An adaptive fuzzy maximum power point tracking for PV systems by a mutant gray wolf optimization algorithm

Abstract

An innovative and adaptive maximum power point tracking (MPPT) based on a fuzzy controller is proposed in this article. The proposed controller is optimized by the mutant gray wolf optimization (MGWO) algorithm, and the objective function uses an integral squared time squared error (ISTSE) index. To indicate the efficiency of the proposed MPPT algorithm (MGWO-AF-MPPT), the simulations for several irradiation conditions are modeled in MATLAB. In addition, two fair comparisons are performed between the simulation results of the proposed MPPT method with an adaptive fuzzy controller that optimized with particle swarm optimization (PSO-AF-MPPT), as well as a Perturbation and Observation method (P&O-MPPT). The comparison results illustrate that the amount of generated electrical power from the PV system for all adaptive MPPT systems is almost close to each other, indicating the high accuracy of both fuzzy-based controllers. However, the PV power generation of the proposed MGWO-AF-MPPT is higher, and also its ripple is lower than PSO-AF-MPPT. Note that, the calculated total harmonic distortion (THD) for the P&O-MPPT, PSO-AF-MPPT, and the proposed MGWO-AF-MPPT methods are calculated at 3.4%, 2.3%, and 2.1%, respectively. A lower THD indicates better power quality generated by the PV system employing the MGWO-AF-MPPT method.

1 Introduction

Recently, such clean, free, and unlimited electricity generated by solar systems has received great attention all around the world. A photovoltaic (PV) system is the most popular method of harvesting electricity power from solar energy. To enhance the efficiency of the PV system and maximize power generation capacity, it is essential to utilize maximum power point tracking (MPPT) circuits [1–3]. The PV system output power in any condition will be maximized if the converter operation point is optimized carefully [4].

A comprehensive review of MPPT methods reveals their classification into four distinct groups [5]. The first group revolves around techniques that rely on the estimation of solar cell model parameters and the establishment of relationships between these parameters and maximum power [6, 7]. Despite its utility, this approach faces challenges in adapting to variations in PV cell characteristics and lacks universality across different solar cell manufacturers [8]. Furthermore, inaccuracies in parameter estimation can significantly impact the performance of MPPT systems [9]. The second group focuses on the interplay between solar cell parameters and the system's operating point. These methods aim to identify the maximum power point (MPP) by leveraging the linear relationship between the panel’s open circuit (OC) voltage and the operating point voltage [10–12]. However, as this relationship is inherently nonlinear, attempts to linearize it may introduce errors. Additionally, frequent switching off of the PV system to measure its OC voltage leads to undesirable power losses and diminished efficiency [13]. In contrast, the third group employs the perturbation and observation (P&O) algorithm, which involves introducing perturbations in PV output voltage and monitoring the resulting changes in output power. If a perturbation reduces output power, the direction of the perturbation is reversed; otherwise, perturbation in the same direction continues [10, 14–17]. Despite its effectiveness, this method tends to introduce constant fluctuations around the MPP, resulting in power ripple [18]. Finally, the fourth group harnesses artificial intelligence techniques such as neural networks, fuzzy logic, and optimization algorithms. These intelligent methods offer the promise of reduced oscillations and higher accuracy [2, 19–21].

This article proposes an innovative and adaptive fuzzy-based MPPT controller. The membership functions (MFs) and their parameters are optimized by a mutant gray wolf optimization (MGWO) algorithm. Note that, this mutant modeling has never been adopted in the gray wolf optimization (GWO) algorithm. High efficiency, low output fluctuations in the steady state, and accurate MPPT in the partial shading conditions are prominent features of the proposed MPPT controller circuit. Its higher power generation and power quality compared to traditional MPPT methods are illustrated in an identical PV system.

In the continuation of the article, the problem formulation is indicated in the second section, including the proposed MPPT controller, objective function, and design constraints. The proposed MGWO is introduced briefly in the third section. In the fourth section, the results obtained from the simulations are discussed. Finally, this article concludes in section five. In this section, a suggestion for future studies is given.

1.1 Problem formulation

To design a MPPT controller circuit and evaluate its performance, a standard PV system similar to Fig. 1 is used to perform the simulation. A standard PV system consists of three main parts of PV cell array, a converter, and the MPPT controller circuit that feeds a battery as a load. The performance of MPPT controller circuits depends on their structure and optimal selection of the controller parameters.

