A Novel Control Strategy in Grid-Integrated Photovoltaic System for Power Quality Enhancement

Abstract

The integration of solar photovoltaic (PV) systems and utility grids has gradually gained significant interest in improving the sustainability of clean power supply for society. However, power quality remains a challenge due to partial shading conditions and harmonics. To overcome these drawbacks, a flexible radial movement optimization based on a dynamic safety perimeter maximum power point tracking algorithm is employed to track maximum power out of a PV system and to ensure that the optimum voltage level at the common DC bus is obtained under partial shading conditions using fixed-tilt installation configuration. Furthermore, a novel inverter control loop system with a double second order generalized integrator phase-locked loop (DSOGI-PLL) is also proposed to mitigate harmonics and improve the power quality of the grid interfacing PV system using MATLAB SIMULINK software. The proposed system has several merits, such as better harmonic suppression capability, control adaptivity, rapid tracking speed, low computational burden and phase and grid synchronization.

1. Introduction

The transition of energy through photovoltaic (PV) panels has gradually gained significant interest due to the continuous hike in oil prices and the environmental pollution caused by hydrocarbons [1]. Solar energy technologies and PV panels have become inexpensive, pollution-free and more reliable. PV solar panels are the most widely used power source due to the benefits offered, such as free energy from the sun, cheap operation and maintenance costs, green energy, noise-free and a reasonably long lifespan. The PV system can be tied to the load as a standalone system or indirectly connected to the grid system. Despite its merit, PV system power predominantly depends on environmental conditions such as solar insolation and ambient temperature, as its output is a nonlinear characteristic. Under partial shading conditions (PSC), the PV system has multiple maximum power points (MPP), namely local and global power peaks. The major challenge of the PV system is the failure to operate at higher efficiency under complex weather conditions. Hence, the requirement for a maximum power point (MPPT) controller to sustain the global maxima power point (GMPP) under weather intermittency.

The MPPT controllers will ensure that the global power is harvested so that the PV system will always operate at higher efficiency. However, most MPPT techniques have difficulty in tracking GMPP under rapidly changing weather conditions. The PSC is usually caused by moving clouds, tall buildings, flying birds, dust, tall trees, etc., or even a damaged PV module. A bypass diode is normally connected in parallel with the PV cell to protect the shaded PV module from hot-spot damage [2]. Nevertheless, PV power is highly affected by using this bypass diode since the PV system power generated from the shaded module is utterly inadequate. Moreover, the current that flows through the bypass diode contributes to conduction losses, ultimately leading to a further retrogression in device performance. In this paper, the hybrid FRMO-DSP is introduced to track the true GMPP under complex weather conditions by scanning the P-V curve.

Numerous MPPT techniques have been proposed by researchers to optimize GMPP for a PV system under varying weather conditions. The conventional MPPT algorithms such as perturb and observe (P&O), hill climbing (HC) and incremental conductance (INC) were proposed in [3] for GMPP tracking under PSC. These methods use duty ratio adjustment to maximize or minimize until GMPP is extracted. However, these algorithms cannot locate a true GMPP and quite often get trapped in local power peaks, resulting in PV system power reduction. This is because they continuously change the direction of the duty ratio under PSC, which results in large oscillations around MPP [4].

Other researchers proposed hybrid MPPT algorithms to overcome the limitations of conventional algorithms. The main aim was to combine the strengths of the two algorithms for performance improvement. The real-time recurrent learning gradient adaptive linear neuron was proposed in [5]. The method has the merit of using high-volume datasets, and unwanted datasets are removed. It adopted the PSC and presented a fast response with a lower training error. However, the proposed technique had fewer bifurcations due to climatic changes. Researcher [6] proposed the hybrid linear tangents (LT) and Neville interpolation (NI) algorithm for GMPP tracking under PSC. The algorithm eliminates the constant regions of the PV power derivative concerning zero volts at the projection of linear tangents at 75% and 90% of the open-circuit voltage. Under normal weather conditions, the MPPT method reached the GMPP within a precise time. However, during rapidly changing climatic conditions, the proposed LT-NI has a delay in response time due to the re-initialization of the DC scheme. In fact, the proposed MPPT technique fails to accomplish GMPP under different weather conditions. Recently, other researchers developed the hybrid artificial neural network (ANN) integrated with Newton Raphson (NR) for high-efficiency optimization for multilevel inverters for PV applications [7]. The ANN is used for training and estimation, while NR is employed for further optimization. The method is efficient; however, the software development is complex and expensive.

Another major challenge is power quality and grid stability issues in a grid-tied PV system. The electrical power quality includes unbalanced current and voltage, current and voltage harmonics, voltage regulation and input power factor. It is caused by tracking MPP under PSC. Another cause is the deployment of power electronic devices such as inverters, converters, etc., and nonlinear loads. These power quality issues produce undesired effects, such as nuisance trips, voltage flicker, overheating of equipment, efficiency reduction, equipment malfunction due to the excessive voltage distortion, life span reduction to the electrical equipment and lowered system power factor resulting in penalties on monthly utility bills. Another challenge is that most of the inverter control strategies work better under steady-state conditions but fail to comply with the IEEE 519 standard and grid requirements under transient state. This paper proposes a novel inverter control loop system with a DSOGI-PLL to calculate the three-phase reference voltage to generate the optimum PWM signals for the grid interfacing inverter to enhance the power quality of the grid integrated PV system.

To monitor the power quality effects as a result of a grid-tied solar PV system, several studies have been executed over the past few years. Various researchers characterized the harmonics effects of a grid-tied PV system along with contemplation of a few nonlinear loads. In [8], the authors proposed that the DC link voltage regulation is optimized to match the reference current resolution to mitigate large harmonics. However, the comprehensive and systematic analysis of the formation process of the harmonic in the PV inverter output current is missing. In [9], the adaptive current regulation is proposed to reduce harmonics in a grid-connected PV system. The algorithm is trained using a recurrent neural network (RNN) to determine the three-phase reference currents for the generation of pulse width modulation (PWM) signals to drive the grid interfacing inverter to enhance the power quality of the system. The proposed system was modeled and simulated using MATLAB Simulink software. This method requires high computation and RNN performance is heavily dependent on the use of efficient training techniques.

Authors in [10] proposed the PI current controller for grid interfacing inverter to determine the output current to follow the desired reference current for harmonics mitigations. The main limitation of the PI controller is its incapacity to work adequately under unbalanced conditions. In general, the proposed scheme fails to determine a pure reference sinusoidal without steady-state error. The authors of [11] recommend the Newton Raphson iterative technique (NRIT) to drive the switching angles of the grid interfacing inverter to reduce the harmonics. The method measured the voltage total harmonic distortion (THD) of 11.8%, which is not well within the IEEE-519 standard. In [12], a recent Bat algorithm is employed to reduce the harmonics by solving numerical global optimization problems. The algorithm is employed to calculate the switching angle of a multilevel inverter for power quality enhancement. However, the technique has some drawbacks, such as high complexity and computational burden.

The authors of [13] proposed two innovative fault-tolerant methods based on fuzzy logic and model predictive control to investigate fault effects owing to common power-loss in PV arrays in the presence of microgrid uncertainty and disturbance. In a hybrid microgrid benchmark, the efficacy of the proposed system is proven and compared under actual fault situations. The suggested approach also considers the sudden couplings and disconnection of dynamic loads, which is a significant difficulty. The techniques do not need a specific collection of fault situations. The suggested approach for imprecise gain scheduling was performed flawlessly. However, the technique is unsuitable for severe power-loss failures. As a result, the suggested model predictive control approach is designed to tolerate and accept more severe failures than its competitors. The authors, however, did not investigate the designing of active fault-tolerant control systems with real-time control reconfiguration, robust problem detection and diagnostics at the microgrid level. The authors of [14] developed a novel control for a dual-stage grid-connected solar PV dataset based on adjoint least mean square (ALMS). The suggested system includes advantages such as a fast rate of convergence, harmonics capabilities and ease of implementation. The proposed scheme is tested on a three-phase grid and local nonlinear loads. The controller is used to improve grid current quality because of imbalanced, nonlinear load and various weather conditions. The system is intended to meet the IEEE 519 standard’s benchmark. However, the current THD level of 9.72% was measured under steady-state performance.

