1 Introduction

Recently, various Metaheuristic Algorithms (MAs) have been presented for tackling real-life optimization problems in multiple domains. These MAs are designed based on inspirations borrowed from the principles of biology, ethology or swarm intelligence, physics, chemistry, etc. [1, 2]. The researchers have put forward significant efforts in building theoretical foundations and establishing experimental results, through employing MHs in wide spread application areas of science and technology, mainly due to implementation flexibility, gradient-free search mechanism, and ability to deal with search space with numerous local optima [3]. These extensive applications include bioinformatics [4], cost-effective emission dispatch [5], feature selection [6], image segmentation [7], task scheduling in cloud computing [8], engineering problems [9], and combinatorial and global optimization problems [10, 11]. Nevertheless, it is the basic requirement for these MAs to maintain tradeoff balance between exploration and exploitation capabilities, in order to avoid producing local solutions [12]. In this vein, several attempts have been made to deal with premature convergence, and opposition-based learning (OBL) [13] has proven to be a useful approach in this regard, as it enhances global exploration, convergence speed, helps avoid local regions, and achieves balanced exploration-exploitation in original MA [14, 15].

Recently, renewable energy resources such as solar energy has been utilized as additional mean to meet the ever-increasing demands, as it also reduces the dependence on fossil fuels which have many issues such as the shortage of fossil fuels, fluctuation of fuel prices, and greenhouse gas emission [16, 17]. In solar energy, solar photovoltaic (PV) modules are used to convert into electrical energy [18]. Each of these PV modules comprises of a number of solar cells that convert sunlight into the required electricity [19]. So, the cell represents the principal construction part of PV systems [20]. A proper mathematical model is needed for implementing the PV solar cells in actual life under all different conditions [21]. There exist different approaches to simulate the PV solar cells, and the single diode solar cell model (SDM) is the starting of the establishment of the PV model [22, 23]. This model is fast and simple as it only has five unknown parameters. One of the significant problems of this model is its insufficient accuracy when employed on open-circuit voltage with low radiation. To overcome this problem, the double diode model (DDM) has been developed by adding another diode. Therefore, DDM gives more accuracy but poses significant complexity because of additional two unknown parameters (totally seven unknown parameters). Meanwhile, a TDM was employed to reach more precision than the SDM and DDM, which has nine unknown parameters [24]. Recently, researchers suggest a new modified model (MTDM) to the TDM to improve the accuracy and the reliability [25].

Many MAs have been proposed to estimate the PV parameters for the SDM, DDM, and TDM [26,27,28]. The conventional methods need to be efficient without heavy computations and able to converge to global solutions rather than local ones [29]. Consequently, meta-heuristic techniques are considered a more reliable solution to avoid the drawbacks of conventional methods [30]. There are several other techniques have been proposed for extracting the PV parameter’s values such as enhanced Lévy flight bat algorithm (ELBA) [31], artificial electric field algorithm (AEFA) [32], modified gradient-based optimizer (MGBO) [33], transient search optimization (TSO) [34], hybrid adaptive TLBO with DE algorithm (ATLDE) [35], supply demand-based optimization (SDO) [36], and improved bonobo optimizer (IBO) [37].

An improved equilibrium optimizer (IEO) by local minima elimination and linear reduction diversity is presented for solar PV model paramter estimation [38]. In this connection, another MA, Harris’ hawks optimization (HHO), is enhanced by incorporating orthogonal learning and general OBL for increased convergence ability through population diversity and better exploitation performance [39]. In [40, 41], tree growth algorithm (TGA) and triple-phase teaching-learning-based optimization (TPTLBO) are repectively utilized for the same purpose considering the single, double, and triple diode models.

In particular, Lévy flight distribution (LFD) is a recent algorithm published in 2020 by [42], which has already proved its search efficiency through solving challenging optimization problems including the one presented in the Congress on Evolutionary Computation 2017 (CEC’17) and others related to engineering designs. Despite its promising results in literature, LFD is still prone to the drawbacks, like slow convergence, local optima entrapment, many hyper-parameters, imbalanced tradeoff between exploration and exploitation, that a common MA may face [3]. In this regard, the researchers in this field have overwhelmingly utilized the OBL approach which not only has proved its usability but also it can effectively deal with shortcomings discussed earlier [13].

