A flexible power point tracking algorithm based on adaptive lion swarm optimization for photovoltaic system

In PV power generation systems, flexible power point tracking control needs to be introduced to regulate the output power on the PV side for meeting the grid connection requirements. However, traditional FPPT control methods tend to cause certain oscillations in the system. This paper proposes an FPPT method based on the adaptive lion swarm optimization algorithm (ALSO). First, an ALSO algorithm is proposed, which adds adaptive scaling parameters to the lion swarm optimization algorithm, so that the number of adult lions can be adjusted adaptively with increasing iterations to coordinate the local exploitation and global search abilities of the algorithm. Then the ALSO is used to selectively optimize the operation of different intervals so as to achieve the FPPT control on the left or right side of the maximum power point. The test experiments for the ALSO algorithm show that the ALSO algorithm has strong local exploitation and global search abilities, which is better than the compared algorithms. Through the simulations on the PV grid-connected system, the validity and superiority of the ALSO-based method over the traditional FPPT algorithm in terms of dynamic and steady-state performance are verified under different operating conditions.


Introduction
Photovoltaic (PV) technology is an indispensable component of renewable energy technologies (Rana et al. 2022;Xie and Wu 2021;REN21 2018) and is an important means to alleviate the current energy crisis and environmental pollution. Due to the continuous improvement of PV technology, the cost of PV panels is gradually decreasing, meanwhile, the installed PV capacity is increasing year by year. However, with the increasing number of grid-connected photovoltaic power plants (GCPVPP) and photovoltaic power generation, too much grid-connected power during the peak generation period may cause some adverse effects on the power system, such as overvoltage or overload. Flexible power point tracking (FPPT) is a new method to address these issues in GCPVPP Tafti et al. 2018a;Gevorgian 2019;Chamana et al. 2018), which increasingly replaces the maximum power point tracking (MPPT) control strategy to provide more grid support functions.
According to recent references on FPPT, the existing FPPT algorithms fall into two main categories. The first type is the modification of the controller of the converter connected to the PV string to regulate the PV side power to its reference power (Mirhosseini et al. 2015;Tafti et al. 2017;Tafti et al. 2018b). This type of FPPT algorithms usually works by adjusting the reference voltage calculated by the MPPT in the ''PV Voltage Control'' block. The second type is the direct calculation of the reference voltage corresponding to its reference power (Tafti et al. 2018a;Sangwongwanich et al. 2017;Tafti et al. 2022). In these algorithms, the voltage value corresponding to the reference power is calculated using the FPPT algorithm, and the ''PV Voltage Control'' block is used to adjust the PV side voltage to its reference value given by the FPPT algorithm. Tafti et al. 2020 experimented with 11 algorithms of the two types. The experimental results show that the second class of algorithms outperforms the first class of algorithms in both dynamic and steady-state performance; meanwhile, these algorithms can switch MPPT or FPPT operating modes independently. Among the 11 algorithms, the general active power control method (GAPC) (Tafti et al. 2018c) works on the left or right side of the MPP in a dynamic environment better than the other 10 algorithms. In addition, it was also found that slight fluctuations in voltage can lead to relatively larger power oscillations due to the sensitivity of voltage to power variations on the right side of the MPP. As a result, the PV power of the above 11 algorithms operating on the right side of the MPP all deviated from their reference values by a relatively large margin. Therefore, how to improve the dynamic performance of the FPPT algorithm and how to improve the stability of GCPVPP remains a sticking point for researchers.
Intelligent optimization algorithms are effective methods for solving engineering problems such as PV generation and parameter identification. For solving complex engineering problems, many classical intelligent optimization algorithms have been developed . Earlier optimization algorithms include particle swarm optimization (PSO) algorithms, genetic algorithms (GA) and differential evolution (DE) algorithms (Eberhart and Kennedy 1995;Holland 1992;Storn and Price 1997), but these algorithms often suffer from slow convergence and tend to trap in a local optima. Mirjalili et al. (2014) presents a gray wolf optimizer (GWO). The algorithm achieves optimal search for objective functions by simulating the leadership rank and hunting mechanism of gray wolves in nature, which has strong ability of local exploitation and global search. Mirjalili et al. (2016) presents a multi-verse optimizer (MVO). The algorithm is mainly inspired