In this paper, a new hybrid flexible radial movement optimization (FRMO) based on the dynamic safety perimeter (DSP) MPPT algorithm is proposed to ensure that the required DC link voltage is obtained under partial shading conditions. Furthermore, a novel inverter control loop system (ICLS) with a double second-order generalized integrator phase-locked loop (DSOGI-PLL) is proposed to improve the power quality of a grid-connected PV system. The PLL is deployed for grid and inverter synchronization using closed feedback. This paper addresses the main components, control strategy, stability analysis and experimental validation. The superiority of the newly proposed grid-tied PV system is supported by MATLAB Simulink software results. The novelty of this research and contributions of the proposed work can be summarized as follows:

• A new FRMO based on the DSP MPPT technique is proposed to track the PV system’s global maximum power under fast-changing weather conditions using a fixed-tilt installation configuration.
• The proposed FRMO-DSP MPPT is integrated with the PID controller to minimize the steady-state error and maintain the optimal voltage level at the DC link bus.
• A novel inverter control loop system with a DSOGI-PLL is proposed to calculate the three-phase reference voltage to generate the optimum PWM signals for the grid interfacing inverter to enhance the power quality of the grid-integrated PV system.
• The proposed system ensures rapid grid synchronization and convergence time to meet the benchmark of the grid requirements.
• The proposed inverter control technique is compared with other recently developed algorithms from the literature in different case studies using the fast Fourier transform (FFT) to measure the power quality of the proposed system.

The remainder of the paper is systematized as follows: Section 2 explains the PSC and power quality. Section 3 introduces the proposed MPPT and inverter control algorithm. Section 4 presents the system setup. Section 5 demonstrates the results and discussion. Ultimately, Section 6 concludes this paper.

2. PSC and Power Quality

2.1. Partial Shading Conditions

When PSC occurs on PV module arrays, their efficiency is lowered due to the decreased output voltage and current. This is caused by shade from flying birds, tall buildings and structures, passing clouds, dust and malfunction of some PV modules [15]. This causes the P-V curve to have multi-power peaks, namely the local and global power peaks [16]. The global power peak is the highest power point, and the local is defined as the lowest power peak under sudden changes in environmental conditions. Hence the need for the MPPT controller to ensure that the PV system operates at high efficiency even under PSC.

2.2. Power Quality

A good quality power is defined as a constant voltage supply that functions within a designed range. It can also be defined as an AC frequency that is close to the fundamental/designed frequency and a pure voltage sinusoidal waveform. Any power quality-related issues are deemed a power quality concern and can reduce the life span of an electrical device. In a generation, it can be described as an ability to produce power at a specified frequency with a small variation. Power quality-related issues are a concern for customers as it lowers the system power factor and ultimately a hike in monthly utility bills. Many Engineering institutions all over the world are focusing on developing the power quality standards suitable for electrical equipment.

The concerning power quality issues are the emerging harmonics. Harmonics are voltages or current sinusoidal having frequencies that are integer multiples of the fundamental/designed frequency [17]. Harmonics, in combination with the designed voltage or current, can result in a distorted waveform. Harmonics in the grid-tied PV system can be caused by the MPPT controller, power electronics devices such as inverters, etc. and nonlinear loads. The distorted waveform is caused by varying current from nonlinear loads and ultimately results in a voltage drop across the system impedance. Several researchers used total harmonic distortion (THD) to gauge the distorted waveform concerning the fundamental waveform. In a perfect world, the pure sinusoidal waveform will have 0% THD. Following the IEEE-519 standard, the THD level shall not exceed 5% in the electrical system [18]. The THD level of a voltage and current is given by Equations (1) and (2), respectively.

$$TH{D}_V=\frac{\sqrt{{{\displaystyle \sum }}_{n=2}^N{V}_{\Pi }^2}}{V_1}\times 100\%$$

(1)

$$TH{D}_I=\frac{\sqrt{{{\displaystyle \sum }}_{n=2}^N{I}_{\Pi }^2}}{I_1}\times 100\%$$

(2)

where $Vn \mathrm{and} In$ are the rms voltage and current at harmonic n, $V1 \mathrm{and} I1$ are the fundamental rms voltage and current and $N$ is the maximum harmonic order to be considered.

Ultimately, the large THD level affects the system power factor, which increases the customer’s monthly utility bill. Equation (3) demonstrates the system power factor, which depends on the THD level.

$$PF=\frac{1}{\sqrt{1+{\left(\frac{THD\%}{100}\right)}^2}}$$

(3)

The other frequency disturbances can be described as a modifier gaging the size of the voltage variation, such as unbalanced voltage, voltage sag, swell, interruption, etc.; the duration can be instantaneous, momentary or temporary. The general characteristics of these variations and limits set by IEEE standards can be viewed in [19].

3. Grid-Tied System Description

Figure 1 demonstrates the proposed grid-integrated PV system power stage and the implementation of the proposed MPPT and inverter control algorithms. The proposed scheme consists of a two-stage PV system supplying the power to the utility grid. The new FRMO based on the DSP MPPT control system is designed to track global maximum power and ensures that the optimal DC link voltage level is achieved even under weather intermittency. A boost DC/DC converter is designed, taking into consideration the DC link capacitor design. It is used to step up the DC link voltage to the required level of the DC/AC inverter across the bus capacitor. The DC-link capacitor is utilized to produce a more stable DC voltage and minimize the voltage ripples across the PV array. The DC-link capacitor is also employed as an energy storage system to provide power to the inverter in the event of transient faults, PSC, etc. The DC/AC inverter is connected to the utility grid through the LC filter. The inverter is deployed to convert DC to AC power, and the LC filter is designed to further suppress the high-order harmonics. The new inverter control loop system is implemented for controlling the grid interfacing inverter to enhance the power quality. This control system consists of a double control loop, namely, inner current and outer voltage control loop. This is achieved by sensing the three-phase inverter currents and grid voltages. These currents and voltages are then transformed to direct, quadrature and zero components in a rotating reference frame using Park’s transformation. Its goal is to determine the three-phase reference voltage to generate a control signal for the PWM generator to drive the inverter (IGBT gates) to a pure reference sinusoidal waveform. The double second-order generalized integrator phase-locked loop (DSOGI-PLL) is deployed for grid and inverter synchronization. The proposed grid-integrated PV system in this paper is connected to the linear and nonlinear load.

4. The Proposed MPPT and Inverter Control Algorithm

4.1. Proposed FRMO Based on DSP MPPT

Numerous MPPT algorithms have been proposed for GMPP tracking under fast-changing climatic conditions, as discussed in the literature. However, most of these algorithms fail to locate the true global power peak and end up trapped in the local power peak and ultimately reducing the PV output power. Some of these techniques have a high complex computational burden and are expensive and unreliable. To overcome the PSC issue, a new FRMO based on the DSP MPPT algorithm is proposed.

• Flexible Radial Movement Optimization

The FRMO is an efficient optimization tool used to solve multiple variable complex problems. The method uses the same approach as the other evolution optimizations, such as particle swarm optimization and differential optimization [20,21]. However, FRMO coefficients commute during the technique process. It initializes the optimization process by disintegrating the multi-particles in the predetermined search space. The disintegrated particles are taken as the solution suggested and the particle’s fitness values are determined utilizing the objective function through each step of the process. Then, the random vector and two pre-eminent fitness values are used to calculate the movement vector. The uniqueness of the FRMO from the other metaheuristic algorithms is the particle’s motion manner. The particle traverses the predefined region around the center of the algorithm. Figure 2 demonstrates the particles moving out with non-identical velocities along the radii of the center. The position of the greatest particle is recognized based on their fitness values explicated by the objective function. The FRMO particles traverse the search space proficiently to circumvent drifting towards the local optima. This method offers a superior and broad inspection ground for the targeted search space. Therefore, the algorithm is fit to work with a very large and complicated search space and needs small storage memory. The particle position is the search space which is represented by a matrix of $nop\times nod$ magnitude.