Keeping in view the No-Free-Lunch theorem [43], it is obvious that there exists no single MA that is the solution for all problems. If a MA produces superior results on one type of problem, it may perform opposite on another type of problem. It is therefore, new and innovative solutions are still needed for cover most of the problems. LFD is also a new addition in this regard, which has already attracted researchers. But, this algorithm still suffers from imbalanced exploration-exploitation capability, and needs further improvement. In this work, we modify LFD by incorporating OBL for enhanced search efficiency. Particularly, we introduce OBL operator in the local search component of LFD, in order to help LFD avoid local optima entrapment. For the rigorous evaluation of the proposed approach, we employ CEC’20 test suite which maintains highly complex problem landscapes. Moreover, we also apply LFD-OBL on parameter estimation of solar cell based on the Three-Diode Photovoltaic model. The results of LFD-OBL are compared with the moth-flame optimization algorithm (MFO) [44], sine-cosine algorithm (SCA) [45], grasshopper optimisation algorithm (GOA) [46], artificial ecosystem-based optimization (AEO) [47], whale optimization algorithm (WOA) [48], Harris’ hawks optimization (HHO) [49], thermal exchange optimization (TEO) [50], Henry gas solubility optimization (HGSO) [1], and the original LFD [42]. Overall experimental results confirmed the superiority of the proposed LFD-OBL compared with other well-known MAs. Mainly, this study contributes to the literature in the following manner:

  • Integrating OBL in LFD for improved search performance.

  • Adapting LFD-OBL for estimating the parameters of TDM and MTDM of solar cell.

  • Evaluating the effectiveness of LFD-OBL on CEC’20 test functions.

  • Comparing LFD-OBL with a number of recent and efficient techniques to verify its effectiveness.

  • The experiments confirm that LFD-OBL has more precise results compared to other recent optimization techniques.

The remainder of the article is organized as follows: Sections 2 and 3 introduce the original LFD and proposed LFD-OBL and its components, i.e., the original LFD and the OBL features. Section 4 explains the Three-Diode model and Modified Three-Diode model problem. Section 5 discusses the results obtained by LFD-OBL and competitive algorithms on the CEC’20 test suite and the TDM and MTDM problems. Section 7 concludes the article.

2 Lévy flight distribution

Lévy flight distribution (LFD) algorithm was first introduced by [42], which solves optimization problems by performing Lévy flight random walk to explore solution space. It is important to mention that this algorithm also utilizes the mechanism of distribution of wireless sensors in a network environment or deployment area (search space). The algorithm measures Euclidean distance (ED) between every two sensors, in order to determine the next position of the sensor, if the sensor needs to placed in any other location or it is okay to remain as is. Another position update related measurement is performed by the Lévy flight model which places a sensor node (candidate solution) in an area where there exists no other sensor node or a neighborhood with few sensor nodes only. This mechanism ensures that the search space is visited effectively, in order to find optimal solution.

For a symmetric and stable Lévy distribution, LFD uses Mantegna algorithm which efficiently moves towards optimum location through appropriately determining step length and search direction. The Mantegna’s algorithm determines step length SL as (1):

$$ \begin{array}{@{}rcl@{}} &&SL = \frac{U}{{|V{|^{1/\beta}}}},\\ \text{ where } &&\beta= 0<\beta \leq 2, \\ &&\mathrm{U} \sim \mathrm{N}\left( 0, \sigma_{\mathrm{u}}^{2}\right), \quad \mathrm{V} \sim \mathrm{N}\left( 0, \sigma_{\mathrm{v}}^{2}\right), \\ &&{\sigma_{u}} = {\left\{ {\frac{{{{\varGamma}}(1 + \beta ) \times \sin (\pi \beta /2)}}{{{{\varGamma}}[(1 + \beta )/2]\times \beta \times {2^{(\beta - 1)/2}}}}} \right\}^{1/\beta }}, \\&& {\sigma_{v}} = 1, \text{ and }\\ &&{\Gamma} (z)=\int\limits_{0}^{\infty }{{{t}^{z-1}}{{e}^{-t}}dt}. \end{array} $$
(1)

where β denotes the Lévy distribution index, σu and σv are standard deviations, and Γ the Gamma function for an integer z.