by three concepts in cosmology: white holes, black holes and wormholes, and the global search and local exploitation of the algorithm are achieved by developing mathematical models of these three concepts. Mirjalili (2015a) presents a moth-flame optimization (MFO) algorithm, which is mainly inspired by the navigation method of moth in nature. Based on the MFO algorithm, a migration-based moth-flame optimization algorithm (M-MFO) is proposed in Nadimi-Shahraki et al. (2022), which improves the diversity of the population and enhances the global search ability of the algorithm by introducing a migration strategy. Liu et al. (2018) presents a lion swarm optimization (LSO) algorithm that searches for the optimal value of objective functions by simulating the habitat and foraging behavior of lions. Askarzadeh (2016) presents a crow search algorithm (CSA) by exploiting the social behavior of crows. Zamani et al. (2019) presents a conscious neighborhood-based crow search algorithm (CCSA), which enhances the balance between local exploitation and global search of the algorithm by introducing neighborhood-based local search, non-neighborhood-based global search and roaming-based search. Zamani et al. (2021) presents a quantum-based avian navigation optimizer algorithm (QANA). The algorithm uses self-adaptive quantum orientation and quantumbased navigation to efficiently explore the search space. Besides, the quantum mutation strategy is introduced to avoid the algorithm trapping into a local optimum, two long-term and short-term memories are introduced to provide meaningful knowledge for partial landscape analysis and a qubit-crossover operator to generate the next search agents.  presents a starling murmuration optimizer (SMO) by simulating the twisting, diving and rotating behavior of starlings to solve complex engineering optimization problems. These algorithms are widely used in various engineering fields (Wang and Liu 2019;Dey et al. 2022;Wu et al. 2017a).
Intelligent optimization algorithm is an important method to solve complex engineering problems. Since the traditional FPPT control method tends to cause large oscillations in the system, we try to achieve more stable FPPT control by using intelligent optimization algorithms. However, for the FPPT control of GCPVPP system, there is little literature applying intelligent algorithms to FPPT control. The reason is that there are two potential operating points (left or right side of the MPP) in FPPT mode. If an intelligent algorithm is applied directly to FPPT, the difference in the output each time will lead to severe oscillations in GCPVPP. Moreover, among the above algorithms, LSO algorithm is a high-performance optimization algorithm, which can be used to design the FPPT control strategy. However, in LSO algorithm, the update processes of lioness and lion cub in the algorithm are the local search mechanism and global search mechanism of the algorithm, respectively, they are executed with a fixed probability. While different optimization problems usually require different local and global search ability of the algorithm, it is difficult to guarantee better search performance by adjusting the parameters. Therefore, it is worthwhile to further improve the LSO algorithm, weighing the local exploration and global search abilities of the algorithm and achieving better FPPT control.
To address the above problems, a novel FPPT method based on adaptive lion swarm optimization (ALSO) is proposed in this paper. The main contributions can be summarized in the following three aspects.
(1) An adaptive proportion factor is introduced in the LSO algorithm. The number of adult lions accounts for a larger proportion in the early stage of the iteration and a smaller proportion in the final stage of the iteration, which can effectively improve the convergence speed and optimization accuracy of the algorithm.
(2) Since there are two potential operating points in the FPPT mode, if the intelligent algorithm is used directly for FPPT, the algorithm will produce different outputs each time, which will cause severe oscillations in the GCPVPP. For this reason, a new objective function has been established for the FPPT mode, which is used for searching the voltage value of the reference power point in a specific voltage range.
(3) An FPPT algorithm is proposed based on the ALSO algorithm and the designed objective function. The FPPT algorithm can ensure the smooth operation of the system on both sides of MPP, which compensates for the fact that the traditional FPPT algorithm cannot operate smoothly on the right side of MPP.
The rest of the paper is structured as follows. The details of LSO and ALSO algorithms are presented in Sect. 2. Section 3 describes the proposed ALSO-based FPPT algorithm in detail. The performance of the ALSO algorithm is tested compared with eight classical algorithms in Sect. 4. Section 5 experimentally compares the performance of the ALSO-based FPPT algorithm and the GAPC algorithm, providing a comprehensive comparison of the two FPPT algorithms. Finally, the conclusion of this paper is given in Sect. 6.

