BECAUSE of the rapid increase in air pollution caused by generating the power using fossil fuels, and greenhouse gas emissions, industrialized countries and governments need to rely on clean sources such as renewable energy sources. Presently, one of the most popular and promising renewable energy sources is solar energy (SE). SE is converted to electrical energy using photovoltaic (PV) systems [1]. PV components are easily deteriorated because they operate in harsh outdoor environments, significantly affecting the efficiency of the use of SE. As a result, the process of designing accurate models of PV systems becomes an important and challenging task due to the behavior evaluation of PV cells. Different mathematical models have been introduced to describe the nonlinear performance of PV systems [2], [3]. The most popular models are the single-diode (SD) and double-diode (DD) models [4], [5]. The accuracy of any model is mainly affected by adjusting the model parameters. However, these parameters have two problems: ① they are usually unavailable and change due to faults, volatile operation conditions, and aging; ② they have a certain degree of uncertainty due to the measurement process. Hence, the creation of an accurate scheme for parameter identification is a crucial and challenging task for the effective evaluation, simulation, and control of PV systems.
Some approaches based on deterministic techniques have been proposed for PV parameter identification based on deterministic techniques. Such approaches were developed based on the Newton approach [6], Lambert formulation [7], and so on [8]. However, some restrictions such as differentiability and convexity that lead to trapping in the local optima are present in the deterministic techniques.
The advent of metaheuristic approaches has provided promising alternative approaches for solving complex optimization tasks [9]-[12] and overcoming the limitations of the deterministic ones as well. Several approaches such as differential evolution (DE) have been introduced for the parameter identification problem [13]. In [14], artificial bee colony (ABC) was developed for identifying solar cell parameters, and in [15], a bacterial foraging approach (BFA) was used for parameter estimation of solar cells and other systems [16]-[18]. Salp swarm algorithm (SSA) is a new optimization approach that was proposed in [19] for solving optimization tasks. This algorithm is based on the behavior of salps, which form a chain through some leaders. Furthermore, through a rapid, coordinated change strategy, this behavior can achieve better convergence. However, as a new approach, SSA has some disadvantages: ① they are only guided by leaders, which leads to unsatisfactory outcomes; ② there is no strategy for the diversity and improving the best location with each generation, which may lead to trapping in the local optima. Besides, to the best of our knowledge, no attempts in the SSA literature have been employed to solve the parameter estimation of PV models.
Identifying the parameters of PV modules is significant for their design and simulation. Because of instabilities in the weather action and land surface of the earth, which cause errors in measuring, a novel fuzzy representation-based PV module is formulated and developed. In this sense, the PV modules are proposed in two aspects: the first aspect addresses crisp PV modules, while the second one concentrates on fuzzy modeling of PV modules in which the input data of terminal (output) voltage and solar cell terminal current are the fuzzy numbers. Through each fuzzy PV module, the terminal (output) voltage as well as the solar cell terminal current is characterized with the membership function. To effectively handle the crisp and fuzzy aspects of the PV modules, this paper hybridizes the SSA with the particle swarm optimization (PSO), in a scheme called the locomotion-based hybrid salp swarm algorithm (LHSSA), to effectively achieve robust identification of the parameters of the PV modules. In this approach, the SSA operates as an explorer tool for the solution vector while PSO is integrated to modify the locations of the leader salps. The performance of the proposed LHSSA is investigated and evaluated through different PV models. Comprehensive results affirm that the LHSSA exhibits superior performance compared with different algorithms, especially on the quality and reliability.
The main contributions of this paper are as follows.
1) An LHSSA-based opposition learning scheme is introduced to refine the quality of leaders.
2) A PSO algorithm-based leading method is incorporated to enhance exploitation capability and avoid trapping in the local optima.
3) Fuzzy representation-based PV models are introduced as a novel sight, and different level schemes are conducted.
4) The efficiency of the LHSSA is investigated by comprehensive experiments and the comparison on different PV models.
The remainder of this paper is organized as follows. Section II describes the formulation of the PV models. Section III provides some basics regarding the methodology. The proposed LHSSA is developed in Section IV. The results and the comparison of the different PV models are provided in Section V. Finally, the conclusions and potential for future study are exhibited in Section VI.
In practice, there are different PV models that describe the characteristics of the current-voltage performance of solar cells and PV modules. The formulation of these models associated with their objective functions is described in this section.
The SD model uses one diode that is parallel to a current source. The structure of this model includes a current source that is parallel to the circuit diode, one series resistor to represent the losses of load current, and a shunt resistor to represent the leakage current. The current of this cell is calculated using:
where is the terminal current of solar cell; is the photo-generated current; is the diode current; is the shunt branch current; is the reverse saturation current; is the terminal (output) voltage; and are the shunt and series resistances, respectively; is the diode ideal factor; is the charge of an electron ; is the Boltzmann constant (); and T is the temperature of the cell. According to (2) and (3), the terminal (output) current in (1) can be rewritten as:
Although the SD contains five unknown parameters , [1] and [2] have shown that this optimization task has high multi-modal and noisy characteristics, thus this fact requires robust search strategies.
The DD model considers two parallel diodes with a shunt resistance and current source. The terminal (output) current can be described as follows:
where and are the first-diode and second-diode currents, respectively; and are the diffusion and saturation currents, respectively; and and are the ideal factors of diffusion and recombination diodes, respectively. In this case, seven unknown parameters need to be estimated to obtain the accurate performance of the solar cell.