Where the $nop \mathrm{and} nod$ denote the number of particles in the design and the number of dimensions, which depends on the number of variables needed to be optimized, respectively. The particle positions are calculated using Equation (4), where the $nop \mathrm{and} nod$ remain constant.

$$X_{j,j}=\left[\begin{array}{cccc}{X}_{1,1}& X_{1,2}& \dots & X_{1,nod}\\ X_{2,1}& \ddots & \dots & ⋮\\ ⋮& ⋮& \ddots & ⋮\\ X_{nop,1}& X_{nop,2}& \dots & X_{nop,nod}\end{array}\right]$$

(4)

$$\mathrm{where} \{\begin{array}{c}i=1,2,3\dots nop;\\ j=1,2,3\dots nod.\end{array}$$

i.

Initialize

First particle positions are randomly allocated inside the boundaries of the search space. The random allocation can be exemplified using Equation (5).

$$X_{i,j}=X_{\min \left(i\right)}+rand\left(0,1\right)\times (X_{\max \left(j\right)}-X_{\min \left(i\right)})$$

(5)

$$\mathrm{where} \{\begin{array}{c}i=1,2,3\dots nop;\\ j=1,2,3\dots nod.\end{array}$$

where $X_{\max \left(j\right)} \mathrm{and} X_{\min \left(i\right)}$ represents the $j^{th}$ size constraints in the design and Gaussian distribution as 0 and 1, utilized for rand (0,1).

ii.

Movement

The particle layout, a straight line route from the radii on the vector $V{e}_{ij}$, can be determined using Equation (6).

$$V{e}_{ij}=rand\left(0,1\right)\times V{e}_{\max \left(i\right)}$$

(6)

$$\mathrm{where} \{\begin{array}{c}V{e}_{\max \left(i\right)}=\frac{Y_{\max \left(i\right)-Y_{\min \left(i\right)}}}{k};\\ i=1,2,\dots ,nop;j=1,2\dots ,nod.\end{array}$$

where $k$ is denoted as the positive integer. Then, the inertia weight $\left(W\right)$ is introduced to solve all the convergence problems. Equation (7) is used to show the relationship between the inertia weight and the iteration number.

$$V{e}^k{}_{ijj}=W_k\times rand \left(0,1\right)\times V{e}_{\max \left(j\right)}$$

(7)

In FRMO, $W_k$ is a constant value between 0 and 1 for the initial 10 iterations and is tuned using Equation (8).

$$W_k=W_{max}-\frac{W_{max}-W_{min}}{Ite{r}_{max}}\times 2Ite{r}_k$$

(8)

where $W_{max}=1$ and $W_{min}=0$. The $Ite{r}_k$ is denoted as the current iteration and $Ite{r}_{max}$ is the highest iteration. Figure 3 illustrates the radial movement of the particles from the center and is determined using Equation (9).

$$X_{ijj}{}^k=C^k+V{e}^k{}_{ijj}$$

(9)

The particles are depicted as black dots and the boundaries of $V{e}_{\max }$ are demonstrated as the dashed circle.

The objective function is utilized to determine the performance of the particles and denote the best fitness/radial value $\left(R_{best}\right).$ This position and its fitness value are stored in this step. The best $R_{best}$ in all iterations is taken as the global best $\left(G_{best}\right).$ The $R_{best }and G_{best }are$ utilized to progressively update the center using Equation (10).

$$C^{new}=C^{old}\times a_1\times \left(G_{best}-R_{best}\right)+a_2\left(R_{best}-c^{old}\right)$$

(10)

where $a_1 \mathrm{and} a_2$ denote the coefficients, which can be determined using Equations (11) and (12) after the initialization of the FRMO process. Where $C$ represents the center.

$$a_1=a_{1min}+\frac{a_{1max}-a_{1min}}{Ite{r}_{max}}\times Ite{r}_k$$

(11)

$$a_2=a_{2min}+\frac{a_{2max}-a_{2min}}{Ite{r}_{max}}\times Ite{r}_k$$

(12)

The $G_{best}$ is continuously compared with the $R_{best}$ to find the greatest solution acquired from all generations. The process is ongoing till the $G_{best}$ is equal to the maximum threshold value. Figure 3 demonstrates the two successful iterations where the updated vector defines a new center.

• Dynamic Safety Perimeter (DSP)

Generally, the design aim of a control system is to attain a defined constrained output [22]. The DSP is a performance indication whose predictor variable is the space from a predefined safety margin. It measures the distance from the system track to a predefined safety margin in the state space at any given time. The DSP’s objective is to ensure that the system operates within a safe boundary all the time, even during transient behavior. In this case, the DSP will ensure the quality of the MPPT controller under varying weather conditions. The safety perimeter-based design is based on the safety border of the operation area, and it is determined by following the experience of the system behavior. The MPPT controller design based on DSP is essential to sustain a predefined safety boundary during transient and disturbance behavior. This will ensure that the PV system operates at higher efficiency and speeds up the performance recovery of the MPPT controller under PSC.

For the explanation of the idea, let X be defined as the state space in $R^n$ and $\varnothing \subseteq X$ as the safe operation area for vital state variables $X\in R^n$ in the state subspace Ø and $m\le n$. State space can be defined by a set of inequalities such as:

$$\varnothing =\left\{x\right|{\phi }_i\left(x\right)\le 0, i=1,\dots ,q\}$$

where ${\phi }_i$:$R^m\to R.{\phi }_i\left(x\right)>0$ is the unsafe region and $\partial \varnothing =\left\{X|{\phi }_i\left(x\right)=0, i=1,\dots q\right\}.$ $\partial \varnothing$ is the border of the safe region.

Where variables $q \mathrm{and} m$ represent the number of specified inequalities and the number of state variables relevant to the safety perimeter. The $m\le n$, where $n$ denotes the size of the state space. The DSP path starts from the initial condition, $x_o$ to the operating point $x_s$ crossing the state space with varying distance to the safety boundary, as shown in Figure 4. Therefore, DSP can be defined as the instant shortest distance $\delta \left(t\right)$ between the current state vector, $x and \partial \varnothing$. DSP is determined by solving the following quadratic programming optimization problem:

$$mi{n}_{xp}|\left|x-x_p\right|{|}_2^2$$

(13)

Subject to $x_p\in \partial \varnothing$ and DSP is:

$$\delta =s\left(t\right)\left|\right|(x-x_{po) }|{|}_2^2$$

(14)

where $x_{po}$ is defined as the solution to the preceding optimization problem and:

$$s\left(t\right)=\{\begin{array}{c}+1, if x is in the area of the safe operation;\\ -1, if x is out of the area of safe operation.\end{array}$$

The state relevant to the safety perimeter, in this case, is defined as the PV current and PV voltage, $x={[I_{PV}{V}_{PV}]}^T$. The safe operation region $\mathsf{\Phi }$ is chosen to be the subspace that includes the corresponding MPPT for all predictors, such as varying solar irradiance and temperature. If the input predictors are below the $\mathsf{\Phi }$, the PV system will not reach the GMPP and will result in poor performance. The DSP of a PV module is determined by the two examples below:

i.

Offline technique requires accurate current and voltage (I-V) data of the PV module characteristic.

ii.

Online technique requires the recorded MPP values acquired from the FRMO without PSC.

