^{1}

^{2}

^{*}

^{3}

^{4}

^{3}

^{1}

To evaluate the performance of a photovoltaic panel, several parameters must be extracted from the photovoltaic. These parameters are very important for the evaluation, monitoring and optimization of photovoltaic. Among the methods developed to extract photovoltaic parameters from current-voltage (I-V) characteristic curve, metaheuristic algorithms are the most used nowadays. A new metaheuristic algorithm namely enhanced vibrating particles system algorithm is presented here to extract the best values of parameters of a photovoltaic cell. Five recent algorithms (grey wolf optimization (GWO), moth-flame optimization algorithm (MFOA), multi-verse optimizer (MVO), whale optimization algorithm (WAO), salp swarm-inspired algorithm (SSA)) are also implemented on the same computer. Enhanced vibrating particles system is inspired by the free vibration of the single degree of freedom systems with viscous damping. To extract the photovoltaic parameters using enhanced vibrating particles system algorithm, the problem can be set as an optimization problem with the objective to minimize the difference between measured and estimated current. Four case studies have been implemented here. The results and comparison with other methods exhibit high accuracy and validity of the proposed enhanced vibrating particles system algorithm to extract parameters of a photovoltaic cell and module.

Many disadvantages affect the availability of fossil fuels: the fluctuating prices, the environmental pollution and the fact that they are not abundant [

To design and assess the operation of a PV system, a PV model should be implemented with appropriate accurateness that one can employ to predict the reliable I-V and P-V output characteristics under normal operation [

Many methods in the literature have been developing to extract PV parameters. These methods can be classified into three categories: analytical methods, numerical methods and evolutionary methods. In the analytical method, a set of transcendental equations is solved to extract parameters from solar cell [

Explicit modeling from current and voltage characteristic is used by [

Numerical extraction techniques based on some algorithm fit the points on the PV characteristic curve. Compared to the analytical method, an accurate result can be attained since the algorithm tries to consider all points on the characteristic curve [

The main drawbacks of numerical techniques such as Newton Raphson are the need for extensive computations for convergence and fail to result in accurate results when the number of parameters to be estimated increases and a close approximation of initial conditions [

Despite the efficiency of the numerical methods, their slow convergence does not always guarantee the best result because they can converge through a local minimum and the choice of the initial condition is not often easy [

To overcome the drawback of analytical and iterative methods, metaheuristic algorithms have been developed. They are nature-inspired algorithms using probabilities to find the best result. They have shown their effectiveness in solving difficult problems. Their main advantage is that they do not need continuity and differentiability of the objective function In the last decade, metaheuristics have been frequently applied for parameter estimation of circuit model parameters of solar PV cells. The main develops in recent research are: genetic algorithm (GA) [

Until today in the research and industry domain, there is no method of extracting PV parameter that has been introduced in the manufacturing of PV. By another, “The no-free-lunch theorem” remarked that: there is no algorithm able to solve all optimization problems, where it is important to propose new algorithms for solving engineering optimization problems [

The rest of this paper is presented as follows: In Section 2, PV cell modeling is presented; Section 3 presents the problem formulation for extracting parameters from single and double diode model; the inspiration and the mathematical model of vibrating particles system are proposed in Section 4; Section 5 presents the different case study with different results and Section 6 is the conclusion.

Many models of PV cell have been developed in the literature; but there are two models mostly used: single and double diode model.

Most of the literature uses this model. The main reason for this widely used is their simplicity and the least number of parameters. In the single diode model, there are five parameters to be extracted.

The current I at the output of a PV module can be expressed using Kirchhoff’s theorem in Equation (1).

I = I r − I d − I p (1)

The diode current can be express as:

I d = I 0 [ exp ( V d n ⋅ V t ) − 1 ] (2)

where

V t = N s ⋅ K ⋅ T q (3)

The current through parallel resistance is

I p = V + I ⋅ R s R p (4)

By replacing Equation (2) & Equation (3) into Equation (1), we have the output current at the output of a PV module.

I = I r − I 0 [ exp ( V d n ⋅ V t ) − 1 ] − V + I ⋅ R s R p (5)

The parameters which characterize this equation are: θ = [ I r , I 0 , n , R s , R p ] . These five parameters can be determined by all the method described in Section 1.

This double-diode model (

The current I, at the output of a PV module can be expressed using Kirchhoff’s theorem in Equation (6).