The PV module model is structured from several solar cells that are linked in series and/or in parallel. The output (terminal) current can be considered as follows:
where and are the numbers of solar cells in series and parallel, respectively. This model contains five unknown parameters that need to be identified.
B. Objective Function for PV Modules: Crisp Aspect
The main aim of the PV models is to minimize the difference between the experimental and estimated data. In this regard, the error function is represented and defined by (7) and (8) for the SD and DD, respectively.
where is the solution set of the unknown parameters.
To objectively evaluate the performance of the proposed methodology, the objective function of the PV modules is formulated by quantifying the root mean square error (RMSE) as:
where is the number of experimental data; and the subscripts min and max represent the minimum and maximum values of related variables, respectively.
The main aim of the optimization process is to minimize the fitness function with respect to the bounds of the parameters. A smaller RMSE output implies a smaller deviation between the experimental and simulated data by the proposed algorithm.
C. Objective Function for PV Modules: Fuzzy Aspect
The electrical energy generated by PV modules involves many controlled parameters such as terminal (output) voltage and solar cell terminal current whose possible values are uncertain and ambiguous due to weather fluctuations or instabilities in the measuring process. Thus, the fuzzy model of the PV modules can be formulated as:
where is the fuzzy objective form of RMSE; is the fuzzy terminal (output) voltage; and is the fuzzy solar cell terminal current. is the variable of the fuzzy parameters that are represented as fuzzy number. Each fuzzy number component of is associated by its own degree of membership function [20].
Definition 1 (fuzzy number): the fuzzy number is a fuzzy subset of the real line , which is associated with the membership function that has the following features:
1) is continuous.
2) .
3) Strictly increasing on .
4) for .
5) Strictly decreasing on .
Therefore, each measured value of the terminal voltage and solar cell terminal current can be represented by a certain “grade of membership” that ranges from grade 0 to 1, where 0 and 1 indicate the lowest and highest grades of membership, respectively. Thus, each fuzzy parameter is expressed by a given interval of real numbers, associated with grades of membership between 0 and 1.
Definition 2 (-level set): the -level set (-cut) of the fuzzy numbers is represented by the ordinary set that contains the values of when the degree of membership functions exceeds a certain level , .
The fuzzy representation shown in Fig. 1 has been extracted according to expert suggestions or from observing weather fluctuations or instabilities in the measuring process. Based on the -cut concept of the fuzzy numbers, the problem with the fuzzy parameters aspect can be converted into a crisp aspect. The membership function which defines the fuzzy parameter is formulated as:
Fig. 1 Fuzzy numbers for PV modules.
According to the membership function and the concept of the α-cut level, the fuzzy parameter is transformed to a crisp scheme through two end points which are induced by the α-cut level, and they represent the upper and lower bounds for the crisp parameter, as shown in Fig. 1, where and are the lower and upper bounds induced by -cut for parameter , respectively. Thus, the optimization task of the PV modules can be converted to a non-fuzzy (crisp) form as:
where superscripts L and U represent the lower and upper bounds of related variables, respectively.
Definition 3 (-global minimum): is defined as the -global minimum to the problem (14) if and only if , where is the feasible region of the search space and is the -level optimal parameter.
The SSA is a meta-heuristic optimization algorithm presented in [19]. It mimics the natural behavior of salps living in the deep parts of oceans. Salps are a type of Salpidae and look like jelly fish with a transparent body. Salps have an intelligent behavior during their navigation and foraging as a salp chain. In this context, the SSA is a recent type of swarm algorithm developed to model the salp chain [19]. In the salp chain, the salp population is split into two categories: the leader salp which is the first salp in the chain, and the follower salps that follow the leader salp while reaching for the food source. The position set of each salp in the search space is ,, where is the number of salps. Thus, the leader salp updates its position as:
where and are the upper and lower bounds in the jth dimension, respectively; is the position of the target source in the jth dimension; and , , and are the algorithm parameters. The first parameter , which is responsible for balancing the exploration and the exploitation mechanisms, is the most important parameter, and is defined as:
where and are the current iteration and maximum iteration, respectively. and are random numbers between 0 and 1 whose values are uniformly generated. Furthermore, each follower salp updates its position based on Newton’s law of motion using the following equation.
where is the position of the ith follower salp in the jth dimension.
PSO is a population-based algorithm introduced in [21]. It is inspired by the cooperative behavior of some birds, fishes, and insects. In PSO, each particle improves its position by considering its best personal and global positions. The position of each particle is updated as:
where and are the current position of the ith particle and its velocity at iteration , respectively; is the best position of the ith particle; is the best solution among all particles; and are the cognitive and social parameters, respectively; is the inertia weight; and and are the random numbers between 0 and 1.
III. Proposed LHSSA Algorithm
The motivation behind developing the LHSSA is to achieve two features: ① improving the locomotion of the leader salps by memorizing the track of the leader; ② enhancing the seeking process using the hybridization-based PSO.
A. Locomotion-based Internal Memory
An internal memory is incorporated for the leader salps to keep track of their positions. During each iteration, the best agents between the salps and particles are saved and denoted by personal leaders and the global leader is denoted by .
