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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 1101692, 18 pages https://doi.org/10.1155/2022/1101692 Research Article Hyper-Spherical Search Algorithm for Maximum Power Point Tracking of Solar Photovoltaic Systems under Partial Shading Conditions 1 1 2 Kamran Eetivand , Ali Zangeneh , and Seyed M. H. Nabavi Electrical Engineering Department, Shahid Rajaee Teacher Training University, Tehran, Iran Electrical Engineering Department, Engineering Institute of Technology University, Perth, Australia Correspondence should be addressed to Ali Zangeneh; a.zangeneh@sru.ac.ir Received 10 February 2022; Revised 25 July 2022; Accepted 29 July 2022; Published 31 August 2022 Academic Editor: Sujin Bureerat Copyright © 2022 Kamran Eetivand et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Maximum Power Point Tracking (MPPT) for photovoltaic systems has widely been studied. However, studying the impact of partial shading (due to buildings, trees, clouds, adjacent arrays, and so on) on the MPPT and achieving the global maximum is still a challenging topic. *is is because not only there is a global maximum power point (GMPP) but also there are several local maximums due to the non-linear nature of the power-voltage curve after partial shading. *erefore, the conventional MPPT methods fail to track the GMPP and are commonly trapped at one of the local maximum power points. *is study utilizes the Hyper-Spherical Search (HSS) algorithm in MATLAB to achieve the GMPP while improving the eﬃciency, convergence speed, dynamic response, and reducing the losses. To track the GMPP using the HSS algorithm, the output power for the array (PPV) is deﬁned as the objective function, and the duty cycle of the DC-DC converter is selected as the particles’ position (control variable). *e performance of the proposed algorithm has been studied in three diﬀerent partial shading patterns, and the simulation results conﬁrm the capability of the algorithm in the rapid tracking of GMPP points. In addition, if the GMPP position changes over time, it will track the new GMPP with minimal oscillations. *e proposed method along with PSO, P&O, ABC, and Dragon algorithms has been applied for various scenarios, and the obtained results using HSS have been compared with the four mentioned al- gorithms, which conﬁrmed the eﬀectiveness of the proposed method. Brieﬂy, the advantages of the HSS algorithm in ﬁnding GMPP can be stated as simple implementation with few parameters, strong exploration, and exploitation during the tracking process, fast-tracking and low ﬂuctuations during tracking, low oscillation at steady-state, high dynamic eﬃciency, and non- convergence to LMPP points. above-mentioned drawbacks, maximum power point 1. Introduction tracking (MPPT) algorithms have been proposed [3, 4]. Due to the increase in energy demand, environmental *e electrical characteristics of PV modules signiﬁcantly pollution, and the cost of non-renewable energy sources, depend on shadings and temperatures. *e maximum power numerous studies have been conducted to replace these point can be tracked appropriately using the conventional resources with renewable energies. Among the renewable MPPT algorithms such as Perturb and Observe (P&O) and energy resources, photovoltaic (PV) has attracted a great Incremental Conductance (INC) if the impact of partial deal of attention due to availability, lack of environmental shading is being neglected [5, 6]. However, in these methods, pollution, low maintenance, and repair . However, there are the velocity and accuracy of the maximum power point some disadvantages, such as low power eﬃciency (15–20%), tracking depend on the turbulence step, which results in high installation cost, power ﬂuctuations, and non-linear disadvantages such as continuous ﬂuctuations of the duty I–V and P–V characteristics [2–4]. To compensate for the cycle in the power perturbation around the MPP, the poor 2 International Transactions on Electrical Energy Systems other evolutionary algorithms for maximum power tracking performance of the MPPT in fast variations of the irradiance, low eﬃciency, low speed, and accuracy in tracking the MPP such as ant colony optimization (ACO) [29], cuckoo search [30], grey wolf optimization [31], and artiﬁcial bee colony [7, 8]. While researchers have relatively solved most of the above problems using a comparative approach and variable (ABC) [32]. While these methods are suﬃciently adequate step size in the duty cycle [9–11], they can also fail to ﬁnd for optimization purposes, their performance depends on MPP in the rapid irradiance changes and resulting in in- many factors, including the initial population, the number of creased power losses [12]. In [13], the P&O method was used repetitions, the setting of parameters, and the mechanism based on a PID controller. However, using genetic and sharing information [33]. cuckoo search algorithms for tuning the parameters of the *ese metaheuristic optimization algorithms use a constant and ﬁxed number of searching components controller make the proposed algorithm more complex. Nonlinear I–V and P–V characteristics along with partial (particles, population, etc.) through all iterations which restrict the exploration (search at the beginning iterations) shading problems make several optimal points (including one global and some local optimal points) that is diﬃcult for and exploitation (search at the end iterations) of the opti- mization algorithm. Eltamaly [34] proposed a novel opti- conventional MPPT algorithms to ﬁnd exact MPP. Intelli- gence-based methods such as Genetic Algorithm (GA) [14], mization algorithm called the musical chairs algorithm with Artiﬁcial Neural Networks (ANN) [15], and Fuzzy Logic