Adaptive lion swarm optimization
This section first describes an LSO algorithm, and proposes an adaptive lion swarm algorithm for designing a novel FPPT control method.

Lion swarm optimization (LSO)
The main principle of the LSO algorithm is described as follows . For all individuals in a population, the individual with the optimal fitness value is set as the ''lion king,'' then some individuals are selected as ''lionesses'' and the remaining individuals are set as ''cubs.'' In the optimization process, several lionesses hunt cooperatively. The cubs move around the adult lions. When the cubs grow up, they are expelled from the lion pack. After each search, the lion king reoccupies the position of the optimal fitness value.
Suppose there are N lions in the search space, the number of adult lions is N ad , and the number of cubs is N À N ad . The position of the ith lion is x i ¼ x i1 ; x i2 ; . . .; x iD , where 1 i N and D is the dimension of x i , and the adult lion with the optimal fitness value is set to ''lion king''; the rest are ''lionesses'' and ''cubs.'' The number of adult lions is where b is the proportion of adult lions in the population, de means rounding up to an integer.
In the algorithm, different species of lions move differently in the optimization process. The lion king moves in a small range to ensure its advantage and updates its position according to Eq. (2).
where c is a normally distributed random variable, k is the current number of iterations, g k is the best position of the kth generation, p k i is the best position of the ith individual of the kth generation, x kþ1 i is the updated position. The lioness needs to cooperate with another lioness to hunt and update the position according to Eq. (3).
where p k c is the position of another lioness randomly selected from the population; step ¼ 0:1 Á ub À lb ð Þrepresents the maximum step size of the cub; ub and lb are the upper and lower boundaries of the search space, respectively; T is the maximum number of iterations; k is the number of current iterations; a f is the disturbance factor for updating the position of lionesses.
There are three ways to update the position of the cub, as shown in Eq. (4).
where a c is the disturbance factor for updating the position of the cubs; g k ¼ lb þ ub À g k is the location where the cubs are driven out of the population; p k m is the position of the lioness followed by the lion cub. And q is a random number between 0 and 1. If q\ 1 3 , the cubs hunt near the lion king g k ; if 1 3 \q\ 2 3 , the cubs move around the lioness p k m ; and if q [ 2 3 , the cubs grow up.

Adaptive lion swarm optimization (ALSO)
In LSO, the behavior of lionesses is a ''global search'' mechanism and the behaviors of cubs are the ''local search'' mechanism and ''avoid trapping in a locally optimal state'' mechanism of the algorithm. Thus, the algorithm has a strong global search capability in the case of a large proportion of adult lions b; in contrast, the algorithm has a better ability to search finely and avoid trapping in the locally optimal solution. Hence, the setting of b has a great impact on the performance of the algorithm. However, a constant b cannot weigh the local exploitation and global search, and may weaken the optimization ability of the algorithm. In order to fully utilize the role of each mechanism and enhance the optimization ability of the algorithm, an ALSO algorithm is proposed.
ALSO algorithm is an improved algorithm of LSO algorithm. Based on the LSO algorithm, the constant b is modified as a parameter adaptively changing with the increase in the number of iterations, as shown in Eq. (5).
where b max and b min are the upper and lower limits of b.
As shown in Eq. (5), in the early iteration, b is larger, thus the ALSO algorithm has a strong global search ability and faster convergence speed. In the late iteration, b is smaller, then the ALSO algorithm can avoid trapping into a local optimum. Therefore, the introduction of Eq. (5) can well weigh the local exploitation and global search and improve the performance of the algorithm.
Combining the above description, the pseudo code of the ALSO algorithm is given in Table 1. In addition, the steps of the ALSO algorithm are summarized as follows.
Step 1 Initialize the population and the parameters of the algorithm.
Step 2 Calculate the number of adult lions according to Eq. (1) and Eq. (5), the historical optimal position of each lion is set to the current position, and the optimal position is set to the lion king.
Step 3 Calculate the position of the lion king using Eq. (2), the position of the lioness using Eq. (3), and the position of the cubs using Eq. (4).
Step 4 Calculate the fitness value of each lion, update the individual historical optimal and population historical optimal positions.
Step 5 If the maximum number of iterations is reached, turn to Step6; otherwise turn to Step2.
Step 6 Output the position of lion king, i.e., the current optimal solution.