B. Hybridization-based Iteration
Hybridization-based iteration is a straightforward approach for executing two algorithms iteratively through a certain sequence to enhance optimization performance [22]. Here, SSA works as an explorer using (14), while PSO is responsible for exploiting the previous leader salps to obtain more refined leaders. Thus, the modified position and velocity equations of PSO are defined as:
where is the personal best for the ith particle of PSO; and is the global best of the swarm in the iteration of the PSO.
C. Experience-based Opposition Learning Scheme
The opposition learning scheme based on the behavior of follower salps is developed to improve swarm diversity with the aim of increasing exploration ability. To be specific, for each follower salp, the opposition solution is defined as in (22).
where is the jth element of the ith opposition solution set on the tth iteration; and and are the dynamic bounds of the jth variable defined as:
and are updated every 50 generations and if the obtained solution does not lie within the bounds, a random solution is generated. The flow chart of the proposed scheme is given in Fig. 2.
Fig. 2 Flow chart of proposed LHSSA.
IV. Experimental Results and Analysis
The performance of LHSSA is investigated to identify the parameters of different PV models such as the SD, DD, and PV module. Thus, the benchmark data set for the solar cells and solar module is utilized [23]. To guarantee accurate comparison, the bounds for the parameters are given in Table I [23]. In this context, the superior performance of the proposed LHSSA is validated by comparing with other well-established algorithms [24] such as PSO, fire fly algorithm (FFA), grey wolf algorithm (GWO), dragonfly algorithm (DA), and standard SSA. For fair comparisons, the same maximum number of iterations in each run with 20 independent runs for every problem is adopted in the comparative algorithms. Further, the parameter configurations of the compared algorithms are listed in Table II and are suggested as in [24].
Table I Parameters Range for SD, DD, and PV Module
Model | Bound | | | | | |
SD |
Lower |
0 |
0 |
0.0 |
0 |
1 |
Upper |
1 |
1 |
0.5 |
100 |
2 |
DD |
Lower |
0 |
0 |
0.0 |
0 |
1 |
Upper |
1 |
1 |
0.5 |
100 |
2 |
PV module |
Lower |
0 |
0 |
0.0 |
0 |
1 |
Upper |
2 |
50 |
2.0 |
2000 |
50 |
Table II Parameter Configurations for Comparative Algorithms
Parameter | Algorithm |
Population size , acceleration coefficients , inertia weight : 0.2-0.9 |
PSO |
, initial attractiveness , randomization parameter , absorption coefficient |
FFA |
, , represents a uniform distribution |
GWO |
, w: 0.2-0.9, separation , alignment , cohesion , food , enemy |
DA |
, , |
SSA |
, acceleration coefficients: , inertia weight w: 0.2-0.9, , |
LHSSA |
In this section, the comparison results of the SD model including the extracted parameters and RMSE are shown in Table III, where the best RMSE value among all comparative algorithms is highlighted in boldface.
Table III Comparisons with Various Algorithms for SD Model
Algorithm | | | | | | RMSE (mA) |
Proposed LHSSA |
0.03640 |
53.7185 |
0.7607 |
0.32300 |
1.48170 |
0.98602 |
PSO |
2.22040×1016 |
1.1489 |
0.8368 |
2.22000×108 |
1.00000 |
19.58100 |
FFA |
0.01540 |
47.0033 |
0.5697 |
3.09000 |
1.89920 |
141.63000 |
GWO |
0.02970 |
32.6211 |
0.7652 |
1.10700 |
1.61860 |
222.86000 |
DA |
2.22040×1016 |
1.1489 |
0.8368 |
2.22000×108 |
1.00000 |
222.86000 |
SSA |
4.73800×104 |
5.6237 |
0.7443 |
11.02000 |
1.98690 |
39.90600 |
IJAYA |
0.03640 |
53.7595 |
0.7608 |
0.32280 |
1.48110 |
0.98603 |
JAYA |
0.03640 |
54.9298 |
0.7608 |
0.32810 |
1.48280 |
0.98946 |
GOTLBO |
0.03630 |
53.3664 |
0.7608 |
0.32970 |
1.48330 |
0.98856 |
LETLBO |
0.03630 |
53.7429 |
0.7608 |
0.32597 |
1.48210 |
0.98738 |
LBSA |
0.03640 |
54.1083 |
0.7609 |
0.32583 |
1.48200 |
0.99125 |
CLPSO |
0.03610 |
54.1965 |
0.7608 |
0.34302 |
1.48730 |
0.99633 |
BLPSO |
0.03590 |
60.2845 |
0.7607 |
0.36620 |
1.49390 |
1.02720 |
DE/BBO |
0.03640 |
55.2627 |
0.7605 |
0.32477 |
1.48170 |
0.99922 |
CMM-DE/BBO |
0.03640 |
53.8753 |
0.7608 |
0.32384 |
1.48140 |
0.98605 |
IADE |
0.03621 |
54.7643 |
0.7607 |
0.33613 |
1.48520 |
0.98900 |
IGHS |
0.03610 |
53.2845 |
0.7608 |
0.34350 |
1.48740 |
0.99306 |
ABSO |
0.03659 |
52.2903 |
0.7608 |
0.30623 |
1.47878 |
0.99124 |
The results of all the comparative algorithms and the results taken from [23] are listed in Table III, where the meaning of the abbreviations can be found in [23]. Regarding Table III, it can be observed that the proposed LHSSA gives the lowest RMSE value (0.98602 mA) compared with the other comparative algorithms. Additionally, to further emphasize the quality of the obtained results, the best extracted parameters of the LHSSA are employed to replot the I-V and P-V characteristic curves as depicted in Fig. 3. The depicted curves show that the calculated data acquired by the LHSSA highly coincides with the measured one based on the voltage range. Additionally, the individual absolute error (IAE) is introduced as a quality index to determine the absolute difference between the experiment data and the simulated one as in Table IV. The obtained values of the IAE are less than 1.62×10-3 and the total sum is 2.58×10-5, which affirms the accuracy of the parameters estimated by the LHSSA.
