Research on MPPT control strategy based on CCAOA algorithm

Photovoltaic (PV) arrays under partial shading conditions (PSC) can lead to multiple peaks in the power-voltage curve of PV system output. The traditional maximum power point tracking (MPPT) algorithm is difficult to solve the multi-peak problem and generally has slow convergence speed and easy fall into local optimality. To address this problem, a collaborative and cosine arithmetic optimization algorithm (CCAOA) was proposed in this paper. The cosine factor was introduced into the mathematical optimization acceleration function in traditional AOA to enhance the global search capability of the algorithm. And the circle chaotic mapping and cross-variance strategy were introduced to increase the diversity and randomness of the algorithm population. Meanwhile, a cooperative search strategy of addition and subtraction is used to strengthen the local search capability of the algorithm, thus accelerate the convergence speed of the algorithm. The effectiveness of the CCAOA is evaluated by using six typical IEEE standard test functions, and the simulation results show that compared with AOA, TSO and PSO algorithms it outperforms other algorithms in terms of convergence speed and accuracy. Appling the CCAOA into the MPPT control, the performance of MPPT control strategy based on CCAOA was verified by simulation. The simulation results illustrate that the CCAOA has better performance in tracking speed, stability and efficiency when comparing with AOA, TSO and PSO algorithms. In conclusion, the MPPT control based on CCAOA can significantly improve the power generation efficiency of PV arrays under PSC.


Introduction
With environmental pollution, depletion of fossil fuel and energy shortage, the exploration of new renewable clean energy has become a research hotspot worldwide.Solar energy, as a renewable and clean energy source, has been paid more and more attention [1,2].However, photovoltaic (PV) systems output is easily affected by the environment (i.e., irradiance and temperature), and especially the output curve of PV arrays under shading conditions exhibits multiple peaks.This makes it very difficult to track the global maximum power point (GMPP) of the PV array [3][4][5].Therefore, it is necessary to use MPPT control algorithm to track the maximum power point of the PV system in order to ensure its maximum output power.
The traditional algorithms (i.e., perturb & observe (P&O) and incremental conductance (IC)) are suitable for tracking the maximum power point under uniform irradiation conditions.Under local shading conditions, these algorithms can easily fall into local maximum power points (LMPP) and unable to solve the GMPPT problem effectively, and it also easily generates large steady-state oscillations, resulting in performance degradation [6][7][8].
To solve those problems, many researchers have proposed many different meta-heuristic algorithms to improve GMPPT performance [9][10][11].The authors (G.Dileep. et al. 2021) proposed an improved particle swarm algorithm (PSO).Applying the algorithm into the MPPT control system, the convergence speed and tracking accuracy of the algorithm were improved by both linearly decreasing the inertia weight and cognitive coefficient, and linearly increasing the social coefficient [12].The authors (AboKhalil et al. 2021) proposed an opposition-learning firefly algorithm (FA) and applied it to the MPPT system.
By adjusting the parameters of the firefly algorithm and using an opposition-based learning (OBL) method it can effectively maintain the diversity of the population and track the GMPP better [13].The authors (AlShammaa et al. 2022) used the cuckoo search algorithm (CSO), which has faster convergence speed and higher precision than the conventional algorithm [14].The authors (Fu Changxin. et al. 2022) applied an improved slime mold algorithm (SMA), which selected individuals with better fitness values by linear decreasing, and introduced a spiral search strategy to expand the search range and improve the global search performance of the algorithm [15].Arithmetic optimization algorithm (AOA) is a metaheuristic algorithm proposed by Abualigah et al. in 2021 based on the idea of arithmetic operation notation in mathematics, which has the characteristics of simple formula structure and fast convergence speed [16][17][18][19][20].
However, being used in the MPPT control system, the algorithm is easy to fall into LMPP and too much oscillation during the search process.To resolve these problems, some researchers put forward a few solutions [21][22][23][24].The authors (Chtita Smail et al. 2023) proposed an MPPT control system using the AOA, the authors proposed an MPPT control system using the AOA, which has higher accuracy and faster convergence speed than the traditional algorithm, but the algorithm is easy tends to fall into LMPP [25].The authors (Thota Rajasekar. et al. 2022) proposed an AOA based on the Levy flight strategy (AOA-LF).The algorithm utilizes the stochastic variable step size of the Levy flight strategy so that the algorithm does not easily fall into LMPP and improves the tracking accuracy of the algorithm [26].
To further improve the performance of the MPPT control system based on AOA, the improved AOA called Collaborative and Cosine AOA(CCAOA) was proposed in this paper.A cosine factor was introduced to avoid the algorithm falling into LMPP.And the circle chaotic mapping and cross-variance strategies were introduced to increase global search ability of the algorithm.Meanwhile, the cooperative strategy was proposed to improve search accuracy and increase the convergence speed of the algorithm.Then the superiority of the algorithm was verified by using IEEE standard functions.Finally, the MPPT control system simulation based on CCAOA was made, and algorithm performance comparison among PSO, TSO, AOA and CCAOA was done.The results show that CCAOA has faster convergence speed and tracking accuracy, and is not easy to fall into LMPP, thus decrease the power loss of the PV power generator system.The expression for the diode current I d is as follows:

