Global maximum power point tracking for solar power systems using the hybrid artificial fish swarm algorithm

1 Introduction

Renewable energy has been increasingly used to solve problems caused by the use of conventional energy, such as the greenhouse effect, price rise, and others.[1].Solar energy is an important type of renewable energy and maximum power point tracking (MPPT) techniques are key in photovoltaic (PV) power generation systems.The PV module P-V (power-voltage) output characteristics and I-V (current-voltage) output characteristics are nonlinear.Under uniform irradiance, there is only one maximum power point on the P-U output curve where the PV module can operate at maximum efficiency and produce maximum output power [2].However, when part of the PV array receives lower solar irradiance due to occlusion by objects such as clouds, trees, and buildings, the resultant condition is known as the non-uniform irradiance conditions (NUIC).Under NUIC, the output of the PV system is affected [3].

The three main factors affecting the maximum power output of a PV system are: irradiance, temperature, and load impedance [4].To ensure the continuous operation of a PV system at the maximum power point, many MPPT algorithms, such as Perturb and Observe (P&O) [5] and Incremental Conductance (INC) [6], have been proposed.Under uniform irradiance, P&O and INC show good tracking efficiency and speed.However, under NUIC, conventional MPPT techniques fail to track the global peak and instead converge into one of the local maximum power points, resulting in the considerable underutilization of PV power [7].Further, the irradiances on various parts of a PV array are different; these differences can cause hot spot effects and damaging PV modules.The bypass diode configuration can alleviate this problem, but it makes the P-U output curve multi-peak under NUIC [8].Manickam et al.demonstrate that under NUIC, the conventional MPPT algorithm may cause a decrease in the output power of a PV array by about 70% [9].Therefore, under NUIC, global maximum power point tracking (GMPPT) technology is crucial to increase the efficiency of the system by operating at GMPP.

To solve the problems of GMPPT under NUIC, an intelligent algorithm, such as the particle swarm optimization (PSO) [10], back propagation (BP) neural network [11], or cat swarm optimization (CSO) [12], is introduced into the GMPPT technology, and GMPPT is achieved using the global search capability of the intelligent algorithm.PSO has been proposed as suitable for use with GMPPT technology based on the flocking behavior of birds [13].In this technique, particles collectively solve a problem by sharing information to determine the best solution.The technique is limited in its implementation by the presence of random variables, and its requirement of several parameters to be defined for each system.Another GMPPT algorithm based on simulated annealing (SA) optimization [14] has been proposed recently.However, this algorithm results in a greater number of PV voltage variations during the search process and needs higher convergence time.A further drawback of these methods is that there is an oscillation around the maximum power point with their use, which leads to extend convergence times during the tracking process.

In this study, the intelligent artificial fish swarm algorithm (AFSA) was introduced into GMPPT technology, to develop a hybrid AFSA (HAFSA) by: (1) using PSO to reformulate AFSA, (2) extending the memory, and communication behaviors of PSO into AFSA, and (3) improving the principal parameters of AFSA, so that value changes can be adapted to suit the parameter requirements at different search stages.The remainder of the paper is arranged as follows.

In the second section, analysis of the establishment of a PV system under uniform irradiance is presented.The third section describes the PSO algorithm and AFSA.The fourth section proposes a new intelligent algorithm and applies it to the GMPPT of a PV system.In the fifth section, the simulation results are presented along with the comparison of the performances of PSO, AFSA, and HAFSA in GMPPT.In the sixth section, the conclusions with respect to the proposed GMPPT algorithm are given.

2 Modeling of PV cells under uniform irradiance

PV cells are made according to the principle of the photovoltaic effect, and the PN junction is the core of a PV cell’s working principle.The external characteristic model of a PV cell can be seen as a parallel circuit consisting of a constant current source and parallel diode.In general, the equivalent circuit model of a PV cell has two forms: singlediode and two-diode.The single-diode model performs well in terms of complexity and accuracy.Its equivalent circuit with series and parallel resistances is shown in Fig.1.

where Iph is the PV current, Id is the current of the parallel diode, Ish is the shunt current, I is the output current of the PV cell, U is the output voltage of the PV cell, Rs is the equivalent series resistance inside the PV cell which is generally less than 1 Ω, and Rsh is the equivalent shunt resistance inside the PV cell which is generally several thousand Ω.