Fig. 3 Comparison between experimental and simulated data obtained by LHSSA for SD model. (a) I-V characteristics. (b) P-V characteristics.
Table IV IAE of LHSSA for Each Measurement on SD Model
Item | Measured voltage (V) | Measured current (A) | Calculated current (A) | IAE |
1 |
-0.2057 |
0.7640 |
0.7640 |
-0.0000872 |
2 |
-0.1291 |
0.7620 |
0.7626 |
-0.0006600 |
3 |
-0.0588 |
0.7605 |
0.7613 |
-0.0008500 |
4 |
0.0057 |
0.7605 |
0.7601 |
0.0003500 |
5 |
0.0646 |
0.7600 |
0.7590 |
0.0009500 |
6 |
0.1185 |
0.7590 |
0.7580 |
0.0009600 |
7 |
0.1678 |
0.7570 |
0.7571 |
-0.0000911 |
8 |
0.2132 |
0.7570 |
0.7561 |
0.0008600 |
9 |
0.2545 |
0.7555 |
0.7551 |
0.0004100 |
10 |
0.2924 |
0.7540 |
0.7536 |
0.0003400 |
11 |
0.3269 |
0.7505 |
0.7513 |
-0.0008900 |
12 |
0.3585 |
0.7465 |
0.7473 |
-0.0008500 |
13 |
0.3873 |
0.7385 |
0.7401 |
-0.0016200 |
14 |
0.4137 |
0.7280 |
0.7273 |
0.0006200 |
15 |
0.4373 |
0.7065 |
0.7069 |
-0.0004700 |
16 |
0.4590 |
0.6755 |
0.6752 |
0.0002200 |
17 |
0.4784 |
0.6320 |
0.6307 |
0.0012400 |
18 |
0.4960 |
0.5730 |
0.5719 |
0.0010700 |
19 |
0.5119 |
0.4990 |
0.4996 |
-0.0006100 |
20 |
0.5265 |
0.4130 |
0.4136 |
-0.0006500 |
21 |
0.5398 |
0.3165 |
0.3175 |
-0.0010100 |
22 |
0.5521 |
0.2120 |
0.2121 |
-0.0001500 |
23 |
0.5633 |
0.1035 |
0.1022 |
0.0012500 |
24 |
0.5736 |
-0.0100 |
-0.0087 |
-0.0012800 |
25 |
0.5833 |
-0.1230 |
-0.1255 |
0.0025100 |
26 |
0.5900 |
-0.2100 |
-0.2084 |
-0.0015200 |
The parameters of the DD model associated with the RMSE of the different methods are recorded in Table V. The results of the compared algorithms are also presented in Table V. It is obvious that the proposed LHSSA outperforms the comparative algorithms because it provides the best RMSE value (0.98249 mA). The characteristic curves for I-V and P-V of the measured data and the one estimated by the LHSSA are depicted in Fig. 4, whereas the IAE values are given in Table VI. Figure 4 shows that the data estimated by the LHSSA are in good congruence with the measured data. From Table VI, the sum of errors is 3.84×10-6 and all the IAE values are smaller than 1.239×10-3, indicating the high accuracy of the identified parameters.