For calculating ED between the two candidate solutions, or in other words, search agents, or in terms of LFD, sensor nodes sk and sl, LFD uses (2):

$$ ED\left( {{s_{k}},{s_{l}}} \right) = \sqrt {{{\left( {{x_{j}} - {x_{i}}} \right)}^{2}} + {{\left( {{y_{j}} - {y_{i}}} \right)}^{2}}} $$
(2)

where (xi,yi) and (xj,yj) respectively represent the position coordinates for sk and sl. The ED is then compared with a threshold value τ until specific number of iterations. In case of less distance than the selected threshold, LFD adjusts the positions of these solutions using (3):

$$ {s_{l}}(\mathrm{t}+1) = LF({s_{l}}(\mathrm{t}),{s_{best}},LB,UB) $$
(3)

where t denotes iteration index, LF performs random walk using Lévy flights, sbest the best sensor node (candidate solution) with lowest number of neighbours, UB and LB the upper and lower bounds of the solution space, respectively. Here, sl is moved towards the sensor with the lowest number of neighbours.

$$ {s_{j}^{t+1}} = LB + (UB - LB)\times rand( ) $$
(4)

where rand() is a random number with uniform distribution [0,1]. This way, LFD is able to explore unvisited areas in solution space.

$$ R=rand (), CSV=0.5 $$
(5)

LFD uses a comparative scalar value R = rand() and comparison threshold CSV = 0.5 for updating sj position. LFD compares the value of R, and in case of R < CSV, LFD executes (3); otherwise, (4). This way, LFD finds more opportunities to effectively discover the search space. Depending on the problem in hand, the value of CSV may vary. The new position for the search agent si is obtained as (6):

$$ \begin{array}{@{}rcl@{}} {s_{i}^{t+1}} && = TP + {\alpha_{1}}\times{TF_{Neighbours}} + {\text{ }}rand () \times \\ && {\alpha_{2}}\times (TP + {\text{ }}{\alpha_{3}}\times{s_{best}})/2 - {\text{ }}s_{i}^{t+1} TF_{Neighbours} \\ &&= \sum\limits_{k=1}^{N N} \frac{d_{k}\times s_{k}}{N N} \end{array} $$
(6)
$$ {s_{new}^{t+1}} = LF\left( {{s_{i}^{t+1},}TP,LB,UB} \right) $$
(7)

The new solution \(s_{i}^{t+1}\) is updated via (6), and it is finalized via (7). Here, TP is the best solution with best fitness value using the objective function, and it is referred to as the target solution. In (6), α1,α2 and α3 are random numbers such that α1 = 10,α2 = 0.00005, and α3 = 0.005. Equation (8) defines how to calculate total target fitness of neighbours TFNeighbours around \({s_{i}^{t}}\), where sk denotes the neighbouring position of \({s_{i}^{t}}\), k the index of the neighbouring solution, NN the number of neighbouring solutions of \({s_{i}^{t}}\), whereas the fitness degree of each neighbouring solution is given by dk.

$$ \begin{array}{@{}rcl@{}} d_{k} &&= \frac{{{\partial_{1}}(V - min(V))}}{{max(V) - min(V)}} + {\partial_{2}} \text{, where}\\ V &&= \frac{{fit({s_{j}}(\mathrm{t}))}}{{fit({s_{i}}(\mathrm{t}))}}, \text{ and} 0{\text{ }} < {\text{ }}{\partial_{1}}, {\text{ }}{\partial_{2}}{\text{ }} \le {\text{ }}1. \end{array} $$
(8)

Algorithm 1 presents the step by step pseudo code of the original LFD algorithm.

figure f

3 Architecture of the proposed LFD-OBL

The main idea for the LFD algorithm is the distribution of nearly agents where the search mechanism depends only the distances between search agents under specific threshold. The search strategy starts when the condition of distances is achieved with ignoring the other case when the agents are far away from each other and the condition of distances are failed. The neighbor agents step need more setting to avoid falling in spurious neighbor. In LFD, the main agents update their position with constant parameter that leads to stagnation at local optimal. To tackle with the aforementioned drawbacks, four enhancement steps were utilized in the proposed LFD-OBL algorithm as follows; 1) the exploration parameter is modified, 2) the parameter settings for neighbor agents is modified, 3) the exploitation phase is modified and 4) Opposition-based learning (OBL) is applied. Firstly, we proposes parameter setting for neighbor agents, modified exploration parameter, present new exploitation search strategy that will applied if the condition of distances not achieve. Then the Opposition-based learning concept is applied on LFD solutions to get opposite solutions, this enhances the distribution of agents in new the regions to escape efficiently from falling into local optimum. The proposed LFD-OBL method has been changed the overall hierarchy of the LFD’s search strategy. An overview for the modification steps of the proposed LFD-OBL is illustrated as follows.

  1. 1.