To demonstrate the idea, the 1STH-215-P PV module is considered with 7.84 A and 36.3 V rated short circuit current and open circuit voltage, respectively. By I-V characteristic analysis, the safety area will be of a convex set, as illustrated in Figure 5. The safety region in this study, the PV Power $\ge 207 \mathrm{W} \mathrm{or} \mathrm{PV} \mathrm{efficiency} \eta \ge 97\%.$

If the PV modules are connected in series and parallel, the safety region is determined by integrating the characteristics of the modules. In a series connection, the safe region is determined using Equation (15).

$${\mathsf{\Phi }}_s=\left\{x_s\right|x_s=n_s{x}_{1}xϵ\varphi \}$$

(15)

where ${\mathsf{\Phi }}_s$ is the total safe region of the series connection of the PV modules, and $\mathsf{\Phi }$ is denoted as a single module. Where $x={[I_{PV }{V}_{PV}]}^T$ denotes the state vector of a module and $x_sϵR^2,$ $x_s={[I_s{V}_s]}^T$ is the state variable of a new string current and voltage. The total safety region ${\mathsf{\Phi }}_p$ of a parallel connected module is determined using Equation (16).

$${\mathsf{\Phi }}_p=\left\{x_p\right|x_p=A_p{x}_{1}xϵ\varphi \}$$

(16)

where $A_p=\left[\begin{array}{cc}1& 0\\ 0& n_p\end{array}\right]$, $x_pϵR^2,$ $x_s={[I_p{V}_p]}^T$ is the state variable of the new parallel current and voltage.

In this study, an online DSP method is used to monitor the safety region of the PV module, the reason being that the FRMO is already using the offline approach for GMPP tracking. As mentioned above, the 1STH-215-P SOLTECH module is used to simulate and validate the performance of the MPPT techniques. The safety region can be defined by the set of inequality constraints in a form of $\mathsf{\Phi }=\left\{x|{\phi }_i\left(x\right)\le 0, i=1,\dots ,4\right\},$ where:

$${\mathsf{\Phi }}_i\left(x\right)=a_i^{T}x-ci\le 0, i=1\dots 4,$$

(17)

where $a_i^TϵR^2$ and $ciϵR$ are constant, ${\phi }_i(.)$ equals 0 is a subspace of state vector region, $\partial \mathsf{\Phi }$. These constraints can be integrated into a matrix format as:

$$Ax\le c$$

(18)

where:

$$A=\left[\begin{array}{c}{a}_1^T\\ a_2^T\\ ⋮\\ a_q^T\end{array}\right], c=\left[\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_q\end{array}\right]$$

The border of the safety perimeter is $\partial \mathsf{\Phi }\left(x_p\right)=(x_p|A{x}_p=c)$. Then:

$$A{x}_p=c$$

(19)

The lowest space between the current state $x$ and a boundary segment number $i\left({\delta }_i\left(t\right)\right)$ is determined by Equation (23).

$${\delta }_i\left(t\right)=\frac{ci-a_i^{T}x\left(t\right)}{|\left|a_i\right|{|}_2^2}\{ \begin{array}{c}\ge 0,if {\phi }_i\left(x\right)<0\\ <0,if {\phi }_i\left(x\right)>0\end{array}$$

(20)

The border distance vector for:

$d\left(t\right)={[{\delta }_1\left(t\right), {\delta }_2\left(t\right),\dots .,{\delta }_q\left(t\right)]}^T$ can be determined using Equation (24).

$$d\left(t\right)=D_{ia}\left(c-Ax\left(t\right)\right)=dc-D_{a}x\left(t\right)$$

(21)

where $D_a=\left[\begin{array}{ccc}\frac{1}{|\left|a_1\right|{|}_2^2}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & \frac{1}{|\left|a_q\right|{|}_2^2}\end{array}\right]$ is a diagonal matrix,

$$dc=D_{a}c, D_a=D_{ia}A,d(.)ϵR^q,cϵR^q,dcϵR^q, D_aϵR^{q\times n}, and D_{ia}ϵR^{q\times q}$$

Considering $\mathsf{\Phi }$ is convex and the border constraints are linear, the safety area is a polytype and DSP, $\delta (.)$ is the lowest element in $d(.),$ such as:

$$\begin{gathered} \delta \left(t\right)=min{\delta }_i\left(t\right) \\ 1\le i\le q \end{gathered}$$

(22)

Consequently, the DSP and safety region, $\mathsf{\Phi }$ of a PV system can be determined using the steps below:

i.

Determine the safety region of a 1STH-215-P PV module.

ii.

Calculate its boundary using Equations (17)–(19).

iii.

Calculate the overall safety region of the series and parallel configuration using Equations (15) and (16).

iv.

Enumerate the online DSP by working out the optimization problem (13), depending on constraints (19) by applying (20) and (22).

v.

Then record the observed MPP under normal conditions and update the safety region border of the third step to obtain control of the approximation in the first and second steps.

• FRMO based on DSP.

The FRMO based on DSP is proposed to solve the PV power optimization problems under sudden changes in climatic conditions. The DSP is employed to overcome the limitations of the FRMO algorithm under PSC. This method will guarantee the tracking of GMPP with a rapid tracking and convergence speed. In fact, it is a very efficient, less expensive and computational burden. This algorithm can work with large and complex search spaces. Another motivation to use FRMO-DSP is that it explains how far the system is from safe mode. Another feature of the system is that the DSP allows for the maintenance of a present margin of safety throughout transient and steady-state safety-critical system operations. Furthermore, it can aid in improving the system’s performance under complex weather circumstances, increasing system reliability.

The FRMO begins the process by starting the particle’s movement over $x-amount$ of consecutive iterations. This is conducted by scanning the P-V curve of the PV system under all conditions. The first iteration explores the large section of the search space to obtain the best fitness value (GMPP) in the region. The radii of the circle are reduced on the second iteration to allow the algorithm to explore a small area of search space and obtain the best fitness value. On the third iteration, the radii of the circle are reduced further to acquire the best GMPP value in a small search space. The preceding center positions and values of $R_{best} \mathrm{and} G_{best}$ identify the position of the new center position. Once the new center position is determined, the best GMPP value is explored around the radii. Once the best fitness value is acquired, it is then compared with the best GMPP value and keeps on updating if an enhancement occurs. The procedure resumes for all iterations till the GMPP region is located, and particles drift towards the true GMPP. In the last iteration, the best region and centers are near each other. Equation (23) is proposed to track the best GMPP when the size of the solar irradiance variation is bigger than the minimum acceptable fluctuation. This is monitored through the PV output power fluctuations under PSC, and it is given by $\Delta P:$

$$\left|\frac{Q\left(xi+1\right)-Q\left(x_i\right)}{Q\left(x_i\right)}\right|>\mathsf{\Delta }P$$

(23)

where $Q\left(x_i\right)$ defines the PV output power based on the $i^{th}$ position $xi$ in the search space. PSC occurs frequently; thus, it is stochastic. In this paper, the threshold $\Delta P$ is 3% of the PV power at MPP. The DSP monitors the PV output power by measuring the current and voltage from the FRMO algorithm. Then the DSP value is utilized to acquire the suitable signal to operate the gate of the DC/DC converter. The alterations of FRMO based on DSP are carried out in accordance with the steps below:

i.

Measure the current and voltage values of the FRMO algorithm and calculate the PV output power.

ii.

The instantaneous DSP value, $\delta (.),$ is determined by the current state of the PV, $x={[V_{PV }{P}_{PV}]}^T$ based on the denoted safety region $\mathsf{\Phi }.$

iii.

The system is normal when the PV state $x={[V_{PV }{P}_{PV}]}^Tϵ\mathsf{\Phi }$ otherwise, PSC has occurred.

iv.

The output of FRMO is scaled based on the value of $\delta (.)$ when the PV state, $x={[V_{PV }{P}_{PV}]}^T\notin \mathsf{\Phi }$ using Equation (24).

$$Duty=Dut{y}_{FRMO}\left[1+sg\left(t\right)\times \delta \left(t\right)\right]$$

(24)

Thus,

$$sg\left(t\right)=\{\begin{array}{c}0, if x is in the area of the safe operation\\ k, if x is out of the area of safe operation\end{array}$$

This can also be represented by Algorithm 1, which demonstrates the pseudo-code of the FRMO-DSP MPPT technique.