Table V Comparisons with Various Algorithms for DD Model
Algorithm | | | | | | | | RMSE (mA) |
Proposed LHSSA |
0.036740 |
55.4824 |
0.76080 |
0.7473 |
0.2259 |
2.0000 |
1.4515 |
0.98249 |
PSO |
2.220400×1016 |
1.1487 |
0.83680 |
2.2204×1010 |
2.2204×1010 |
1.0000 |
1.0000 |
222.86000 |
FFA |
2.220400×1016 |
1.1904 |
0.86410 |
2.2204×1010 |
2.2204×1010 |
1.8384 |
1.3378 |
226.05000 |
GWO |
0.048295 |
22.6562 |
0.76170 |
8.3152×103 |
2.4189×1010 |
1.1862 |
1.5192 |
6.61280 |
DA |
2.220400×1016 |
1.1489 |
0.83680 |
2.2204×1010 |
2.2204×1010 |
1.0000 |
1.1676 |
222.86000 |
SSA |
5.613300×104 |
1.9814 |
0.83910 |
2.2204×1010 |
3.9483 |
1.9963 |
1.8427 |
82.60300 |
IJAYA |
0.037600 |
77.8519 |
0.76010 |
0.0050 |
0.7509 |
1.2186 |
1.6247 |
0.98293 |
JAYA |
0.036400 |
52.6575 |
0.76070 |
0.0061 |
0.3151 |
1.8436 |
1.4788 |
0.98934 |
GOTLBO |
0.036500 |
53.4058 |
0.76080 |
0.1389 |
0.2621 |
1.7254 |
1.4658 |
0.98742 |
LETLBO |
0.036500 |
54.3021 |
0.76080 |
0.1739 |
0.2266 |
1.6585 |
1.4578 |
0.98565 |
LBSA |
0.036500 |
56.0524 |
0.76070 |
0.2487 |
0.2744 |
1.8817 |
1.4682 |
0.98751 |
CLPSO |
0.036700 |
57.9422 |
0.76070 |
0.2584 |
0.3862 |
1.4625 |
1.9435 |
0.99894 |
BLPSO |
0.036600 |
61.1345 |
0.76080 |
0.2719 |
0.4351 |
1.4674 |
1.9662 |
1.06280 |
DE/BBO |
0.038500 |
58.4018 |
0.76060 |
0.0012 |
0.3722 |
1.8791 |
1.4956 |
1.02550 |
CMM-DE/BBO |
0.036000 |
57.9882 |
0.76070 |
0.3537 |
0.0256 |
1.4907 |
1.8835 |
1.00880 |
IGHS |
0.036900 |
53.8368 |
0.76080 |
0.9731 |
0.1679 |
1.9213 |
1.4281 |
0.98635 |
ABSO |
0.036600 |
54.6219 |
0.76077 |
0.2671 |
0.3819 |
1.4651 |
1.9815 |
0.98344 |
Fig. 4 Comparison between experimental and simulated data obtained by LHSSA for DD model. (a) I-V characteristics. (b) P-V characteristics.
Table VI IAE of LHSSA for Each Measurement on DD Model
Item | Measured voltage (V) | Measured current (A) | Calculated current (A) | IAE |
1 |
-0.2057 |
0.7640 |
0.7639 |
0.0000165 |
2 |
-0.1291 |
0.7620 |
0.7626 |
-0.0006200 |
3 |
-0.0588 |
0.7605 |
0.7613 |
-0.0008400 |
4 |
0.0057 |
0.7605 |
0.7602 |
0.0003300 |
5 |
0.0646 |
0.7600 |
0.7591 |
0.0008900 |
6 |
0.1185 |
0.7590 |
0.7581 |
0.0008800 |
7 |
0.1678 |
0.7570 |
0.7572 |
-0.0001900 |
8 |
0.2132 |
0.7570 |
0.7562 |
0.0007600 |
9 |
0.2545 |
0.7555 |
0.7552 |
0.0003200 |
10 |
0.2924 |
0.7540 |
0.7537 |
0.0002800 |
11 |
0.3269 |
0.7505 |
0.7514 |
-0.0009100 |
12 |
0.3585 |
0.7465 |
0.7473 |
-0.0008000 |
13 |
0.3873 |
0.7385 |
0.7400 |
-0.0015100 |
14 |
0.4137 |
0.7280 |
0.7272 |
0.0007500 |
15 |
0.4373 |
0.7065 |
0.7069 |
-0.0003500 |
16 |
0.4590 |
0.6755 |
0.6752 |
0.0002900 |
17 |
0.4784 |
0.6320 |
0.6308 |
0.0012400 |
18 |
0.4960 |
0.5730 |
0.5720 |
0.0010100 |
19 |
0.5119 |
0.4990 |
0.4998 |
-0.0007100 |
20 |
0.5265 |
0.4130 |
0.4137 |
-0.0007300 |
21 |
0.5398 |
0.3165 |
0.3175 |
-0.0010500 |
22 |
0.5521 |
0.2120 |
0.2121 |
-0.0001200 |
23 |
0.5633 |
0.1035 |
0.1022 |
0.0013400 |
24 |
0.5736 |
-0.0100 |
-0.0088 |
-0.0012100 |
25 |
0.5833 |
-0.1230 |
-0.1255 |
0.0025400 |
26 |
0.5900 |
-0.2100 |
-0.2084 |
-0.0016300 |
C. Results of PV Module Model
In this model, five parameters are estimated and the RMSE values are obtained and reported in Table VII.
Table VII Results Among Comparative Techniques on PV Module
Algorithm | () | () | (A) | | | RMSE (mA) |
Proposed LHSSA |
0.0334 |
27.2773 |
1.0305 |
3.4822 |
1.3517 |
2.4250 |
PSO |
0 |
2000.0000 |
1.1741 |
3046.3000 |
2.8761 |
77.5850 |
FFA |
0 |
586.7000 |
1.4650 |
226590.0000 |
9.1139 |
220.4100 |
GWO |
0.0005 |
1064.5000 |
1.0467 |
680.5200 |
2.3241 |
23.5320 |
DA |
0 |
2000.0000 |
1.0484 |
765.8700 |
2.3616 |
23.8810 |
SSA |
0 |
1931.5000 |
1.3032 |
54173.0000 |
5.3692 |
144.9300 |
IJAYA |
1.2016 |
977.3700 |
1.0305 |
3.4703 |
48.6298 |
2.4251 |
JAYA |
1.2014 |
1022.5000 |
1.0302 |
3.4931 |
48.6531 |
2.4278 |
GOTLBO |
1.1995 |
969.9300 |
1.0307 |
3.5124 |
48.6766 |
2.4266 |
LETLBO |
1.2015 |
974.6100 |
1.0306 |
3.4705 |
48.6301 |
2.4251 |
LBSA |
1.2010 |
987.7800 |
1.0305 |
3.4901 |
48.6513 |
2.4252 |
CLPSO |
1.1978 |
1017.0000 |
1.0304 |
3.6131 |
48.7847 |
2.4281 |
BLPSO |
1.2002 |
992.7900 |
1.0305 |
3.5176 |
48.6815 |
2.4252 |
DE/BBO |
1.1969 |
1015.1000 |
1.0303 |
3.6172 |
48.7894 |
2.4283 |
CMM-DE |
1.2013 |
981.9800 |
1.0305 |
3.4823 |
48.6428 |
2.4251 |
PS |
1.2053 |
714.2800 |
1.0313 |
3.1756 |
48.2889 |
11.8000 |
SA |
1.1989 |
833.3300 |
1.0331 |
3.6642 |
48.8211 |
2.7000 |
The results of the proposed LHSSA are compared with different algorithms, some of which are taken from literature for comparison [23]. The overall IAE values are given in Table VIII.