Flexible power point tracking method based on ALSO algorithm
This section first introduces the GCPVPP system as well as FPPT control, and explains the shortcomings of current FPPT control methods. Besides, an FPPT control method based on ALSO algorithm is proposed for GCPVPP system.
3.1 Flexible power point tracking for GCPVPP system Figure 1 presents the structural diagram of the GCPVPP system (Fu et al. 2017). As shown in Fig. 1, the system consists of a PV string, a boost converter, a full-bridge inverter, an LCL filter, a grid, a ''PV side control'' block, and a ''grid-side control'' block. In this structure, the PV string is connected to the fullbridge inverter via a boost converter, and the ''PV-side control'' block adjusts the duty cycle of the boost converter according to reference power (P ref ) to control the PV side voltage. The ''grid side control'' block uses P/Q, current dual closed-loop control and SVPWM for pulse modulation. Grid support section provides frequency support and fault ride-though capability. Grid synchronization can be achieved by phase-locked loops, which can calculate the grid voltage angle to synchronize the inverter current with the grid. This paper focuses on the PV side control. For the GCPVPP system, during the peak generation period, excessive grid-connected electricity may cause some adverse effects on the power system, such as overvoltage or overload. To attenuate the adverse effects of grid-connected PV on the power system, it is essential to introduce FPPT and MPPT control strategies to ensure the smooth operation of the system. Figure 2 shows the operating points of the PV strings under different irradiance conditions.
As shown in Fig. 2a, when the maximum power of the PV string is less than the grid-connected reference power P ref , the system always operates at MPP 2 controlled by the MPPT algorithm. As the irradiance increases, the available power gradually increases. If the maximum power is greater than P ref , the operating point will move to the left or right side of the MPP, running at FPP L or FPP R . It is worth noting that, as shown in Fig. 2b, when the system is running at FPP L , the changes in voltage will cause small power oscillations. Conversely, when the system is running at FPP R , slight fluctuations in voltage will result in relatively larger power oscillations. As a result, most current FPPT algorithms makes the system more stable when operating at FPP L than at FPP R .