According to the equivalent circuit shown in Fig.1 and the characteristics of a semiconductor’s PN junction, five parameters(Iph, Id, Rs, Rsh, n) are required to establish a PV cell model.The relationship between the output current and the voltage of a PV cell is described as:

where I0 is the reverse saturation current of PV cell, q is theamount of electric charge (1.6 × 1 0- 19C) contained by an electron, K is the Boltzmann constant (1.38 ×10- 23J / K), T is the temperature of the PV cell (K), and n is the ideality factor of the PV cell( 1 5)= -n .

When the irradiance is strong, the PV current is much greater thanTherefore, Equation (1) can be simplified as:

3 Particle swarm optimization and artificial fish swarm algorithm

3.1 Standard particle swarm optimization algorithm

The standard PSO algorithm is shown as follows:

where t is the index of iteration, is the speed of the particle in the tth iterative process, is the position of the particle in the tth iterative process, Pbest is the current local extreme value point of the particle in the tth iterative process, Gbest is the current global extreme value point of the population in the tth iterative process, ω is the inertia weight, C1 and C2 are the acceleration coefficients and can have values between 0.1 and 2,and rand (0,1).These parameters determine the contribution of social and cognitive factors to the final particle velocity for equation (3).

3.2 Artificial fish swarm algorithm

The artificial fish swarm algorithm is a type of swarm intelligence algorithm based on animal behavior.it was put forward by Li Xiaolei et al.in 2002 [15].Its principle is the simulation of the foraging, clustering, and collision behaviors of fish and the mutual assistance in a fish swarm so as to realize a global optimal point.

The greatest distance traversed in the artificial fish algorithm is defined as the Step, the perceived distance traversed by the artificial fish is defined as the Visual, the retry number is the Try_Number and the factor of the crowd degree is η.The position of an individual artificial fish can be described as the resultant vector X =( X 1 , X 2,..., Xn), and the distance between artificial fish i and artificial fish j is d i j = .The behavioral functions for the artificial fish are defined as: prey, swarm, follow, and random.

(1) Prey

Consider that the fish perceive their food with their eyes, the current position is Xi, and a randomly selected position is Xj within their perception range.Then,where rand (0～1) is a random number between 0 and 1.If Yi＞Yj, then the fish move forward in this direction.Otherwise, the algorithm randomly chooses a new position Xj to judge whether it satisfies the movement conditions.If it does, then:

If it does not even after Try_Number times, then a random move is made as such:

(2) Swarm

To avoid overcrowding, an artificial current position Xi is set.Then the number of fish in its company nf and the center Xc in the area (namely dij＜Visual) are determined.If Yc /n f ＜ η× Yi, the location of the companions indicates a good amount of food and low crowd.Then the fish can move towards its companions’ area center location as such:

Otherwise it begins to carry out the behavior of a prey.

(3) Follow

The current position of an artificial fish swarm is defined as Xi.The swarm determines its biggest company Yj as Xj in the area (namely d i j＜ Visual ).If Y j /n f ＜ η× Yi, the location of this company indicates a good amount of food and less crowding.Then the swarm can move towards Xj as such:

(4) Random

The behavior of randomization allows artificial fish to obtain food and company over a larger area.A position is random selected, and artificial fish move towards it.

Within the search space of a dimension D, the maximum possible distance between two artificial fishes is used to dynamically limit the Visual and Step of the artificial fish.This is defined as MaxD:

where xmax and xmin are the upper and lower bounds of the optimization range respectively, and D is the dimension of the search space.

4 Hybrid Artificial Fish Swarm Algorithm and Its Application to GMPPT

The study modifies the AFSA in two aspects: (1) itintroduces several features such as the velocity inertia factor, memory factor, and communication factor of the PSO into the AFSA to reduce the blindness of the searching process; (2) it optimizes the principal parameters (ω,Visual, Step) of the algorithm such that the changes in values are adapted to the parameter requirements in different search stages and the accuracy of the optimal solution of the algorithm is improved.

