2.1.

Double diode model of PV

In double diode model circuit of PV, there are a current source, two diodes and two resistances. The current source represents the photocurrent. One diode represents the diffusion phenomena, and the other diode represents the recombination loss in the depletion region. This model can give more accurate output characteristic of PV cell than single diode model in low irradiance and ambient temperature. Figure 1 shows the double diode circuit model of PV.

Figure 1.

Double diode model of PV.

From the circuit, the current equation can be obtained as:

In equation (1), I_d1 and I_d2 are currents of diode D1 and D2 respectively in Figure 1, they can be descried by diode current. So equation (1) can be written as:

where I_sd1 and I_sd2 are reverse saturation currents of two diodes respectively. m₁ and m₂ represent ideal factors of diodes. V_T is thermal voltage which can be described by electron charge q, Boltzmann constant k and cell temperature T as:

Equation (2) describes the output characteristic I-V of PV cell. If equation (2) is used to describe the characteristics output of PV, there are seven parameters to be determined. They are photocurrent I_pv, saturation currents I_sd1 and I_sd2 of diode D1 and D2, ideality factors m₁ and m₂ of diode D1 and D2, series resistance R_s and parallel resistance R_p.

2.2.

Double diode model parameters identification problem

From the description above, there are seven parameters to be identified which can be listed as vector:

Actual measurement voltage and current output of PV cell are used to determine the vector U. Figure 2 shows a set of voltage and current data of PV cell.

Figure 2.

Output characteristic of PV cell.

In order to determine seven parameters using intelligent algorithm, the initial values of seven parameters are estimated at first, and based on equation (2), the error between actual and estimated can be calculated. The objective is to minimize the error between the actual data and estimated data. Based on equation (2), the objective function can be constructed as equation (4).

In the evolution of intelligent optimization, every actual measurement I-V output and estimated vector U are substituted into equation (4) and the error will be calculated and evaluated. In order to obtain the optimal solution of parameters, the sum of square root error is selected as follows:

where n is the number of measurement pair of voltage and current. According to equations (4) and (5), it will guide the evolution to better solution.

3.1.

Overview of DE algorithm

Differential Evolution was proposed in 1997. It is a heuristic and parallel search algorithm which belongs to a swarm intelligence search algorithm^{13, 14}.

DE randomly generates the initial value and finds the optimal solution through difference calculation and iteration. For the problem of population NP and dimension D, it randomly generates initial population:

where i = 1,2,…NP represents the ith individual, j = 1,2,…D represents the jth dimension.

In each iteration of the algorithm, mutation, crossover and selection are performed.

In DE algorithm, there are five mutation strategies to generate mutation vector V_i,G+1. Among them, the DE/best/1 strategy is described as:

where F is the difference vector scale factor. X_best,G represents the best individual in generation G. X_r1,G and X_r2,G are two different vectors which are chosen from the population.

In DE algorithm, there are two crossover methods. Binomial crossover is described as:

where rand(0,1) is a random number between 0 and 1. It is independently generated for each i and j.

According to the fitness values, the better vector between X_i,G and trial vector U_i,G+1 will be selected. The selection operation is defined by:

At the end of each iteration, the evolution stop condition will be checked. If the condition is not satisfied, the next iteration will be carried out.

3.2.

Overview of fireworks algorithm

Fireworks algorithm was first published in 2010 which simulated the explosion of fireworks. By introducing random strategy and selection strategy, FWA controls the direction of fireworks explosion and achieves global optimization^15,16.

FWA randomly generates NP fireworks as an initial population and calculates their fitness value. It includes three steps in each iteration: explosion, mutation and selection.

Every individual in population explodes and generates new fireworks. The number of fireworks that the ith individual generates is S_i and the explosion amplitude is A_i. S_i and A_i can be described as equations (10) and (11) respectively.

In equations (10) and (11), S_total is the largest amount of fireworks produced. f (x_i) is the fitness value of the ith individual. A_max indicates the maximum explosion amplitude. Y_max and Y_min are the best fitness and the worst fitness respectively. ε is a minimum to avoid zero numerator or zero denominator.

FWA uses Gaussian mutation to produce new fireworks. It randomly selects an individual fireworks and makes Gaussian mutation. Gaussian mutation can be described as:

where g is a Gaussian random number which has mean 1 and variance 1.

NP fireworks will be selected from new fireworks which are produced through explosion and mutation. The individual with the best fitness value will be preserved and the remain NP-1 individuals will be selected with probability p(x_i) :

After the three steps mentioned above, the population of generation G+1 is determined.

3.3.

FWADE algorithm realization

DE is an effective algorithm in search space and has better global search ability. But the parameters of DE are difficult to set and the algorithm is easy to achieve local optimum. FWA is an algorithm with excellent performance and wide adaptability. But it is not enough to exploit the local space. In this paper, the two algorithms are combined to make full use of their advantages. FWADE is described as follow:

Step 1: Initial population

Produce initial population of the hybrid algorithm and the number is NP. Divide the initial population into two parts. One part is the initial population of DE which number is NP1. The other part is the initial population of FWA which number is NP2.

Step 2: Evolution

The evolution consists of DE evolution and FWA evolution.

(1) DE evolution.

Firstly, for the NP1 individuals of DE population, the mutation operation is carried out. The mutation strategy is equation (7) and the corresponding mutation vectors V_i,G+1 are generated. Then, the crossover operation is carried out to produce NP1 trial vectors U_i,G+1. The crossover strategy adopts binomial crossover which is described in equation (8). Finally, the better vector between the original population X_i,G and the trial vectors U_i,G+1 is selected through equation (9). Through the three operations, the individual X_i,G of population is renewed to X_i,G+1 and the population is renewed which has better fitness. So the NP1 individuals of the next generation are determined.

(2) FWA evolution.

Firstly, NP2 individuals of FWA population explode to generate new fireworks. The number of the ith individual and the explosion amplitude are shown in equations (10) and (11) respectively. Then the new fireworks are generated based on equation (12). Finally, in all new fireworks, the individual with best fitness is kept and the other NP2-1 individuals are selected with probability (13). So the NP2 fireworks of the next generation are determined.

Step 3: Mixing and distribution

Randomly mix NP1 individuals of DE and NP2 individuals of FWA that produce through step2 above to obtain the new NP population of hybrid algorithm. Then the population is randomly divided into DE population with NP1 individuals and FWA population with NP2 individuals. Finally, in each iteration, the stop condition of evolution will be check. If the stop condition is not satisfied, the step 2 will be carried on until the stop condition is satisfied.

The flow chat of FWADE is shown as Figure 3.

Figure 3.

Hybrid algorithm flow chat.