superior performance in tracking the global MPP compared Controllers (FLC) [16, 17] can distinguish between the to many metaheuristic optimization algorithms under global and local MPPs due to their ability to solve non-linear partial shading conditions. *e proposed optimization al- and complex problems. However, the GA method needs a gorithm uses high numbers of searching components in the long tracking time to achieve convergence [2], on the other initial steps of optimization and decreases it gradually to the requires lower numbers of searching agents. Yang et al. [35] hand, ANN and FLC methods require extensive training data and memory, prior knowledge, and complex calcula- presented a novel dynamic leader-based collective intelli- gence with multiple sub-optimizers for maximum power tions [3, 18]. In recent years, particle swarm optimization (PSO) has attracted much attention due to its ability to point tracking of PV systems under partial shading con- ditions. *e proposed algorithm can achieve a much wider converge to the global MPP (GMPP) in the partial shading conditions [19]. A direct control based on the PSO algorithm exploration by using various searching mechanisms instead is used in [20] to directly determine the duty cycle instead of of a single searching mechanism. Furthermore, to have setting the PI controller in the MPPT algorithm. Liu et al. deeper exploitation, the sub-optimizer with the current best [21] applied the PSO algorithm with a new formulation and solution is chosen as the dynamic leader for eﬃcient linear adjustment of parameters to track GMPP in the searching guidance to other sub-optimizers. It can oﬀer an shaded conditions. *e modiﬁed PSO is used in [3] with an enhanced searching ability and a more stable convergence compared to that of conventional metaheuristic algorithms emphasis on initial values for the sake of quick tracking. In [22], a deterministic PSO algorithm was used to omit the but in a higher computational complexity. Similarly, a modiﬁcation to the original salp swarm optimization is random factor in the speed equation and simplify the structure. *ese methods also require periodic adjustment presented in [36] with multiple salp chains named memetic and suitable values to ﬁnd the optimal duty cycle. Likewise, salp swarm optimization. *e proposed algorithm can im- an inappropriate setting of parameters also speeds up par- plement a wider exploration as well as deeper exploitation ticle updates and reduces diversity [23]. To improve the under the memetic computing framework. Besides, a virtual performance of MPPT, some researchers have used two- population-based regroup operation is used for the global stage methods as a combination of PSO and P&O algo- coordination between diﬀerent salp chains to enhance rithms. First, the PSO algorithm initially determines the convergence stability. GMPP location, and the tracking is then continued through One of the other drawbacks of the MPPT algorithms is the oscillation around the operating point, which can be miniﬁed the P&O method [24]. Krishna Mathi and Chinthamalla [25] proposed a two-stage method to increase the speed and by sliding mode control methods. In [37], a sliding surface is designed to set the operating point. *e MPP can be tracked prevent unnecessary searches. In the ﬁrst stage, the region including global point (GP) is determined using adaptive smoothly by changing the duty cycle of the converter under butterﬂy PSO (ABF-PSO) and then in the second stage, the various conditions. *e proposed method has low transient exact location of GP is tracked using P&O with variable under sudden variation as well as faster convergence with steps. Despite the advantages of the proposed method in respect to the P&O algorithm. Bianconi et al. [38] have speed and convergence, it has too many adjustment pa- presented a sliding mode control technique based on ca- rameters that make it complicated and require high memory. pacitor current sensing instead of PV voltage sensing. *e proposed technique requires few components, tracks un- In [26, 27], the PSO algorithm is used with a fuzzy controller and diﬀerential evolution to ﬁnd GMPP in the partial usually fast irradiance variations, and rejects the low-fre- quency disturbances aﬀecting the bulk voltage in grid- shading conditions. *e computation and implementation of these methods are time-consuming and complicated. connected applications and back-propagating toward the PV generator. Sliding mode control avoids the need of having Authors of [28] formed four control levels using state-de- pendent Riccati equation and fuzzy sliding mode control to exact knowledge of the system parameters. Besides, it provides improve dynamic response and power ﬂuctuation of the good performance against no modeled dynamics, insensitivity MPPT in the shaded and non-shaded conditions. *ere are to parameter variations, and excellent external disturbance International Transactions on Electrical Energy Systems 3 rejection. In [39], an MPPT algorithm based on hill climbing Application of Hyper-Spherical Space (HSS) optimi- or perturbation–observation (P&O) and a new sliding mode zation algorithm in the MPPT problem considering partial shading conditions. controller are used to regulate the desired inverter voltage. Mostly global MPPT algorithms analyze partial shading Implementation of four benchmark algorithms (P&O, for a standalone PV system and not for a grid-connected PV PSO, ABC, and Dragon Algorithms) to assess the system. Lodhi et al. [40] suggested a dragonﬂy optimization- performance of the HSS algorithm in ﬁnding MPP. based MPPT algorithm to overcome these issues. A dual- Comparison of the performance of the HSS algorithm level interfacing scheme including a boost converter and in various