4.1 Hybrid Artificial Fish Swarm Algorithm (HAFSA)

The AFSA has strong searching ability in the initial stage.However, in the later stage, its searching ability decreases, and it can easily fall into a local optimum, thus lowering the accuracy of the optimal solution.To improve the convergence speed and accuracy of the algorithm, the paper introduces several features such as the velocity inertia factor, memory factor, and communication factor of the PSO into the AFSA.The HAFSA algorithm makes the artificial fish move with the velocity inertia characteristic, and the behavior patterns of the artificial fish are divided into memory behavior and communication behavior.The HAFSA algorithm also reduces the blindness in the artificial fish searching process.

The study uses the formulation of the PSO to reformulate the AFSA.The introduction of velocity inertia weight can reduce the blindness of the artificial fish movement.Taking the update of swarm behavior as an example, if Yc /n f ＜ η× Yi, the update equations (11) and (12) are

The equations for follow and prey behavior of the AFSA are similar to that of swarm behavior, so no extra enumeration is required.

Equation (13) and (14) introduce the memory and communication factors of the PSO into the AFSA to add memory and communication behavior.The two new behaviors further reduce the blindness of artificial fish movement.Before utilizing the new behaviors, the artificial fish must determine if overcrowding conditions exist.

First, the algorithm introduces the memory behavior pattern.The memory behavior pattern is the optimal position that the artificial fish can refer to during motion.If Yp best /n f ＜ η× Yi, it indicates that the location of its company has a great amount of food and is not crowded.Then, artificial fish can move in this direction.Equation (13) is updated as follows:

where Xtpbest is the best location vector of the artificial fish at the tth iteration.

Second, the communication behavior pattern is the optimal position of the entire fish swarm that the artificial fish can refer to during motion.If Yg best /n f ＜ η× Yi, it indicates that the location of its company has a great amount of food and is not crowd.Then, artificial fish can move in this direction.Equation (14) is updated as follows:

where is the best location vector of all artificial fishes on the bulletin board for an iteration.

The above mentioned behaviors are mutually converted under different conditions, and the artificial fish selects a current optimal behavior by evaluating which behavior would lead to the location with a higher where the food concentration.

4.2 Optimization of Principal Parameters in HAFSA

Reasonable parameter settings can maximize the performance of the algorithm.The three most significant parameters affecting the HAFSA optimization performance are ω, Visual, and Step.The inertia weight ω keeps the artificial fish in motion inertia, giving it the tendency to expand the search space and the ability to explore new areas.During the early stage of the search, a greater inertia weight can improve the comprehension of the search space to determine the location of the optimal solution.In contrast, during the later stage of the search, when the algorithm is converging to the optimal solution, a smaller inertia weight can help to find the global optima more efficiently.Visual determines the perception and search range of the artificial fish.During the early stage of the search, a greater Visual increases the artificial fish’s perception range, and the probability of finding a higher food concentration area is also higher.Meanwhile, if using a greater Step, the artificial fish can be aggregated to the global optimal solution and the local optimal solution neighborhood quickly.During the later stage of the search, if Visual is large, the artificial fish cannot forage effectively, resulting in oscillation at the optimal solution.During the early stage of the search, using a larger Step to increase the range of artificial fish movement is conducive to the rapid convergence of the algorithm.However, during the later stage of the search, the artificial fish may cross the global optimal point which is not conducive to the convergence of the algorithm; During the early stage of the search, Step is small, which is beneficial for the accurate search of the algorithm, but if Step is too small, it will make the algorithm fall into local optimum.

Based on the above analysis, the parameters of the HAFSA are optimized as follows:

To meet the requirement that the fish swarm can run at high speed in the early stage and understand the search space effectively while accurately searching at low speed within the optimal solution neighborhood in the later stage, the paper proposes a new nonlinear decrement method based on the inertia weight ω with linear reduction.As shown in equation (15),

where t is the number of algorithm iterations, tmax is the maximum number of iterations, ωmin and ωmax are the upper and lower limits of the inertia weight range, respectively, and k is the order ( k = 1,2,3,4..., the value of k is selected according to the specific application of the algorithm).Fig.2 shows the relationship between inertia weight ω and the number of iterations when k takes different values.

