# Artificial Intelligence of Things-Based Optimal Finite-Time Terminal Attractor and Its Application to Maximum Power Point Tracking of Photovoltaic Arrays in Smart Cities

Authors: En-Chih Chang; Chun-An Cheng; Rong-Ching Wu
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

The combination of artificial intelligence of things (AIoT) and photovoltaic power generation can save energy and reduce carbon emissions and further promote the development of smart cities. In order to obtain the maximum power output from photovoltaic (PV) arrays, we can use optimal maximum power point tracking (MPPT) technique with AIoT sensing to improve system efficiency. The optimal MPPT technique is the finite-time terminal attractor (FTTA) based on the gradient particle swarm optimization (GPSO), which can be applied to track the maximum power of a PV array system. The FTTA not only provides fast finite-time convergence but also attenuates steady-state errors, making it ideal for nonlinear system applications. The GPSO is used to search the control parameters of the FTTA, which is able to find the global best solution. This avoids unmodeled dynamic behavior of the system excited by the quiver, which slows down the control convergence and prematurely traps the system into a local optimum. The MATLAB computer software is used to simulate the proposed PV maximum power point tracking system. The results show that more accurate and better tracking control of the PV array can be produced under partial shading conditions and then improve the steady-state and transient performance.


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## Body

## 1. Introduction

With advances in technology, solar energy is becoming the most cost-effective long-term investment [1–3]. Like any equipment, solar systems must be managed effectively to optimize energy and power production. With solar smart cloud monitoring, users can get real-time information on PV power generation through the Internet and collect environmental information from the site to ensure the efficiency of the PV array system. Therefore, how to maximize the performance of solar cells has been one of the most important development projects in various countries around the world and is also the most important issue in solar energy-related technology. With the aim of attaining PV array systems with maximum power tracking, a DC-DC power converter including maximum power tracking control method modulates the solar cell output to deliver maximum power [4–7]. There are many research works presenting all kinds of maximum power point tracking (MPPT) alternatives, such as incremental conductivity, perturbation observation, most valuable player methodology, and fuzzy logic [8–11]. There are couplings among sunlight brightness as well as atmospheric climate and the changes in PV maximum output power. The majority of the MPPT algorithms fails to perform a rigorous evaluation of both convergence and reliability and even cannot preserve stable pursuit of the maximum power point quickly. The sliding mode control (SMC) has simplified architecture, allows effortless design, and gives the system robustness against changes in internal parameters as well as external perturbations [12–18]. There are lots of useful SMC publications for controlling PV array systems [19–22]. However, photovoltaic array systems controlled by the SMC are subject to uncertainties. The state convergence time of such a system is not limited and quiver arises, thus compromising system performance in the steady-state and transience. The quivering can be particularly problematic in practice, as it not only implies excessive energy consumption but also can provoke unmodeled high-frequency plant dynamics. There have been suggested solutions to ameliorate the quiver problem, such as estimator and adaptive methods. Even though they have improved the quiver problem and also enhanced the transience behavior of the system in the presence of unspecified disturbances, the mathematically complex and computationally time-consuming exist [23–25]. With explicit convergence and stability analysis, the finite-time terminal attractor (FTTA) not only provides the design approach for the robustness of the conventional sliding law, but more specifically, it provides a finite system state convergence time, i.e., a finite time for the system trajectory to reach the sliding mode zone in the presence of uncertainties. This methodology further strengthens the dynamical characteristics of the system and minimizes steady-state errors as well as quiver problems [26–29]. However, what is more important to note is that even though the FTTA allows the PV system to achieve the desired control effect, PV arrays may be partially shaded by buildings, wood, pollution, etc., degrading power output and leading to energy loss. There will be no regular fluctuation in the PV output power which resides in multiple local extremes. If the conventional MPPT methods described previously (e.g., disturbance observation, incremental conductivity, most valuable player methodology, and fuzzy logic) are used, they will be traced to the local extremes (local maximum power points) rather than the global extremes (global maximum power points). There have been several approaches tried to tackle multiple local extrema, such as the grey wolf optimizer and the brute force algorithm [30–32]. The grey wolf optimizer is fast, but it is constrained to local searching and incapable of searching globally. The brute force algorithm gives a superior ability to seek the best solution, but it has the weakness in needing a longer seeking time which stagnates easily on a certain solution. The gradient particle swarm optimization (GPSO) algorithm is based on the concept of gradient mechanism, showing simplified computation and enhanced population diversity that can demonstrate global domain search capability and has been widely used to solve many optimization problems [33–37]. It can improve the drawback of traditional PSO algorithm which tends to prematurely converge into local extrema [38–42]. The GPSO is therefore used to calculate the voltage reference value of the maximum power point of the PV array in the event of partial shading. In this paper, a GPSO is used to find the global extrema of the PV array under partial shading, and then, a FTTA is employed for its tracking control, providing good control power output to maintain the highest conversion efficiency of the PV array system. As a result, the FTTA based on the GPSO will improve the quiver problem, lessen the steady-state error, shorten the system state convergence time, solve the multipeak phenomenon (local maximum power point), and then render the PV MPPT system to have good steady state and dynamic response at the same time. The proposed controller with simplicity and high speed convergence as well as easy programming permits higher accuracy and stable following control. Computer simulations display that the proposed controller will provide the PV MPPT system to improve steady-state performance and transient tracking speed in the presence of partial shading or under highly uncertainty conditions. The proposed system is also compared with a conservative sliding mode controlled PV MPPT system to show the superior performance and theoretical applicability of the proposed system.


