A Novel Hybrid Maximum Power Point Tracking Technique for PV System under Complex Partial Shading Conditions in Campus Microgrid

Abstract

Solar generation has become increasingly important in grid applications. In order to improve the energy efficiency of the photovoltaic array (PV), factors such as temperature, nonlinear characteristics, and partial shadow conditions (PSCs) of the PV must be fully considered. An excellent maximum power point tracking (MPPT) control strategy can effectively improve the energy utilization efficiency of photovoltaic cells and provide strong support for the construction of smart campuses in terms of environmental protection and energy saving. A traditional method such as Perturb & Observe (P&O) and incremental conductance (INC) will fall into the local maximum power point (LMPP). In the past decade, researchers have proposed many MPPT methods to solve the difficulties of the PV system. However, they have failed to fully consider dynamic changes in irradiance conditions. Changes in the irradiance of photovoltaic arrays can lead to an extension of the convergence time and an increase in the oscillation amplitude. Many current MPPT methods have shortcomings such as requiring a long convergence time, large oscillation amplitude, and being prone to falling into LMPP. In order to reduce the oscillation amplitude and improve the convergence speed, a novel Multi-strategy Improved Tuna Swarm Optimization hybrid INC (ITSO-INC) method is introduced in this article. This strategy involves improving the Tuna Swarm Optimization (TSO) through Levy Flight and a linear weight coefficient. In addition, the INC method is added in the later stage to improve the accuracy of MPPT tracking. The proposed algorithm can extract the global maximum power point under different partial shading. In order to verify the effectiveness of the proposed method, the proposed method was compared with other metaheuristic algorithms such as Cuckoo Search (CS) and TSO. The proposed ITSO-INC technique was tested over four different patterns of partial shading conditions. Modulation was performed by tracking the sudden change in the shadow pattern of the MPP. These simulation results confirm that the proposed method has fast convergence, high accuracy, zero steady state oscillation, and a rapid response to dynamic change.

1. Introduction

Large amounts of greenhouse gas emissions have a great negative impact on global environmental issues. The development and utilization of renewable energy is an urgent issue. Solar energy as a sustainable energy source has the advantages of high conversion efficiency, abundant reserves, and low pollution [1]. The development and utilization of solar energy are of great concern to governments of all countries, and photovoltaic power generation is considered to be the most promising technology in the field of renewable energy. Solar energy has been widely developed and used in recent years, and it occupies a very important position in the global energy structure [2]. In the construction of a smart campus, energy conservation and the use of green energy are the key points.

An analysis of the output characteristics of PV cells shows that changes in light intensity and temperature can cause changes in the maximum power point (MPP), but there is always a GMPP for a particular light intensity and temperature [3]. The task of MPPT is to find that point and keep the PV cell in that state for continuous operation and maximum external power output to improve PV cell utilization.

Traditional MPPT methods have good optimization finding characteristics under stable conditions such as constant illumination and constant temperature. However, local shadows, for example from birds and trees, often disturb PV modules when they are arranged [4]. These partial shadow conditions (PSCs) can lead to different levels of solar irradiance for each part of the PV module. These methods often suffer from the continuous oscillation of the P-U characteristic curve and tend to fall into local power maxima when dealing with complex shading changes, thus reducing the PV system output power and affecting the system efficiency. A metaheuristic algorithm ensures that the PV module operates at the GMPP, minimizing the effect of shadows on the PV cells. At the same time, the algorithm has good adaptability in dealing with multi-peak and non-linear function-seeking problems, and has been widely studied and applied by many scholars in the MPPT control of photovoltaic power generation [5].

Traditional MPPT methods include perturbation observations (P&O) and conductance increment algorithms (INC), which are widely used because of their simple design and easy implementation. However, P&O generates large oscillations near the MPP, and the solution accuracy is poor. The INC method does not easily converge to a stable MPP, generating continuous oscillations when disturbed [6].

In recent years, researchers have proposed many new metaheuristic algorithms to solve the problem of parameter identification, which are similar to the Tuna Swarm Optimization (TSO) [7], Flower Pollination Algorithm, (FPA) [8], Bat Algorithm (BA) [9], Harmony Search (HS) [10], Squirrel Search Algorithm (SSA). All of these algorithms have been added to the MPPT application. Ref. [11] combined an optimized neural network trained through Deep Reinforcement Learning (DRL) with a classical closed-loop control and proved that this method can effectively improve the performance of MPPT. Literature [12] applied the FPA method to MPPT and achieved good results. In addition, there is a new trend in improving metaheuristic algorithms to solve the problem of multimodal objective functions. For example, [13] proposes to improve the performance of MPPT by reducing the random number in the CS. An MPPT algorithm for fast-changing shadows was proposed by [14], which showed good performance in reducing the algorithm oscillation frequency. Ref. [15] proposed to improve the Grey Wolf Algorithm by adding a Levy flight strategy. Ref. [16] proposed the Tunicate Swarm Algorithm (TSA) in an MPPT Control. Ref. [17] proposed the novel Musical Chairs Algorithm (MCA), which performs well in convergence time and reducing steady-state oscillations. In the study [18], by improving the tracking method of the CS algorithm, the tracking accuracy of the MPPT was effectively improved and the oscillation amplitude was reduced.