The PV cell from SunPower Co. with nominal capacity of 135 W is utilized in this modeling. Table 1 lists other specifications of this array [22]. The current and voltage of the PV array at the MPP are 7.71A and 17.5 V, respectively. Moreover, the OC voltage is 22.3 V and the short circuit (SC) current of this array is 8.2A.

1.2 The proposed adaptive MPPT controller

In this paper to solve classical MPPT controller problems in different working conditions as well as increasing the generated power of PV panels, an optimized adaptive fuzzy-based controller is adopted for the MPPT. Figure 2 illustrates the block diagram of the proposed adaptive fuzzy MPPT (AF-MPPT) controller circuit.

The proposed AF-MPPT consists of two main and adaptive mechanisms; fuzzy logic is used in both sections. In case of operation condition changes, the output of the AF-MPPT is tuned by the adaptive section. The fuzzy controller's input signals are the power derivative and its deviation. The controllers’ output is given to the pulse generator in order to provide a suitable VMPP by the converter. The initial output and input MFs of the fuzzy controllers are shown in Fig. 3.

Figure 3 shows five MFs that are considered as output and inputs of the fuzzy controller. NS and PS acronyms depict negative small and positive small, respectively; Z denotes zero variations; NB and PB represent negative big and positive big, respectively. Table 2 gives the 25 rules for the fuzzy controllers.

This article uses a mutated version of GWO for designing the proposed MPPT controller and also the optimal estimating of the controller coefficients and MFs. Therefore, a suitable objective function is necessary to optimize the proposed MPPT controller.

1.3 Objective function and constraints

The performance of fuzzy-based controllers depends on the design of their MFs. The integral squared time squared error (ISTSE) is selected as an objective function to minimize the difference between maximum and output power and the maximum power of the PV panels in the fastest time. The ISTSE can be planned as Equation (1):

where t_sim and t are the simulation time and the time operator, respectively; P_max and P are the maximum and output active power of the PV system, respectively. The controller designing constraints are given in Table 3.

2 The proposed MGWO algorithm

Gray wolves live in groups with a social hierarchy shown in Fig. 4. The leader of the group is called the ‘alpha’. The alpha decides about the location and the time for sleeping, hunting, waking, etc. The second hierarchy level of the gray wolf is the so-called ‘beta’. The betas are subordinate wolves that help alpha to decide on other tasks of the group. They boost alpha’s command and give feedback to alpha. The third group is ‘delta’ wolfs, which take orders from alpha and betas. The last group of gray wolfs is the so-called ‘omega’. They play a victim role that must surrender to others in the group [23].

The main phase of gray wolf hunting is as follows:

• Search, run, and approach hunting.

• Chase, besiege, and bore the hunt, till stopping it.

• Attack on their prey.

In fact, the wolves surround the prey during hunting process. Equations (2) and (3) create a mathematical model of their siege behavior as follows [23]:

where t denotes iteration number; A and C are vector coefficients; D indicate the distance between the prey and wolf; X_p and X represent the position of the prey and the gray wolf, respectively. Equations (4) and (5) define the vectors C and A:

where |$\overrightarrow{k}$| vector reduces linearly from 2 to 0 during the iterations; vectors r₁ and r₂ have random values in the range of [0 1]. The values and timing of the vectors |$\overrightarrow{A}$| and |$\overrightarrow{C}$| can be adjusted to obtain various locations around the best position. Using r₁ and r₂, permit the wolves to move anywhere within the search area. Usually, the alpha leads the hunting process. Also, delta and beta contribute for hunting occasionally. In order to model the hunting process of gray wolves mathematically, the best solution is assumed as alpha (α), while beta (β) and delta (δ) also carry useful data about the potential position of the prey. Hence, three of the best solutions are stored. Then, the other search agents (including omega) are forced to update their positions among the position of the best solutions. According to this model, the following formulas have been suggested:

A random position within the search space of delta, beta, and alpha positions defines the final position. Delta, beta, and alpha estimate the prey’s position, then the other wolves randomly determine their positions around the prey.

