2.1 Artificial Bee Colony (ABC)

Artificial Bee Colony was proposed by Karaboga [²⁷] inspired by the behavior of honeybee swarm. The ABC employs a population Sks1k,s2k,…sNk of N food sources randomly distributed from an initial point k = 0 to a total number of iterations k = iterations. In ABC, each food source siki∈1,…,N is represented as a m-dimensional vector si,1k,si,2k,…,si,mk, where m is the number of decision variables of the optimization problem. After initialization, the food source quality is evaluated considering a fitness function that determinates if is a feasible solution or not. If the solution sik is a candidate, the operators of ABC evolved this candidate to generate a new food source v _i that is defined as follows:

where sjk is a random food source that satisfies and ϕ is a random scale factor between [-1,1]. The fitness function for a minimization problem assigned to a candidate solution can be defined as follow:

where f(•) is the fitness function to be minimized. When a new food source is computed, a greedy selection handle, if the new food source v _i is better than the actual sik, the actual food source sik is replaced for the new one v _i , otherwise the actual food source sik reminds.

2.2 Differential Evolution (DE)

Storn and Price developed Differential evolution algorithm [²⁸], which is a stochastic vector-based evolutionary technique, which utilizes m-dimensional vectors defined as follow:

where xik is the i-th vector at iteration k. The DE uses the weighted difference between two vectors to generate a third vector; this process is known as mutation and is described below:

where F is a constant that controls the magnitude of differential variation within the interval [0, 2]. On the other hand, to increment the diversity of the mutated vector, a crossover is incorporated. It is represented as follows:

Finally, the vector selection produces the final value comparing the fitness values between the can didate vector against the original as follows:

2.3 Harmony Search (HS)

Harmony search algorithm was introduced by Geem [²⁹]. This optimization method is particularity based on the improvisation process taking place in jazz music. HS defines a harmony memory with a population of N individuals represented as HMkH1k,H2k,…,HNk. Each harmony Hik represents a m-dimensional vector of the decision variables. After initialization, HS generates new solutions by considering a pitch adjustment or with a random re-initialization.

When a new harmony is produced, a uniform random number between [0,1] is generated r ₁ , if this number is less than the harmony-memory consideration rate (HMCR), the new harmony is generated with memory considerations, otherwise, the new harmony is re-initialized with random values between bound limits. The generation process of a new harmony is described below:

To determinate which element should be adjusted by the pitch process, further examinations must be considered. For this operation two parameters are defined: The pitch-adjusting rated (PAR) and bandwidth factor (BW). PAR considers the frequency of adjustment while BW controls the magnitude of the local search process. This complete operation can be described as follows:

2.4 Gravitational Search Algorithm (GSA)

Rashedi proposed Gravitational Search Algorithm [³⁰] which is based on the laws of gravity. This technique emulates the candidate solutions as masses, which are attacked one each other by the gravitational forces. Under this approach, the quality (mass) of an individual is determined by its fitness value. The GSA uses a population of N individuals that represent an m-dimensional vector xikx1,x2,…,xN, where the dimension is the number of decision variables. A force from a mass i to a mass j in a defined time t is determined as follows:

where Ma _j is the active gravitational mass related to individual j, Mp _i . is the passive gravitational mass of individual i, G(t) is the gravitational constant, ε is a constant, and R _ij is the Euclidian distant between i and j individuals. In GSA, the balance between exploration and exploitation is made by modifying G(t) through the iterations. The sum of all forces acting on individual i is expressed bellow:

the acceleration of each individual at time t is given bellow:

where Mn _i is the inertia mass of individual i, with this, the velocity and position for each individual are computed as follows:

After evaluating the fitness of each individual, their inertia, and gravitational masses are updated, where a heavier individual means a better solution, fort this, GSA uses the follows equations:

where are the fitness function and best (t) and worst (t) represent the best and worst solution of the complete population at time t.

2.5 Particle Swarm Optimization (PSO)

Particle Swarm Optimization, introduced by Kennedy [³⁶], is based on the behavior of birds flocking. The PSO uses a group of N particles Pkp1k,p2k,…,pNk which after being evaluated by a cost function, the best positions are storage pi,j*. To calculate the velocity of each candidate solution for the next iteration, the following model is used:

where ω is the inertia weighs used to control the velocity (i = 1,2,..,N), (j = 1,2,...,p). c ₁ and c ₂ are the acceleration coefficients that adjust the movement of each individual in the positions g and p ^* respectively. r ₁ and r ₂ are two random numbers between [0,1]. Therefore, to calculate the new position of the individuals used the following equation is employed:

Once the new position is computed, it is evaluated by a cost function. If the new solution is better that the last one, then it is replaced, otherwise it remains.