Table VIII IAE of LHSSA for Each Measurement on PV Module
Item | Measured voltage (V) | Measured current (A) | Calculated current (A) | IAE |
1 |
0.1248 |
1.0315 |
1.02912 |
0.0023800 |
2 |
1.8093 |
1.0300 |
1.02738 |
0.0026200 |
3 |
3.3511 |
1.0260 |
1.02574 |
0.0002600 |
4 |
4.7622 |
1.0220 |
1.02411 |
-0.0021100 |
5 |
6.0538 |
1.0180 |
1.02229 |
-0.0042900 |
6 |
7.2364 |
1.0155 |
1.01993 |
-0.0044300 |
7 |
8.3189 |
1.0140 |
1.01636 |
-0.0023600 |
8 |
9.3097 |
1.0100 |
1.01049 |
-0.0005000 |
9 |
10.2163 |
1.0035 |
1.00063 |
0.0028700 |
10 |
11.0449 |
0.9880 |
0.98455 |
0.0034500 |
11 |
11.8018 |
0.9630 |
0.95952 |
0.0034800 |
12 |
12.4929 |
0.9255 |
0.92284 |
0.0026600 |
13 |
13.1231 |
0.8725 |
0.87260 |
-0.0001000 |
14 |
13.6983 |
0.8075 |
0.80728 |
0.0002200 |
15 |
14.2221 |
0.7265 |
0.72834 |
-0.0018400 |
16 |
14.6995 |
0.6345 |
0.63714 |
-0.0026400 |
17 |
15.1346 |
0.5345 |
0.53622 |
-0.0017100 |
18 |
15.5311 |
0.4275 |
0.42951 |
-0.0020100 |
19 |
15.8929 |
0.3185 |
0.31877 |
-0.0002700 |
20 |
16.2229 |
0.2085 |
0.20739 |
0.0011100 |
21 |
16.5241 |
0.1010 |
0.09616 |
0.0048400 |
22 |
16.7987 |
-0.0080 |
-0.00833 |
0.0003300 |
23 |
17.0499 |
-0.1110 |
-0.11095 |
-0.0000538 |
24 |
17.2793 |
-0.2090 |
-0.20926 |
0.0002600 |
25 |
17.4885 |
-0.3030 |
-0.30088 |
-0.0021200 |
It is clear that the proposed LHSSA outperforms the other algorithms as it gives the best RMSE value (2.42507×10-3) among all the compared algorithms. The comparison shows that the LHSSA performs well. Due to space limitations, the I-V and P-V characteristics are not depicted. The obtained IAE values are all smaller than 4.837×10-3 and the total sum of error is 4.3172×10-5. The parameters with high accuracy are achieved again by the LHSSA.
D. Statistical Measures and Convergence Behavior
In this subsection, statistical measures are calculated for the proposed and comparative algorithms over 20 independent runs and are recorded in Table IX. These measures include the mean RMSE that quantifies the average accuracy and also confirms the stability of the algorithm runs, and St.dev that represents the standard deviation of the RMSE values that defines the reliability of the parameter estimation. For each model, the overall best RMSE values among the comparative algorithms are highlighted in boldface. Table IX demonstrates that the proposed LHSSA performs much better than all the other comparative algorithms for all the models in terms of reliability and accuracy. In this regard, the convergence curves for the comparative algorithms are depicted in Fig. 5 and the box-plot representations are used to show the distribution of the results obtained by those algorithms over 20 independent runs, as shown in Fig. 6. It is noted that the LHSSA has a faster convergence rate than the other algorithms in all models.