## 2. Modeling of PV MPPT System

A solar energy system usually consists of a PV array, a DC-DC converter, a DC-AC inverter, and the attached load. With changes in brightness as well as temperature, there will be a reference value for the voltage corresponding to the maximum power point of the PV system. Therefore, an interleaved Boost DC–DC converter (Figure1) is used to adjust the PV array voltage (for the maximum power point voltage). The dynamic equation of the interleaved Boost DC–DC converter is derived from the state-space averaging method for the PV system as follows:
(1)i̇db=−2LdbL−1vdbo⋅1−u−idb⋅RdbL−vdbi,v̇dbo=Cdb−1idb⋅1−u−vdbo⋅Rdb−1,where idb=idb1+idb2, vdbi is the PV array output voltage, idbo indicates the PV array output current, idbo denotes the output load current, vdbo means DC output voltage, u symbols control input, LdbL=LdbL1=LdbL2 and RdbL=RdbL1=LdbL2, here RdbL1 and RdbL2 refer to the equivalent series resistance of inductors Ldb1 and Ldb2, respectively. Let vdem be the voltage reference of the maximum power point calculated by the GPSO algorithm, and in order to make vdbo follow vdem, the FTTA closed-loop control technology is necessary to achieve such a result. In other words, in a PV system, the error in the output voltage can be defined as the state variable x~2=vdbo−vdem, which is regulated from the state variable x~1=idb feed current. Our goal is to design the control law u properly. If it is well designed, (1) will be stable and the error x~2 quickly converges to the balance point. The PV output voltage will be the same as the required reference voltage. Even if the PV MPPT system is partially shaded or malfunctioning or under nonmatching uncertainty, the tracking control can still be fast, accurate, and robust.Figure 1
Interleaved Boost DC–DC converter.Then, a single-phase PWM DC–AC inverter is used to convert the generated DC power into AC power for transmission to the grid. A typical single-phase PWM inverter is shown in Figure2, where four semiconductor switches, LC filters, and loading (resistive loading or capacitive input rectifier loading) are considered as the plant to be handled. Owing to the diversity of the loading connected to the inverter, it is not possible to develop a general model that is representative of every type of load. In this case, a nominal load is defined to derive a linear model, where load changes and model uncertainties are regarded as disturbances. Furthermore, in view of the assumption that the switching frequency is much higher than the modulation frequency of the inverter, the dynamics of the single-phase PWM DC–AC inverter is mainly handled by its LC filter and the attached load, which can be modeled as a second-order linear system. Based on Kirchhoff’s voltage and current laws, the state equations of the inverter can be described as follows:
(2)Li̇L+vo=uin,(3)iL=Cv̇o+voR,where uin is equal to Vs⋅d (time-averaging technique), where d indicates the duty ratio of switching ranging from negative one to positive one and Vs represents the DC link voltage.Figure 2
Single-phase PWM DC–AC inverter.The differential equation is obtained by replacing (3) in (2) in the following.
(4)v¨o+1RCv̇o+1LCvo=1LCuin.The state variables are specifiedx1=vo and x2=v̇o, whereupon (4) can be formulated as the result of the state-space form below:
(5)ẋ1ẋ2=01−1LC−1RCx1x2+01LCuin.The design problem of the inverter is equivalent to a typical tracking control issue since the output voltage of the inverter is a sinusoidal waveform. Allowvref to be the desired sinusoidal waveform that vo will have to follow. Through the definition of the error state variables e1=x1−vref and e2=ẋ1−v̇ref, it is possible to acquire the tracking control error state equation as follows:
(6)ė1ė2=01−a1−a2e1e2+0buin+01w,where a1=1/LC, a2=1/RC, and b=1/LC and w=−a1vref−a2v̇ref−v¨ref denotes the disturbance, with the variable e1 being a difference measurement between vo and vref. The system (6) will become stable where e1 eventually converges to zero with the control signal uin being completely designed so that the output of the inverter is kept at the same level as the desired vref. In order to allow the error states converged to zero, the control law uin can be designed via discrete proportional–integral-differential scheme. The significance of the inverter in the system needs to be specifically pointed out as follows. The inverter is the key equipment of a PV generation system, mainly serving as the interface between the PV arrays as well as the power grid. The absence of an inverter will not create a completely operational and autonomous PV generation system for residential or commercial use. In this system configuration, there are two power-converting stages. The DC-DC converter handles maximum power point tracking (MPPT) as well as adjusts the DC loading voltage. While the grid-connection occurs, the power is produced by PV arrays which are converted by the DC-AC inverter into high-quality AC output. Also, it is worth observing that the sampling rate (1/sampling time) acts as a significant deciding point for the performance of the control. It is advisable to adopt a high sampling rate (which leads to a less total harmonic distortion, even at rectified loads) in order to achieve a superior level of control. It implies the possibility of using a digital control system with the maximum frequency, alternatively an analogue design reflecting the algorithm of control [43–45].