Another noteworthy method is to mix different algorithms to improve the efficiency of MPPT. Literature [19] proposed a new hybrid TSA-PSO algorithm that combines the Tunicate Swarm Algorithm (TSA) with the particle swarm optimization (PSO) technique for efficient maximum power in PSCs. Ref. [20] combined the Cuckoo Algorithm with Grey Wolf Optimization, and the tracking accuracy of the MPPT was improved successfully. Ref. [21] presents a new hybrid MPPT technique by combining the Artificial Bee Colony (ABC) and P&O algorithms. This technique accelerates the convergence speed of the algorithm. In the article [22], Considering the Sliding mode control structure, the improved PSO (MPSO) algorithm is combined with P&O to effectively improve the convergence speed and stability of the tracking process. Research in [23] combines an improved Artificial Bee Colony algorithm with the Multiple Heat Transfer Search algorithm, greatly improving the accuracy of algorithm tracking. The noteworthy MPPT algorithm features found in a literature review are shown in

Table 1. Research summary of published MPPT algorithms.

Authors	Years	Algorithms	Control Parameter	Remark
[12]	2019	Improve FPA	Duty cycle	The proposed technique merged fractional chaos maps with the Flower Pollination Algorithm (FPA) and compared the accuracy between different chaotic variants.
[13]	2021	CS	Duty cycle	The proposed algorithm avoids spreading the initial particles among the whole curve to predict shading patterns.
[14]	2017	FPA	Duty cycle	FPA was added to the MPPT application and the efficiency of FPA was compared with P&O and INC.
[15]	2021	Improve GWO	Duty cycle	Levy flight strategy was added to GWO to improve MPPT tracking accuracy.
[16]	2021	TSA	Voltage	The TSA algorithm was used to solve the partial shadow problem in MPPT, and a search and skip scheme (SAS) was added to minimize the search range.
[17]	2021	MCA	Duty cycle	Compared to most basic metaheuristic algorithms, the MCA algorithm has significant advantages in tracking speed and reducing vibration amplitude.
[18]	2021	ICS	Duty cycle	By improving the CS algorithm, the problems of high oscillation, high failure rate, and long convergence time in the basic CS algorithm were solved.
[19]	2022	Hybrid TSA-PSO	Duty cycle	The effectiveness of TSA-PSO was proved by simulation experiments, and the tracking accuracy reached 97.64%. Two nonparametric tests were used to prove the reliability of the algorithm.
[20]	2022	Hybrid CS-GWO	Duty cycle	By combining GWO and CS in MPPT, the efficiency of the hybrid GWO-CS algorithm was higher than that of the separate GWO and CS.
[21]	2021	Hybrid ABC-P&O	Duty cycle	The ABC-P&O algorithm has a reasonable computational cost. The experiment proved that ABC-P&O has a good performance under different PSCs.
[22]	2022	MPSO-MP&O	Duty cycle	This paper combined the improved PSO algorithm with an improved P&O method by analyzing two sliding mode control structures. The stability of MPP was achieved under changes in physical parameters of different photovoltaic systems.
[23]	2022	Hybrid IABC-SHTS	Duty cycle	The combination of an improved Artificial Bee Colony algorithm and multiple heat transfer search algorithms effectively reduced the subjectivity of manually setting parameters in the algorithm, ensuring the universality and accuracy of the search process.

.