2.6 Cuckoo Search (CS)

The cuckoo search was proposed in 2009 by Deb and Yang [³¹]. This technique emulates the parasite behavior of cuckoo birds through the use of the Lévy flights [³⁷]. CS uses a population of Eke1k,e2k,…,eNk individuals (eggs) in a determined number of generations (N = gen), where each individual eiki=1,2,…,N represent a m-dimensional vector ei,1k,ei,2k,…,ei.mk. To improve the exploration of the search space, CS includes the Lévy flights which perturb each individual with a position c _i using a random step s _i generated with a symmetric distribution computed as follows:

where uu1,u2,…,un and vv1,v2,…,vn are n-dimensional vectors and β = 3/2. The elements of u and v are calculated by the following normal distribution:

where Γ(•) is the Gamma distribution. Once s _i is calculated, c _i is computed as follows:

where ⊕ is the element-wise multiplications. Ones are obtained, the new candidate solution is estimated as bellow:

2.7 Differential Search Algorithm (DSA)

Differential search algorithm [³²] imitates the Brownian-like random-walk movement used by an organism to migrate. In the DSA a population Xkx1k,x2k,…,xNk of individuals (artificial organisms) is initialized randomly through the search space. After the initialization, a stopover vector is generated described by the Brownian-like random-walks for each element of the population; this stopover is described below:

where ri ∈ [1, NP] is a random integer within the population range and ri ≠ i. A is a scale factor that regulate the position changes of the individuals. For the search process, the stopover position is determined as below:

where j = [1,…, d] and r _i,j can take the value of 0 or 1. After the selection of the candidate solution, each individual is evaluated by a cost function f(•) to determinate their quality, then a criterion of selection is used which is described as follows:

2.8 Crow Search Algorithm (CSA)

The crow search algorithm was originally proposed by Askarzadeh [³³], based on the intelligent behavior of crows. CSA uses a population of ckc1k,c2k,…,cNkN individuals (crows), where each individual represents a m-dimensional vector ci,1k,ci,2k,…,ci,mk. The search strategy of CSA can be summarized in two steps. In the first one, it is when a crow is aware that is being followed for another crow while the second is when is not aware. The states of each crow are determined by a probability factor APik. Therefore, the new candidate solution is computed as follows:

where r _i and r _j are random numbers between [0,1], fl is a parameter that controls the flight length. mjk is the memory of the crow j which stores the best solution at iteration k.

2.9 Covariant Matrix Adaptation with Evolution Strategy (CMA-ES)

CMA-ES [^35,36] is an evolutionary algorithm proposed by Hansen based on the estimation of the covariant matrix on the search data. The CMA-ES uses a population of Xkx1k,x2k,…,xNkN individuals, which are randomly initialized. In the each generation, λ individuals are selected to be updated using the following equation:

where N(μ, C) is a normally distributed random vector with a mean μ and a covariance matrix C. The next weighted mean xwk selected as the best interval is computed as follows:

where ∑i=1μwi=1. To carry out the modification in the parameters, the CMA-ES uses two different adaptations, on the covariance matrix C ^k and on the global step size σ ^k . For the covariance matrix adaptation case, an evolution path Pck+1 is used, which depends on the parent's separation with xwk and the recombination points xwk+1 as is shown below:

where μeff=∑i=1μwi2/∑i=1μwi is the effective variance selection and ccov≈min1,2μeff/n2 is the learning rate. For the global step size adaptation a parallel path is used to modify σ ^k , this process is described below:

where B ^k is an orthogonal matrix and D ^k is a diagonal matrix. The adaptation of global step size for the next generation is given by the following equation:

where E∥N(0,1)∥=2Γn+1/2/Γn/2≈n1-1/4n+1/21n2 is the length of p _σ

3.1 Single Diode Model (SDM)

The single diode model is the basic and most used model for the representation of the solar cell behavior. This model uses a diode connected in parallel with the source of current. Fig. 1 presents a representation of this model. Considering the circuit theory, the total current of single diode model is calculated as follows:

where k is the Boltzmann constant, q is the electron charge, I _SD is the diffusion current, V _cell is the terminal voltage, R _p and R _S are the parallel and serial resistances. For the single diode model, the parameters that determinate its performance is given by five parameters; R _S , R _p , I _L , I _SD and n.

3.2 Double Diode Model (DDM)

The double diode model is another alternative to characterize the solar cell behavior. Under this circuit, instead of using only one diode, it involves two diodes in a parallel array as is shown in Fig. 2. Therefore, the total current of this model is calculated as follows:

where the diodes and leakage currents are calculated as follows:

In the double diode model, the elements that must be determined are R _s , R _p , I _L , I _SD1 , I _SD2 , n ₁ and n ₂ .

