 62.5 Hz. It can be observed that the fundamental peak occurs at 50 Hz, while the third, fifth, and seventh harmonics are less prominent. Therefore, the % THD of i g at 700 W/m 2 is measured to be ≈2.5%. There appears to be an injection of p g electrical power into the utility grid with a unity power factor. In this scenario, the variation in the injected p g , which is kept within the IEEE 519-2014 standard limit of 5%, is more noticeable since i g has a lower % THD.
Scenario-2: In this scenario, the evaluation of the proposed LFBC for a single-phase two-stage PV-GFI system is performed while the estimated parameters differ from the actual parameters by 15%. During this scenario, a linear change in solar irradiation is observed, and the results are verified in Fig. 11. The resilience and reliability of the proposed LFBC are observed in the worst case scenario where all the parameter fluctuations exist simultaneously. Despite these parameter variations, i pv decreases linearly with small variations in v pv during the linear change in solar irradiation from 1000 to 500 W/m 2 , as shown in Fig. 11(a). Moreover, the proposed LFBC with the MPPT algorithm tracks the maximum power p pv from the PV panel. The dc-link voltage v dc remains constant during this scenario. In addition, the reference current i g decreases with a reduction in PV power. The proposed LFBC control can dampen out the resonance effect in grid current i g in the case of parameter variations, as shown in Fig. 11(b). Furthermore, the grid current i g maintains unity power factor with respect to grid voltage v g . As a consequence, the proposed controller guarantees stability even in the presence of parameter variations. Fig. 11(b) (bottom) shows an FFT analysis of i g at 312.5 Hz/div. The fundamental peak is observed at fundamental frequency, and low-order harmonics have significantly less effect under Scenario 2. The % THD is determined to be 1.5% at 1000 W/m 2 , which is substantially below the IEEE 519-2014 standard limit of 5%.
Scenario-3: The dynamic performance of the proposed controller under sag/swell of the grid voltage is shown in Fig. 12. During this scenario, the PV panel is operated at constant irradiation and temperature that maintains v pv , i pv , Fig. 13. Experimental results of system performance under dynamic loads. (a) v pv , i pv , v dc and P pv waveforms. (b) i g , i L and i inv waveforms along with FFT of i g . (c) P pv , P g and P L waveforms. and p pv to its rated value, as shown in Fig. 12(a). The maximum amplitude of v g is used to estimate the referencev dc using a loss factor µ. Hence, the adaptive v dc decreases during the sag condition and increases in the swell condition of v g , as seen in Fig. 12(a). Since LFBC operates in constant power mode, the reference grid currentî g increases as v g decreases or vice versa. The proposed LFBC has the ability to track the reference current efficiently and also inject the grid current into the grid. Fig. 12(b) shows an increase in i g in a voltage sag scenario. When v g is restored from 195 to 230 V, i g maintains the previous value. When v g increases from 230 to 265 during the swell condition, the grid current i g decreases further. Fig. 12(b) illustrates injection of p g into the grid at Comparative experimental results for a sudden change in solar irradiation using [33]. (a) v pv , i pv , v dc and P pv waveforms from top to bottom. (b) v g , i g , P g and cos φ g waveforms along with FFT of i g . almost unity power factor. Fig. 12(b) (bottom) indicates the FFT analysis of i g at 62.5 Hz/div under the 0.15 p.u. voltage sag that occurs in grid voltage. In Scenario 3, the % THD of i g is approximated to 3.4% with the proposed LFBC technique. The proposed control approach maintains % THD of i g below 5% according to the IEEE 519-2014 standard under voltage sag/swelling condition.
Scenario-4: The dynamic performance of a single-phase two-stage PV-GFI system under load perturbations at constant solar irradiation is shown in Fig. 13. A diode bridge rectifier with the R L load is considered as a nonlinear load for this case. It can be observed from Fig. 13(a) that v pv , i pv , and p pv remain in the rated value of the PV panel during load perturbations. Fig. 13(b) depicts the performance of i i , i L , and the injected current during linear and nonlinear load conditions. It can be observed that i i acts as a harmonic compensator for the nonlinear load. Moreover, the proposed LFBC ensures injection of sinusoidal current, thereby minimizing harmonics. Thus, the GFI acts as a harmonics compensator and injects the active power generated from the PV array into the grid. The power flow of a PV-GFI system is shown in Fig. 13(c). It shows the power flow between the PV, local load, and the grid. It can be concluded that PV power maintains its rated value of 3.32 kW while the load fluctuates. The injected power is the sum of the PV and load power vectors, considering the additional losses in the PV-GFI system. When the load demand Fig. 15. Comparative experimental results with 15% variation in LC L filter parameters using [33]. (a) v pv , i pv , v dc and P pv waveforms. (b) v g , i g , P g and cos φ g waveforms along with FFT of i g .
increases, the active power injected into the grid decreases. The observed % THD of i g is 2.3%, as shown in Fig. 13(b). The proposed controller can maintain a sinusoidal grid current without distortion even when the nonlinear load is connected or disconnected.