TSO, as a new metaheuristic algorithm with a fast dynamic response, has been used in MPPT. The TSO algorithm has the advantages of good stability and relatively few algorithm parameters. Moreover, the algorithm has great potential for improvement in tracking speed and convergence time. Improving the TSO algorithm has the potential to effectively reduce the oscillation amplitude and improve the algorithm convergence speed. The TSO algorithm also has the potential to further optimize the dynamic response speed. In this paper, the Levy flight strategy was added to the TSO algorithm to improve the ability of the algorithm to jump out of the previous LMPP. Due to the fixed weight coefficient of the TSO algorithm, the global stable convergence cannot be considered. The introduction of an adaptive weight coefficient can effectively improve the local convergence performance of the TSO algorithm. During the algorithm initialization stage, the search agents were initialized by equally dividing them to increase the probability of the initial position being near the MPP. In the later stage, a variable step formula was added to improve the convergence speed of the algorithm. Combining the improved ITSO algorithm with INC can effectively overcome the problem of conventional MPPT algorithms falling into LMPP and further accelerate the convergence speed of the algorithm. The ITSO-INC algorithm runs in two stages. The first stage uses the ITSO algorithm because the improved TSO algorithm has a large search range and the ability to jump out of LMPP. The algorithm transitions to the next stage when the number of iterations meets the specified conditions. This second stage uses the INC algorithm for accurate tracking. ITSO is used over a large search range in the early stage, and INC is used for small step tracking over a small range. This two-stage approach can effectively prevent the algorithm from falling into sustained oscillations. The proposed algorithm improves tracking accuracy and search efficiency in the later stage, ensuring that the algorithm can effectively track changes in irradiance by setting algorithm restart conditions. When the irradiance of the photovoltaic array changes significantly, the algorithm is restarted.

The main contributions of the article are:

• In this paper, the author introduces a novel multi-strategy method to improve the TSO algorithm for MPPT tracking.
• A simulated battery model was established, and the effectiveness of the proposed method was verified using three different shadow conditions. To verify the effectiveness of the algorithm under changing shadow conditions, the photovoltaic array irradiance was set to suddenly change from pattern2, pattern3 to pattern4.
• During the sudden change of radiation conditions, the robustness of the algorithm in tracking GMPP was evaluated.
• By comparing the new algorithm with the basic TSO algorithm and CS algorithm, the superiority of the new algorithm in fitness convergence performance was verified.
• The simulation experiment of MPPT was carried out using a photovoltaic cell model. Compared with the basic TSO, CS, and INC, the results show that the new algorithm has more advantages in MPPT’s fast response and anti-interference.

2. Distributed PV System and Proposed MPPT Algorithm

In this section, the distributed PV system and the proposed algorithm will be explained in detail, including the patterns and characteristic curves of PV cells. The specific steps of the ITSO-INC method in MPPT application will also be introduced.

2.1. Distributed PV System

The system in the current investigation is shown in Figure 1. The system consists of one PV module, one Module-level power electronics (MLPE), and a DC-DC boost converter connected to a resistive load. The MPPT algorithm controls the duty ratio of the MLPE. The described models in the system are presented as follows.

2.1.1. PV Module

The equivalent model of a photovoltaic cell is shown in Figure 2. The photovoltaic module consists of an ideal current source (Iph), two internal resistors (Rsh, Rs), and a diode (D).

The equivalent circuit of photovoltaic cells and Kirchhoff’s law, as well as the output current equation can be written as shown in in Equation (1):

$$I=I_{ph}-I_D-I_{sh}$$

(1)

The current flowing through the diode can be written as shown in Equation (2):

$$I_D=I_0[\exp (\frac{q(U+I{R}_s)}{AKT})-1]$$

(2)

The total current generated by the PV module can be estimated by applying KCL to the circuit, and the final current equation can be written as shown in Equation (3):

$$I=I_{ph}-I_0[\exp (\frac{q(U+I{R}_s)}{AKT})-1]-\frac{U+I{R}_s}{R_{sh}}$$

(3)

In order to further calculate the total current of the PV module under partial shading, the formula in Equation (3) can be rewritten as [24]:

$$I=I_{p\mathrm{h}}{N}_{pp}-I_0{N}_{pp}\left[\exp \left(\frac{\left(U+I{R}_s\left(N_{s\mathrm{s}}/N_{pp}\right)\right)}{a}\right)-1\right]-\frac{\left(U+I{R}_s\left(N_{ss}/N_{pp}\right)\right)}{R_{\mathrm{sh}}\left(N_{s\mathrm{s}}/N_{pp}\right)}$$

(4)

$$a=\frac{N_{ss}AKT}{q}$$

(5)

where $N_{pp}$ and $N_{ss}$ represent the number of parallel and series-connected PV modules, respectively.

$I$ is the output current of the photovoltaic cell, $I_{ph}$ is the photocurrent, $I_0$ is the reverse saturation current, $q$ is the unit charge, IRs is the equivalent series resistance, Rsh is parallel resistance, $K$ is the Boltzmann constant, $A$ is the quality factor of the diode, and $T$ is the absolute temperature [25].