Table IX Statistical Measures of Different Techniques for Three Models
Model | Algorithm | RMSE (mA) |
---|
Min | Mean | Max | St.dev |
SD |
LHSSA |
0.98602 |
0.98602 |
0.98602 |
6.269700×10-10 |
PSO |
19.58100 |
212.69000 |
222.86000 |
45.454000 |
FFA |
141.64000 |
225.21000 |
241.84000 |
20.333000 |
GWO |
222.86000 |
222.86000 |
222.87000 |
4.254900×10-3 |
DA |
222.86000 |
222.86000 |
222.86000 |
1.042400×10-13 |
SSA |
39.90600 |
121.35000 |
222.86000 |
44.665000 |
IJAYA |
0.98603 |
0.99204 |
1.06220 |
1.403300×10-2 |
JAYA |
0.98946 |
1.16170 |
1.47830 |
0.187960 |
GOTLBO |
0.98856 |
1.04500 |
1.20670 |
0.502180 |
LETLBO |
0.98738 |
1.03330 |
1.15930 |
0.469460×10-2 |
LBSA |
0.99125 |
1.14660 |
1.48620 |
0.134820 |
CLPSO |
0.99633 |
1.05810 |
1.31960 |
7.485400×10-2 |
BLPSO |
1.02720 |
1.31390 |
1.79280 |
0.211660 |
DE/BBO |
0.99922 |
1.29480 |
2.22580 |
0.250740 |
CMM-DE/BBO |
0.98605 |
1.04860 |
1.34750 |
8.167900×10-2 |
DD |
LHSSA |
0.98249 |
0.98337 |
0.98602 |
1.494500×10-3 |
PSO |
222.86000 |
230.57000 |
299.95000 |
24.378000 |
FFA |
226.06000 |
244.84000 |
266.29000 |
11.980500 |
GWO |
6.61290 |
159.45000 |
222.87000 |
102.160000 |
DA |
222.86000 |
222.86000 |
222.86000 |
1.072200×10-4 |
SSA |
82.60300 |
147.24000 |
165.79000 |
24.236000 |
IJAYA |
0.98293 |
1.02690 |
1.40550 |
0.098325 |
JAYA |
0.98934 |
1.17670 |
1.47930 |
0.193560 |
GOTLBO |
0.98742 |
1.14750 |
1.39470 |
0.113300 |
LETLBO |
0.98565 |
1.08690 |
1.48700 |
0.153600 |
LBSA |
0.98751 |
1.25450 |
1.73430 |
0.222360 |
CLPSO |
0.99894 |
1.14580 |
1.54940 |
0.143670 |
BLPSO |
1.06280 |
1.48210 |
1.74110 |
0.177890 |
DE/BBO |
1.02550 |
1.55710 |
2.40420 |
0.362970 |
CMM-DE/BBO |
1.00880 |
1.54870 |
2.05890 |
0.294130 |
PV module |
LHSSA |
2.42500 |
2.42500 |
2.42500 |
3.525700×10-9 |
PSO |
77.58500 |
202.35000 |
274.25000 |
87.493000 |
FFA |
220.41000 |
242.59000 |
261.92000 |
15.209000 |
GWO |
23.53200 |
99.18200 |
274.29000 |
120.820000 |
DA |
23.88100 |
213.11000 |
274.25000 |
96.186000 |
SSA |
144.93000 |
186.23000 |
246.43000 |
37.828000 |
IJAYA |
2.42510 |
2.42890 |
2.43930 |
3.775500×10-3 |
JAYA |
2.42780 |
2.45370 |
2.59590 |
3.456300×10-2 |
GOTLBO |
2.42660 |
2.47540 |
2.56380 |
2.938800×10-2 |
LETLBO |
2.42510 |
2.44070 |
2.58210 |
2.949000×10-2 |
LBSA |
2.42520 |
2.46740 |
2.53440 |
2.910900×10-2 |
CLPSO |
2.42810 |
2.45490 |
2.54330 |
2.581000×10-2 |
BLPSO |
2.42520 |
2.43790 |
2.48830 |
1.372400×10-2 |
DE/BBO |
2.42830 |
2.46160 |
2.52560 |
2.925100×10-2 |
CMM-DE/BBO |
2.42510 |
2.42520 |
2.42680 |
3.554800×10-4 |
Fig. 5 Convergence curves for two models. (a) SD model. (b) DD model.
Fig. 6 Boxplot representations of RMSE for SD model and DD model. (a) SD model. (b) DD model.
E. Study of Fuzzy Representation
The imprecise descriptions of the solar cell models are often caused by weather fluctuations or instabilities in the measuring process. Thus, new insight from the operating point of view is presented by incorporating this impreciseness using the fuzzy concept in the solar cell models. The fuzzy number representation is illustrated in Section II. Additionally, the fuzzified value is transformed into a crisp value based on the -cut level, using upper and lower bound values.
A searching process is carried out to identify the optimal values for the parameters in terms of the level. This strategy is investigated on the SD, DD and PV module models at different levels, but the results are reported for only due to space limitation. The estimated parameters are recorded in Tables X-XII. The convergence curves and box plot are demonstrated for the DD model only due to space limitation in Figs. 7 and 8, respectively. The statistical measures for these models are also listed in Table XIII.
Fig. 7 Convergence curves for DD model at .
Fig. 8 Box plot for RMSE over 20 runs for DD .