The uniform random distribution coefficients Step and Visual of artificial fish contribute to the stable convergence of the algorithm and the improvement of the optimal solution accuracy.However the changes in Step and Visual are the same during the entire process of the algorithm, so there is no significant difference between them in the early and later stages.Therefore, to further improve the performance of the algorithm, this paper proposes an improved approach to meet the expectations of changes in Step and Visual.As shown in equations (16) and (17),

where, VISmin and VISmax are the upper and lower limits of the Visual respectively, and

where X is the current position of artificial fish, XE is the next position that artificial fish X explores in various behaviors, Y and YE are the fitness value corresponding to positions X and XE, respectively.

Fig.3 shows the relationship between Visual and the number of iterations.

4.3 HAFSA Applied to Global Maximum Power Point Tracking of PV System

Under NUIC, the P-U output curve of the PV system becomes multi-peak.To perform the simulations, a PV array was built with three series-connected PV modules, and the system configuration was tested under three different shade conditions (pattern1: G1=1000 W/m2, 1000 W/m2, 1000 W/m2; pattern2: G2=1000 W/m2, 600 W/m2, 600 W/m2; pattern3: G3=1000 W/m2, 800 W/m2, 400 W/m2), corresponding I-U and P-U curves are shown in Fig.4 and Fig.5, respectively.Under NUIC, the I-Ucurve of the PV array becomes multi-staircase while the P-U curve becomes multi-peak.The corresponding curves for pattern2 and pattern3 are shown in Fig.5.The PV output curve has three peaks, especially for pattern 3, which results in serious power loss under NUIC.

In this study, the proposed algorithm is applied to the GMPPT technology.The GMPP can be accurately tracked with high efficiency, and the output of the PV system is considerably improved.The study applies HAFSA to track the GMPP as follows:

(1) The principal parameters of the algorithm are listed in Table 1:

(2) Artificial fish individual: the output current of the PV system is used as the optimal value component X.

(3) The fitness function: two PV modules in series are selected as an example.Assuming that PV module1 is shadowed but the PV module2 is not shadowed, PV module2 will get stronger irradiance than PV module1.Therefore, I ph1 ＜ Iph2:

where nb is the diode influence factor, and I0b is the saturation leakage current of the bypass diode under standardized testing conditions.

(4) Algorithm restart strategy: when the external conditions change, the operating point of the PV system operating point changes accordingly; therefore, the PV system will not always operate at the maximum power point.The change in the irradiance conditions has a certain impact on the output voltage and current of the PV system.The algorithm judges whether the environment has changed by monitoring the voltage and current in two iterations.As shown in the following equations,

where t is the number of iterations, U and I are the output voltage and current of the PV system, respectively; they were set as 0.2 and 0.1 in the experiment [16].

5 Simulation result of HAFSA GMPPT algorithm

The simulation results are summarized in this section, and the performance of HAFSA is compared with that of the PSO and the AFSA under the same condition.

In the experiment, a PV array with three modules connected in series is taken as an example.Under the standardized conditions, G=1000 W/m2, T=25 ℃, the parameters of a single PV module are listed in Table 2.

When all modules in the PV array are under thecondition G=1000 W/m2 and T=25 ℃, the P-I output characteristic curve of the PV array is shown in Fig.6.When the ambient temperature is constant, and the irradiance of the three modules are 1000 W/m2, 800 W/m2, and 400 W/m2, the P-I output characteristics of the PV array are shown in Fig.7.

From Fig.6 and Fig.7, it can be seen that when G=1000 W/m2, the P-I output curve of the PV array has a single peak of Pmax =894.3010 W when I mpp= 8.23A.When G=1000 W/m2, 800 W/m2, 400 W/m2, the PV array output curve has three peaks, and the maximum power as well as the corresponding current change, the maximum power is Pmax=503.7598 W when Impp = 6.77 A.Under the second kind of irradiance condition, the GMPP of the PV array is tracked by the proposed HAFSA GMPPT algorithm.The tracking process is shown in Fig.8.

From Fig.8, it can be seen that the HAFSA GMPPT algorithm can accurately track the GMPP with high efficiency.After the 21st iteration of the algorithm, the result tends to be smooth.The GMPP is Pmax=503.7598 W when T=25 ℃.Therefore, when the PV array is in NUIC, the proposed algorithm can effectively track the GMPP and maintain the output state of the PV array to reduce energy loss.