criteria: tracking time, tracking of global three-phase VSI has been applied to connect the PV system maximum power, and static and dynamic eﬃciency to the grid. under various partial shading conditions. A comprehensive review has been performed by Yang et al. [41] to systematically study an discuss various MPPT *e rest of the study is organized as follows. In Section 2, algorithms utilized in PV systems under partial shading PV system modeling and its characteristics are described in conditions. Moreover, they are categorized into seven the shaded conditions. Section 3 describes the HSS algo- groups, e.g., conventional algorithms, metaheuristic algo- rithm, and its implementation in the MPPT problem is rithms, hybrid algorithms, mathematics-based algorithms, described in Section 4. *e numerical studies are analyzed artiﬁcial intelligence (AI) algorithms, algorithms based on for various case studies in Section 5, and ﬁnally, conclusions the exploitation of characteristic curves, and other algo- are presented in Section 6. rithms. In [42], the main MPPT algorithms for PV systems are reviewed and divided into three groups according to 2. Model and Characteristics of the their control theoretic and optimization techniques: (1) Prototype PV Traditional MPPT algorithms, (2) MPPT algorithms based on intelligent control, and (3) MPPT algorithms under In this study, it is assumed that the prototype PV has two partial shading conditions. *e advantages and disadvan- arrays with 7 series modules in each (Figure 1). *e electrical tages of these algorithms are compared and analyzed. Be- equivalent circuit of one of the PV arrays is presented in sides, possible future research directions for MPPT are Figure 2. In order to prevent the negative impacts of partial discussed. A brief comparison among conventional algo- shading on the PV array, bypass diodes are used, in inverse rithms to ﬁnd MPPT is presented in Table 1. parallel with each module. While due to the lack of uniform Considering the problems mentioned above, in this irradiance in the partial shading caused by clouds, trees, study, an evolutionary Hyper-Spherical Search (HSS) al- buildings, or adjacent arrays, the bypass diodes in the in- gorithm [43] is applied for tracking global maximum power ternal structure of the PV module create several local MPPs point. *e algorithm is inspired by the spherical space with a single global MPP in the P–V curve. (1) presents the structure where the population consists of two groups: current for each module consisting of N series cells, as Particles and Centers of spheres. Each particle is assigned to follows:Where the I (G) photovoltaic current without loss ph one sphere space, moves toward the center of the sphere, and and this current depends on irradiance and the temperature looks for the best center. *is algorithm’s performance has of the solar cell, I represents reverse saturation current, q been studied in extensive partial shading patterns, and the means the electron charge, K is the Boltzmann constant, T simulation results conﬁrm the algorithm’s capability in the is the temperature of the p-n junction, and R , series rapid tracking of GMPP points. Besides, tracking the new resistance. position of the GMPP will be with minimal oscillations. In summary, the features of this algorithm are as follows: q V + I R N PVm PVm s s I � I (G) − I exp − 1, PVm ph s N AKT s k (1) Simple implementation and low conﬁguration (1) parameters V + I R N PVm PVm s s (2) Ability to exploration and exploitation during the − . tracking process (3) Fast-tracking and low ﬂuctuations during tracking In the following section, the current and voltage equa- (4) Low oscillation at steady-state tions of the array for each of the irradiance patterns are (5) High eﬃciency and non-convergence to LMPP presented. points To examine the HSS algorithm’s eﬀectiveness in the 3. Hyper-Spherical Search Algorithm MPPT problem, three diﬀerent shaded patterns with rapid irradiance changes are simulated, and the obtained results *e HSS algorithm was ﬁrst proposed by Karami et al. [43]. are compared with the results of P&O, PSO, ABC, and Similar to the other evolutionary algorithms, this algorithm Dragon algorithm (DA). It should be noted to keep the same begins with an initial population, which consists of two conditions for comparing the results; all four optimization groups: Particles and Centers of the spheres. In this algo- algorithms have been coded using MATLAB. *e main rithm, the search process is carried out within the space of contributions of this study are highlighted as follows: each sphere using its center and associated particles inside 4 International Transactions on Electrical Energy Systems Table 1: A comparison among conventional MPPT algorithms. Criterion PSO [19] FLC [16] P&O [5] ABC [32] Cuckoo [30] ACO [29] Tracking speed Moderate Moderate Low Fast Fast Fast Steady-state oscillation Zero Moderate High Zero Zero Zero Exploration process ✓ ✓ — ✓ ✓ ✓ Convergence to local peak Low Moderate High Low Low Low Diﬃculty in parameter tuning Moderate High Low Moderate Moderate Moderate Performance under PSC Moderate Moderate Low Moderate High High Eﬃciency Moderate Moderate Low Moderate High High Array 1 Array 2 I pva pvm1 I sm PV ph (G1) R D shm by1 module1 pvm3 PV PV M1 M1 sm ph (G3) R pva D by3 shm module3 3 PV PV M2 M2 pvm7 sm ph (G7) PV PV R D D shm M3 7 by7 M3 module7 Figure 2: Equivalent circuit of one of the parallel PV arrays with PV PV M4 M4 seven series modules. where OFD is the diﬀerence between the objective function PV PV M5 of each candidate sphere center and the maximum objective M5 function at each interval, and D is the normalized SC dominance of each sphere center. *en, the number of particles belonging to each center of the spheres is being PV PV M6 M6 determined as follows: N � roundD × N − N , (4) B SC POP SC PV PV M7 M7 where N represents particles belong to each center spheres, N is the number of initial population, and N is the POP SC number of hyper-sphere centers. Figure 1: *e prototype PV with two parallel arrays and seven modules in each array. 