The characteristic curves of the photovoltaic module drawn according to Equation (4) are shown in Figure 3. These curves have several variables worthy of attention, including The short-circuit current (ISC), open-circuit voltage (VOC), maximum power point (MPP), voltage at MPP (VMPP), current at MPP (IMPP), and power at MPP.

2.1.2. Description of DC–DC Boost Converter

This article uses a boost circuit, as shown in Figure 4.

When the circuit operates stably, the energy stored by the inductor, L in one cycle T, is equal to the energy released by it. This relationship can be represented by the following equation:

$$E{i}_1{t}_{on}=(U_{\mathrm{out}}-E)i_1{t}_{off}$$

(6)

The output voltage $U_{\mathrm{out}}$ is:

$$U_{\mathrm{out}}=\frac{t_{on}+t_{off}}{t_{off}}E=\frac{T}{t_{off}}E$$

(7)

From Equation (7), it can be determined that:

$$\frac{U_{\mathrm{out}}}{E}=\frac{T}{t_{off}}=\frac{T}{T-t_{on}}=\frac{1}{1-D}$$

(8)

where, $D$ is the duty ratio, $D=t_{on}/T$.

According to Equations (7) and (8), the equivalent internal resistance of the boost circuit is:

$$R_s=\frac{E}{i_1}=\frac{U_{\mathrm{out}}(1-D)}{\frac{I_0}{1-D}}=R{(1-D)}^2$$

(9)

The output capacitance equivalent formula is:

$$C_{\mathrm{out}}=\frac{D}{R_o{\gamma }_{V_{\mathrm{mp}}}f}$$

(10)

$$C_{i\mathrm{n}}=\frac{D}{8L{\gamma }_{V_{mp}}{f}^2}$$

(11)

where $R_o$ is output resistance, ${\gamma }_{V_{mp}}$ is the output voltage ripple factor, and $f$ is the switching frequency.

A DC/DC boost converter was applied to the MPPT system, and the corresponding data are presented in

Table 2. DC/DC boost converter parameters.

Item	Value
Frequency of switching	50 Hz
C	202.55 × 10−6 F
L	8.578 × 10−3 H
R	20 Ohm

.

2.1.3. Partial Shading and Its Effects

Under ideal operating conditions, the characteristic curve of photovoltaic cells is shown in Figure 3a. However, the operation of photovoltaic cells under actual working conditions will be affected by various external factors. PV cells are usually partially or completely covered by adjacent buildings, trees, and clouds. Under PSCs, the PV curve of photovoltaic cells will become very complex and have multiple peaks. One peak is called the global MPP (GMPP), while the other peaks are called local MPPs (LMPP) [26]. The photovoltaic cell JY-210-A parameters are shown in

Table 3. PV cell JY-210-A parameters.

Parameters	PV Modules
Maximum Power	213.15 W
Peak power voltage (Vmpp)	29 V
Peak power current (Impp)	7.35 A
Open circuit voltage (Voc)	36.3 V
Short circuit current (Isc)	7.84 A
Current temperature coefficient (ki)	−0.36099 %/°C
Voltage temperature coefficient (kv)	0.102 %/°C
No. of cells	48

.

In this simulation, a PV consisting of four rows and three columns is built, and each PV module is composed of four PV cells. The PV model and P-V curves are shown in Figure 5. PV model temperature is set to either 25 °C or 30 °C. Each photovoltaic module is inserted into the anti-counterattack diode to ensure the stable operation of the battery.

In order to verify the proposed method, constant irradiance (pattern 1) and variable PSCs (pattern 2, pattern 3, and pattern 4) were set. The irradiance settings for three patterns are shown in

Table 4. Photovoltaic array irradiance settings.

Type of Pattern	Irradiance			Maximum Peak Power Point (MPP)/W
	W/m2
Pattern 1	G11 = 600	G12 = 600	G13 = 600	GMPP = 6166.58
	G21 = 600	G22 = 600	G23 = 600
	G31 = 600	G32 = 600	G33 = 600
	G41 = 600	G42 = 600	G43 = 600
Pattern 2	G11 = 1000	G12 = 1000	G13 = 1000	GMPP = 6470.72
	G21 = 1000	G22 = 800	G23 = 1000	LMPP1 = 6159.16
	G31 = 800	G32 = 800	G33 = 800	LMPP2 = 4750.62
	G41 = 600	G42 = 600	G43 = 400	LMPP3 = 2641.03
Pattern 3	G11 = 1000	G11 = 1000	G13 = 1000	GMPP = 4725.83
	G21 = 1000	G22 = 800	G23 = 1000	LMPP1 = 4578.65
	G31 = 400	G32 = 400	G33 = 400	LMPP2 = 4577.86
	G41 = 600	G42 = 600	G43 = 400	LMPP3 = 2641.03
Pattern 4	G11 = 1100	G11 = 1300	G13 = 1200	GMPP = 5188.57
	G21 = 1100	G22 = 400	G23 = 1200	LMPP1 = 4867.55
	G31 = 400	G32 = 800	G33 = 400	LMPP2 = 4609.46
	G41 = 700	G42 = 600	G43 = 400	LMPP3 = 3115.19

. By setting different irradiance levels for each photovoltaic module (G11, G12, G13, etc.), sudden changes in irradiance were controlled through step functions.