I = I r − I d 1 − I d 2 − I p (6)

The diodes currents can be express as:

I d 1 = I 01 [ exp ( V d n 1 ⋅ V t ) − 1 ] (7)

I d 2 = I 02 [ exp ( V d n 2 ⋅ V t ) − 1 ] (8)

where

V t = N s ⋅ K ⋅ T q (9)

The current through parallel resistance is

I p = V + I ⋅ R s R p (10)

By replacing Equations (7)-(10) into Equation (6), we have the output current at the output of a PV module:

I = I r − I 01 [ exp ( V d n 1 ⋅ V t ) − 1 ] − I 02 [ exp ( V d n 2 ⋅ V t ) − 1 ] − V + I ⋅ R s R p (11)

The parameters which characterize this equation are:

θ = [ I r , I 01 , I 02 , n 1 , n 2 , R s , R p ] .

These seven parameters can be determined by all the methods described in Section 1.

To extract the PV parameters using EVPS, the problem can be set as an optimization problem with the objective to minimize the difference between measured and estimated current. The objective function (OF) is defined as the root mean square error (RMSE) where the error function is defined as the difference between estimated and experimental currents. It’s expressed as follows:

Min ( F ( θ ) ) = 1 N ∑ i = 1 N ( I i , m e s − I i , e x t ( θ ) ) 2 (12)

where:

F ( θ ) : is the objective function to minimize;

N: is the number of points (V_{i}, I_{i}) measured;

I i , m e s : is the measured current;

I i e x t ( θ ) : is the estimated current;

θ = [ I r , I 01 , I 02 , n 1 , n 2 , R s , R p ] : Parameters to estimate.

For a single diode model, the objective function is expressed as:

Min ( F ( θ ) ) = 1 N ∑ i = 1 N ( I i , m e s − I r + I 0 [ exp ( V i , m e s + I i , m e s ⋅ R s n . V t ) − 1 ] + V i , m e s + I i , m e s ⋅ R s R p ) 2 (13)

with θ = [ I r , I 0 , n , R s , R p ] the parameters to estimate.

For double diode model, the objective function is:

Min ( F ( θ ) ) = 1 N ∑ i = 1 N ( I i , m e s − I r + I 01 [ exp ( V i , m e s + I i , m e s ⋅ R s n 1 ⋅ V t ) − 1 ] + I 02 [ exp ( V i , m e s + I i , m e s ⋅ R s n 2 ⋅ V t ) − 1 ] + V i , m e s + I i , m e s ⋅ R s R p ) 2 (14)

with θ = [ I r , I 01 , I 02 , n 1 , n 2 , R s , R p ] the parameters to estimate

In this paper, EVPS algorithm is used to minimize Equation (13) and Equation (14).

The inspiration of vibrating particles system comes from the free vibration of a single degree of freedom system with viscous damping. The VPS contains a number of population solutions that represent the particle system. The particles are randomly initialized in an n-dimensional search space and step-by-step, they approach their equilibrium positions [

As other meta-heuristic algorithms, VPS has population particles which are considered as the parameters of the problem. The initial positions of particles are firstly generated randomly.

x j i = x min − r a n d ( x max − x min ) (15)

x j i represents the jth position of the ith particle; x min and x max are respectively the initial and the final position r a n d is a random number between [0, 1].

Three equilibrium positions affected by different weights are defined for each particle. During each generation, the particle positions are updated by learning from them. The equilibrium positions are:

- HB: Historically best location;

- GP: Good particle;

- BP: Bad particle.

To include the effect of the damping level in the vibration, a descending function is introduced:

D = ( i t e r i t e r max ) − α (16)

i t e r , i t e r max and α : represent respectively the current iteration, the maximum iteration and a constant.

The next position is updated by the following equations:

x j i = w 1 [ D ⋅ A ⋅ r a n d 1 + H B j ] + w 2 [ D ⋅ A ⋅ r a n d 2 + G P j ] + w 3 [ D ⋅ A ⋅ r a n d 3 + B P j ] A = [ W 1 ( H B j − x i j ) ] + [ W 2 ( G P j − x i j ) ] + [ W 3 ( B P j − x i j ) ] W 1 + W 2 + W 3 = 1 (17)

x j i : represent the jth position of the ith particle;

W 1 , W 2 , W 3 : parameters to measure the best value of HB, GP, BP;

r a n d 1 , r a n d 2 , r a n d 3 : random numbers between [0, 1].