Proposed flexible power point tracking algorithm
Most of the current FPPT algorithms have certain oscillations in the steady-state process, and the system oscillations are more violent when the system is operating on the right side of the MPP. To improve the steady-state performance of the system, an ALSO-based FPPT control method is proposed.
Considering that there are two potential operating points in the FPPT operation mode as shown in Fig. 2, if the ALSO algorithm is used directly for the FPPT operation, the output of the algorithm will be different, which will cause severe oscillations in the GCPVPP. For the above reasons, it is necessary to establish a reasonable objective function to ensure the smooth directional operations of the system.
First, by comparing the reference power and maximum output power of the system, the operation mode is divided into two types: MPPT mode and FPPT mode. When the reference power is larger than the maximum output power, the system operates in MPPT mode; when the reference power is smaller than the maximum output power, the system operates in FPPT mode. The two operation modes are designed with different objective functions.
When the system operates in MPPT mode, the system always operates at MPP, so the objective function can be defined as: where and v oc is the open circuit voltage. I v ð Þ is the output current of the PV cell calculated according to the engineering model of PV cell Eq. (7) (Wu et al. 2020).
where v m , I m and I sc are the peak voltage, peak current and short-circuit current of the PV cell, respectively.
When the system operates in FPPT mode, the system will operate at FPP L or FPP R (as shown in Fig. 2), and F 1 (v) is no longer applicable to the FPPT mode. Therefore, the new objective function is designed as follows.
The function curve of f 2 (v) is shown in Fig. 3.
As shown in Fig. 3, the design principle of F 2 is that the part greater than P ref is folded down and 0; v oc ½ is divided into two parts, v 2 0; v MPP ½ and v 2 v MPP ; v oc ½ , which are the different intervals with operating on the left or right side of MPP.
So far, the design for the objective functions in both MPPT and FPPT modes is completed.
According to different operation modes, the optimal solution of the objective function Eq. (6) or Eq. (8), i.e., the voltage of the reference power, is searched using the ALSO algorithm. Then a voltage controller is used to regulate the system output voltage to the voltage, which enables FPPT control.
The detailed steps of the proposed FPPT algorithm are shown below.
Step 1 Acquire system signals: output voltage v, output current I, temperature T, irradiance S and reference power P ref .
Step 2 Using the ALSO algorithm with Eq. (6) as the objective function to search the maximum available power P MPP and its corresponding voltage value v MPP in v 2 0; v oc ½ .
Step 3   Step 4 If the system is operating on the left side of MPP, the system operates in MPPT mode, let v 2 0; v MPP ½ ; otherwise, the system operates in FPPT mode, let v 2 v MPP ; v oc ½ .
Step 5 Using the ALSO algorithm with Eq. (8) as the objective function to search in the voltage interval obtained from Step 4, finding the voltage v FPP of the reference power point.
Step 6 If the system operates in MPPT mode, the output voltage of the system is regulated to v MPP using the PI controller; conversely, the output voltage of the system is regulated to v FPP .
For clearly describe the proposed FPPT algorithm, Fig. 4 presents the schematic of the ALSO-based FPPT algorithm.
As shown in Figs. 3 and 4, the workflow of the system is as follows. First, the P ref , PV side current and PV side voltage are entered into the ''PV side control'' block, the ALSO algorithm is used for searching in the interval v 2 0; v oc ½ to obtain the maximum available power P MPP and its corresponding voltage value v MPP . The objective function is Eq. (6) and the detailed search procedure is shown in the algorithm steps in Sect. 2.2. Then, if the available power is less than P ref , v MPP will be considered as the reference voltage for controlling the operation of the system at that point. If the available power is greater than P ref , then ALSO will be used to optimize the output voltage of , outputting the optimal voltage v FPP ð Þ, and v FPP will be considered as the reference voltage for controlling the operation of the system at that point.

Experimental evaluation of ALSO algorithm
In this section, the performance of the ALSO algorithm is evaluated by multiple experiments. Eight functions in the benchmark function CEC2017 (Wu et al. 2017b) are selected for experimental analysis as shown in Table 2, where f 1 -f 4 are unimodal functions, f 5 -f 8 are multimodal functions, and f min in the table indicates the optimal solution. In all experiments, the performance of the ALSO algorithm is compared with eight classical optimization algorithms: PSO (Eberhart and Kennedy 1995), GWO (Mirjalili et al. 2014 All experiments are performed on a personal computer with the following performance: Intel(R) Core(TM) i7-10510U CPU, 2.30 GHz, 16 GB RAM, Windows 7, 64-bit operating system. All algorithms are programmed using MATLAB 2019b.
Next, the experimental results of each algorithm will be analyzed and compared in four aspects: quantitative analysis, qualitative analysis, nonparametric statistical test analysis and validity analysis. Table 3 shows the experimental results of each algorithm in three dimensions, in which F denotes the function, D denotes the dimension, Avg denotes the average value, SD denotes the standard deviation, Min denotes the minimum value, W denotes win, T denotes tie, and L denotes worse than other algorithms.