Pattern 1 has only one GMPP and no LMPP. It is worth noting that there are three LMPPs and one GMPP in pattern 2, three LMPPs and one GMPP in pattern 3, and three LMPPs and one GMPP in pattern 4.

2.2. Multi-Strategy Improved Tuna Swarm Optimization (ITSO)

Tuna Swarm Optimization was proposed by Lei Xie et al. in 2021. The Tuna Swarm Optimization algorithm has a large search range and high accuracy in solving single-objective problems, and it is an efficient meta-heuristic algorithm. The algorithm was inspired by the cooperative foraging behavior of tuna, and it includes two strategies: spiral foraging and parabolic foraging [27].

2.2.1. Initializing

The ITSO algorithm randomly and uniformly generates the initial population in the search space. The initialization is represented mathematically as shown in Equation (12):

$$X_i^{}=rand\cdot (ub-lb)+lb,\hspace{1em}i=1,2,\dots ,NP$$

(12)

where $X_i^{}$ is the first initial individual, and the upper and lower boundaries of search space $NP$ are the tuna population.

In this study, the initialization conditions for CS and TSO algorithms were improved by equally dividing the search agents, as shown in Equation (13) [20]:

$$X_0^k=k/(n+2)$$

(13)

where $n$ is the swarm-size and $X_0^k$ is the initial individual (duty cycle).

2.2.2. Spiral Foraging

According to the standard TSO, the weight coefficients α1 and α2 control the global exploration and local development ability of the algorithm. Basic weight coefficients are represented mathematically as shown in Equations (14) and (15).

$${\alpha }_1=a+(1-a)\cdot \frac{t}{t_{\max }}$$

(14)

$${\alpha }_2=(1-a)-(1-a)\cdot \frac{t}{t_{\max }}$$

(15)

A larger weight coefficient is beneficial to the algorithm in obtaining a larger search space in the pre-iteration and mid-iteration. When the weight coefficient is smaller, the algorithm can better carry out local development in the later iteration and accelerate the convergence of the algorithm. The linearly changing weight coefficient will easily lead to the algorithm falling into the local optimum at the early stage of iteration, and the slow decline at the later stage will make the local development ability of tuna individuals insufficient.

To balance the global and local search capabilities of TSO, α1, and α2 make nonlinear improvements, and the weight coefficient after improvement is shown in Formulas (16) and (17). The trend in the original weight coefficient and the improved weight coefficient with the number of iterations is shown in Figure 6. It can be seen in Figure 6 that the improved weight coefficient changes nonlinearly with α1 and α2. The range of change in the first half of α2 enhances the global exploration ability of the algorithm. When the iteration reaches the later stage, α1 and α2 rapidly change to a stable value, making the individual quickly approach the optimal individual and fully complete the local development. Therefore, the improved weight coefficient can further balance the global and local search process [7].

$${\alpha }_1=1-0.3\sin {\left(\frac{t}{t_{\max }}\times \frac{\pi }{2}\right)}^2$$

(16)

$${\alpha }_2=0.3\sin {\left(\frac{t}{t_{\max }}\times \frac{\pi }{2}\right)}^2$$

(17)

The mathematical equation of the final spiral feeding is as follows:

$$X_i^{t+1}=\{\begin{array}{l}{\alpha }_1\cdot \left(X_{rand}^t+\beta \cdot \left|X_{rand }^t-X_i^t\right|\right)+{\alpha }_2\cdot X_i^t,\hspace{1em}i=1,\hfill \\ \begin{array}{l}{\alpha }_1\cdot \left(X_{rand}^t+\beta \cdot \left|X_{rand}^t-X_i^t\right|\right)+{\alpha }_2\cdot X_{i-1}^t,\hspace{1em}i=2,3,\dots ,NP,\hspace{1em}, \mathrm{if} rand<\frac{t}{t_{\max }}\\ \begin{array}{l}{\alpha }_1\cdot \left(X_{\mathrm{best}}^t+\beta \cdot \left|X_{\mathrm{best}}^t-X_i^t\right|\right)+{\alpha }_2\cdot X_i^t,\hspace{1em}i=1,\hfill \\ {\alpha }_1\cdot \left(X_{\mathrm{best}}^t+\beta \cdot \left|X_{\mathrm{best}}^t-X_i^t\right|\right)+{\alpha }_2\cdot X_{i-1}^t,\hspace{1em}i=2,3,\dots ,NP\hfill \end{array}\hspace{1em}, \mathrm{if} rand \ge \frac{t}{t_{\max }}\end{array}\hfill \end{array}$$