    Modified exploration parameter (α2): The main factors of any population-based optimization algorithm depend on exploration and exploitation. The exploration meaning the global part of the algorithm that explore different parts in the given search space with large step while in the exploitation; the step of movement is small to get the local optimum. The balance between them reflects the success of the algorithm. This achieve by many factors, one of them is the tuning parameters that transfer the agents’ movement from two phases gradually. In LFD, the agents update as (9):

    $$ {s_{i}^{t+1}}=\mathit{TP}+{{\alpha }_{1}}\times \mathit{T}{{F}_{\mathit{Neighbours}}}+\mathit{rand}\times {{\alpha }_{2}}\times \left( \frac {\mathit{TP}+{\alpha }_{3}\times{s_{l}}}{2-{s_{i}^{t}}}\right) $$
    (9)

    where, α2 parameter is constant which is illogical as the agent’s position must changes exponentially with time. This leads to fall the agents in specific area. In LFD-OBL, to solve this issue, the factor α2 will consider the exploration-exploitation parameter that decreases exponentially with time, given by:

    $$ {{\alpha}_{2}}=\exp \left( -\frac{t}{T}\right) $$
    (10)
  2. 2.

    Parameter settings for neighbor agents : In LFD, if the Euclidean distance between agents are less than threshold, the algorithm calculates fitness degree dk for each neighbor using:

    $$ d_{k}=\frac{{{\partial }_{1}}(V-{{min(V)}})}{{{max(V)}}-{{min(V)}}}+{{\partial }_{2}}, 0< {{\partial }_{1}}, {{\partial }_{2}}\le 1 $$
    (11)
    $$ V=\frac{fit({s_{j}^{t}})}{fit({s_{i}^{t}})} $$
    (12)

    Then calculate the total target fitness of neighbors TFNeighbours via (6).

    For the main agents, the positions are updated using (9), and for adjacent agents, the position are updated based on the comparative scalar value (CSV):

    $$ {s_{j}^{t+1}}= \begin{cases} LF({s_{j}^{t}},s_{leader},LB,UB) &\text{ if } CSV> rand() \\ LB+rand(UB-LB) &\text{ otherwise } \\ \end{cases} $$
    (13)

    where LF denotes the levy flight function, UB and LB the bounds of the search space, respectively. sleader is the random agent. In many cases, the fitness degree (dk) in (11) has zero value, this gives zero also in (12) that reflects on incorrect movement of the main agents in (9). In LFD-OBL, the neighbor condition is applied that ensures the right movement of main agents and adjacent agents by excluding the spurious neighbors as follows: where dk > 0, so \(s_{i}^{t+1}\) updated based on (9) otherwise \(s_{j}^{t+1}\) updated based on (13).

  3. 3.

    Exploitation phase: The LFD ignored the exploitation phase and the agents update their positions according to (9), which depends mainly on randomization, this leads to fail the balance between exploration-exploitation phases. In LFD-OBL, (14) is presented to transfer the agents to the exploitation phase. Then \(s_{i}^{t+1}\) updates its position using (7).

    $$ {s_{i}^{t+1}}={s_{i}^{t}}+F\times rand()\times (TP-{x_{i}^{t}}) \text{, where } F= \pm 1 $$
    (14)
  4. 4.

    Opposition-based learning (OBL): In order to find better optimal solutions, OBL provides opposite solutions obtained through optimization process using (15). For greater detail on OBL, the reader is encouraged to refer to [51].

    $$ OB{L_{i}^{t}}=U+L-{s_{i}^{t}}, i\in [1 ... n] $$
    (15)

The step by step schema of LFD-OBL is presented in Algorithm 2.

3.1 Complexity of the proposed LFD-OBL

Computing any metaheuristics algorithm computational complexity is a crucial task as it determines its run-time. In general, the complexity of any algorithm is depends on its structure. So, it will be depends on maximum number of iteration, dimensions of a specific task, and the number of individuals. The calculation of our proposed algorithm can be calculated as follows: O(LFD − OBL) = O (Updating individuals positions) + O(Updating individuals positions during opposition phase ) + O(Comparing and selecting destination ). O(LFD − OBL) = O(T × (D × N + N)) = O(TDN + TN) = O(TDN). Similarity the complexity of the original algorithm O(TDN + TN) = O(TDN) where D refers to the dimension number, N refers to number of individuals, and T refers to maximum iteration.

figure g
Fig. 1
figure 1

TDM mathematical model

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4 Mathematical model of solar cell