EVPS algorithm has been initially developed by [

In EVPS, we introduced two new parameters: “memory” and “OHB (one of the best historically locations in the whole population)” [

x j i = { [ D ⋅ A ⋅ r a n d 1 + O H B j ] [ D ⋅ A ⋅ r a n d 2 + G P j ] [ D ⋅ A ⋅ r a n d 3 + B P j ] A = { ( ± 1 ) ( O H B j − x i j ) ( ± 1 ) ( G P j − x i j ) ( ± 1 ) ( B P j − x i j ) W 1 + W 2 + W 3 = 1 (18)

(±1) are applied randomly. BP, GP and OHB are independently determined for each particle [

The steps to compute EVPS algorithms are described as follows:

Step 1: Initializing of EVPS’s parameters.

- Initialize VPS parameters (size of the population, number of optimization variables, memory size, maximum number of iterations, lower and upper bound of the variables, parameters for handling the side constraints, w_{1} and w_{2}).

- Initializing particles positions using Equation (19).

x j i = x min − r a n d ( x max − x min ) (19)

Step 2: Search.

- Evaluate the objective function for each particle.

- For each particle, select “memory” and “OHB (one of the best historically locations in the whole population).

- Update particle’s position.

D = ( i t e r i t e r max ) − α (20)

Updating the next position by

x j i = { [ D ⋅ A ⋅ r a n d 1 + O H B j ] [ D ⋅ A ⋅ r a n d 2 + G P j ] [ D ⋅ A ⋅ r a n d 3 + B P j ] A = { ( ± 1 ) ( O H B j − x i j ) ( ± 1 ) ( G P j − x i j ) ( ± 1 ) ( B P j − x i j ) W 1 + W 2 + W 3 = 1 (21)

A parameter like k in a range of [0, 1] must be defined to specify if BP must be considered in the new position. For each population, k is compared with a random number (rand) uniformly distributed in the range of [0, 1]; if k < rand, then w_{3} = 0 and w_{1} = 1 − w_{2}.

Step 3: Handling the Side Constraints.

If a particle went out of the boundary, it must be updated by harmony search-based side constraints handling approach. The method consists to determine if the violating particle should be updated either by the best historically particle or randomly in the search space.

Step 4: Out memory and best positions.

The goal is to determine the global optimum, which is the best value of the OF (RMSE). If we replace memory (OHB) by the global optimum and the positions ( x j i ) by the estimated parameters, therefore ( θ i ), the best position automatically moves towards it. However, the problem is that the global solution of the optimization problems is unknown. In this case, the optimal solution obtained is the global optimum and presumed as the best selection of the memory.

The objective function of Equation (22) is used to find the best parameters of single and double diode model; the difference between single and double diodes model is the numbers of parameters (5 to single diode model and 7 to double diodes model). The formulation is:

Find ⊳ { x } = [ x 1 , x 2 , x 3 , ⋯ , x N ] Tominimize F ( x ) = 1 N ∑ i = 1 N ( I i , m e s − I i , e x t ( x ) ) 2 Subjectedto { g j ( { x } ) ≤ 0 , j = 1 , 2 , 3 , ⋯ , N c x i min ≤ x i ≤ x i max (22)

where [ x ] is the best parameters; ng is the number of parameters; F ( x ) the RMSE; N the number of points ( V i , I i ) measured; I i , m e s the measured current; I i , e x t ( x ) the estimated current; x i min is the lower bound, and x i max the upper bounds; g j ( { x } ) the design constraints and Nc the number of constraints.

The objective function of each model is formulated as follows:

- Single diode model

For a single diode model, the objective function is expressed as:

Min ( F ( x ) ) = 1 N ∑ i = 1 N ( I i , m e s − x 1 + x 2 [ exp ( V i , m e s + I i , m e s ⋅ x 4 x 3 ⋅ V t ) − 1 ] + V i , m e s + I i , m e s ⋅ x 4 x 5 ) 2 Subjectedto : x i min ≤ x i ≤ x i max (23)

with x = [ x 1 , x 2 , x 3 , x 4 , x 5 ] the five estimated parameters which correspond respectively to θ = [ I r , I 0 , n , R s , R p ] .