(18)

where $X_{\mathrm{rand} }^t$ is a randomly generated reference point in the search space, $X_i^{t+1}$ is the position after iteration, $t_{\max }$ is the maximum number of iterations, $t$ is the current number of iterations, and $X_{best}^t$ is the current best location.

2.2.3. Parabolic Foraging

The basic TSO parabolic foraging behavior is shown in Equation (19).

$$X_i^{t+1}=\{\begin{array}{l}{X}_{\mathrm{best}}^t+rand\cdot \left(X_{\mathrm{best}}^t-X_i^t\right)+TF\cdot p^2\cdot \left(X_{\mathrm{best} }^t-X_i^t\right),\mathrm{if} rand <0.5\hfill \\ TF\cdot p^2\cdot X,\mathrm{if} rand\ge 0.5\hfill \end{array}$$

(19)

where $TF$ is a random number, ranging from −1 to 1.

For basic TSO, the most critical issue is how to eliminate the attraction of the local best. The introduction of the Levy flight strategy into the parabolic foraging strategy can enhance the global search ability of the algorithm. Therefore, this article introduces the Levy flight strategy to modify the TSO’s group update strategy. Equation (19) is updated to (20),

$$X_i^{t+1}=\{\begin{array}{l}{X}_{best}^t+\mathrm{rand}\cdot \left(X_{best}^t-X_i^t\right)+TF\cdot p^2\cdot \left(X_{\mathrm{best} }^t-X_i^t\right),\mathrm{if} \mathrm{rand} <0.5\hfill \\ TF\cdot p^2\cdot [X_i^t+\alpha \oplus L(\lambda )],\mathrm{if} \mathrm{rand}\ge 0.5\hfill \end{array}$$

(20)

$$\alpha \oplus \mathrm{L}(\lambda )~0.01\frac{\mathrm{u}}{|\mathrm{v}{|}^{-\beta }}\left(\overrightarrow{\mathrm{X}}(\mathrm{t})-{\overrightarrow{\mathrm{X}}}_{\alpha }(\mathrm{t})\right)$$

(21)

where u and v obey normal distribution.

$$\mathrm{u}~\mathrm{N}\left(0,{\sigma }_{\mathrm{u}}^2\right),\mathrm{v}~\mathrm{N}\left(0,{\sigma }_{\mathrm{v}}^2\right)$$

(22)

$${\sigma }_{\mathrm{u}}={\left[\frac{\Gamma (1+\beta )\sin \left(\frac{\pi \beta }{2}\right)}{\Gamma \left(\frac{1+\beta }{2}\right)\beta \times 2^{\frac{\beta -1}{2}}}\right]}^{\frac{1}{\beta }},{\sigma }_{\mathrm{v}}=1$$

(23)

2.2.4. Step Length and Restart

The improved algorithm can consider the global search efficiency. In order to ensure the convergence speed of the later algorithm, the MPPT late step formula is changed to Equation (24):

$$d_i=D\times \frac{\Vert I_{\mathrm{best} }-I_i\Vert }{d_{\max }}$$

(24)

where $d_i$ the current step size of the algorithm, D is the step amplification factor, $I_{\mathrm{best} }$ is the maximum step length, $I_i$ is the current number of iterations, and $d_{\max }$ is the maximum number of iterations.

The algorithm control restart conditions are:

$$\left|P_d-P_s\right|>0.05P_s$$

(25)

The flowchart of ITSO implementation in problem-solving is demonstrated in Figure 7.

2.2.5. Algorithm Test

Four groups of classical test functions were selected for testing to verify that the improved algorithm had improved performance. The functions used for testing are shown in

Table 5. Benchmark functions.