4.1 Three-diode model

In this article, The three-diode model (TDM) was applied to achieve more accuracy than the SDM and DDM. The TDM for solar cells is depicted in Fig. 1. In TDM, there is photo-generated current IPh averted with three diodes. It is linked with shunt resistance Rsh and series resistance RS. The current ID1 flows through the diode D1, the current ID2 flows through the diode D2, whereas the current ID3 passes through the diode D3. In TDM, total current I is calculated as (16):

$$ I=I_{Ph}-I_{D1}-I_{D2}-I_{D3}-I_{sh} $$
(16)

where Ish denotes the shunt resistor current. The TDM formation is denoted as (17):

$$ \begin{array}{@{}rcl@{}} I=\!\!\!\!\!{~}&& I_{Ph}-I_{sd1}\left[exp\left( \frac{q(V+IR_{S}}{N_{1} KT}\right)-1\right]\\ &&-I_{sd2}\left[exp\left( \frac{q(V+IR_{S}}{N_{2} KT}\right)-1\right]\\ &&-I_{sd3}\left[exp\left( \frac{q(V+IR_{S}}{N_{3} KT}\right)-1\right]-\frac{V+IR_{S}}{R_{sh}} \end{array} $$
(17)

where

  • Isd1, Isd2, and Isd3 denote the reverse saturation current of D1, D2, and D3, respectively.

  • q = 1.60217646 × [10](− 19) C denotes the electron charge.

  • V denotes the voltage.

  • K = 1.3806503 × [10](− 23) (J/k) denotes the Boltzmann’s constant.

  • T denotes the Kelvin temperature.

  • N1, N2, and N3 denote the ideality factor of D1, D2, and D3, respectively.

4.2 Modified three-diode model

The modified three-diode model (MTDM) for solar cells is illustrated via Fig. 2. Series resistance with diode 3 is The difference between the MTDM and TDM model which is employed to represent the losses in the space charge region. The total current generated from MTDM is given as (18), where the defect region losses are represented via series resistance Rm with the third diode D3.

$$ \begin{array}{@{}rcl@{}} I={~}\!\!\!\!\!\!\! &&I_{Ph}-I_{sd1}\left[exp\left( \frac{q(V+IR_{S})}{N_{1} KT}\right)-1\right]\\ &&-I_{sd2}\left[exp\left( \frac{q(V+IR_{S})}{N_{2} KT}\right)-1\right]\\ &&-I_{sd3}\left[exp\left( \frac{q(V+IR_{S})-I_{sd3}R_{m}}{N_{3} KT}\right)-1\right]\\ &&-\frac{V+IR_{S}}{R_{sh}} \end{array} $$
(18)
Fig. 2
figure 2

MTDM mathematical model

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4.3 The objective function

An objective function is optimized by finding the best values for optimization parameters. The objective function for TDM and MTDM is presented in [20], and defined as (19) and (20), respectively:

$$ \begin{array}{@{}rcl@{}} f_{TD}\left( V,I,x\right)=\!\!\!\! &&I-x_{3}+x_{4}\left[exp\left( \frac{q(V+Ix_{1})}{x_{6} KT}\right)-1\right]\\ &&-x_{5}\left[exp\left( \frac{q(V+Ix_{1})}{x_{7} KT}\right)-1\right]\\ &&-x_{8}\left[exp\left( \frac{q(V+Ix_{1})}{x_{9} KT}\right)-1\right]\\ &&-\frac{V+Ix_{1}}{x_{2}} \end{array} $$
(19)
$$ \begin{array}{@{}rcl@{}} f_{MTD}\left( V,I,x\right)=\!\!\!\! &&I-x_{3}+x_{4}\left[exp\left( \frac{q(V+Ix_{1})}{x_{6} KT}\right)-1\right]\\ &&-x_{5}\left[exp\left( \frac{q(V+Ix_{1})}{x_{7} KT}\right)-1\right]\\ &&-x_{8}\left[exp\left( \frac{q(V+Ix_{1}-x_{8}x_{10}}{x_{9} KT}\right)-1\right]\\ &&-\frac{V+Ix_{1}}{x_{2}} \end{array} $$
(20)

where

  • V and I denote the values taken from a solar cell.

  • x = [x1,x2,…xn] denote the parameters vector for each solar cell model and can be defined as:

    $$ - x = [R_{s}, R_{sh}, I_{ph}, I_{sd1}, I_{sd2}, I_{sd3}, n_{1}, n_{2}, n_{3}] $$

    for TDM.

    $$ - x = [R_{s}, R_{sh}, I_{ph}, I_{sd1}, I_{sd2}, I_{sd3}, n_{1}, n_{2}, n_{3},R_{m}] $$

    for MTDM.