- Double diode model

For single diode model, the objective function is expressed as:

Min ( F ( θ ) ) = 1 N ∑ i = 1 N ( I i , m e s − x 1 + x 2 [ exp ( V i , m e s + I i , m e s ⋅ x 6 x 4 ⋅ V t ) − 1 ] + x 3 [ exp ( V i , m e s + I i , m e s ⋅ x 6 x 5 ⋅ V t ) − 1 ] + V i , m e s + I i , m e s ⋅ x 6 x 7 ) 2 Subjectedto : x i min ≤ x i ≤ x i max (24)

with x = [ x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ] the seven estimated parameters which correspond respectively to θ = [ I r , I 01 , I 02 , n 1 , n 2 , R s , R p ] .

The flowchart algorithm is represented in

This section presents different results and implementation of the algorithm. Four case studies have been implemented. The first two cases of study have been implemented in Matlab 2017a. In the first case study; Photowatt-PWP201 PV which has 36 polycrystalline silicon cells, all connected in series, the irradiance is 1000 W/m^{2} and temperature 45˚C. The second case refers to the RTC France commercial silicon PV, irradiance is 1000 W/m^{2} and temperature 33˚C. These two cases were for the first time initiated by [

A single diode model has been implemented in this case study to extract the five parameters of the Photowatt-PWP201 PV which is a 11.5 W PV module. It has 36 cells connected in series. Irradiance is 1000 W/m^{2} and temperature 45˚C. It’s widely used in literature by many researchers. The manufacturer’s characteristics of the PV module at STC are listed in

The initial parameters of EVPS are: number of search agents = 50; maximum number of iterations = 1000; alpha = 0.05; w_{1} = 0.3; w_{2} = 0.3.

^{−3}. In

Parameters | Values |
---|---|

I_{sc} (A) | 1.0317 |

V_{oc} (V) | 16.7785 |

I_{mp} (A) | 0.9120 |

V_{mp} (V) | 12.6490 |

N_{s} | 36 |

K_{i} for I_{sc} | 0.0360%/C |

Parameters | Lower Bound | Upper Bound |
---|---|---|

I_{r} (A) | 0 | 10 |

I_{o} (A) | 1e−12 | 1e−5 |

n | 0.5 | 2.5 |

R_{s} (Ω) | 0.001 | 2 |

R_{p} (Ω) | 0.001 | 5000 |

Parameters | Best Solutions |
---|---|

I_{r} (A) | 1.0318 |

I_{o} (A) | 3.2679 |

n | 1.3445 |

R_{s} (Ω) | 1.2066 |

R_{p} (Ω) | 845.759 |

OF (RMSE) | 2.4267 × 10^{−3} |

Best | Parameters | WOA [ | SSA [ | MVO [ | GWO [ | MFO [ | EVPS |
---|---|---|---|---|---|---|---|

I_{r} (A) | 1.0320 | 1.0452 | 1.0329 | 1.0324 | 1.0325 | 1.0318 | |

I_{o} (µA) | 7.5936 | 4.7508 | 1.7849 | 4.6572 | 2.7228 | 3.2695 | |

n | 1.4414 | 1.3868 | 1.2833 | 1.3832 | 1.3254 | 1.3446 | |

R_{s} (Ω) | 1.0444 | 1.0950 | 1.2775 | 1.1354 | 1.2255 | 1.2064 | |

R_{p} (Ω) | 738.6218 | 406.5791 | 658.9906 | 820.9687 | 742.2522 | 843.7385 | |

RMSE × 10^{−3} | 6.3860 | 7.6817 | 3.1162 | 3.9851 | 2.4842 | 2.4267 | |

Worst | I_{r} (A) | 1.3568 | 0.9884 | 1.0489 | 1.0960 | 1.0856 | 1.0329 |

I_{o} (µA) | 9.6098 | 8.6491 | 8.635 | 1.7016 | 6.0648 | 6.5761 | |

n | 1.5413 | 1.4705 | 1.4586 | 1.3057 | 1.4235 | 1.4231 | |

R_{s} (Ω) | 0.2965 | 0.1955 | 1.0475 | 0.2845 | 0.8900 | 1.1183 | |

R_{p} (Ω) | 23.0796 | 549.0804 | 337.8831 | 61.4542 | 118.6442 | 999.5245 | |

RMSE × 10^{−3} | 141.6485 | 71.9332 | 9.0679 | 67.7101 | 24.9788 | 3.4707 | |

Average | RMSE × 10^{−3} | 44.1232 | 28.2501 | 5.4726 | 19.0572 | 3.9637 | 2.7877 |