3.2. Search Process. A particle looks for a better solution by searching in its sphere space, formed by its center (SC) and the sphere, where eventually, all the particles converge to a radius (r) as the distance between the particle and the SC. sphere center with the best position. *e HSS algorithm has Each particle is presented with its parameters, i.e., radius (r) been applied in four steps, as shown in below sections: and angles (N-1 angles in an N-dimensional space), and the search process will be performed by changing these param- 3.1. Initialization and Particles Distribution. *e algorithm eters. Each angle is changed by α radians with the probability begins with generating N random solutions (particles) in of Pr . Parameter α is selected randomly in each repetition POP angle the feasible region. *en, based on the domination criterion with a uniform distribution between (0, 2π). After changing presented in (2) and (3), N particles are selected as sphere SC the particle angels, the distance (r) between the particle and centers, and the remaining particles are distributed among the corresponding sphere center (SC) is randomly chosen in the spheres [37]: the interval [r , r ] using (5) in an N-dimensional sphere. min max *e possible positions of the particle are shown as the hatched OFD SC D � , (2) space for a three-dimensional case in Figure 3. SC SC OFD i�1 r � P − P , (5) max i,center i,partile OFD � f − f, (3) SC SC SC i�1 s International Transactions on Electrical Energy Systems 5 Searching sphere aer “dummy particle recovery” Particle max New SC min Center Dummy particle Figure 3: Possible positions of the particle after searching in the sphere space [43]. where P represents position the center in the dimen- i,center Old SC sion i represents the particles belong to each center spheres, N represents the number of initial population, and N Searching sphere before POP SC “dummy particle recovery” represents the number of hyper-sphere centers. *e particle may achieve a position that has a better objective function value than SC after searching in its own space. In this case, the labels of this particle and SC will be Figure 4: Search space change [43]. replaced. To make the space more ﬂexible for a better search, N , the worst particles will be replaced with the same newpar 3.3. New Search Space Allocation. Considering the fact that in number of new particles. If the N is more than 5% of newpar the search space, there will be dummy particles (worst set), it is the number of initial population, it will be capped at 5% of required to improve the eﬀectiveness of the search algorithm. the number of the initial population. *ese particles are *e following steps present the required actions: assigned to the SCs using equation (2). (a) Find the worst set by sorting the particles based on their objective function (SOF) value. Since the value of 4. Implementing HSS for MPPT particles’ objective function is less important than the objective function of the sphere center, the parameter c In this study, to track the GMPP using the HSS algorithm, (equal to 0.1) is applied in the deﬁnition of SOF as the output power for the array (P ) is deﬁned as the ob- PV follows: jective function, and the duty cycle of the DC-DC converter SOF � f + cmeanf . (6) is selected as the particles’ position (control variable). Fig- i SC particles of SC ure 5 illustrates the ﬂowchart of the proposed algorithm. *e (b) Determination of the diﬀerence between the set of details for implementing the proposed HSS method are objective functions (DSOF) using presented in 4.1 to 4.7. DSOF � SOF − max SOF of groups. i i (7) groups 4.1. Step 1: Initialization. A solution set (N � ) of POP module (c) Assigning the particles to one of the SCs based on the the duty cycles is generated randomly in the feasible region calculated DSOF and the probability function for (D ), and the initial variables have been selected as follows: each SC, as follows [43]: N � N , N � 2, r � 0, r � 1, N � 1Iter � 1, POP module SC min max new DSOF i AP � , Di � D , D , i � 1, 2, . . . N . (8) i min max POP SC DSOF i�1 (9) where vector AP � [AP , AP , . . . , AP ] is formed 1 2 N SC to distribute the particles among the SCs. It should be noted that the particles with inappropriate search 4.2. Step 2: Objective Function Evaluation. A constant duty space will lose their worst set. cycle is applied to the power converter to calculate the (d) *e particles will search in the new SCs based on AP photovoltaic system’s output current, voltage, and output values (Figure 4). At the end of assigning, if an SC power (objective function). *en, N of particles (with the SC does not have any particle, it will be changed to a highest objective functions) are selected as the sphere particle. centers. *e rest of the particles are distributed among the (e) Hence, new particles are generated. sphere centers, using equation (2). 