Where the duty denotes the duty cycle of the PWM to operate the MOSFET gate of the BOOST DC/DC converter of the PV system, where $kϵ\left[0,1\right]$ defines the alteration gain and $\delta \left(t\right)$ is the instantaneous DSP. Under PSC, the operating point drifts towards abnormal conditions. The DSP ensures the improvement of the FRMO to drift towards the GMPP identified in $\mathsf{\Phi }.$

Algorithm 1. Pseudo-code of FRMO-DSP.

Define the set of inequalities constraints in a form of $\mathsf{\Phi }=\left\{x|{\phi }_i\left(x\right)\le 0, i=1,\dots ,4\right\}$

Input: the set of current (I) and voltage (V) historical dataset, $dataHistory$;

Output: The prediction of PV output power,$prediction$;    1. $n=dataset$ History;    2. $\mathrm{For} i=0 to n-1$ do    3. $x\left[i\right]-i+1$    4. $y\left[i\right]=dataHistory \left[i\right];$    5. $Measure the values of I and V$    6. $Determine the output power x={[V_{PV}{P}_{PV}]}^T$    7. $if \left|\frac{Q\left(xi+1\right)-Q\left(x_i\right)}{Q\left(x_i\right)}\right|<3\%$, terminate; otherwise;    8. $\mathrm{Scale} \mathrm{the} Duty=Dut{y}_{FRMO}\left[1+sg\left(t\right)\times \delta \left(t\right)\right]$ such that    9. $x={[V_{PV}{P}_{PV}]}^Tϵ\mathsf{\Phi }$    10. Terminate if    11. Terminate for    12. Update y    13. End while    14. Return the current equation;

4.2. Proposed Novel Inverter Control Loop System (ICLS) with a DSOGI-PLL

The inverter control loop system has a double loop power flow algorithm. The proposed system consists of the inner current and outer voltage control loops. The three-phase AC currents and voltages are sensed and transformed into DC components through direct, quadrature transformation. The transformation procedure consists of two stages: namely, three-phase $ABC$ frame to $\alpha \beta$ transformation using Clarke’s transformation and from $\alpha \beta$ frame to the $dq$ components using Park’s transformation. These transformations are used to make computation much easier. The three-phase inverter output voltage can be expressed using Equations (25)–(27).

$$V_{ia}=V_{amp}\sin \left(\omega t\right)$$

(25)

$$V_{ib}=V_{amp}\sin \left(\omega t-\frac{2\Pi }{3}\right)$$

(26)

$$V_{ic}=V_{amp}\sin \left(\omega t+\frac{2\Pi }{3}\right)$$

(27)

where $V_{amp}$ represents the voltage amplitude, $\omega$ denotes angular frequency, and $V_{ia}, V_{ib},V_{ic}$ depicts the three-phase inverter voltages. The single-phase formula can be derived as:

$$V_i=V_L+V_R+V_g$$

(28)

$$V_i=L\frac{di}{dt}+I_R+V_g$$

(29)

$$V_g=V_i-L\frac{di}{dt}-I_R$$

(30)

The matrix formula for the three-phase stationary frame is expressed using Equation (31).

$$\left[\begin{array}{c}{V}_{ga}\\ V_{gb}\\ V_{gc}\end{array}\right]=\left[\begin{array}{c}{V}_{ia}-L\frac{dia}{dt}-R_{ia}\\ V_{ib}-L\frac{dib}{dt}-R_{ib}\\ V_{ic}-L\frac{dic}{dt}-R_{ic}\end{array}\right]$$

(31)

Then, the three-phase stationary frame ($abc$) is transformed into two stationary frames $(\alpha \beta$).

$$\left[\begin{array}{c}{V}_{g\alpha }\\ V_{g\beta }\end{array}\right]=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left[\begin{array}{ccc}1& \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& -\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]\left[\begin{array}{c}{V}_{ga}\\ V_{gb}\\ V_{gc}\end{array}\right]$$

(32)

$$\left[\begin{array}{c}{V}_{g\alpha }\\ V_{g\beta }\end{array}\right]=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left[\begin{array}{ccc}1& \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& -\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]\left[\begin{array}{c}{V}_{ia}-L\frac{dia}{dt}-R_{ia}\\ V_{ib}-L\frac{dib}{dt}-R_{ib}\\ V_{ic}-L\frac{dic}{dt}-R_{ic}\end{array}\right]$$

(33)

$$\left[\begin{array}{c}{V}_{g\alpha }\\ V_{g\beta }\end{array}\right]=\left[\begin{array}{c}{V}_{i\alpha }-L\frac{di\alpha }{dt}-R_{i\alpha }\\ V_{i\beta }-L\frac{di\beta }{dt}-R_{i\beta }\end{array}\right]$$

(34)

Then, the αβ is converted to the synchronous $dq$ component.

$$\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]=\left[\begin{array}{cc}cos\omega t& sin\omega t\\ -sin\omega t& cos\omega t\end{array}\right]\left[\begin{array}{c}{V}_{g\alpha }\\ V_{g\beta }\end{array}\right]$$

(35)

$$\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]=\left[\begin{array}{cc}cos\omega t& sin\omega t\\ -sin\omega t& cos\omega t\end{array}\right]\left[\begin{array}{c}{V}_{i\alpha }-L\frac{di\alpha }{dt}-R_{i\alpha }\\ V_{i\beta }-L\frac{di\beta }{dt}-R_{i\beta }\end{array}\right]$$

(36)

$$\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]=\left[\begin{array}{c}Vid\\ Viq\end{array}\right]-R\left[\begin{array}{c}id\\ iq\end{array}\right]-L\left[\begin{array}{cc}cos\omega t& sin\omega t\\ -sin\omega t& cos\omega t\end{array}\right]\left[\begin{array}{c}\frac{di\alpha }{dt}\\ \frac{di\beta }{dt}\end{array}\right]$$

(37)

The resistance can be practically ignored.

$$\left[\begin{array}{cc}cos\omega t& sin\omega t\\ -sin\omega t& cos\omega t\end{array}\right]\left[\begin{array}{c}\frac{di\alpha }{dt}\\ \frac{di\beta }{dt}\end{array}\right]=\frac{d}{dt}\left[\begin{array}{c}id\\ iq\end{array}\right]+w\left[\begin{array}{c}-id\\ iq\end{array}\right]$$

(38)

The electrical dynamics of the inverter in the $dq$ synchronous frame are expressed as:

$$\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]=\left[\begin{array}{c}Vid\\ Viq\end{array}\right]-R\left[\begin{array}{c}id\\ iq\end{array}\right]-L \frac{d}{dt}\left[\begin{array}{c}id\\ iq\end{array}\right]+wL\left[\begin{array}{c}-id\\ iq\end{array}\right]+R\left[\begin{array}{c}id\\ iq\end{array}\right]$$

(39)

$$\left[\begin{array}{c}{V}_{id}\\ V_{iq}\end{array}\right]=\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]+L \frac{d}{dt}\left[\begin{array}{c}id\\ iq\end{array}\right]+wL\left[\begin{array}{c}-id\\ iq\end{array}\right]+R\left[\begin{array}{c}id\\ iq\end{array}\right]$$

(40)

The difference between Equations (39) and (40) is represented by Equation (41), where the resistance is practically neglected.

$$\frac{d}{dt}\left[\begin{array}{c}id\\ iq\end{array}\right]=\frac{1}{L}\left[\begin{array}{c}Vid-V_{gd}\\ Viq-V_{gq}\end{array}\right]-w\left[\begin{array}{c}-id\\ iq\end{array}\right]$$

(41)

Equation (41) explains that the $d-q$ axis is reliant on each other. The reason being the $d-axis$ flux produces the $q-axis$ back EMF in electrical machinery. Then, the PI controller is utilized to control the currents $id and iq.$ The inverter reference voltage can be determined using Equation (42).

$$\left[\begin{array}{c}{V}_{id\_ref}\\ V_{iq\_ref}\end{array}\right]=\left[\begin{array}{c}{V}_{gd}\\ V_{gq}\end{array}\right]+\left[\begin{array}{c}\left(Kp+\frac{k_i}{s}\right)-\left(i_{d_{ref}}-id\right)\\ \left(Kp+\frac{k_i}{s}\right)-\left(i_{q_{ref}}-iq\right)\end{array}\right]+w\left[\begin{array}{c}-id\\ iq\end{array}\right]$$

(42)

The inverter reference voltage is then transformed back to three-phase AC reference voltages to generate a signal for the PWM. The PWM will then send an optimum control signal to drive the IGBT’s gate of the inverter. This is achieved by adjusting the firing angles of the inverter and injecting the three-phase anti-harmonics current/voltages into the grid for power quality enhancement.