C. Comparative Performance
Figs. 14-16 show the comparative performance evaluation of the GFI system using the finite-control-set model predictive control (FCS-MPC) [33] technique. The proposed method has less undershoot in i pv compared with the FCS-MPC method in PV voltage, as shown in Fig. 14(a). At the bottom of Fig. 14(b), the FFT analysis of i g is presented at a frequency per division of 62.5 Hz/div. In Scenario 1, the fundamental peak is at 50 Hz, and the low-order harmonics are less evident. The % THD of i g is 4.5% corresponding to solar irradiance of 700 W/m 2 . Similarly, the % THD of i g can be achieved as 7.5% at 500 W/m 2 and 4.43% at 0.15 voltage sag under scenarios 2 and 3, respectively, as shown in Figs. 16(b) (bottom) and 15(b) (bottom). Small peaks at low-order harmonics are seen in the FFT analysis of Fig. 15  Comparative experimental results with grid voltage sag/swell conditions using [33]. (a) v pv , i pv , v dc and P pv waveforms. (b) v g , i g , P g and cos φ g waveforms along with FFT of i g . the performances of the FCS-MPC method have less accuracy under parameter variation, since this method is completely parameter-dependent.
The proposed LFBC along with the FCS-MPC [33] and PI [5] controller is used for the dynamic study of BC during the change in solar irradiance and solar temperature. The performance evaluation of the undershoot and settling time t s of i pv at the stochastic PV system behavior is shown in Fig. 17. During the simulation study, identical operating circumstances are maintained with the same rating of the PV-GFI system. The proposed LBFC approach has less undershoot in i pv than the existing methods, with a setting time of 30 ms under the change in solar irradiance. Compared with the FCS-MPC and PI methods under change in temperature, the proposed LBFC technique has a fast settling time and a minimum undershoot in i pv . The assessment of % THD of i g is performed for the GFI system during the change in solar irradiance and the change in the grid voltage. This comparative evaluation is done for innerloop current controllers such as the proposed LFBC, FCS-MPC [33], PR [14], and PI [14] controller, as shown in Fig. 18. Comparative performance evaluation for the GFI system is made using the FCS-MPC controller [33], PR [14], PI [14], and the proposed methods. Fig. 18(a) and (b) shows the % THD of i g with the change in irradiance and grid voltage, respectively. The comparative study validates the superiority of the proposed LFBC over the existing control strategies FCS-MPC, PR, and PI method. Furthermore, the proposed LBFC approach keeps the % THD of i g within the IEEE 519-2014 limit and lowers the % THD when there is a change in irradiance and grid voltage.
Incorporating the grid-integrated rooftop PV system, Table III shows comparison of the proposed LFBC methodology with the existing methods [14], [17], [18], [25], [33]. Compared with the existing methods, the proposed technique shows good performance even when the system parameters vary. However, FCS-MPC has limited accuracy under parameter uncertainty and unknown disturbances. The proposed techniques have inherent load rejection capability, which eliminates the need for an additional filter. The linear controller [14], [17], [18] can ensure local stability at a specific operating point. However, it cannot guarantee stable operation in all the situations caused by significant changes in the operating point. The proposed LBFC approaches allow for a quick dynamic response and guarantee the global stability of the closed-loop system.

D. Performance Analysis of the Proposed Controller in Three-Phase System
To evaluate the performance of the proposed control techniques, the GFI system used specific parameters, including inverter-side resistance per phase of 0.17 , grid-side resistance per phase of 0.076 , inverter-side inductor per phase of 1.436 mH, grid-side inductor per phase of 0.6867 mH, filter capacitor per phase of 50 µF, and grid phase voltage (v a g ) of 110 V. The parameters for the PV array are also used, as specified in Table I, with the number of series and parallel panels being N se = 6 and N p = 1, respectively.
The effectiveness of the proposed control techniques for the three-phase two-stage PV-GFI system under changes in the irradiance of the PV array is shown in Fig. 19. A sudden variation in solar irradiation can cause a change in  19.
Dynamic performance of the proposed LBFC controller for a three-phase PV-GFI system under change in solar irradiance. (a) v a g , i a g , i b g , and i c g . (b) v pv , i pv , v dc , and p g .
the PV current (i pv ), as depicted in Fig. 19(b). To achieve maximum power output, a corresponding slight variation in PV voltage (v pv ) occurs. In addition, the dc-link voltage (v dc ) remains constant during this event. In accordance with the reference generated by the outer loop controller, the grid current decreases, as shown in Fig. 19(a). As a consequence, a reduction in injected power ( p g ) occurs with a decrease in solar irradiation. In this situation, the grid phase voltage (v a g ) and grid phase current (i a g ) are maintained at unity power factor. In this scenario, it is observed that i a g exhibits a THD of less than 5% and retains a sinusoidal waveform.

V. CONCLUSION
The LFBC controller design and robust performance have been validated through an experimental prototype for a 3.3-kW single-phase PV-GFI system with an LC L filter. The controller uses an adaptive control method with decreasing error accumulation for PI parameter tuning, which increases the power transfer capability under stability constraints. The proposed LFBC controller guarantees global stability and has substantial resonance damping to improve power quality, making it suitable for applications such as rooftop and water pumping systems. Surya Prakash (Graduate Student Member, IEEE)