Quantitative analysis
f 1 -f 4 are unimodal functions, which can be used to test the local development of the algorithm. f 5 -f 8 are multimodal functions, and there are multiple local optima, which can be used to test the global search ability of the algorithm. As presented in Table 3, compared with other algorithms, the ALSO algorithm achieves the best results on all eight test functions under all three dimensions, except for the result of f 2 with D = 30, which is slightly larger than that of the GWO algorithm. In particular, for the multimodal function f 8 , only the ALSO algorithm obtains the global optimum, while all other algorithms traps in the local optimum.
The main reason is that, in ALSO, the collaborative hunting between lionesses, a local search mechanism, can efficiently use the information of each individual's location to move closer to the optimal value. The process of lion cubs being driven away from the population is the global search mechanism of the algorithm, which prevents the algorithm from trapping in local optima. Moreover, the adaptive mechanism (Eq. 5) allows the ALSO algorithm to have a larger proportion of adult lions in the early iterations, which accelerates the local exploitation, and the global search ability of the algorithm can be increased by increasing the proportion of lion cubs in the late iteration.
This provides the ALSO algorithm with a strong local development and global exploration capability.

Qualitative analysis
In this section, the convergence speed of the ALSO algorithm is evaluated and analyzed compared with other algorithms. Figure 5 presents the convergence curves of the minimum fitness values obtained for the nine algorithms in three dimensions. Fig. 4 The schematic of the FPPT algorithm based on ALSO  -5.12, 5.12] 0  As shown in Fig. 5, the ALSO algorithm has faster convergence speed on most of the functions at the beginning of the iteration. In particular, the ALSO algorithm obtains the optimal fitness on f 1 , f 3 , f 5 and f 7 in the middle of the iterations. In addition, compared with the convergence curves in case of D = 30, the eight compared algorithms with D = 100 are obviously affected by the increasing dimension, and their convergence speed are significantly slower than that of the ALSO algorithm. While the ALSO algorithm still has a faster convergence speed and higher convergence accuracy. The reason is that the ALSO algorithm can coordinate global search and local exploitation well through the adaptive mechanism.

Nonparametric statistical test analysis
This subsection presents a numerical comparison of the nine algorithms analyzed by five statistical analyses: Wilcoxon signed-rank sum, Friedman test, analysis of variance test, root mean square error and mean absolute error.

Wilcoxon signed-rank test
Wilcoxon signed-rank is a powerful statistical test in nonparametric statistical tests that shows the statistically significant differences in the results of each algorithm by analyzing the magnitude of the differences between the obtained results. In this test, p-values (a ¼ 0:05) with 95% significance level are calculated. Table 4 presents the results of the Wilcoxon signed-rank test for the ALSO algorithm and other algorithms for unimodal and multimodal functions in dimensions 30, 50 and100. As shown in the table, the symbol ' [ ' indicates that the ALSO algorithm is significantly better than the compared algorithm, and the overall results prove the effectiveness of the ALSO algorithm.

Friedman test
In this subsection, the Friedman test is used to rank the differences between the ALSO algorithm and the compared algorithms. The Friedman test first finds the rank of each algorithm and then calculates the average rank of the considered problem by using Eq. (9).
where n denotes the number of functions, m denotes the number of algorithms, and R j denotes the average rank of the jth algorithm. Table 5 shows the results of the Friedman test for each algorithm, which shows that the ALSO algorithm The bold values indicate that the value is better than that of other compared algorithms A flexible power point tracking algorithm based on adaptive lion swarm optimization for photovoltaic… 4963 outperforms the other eight compared algorithms in all dimensions.

The analysis of variance test (ANOVA)
In this subsection, the ANOVA test is performed on the data obtained from 20 runs of the nine algorithms, and the analysis results are presented in Fig. 6. As can be observed from the figure, the ALSO algorithm can reach its theoretical optimal value for different benchmark functions without trapping in the local optimum, and the range of the obtained objective values is relatively concentrated with better stability, which indicates that the ALSO algorithm is better than the eight compared algorithms.

Root mean square error (RMSE) and mean absolute error (MAE)
In this subsection, the RMSE and MAE of the results obtained from the nine algorithms for 20 runs are statistically analyzed to determine the difference between the optimal solution obtained by each algorithm and the true optimal solution. The RMSE and MAE are calculated using Eqs. (10) and (11), respectively, and the results are presented in Table 6.
where f oj denotes the obtained optimal solution of the jth function and f min j denotes the true optimal solution of the jth function. As shown in Table 6, the ALSO algorithm outperforms the other eight algorithms in all cases, except for that the RMSE and MAE of the ALSO algorithm for unimodal functions with D = 30 are slightly larger than those of the GWO algorithm.