Name	Function	Dim	Range	Fmin
Step	$F_1(x)={\displaystyle \sum _{i=1}^D{\left(\lfloor x_i+0.5\rfloor \right)}^2}$	30	[−100, 100]	0
Schwefel 2.26	$F_2(x)={\displaystyle \sum _{i=1}^D-}{x}_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	−418.98 × D
Branin	$\begin{array}{l}{f}_3(x)={\left(x_2-\left(5.1/4{\pi }^2\right)x_1^2+(5/\pi )x_1-6\right)}^2\\ +10\left(1-(1/8\pi )\cos x_1+10\right)\end{array}$	30	[−5, 10]	0.398
Goldstein price	$\begin{array}{l}{f}_4(x)=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]\times \\ \left[30+{\left(2x_1-3x_2\right)}^2\left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]\end{array}$	30	[−5, 5]	3
Rosenbrock	$f_5(x)={\displaystyle \sum _{i=1}^{D-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]}$	30	[−30, 30]	0
Ackley	$\begin{array}{l}{f}_6(x)=20+e-20\exp \left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D{x}_i^2}}\right)\\ -\exp \left(\frac{1}{D}{\displaystyle \sum _{i=1}^D\cos }(2\pi x_i)\right)\end{array}$	30	[−32, 32]	0

. And the function image is shown in Figure 8. The ITSO algorithm was compared with the basic TSO and CS. To avoid accidental interference, all algorithms were repeated 30 times, and the maximum number of iterations was limited to 500. The experimental results are shown in

Table 6. Experimental results in 30 dimensions.

Function	Performance	ITSO	TSO	CS
F1	Mean	2.13 × 10−10	1.21 × 10−9	5.45 × 10−7
	Std	2.29 × 10−10	8.74 × 10−6	6.25 × 10−8
F2	Mean	−1.195 × 104	−9.925 × 103	−6.128 × 103
	Std	7.583 × 102	3.153 × 103	8.997 × 102
F3	Mean	3.98 × 10−1	3.99 × 10−1	3.96 × 10−1
	Std	0.00 × 100	6.82 × 10−6	4.78 × 10−7
F4	Mean	3.01  × 100	3.00  × 100	4.20 × 100
	Std	2.81 × 10−15	7.75 × 10−16	2.09 × 101
F5	Mean	2.759 × 101	2.896 × 101	2.226 × 105
	Std	1.058 × 100	1.163 × 100	9.868 × 103
F6	Mean	9.203 × 10−7	4.392 × 10−6	1.895 × 101
	Std	5.295 × 10−6	5.961 × 10−5	3.459 × 100

.

3. Simulation Results

In this section, the main content is to apply the proposed ITSO to MPPT for simulation experiments, and to compare and analyze the performance of four methods: ITSO-INC, TSO, CS, and INC in MPPT. The swarm size for all metaheuristic algorithms is six. The parameters used in these optimization techniques shown in

Table 7. The parameters used in this study.

Technique	Parameters
ITSO-INC	z = 0.5, β = 1.5
CS	β = 1.5, α = 0.8, pa = 0.25
TSO	z = 0.05, a = 0.8

have been frequently used in previous studies; therefore, they have good performance in MPPT applications.

3.1. Result of Pattern 1

The ITSO-INC performance in solving MPPT problems was evaluated for pattern 1. The results of ITSO-INC were also compared with CS, TSO, and INC. The PV power curves obtained using the TSO, ITSO-INC, CS, and INC are shown in Figure 9. Pattern 1 had a peak power of 6166.24 w. Obviously, the steady-state fluctuation of MPPT based on ITSO-INC was very low, reaching a maximum power of 6166.72 w. Compared with INC, TSO, and CS, ITSO-INC reached the maximum power point at an extremely fast speed, with a tracking efficiency of 99.95%. The peak power of TSO, CS, and INC MPPT methods in pattern 1 were 6165.51 w, 6153.07 w, and 6145.29 w, and the tracking efficiency was 99.95%, 99.75%, and 99.58%, respectively. In terms of tracking efficiency, these three basic methods required 0.34 s, 0.275 s, and 0.41 s, respectively. ITSO-INC only needed 0.048 s to reach GMPP, which was the fastest speed of the four methods. As shown in

Table 8. Numerical results of TSO, ITSO-INC, CS, and INC methods in pattern 1.

Parameter/Method	TSO	ITSO-INC	CS	INC
Extracted power (w)	6165.51	6166.72	6153.07	6145.29
Tracking efficiency (%)	99.97	99.99	99.77	99.62
Tracking time (s)	0.340	0.048	0.275	0.41

, ITSO-INC had a higher accuracy and tracking efficiency in MPPT than CS, INC, and TSO.

The PV duty cycle curves obtained using TSO, ITSO-INC, CS, and INC are shown in Figure 10. It can be seen in the figure that the four methods can maintain zero stable state occupation after a certain period of time. Compared with TSO and CS, the new method can quickly maintain stability after small frequency shocks and converge to the maximum power point. It is proved that the stability of the ITSO-INC algorithm was better than CS and TSO algorithms in the case of pattern 1.