Table XIII Statistical Measures Among Different Techniques for Three Models Under Fuzzy Environment ()
Model | Algorithm | RMSE (mA) |
---|
Min | Mean | Max | SD |
SD |
PSO |
220.77000 |
231.40000 |
300.55000 |
24.33200 |
FFA |
189.52000 |
227.43000 |
242.79000 |
146.71000 |
GWO |
217.92000 |
218.65000 |
219.02000 |
0.36142 |
DA |
228.64000 |
273.31000 |
295.14000 |
23.96100 |
SSA |
116.52000 |
161.67000 |
208.19000 |
23.10600 |
LHSSA |
0.10029 |
0.25622 |
0.34823 |
0.07818 |
DD |
PSO |
29.05000 |
219.53000 |
302.33000 |
74.49600 |
FFA |
222.72000 |
238.44000 |
251.02000 |
9.48900 |
GWO |
8.65650 |
118.91000 |
219.85000 |
105.79000 |
DA |
251.53000 |
11504.00000 |
111420.00000 |
35109.00000 |
SSA |
65.54200 |
137.93000 |
198.67000 |
45.38300 |
LHSSA |
0.11987 |
0.27267 |
391.31000 |
0.07176 |
PV module |
PSO |
91.95800 |
213.75000 |
276.93000 |
720.69000 |
FFA |
212.37000 |
246.50000 |
264.59000 |
14.30600 |
GWO |
19.14200 |
47.80500 |
266.10000 |
76.81400 |
DA |
194.71000 |
354.88000 |
441.70000 |
96.89600 |
SSA |
136.25000 |
196.20000 |
254.81000 |
51.95600 |
LHSSA |
0.49438 |
0.69770 |
0.90773 |
0.13615 |
Table X Estimated Parameters in Fuzzy Environment (SD Model) when
Algorithm | () | () | (A) | (A) | | RMSE (mA) |
PSO |
2.2204×1016 |
1.1058 |
0.841825 |
2.2204×1010 |
2.0000 |
220.77000 |
FFA |
0.0181 |
75.4861 |
0.949916 |
8.5959 |
1.9143 |
189.52000 |
GWO |
4.3206×1016 |
1.1224 |
0.832974 |
1.7551×109 |
1.9233 |
217.92000 |
DA |
2.6996×103 |
1.0181 |
0.899220 |
4.8877×1010 |
1.0000 |
228.64000 |
SSA |
0.0269 |
1.6612 |
0.863027 |
4.3412 |
1.8747 |
116.52000 |
LHSSA |
2.2204×1016 |
36.5013 |
0.766241 |
14.1930 |
1.9995 |
0.10029 |
Table XI Estimated Parameters in Fuzzy Environment (DD Model) when
Algorithm | | | | | | | | RMSE (mA) |
PSO |
2.2204×1016 |
100.0000 |
0.7426 |
13.238 |
2.220×1010 |
2.00 |
2.00 |
29.05 |
FFA |
2.2204×1016 |
1.1034 |
0.8331 |
2.220×1010 |
2.220×1010 |
1.50 |
1.11 |
222.72 |
GWO |
1.4317×102 |
73.9081 |
0.7673 |
12.968 |
8.489×1010 |
2.00 |
1.43 |
8.66 |
DA |
2.3572×1016 |
2.7496 |
0.6802 |
3.754×1010 |
2.818×1010 |
1.06 |
1.04 |
251.53 |
SSA |
1.6169×102 |
4.6353 |
0.8661 |
4.672×103 |
12.827 |
1.54 |
1.97 |
65.54 |
LHSSA |
2.2561×102 |
65.3882 |
0.7539 |
2.024 |
2.220×1010 |
1.71 |
1.00 |
0.12 |
Table XII Estimated Parameters in Fuzzy Environment (PV Module) when
Algorithm | | | | | | RMSE (mA) |
PSO |
2.2204×1016 |
2000.0000 |
1.2169 |
6938.000 |
3.3448 |
91.95800 |
FFA |
2.2204×1016 |
548.3519 |
1.1426 |
88576.000 |
7.3094 |
212.37000 |
GWO |
2.2204×1016 |
905.2461 |
1.0513 |
795.350 |
2.3831 |
19.14200 |
DA |
2.2204×1016 |
1659.3036 |
1.0943 |
76921.000 |
6.7461 |
194.71000 |
SSA |
1.3802×104 |
1703.3602 |
1.3011 |
38435.000 |
4.8604 |
136.25000 |
LHSSA |
0.0220 |
1999.9620 |
1.0324 |
65.662 |
1.7777 |
0.49438 |
F. Effects of α-level Schemes on RMSE
In order to clear the effects of the -level schemes on RMSE, three cases are considered () as in Fig. 9, where represents the crisp case. Based on the obtained results, the RMSE is affected by the vagueness aspect induced by the α-level cut. Finally, we hope that this paper will inspire researchers in studying the uncertainty aspect of different solar cell performances, which is caused by various factors such as shading, weather changes and so on.
Fig. 9 Effects of α-level schemes on RMSE.
In this paper, a novel LHSSA is presented to accurately estimate the parameters of PV models. In the LHSSA, the standard SSA is conducted to search globally and explore the different areas in the search space. Afterwards, PSO is employed to guide the SSA leaders with the aim of eliciting the promising area. Additionally, a learning scheme based on the follower behavior is introduced with the aim of improving the population diversity. In this regard, the SSA emphasizes on the diversification while PSO focuses on the intensification.
The proposed LHSSA is investigated on different PV models, i.e., SD, DD, and PV module models. Comprehensive results affirm that LHSSA is able to obtain highly competitive performance in comparison with other algorithms, especially in terms of quality and reliability. In future work, we will study the effect of shading on the performance of the PV modules. Additionally, some other modification will be developed, dealing with more complex renewable energy problems and studying the use of the rough set theory for dealing with different PV models. Finally, we hope that this paper will inspire researchers in studying the uncertainty aspect of solar cell performances.
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