Validity analysis
In this subsection, the overall effectiveness OE (Nadimi-Shahraki et al. 2020) of each algorithm is calculated using Eq. (12), and the results are shown in Table 7.
where L i denotes the number of test functions for which the ith function is a loser. As evident from the table, 95.83% of the tested functions with ALSO algorithm can find the best solution, which proves the effectiveness and superiority of ALSO algorithm.

Simulation results
In order to evaluate the effectiveness and accuracy of the proposed FPPT algorithm, the GCPVPP system is modeled and simulated using Matlab/Simulink, as shown in Fig. 1, and the parameters of the system are listed in Table 8. The performance of the algorithm is compared with the GAPC reported in Tafti et al. 2018c, and the parameter values of the two algorithms are shown in Table 9.
In order to accurately evaluate the response performance of the system, the cumulative tracking error (TE) is introduced to measure the steady-state and dynamic performance of the two algorithms in FPPT mode, which is defined as follows: where TE ss is the steady-state tracking error and TE tr is the transient tracking error. The simulations are performed under two conditions, a rapid change in irradiance and a change in reference power. In addition, both algorithms were simulated in the left and right operating regions.

Fast change of the irradiance
In this section, the irradiance varies with time, as shown in Fig. 7. Three different reference values are used for the simulation of the two algorithms, i.e., 80% (Case 1, P ref-= 400kw), 50% (Case 2, P ref = 250kw), and 20% (Case 3, P ref = 100kw) of the MPP of the PV strings, respectively.
The response performances of the two algorithms is shown in Figs. 8 and 9. In Figs. 8 and 9, P avai is the available power value; P ref is the reference power value; P pv_1 , P pv_2 and P pv_3 are the PV side power of Case1, Case2 and Case3, respectively, and Irr is the irradiance value. The response performance of the two algorithms when running on the left side of the MPP is shown in Fig. 8. It can be seen from Fig. 8a 1 that ALSO can adjust the PV side power to P ref under three cases when running on the left side of MPP. Both the dynamic and steady-state processes operate smoothly. The tracking error is small in all cases, TE 1 = 0.26%, TE 2 = 0.39% and TE 3 = 2.62%, respectively. Figure 8a 2 shows the operating points of the PV strings in the three cases, and the ALSO can approach the P ref steadily and accurately. Figure 8b 1 and b 2 shows the performance of the GAPC in the three cases when running on the left side of the MPP. Although the steadystate process of GAPC is similar to that of ALSO, the dynamic process of GAPC has some oscillations in all cases. And its tracking error is larger than ALSO with TE 1 = 1.85%, TE 2 = 2.82% and TE 3 = 5.89%, respectively.
The response performance of the two algorithms when operating on the right side of MPP is shown in Fig. 9. The system with the ALSO algorithm still maintains smooth dynamic and steady-state processes in all three cases with tracking errors that are not much different from those operating on the left side of the MPP. For GAPC, on the other hand, the PV side power deviates significantly from its references values in both steady-state and dynamic operation; the error is relatively large in all cases.

Varying reference power
In this section, the irradiance is set to a constant (Irr = 800 W/m 2 ) and the reference power varies with time, as shown in Fig. 10. Figures 11 and 12 show the performance of the two algorithms when running on the left or right side of the MPP.
It can be seen from the graphs that P avai is smaller than P ref in the interval from t = 0 s to t = 0.5 s and from t = 1.5 s to t = 2 s, during which the system works on MPP. Both algorithms perform well for MPPT, while the ALSO algorithm has a significantly shorter tracking time than GAPC. In the interval from t = 0.5 s to t = 1.5 s, P ref decreases and is smaller than P avai , as shown in Figs. 11a and 12a, and the tracking error of ALSO is exceptionally small during steady state. At t = 1 s, the P ref changes from 300 to 200 kW, and accordingly, the PV side power drops to 200 kW. The output power of ALSO quickly tracks to its reference value after a short oscillation, and the tracking error is TE = 0.76% for operation at left side of MPP and TE = 0.79% for operation at right side of MPP, respectively. The steady-state error of the GAPC algorithm when operating on the left side of MPP is also small, while working on the right side of the MPP, there is a relatively large deviation from its reference value, as shown in Figs. 11b and 12b.