The optimization parameters are to be found from a search space with certain lower and upper bounds which are presented in Table 1.

Table 1 The lower and upper bounds for photovoltaic parameters
Full size table

In this article, we use root mean square error (RMSE), as in (21), to minimize the error between the estimated and the measured data of the diode model.

$$ RMSE=\sqrt{\frac{1}{N} \sum\limits_{i=1}^{N}f_{h} (V,I,x)^{2}} $$
(21)

where h and N denote the solar cell model to be utilized and the sample size, respectively (Table 2).

Table 2 Parameter settings of LFD-OBL and the other counterparts
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5 Performance evaluation of LFD-OBL on CEC’20 test suite

In order to evaluate optimization efficacy of the proposed LFD-OBL, we initially employed CEC’20 test suite (Table 3), which was introduced in The IEEE Congress on Evolutionary Computation (CEC) [52]. Among these functions are, unimodal, multimodal, hybrid, and composition functions. To help understand the differences and the nature of these problems, Fig. 3 illustrates the problem landscapes.

Table 3 CEC’20 test suite
Full size table

5.1 Parameter settings

The parameter settings employed in this study are reported in Table 2, here we preferred default settings for each algorithm, as suggested by [53]. For fair comparison, each algorithm was assessed on each optimization problem for 30 times, with maximum number of iterations 1000.

5.2 CEC’2020 test suite description

This section presents the process of evaluating the proposed modified version of the Lévy Flight Distribution (LFD) called LFD-OBL. Consequently, The IEEE CEC’2020 [52] benchmarks were selected as the test problems to assess the proposed approach’s performance. The CEC’2020 test suite has 10 test functions that have uni-modal, multi-modal, hybrid, and composition functions. Table 3 shows the properties and mathematical formulas of CEC’2020 benchmark tests; where ‘Fi*’ represents the optimum global value.

Figure 3 shows the 2D visualization of the CEC’2020 test suite for ease understanding the nature of each one.

Fig. 3
figure 3

CEC’20 problems landscape

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5.3 Statistical analysis on CEC’2020 test suite

The statistical results obtained from the proposed LFD-OB are evaluated against the results reached from nine counterpart methods including moth-flame optimization algorithm (MFO) [44], grasshopper optimisation algorithm (GOA) [46], thermal exchange optimization (TEO) [50], sine-cosine algorithm (SCA) [45], whale optimization algorithm (WOA) [48], Harris hawks optimization (HHO) [49], artificial ecosystem-based optimization (AEO) [47], Henry gas solubility optimization (HGSO) [1], and the original LFD [42]. The algorithm performance is evaluated with computing the mean and the standard deviation (STD) metrics of the best-so-far solutions reached in each run.

Table 4 reports (mean and STD) the results of the algorithms employed on problems of CEC’20 with Dim = 20. The algorithm with best performance is highlighted in bold face. From the results, it is easy to interpret that LFD-OBL outperforms other methods on almost all the functions.

Table 4 Results of the CEC’20 test suite with D = 10
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5.4 Boxplot behavior analysis

Apart from statistical information, it is important to also consider boxplots for depicting data distributions into quartiles with a Dim = 10. Figure 4 shows that the boxplots of the proposed LFD-OBL algorithm are narrow compared to distributions in other algorithms, except on F2. Moreover, it is also critically important to analyze the performance of the proposed method in terms of convergence ability. It is observed that, the distribution of boxplots obtained by the introduced LFD-OBL algorithm are, for most test methods, narrower and achieving the minimum values compared to the other algorithms distribution. The graphical results of boxplots confirms the consistency of the proposed LFD-OBL algorithm to search the optimum regions for the considered test problems.

Fig. 4
figure 4

Boxplots of the proposed LFD-OBL and the competitor algorithms

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5.5 Convergence performance analysis

Figure 5 illustrates how the selected algorithms converged to final solution or an stable point. These convergence graphs suggest that the proposed algorithm converges to the best so-far solutions in quick time as compared to other algorithms. Here, it can be noticed that LFD-OBL proves to be an efficient optimization algorithm for solving with difficult problem landscapes. It is observed that the LFD-OBL algorithm get a stable point for most test functions. This explains that the proposed LFD-OBL converges properly towards and nearer the optimum solution. Additionally, the introduced LFD-OBL achieves the lowest mean of the best found solutions and a proper fast convergence on most test problems, compared to the other competitors. This reflects the stable performance of the LFD-OBL to converge near/optimal solution as a proper tool to optimize the recent complex problems.