6 International Transactions on Electrical Energy Systems HSS initialization: N = 6, N = 2, r = 0, r = 1, N = 1 POP SC min max new Iter = 1 i=1 Calculate the PV output power using the speciﬁed value of the duty cycle (D) and calculate voltage and current P = V × I PV i PV i PV i Yes P > P PV i SC i No No i > N D = D i = i + 1 POP SC i i Yes Particle i moves toward its sphere center (Eq. 18) Selection of particles with inappropriate search space and look for new SC using AP values according to Eq. 14-16 No Change the SC as If SC has any particle particles? Yes Generate new particles (N ) new Assign new particles (N ) to the SCs using Eq. 10 and 11 new No Convergence criteria met? Yes Set the output as D SCbest No Yes Shading pattern changed? Figure 5: Flowchart for the HSS implemented for MPPT. 4.3. Step 3: Particles Updates. In this step, it is assumed that r � D − D , (10) max i,center i,particle the particles can move toward their assigned sphere centers. *e distance between the particle and the sphere center is chosen from the interval [r , r ], r is calculated using where D is the center duty cycle, D is the particle min max max i,center i,particle the equation. (19), and r is considered equal to zero. duty cycle. min International Transactions on Electrical Energy Systems 7 PV Array 4.4. Step 4: Search Space Update. As mentioned in Section Diode pv 3.3, the sphere will lose dummy duty cycles with inappro- priate search space, and the search sphere will be updated if C V 2 some of the duty cycles have low output power (objective 1 pv SW Load function). 4.5. Step 5: Generation of New Duty Cycles. To enhance the Current Voltage Sensor Sensor performance of the search, Nnewpar (�1 in this study) MPPT Controller number of the worst duty cycles in each repetition is being pv deleted, and the same number of new duty cycles will be Figure 6: PV array is connected to a DC-DC converter and load generated and replaced. with an MPPT controller. 4.6. Step 6: Convergence Criteria. After successive repeti- Table 2: Parameters designed for the DC-DC boost converter. tions, all SCs, except the best one with maximum output power, will be eliminated, and the other duty cycles (the Parameters Value particles) will be assigned to the best SC. In this case, there is Capacitor C 25 μF not much diﬀerence between the SC and the other duty Capacitor C 6 μF cycles with the same amount of power and position. *e Inductance L 2 mH convergence criterion is deﬁned based on the position of Load resistance R 100Ω duty cycles as (11), where the value of ε is the maximum Switching frequency 25 kHz tolerance between the duty cycles. k k D − D ≤ ε, i, j � 1, 2, . . . N i≠ j. (11) i j POP Table 3: Speciﬁcation of the MSX60 module. Variable Value 4.7. Step 7: Initialization. Since the GMPP is dynamic and Maximum power point (P ) 60 W MPP depends on the environmental conditions, the optimization Voltage at P (V ) 17.1 V MPP MPP process must be restarted by initializing the new population Current at P (I ) 3.5 A MPP MPP to obtain a new duty cycle corresponding to the new GMPP Open-circuit voltage (V ) 21.1 V OC Short-circuit current (I ) 3.8 A position. *erefore, by changing the irradiance and shading SC patterns, the process of initializing the parameters and search for the new GMPP position (Steps 1–6) should be In this study, a prototype PV with two parallel array and performed again. seven series modules in each array (7 × 60 W) is considered When the photovoltaic array is subjected to a non- to conﬁrm the eﬀectiveness of the proposed algorithm under uniform shade, its current-voltage curve is step-shaped, and various environmental conditions. A bypass diode (MS × 60) thus its power-voltage curve has multiple peak points. In is installed in parallel with each module (Figure 1). *e order to distinguish between the irradiance variation and speciﬁcations of the diode are presented in Table 3. partial shading conditions, continuous sampling of the *e simulation results obtained by the proposed HSS voltage and current in diﬀerent repetitions can be performed algorithm are compared with the P&O, PSO, ABC, and using the following equations [44]: Dragon methods in terms of quantitative criteria such as I(k) − I(k − 1) tracking time, static eﬃciency, dynamic eﬃciency, and ≥ 0.1, (12) I(k) output power. Two diﬀerent types of performances (Static and dy- V(k) − V(k − 1) namic) have been considered in this paper. *e static eﬃ- ≥ 0.2. (13) V(k) ciency (14)represents the steady-state performance, while the dynamic eﬃciency (15)represents both the transient and If (12) and (13) are fulﬁlled, partial shading conditions or steady-state performance [46]. changes in the shading pattern have occurred. In these equations, k is the repetition number, and values 0.1 and 0.2 MPPT η � × 100, (14) static are determined using the trial and error method. max 5. Simulation Results P dt PV η � × 100, (15) dynamic An MPPT controller includes the power-electronic interface P dt max (DC-DC boost converter), load, and photovoltaic array for a standalone application shown in Figure 6 [36]. *e pa- where η is the static eﬃciency, η is the dynamic static dynamic rameters of the DC-DC boost converter are designed for a eﬃciency, P is the MPP obtained in the steady-state, MPPT continuous conduction mode (Table 2) [45]. and P is the maximum available power of the array. To max 8 International Transactions on Electrical Energy Systems Table 4: Parameters of P&O, PSO, Dragon, and HSS algorithms. P&O ABC PSO DA HSS D � 0.005 NB � 6 C � 1.2 Alignment weight: S � 0.5 × rand × M N � 2 1 SC D � 0.85 SN � 3 C � 1 Alignment weight: A � 0.5 × rand × M N � 1 1min new — MCM � 30 C � 1.6 Cohesion weigh: C � 1.5 × rand × M r � 0 2 min — C � 1 Food factor: F � 1.5 × rand × M r � 1 2max max — W � 0.4 Enemy factor: E � 0.5 × M — min 0.2×iter — — W � 1 M � 0.1 − — max Max iter 0.5×iter — — — Inertia weight: W � 0.5 − — Max iter Table 5: Radiation in pattern 1. Module Array 2 2 2 2 2 2 2 M1 (m /W) M2 (m /W) M3 (m /W) M4 (m /W) M5 (m /W) M6 (m /W) M7 (m /W) 1 1000 800 600 400 300 200 100 2 1000 800 600 400 300 200 100 maintain uniformity and make a comparison, the sampling 5.1.2. Second Pattern (G > G > G > G > G > G > G ). 