## 3. Control Design

The problem statement for nonlinear systems using the classical TA is briefly summarized, and then, an improved technique is designed. Consider the following classical TA sliding surface as(7)Λ=ẋ1+ηx1m1/m2,where x1 is the system variable, η signifies positive constant, and 0<m1/m2<1. An SMC law u=uupperx, ulowerx can be used, representing Λ>0, Λ<0, which is expressed as driven Λ to the sliding mode Λ=0 during finite time. As such, it is possible to control the dynamics of the system as
(8)Λ=0=ẋ1+ηx1m1/m2.There is a finite timeTf for getting from the starting state x10 to zero that arises from the following:
(9)Tf=η−1x101−m1/m21−m1/m2.In this case, it is revealed that the system state is converged to zero for a finite time duration. Taking into account the Jacobian matrix =∂ẋ1/∂x1, there is a convergence of the system states to zero around the balancing point x1=0 at finite time. The eigenvalues of the Jacobian matrix are derived to be negatively infinite, as x1 tends to zero. As the speed of the system trajectory towards the balancing point becomes infinite, this causes time reachability to be limited. Yet, the equivalent control section suffers from the issues listed below. In the case of ẋ1≠0, as well as x1=0 and 0<m1/m2<1, the control law probably gives rise to a singularity. An unknown control signal is generated by this singularity, which induces an unstable feedback system. (ii) For the condition 0<m1/m2<1, it is possible to have an imaginary number x1m1/m2−1.Therefore, the FTTA is proposed to improve the conservative TA, and its sliding function based on the system dynamics (1) is represented as
(10)Λ=x~2+ρ1−1x~2κ+1+ρ2−1x~̇2m,where κ1>0, ρ1>0, ρ2>0, and 1<m<2. Simultaneously, it is recommended that the sliding mode reaching law, i.e.,
(11)Λ̇=−σ1Λ−σ2ΛksigmΛ,where σ1>0, σ2>0, 0<k<1, and sigmΛ=2/1+e−Λ/ε−1 is a smooth sigmoidal function with a positive constant ε.Recapitulating (1), the dynamics can be written as x~̇=px~t+qx~tu; here, p and q denote the functions. Then, it is possible to derive the control law of FTTA to be
(12)u=−q−1Kex~+ρ2m−1x~̇22−m1+κ+1x~2κρ1−q−1σ1Λ+σ2ΛksigmΛ,where Ke represents the feedback gain of equivalent control, which creates the desired sliding mode while system uncertainties equal to zero. Equation (12) leads to the system state will be compelled towards Λ=0, which converges in a finite amount of time.
(13)νj+1=Wνn+ℓ1RAN1χpb,j−χj+ℓ2RAN2χgb,j−χj,(14)χj+1=χj+νj+1,where νj+1 represents present flying speed, χj indicates present position, χpb shows individual best position, χgb stands for global best position, ℓ1 and ℓ2 signify learning factor, RAN1 and RAN2 are random number amidst zero and one, and W denote inertia weight. Then, let the function Gχ advance in the direction of its fastest descent, namely, the direction of the negative gradient −∇Gχ. It will gradually move towards the optimum so that the negative gradient arises from the condition of j−1 generation to j generation as follows:
(15)−∇Gχ=Gχj−1−Gχjνj,where the symbol ∇ stands for gradient operator. One may choose carefully, W is positive in case Gχj−1 is greater than equal to Gχj; otherwise, W is negative. The proposed algorithm converges fast as well as the searching of the gradient at the global best position. Because there is a constant or minor variation in its fitness value depending on a prescribed iteration number, the computational burden of the proposed algorithm diminishes [46–48]. It is important to note that the voltage, current, and power of the PV array vary with the strength of daylight as well as climate, which impacts both the power output and energy efficiency of the system. As the environmental temperature rises, there will be a reduction in the maximum power of the PV array. However, increasing irradiation levels result in potentially greater maximum PV power. To allow the PV array system to be more efficient, a MPPT function becomes imperative. Once the solar cell module approaches the maximum power point, the derivative of power with respect to voltage becomes zero. Thus, one always operates the PV array at the maximum power point through power converters together with the MPPT algorithm. More precisely, the design of the MPPT controller can be summarized as follows. (i) The overall number of particles and particle locations as well as velocities are initialized and the fitness values can be estimated. (ii) Determine each particle voltage relative to the PV array output power, as well as initialize the group best voltage. (iii) Renew the particles and execute the gradient search in accordance with (13)–(15). (iv) Allow the voltage of all particles in the group to be renewed for one-round iteration g; afterwards, the g-th particle voltage should be renewed as well as estimate the output power of the PV array working at such a voltage. (v) Update the best voltages of the particles themselves and the group. (vi) Check whether the convergence criterion has been fulfilled. In case of yes, the best solution can be obtained; otherwise, step (iii) is continued.Proof.
The definition of a Lyapunov candidate gives the followingV=Λ2/2, and based on the dynamical system trajectory along from the control law (10) with the use of the Lyapunov candidate, time derivative of V becomes V̇=ΛΛ̇=Λx~2+ρ1−1x~2κ+1+ρ2−1x~̇2m′. Because m is fractional satisfying 1<m<2, and when x~̇2≠0, x~̇2m−1>0 is valid. For ρ2>0, therefore V̇≤0 holds. Under the condition x~̇2≠0, it is proven that the system fulfills the Lyapunov stability condition which allows it to quickly arrive at the sliding surface within a finite time. Yet, there is quiver or steady-state error in the FTTA. This is caused by drastic variations or great nonlinearities in the system load, which prevent the eventual system output from following the reference sine wave, rendering the tracking behavior imprecise. Aiming to acquire the global best solution, the GPSO algorithm can be used. Equations (13) and (14) represent the model of particle evolution, which can then update the velocity and location of each particle as it flies to its target.