Fig. 5
figure 5

Convergence ability of LFD-OBL and the competitor algorithms

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5.6 Wilcoxon rank test analysis

Wilcoxon’s rank-sum test is a non-parametric test that is performed statistically to illustrate the significance of an algorithm’s resulting data. Additionally, the wilcoxon test depicts that the algorithm behavior is consistent and not random. Although MAs have a stochastic nature, the probable performance should be accurate enough. For more details about Wilcoxon’s test, interested reader can refer to [54]. To be able to prove the superiority of the proposed method, a nonparametric test called Wilcoxon rank-sum is proposed by Derrac et al. [54] is used with 5% significant level. From Table 5, it’s noted that LFD-OBL win in all comparison between it and other compared algorithms except when compared with MFO. As it wins in only 7 functions out of 10.

Table 5 The calculated p-values from the Wilcoxon signed-rank test
Full size table

5.7 Advantages and limitations of LFD-OBL

Even though many simulation and experimental results demonstrate that metaheuristic algorithms (MAs) are efficient optimization tools, MAs may suffer from some limitations such as premature convergence, improper exploration-exploitation balance, and getting stuck in an optimum local region [55]. Therefore, to develop a more effective algorithm to solve TDM and MTDM Problems, we have studied recent algorithms and features. In particular, Lévy flight distribution (LFD) is a recent algorithm published in 2020 by [42], which has already proved its search efficiency through solving challenging optimization problems including the one presented in the Congress on Evolutionary Computation 2017 (CEC’17) and others related to engineering designs.

Although the original LFD has shown well competitive performance with other optimization algorithms, it still suffers from some drawbacks such as:

  • slow convergence,

  • local optima entrapment,

  • imbalanced tradeoff between exploration and exploitation.

In this regard, the researchers in this field have overwhelmingly utilized the OBL approach which not only has proved its usability but also it can effectively deal with shortcomings discussed earlier. In this paper, we modify LFD by incorporating OBL for enhanced search efficiency. Particularly, we introduce OBL operator in the local search component of LFD, in order to help LFD:

  • improve convergence speed of LFD,

  • enhance the exploration ability of LFD,

  • decrease the possibility of LFD falling into the local optima.

For the rigorous evaluation of the proposed approach, we employ CEC’20 test suite which maintains highly complex problem landscapes. Moreover, we also apply LFD-OBL on parameter estimation of solar cell based on the Three-Diode Photovoltaic model. Finally, the superiority and practicability of the presented LFD-OBL are comprehensively verified by comparing with several counterparts. The comparison results demonstrate that the proposed LFD-OBL is a powerful and attractive alternative for solving engineering optimization problems and the parameter estimation of solar cell.

Besides its benefits, the proposed LFD-OBL is also exposed to some limitations which are discussed below:

  • incorporating the OBL operator would significantly increase the computational cost.

  • Referring to the No Free Lunch (NFL) that supports the idea of no superior optimization algorithm can work well at all the optimization problems. So, the authors believe that the LFD-OBL algorithm, like the others MAs methods, obeys the same rule however, it outperforms many other recent and well-known algorithms.

6 Performance evaluation of LFD-OBL on TDM and MTDM problems

The numerical simulation of the proposed LFD-OBL algorithm for identifying parameters of TDM and MTDM is illustrated in this section. The seven recent techniques HHO [56], MFO [57], equilibrium optimizer (EO) [58], supply-demand-based optimization (SDO) [59], marine predators algorithm (MPA) [60], and tunicate swarm algorithm (TSA) [61], as well as the conventional LFD [42] are used for the comparison. All the mentioned techniques have been assessed 20 times, in order to avoid performance bias. We simulated the computing environment using MATLAB 2016a with an Intel core i5-4210U CPU, 1.70 GHz with 8GB of RAM.

6.1 Case 1: Three-diode model

Here, we investigate parameter estimation capability of the proposed algorithm for TDM of PV cells. In comparison with recent algorithms, the LFD-OBL achieves the least RMSE as provided in Table 6. Table 6 presents also the nine identified parameters of the TDM using the proposed LFD-OBL and other techniques. Figure 6 shows the convergence curve for this case study using LFD-OBL and other algorithms for the TDM. Obviously, LFD-OBL algorithm has a faster convergence rate than other techniques for the triple diode model. Table 7 reports the experimental results for all algorithms, and it is easy to suggest that LFD-OBL generated robust solutions among others. Also, Fig. 7 shows the graphical plot of the statistical results in the form of boxplots for the three diodes, illustrating the distribution of results achieved by various techniques in 20 individual runs. In addition, Table 8 reports the value of individual absolute error between the experimental measurements and the simulated data of the current, voltage, and power. The integral absolute error of the simulated current (IAEI) and the integral absolute error of the simulated power (IAEP) of the TDM using the proposed LFD-OBL are graphically displayed in Fig. 8. Figure 9 depicts The coincidence between the experimental measurements and simulated data points for the I-V and P-V curves.