Parameters | TVAPSO [ | LI [ | IADE [ | PS [ | SA [ | RF [ | GCPSO [ | GAMS [ | EVPS |
---|---|---|---|---|---|---|---|---|---|

I_{r} (A) | 1.031435 | 1.0334 | 1.0311 | 1.0313 | 1.0331 | 1.032375 | 1.0323823 | 1.032015 | 1.0318 |

I_{o} (µA) | 2.638610 | 2.4424 | 3.6642 | 3.1756 | 3.6642 | 2.518884 | 2.512922 | 3.26812 | 3.2679 |

n | 47.556652 | 1.2975 | 1.3561 | 1.3413 | 48.8211 | 1.239018 | 1.31730 | 1.344574 | 1.3445 |

R_{s} (Ω) | 1.235611 | 1.2975 | 1.1989 | 1.2053 | 1.1989 | 1.317400 | 1.239288 | 1.206210 | 1.2066 |

R_{p} (Ω) | 821.59514 | 603.4037 | 921.85 | 714,285 | 833.3333 | 745.6431 | 744.7166 | 828.292864 | 845.759 |

RMSE × 10^{−3} | 6.9665 | 2.477 | 2.4 | 11.8 | 2.7 | 2.7001 | 2.5915 | 2.442689 | 2.4267 |

show the superiority of the algorithm. ^{2} irradiance and 45˚C temperature. In

In this case, seven parameters of RTC France PV have been extracted. The irradiance of the RTC France PV is 1000 W/m^{2} and temperature 33˚C. The typical electrical characteristics of the PV cell at STC are listed in

^{2} and 33˚C. In

The case study 3 consists of the implementation of a Matlab/Simulink model at

Parameters | Values |
---|---|

I_{sc} (A) | 0.760 |

V_{oc} (V) | 0.573 |

I_{mp} (A) | 0.691 |

V_{mp} (V) | 0.450 |

N_{s} | 1 |

K_{i} for I_{sc} | 0.0350%/C |

Parameters | Lower Bound | Upper Bound |
---|---|---|

I_{r} (A) | 0 | 10 |

I_{o}_{1} (A) | 1e−12 | 1e−5 |

I_{o}_{2} (A) | 1e−12 | 1e−5 |

n_{1} | 0.5 | 2.5 |

n_{2} | 0.5 | 2.5 |

R_{s} (Ω) | 0.001 | 2 |

R_{p} (Ω) | 0.001 | 5000 |

Parameters | Best Solutions |
---|---|

I_{r} (A) | 0.7607 |

I_{o}_{1} (µA) | 0.29749 |

I_{o}_{2} (µA) | 0.2504 |

n_{1} | 1.4749 |

n_{2} | 1.9726 |

R_{s} (Ω) | 0.0363 |

R_{p} (Ω) | 55.8827 |

OF (RMSE) | 9.8510e−4 |

Best | Parameters | WOA [ | SSA [ | MVO [ | GWO [ | MFO [ | EVPS |
---|---|---|---|---|---|---|---|