A double second-order generalized integrator phase-locked loop (DSOGI-PLL) is used for grid synchronization and rapid and precise approximation of the frequency/phase. To be more precise, the role of the DSOGI-PLL is to approximate the angle gaging the instantaneous voltage waveform. It uses the second-order generalized integrator-based filter. It is defined by the second-order transfer functions.

$$G_1\left(s\right)=\frac{E^{\prime }}{E}=\frac{k\omega r}{s^2+k\omega s+{\omega }^2}$$

(43)

$$G_1\left(s\right)=\frac{E^{″}}{E}=\frac{k\omega r}{s^2+k\omega s+{\omega }^2}$$

(44)

where ω denotes grid voltage angular frequency and $k$ is the damping factor. The smaller the $k$, the better harmonic reduction. The phase estimation can be determined by using ${\tan }^{-1}$ utilizing Equations (43) and (44) by introducing the shift signal by $-{90}^{°}$.

The final step is to execute the PWM subroutine once the grid angle is obtained by the DSOGI-PLL. The PWM subroutine is utilized to determine the modulation index ($m)$ and the time interval $t1,t2,t0$. This is calculated using Equations (45)–(48).

$$m=\frac{Vref}{\raisebox{1ex}{$Vdc$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(45)

$$t1=\frac{\sqrt{3}\cdot T_s m \sin \left(\frac{\pi }{3}-\theta \right)}{2}$$

(46)

$$t2=\frac{\sqrt{3}\cdot T_s m \sin \left(\theta \right)}{2}$$

(47)

$$t0=T_s-t1-t2$$

(48)

where $Vref$ is denoted as the fundamental amplitude, $Vdc$ as the DC-bus voltage, $T_s$ as the sampling time and $s$ as the grid voltage phase angle obtained by the DSOGI-PLL. The PWM procedure regulates the sector from the DSOGI-PLL angle by differentiating the extracted grid phase angle with the angle of each zone, as shown in Figure 6.

Table 1. Duty ratio in zones.

Zone	SW 1	SW 2	SW 3
1	$-tr+tl+\frac{to}{2}$	$tl+\frac{to}{2}$	$\frac{to}{2}$
2	$tr+\frac{to}{2}$	$tr+tl+\frac{to}{2}$	$\frac{to}{2}$
3	$\frac{to}{2}$	$tr+tl+\frac{to}{2}$	$tl+\frac{to}{2}$
4	$\frac{to}{2}$	$tr+\frac{to}{2}$	$tr+tl+\frac{to}{2}$
5	$tl+\frac{to}{2}$	$\frac{to}{2}$	$tr+tl+\frac{to}{2}$
6	$tr+tl+\frac{to}{2}$	$\frac{to}{2}$	$tr+\frac{to}{2}$

shows the duty ratio of the three inverter switches in six different zones.

5. System Setup

Three case studies were undertaken to assure the genuine performance of the proposed system utilizing MATLAB SIMULINK and Fast Fourier analysis (FFT) to test, evaluate and validate the performance of the proposed FRMO based on the DSP algorithm and ICLS with DSOGI-PLL, as shown in

Table 2. Specification of the proposed system.

PV Model	1STH-215-P
PV rated Power	213.15 W
Vmpp	29 V
Impp	7.35 A
Number of cells in series (Ns)	60
Series resistance (Rs)	0.39383 Ω
Shunt resistance (Rsh)	313.3991 Ω
L1 (DC link inductor)	100 mH
C1	100 µF
C2	100 µF
Ro	20 Ω
C3 (DC Link Capacitor)	$1100 \mathsf{\mu }\mathrm{F}$
DC link Voltage	24–38 VDC
LC Filter	$L=4.8 \mathrm{mH};C=4.3 \mathsf{\mu }\mathrm{F}$
Controller sampling frequency	20 kHz
Inverter input VDC	24–40 Vdc
Inverter Output VAC	380 Vac (±5%)—3 phase
Grid nominal voltage	380 Vac
Grid nominal current	25 A
Grid local power	0.2 kW
Grid nominal Frequency	50 Hz

. Figure 7 depicts the proposed system’s implementations. The specs of the Soltech 1STH-215-P PV panel, Boost DC/DC converter, three-phase inverter and LC filter are shown in

Table 3. Case studies conducted.

Weather Conditions	$\mathbf{Irradiance} (\mathrm{W}/{\mathrm{m}}^2)$	Temperature (°C)	DC-Link Inductor	Load
Case Study 1	G1 = 1000	T1 = 25	L = 100 mH	100 W
	G2 = 850	T2 = 31
	G3 = 900	T3 = 27
	G4 = 920	T4 = 19
Case Study 2	G1 = 1000	T1 = 25	L = 80 mH	100 W to 150 W
	G2 = 690	T2 = 27
	G3 = 835	T3 = 21
	G4 = 740	T4 = 30
Case Study 3	G1 = 950	T1 = 25	L = 50 mH	100 W to 180 W
	G2 = 750	T2 = 22
	G3 = 670	T3 = 32
	G4 = 800	T4 = 17

.

Case study 1 demonstrates the step change in solar irradiance from $\mathrm{G}1=1000 \mathrm{W}/{\mathrm{m}}^2$ to $\mathrm{G}2=850 \mathrm{W}/{\mathrm{m}}^2$, $\mathrm{G}3=900 \mathrm{W}/{\mathrm{m}}^2$ and $\mathrm{G}4=920 \mathrm{W}/{\mathrm{m}}^2$ and module varies from temperature $\mathrm{T}1=25 °\mathrm{C}, \mathrm{T}2=31 °\mathrm{C}, \mathrm{T}3=27 °\mathrm{C}, \mathrm{and} \mathrm{T}4=19 °\mathrm{C}$ to study the performance of the proposed MPPT technique under PSC. The DC link inductor was kept at $100 \mathrm{mH}$ for faster dynamic response and to provide stable DC current within specified limits for the inverter. The proposed ICLS with DSOGI-PLL’s performance is studied with a linear load of 100 W to measure the grid’s total harmonic distortion (THD). The Fast Fourier analysis (FFT) is used to measure the THD level following the IEEE 519 standard [23,24]. Ultimately the system power factor is determined using Equation (3) in accordance with the IEEE standard 1459 [25,26]. The proposed inverter control strategy is compared with the recent Newton Raphson iterative technique (NRIT) from the literature for power quality analysis.

Case study 2 shows the PSC pattern 2, where the solar irradiance levels are varied from $\mathrm{G}1=1000 \mathrm{W}/{\mathrm{m}}^2$, $\mathrm{G}2=690 \mathrm{W}/{\mathrm{m}}^2$, $\mathrm{G}3=835 \mathrm{W}/{\mathrm{m}}^2$, and $\mathrm{G}4=740 \mathrm{W}/{\mathrm{m}}^2$ and the module temperature changes from $\mathrm{T}1=25 °\mathrm{C}, \mathrm{T}2=27 °\mathrm{C}, \mathrm{T}3=21 °\mathrm{C}, \mathrm{and} \mathrm{T}4=30 °\mathrm{C}$ to validate the performance of the proposed FRMO-DSP under PSC. The DC link inductor is reduced to 80 $\mathrm{mH},$ and a nonlinear load (changing from 100 W to 150 W at $\mathrm{t}=0.3-0.6 \mathrm{s}$) is connected to the system to gauge the THD level of the proposed ICLS with DSOGI-PLL with reference to the IEEE 519 standard using FFT. The system power factor is also measured with reference to the IEEE 1459 international standard for power quality analysis.