Comparison of ALSO and GAPC
In order to visually compare and analyze the performance of the two algorithms, the tracking errors were calculated for all cases in 5.1 and 5.2 above and are listed in Table 10.
In Table 10, it should be noted that the TE ss , TE tr and TE of the ALSO algorithm are much smaller than those of the GAPC algorithm in all above cases. Compared to the GAPC algorithm, the tracking error of the ALSO algorithm is at most 40% of the GAPC algorithm (Case3, working on the left side of the MPP) and at least 8% of the GAPC algorithm (Case2, working on the right side of the MPP). Second, the smaller the reference power, the larger the tracking error when the system is operating at rapidly changing irradiance. Obviously, the TE of the ALSO The bold values indicate that the value is better than that of other compared algorithms The bold values indicate that the value is better than that of other compared algorithms Inverter switching frequency f sw-inv 10 kHz Table 9 Parameters of ALSO and GAPC Algorithm Parameter ALSO N = 20, T = 10,b max ¼ 0:8,b min ¼ 0:05 GAPC Left: V step = 3.5, T step = 0.001, k 1 = 0.001, k 2 = 0.085 Right: V step = 1.5, T step = 0.001, k 1 = 0.006, k 2 = 0.05 algorithm differs little when operating on the left and right sides of the MPP. From the above experiment results, it can be concluded that the proposed ALSO-based FPPT algorithm has better stability when running on both sides of the MPP, and the tracking error is much smaller than that of the GAPC algorithm. The reason is that the GAPC algorithm is a hillclimbing algorithm that approaches toward the reference power point step by step without knowing the exact voltage value of the reference power point. Consequently, the algorithm tracks the reference power point slowly and causes large oscillations in the system, which is seen in Figs. 9 and 11. As for the proposed ALSO-based FPPT algorithm, it employs ALSO algorithm to directly search the voltage value of the reference power point. The ALSO algorithm converges quickly and has strong global search and local exploitation abilities, which are confirmed in Sect. 4, the accurate voltage value can be obtained. Thus it enables the system to track to the reference power point quickly and accurately, thereby providing high dynamic and steady-state performance for the system.

Conclusion
This paper proposes a novel FPPT method based on ALSO for ensuring the smooth operation of GCPVPP. First, an ALSO algorithm is proposed by introducing adaptive proportion factor into the LSO algorithm, which can make the algorithm have a strong coordination ability for global search and local development. Besides, the objective function for FPPT control is designed, and the ALSO algorithm is used to search the voltage value of the reference power point in a determined voltage interval, thereby realizing FPPT control. The performance test of the LSO algorithm shows that the ALSO algorithm has better search accuracy and convergence speed, which is significantly better than the eight compared algorithms. The nonparametric test analysis of the ALSO algorithm also proves the effectiveness and superiority of the algorithm. Simulations on PV system show that, compared with GAPC, the ALSObased FPPT algorithm can ensure the smooth operation of the system on both sides of MPP in a dynamic environment, which compensates for the fact that the traditional (a1) (a2) (b1) (b2) Fig. 9 The response performances of the two algorithms under fast changes of irradiance with operating at the right side of MPP FPPT algorithm cannot operate smoothly on the right side of MPP. In particular, the tracking error is reduced by 60-92% compared with GAPC, which proves the effectiveness and superiority of the proposed FPPT method for GCPVCC. In future work, the partial shading condition (PSC) in PV systems will be further considered and the FPPT (a) (b) Fig. 11 The response performances of the two algorithms under fast changes of reference power with operating at the left side of MPP (a) (b) Fig. 12 The response performances of the two algorithms under fast changes of reference power with operating at the right side of MPP The bold values indicate that the value is better than that of other compared algorithms control method under PSC will be designed and implemented using the ALSO algorithm.