I_{r} (A) | 0.7646 | 0.7637 | 0.7606 | 0.7635 | 0.7627 | 0.7607 | |

I_{o}_{1} (µA) | 1.7641 | 0.9880 | 3.6662 | 0.3639 | 0.0009 | 0.2975 | |

I_{o}_{2} (µA) | 2.0948 | 0.8796 | 0.1880 | 0.0012 | 3.2555 | 0.2504 | |

n_{1} | 1.6839 | 2.2119 | 2.1225 | 1.4942 | 1.1596 | 1.4749 | |

n_{2} | 2.1583 | 1.5932 | 1.4417 | 2.1433 | 1.7848 | 1.9726 | |

R_{s} (Ω) | 0.0277 | 0.0307 | 0.0354 | 0.0348 | 0.0293 | 0.0363 | |

R_{p} (Ω) | 64.2652 | 42.7318 | 94.5515 | 30.7060 | 100.0000 | 55.8827 | |

RMSE × 10^{−3} | 53.6880 | 35.4087 | 14.8497 | 26.2566 | 45.8807 | 9.8510 | |

Worst | I_{r} (A) | 0.7457 | 0.7765 | 0.7756 | 0.7669 | 0.7589 | 0.7606 |

I_{o}_{1} (µA) | 2.5483 | 3.2039 | 4.5604 | 0.9919 | 10.0000 | 1.262e-03 | |

I_{o}_{2} (µA) | 3.2766 | 5.8725 | 0.9121 | 9.9708 | 10.0000 | 0.5005 | |

n_{1} | 1.7422 | 2.0224 | 1.8147 | 2.0889 | 2.1160 | 1.2067 | |

n_{2} | 2.2150 | 1.8891 | 2.4204 | 1.9505 | 2.0254 | 1.5392 | |

R_{s} (Ω) | 0.0014 | 0.0135 | 0.0181 | 0.0015 | 0.0010 | 0.0358 | |

R_{p} (Ω) | 87.5855 | 13.7374 | 13.8209 | 15.8790 | 100.0000 | 63.4186 | |

RMSE × 10^{−3} | 322.0510 | 150.5212 | 135.4657 | 200.7120 | 166.5394 | 11.19 | |

Average | RMSE × 10^{−3} | 164.3665 | 82.9360 | 93.3376 | 107.7185 | 86.5089 | 10.083 |

Parameters | MBA [ | SA [ | HS [ | CPSO [ | ABCO [ | ABC [ | IGHS [ | EVPS |
---|---|---|---|---|---|---|---|---|

I_{r} (A) | 0.7605 | 0.7623 | 0.7616 | 0.762321 | 0.7608 | 0.7609 | 0.76079 | 0.7607 |

I_{o}_{1} (µA) | 0.4513 | 0.4767 | 0.12546 | 0.297108 | 0.0407 | 2.6900 | 0.97310 | 0.29749 |

I_{o}_{2} (µA) | 1.1846 | 0.0100 | 0.25470 | 0.710454 | 0.2874 | 2.8198 | 0.16791 | 0.2504 |

n_{1} | 1.5920 | 1.517 | 1.49439 | 1.476035 | 1.4495 | 1.4670 | 1.92126 | 1.4749 |

n_{2} | 1.8450 | 2.000 | 1.49989 | 1.998103 | 1.4885 | 1.8722 | 1.42814 | 1.9726 |

R_{s} (Ω) | 0.0314 | 0.0345 | 0.03562 | 0.035601 | 0.0364 | 0.0364 | 0.03690 | 0.0363 |

R_{p} (Ω) | 493.7200 | 43.10 | 46.8269 | 45.547533 | 53.7804 | 55.2307 | 56.8368 | 55.8827 |

RMSE × 10^{−4} | 0.1092 | 0.01667 | 12.6 | 13.0565 | 9.861 | 10 | 9.8635 | 9.8510 |

different irradiance conditions. The case study 3 refers to the polycrystalline SW255. The using manufacturer data at STC is reported in ^{2}) = 1000, 800, 600, 400, 200) at 25˚C temperature are exported to Matlab via the blocks Workspace.

One diode model has been implemented in this last case to extract the five parameters based on experimental data of the Sharp ND-R250A5 PV module. The PV has 60 cells in series. Irradiance is 1040 W/m^{2} and temperature 59˚C. The typical electrical characteristics of the Sharp ND-R250A5 PV module at STC are listed in