## 4. Results and Discussion

In order to verify the effectiveness of the proposed controller, the system is tested under steady-state response for full resistive loads and rectifier loads and transient response for step load changing. Figure3 shows the output voltage (%THD=0.05%) of the PV array system with a resistive load R=12Ω for the conservative SMC. It can be seen that the output voltage is very close to a sine wave. Figure 4 shows the output voltage (%THD=0.02%) of the proposed controller for the PV array system with a resistive load R=12Ω. Similarly, the output-voltage waveform is very close to a sine waveform. Figure 5 shows the output voltage of a PV array system controlled by the conservative SMC at a trigger angle, changing from no load to full load (R=12Ω). The figure shows that the transient voltage drop does not recover quickly and the controller is not able to compensate well. Figure 6 shows the output voltage of the PV array system using the proposed controller at trigger angle, changing from no load to full load (R=12Ω). The graph shows that the transient voltage drop is recovered within a very short period of time with excellent compensation capability. Figure 7 shows the output voltage of the PV array system under rectifier-type load with the conservative SMC. It can be seen that the output voltage is a distorted sine waveform with a high %THD value of 12.36%. Figure 8 shows the output waveform of the PV array system controlled by the proposed controller under rectified load. The output-voltage waveform is very close to the required sinusoidal reference voltage (low %THD value of 0.08%). It can be seen that the proposed PV array system has a better performance than the conservative sliding mode controlled PV array system. A comparison is listed in Table 1 of the voltage drops and THD values of the conservative SMC and the proposed controller for transient loading and steady-state loading. The proposed controller reveals small voltage drop and low THD values with the voltage waveforms close to the required sinusoidal reference voltages. However, in the case of rectified loads, the output voltage of the PV array system controlled by the conservative SMC has a distortion rate of more than 5%, while the proposed PV array system has a distortion rate of less than 5%, which is better than the 519 harmonic control standard set by the American Institute of Electrical and Electronics Engineers. The results are compared with existing work hereunder. A SMC is designed for active and reactive power control of the renewable energy generations, while the PSO algorithm allows optimizing the SMC parameters. The system can generate real power within 1% of the required power at a fast time, indicating stable power. Although the proposed method is capable of producing acceptable transience and steady-state actions, the speed of the system state driving to the sliding surface remains to be refined [49]. The single-phase inverter based on the SMC is connected to the grid with the purpose of establishing optimized PV delivery. Meanwhile, the overall circuit structure uses a boost converter with a PSO algorithm optimized fuzzy logic scheme, which yields the MPPT. The presented inverter provides quick dynamic response; nevertheless, the steady state should be investigated under severely nonlinear loading perturbations [50]. A double feedback controller realized by the SMC and proportional-integral control has been applied to the microsource inverter, where a PSO algorithm is employed for the optimization of the control parameters. The sliding-mode reaching law does not precisely strike the requisite sliding manifold, so the resulting waveforms still suffer from considerable distortion [51]. A brushless DC motor actuated water pumping system dependent on the PV supply has been proposed. This system offers several features such as no additional DC-DC converter hardware, sliding mode controller implementing sensorless speed controller, and hybrid whale optimization-perturbation and observation algorithm getting the maximum available power either in partially shaded and normally operating conditions. Because the applied