Photovoltaic Cell Model
Where I 0 is the diode saturation current; q is the unit charge; K is the Boltzmann constant; T is the Kelvin temperature; η is the ideal factor of the dipole, generally taking the value of 1~2, and the ideal diode factor is 1.
The diode saturation current I 0 is expressed as follows: Where I RS is the reverse saturation current of the diode; E g is the bandgap energy of the diode; T r is the ambient temperature reference value, which takes the value as 25°.
The photogenerated current I ph is influenced by the solar irradiance G and ambient temperature T. The expression for I ph is as follows: ( ) Where I scr is the short-circuit current; K i is the ambient temperature influence factor.
A PV cell array usually consists of several PV cells connected in series and parallel.Assuming that the PV array consists of M * N PV cells, where M cells are connected in series to form a PV battery string, and such N battery strings are connected in parallel, the PV array output current I sm is expressed as follows: ( )

Output characteristic curve of PV array under partial shading
In engineering applications, as some PV cells are in indifferent irradiation, for example caused by shading, they easily become loads and consume energy, resulting in high PV cell temperature and the "hot spot effect".So that a bypass diode is usually connected to each PV cell in parallel to prevent the PV cells from being damaged.As shown in Fig. 2 The algorithm involves three specific phases: initialization, exploration, and exploitation.

Initialization
During the initialization phase, the AOA randomly generates N initial candidate solutions in the search space, each of which converges towards the global optimal solution using mathematical models based on arithmetic operators.
Then before performing search, the algorithm determines whether to perform global exploration or local exploitation by comparing the size of r 1 and the value of math optimizer accelerated (MOA) function.If r 1 >MOA (r 1 is a random number within [0,1]), the exploration phase begins.Otherwise, the exploitation phase starts.The MOA function is given by Eq.6: Where C_Iter is current iteration number, M_Iter is the maximum number of iterations, Min, Max are the minimum and maximum values of acceleration function, and Min value is 0.2, Max value is 1, accordingly.

Exploration
When the exploration phase starts, the D and M operators perform exploration due to their highly decentralized nature, which can explore the search field extensively, thus avoid local optimal solutions.If r 2 < 0.5 (r 2 is a random number within [0,1]), the D is executed.
Otherwise, the M is carried out.The position updated is expressed by Eq.7: ( ) Where  is a small integer number, best(x) is the position of the current optimal individual x, and UB and LB are the upper and lower search limits respectively.µ is a control parameter to adjust the search process and its value is 0.5.
Math Optimizer Probability (MOP) is a probability coefficient, and its value is calculated by Eq.8 as follows:  ( ) ) Fig. 3 illustrates the search mechanism of AOA to find the global optimal solution in a two-dimensional search space.As shown in Fig. 3, the D, M, S, and A operators are used in the search space to update the locations of candidate solutions, aiming for obtaining the global optimal solution.

CCAOA
In GMPP tracking, the conventional AOA has some limitations that lead to poor exploration.This is related to three factors: (i) In the actual search process, the choice of global exploration and local exploitation varies nonlinearly, while the MOA function is a linearly varying curve, so it leads to poorer search results.(ii) control parameter µ and boundary conditions are all constants throughout the iterations (i.e., more than half of the variables in the equations are constants, resulting in lack of randomness), so it is easy to diverge from the global optimal region.(iii) During local exploitation, the algorithm does not effectively utilize the A and S operators, thus reducing the individual search capability and leading to poor accuracy.
To solve these problems, an improvement AOA called collaborative and cosine AOA (CCAOA) was proposed.In CCAOA, MOA is amended, control parameter µ is modified, crossover variation strategy is used in global exploration, and collaborative exploitation of the A and S operators is applied in local exploitation.According to Eq.10 and Eq.6, the curve of AMOA function and MOA function can be drawn as Fig. 4 shows.

Control parameter µ modification
During the iteration process, the position update of the candidate solutions is related to the constant control parameter µ.In the original AOA, µ takes the value of 0.5, which is obviously not beneficial to the diversity of population position.In this work, the chaotic circle mapping is introduced into µ to increase the diversity of population.The modified mathematical model of µ is given by Eq.11.