^{−3}. ^{2} irradiance and 59˚C temperature. In

Parameters | Values |
---|---|

I_{sc} (A) | 8.8 |

V_{oc} (V) | 38.0 |

I_{mp} (A) | 8.32 |

V_{mp} (V) | 30.9 |

P_{mp} (W) | 255 |

N_{s} | 60 |

K_{i} [A/K] | 0.051 |

K_{v} [A/K] | −0.31 |

Parameters E (W/m^{2}) | 1000 | 800 | 600 | 400 | 200 |
---|---|---|---|---|---|

I_{r} (A) | 8.9005 | 7.1156 | 5.3376 | 3.5635 | 1.7760 |

I_{o} (A) | 2.376e−07 | 6.8846e−08 | 6.8962e−08 | 6.8596e−08 | 6.8830e−08 |

n | 1.3632 | 1.3212 | 1.3222 | 1.3232 | 1.3264 |

R_{s} (Ω) | 0.2109 | 0.2169 | 0.2110 | 0.1937 | 0.1942 |

R_{p} (Ω) | 6709.6 | 4428.5 | 6065 | 2177.7 | 2742.4 |

RMSE × 10^{−3} | 19.4628 | 9.6298 | 7.6950 | 6.6091 | 9.8569 |

E (W/m^{2}) Parameters | 1000 | 800 | 600 | 400 | 200 |
---|---|---|---|---|---|

I_{r} (A) | 8.8928 | 7.1234 | 5.3394 | 3.5655 | 1.7817 |

I_{o}_{1} (A) | 6.9030e−08 | 7.6854e−08 | 3.4988e−08 | 7.6950e−08 | 7.3365e−08 |

I_{o}_{2} (A) | 5.9342e−08 | 2.7425e−07 | 4.1686e−07 | 1.1246e−07 | 2.1394e−07 |

n_{1} | 1.3213 | 1.3323 | 1.2802 | 1.9997 | 1.3338 |

n_{2} | 1.7503 | 1.7486 | 1.7352 | 1.3611 | 1.7746 |

R_{s} (Ω) | 0.2200 | 0.2099 | 0.2154 | 0.1743 | 0.1150 |

R_{p} (Ω) | 7114.9 | 3257.3 | 4515.6 | 2.9635 | 5.8048 |

RMSE × 10^{−3} | 12.6607 | 14.7437 | 7.7546 | 8.7455 | 4.3193 |

E (W/m^{2}) | Algorithm | I_{r} (A) | I_{0} (A) | n | R_{s} (Ω) | R_{p} (Ω) | RMSE × 10^{−3} |
---|---|---|---|---|---|---|---|

1000 | EVPS | 8.9005 | 2.376e−07 | 1.3632 | 0.2109 | 6709.6 | 19.4628 |

SSA | 8.9273 | 8.006e−07 | 1.5197 | 0.17128 | 4693 | 46.829 | |

WAO | 8.9035 | 5.1465e−07 | 1.4798 | 0.18597 | 5992.3 | 44.717 | |

MVO | 8.9045 | 1.7761e−07 | 1.3908 | 0.20437 | 7999.1 | 24.565 | |

GWO | 8.9465 | 3.4037e−07 | 1.4437 | 0.19151 | 704.46 | 38.846 | |

MFO | 8.9046 | 1.8009e−07 | 1.3919 | 0.2036 | 8000 | 24.706 |

800 | EVPS | 7.1156 | 6.8846e−08 | 1.3212 | 0.2169 | 4428.5 | 9.6298 |
---|---|---|---|---|---|---|---|