technology fails to remove the chattering effect, there is a steady-state error around the state trajectory [52]. The passive-based control has been developed for grid-connected inverters. The PSO algorithm and Kalman filter observer give the simplification of the passive-based control parameters design and the diminution of sensors number, respectively. It has revealed an improvement in the total harmonic distortion and dynamic behavior both during steady state and transient loading; however, a little oscillation in the output waveform appears due to the potential for premature convergence with conventional PSO [53]. In spite of the fact that the proposed system’s performance does not greatly overtake the THD results from the latest reports, the proposed controller is unsophisticated, straightforward to comprehend, and quick to converge, thus affording better target tracking control. It can be a valuable referral for readers concerned with renewable energy converters as well as AIoT applications.Figure 3
Output voltage of a PV array system for the conservative SMC under resistive load (vertical: 50 V/div; horizontal: 5 ms/div).Figure 4
Output voltage of a PV array system for the proposed controller under resistive load (vertical: 50 V/div; horizontal: 5 ms/div).Figure 5
Output voltage of a PV array system for the conservative SMC under step load changing (vertical: 50 V/div; horizontal: 5 ms/div).Figure 6
Output voltage of a PV array system for the proposed controller under step load changing (vertical: 50 V/div; horizontal: 5 ms/div).Figure 7
Output voltage of a PV array system for the conservative SMC under rectifier load (vertical: 50 V/div; horizontal: 5 ms/div).Figure 8
Output voltage of a PV array system for the proposed controller under rectifier load (vertical: 50 V/div; horizontal: 5 ms/div).Table 1
Output voltage drop and %THD.
Proposed controllerConservative SMCStep loads Voltage drop6.67Vmax51.3VmaxFull resistive load %THD0.02%0.05%Rectifier load %THD0.08%12.36%


## 5. Conclusions

With the proposed controller, it is possible to employ a gradient particle swarm optimization algorithm to detect the global maximum power point of the photovoltaic array in the presence of partial shading while using the unique advantages of the finite-time terminal attractor to offer fast convergence of the system state under uncertain interferences, quiver, and reduction of steady-state errors for tracking control. The simulated operating conditions of the photovoltaic arrays, such as voltage, current, and power, are also stored, collated, monitored, and analysed by the IoT platform to enhance the reliability and integrity of the system. The controller developed in this manner can create the maximum power output from the solar panel and retain the highest energy conversion efficiency in the event of partial shading. The proposed controller has good steady-state and transient response behavior in terms of total harmonic distortion and momentary voltage drop under rectifier loading and step load varying conditions, displaying the waveforms are close to the required sinusoidal reference voltage. However, the output voltage of the conservative SMC of the PV array system suffers from higher than 5% THD and greater voltage drop for the same testing load conditions. As a result, the proposed PV array system actually yields promising performance when subjected to partial shading circumstances. This paper has innovative highlights which can be summarized as follows. (i) The proposed controller can be used not only for interleaved Boost DC–DC converter but also directly for Cuk, Sepic, and Zeta DC-DC converters, which provides the advantage of extended applications. (ii) The proposed controller incorporating AIoT allows the information analysis of the voltage, current, and power in the PV system, which will be the future trend of smart city development. (iii) The proposed controller is capable of yielding high-quality AC supply, which can be fully adopted in the future for various renewable energy sources, such as wind and hydrogen energies.



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*Source: 4213217-2022-04-07.xml*