1 2 3 4 5 6 7 time is considered to be 20 milliseconds. *e selected of In this pattern, according to Table 6, diﬀerent radiation various applied algorithms for the comparison are presented conditions have been applied and each module works at a in Table 4.All the algorithms were coded and executed in the lower and diﬀerent radiation level than pattern 1. In this matlab platform on a system having 8 GB RAM supported pattern, there are also 7 maximum points in the power- with INTEL i7 processor voltage curve. *e voltage and current equations are rep- resented by (19)–(21). I G > I G > . . . > I G , (19) 5.1. Partial Shading Patterns. *e PV array is tested under ph 1 ph 2 ph 7 three steady-state shading conditions as follows. I , I < I G , ⎧ ⎪ pvm1 pva ph 1 ⎪ I , I < I G , 5.1.1. First Pattern (G > G > G > G > G > G > G ). In pvm2 pva ph 2 1 2 3 4 5 6 7 ⎪ this pattern, the received radiation of each module is ⎪ I , I < I G , pvm3 pva ph 3 according to Table 5, which causes the creation of 7 max- I , I < I G , I � 2 × (20) pva pvm4 pva ph 4 imum points in the power-voltage curve shown in Figure 2. ⎪ I , I < I G , pvm5 pva ph 5 *e voltage and current equations of the PV arrays are as ⎪ (16)–(18): I , I < I G , ⎪ pvm6 pva ph 6 I , I < I G . I G > I G > . . . > I G , (16) ph 1 ph 2 ph 7 pvm7 pva ph 7 V , I < I G , I , I < I G , ⎪ PVm1 pva ph 1 ⎧ ⎪ pvm1 pva ph 1 ⎧ ⎪ V + V , I < I G , I , I < I G , ⎪ PVm1 PVm2 pva ph 2 ⎪ pvm2 pva ph 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V + V + V , I < I G , I , I < I G , ⎪ ⎪ PVm1 PVm2 PVm3 pva ph 3 pvm3 pva ph 3 I V V + V + V + V , I < I G , I � 2 × , I < I G , (17) � PVm1 PVm2 PVm3 PVm4 pva ph 4 pvm4 pva ph 4 pva pva ⎪ ⎪ ⎪ ⎪ ⎪ I , I < I G , V + V + · · · + V , I < I G , ⎪ ⎪ PVm1 PVm2 PVm5 pva ph 5 pvm5 pva ph 5 ⎪ ⎪ I , I < I G , ⎪ V + V + · · · + V , I < I G , ⎪ PVm1 PVm2 PVm6 pva ph 6 pvm6 pva ph 6 ⎪ ⎩ ⎩ I , I < I G . V + V + · · · + V , I < I G , PVm1 PVm2 PVm7 pva ph 7 pvm7 pva ph 7 (21) V , I < I G , ⎧ ⎪ PVm1 pva ph 1 V + V , I < I G , PVm1 PVm2 pva ph 2 V + V + V , I < I G , ⎪ PVm1 PVm2 PVm3 pva ph 3 5.1.3. ;ird Pattern (G > G > G > G > G > G > G ). 3 2 7 5 1 6 4 Despite the previous patterns with regular decreasing ra- V + V + V + V , I < I G , V � PVm1 PVm2 PVm3 PVm4 pva ph 4 pva ⎪ diation, a random pattern is considered for further inves- V + V + · · · + V , I < I G , PVm1 PVm2 PVm5 pva ph 5 tigation in Table 7. In this pattern, there will be 7 maximum ⎪ V + V + · · · + V , I < I G , PVm1 PVm2 PVm6 pva ph 6 ⎪ points with diﬀerent positions in the power-voltage curve, where the voltage and current equations are shown with V + V + · · · + V , I < I G . PVm1 PVm2 PVm7 pva ph 7 (22)–(24). (18) International Transactions on Electrical Energy Systems 9 Table 6: Radiation in pattern 2. Module Array 2 2 2 2 2 2 2 M1 (m /W) M2 (m /W) M3 (m /W) M4 (m /W) M5 (m /W) M6 (m /W) M7 (m /W) 1 650 550 450 350 250 150 50 2 650 550 450 350 250 150 50 Table 7: Radiation in pattern 3. Module Array 2 2 2 2 2 2 2 M1 (m /W) M2 (m /W) M3 (m /W) M4 (m /W) M5 (m /W) M6 (m /W) M7 (m /W) 1 500 800 900 200 600 400 700 2 500 800 900 200 600 400 700 I G > I G > I G > I G > I G > I G > I G . (22) ph 3 ph 2 ph 7 ph 5 ph 1 ph 6 ph 4 I , I < I G , ⎧ ⎪ pvm3 pva ph 3 ⎪ I , I < I G , pvm2 pva ph 2 I , I < I G , pvm7 pva ph 7 I , I < I G , I � 2 × (23) pva pvm5 pva ph 5 I , I < I G , ⎪ pvm1 pva ph 1 I , I < I G , pvm6 pva ph 6 I , I < I G , pvm4 pva ph 4 V , I < I G , ⎪ PVm3 pva ph 3 V + V , I < I G , ⎪ PVm2 PVm3 pva ph 2 V + V + V , I < I G , ⎪ PVm2 PVm3 PVm7 pva ph 7 V + V + V + V , I < I G , V � (24) PVm2 PVm3 PVm7 PVm5 pva ph 5 pva V + V + V + V + V , I < I G , PVm2 PVm3 PVm7 PVm5 PVm1 pva ph 1 ⎪ V + V + V + V + V + V , I < I G , PVm2 PVm3 PVm7 PVm5 PVm1 PVm6 pva ph 6 V + V + · · · + V , I < I G . PVm1 PVm2 PVm7 pva ph 4 5.2. Case Studies seconds (Figure 10(b)). *e results in both Figures 9 and 10 show that the HSS algorithm performs better in terms of 5.2.1. Case 1. In this pattern, the partial shadow of the global peak point tracking time and accuracy. For a quan- module radiation is randomly selected (Figure 7). *is titative comparison of this algorithm with other simulated pattern leads to 7 peaks in the P–V curve (Figure 8). *e algorithms, a summary of these results in terms of pa- powers at the local peaks are 86.63 W, 178.83 W, 221.10 W, rameters such as dynamic eﬃciency, static eﬃciency and 254.52 W, 238.37 W, and 87.53 W, respectively, and the peak extraction time is presented in Table 8. power is 261.03 W. Figure 9 shows the power, voltage, and duty cycle curves corresponding to the PSO, ABC, and P&O algorithms. 5.2.2. Case 2. Similar to Case 1, in this case, the radiation According to the results using PSO, the proposed PSO al- pattern on the modules is assumed to be random, as depicted gorithm has tracked the global maximum point (261.04 W) in Figure 11. *is pattern also produces 7 peaks in the P–V after 2.52 seconds. In addition, the ABC algorithm tracks the curve as shown in Figure 12. *e powers at the local peaks same global point in 1.20 seconds, while the P&O algorithm are 54.99 W, 119.13 W, 159.83 W, 158.62 W, 113.58 W, is trapped in the second local peak (187.87 W). 