SSA | 7.1487 | 1.0559e−06 | 1.55 | 0.14622 | 2741.4 | 41.995 | |

WAO | 7.1607 | 3.6348e−06 | 1.6818 | 0.10242 | 5347 | 57.411 | |

MVO | 7.1496 | 7.078e−07 | 1.5116 | 0.15786 | 1563.7 | 37.947 | |

GWO | 7.1452 | 8.0295e−07 | 1.5233 | 0.15771 | 5559.5 | 38.485 | |

MFO | 7.13 | 3.1117e−07 | 1.4383 | 0.18167 | 8000 | 26.455 | |

600 | EVPS | 5.3376 | 6.8962e−08 | 1.3222 | 0.2110 | 6065 | 7.6950 |

SSA | 5.3434 | 1.5209e−07 | 1.3822 | 0.18597 | 7677.2 | 14.248 | |

WAO | 5.3634 | 3.995e−06 | 1.7014 | 0.018556 | 2647.7 | 54.056 | |

MVO | 5.3595 | 7.7029e−07 | 1.5239 | 0.12557 | 7461.4 | 29.299 | |

GWO | 5.314 | 9.2173e−09 | 1.1905 | 0.2572 | 3847.2 | 14.454 | |

MFO | 5.3513 | 3.5134e−07 | 1.452 | 0.15616 | 6324.5 | 21.965 | |

400 | EVPS | 3.5635 | 6.8596e−08 | 1.3232 | 0.1937 | 2177.7 | 6.6091 |

SSA | 3.5633 | 1.9988e−07 | 1.4076 | 0.14348 | 7882.9 | 11.392 | |

WAO | 3.6325 | 2.5137e−06 | 1.6575 | 0.06760 | 3564.3 | 45.294 | |

MVO | 3.5721 | 1.7561e−07 | 1.3968 | 0.14598 | 1496.1 | 12.517 | |

GWO | 3.5808 | 1.5835e−06 | 1.6058 | 0.018474 | 7752.8 | 24.66 | |

MFO | 3.5641 | 1.9926e−07 | 1.4072 | 0.14678 | 8000 | 11.277 | |

200 | EVPS | 1.7760 | 6.8830e−08 | 1.3264 | 0.1942 | 2742.4 | 9.8569 |

SSA | 1.7846 | 2.2927e−07 | 1.4259 | 0.012545 | 4636.4 | 7.0205 | |

WAO | 1.7692 | 1.5472e−07 | 1.3919 | 0.0010176 | 5259 | 11.009 | |

MVO | 1.7897 | 6.1994e−08 | 1.3178 | 0.12359 | 1323.6 | 6.1192 | |

GWO | 1.808 | 6.1509e−07 | 1.5204 | 0.034513 | 1733.9 | 18.224 | |

MFO | 1.7897 | 8.3885e−08 | 1.3415 | 0.001 | 898.89 | 14.321 |

Parameters | Values |
---|---|

I_{sc} (A) | 8.68 |

V_{oc} (V) | 37.6 |

I_{m}p (A) | 8.10 |

V_{mp} (V) | 30.9 |

N_{s} | 60 |

Parameters | Lower Bound | Upper Bound |
---|---|---|

I_{r} (A) | 0 | 10 |

I_{o} (A) | 1e−12 | 1e−5 |

n | 0.5 | 2.5 |

R_{s} (Ω) | 0.001 | 2 |

R_{p} (Ω) | 0.001 | 5000 |

Parameters | EVPS |
---|---|

I_{r} (A) | 9.146656 |

I_{o} (µA) | 1.094195 |

n | 1.213629 |

R_{s} (Ω) | 0.589391 |

R_{p} (Ω) | 4999.9999 |

OF (RMSE × 10^{−3}) | 11.252719 |

In this paper, we have presented a novel bio-inspired optimizer of a very recent heuristic-based on technique, namely enhanced vibrating particles system to extract the best values of parameters of a photovoltaic cell. The particles are randomly initialized in an n-dimensional search space and Step-by-Step, they approach their equilibrium positions.

To show the performance of the algorithm, many cases have been implemented from one and two diodes model. The current-voltage and power-voltage characteristic of measured and estimated data show the best accuracy of the method. The simulations result and comparisons with another method exhibit high accuracy and validity of the proposed Enhanced Vibrating particles system to extract parameters of a photovoltaic cell and module. Thus, enhanced vibrating particles system can be recommended as an efficient method not only to extract the best parameters of a PV cell and module, but also to solve optimization problems in power systems. As every algorithm, enhanced vibrating particles system has some drawback like the variability of the result at each independent test and the limit of the algorithm to solve the only mono-objective problem. In the future work, the stability of the enhanced vibrating particles system should be improved and other parameters should be added to permit the algorithm to solve multi-objective optimization in power systems.

The authors declare no conflicts of interest regarding the publication of this paper.

Gnetchejo, P.J., Essiane, S.N., Ele, P., Wamkeue, R., Wapet, D.M. and Ngoffe, S.P. (2019) Enhanced Vibrating Particles System Algorithm for Parameters Estimation of Photovoltaic System. Journal of Power and Energy Engineering, 7, 1-26. https://doi.org/10.4236/jpee.2019.78001

E: Solar irradiance;

F ( θ ) : objective function to minimize;

I: cell output current [A];

I i e x t ( θ ) : is the estimated current;

I i , m e s : measured current [A];

I i e x t ( θ ) : estimated current [A];

I 0 , I 01 , I 02 : Diode reverse saturation currents [μA];

I d , I d 1 , I d 2 : diode currents [A];

I m p current at the maximum power point [A];

I r photoelectric current [A];

I s c short-circuit current [A];

k: Boltzman constant [J/K];

k i : temperature coefficient of Isc [A/K];

n , n 1 , n 2 : Diode ideality factors;

N: number of the experimental I-V data pairs;

N S : number of cells connected in series;

OF: objective function;

q: electron charge [C];

r a n d 1 , r a n d 2 , r a n d 3 : random numbers between [0, 1];

R P : parallel resistance [Ω];

R S : series resistance [Ω];

RMSE: root mean square error;

STC: Standard testing condition (1000 watts/m^{2}, 25˚C);

T: Temperature [K];

V: cell output voltage [V];

V i , m e s : measured voltage [V];

V m p : voltage at the maximum power point [V];

V o c : open-circuit voltage [V];

V t : thermal voltage [V];

w 1 , w 2 , w 3 : parameters;

x j i : position of the particle;

θ : Parameters to estimate.