When T=25 ℃ and G=1000 W/m2 , 800 W/m2, 400 W/m2, PSO, AFSA, and HAFSA are used for GMPPT.The tracking results are compared in Fig.9.

From the tracking process shown in Fig.9, it can be seen that the three algorithms can track the GMPP, and the proposed algorithm can track the GMPP with fewer iterations.In the early stage of tracking the GMPP, the AFSA algorithm converges to the GMPP neighborhood more quickly than the PSO algorithm.In the later stage of tracking the GMPP, the AFSA algorithm shows significant oscillation at the GMPP.From Fig.9, it can be seen that the HAFSA algorithm shows better convergence speed andstability than the other two algorithms when tracking the GMPP.

Fig.11 and Fig.12 show the population distribution of the three algorithms after the 10th and 30th iterations respectively.The black rectangle and blue rectangle regions in Fig.10 represent the range of population distribution after the 10th and 30th iterations respectively.From Fig.11, it can be seen that the PSO algorithm converges more slowly than AFSA algorithm and HAFSA algorithm after the 10th iteration.After the 30th iteration, the populations of the three algorithms are distributed in the optimal solution neighborhood, and the distribution of HAFSA population is closer to GMPP.Therefore, the proposed algorithm demonstrates better performance in terms of convergence speed and stability.

To further compare the performance of the proposed algorithm with other GMPPT algorithms, this study sets the same simulation environment, using PSO, AFSA and the proposed algorithm to track the GMPP 1000 times.The tracking results are summarized in Table 3.

From the statistical tracking results in Table 3, the average tracking time of the HAFSA algorithm is reduced by 34.74% and 21.26% compared with PSO and AFSA, respectively.The performance of the HAFSA algorithm in tracking GMPP with respect to tracking time, accuracy and number of successful tracking was better than other intelligent algorithms.

Fig.13 to Fig.15 shows the effect of the three parameters on the convergence time of the HAFSA algorithm.Each simulation controls the remaining parameters to maintain a constant classical value, and only the relevant parameter is changed mode.Tracking was performed fifty times for the GMPP of the PV array.The tracking results are summarized in Fig.13 to Fig.15.

The decreasing times of ω for constant, linear decrement, and nonlinear decrement averaging time are 0.434, 0.305 and 0.280 s, respectively.The convergence time of the nonlinear method for relative constant and linear decrement method was reduced by 35.48% and 8.19%.Similarly, the average time of Visual for constant and nonlinear decrement was 0.423 s and 0.380 s, respectively.The convergencetime of the nonlinear method in this study is 10.16%.The average time of Step for constant and nonlinear decrement was 0.425 s and 0.349 s, respectively.The convergence time of the nonlinear method in this study is 17.88%.The tracking process shows that the optimization of the key parameters of the HAFSA algorithm has a stronger ability to determine the GMPP region, and the convergence speed is improved.

6 Conclusions

A GMPPT algorithm for PV arrays under NUIC was proposed based on the study of the MPPT algorithm.Several factors of the PSO algorithm were introduced into the AFSA, which reduced the blindness of artificial fish movement.The foraging behavior of artificial fish swarm also expands two behaviors, memory and communication, which further reduces the blindness of artificial fish movement.Moreover, the study derives equations for the principal parameters of the proposed algorithm to adapt the changes on parameter requirements for different search stages.Under NUIC, the average convergence time of PSO, AFSA, and HAFSA algorithm after 1000 simulations were 0.4830, 0.4003, and 0.3152s.The HAFSA value is reduced by 34.74% and 21.26%, compared with the PSO and AFSA algorithm, respectively.The tracking process shows that the proposed algorithm has a stronger ability to determine the GMPP region, and the convergence speed and accuracy are improved.The optimization of key parameters of the algorithm also provided good results.The HAFSA algorithm is considered from the whole, and the tracking accuracy and speed of the algorithm are improved to some extent.Future work will consider the individual level, and reduce the number of iterations from the optimal solution area to simplify the algorithm and further improve the performance of the algorithm.

Acknowledgements

This work was supported by National Natural Science Foundation of China (No.61501106), Science and Technology Foundation of Jilin Province (No.20180101039JC and JJKH20170102KJ).