Crossover variation strategy
In global exploration, if the change in population position satisfies the Eq.12 and Eq.13.Then it indicates that the individual position updated exceeds the boundary range of the search, which not satisfying the algorithm's requirements.
( ) Where UB is the upper limit of the search region, LB is the lower limit of the search region, k is a control threshold, which is taken as 0.05 in this work.
To solve the problem, the cross-variance strategy is introduced to update the individual position by exchanging information between the optimal and random individuals.
The mathematical model of update the position is given by Eq.14.
( _ Where Q rand is a random number in [0,1], p is a random number within [-1,1] for the coefficient of variation, t is a random number within [0,1] for the crossover coefficient.
This strategy increases the randomness of the population and improves the algorithm exploration capability.

Collaborative exploitation of the A and S
In exploitation, the AOA chooses the execution of the ( ) Where sign(x) is the sign function, f(best(x)) is the fitness value of the current population optimal individual, f(x) is the fitness value of the current individual, best(x) is the current population optimal individual position, and x is the current individual position, µ is given by Eq.11.

Algorithm performance verification
In order to verify the superiority of CCAOA, six IEEE standard test functions are used to verify the algorithm performance, and performance comparison simulations ware done with AOA, PSO and TSO respectively.The test function expressions are shown in Table 1.In the single-peak test functions, the optimization effect of AOA is better than that of TSO, and in the multi-peak test functions, the optimization effects of AOA and TSO are similar.But the overall convergence speed is relatively slow.And the CCAOA has the best optimization finding effect for both single-peak and multi-peak test functions, and it can find the optimal solution more quickly in most of the test functions.Therefore, CCAOA is more capable of finding the optimal solution than AOA, TSO and PSO, and the convergence speed is faster.

MPPT control system consists of PV arrays and
Boost circuit with a resistive load.The circuit schematic diagram of the MPPT control system is shown in Fig. 7.

Flow chart of MPPT based on CCAOA
When the weather, dust and other environmental factors lead to changes in irradiation and temperature, the GMPP of the PV array will also change.The MPPT control system should respond quickly in this situation and quickly restart tracking the new GMPP.The restart condition is given by Eq.16.
Where P pv is the current output power of the system, P m is the maximum power point power, β is the termination threshold, which is set to 0.05 in this work.
The procedure for the MPPT control based on CCAOA is as follows: Step-1: Initialize the CCAOA parameter information, individual position is duty cycle in MPPT systems.Where the initial parameters Max=1, Min=0, α =5.
Step-2: Calculate the fitness value at the current position using Eq.17.
pv pv pv P V I = Step-3: if P(C_iter)>P(best), update the fitness value of the optimal individual using Eq.18.
Step-5: If r 1 > MOA, CCAOA performs global exploration.Use Eq.7 to update the position, and if the position updated out during this exploration satisfies Eq.12 and Eq.13, the Eq.14 can be used to update the position again.
Else, execute the exploitation phase and update the position using Eq.15.
Step-6: If Eq.19 is satisfied, the optimal position is output and the search is stopped, Q is taken as 0.01.
Step-7: Repeat step 2 to step 7 until the end of the iteration.
Step-8: Check if the iteration reaches the maximum value.If yes, send the best position and terminate the search.
Step9: If Eq.16 is satisfied, the algorithm starts step 1 and restarts.
The MPPT control flow chart based on CCAOA is shown in Fig.In this work, the suggested four algorithms (CCAOA, AOA, PSO, and TSO) was applied to PV system, and their performances were evaluated using five distinct sets of irradiance patterns given in Table 2 and the temperature is set as 25°.Among them, G1 is uniform irradiation, G2 and G3 are static partial shading conditions, G4 and G5 are dynamic partial shading conditions (after 2s the irradiation changes).

Under static conditions
MPPT performance comparison simulation of four algorithms (CCAOA, AOA, PSO, and TSO) were conducted under three different static irradiance levels, labeled as G1-G3.The simulation results are presented in Fig. 9-Fig.11.As shown in Fig. 10, under G2 condition, TSO gets trapped in the LMPP.AOA and PSO take approximately 1.02s and 1.31s, respectively, to converge near the GMPP.
In contrast, CCAOA efficiently finds the GMPP in just 0.35s, achieving a tracking efficiency of 99.98%.
As shown in Fig. 11, under G3 condition, PSO fails to track the GMPP.AOA becomes trapped in the LMPP.
While both CCAOA and TSO successfully track the GMPP with values of 561.91W and 561.35W, respectively.
However, CCAOA exhibits a higher tracking efficiency of 99.98%.Furthermore, the tracking time for TSO is 0.71s, while CCAOA's tracking time is only 0.42s, which is faster in speed.