Finally, case study 3 shows the worst PSC pattern where the solar panel receives the changing solar insolation of $\mathrm{G}1=950 \mathrm{W}/{\mathrm{m}}^2$, $\mathrm{G}2=750 \mathrm{W}/{\mathrm{m}}^2$, $\mathrm{G}3=670 \mathrm{W}/{\mathrm{m}}^2$ and $\mathrm{G}4=800 \mathrm{W}/{\mathrm{m}}^2$ and the module temperature varies from $\mathrm{T}1=25 °\mathrm{C} \mathrm{to} \mathrm{T}2=22 °\mathrm{C}, \mathrm{T}3=32 °\mathrm{C} \mathrm{and} \mathrm{T}4=17 °\mathrm{C}$. The DC link inductor is reduced to 50 $\mathrm{mH}$ and the nonlinear load is connected to validate the performance of the proposed MPPT and inverter control strategy for power quality enhancement using the FFT toolbox.

6. Results and Discussion

The investigated three-phase two-stage grid integrated PV system was simulated using MATLAB SIMULINK and the FFT toolbox to determine the THD level. The effectiveness of the proposed MPPT and inverter control system algorithms were evaluated in terms of three case studies, which are tabulated in

Table 4. Case studies tabulated results for the MPPT algorithm.

	FRMO-DSP MPPT
Case Study	DC Link Voltage (V)	PV Power (W)	Tracking Time (s)	PV Efficiency (%)
1	28.9	212.89	0.02	99.88
2	28.4	208.93	0.12	98.02
3	29.1	207.9	0.13	97.54

and

Table 5. Case study results for the proposed inverter control algorithm.

	ICLS Based on DSOGI-PLL			NRIT
Case Study	THD (%)	System PF	Settling Time (s)	THD (%)	System PF	Settling Time (s)
1	0.46%	0.999	0.18	5.86%	0.998	0.32
2	1.18%	0.999	0.25	9.66%	0.995	0.47
3	1.83%	0.997	0.27	14.42%	0.979	0.51

. A comprehensive study was carried out to compare the proposed inverter control strategy with the recent Newton Raphson iterative technique under different weather conditions, linear load and variable load.

6.1. Case Study 1

In case study 1, the proposed FRMO based on the DSP MPPT method successfully tracked the PV power of 212.89 W with an efficiency of 99.88% and energy losses of 0.12%. The proposed MPPT algorithm obtained the PV current of 7.37 A and the optimal DC link voltage of 28.9 V, respectively. The proposed MPPT technique managed to produce a stable optimal DC link voltage to the input side of the inverter. The switching frequency of the Boost DC-DC converter was kept at 10 kHz. The proposed technique proved to be a fast-tracking algorithm by being able to locate a true GMPP in 0.02 s after the PSC occurred. The GMPP was located at the utmost right side of the P-V curve scan. The tracking of the GMPP process started for the FRMO-DSP algorithm by initializing the multi-particles in the predetermined search space for the duty cycle, and then by running the technique, the duty ratio was revised. The duty ratio stabilized at $D=0.55 \mathrm{at} \mathrm{GMPP}.$ The graphical results of the PV power, PV current and DC link voltage are demonstrated in Figure 8. The proposed approach produced reduced oscillation during the tracking process in the simulation results.

The proposed ICLS based on the DSOGI-PLL algorithm’s performance is evaluated under PSC and a linear load. Figure 9 and Figure 10 depict the grid voltages, current and %THD content for the ICLS based on DSOGI-PLL during linear load. The proposed scheme recorded a THD of 0.46%, which is well within the IEEE 519 standard. With the large DC link inductor of $100 \mathrm{mH}$, minimal PV power ripple was experienced. The settling time of the proposed scheme is 0.18 s after GMPP is tracked. The system power factor of 0.999 (close to unity) is recorded due to the low THD level, which is well within the IEEE 1459 international standard for power quality analysis.

When studying the grid THD analysis, the third- and fifth-order harmonics are the dominant harmonics in the grid current waveform spectrum. Hence, the proposed inverter control was designed and tuned to reduce the harmonics at these frequencies.

Figure 11 and Figure 12 demonstrate the grid voltages, current and %THD level of the NRIT during linear load. The technique recorded a bit high THD of 5.86%, which exceeds the limits of the IEEE 519 standard. The settling time of the NRIT is 0.32 s after the GMPP is tracked. The harmonics content is much higher when compared to the proposed ICLS based on the DSOGI scheme. The NRIT has a system power factor of 0.999, which is slightly lower than the proposed scheme. It is validated that the proposed inverter control strategy has better stability and enhanced power quality than the NRIT. Figure 13 and Figure 14 show grid voltages, current and THD levels without using the harmonic compensator strategy. The high harmonics and distorted waveforms are experienced, as shown in Figure 14. The THD level of 41.57% is recorded, which is not in compliance with the IEEE 519 standard. This highlights the importance of employing the inverter control scheme to mitigate harmonics and power quality improvement.

6.2. Case Study 2

To validate and prove the stability of the proposed FRMO-DSP MPPT algorithm for PV power optimization under climatic change, a rapid change in solar irradiance and module temperature has been introduced in case study 2. Furthermore, the DC link inductor was reduced and the load variation was established. The proposed algorithm effectively tracked the GMPP of 208.93 W in only 0.12 s. It recorded a PV efficiency of 98.02% and energy losses of 1.98%, as shown in Table 4. The proposed MPPT technique obtained the PV current of 7.357 A and the optimal DC link voltage of 28.4 V, respectively. The proposed FRMO-DSP MPPT algorithm managed to produce a stable optimum DC link voltage to the input side of the inverter. The optimal duty ratio was set around $\mathrm{D}=0.56$. It is observed from Figure 15 that the proposed method managed to distinguish between the LMPP and a true GMPP. The technique has a fast convergence, settling time and fewer oscillations around MPP. From the simulation results, it is observed that the proposed FRMO-DSP MPPT technique will work executively under PSC and solve PV power optimization problems because of its dynamic rapid response, tracking speed, convergence and certainty.

Figure 16 and Figure 17 depict the grid voltages, current and %THD level of the proposed ICLS based on DSOGI-PLL during variable load and reduced DC link inductor of $80 \mathrm{mH}.$ The proposed strategy obtained a THD of 1.18% under dynamic load change, which complies with the IEEE 519 standard, as shown in Table 5.

The proposed algorithm recorded an impressive system power factor of 0.999, which complies with the IEEE 1459 standard. However, when the DC link inductor was reduced, the settling time increased to 0.25 s after the GMPP was tracked. The THD content of the algorithm was found satisfactory compared with the NRIT. Its superior performance ensures better stability and improved power quality.

Figure 18 and Figure 19 demonstrate the grid voltages, current and %THD level of the NRIT during load variation and reduced DC link inductor. The algorithm recorded a high THD of 9.66%, which exceeds the limits of the IEEE 519 standard. The settling time of the NRIT is 0.47 s after the GMPP is tracked. The harmonics content is much higher when compared to the proposed ICLS based on the DSOGI scheme. The NRIT has a system power factor of 0.995, which is slightly lower than the proposed scheme. It is proved that the proposed inverter control strategy has better stability and improved power quality than the NRIT.

6.3. Case Study 3

To substantiate the performance of the proposed MPPT technique, the partial shading conditions were changed by varying the solar insolation and module temperature in case study 3. The proposed technique successfully extracted the GMPP of 207.9 W with a dynamic tracking speed of 0.15 s. It delivered the desired DC link voltage of 29.1 V to the input side of the inverter and recorded the PV current of 7.1 A. With the multiple power peaks phenomenon observed, the proposed algorithm recorded the PV efficiency of 97.54% and the energy losses of 2.46% under extremely fast-changing environmental conditions. The PV system power was optimized at around $D=0.63$.