In mathematical modeling, the value of k decreases when approaching the prey. Note that, the changing range of A also decreases with k. The vector A contains random values between [−k, k] that decently change from 2 to 0 during the iterations. While A varies randomly between [−1, 1], the search agent’s next position will be assigned between the hunting and current position. Gray wolves usually search for the location of delta, beta, and alpha. First, they diverge, in order to have more chance for finding a new prey, and then they converge again to hunt the prey. To create a mathematical model of their diverging, a random number more than 1 and less than −1 is used for |$\overrightarrow{A}$| to force the wolves to move away from the prey. This technique improves the identification process and lets the GWO algorithm scan the search space globally. C is another component of GWO that is for searching and identifying. |$\overrightarrow{C}$| is a random vector that varies between [0, 2] so that it provides random weights for prey depending on the distance detection of prey, as significant (C > 1) or insignificant (C < 1). This provides a more random search for GWO during the optimization process, to avoid finding local optimal solutions. It should be noted that, |$\overrightarrow{C}$| also reduces linearly according to A. In fact, C must present random values to emphasize identification during the iteration. This component is a significant factor in final iterations to avoid sticking to a locally optimal solution. Vector C plays various roles in modeling wolf hunting in nature, such as a barrier effect to approaching prey and obstacles to avoid fast collisions owing to any unexpected speed. Due to a wolf's position, it may reach prey and may accidentally move away from it at any moment, which makes hunting conditions more difficult, or vice versa.

A genetic mutation for omega wolves is considered for the modified version of the grey wolf. This model can be designed based on genetic mutation with differential evolution algorithms or regular genetic algorithms. The location of omega wolves is updated by Eq. (9):

where r is a random value with normal distribution, Max_val and Min_val are variable upper and lower bounds, respectively, and M_f is the mutation rate. The proposed mutant GWO for the optimal designing of the fuzzy MPPT controller is illustrated with a flowchart in Fig. 5.

3 Simulation results

The simulation results are segmented into two parts. Firstly, the performance of the proposed MGWO algorithm is assessed by optimizing standard benchmark functions and comparing its results with those of other optimization algorithms. Secondly, following the optimal design of the proposed MPPT circuit, its performance is evaluated under various operating conditions. Subsequently, the simulation results are scrutinized and analyzed.

3.1 Performance evaluation of the proposed algorithm

In this section, the efficiency and precision of the MGWO algorithm are assessed through optimization conducted on 30 standard benchmark functions (CEC 2014 test functions) [24]. The test system comprises a range of functions: F1–F3 represent unimodal functions, F4–F16 are multimodal, and F17–F22 are hybrid functions and finally F23–F30 are Composition functions. To demonstrate its stability and convergence characteristics, the proposed algorithm is executed 30 times for each test function. The optimization results are compared with gray wolf optimization algorithm (GWO) [25], EPSO [26], and bald eagle search (BES) algorithm [27]. The optimization results are accumulated in Table 4.

The proposed MGWO algorithm exhibits superior accuracy in comparison to the GWO, EPSO, and BES algorithms. The confirmation of this claim is supported by the lower values of mean and standard deviation observed in various implementations of the optimization algorithms.

3.2 The proposed MPPT performance evaluation

To determine the optimal parameters of the proposed adaptive fuzzy-based MPPT controller, particle swarm optimization (PSO) and MGWO algorithms are utilized and compared. The main parameters of the algorithms are denoted in Table 5.

A comparison between the convergences of both algorithms is done, which has been indicated in Fig. 6.

The comparison results show that after 50 iterations the per unit value of ISTSE for the proposed MGWO is 0.57 compared to 0.63 for the PSO, which shows better performance and higher speed ratio of the MGWO algorithm. Because, The simulation execution times for PSO and MGWO algorithms are calculated as 137 seconds and 124 seconds, respectively. In Figs 7 and 8, the optimized forms of the MFs are shown.

The optimal values for K₁, K₂, K₃, K₄, and K₅ parameters have been found with both PSO and MGWO algorithms. The results are revealed in Table 6.

In this section, a MATLAB simulation is modeled to evaluate the performance of the proposed adaptive fuzzy-based MPPT that is optimized by the proposed mutant GWO Algorithm (MGWO-AF-MPPT). The results are compared with a similar PV system by utilizing a Perturbation and Observation (P&O) MPPT algorithm, and another adaptive fuzzy that is optimized by the PSO algorithm (PSO-AF-MPPT). In the simulations, the temperature is considered constant at standard 25°C, and irradiation is changed according to Fig. 9.

In the conditions of irradiation changes for these three methods, the extracted electrical power of the utilized PV array is shown in Fig. 10.