Under dynamic conditions
Under two different dynamic conditions, labeled as G4&G5, the MPPT performance comparison simulation results of four algorithms are shown in Fig. 12-Fig.13.    3.

Experimental setup
To verify the proposed CCAOA-based MPPT control method, the hardware experiment platform is constructed.
The platform includes STM32F334 controller chip, and ITECH's IT6005C-80-150 programmable PV simulator, the Boost main circuit and its control circuit which forms a board and a resistor used as the output load.The structure of the experimental system is shown in Fig. 14 This experiment is validated in three different environments and the results are shown in Fig. 15 , Fig. 16 and Fig. Future studies could be carried out in considering more accurate mathematical model of PV cell.

Fig. 1 Fig. 1
Fig.1 shows the mathematical model of a single PV cell.I ph is the photo current generated by the PV cell, I d is the current flowing through the diode, V DS is the voltage across the diode, and I p is the current through the parallel resistor R p .I is the output current of the cell and V is the output voltage.When the ambient temperature and sunlight irradiation are constant, the current I ph remains constant and can be regarded as a constant current source [Error!Reference source not found.].The expression for the output current I of a photovoltaic cell is as follows:

Fig. 2
Fig.2 a PV Array b P-V Characteristic Curves (a), a set of [4×1] PV arrays under partial shading conditions is used as the study object.The individual PV cell parameters are: maximum power P m =264.75W;open circuit voltage U oc =44.6V; short circuit current I sc =8.15A; maximum power point voltage U m =35.3V; maximum power point current I m =7.5A.When the PV array is in the shadow state, there are multiple peaks in the P-V characteristic curve of the PV array, as shown in Fig.2(b).Moreover, the location of the global peak is heavily dependent on factors such as the environment.The use of conventional MPPT algorithms leads to high power losses, so the use of meta-heuristic MPPT algorithms is one of the most efficient ways to solve the problem of multi-peaks of PV arrays.uses the four mathematics operations to solve math multipolar problem.These four operators are division (D), multiplication (M), subtraction (S), and addition (A).D and M are used for global exploration, while S and A are for local exploitation.

8 )
Where α represents the sensitivity parameter for iterative development accuracy, which is set to 5 in AOA.

Fig. 3
Fig.3 Position updating model in AOA3.1.3ExploitationWhen the exploration phase starts, the S and A operators perform exploitation phases owing to their highdensity nature, which can speed up the search of the population towards the target.If r 3 < 0.5 (r 3 is a random number within [0,1]), the position is updated by operator S.Else, A operator is responsible for updating the position.The mathematical model of position update is represented by Eq.9.

Where
MOA function value is used to randomly determine whether to execute global exploration or local exploitation.The larger the value of MOA function, the greater the likelihood of conducting a local search.While the smaller the value of MOA function, the greater the possibility of executing a global search.In the AOA, the curve of MOA function described in Eq.6 is a straight line that grows linearly from 0.2 to 1 with the number of iterations.It shows the probability of MOA function values over 0.5 is higher, and below 0.5 is lower, which decreases the likelihood of global exploration and increases the possibility of local exploitation, which cannot keep the balance of global exploration and local exploitation.In this work the MOA function was reconstructed into a nonlinear function by introducing a cosine factor.The amended MOA (AMOA) function is given by Eq.10.C_Iter is current iteration number, M_Iter is the maximum number of iterations, Min, Max are the minimum and maximum values of acceleration function, which are 0.2 and 1, respectively.

2 ,
and x 1 is a random number within [0,1].According to Eq.11, the distribution of the values of control parameter µ changing with iteration number is shown in Fig.5.

Fig. 5
Fig.5 Circle Chaos map diagram In Fig.5, it can be seen that the values of control parameter µ are distributed between 0 and 1, most of them are concentrated in the range of 0.4 to 0.6.Thus, the diversity of the population can be increased and the superiority of the original algorithm parameters can be inherited, which avoid falling into the local optimal solution.

A
or S operator by a random number r3.This strategy is randomized and does not retain information about previously excellent individuals.Therefore, in this paper, a sign function is used to guide the position update through comparing the information of the previous excellent individual and the current individual, which helps to enhance the convergence speed of the MPPT control algorithm.The update of the position is given by Eq.15.

Fig. 14 a
Fig.14 a Experimental system structure b The experimental setup (a), The experimental equipment used in this work is shown in Fig.14(b).

Fig. 15 aFig. 16 aTimeFig. 17 a 7 Conclusion
Fig.15 a P-V characteristic curve under uniform irradiation b Output curves based on uniform irradiation condition

Table 2
Photovoltaic irradiance patterns and GMPP values of different patterns

Table 3
Performance comparison of four algorithms in five cases.