1. Introduction

2. System Network Architecture Diagram

3. Hybrid Energy Storage Power Allocation Strategy

3.3. Principles of the Secretary Bird Optimization Algorithm

The Secretary Bird Optimization Algorithm (SBOA), introduced in 2024, is a novel swarm optimization technique that models the hunting and escaping behaviors of secretary birds in two distinct stages. During these stages, each member of the secretary bird group is updated accordingly. The hunting behavior is generally divided into three processes: searching for prey, attacking prey, and consuming prey. Meanwhile, escaping behavior is classified into two types: environmental camouflage and rapid movement, such as flying or running swiftly.

The first stage involves the secretary bird’s hunting strategy (exploration phase), where the hunting process is divided into three equal time intervals. This approach enhances the algorithm’s diversity and global search ability. The position update formula for this first stage is provided below:

$$X_{i,j}^{new,P1}=\left\{\begin{array}{c}{X}_{i,j}+(X_{random\_1}-X_{random\_2})\times R_1(\mathrm{t}<{\displaystyle \frac{1}{3}}\mathrm{T})\\ X_{best}+e^{{\left({\displaystyle \frac{t}{T}}\right)}^4}\times (\mathrm{RB}-0.5)\times ({\mathrm{X}}_{best}-X_{i,j}\left)\right({\displaystyle \frac{1}{3}}\mathrm{T}<\mathrm{t}<{\displaystyle \frac{2}{3}}\mathrm{T})\\ X_{best}+{(1-{\displaystyle \frac{t}{T}})}^{2\times {\displaystyle \frac{t}{T}}}\times X_{i,j}\times \mathrm{RL}(\mathrm{t}>{\displaystyle \frac{2}{3}}\mathrm{T})\end{array}\right.$$

(21)

$$X_i=\left\{\begin{array}{c}{X}_i^{new,P1}(F_i^{new,P1}<{\mathrm{F}}_{\mathrm{i}})\\ X_i\mathrm{else}\end{array}\right.$$

(22)

$$RB=randn(1,j)$$

(23)

$$RL=0.5\times s\times {\displaystyle \frac{u\times \sigma }{|\nu {|}^{\frac{1}{\eta }}}}$$

(24)

$$\sigma ={\left({\displaystyle \frac{\Gamma (1+\eta )\times sin\left(\frac{\pi \eta }{2}\right)}{\Gamma \left(\frac{1+\eta }{2}\right)\times \eta \times 2\left(\frac{\eta -1}{2}\right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}$$

(25)

where j represents the dimension of the variable; X denotes the positions of the secretary bird group; $X_i$ indicates the position of the ith secretary bird; $X_{i,j}$ represents the value of a variable for the ith secretary bird in the jth dimension; $X_{i,j}^{new,P1}$ denotes the new position of the ith secretary bird in the jth dimension during the first stage; $X_{random\_1}$ and $X_{random\_2}$ are random candidate solutions during the iteration; $X_{best}$ is the best historical position of the secretary bird; $R_1$ represents an array with dimensions $1\times j$ randomly generated in the interval [0,1]; $F_i$ represents the objective function value obtained by the ith secretary bird, and $F_i^{new,P1}$ represents the new objective function value obtained by the ith secretary bird; t and T represent the current iteration count and the maximum iteration count, respectively; RB is a $1\times j$ dimensional array randomly generated from the standard normal distribution; RL indicates the Levy flight strategy, where $s=0.01$ and $\eta =1.5$; $\mu$ and $\nu$ are random numbers within the interval [0,1]; $\Gamma$ denotes the gamma function.

The second stage is the secretary bird’s escaping strategy (exploitation phase), where both escaping strategies can be represented by Equation (25). The position update formula for the second stage is shown below:

$$X_{i,j}^{new,P2}=\left\{\begin{array}{c}{X}_{best}+(2\times RB-1)\times {(1-{\displaystyle \frac{t}{T}})}^2\times X_{i,j}(\mathrm{r}<0.5)\\ X_{i,j}+R_2\times (X_{random}-K\times X_{i,j})\mathrm{else}\end{array}\right.$$

(26)

$$X_i=\left\{\begin{array}{c}{X}_i^{new,P2}(F_i^{new,P2}<{\mathrm{F}}_{\mathrm{i}})\\ X_i\mathrm{else}\end{array}\right.$$

(27)

$$K=round(1+rand(1,1\left)\right)$$

(28)

where $X_{i,j}^{new,P2}$ represents the new position of the ith secretary bird in the jth dimension during the second stage; $F_i^{new,P2}$ represents the new objective function value obtained by the ith secretary bird during the second stage; $R_2$ denotes an array of dimension $1\times j$ randomly generated from a normal distribution; $X_{random}$ represents the random candidate solution for the current iteration; K denotes a random value of either 1 or 2; $rand(1,1)$ is a random number generated between 0 and 1.