35.55 W, respectively, and the power at the global peak is Figure 10 shows the power, voltage, and duty cycle 173.81 W. curves corresponding to the HSS and DA algorithms. In Figure 13 shows the power, voltage, and duty cycle Figure 10(a) it can be seen that the HSS algorithm has curves for PSO, ABC, and P&O algorithms. *e PSO al- tracked the global peak after 0.90 seconds while the DA gorithm detects the power global peak point (173.693 W) in algorithm has tracked the global peak point after 1.18 1.80 seconds (Figure 13(a)) while the ABC algorithm detects 10 International Transactions on Electrical Energy Systems Array 1 Array 2 PV 1000 1000 M1 M2/W M1 M2/W M2 M2 M2/W M2/W 600 600 M2/W M2/W M3 M3 500 500 M4 M2/W M2/W M4 M5 M2/W M2/W M5 300 300 M2/W M2/W M6 M6 M2/W M7 M2/W M7 Figure 7: *e received radiation for each module. Global Peak=261.05 W Local Peak 4 Local Peak 5 Local Peak 3 Local Peak 2 Local Peak 1 Local Peak 6 0 50 100 150 Voltage (V) Pattern 1 Figure 8: P–V curve of the partial shadow pattern in Case 1. the global peak point in 1.02 seconds (Figure 13(b)). It can be Figure 14 shows the power, voltage, and duty cycle seen that for this shadow pattern, the P&O algorithm has curves corresponding to the HSS and DA algorithms. *e been trapped in the ﬁrst local peak due to the lack of rec- HSS algorithm tracks the global peak power of 173.694 W in ognition between the global peak and the local peak, and 0.80 seconds, while the DA algorithm tracks the same point tracks 54.61 W (Figure 13(c)). after 1.08 seconds (Figure 14(b)). *e tracking time for the Power (w) International Transactions on Electrical Energy Systems 11 400 400 GMPP Traked GMPP Traked LMPP Traked X 1.68 X 1.68 200 X 1.68 Y 261.046 Y 261.04 Y 178.873 Ts=1.20 s Ts=2.52 s Ts=0.35 s 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) 50 50 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) 0.5 0.5 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) Prel Prel Prel Vrel Vrel Vrel (a) (b) (c) Figure 9: Simulation results for Case 1 using (a) PSO, (b) ABC, and (c) P&O. 400 400 GMPP Traked GMPP Traked X 1.9 X 1.9 Ts=0.90 s Ts=1.18 s Y 261.042 Y 261.047 200 200 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) X 1.94 X 1.34 Y 0.431641 Y 0.431497 0.5 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) Prel Prel Vrel Vrel D D (a) (b) Figure 10: Shadow pattern simulation results for Case 1: (a) HSS and (b) DA. Table 8: Radiation pattern for modules for Case 1. Power from MPP curve Power at MPP Duty cycle at Tracking time Static eﬃciency Dynamic eﬃciency Pattern Method (W) (W) MPP (s) (%) (%) HSS 261.047 0.43 0.90 99.99 98.59 DA 261.042 0.43 1.18 99.99 96.71 (1) ABC 261.05 261.046 0.43 1.20 99.99 98.30 PSO 261.040 0.43 2.52 99.99 95.16 P&O 178.873 0.76 0.35 68.52 66.48 Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) 12 International Transactions on Electrical Energy Systems Array 1 Array 2 PV M2/W M1 M2/W M1 M2/W M2 M2/W M2 M2/W M3 M2/W M3 M2/W M4 M2/W M4 M2/W M5 M2/W M5 M2/W M6 M2/W M6 M2/W M7 M2/W M7 Figure 11: Radiation of each module in Case 2. Global Peak=173.81 W Local Peak 3 Local Peak 4 Local Peak 2 Local Peak 5 Local Peak 1 Local Peak 6 0 50 100 150 Voltage (V) Pattern 2 Figure 12: P–V curve of the partial shadow pattern in Case 2. PSO, ABC and DA algorithms decreased by 55.55%, 21.56% 5.2.3. Case 3: Random Non-uniform Irradiance Based on and 25.92%, respectively. Table 9 provides a quantitative Pattern 3. In this part, a complicated shadow pattern is comparison of the proposed algorithm with other simulated selected. *e radiation of the modules is shown in Figure 15. algorithms. *e comparison is based on dynamic eﬃciency, According to the P–V curve in Figure 16, the local peaks for static eﬃciency, and tracking time. the power are 77.65 W, 175.25 W, 251.30 W, 301.97 W, Power (W) International Transactions on Electrical Energy Systems 13 GMPP Traked Ts=1.02 s LMPP Traked GMPP Traked Ts=1.80 s X 2.02 200 Ts=0.06 s X 2.02 Y 54.6171 X 2.2 Y 173.694 Y 173.693 time (S) 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 150 01 2 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) 0.5 0.5 0.5 0 0 0 0 0.5 1 1.5 2 2.5 3 0123 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) Prel Prel Prel Vrel Vrel Vrel D D (a) (b) (c) Figure 13: Simulation results for Case 2 using (a) PSO, (b) ABC, and (c) P&O. 400 400 GMPP Traked Ts=0.80 s GMPP Traked Ts=1.08 s X 1.8 X 1.8 Y 173.694 Y 173.694 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 150 150 100 100 50 50 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 0.5 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) Prel Prel Vrel Vrel D D (a) (b) Figure 14: Shadow pattern simulation results for Case 2: (a) HSS, (b) DA. Table 9: Radiation pattern for modules for Case 2. Power from MPP curve Power at MPP Duty cycle at Tracking time Static eﬃciency Dynamic eﬃciency Pattern Method (W) (W) MPP (s) (%) (%) HSS 173.694 0.46 0.80 99.93 97.61 DA 173.694 0.46 1.06 99.93 96.75 (2) ABC 173.810 173.694 0.46 1.02 99.93 97.53 PSO 173.693 0.46 1.80 99.92 95.16 P&O 54.617 0.83 0.06 31.42 31.87 Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) 14 International Transactions on Electrical Energy Systems Array 1 Array 2 PV 500 500 M1 M2/W M1 M2/W 800 800 M2 M2 M2/W M2/W 900 900 M3 M3 M2/W M2/W 200 200 M4 M4 M2/W M2/W M5 M5 M2/W M2/W 400 400 M6 M6 M2/W M2/W M2/W M7 M2/W M7 Figure 15: Radiation of each module for Case 3. Local Peak 5 Local Peak 4 Local Peak 3 Local Peak 2 Local Peak 6 Local Peak1 Global Peak=325.25 W 0 50 100 150 Voltage (V) Figure 16: P–V curve of the partial shadow pattern in Case 3. 319.02 W, and 187.77 W, respectively, where the global peak seconds 323.844 W (Figure 17(b)). It should be noted that power is 325.25 W. global peak power using the ABC is slightly smaller than Figure 17 shows the power, voltage, and duty cycle the actual global peak power. In addition, in this shadow curves corresponding to the PSO, ABC, and P&O algo- pattern, the P&O algorithm is trapped in the second local rithms. *e PSO algorithm detects the global peak power peak due to the lack of recognition between the global point (325.18 W) in 1.46 seconds (Figure 17(a)), while the peak and the local peak, and detects 175.26 W ABC algorithm detects the global peak power point in 1.04 (Figure 17(c)). Power (W) International Transactions on Electrical Energy Systems 15 GMPP Traked 400 400 400 Ts=1.46 s GMPP Traked Ts=1.04 s LMPP Traked Ts=0.34 s X 1.64 X 1.64 Y 325.189 