3.3 Three Diode Model (TDM)

Recent advances in solar cell systems have conducted to the development of more accurate models than those produced by the electronic circuits of one or two diodes. One example of these new models is the configuration of three diodes proposed in [⁷]. The three diode model is a representation of the solar cell models which include a third diode in parallel with the original two diodes. The model considers the effects of a new current ID ₃ and factor n ₃ that allow improving the modeling accuracy. Similarly to the two diode model, the total current is calculated as:

In the case of three diode model, the parameter R _S is replaced by R _SO (1+KI) to find the variation of R _S with I _cell . Where I _cell is the load current and K is a parameter that must be determined as the other parameters, for this, the parameters to be tuned are R _SO , R _p , I _L , I _D1 , I _D2 , I _D3 , n ₁, n ₂, n ₃ and K.

The solar cells can be configured as modules [^42,43], which are an array of individual solar cells connected in serial and parallel. When the cells are connected to serial the voltages increases N _S times, in the case of cells connected in parallel only the current components increases in N _P times. So that, the output of a module of N _s × N _p cells is computed as follows:

3.4 Solar Cells Parameter Identification as an Optimization Problem

The identification of solar cells can be faced as an optimization problem. Under this approach, the objective is the correct approximation of the I - V output between the true model and the equivalent circuit model. With this results, each optimization technique is evaluated by a cost function to determinate the quality of the approximation. For the identification process, the equations (23, 24) and (28) are rewritten to reflex the difference of the experimental data as follows:

For the three models x represents the parameters to be estimated, for the single diode model (SDM) x = [R _s , R _p , I _L , I _SD , n], in the double diode model (DDM) x = [R _s , R _p , I _L , I _SD1 , I _SD2 , n ₁, n ₂] and for the three diode model (TDM) x = [R _so , R _p , I _L , I _SD1 , I _SD2 , I _SD3 , n ₁, n _2, n _3. K]. In order to evaluate the quality of each candidate solution, the root means square error (RMSE) is considered:

During the optimization process, the parameters are adjusted to minimize the cost function until a stop criterion is reached. Since the data acquisition is in variant environmental conditions, the objective function presents noisy characteristics and multimodal properties [⁴⁴]. Under such circumstances, the estimation process is considered as a complex task [⁴⁵].

4.1 Identification of C60 Mono-Crystalline Solar Cell

The first experiment involves the parameter estimation of a solar cell C60 Mono-crystalline, considering four different sun irradiations: (1000 W/m²), (800 W/m² ), (500 W/m²) and (300 W/m²) at T = 25°C. In the estimation process, the SDM, as well the DDM models, are used. Table 2, 3, 4 and 5 present the results for the cases of (1000 W/m²), (800 W/m²), (500 W/m²) and (300 W/m²), respectively.

Experimental results from Tables 2, 3, 4 and 5 show that the CMA-ES presents the best possible performance; however, CSA and DSA maintain also very good index values. The rest of the algorithms produce results with different performance levels.

One exception, in the results, was the standard deviation in which the lower values have been reached by the CS in the single diode at (1000 W/m²), the DE in the double diode at (800 W/m²), the CS and DSA at (300 W/m²) for the single and double diode, respectively.

4.2 Identification of multi-crystalline solar cell D6P

The second experiment considers the parameter estimation of a multi-crystalline solar cell D6P. The identification examines two different solar irradiations (1000 W/m²) and (500 W/m²) at T = 25°C, for the Single Diode Module (SDM), double Diode Model (DDM) and the Three Diode Model (TDM). Table 6 and 7 present the results for the cases of (1000 W/m²) and (500 W/m²), respectively.

According to the Tables, the CMA-ES obtains the best results in comparison with the others techniques for the SDM, DDM and TDM models. In case of the standard deviation, the best results were obtained by the ABC in the single diode at (1000 W/m²), DE in the three-diode model at (1000 W/m²) and CS in the single and double diode at (500 W/m²).

4.3 Identification of Multi-Crystalline Module KC200GT

Finally, the third experiment analyzes the parameter estimation of a Multi-crystalline module KC200GT. The identification examines the Single Diode Model (SDM) and the Double Diode Model (DDM) for the conditions of (1000 W/m²), (800 W/m²), (600 W/m²), (400 W/m² ) and (200W/m²). Table 8, 9, 10, 11 and 12 present the results of each algorithm for the cases of (1000 W/m²), (800 W/m²), (600 W/m² ), (400 W/m²) and (200W/m²), respectively. Tables 8 and 9 demonstrate that CMA-ES shows competitive results finding the minimum RMSE value in the most of the cases; however, the CS and DE are also able to find a good solution with considerable precision. In the experiments, the CMA-ES obtains the best result for almost all cases, except the standard deviation in several cases such as CS in the single diode model at (800 W/m²) and (600 W/m²). On the other hand, the DSA obtain the lower standard deviation for the single and double diode model at (400 W/m² and (200 W/m²).