Table 6 Calculated parameter in case of the TDM obtained by LFD-OBL and the counterparts
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Fig. 6
figure 6

Convergence curves of LFD-OBL and the counterparts on TDM problem

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Table 7 Statistical results of LFD-OBL and the counterparts for TDM problem
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Fig. 7
figure 7

Best RMSE boxplots of LFD-OBL and the counterparts for TDM problem

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Table 8 Measured and simulated data of voltages, currents, and power and the absolute errors values using LFD-OBL for TDM
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Fig. 8
figure 8

Individual absolute errors for current and power using LFD-OBL for TDM

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Fig. 9
figure 9

Comparisons between experimental data and simulated data obtained by LFD-OBL for TDM

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6.2 Case 2: Modified three-diode model

This case introduces the usage of the proposed LFD-OBL in the estimation of the MTDM parameters. The best-obtained parameters of the MTDM using LFD-OBL and other comparative techniques are tabulated in Table 9. The LFD-OBL reaches the lowest RMSE in comparison with the other compared algorithms as provided in Table 9 and Fig. 10. Also, the proposed LFD-OBL achieved the RMSE value in this case (0.0008814) lower than its value in the previous case (0.00098255). These obtained results prove that the MTDM gives more precision than the traditional TDM. Additionally, the statistical results in Table 10 confirm the superiority of the proposed technique over the other comparative algorithms. Figure 11 shows the box plot of the RMSE values for MTDM in 20 individual runs using the LFD-OBL and other recent algorithms. Table 11 tabulates the IAE of the measured and estimated current and power for the MTDM of PV. Furthermore, the IAEI and IAEP of the MTDM using the proposed LFD-OBL are shown in Fig. 12. Besides, the matching between the experimental and simulated I-V and P-V curves is presented in Fig. 13. The above-mentioned comparisons illustrate that the LFD-OBL algorithm has better searching accuracy, reliability, and a more speed convergence rate for defining the parameters extraction problems of TDM and MTDM, and its performance is superior or competitive in contrast with all other well-known algorithms.

Table 9 Calculated parameter in case of the TDM obtained by the proposed algorithm and other recent techniques
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Fig. 10
figure 10

Convergence curve of the LFD-OBL and other recent algorithms for the MTDM

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Table 10 Statistical results the proposed LFD-OBL algorithm and other recent algorithms in the case of the MTDM
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Fig. 11
figure 11

Best RMSE boxplot in 20 runs of the LFD-OBL and other recent algorithms for the MTDM

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Table 11 Measured and simulated data of voltages, currents, and power and the absolute errors values using LFD-OBL for MTDM
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Fig. 12
figure 12

Individual absolute errors for current and power using LFD-OBL for MTDM

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Fig. 13
figure 13

Comparisons between experimental data and simulated data obtained by LFD-OBL for MTDM

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7 Conclusion and future work

In this paper, we improved the Lévy flight distribution (LFD) algorithm using the opposition-based learning (OBL) approach, hence called LFD-OBL, in order to effectively solve global optimization problems, especially the parameters estimation of the modified three-diode photovoltaic model. The proposed approach of incorporating OBL operators in LFD-OBL enhances the exploitation ability of the algorithm, in order to avoid local optima and improve the local and global search. We assessed the performance of LFD-OBL by employing ten difficult optimization problems exist in CEC’20 test suite, and it was found that LFD-OBL achieved better or similar results than MFO, GOA, SCA, WOA, TEO, HGSO, HHO, AEO, and the original LFD. On another problem set, the proposed method estimated the unknown parameters of the TDM and MTDM for the solar cells, much better than well-known algorithms such as HHO, MFO, EO, SDO, MPA, as well as the original LFD algorithm. The results of the comparison showed the superiority and competitiveness of our algorithms for the TDM and MTDM.

In future, we intend to apply LFD-OBL on solving more challenging engineering optimization problems including multi-objective problems and feature selection, image segmentation, and optimize the parameters of machine learning classifiers.