Y 323.844 200 200 200 X 1.64 Y 175.264 0 0 0123 0 0.5 1 1.5 2 2.5 3 0 123 time (S) time (S) time (S) 150 150 150 50 50 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) time (S) 1 1 0.5 0.5 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 1 2 3 time (S) time (S) time (S) Prel Prel Prel Vrel Vrel Vrel (a) (b) (c) Figure 17: Simulation results for Case 3 using (a) PSO, (b) ABC, and (c) P&O. GMPP Traked 400 GMPP Traked Ts=0.96 s 400 Ts=0.98 s X 1.9 X 1.9 Y 325.19 Y 325.19 200 200 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 50 50 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) 1 1 0.5 0.5 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (S) time (S) Prel Prel Vrel Vrel (a) (b) Figure 18: Shadow pattern simulation results for Case 3: (a) HSS and (b) DA. Table 10: MPPT results of diﬀerent algorithms in Case 3. Power from MPP curve Power at MPP Duty cycle at Tracking time Static eﬃciency Dynamic eﬃciency Pattern Method (W) (W) MPP (s) (%) (%) HSS 325.190 0.49 0.96 99.98 98.06 DA 325.190 0.49 0.98 99.98 97.11 (3) ABC 325.25 323.844 0.49 1.04 99.56 97.57 PSO 325.189 0.49 1.46 99.98 97.41 P&O 175.264 0.77 0.34 53.88 52.52 Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) Duty Voltage (V) Power (W) 16 International Transactions on Electrical Energy Systems Figure 18 shows the power, voltage, and duty cycle details, initially, the HSS algorithm is being executed to curves corresponding to the HSS and DA algorithms. *e pass through the local peaks, then the algorithm is switched to the “Incremental Conductance” method to track the HSS algorithm detects the global peak at 325.19 W in 0.96 seconds, while the DA algorithm tracks the same point at global peak. *e optimization change method avoids ad- 0.98 seconds. Comparing the results illustrate that the ditional searches until reaching the exact value of the global tracking time of the proposed algorithm has been improved peak and increases the eﬃciency and speed of optimization. by 34.24%, 7.6%, and 2.04% compared to PSO, ABC, and DA algorithms, respectively. Table 10 provides a quantitative Abbreviations comparison of the proposed algorithm with other simulated algorithms. *is comparison is based on parameters dy- I : Photovoltaic module current PVm namic eﬃciency, static eﬃciency, and tracking time. It V : Photovoltaic module current PVm should be noted that in all simulated models, the power I : Light-generated current ph ﬂuctuations in the MPP point tracking process for the I : Saturation current proposed algorithm (HSS) are less than in other algorithms. q: Electron charge K: Boltzmann’s constant A: Diode ideality factor 6. Conclusion T : Temperature (Kelvins) *is study presents the application of a robust optimization N : Number of series cells algorithm, Hyper-spherical search (HSS), for tracking MPP P : Possible position of a particle under shading conditions. *e simulation results indicate that R : Series resistance the HSS algorithm fulﬁlls a high tracking accuracy and speed c: Constant coeﬃcient and has good dynamic performance in the irradiance change D : Duty cycle at P MPP MPP conditions. Besides, HSS is compared with four methods: P&O, R : Load resistance PSO, ABC, and Dragon algorithms. *e P&O is not eﬀective in P : Maximum power point MPP partial shading conditions and, in most cases, converges to the V : Voltage at P MPP MPP LMPP. Although the PSO and ABC algorithm has powerful I : Short-circuit current SC performance, it faces a few problems in tracking the GMPP η : Dynamic eﬃciency dynamic point as the shading conditions get complicated. *e simu- C , C : Learning coeﬃcients 1 2 lation results show that all the search-based algorithms (HSS, SOF: Set objective function DA, ABC, and PSO) have signiﬁcantly better performance to AP: Assigning probability ﬁnd GMPP than the P&O conventional method. Although the MPP: Maximum power point diﬀerence between these approaches is less in some criteria GMPP: Global maximum power point such as tracking of MPP, duty cycle, and static eﬃciency, the PSO: Particle swarm optimization proposed HSS algorithm has the lowest tracking time (0.9, 0.8, D : Dominance of each sphere center SC and 0.96 in case studies 1, 2, and 3 respectively) and highest OFD: Objective function diﬀerence dynamic eﬃciency (98.59%, 97.61% and 98.06% in case study 1, N : Number of initial population POP 2 and 3 respectively) with respect to the others. Since the HSS N : Number of hyper-sphere centers SC method easily tracks the GMPP points in all critical shade N : Number of new particles newpar conditions. It conﬁrms that it can be an eﬀective alternative to N : Number of modules module standard and evolutionary MPPT methods. N: Dimensional space *e advantages of the proposed HSS algorithm can be r: Sphere radius summarized follows: r : Minimum radius of the sphere min r : Maximum radius of the sphere max *e HSS algorithm tracks the MPP point faster than the N : particles belong to each center spheres other algorithms and imposes fewer losses on the D: Switch duty cycle system. D : Minimum duty cycle min Under dynamic conditions such as rapid radiation D : Maximum duty cycle max changes, the HSS algorithm tracks new MPP points V : Open-circuit voltage OC with fewer ﬂuctuations. I : Current at P MPP MPP *e static and dynamic eﬃciencies of the proposed HSS η : Static eﬃciency static D: Step-change in duty cycle algorithm is better than other algorithms. W: Inertia weight factor Detects GMPP for various radiation patterns and DSOF: Diﬀerence of SOF shadow conditions with higher reliability. 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International Transactions on Electrical Energy Systems – Hindawi Publishing Corporation
Published: Aug 31, 2022
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