As shown in Figure 20, the proposed system has the capability of locating the true GMPP instead of being trapped at LMPP. With its excellent performance verified, the proposed MPPT algorithm guarantees high efficiency under any type of PSC. It is verified that adding the DSP technique ensures that the system operates within a safe boundary all the time, even during transient behavior. The true GMPP is always guaranteed with the deployment of the DSP on FRMO.

Figure 21 and Figure 22 demonstrate the grid voltages, current and %THD level of the proposed ICLS based on DSOGI-PLL during variable load and reduced DC link inductor of 50 mH. The proposed strategy obtained a THD of 1.83% under dynamic load change, which complies with the IEEE 519 standard. The proposed algorithm recorded an impressive system power factor of 0.997, which complies with the IEEE 1459 standard. However, when the DC link inductor was reduced, the settling time increased to 0.27 s after the GMPP was tracked. The THD content of the algorithm was satisfactory compared with the NRIT. Its superior performance ensures better stability and enhanced power quality.

Figure 23 and Figure 24 demonstrate the grid voltages, current and %THD level of the NRIT during load variation and reduced DC link inductor. The algorithm recorded a high THD of 14.42%, which exceeds the limits of the IEEE 519 standard. The settling time of the NRIT was 0.51 s after the GMPP was tracked. The harmonics content was larger when compared to the proposed ICLS based on the DSOGI scheme, as shown in Figure 24. The method also suffers from phase imbalance because of low DC link inductor and nonlinear load. The NRIT has a system power factor of 0.979, which is slightly lower than the proposed scheme. It is proved that the proposed inverter control strategy has better stability and improved power quality than the NRIT.

6.4. Summary

The proposed ICLS-DSOGI-PLL has various advantages, including improved harmonic suppression capabilities, adaptive control, reduced computing burden and phase and grid synchronization. Grid-tied inverters must generate a consistent, sinusoidal AC waveform that matches grid voltage and frequency according to utility requirements when driving electricity to the grid. Poor synchronization can cause load imbalances, damage to linked equipment, grid instability and even power outages on the grid. In this case, DSOGI-PLL was used to speed up grid synchronization. It identifies phase errors by producing an orthogonal signal and using the Park Transformation. With this method, the orthogonal signal generator is tuned to reject all frequencies and save the grid frequency. Other variants of this technique may identify phase and frequency with high accuracy, even in the presence of low-order harmonics near the grid fundamental frequency. The proposed method was compared with NRIT-PLL, as demonstrated in

Table 6. Comparison of NRIT-PLL and proposed ICLS-DSOGI-PLL.

Parameters	NRIT-PLL	Proposed DSOGI-PLL
Settling time	0.17 s	0.06 s
Frequency overshoot	36%	5%
Phase overshoot	30°	8°
Grid synchronization speed	5 s	0.05 s
Steady-state convergence	0.028 s	0.011 s
Sampling time	50 μs	50 μs
Average system TDD of grid currents	9.98%	1.16%

. The proposed ICLS-DSOGI-PLL had a high grid synchronization speed and settling time of 0.05 s and 0.06 s, respectively. It recorded an impressive frequency and phase overshoot of 5% and 8° distinctly. Finally, the proposed system recorded steady-state convergence and an average system total demand distortion (TDD) of 1.16% in the worst-case scenario (case study 3). With the impressive results, it is noteworthy to mention that the good power quality will bring stability and reliability to the grid-tied PV system. The proposed system fulfills the grid code requirement in terms of THD and TDD. Furthermore, it has the best adaptive control capabilities as it can compensate for the variations caused by faults, voltage sag and swell and rapid changes in grid demand. The NRIT-PLL demonstrated a slow settling time and grid synchronization speed of 0.17 s and 5 s, respectively. Furthermore, the frequency and phase overshoot were larger than the proposed system, as indicated in Table 6. It recorded a TDD of 9.98% and does not comply with the IEEE 519 standard and grid requirements.

The proposed inverter control strategy is compared with the recent NRIT from the literature using practical PV panel parameters embedded in the Simulink file. Hence, the simulation results are satisfactory.

Table 7. Comparison table between the proposed inverter control and recent NRIT from the literature.

Characteristics	NRIT	ICLS-DSOGI-PLL
Type of filter	Adaptive	Adaptive
Oscillations	Moderate	Very less
Complexity	Medium	less
Amplitude tracking	Good	Excellent
Grid Synchronization	Yes	Yes
Accuracy	Moderate	Better
Frequency tracking	Good	Better
Settling time	high	less
THD under linear load	Moderate	Very less
THD under nonlinear load	Exceeds IEEE 519 limits	Very less

is utilized for comparison between the proposed method and NRIT using different characteristics. It is validated that the proposed scheme overperformed the NRIT as it complies with the IEEE 519 standard limits under linear and nonlinear load. The proposed method has advantages, such as high performance, low cost, easily tuned compensator, double loop harmonic mitigation, very accurate and less settling time. However, extensive care must be taken when selecting the DC link inductor value at a very low modulation index. The NRIT is struggling to mitigate the THD level and exceeds the IEEE 519 standard under nonlinear load. The method has advantages such as less complexity and good amplitude tracking. However, it has more advantages such as large harmonics and oscillations, voltage spikes and imbalance and a high settling time.

7. Conclusions

An intelligent FRMO based on the DSP MPPT algorithm was developed to harvest maximum power out of a PV system under fast-changing climatic conditions and maintain stable DC bus voltage. The DSP is utilized to validate the PV system operations under either normal or abnormal conditions (shading) by determining the safe operating region. From the simulation test results, the proposed FRMO based on DSP can track the GMPP with a rapid tracking speed. The method has a fast response and convergence time under any type of PSC. The proposed MPPT algorithm recorded a rapid 0.2 s and 0.13 s tracking time at light PSC and worst PSC, respectively. It has eliminated the power oscillations around the MPP and guaranteed very high PV efficiency and optimum DC link voltage to the input side of the inverter. The proposed MPPT algorithm has advantages such as GMPP tracking capabilities, fast convergence time, rapid tracking and settling speed and fewer oscillations around MPP. With the magnificent performance of the proposed algorithm, it can solve the PV system optimization problems under PSC.

Furthermore, the ICLS based on DSOGI-PLL is proposed for harmonics mitigation and power quality enhancement. The proposed control strategy is compared with the recent NRIT from the literature for performance validation under different case studies. The proposed control strategy judiciously compensated the grid harmonics by adaptively assessing the fundamental amplitude and frequency under linear and nonlinear load. The performance of the proposed system has been found satisfactory as it is adapted to the IEEE 519 standards limits. The simulation results exhibited the excellent performance of the proposed inverter control over NRIT in terms of stability and power quality. The proposed system recorded a TDD of 1.16%, which fulfills the grid requirements. Furthermore, it has the best adaptive control capabilities as it can compensate for the variations caused by faults, voltage sag and swell and rapid changes in grid demand. It is worth mentioning that the proposed inverter control scheme has benefits such as better harmonic suppression capability, control adaptivity, robustness, system stability and reliability, phase and grid synchronization speed and good quality of power.

The proposed algorithms are highly recommended in real PV system execution due to their precision, dynamic response time, ability to track GMPP under PSC, and, more importantly, power stability, reliability and quality improvement.

As with most research, not all research-affiliated aspects can be discussed in a single study. In the future, other optimization algorithms such as lion optimizer algorithm (LOA), jellyfish swarm optimization (JSO), coyote optimization algorithm (COA) and spider monkey optimization (SMO) should be considered as they can solve sophisticated problems. This future work can also investigate the performance of the trending state-of-art methods such as generative adversarial networks (GAN), self-organizing maps (SOM) and autoencoders.