In order to evaluate the performance of the proposed MGWO-AF-MPPT method, the maximum generated power of the PV system at the various intensities (1000, 800, 600, and 400 W/m²) for the three MPPT methods are shown in Fig. 11 in the form of bar charts.

It is obvious that at all the solar irradiations, the maximum generated power for the PV system with the proposed adaptive fuzzy MPPT optimized by the mutant gray wolf algorithm is more than the two other MPPT methods. For instance, the maximum power generated by the PV system using the MGWO-AF-MPPT at 600 W/m² is equal to 99.32 W, whereas the maximum generated power at 600 W/m² for an identical PV system with PSO-AF-MPPT and P&O MPPT are about 97.77 W and 95.07 W, respectively. Increasing solar radiation intensity will increase the electrical output power of the PV system. At 1000 W/m² solar irradiation, the maximum power generation becomes about 134.9 W with MGWO-AF-MPPT, while it is about 134.5 W with PSO-AF-MPPT and about 131.2 W with P&O MPPT. Although the maximum power generation difference for the MPPT methods is small, but significant difference is resulted in generated electrical energy for an extended period.

In order to obtain a comprehensive comparison between the performances of the MPPT methods, the values of total harmonic distortion (THD) are calculated for all the methods. The percentages of THD for the PV system with P&O-MPPT, PSO-AF-MPPT, and MGWO-AF-MPPT are about 3.4%, 2.3%, and 2.1%, respectively. The results show the lowest THD for the proposed MGWO-AF-MPPT method. Additionally, it generally found that the adaptive fuzzy MPPT methods have lower THD compared to the P&O method.

4 Conclusion

This article proposes an innovative and adaptive fuzzy-based control that is optimized by the MGWO algorithm for the MPPT of PV systems. The ISTSE index is also considered as an objective function. The performance of the proposed MPPT controller circuit (MGWO-AF-MPPT) is evaluated by comparing the simulation results with an adaptive fuzzy controller optimized by the PSO algorithm (PSO-AF-MPPT) and Perturbation and Observation method (P&O-MPPT). In the simulation, the irradiation is changed instantaneously and slowly to evaluate the performance of the MPPT methods in harsh conditions. The PV power generation for both optimized adaptive maximum fuzzy power search circuits is almost close to each other, indicating the high accuracy of fuzzy-based controllers. However, the output PV generation is higher and perturbation is fewer when MGWO-AF-MPPT is used. The maximum electrical power difference with the highest irradiation with the MGWO-AF-MPPT method is about 2.82% higher than the P&O-MPPT method. In addition, the calculated THD for P&O-MPPT, PSO-AF-MPPT, and MGWO-AF-MPPT are about 3.4%, 2.3%, and 2.1%, respectively. The lowest THD indicates high power quality of the generated power by the proposed method. Eventually, the simulation results reveal the better performance of the proposed MGWO-AF-MPPT controller.

Author contributions

Yahya Gholami Omali (Conceptualization [equal], Data curation [equal], Formal analysis [equal], Funding acquisition [equal], Investigation [equal], Methodology [equal], Resources [equal], Software [equal], Validation [equal], Visualization [equal]), Hassan Shokouhandeh (Conceptualization [equal], Data curation [equal], Formal analysis [equal], Funding acquisition [equal], Investigation [equal], Methodology [equal], Project administration [equal], Resources [equal], Software [equal], Supervision [equal], Validation [equal], Visualization [equal], Writing—original draft [equal], Writing—review and editing [equal]), Mehrdad Ahmadi Kamarposhti (Conceptualization [equal], Data curation [equal], Formal analysis [equal], Funding acquisition [equal], Investigation [equal], Methodology [equal], Project administration [equal], Resources [equal], Software [equal], Supervision [equal], Validation [equal], Visualization [equal], Writing—original draft [equal], Writing—review and editing [equal]), Mohsen Sedighi (Conceptualization [equal], Formal analysis [equal], Funding acquisition [equal], Methodology [equal], Resources [equal], Software [equal], Validation [equal], Visualization [equal], Writing—review and editing [equal]), and Jae-Yong Hwang (Data curation [equal], Formal analysis [equal], Investigation [equal], Methodology [equal], Resources [equal], Validation [equal], Writing—review and editing [equal]).

Funding

None declared.

References