3.2. Result of Pattern 2, Pattern 3 and Pattern 4

In this section, the design of the PV system to accommodate dynamic changes in irradiation (from pattern 2 to pattern 3 to pattern 4) is discussed. The comparison results of the four algorithms are described below.

For pattern 2, it can be seen in Figure 11 that CS, TSO, and ITSO-INC were able to reach the maximum power of the PV system and maintain a zero state occupation, with maximum power values of 6468.86 w, 6470.66 w, and 6265.27 w, respectively. INC failed to maintain GMPP due to the large amplitude of oscillation, resulting in an average power value of 6044.33 w. These results indicate that the efficiency of INC was the lowest. In terms of algorithm optimization time, the minimum time of INC was 0.022 s, but under PSCs, INC could not maintain zero steady-state optimization.

The time of TSO, ITSO-INC, and CS algorithms was 0.272 s, 0.095 s, and 0.188 s, respectively. On the premise of ensuring the efficiency of the algorithm, the ITSO-INC method took the shortest time and had a small frequency of oscillation, which could ensure stable operation in GMPP.

In the case of dynamic change in irradiation in pattern 3, when irradiance suddenly changed at 0.4 s, INC fell into LMPP with a value of 4489.89 w. TSO, ITSO-INC, and CS reached GMPP again after the disturbance. The recovery time of ITSO-INC after the disturbance was 0.054 s, which shows that the improved algorithm has better robustness. In terms of algorithm efficiency, the maximum power of the four methods in pattern 3 were 4352.33 w, 4701.45 w, 4702.89 w, and 4489.89 w.

When the irradiance suddenly changed at 0.8s, TSO, CS, and INC all exhibited significant oscillations. The ITSO-INC method proposed in this study had a lower oscillation amplitude and frequency, resulting in better performance. In terms of tracking efficiency, the proposed algorithm outperformed the other two metaheuristic algorithms. The INC ethod could not be maintained near MPP. Compared with TSO and CS, the ITSO-INC method improved tracking time by 26.89% and 60.29%.

It can be concluded from

Table 9. Numerical results of TSO, ITSO-INC, CS, and INC methods in pattern 3 and pattern 4.

Parameter/Method		TSO	ITSO-INC	CS	INC
Pattern 3	Extracted power (w)	6468.86	6470.66	6265.27	6044.33
	Tracking efficiency (%)	99.97	99.99	99.94	96.82
	Tracking time (s)	0.272	0.095	0.188	0.022
Pattern 4	Extracted power (w)	4352.33	4701.45	4702.98	4489.89
	Tracking efficiency (%)	92.09	99.48	99.51	95.00
	Tracking time (s)	0.141	0.054	0.269	0.024
Pattern 5	Extracted power (w) Tracking efficiency (%) Tracking time (s)	5110.89 98.50 0.145	5114.33 98.56 0.106	5106.86 98.41 0.267	4672.36 91.08 0.006

that ITSO-INC has better robustness, algorithm efficiency, and algorithm tracking speed.

As can be seen from Figure 12, TSO, ITSO-INC, and CS could maintain zero steady-state dispersion before and after disturbance in the four comparison algorithms. Although the INC algorithm had fast tracking speed, it could not maintain stable operation, and the curve continued to oscillate in constant amplitude. The ITSO-INC algorithm had the fastest tracking speed and the highest algorithm efficiency among the algorithms that could estimate zero steady state.

4. Conclusions

This article proposes a new method that combines the multi-strategy improved TSO and INC methods to track the GMPP of a PV system. The proposed new method performs MPPT tracking by optimizing the converter duty cycle. In the simulation, the PV system was composed of PV array, boost converter, and an MPPT control module. The results were obtained by comparing the proposed ITSO-INC with INC, CS, and TSO through the MATLAB simulation platform. Under three different shadow conditions, ITSO-INC had good tracking speed, robustness, and tracking efficiency. ITSO-INC had good performance under constant and variable PSCs. Compared with TSO and INC, the tracking efficiency of the new method increased by 7.39% and 4.48% under pattern 4. In terms of MPPT tracking rate, ITSO-INC was greatly improved compared with the other three methods. Under the condition of variable PSCs, the new method increased the tracking rate by 26.89% and 60.29%. These results show that ITSO-INC is a competitive and robust high-speed tracking method. Therefore, it can be concluded that ITSO-INC gives the best solution to track MPP for any sort of irradiation changes.