3.3.1. SBOA-Optimized VMD and SVMD Algorithm

When the SBOA algorithm searches for the optimal parameters for SVMD and VMD decomposition, it is necessary to define a fitness function to evaluate whether the selected parameters are optimal. The quality of the chosen fitness function significantly impacts the effectiveness of the SBOA in parameter selection. The average envelope entropy can effectively evaluate the quality of decomposition, filter useful modes, and capture the nonlinear characteristics of the signal, thereby helping to improve the decomposition performance of the wind power signal. The calculation method for average envelope entropy is as follows:

$$<\widehat{k},\widehat{\alpha }>=\underset{(k,\alpha )}{argmin}\left\{{\displaystyle \frac{1}{\widehat{k}}}\sum _{i=1}^{\widehat{k}}{E}_p\left(i\right)\right\}$$

(29)

$$E_p\left(i\right)=-\sum _{i=1}^N{p}_{i}lg{p}_i$$

(30)

$$p_i={\displaystyle \frac{a\left(i\right)}{\sum _i^{N}a\left(i\right)}}$$

(31)

where $\widehat{k}$ and $\widehat{\alpha }$ represent the optimal combination; $E_p\left(i\right)$ is the envelope entropy of each modal component after Hilbert demodulation; i denotes the number of sampling points; $p_i$ is the normalized form of $a\left(i\right)$; $a\left(i\right)$ represents the envelope signal.

This study selects the average envelope entropy as the fitness function, with the minimum value of the average envelope entropy as the optimization objective. This approach results in obtaining the optimal parameters for $[k,\alpha ]$ as well as the maximum parameter $\alpha$. Based on the optimal parameters, the original signal is decomposed, with the boundary frequency for the energy storage units set at ${10}^{-3}Hz$. This meets the requirement that the 10-minute grid-connected fluctuation amount is typically $5\%$ to $7.5\%$ of the installed wind power capacity. The flowchart for the initial power distribution strategy of wind power is shown in the figure below:

Figure 2. Flowchart of the initial power allocation using VMD and SVMD.

Figure 2. Flowchart of the initial power allocation using VMD and SVMD.

3.4. Hybrid Energy Storage Multi-Fuzzy Control Power Secondary Distribution Strategy

Due to differences in installed capacity, aging progress, and manufacturing processes, the initial state of charge (SOC) and capacity of hybrid energy storage systems (HESS) often vary from one another. If the discharge rate of the hybrid energy storage system (HESS) varies during the discharge process, HESS units with lower states of charge (SOC) will complete their discharge first. This inconsistency can reduce the capacity utilization and lifespan of the HESS. Therefore, balancing the state of charge (SOC) of a non-isolated hybrid energy storage system (HESS) is crucial. In the secondary power distribution process, HESS units with higher SOC will output more power or absorb less, while those with lower SOC will absorb more power or output less. This approach ensures that the HESS operates within its normal working range. The control structure diagram is shown in Figure 3.

Figure 4, $P_{wind}$ represents the wind power signal; $P_G$ denotes the grid-connected power; $SO{C}_{SC\_k}$ and $SO{C}_{Bat\_k}$ indicate the SOC states of the kth supercapacitor and battery, respectively; $SO{C}_{SC\_m}$ and $SO{C}_{Bat\_m}$ represent the average SOC of the supercapacitor and battery group; $P_{SC\_k}$ and $P_{Bat\_k}$ denote the smoothing power of the kth supercapacitor and battery; $P_{SC\_k}^{\prime }$ and $P_{SC\_k}^{″}$ illustrate the power adjustment by the fuzzy control for the supercapacitor and its actual power; $P_{Bat\_k}^{\prime }$ and $P_{Bat\_k}^{″}$ depict the power adjustment by the fuzzy control for the battery and its actual power; $P_{SCref\_k}$ and $P_{Batref\_k}$ represent the reference power of the kth supercapacitor and battery, respectively. The power regulator and PWM controller generate pulse signals to control the three-phase inverter, which regulates the power output of the energy storage units. Each energy storage unit is analogous to an individual agent, which communicates through equations (32) and (33) to obtain the global average state of charge (SOC).

$$SO{C}_i(k+1)=Z_i\left(k\right)+\alpha \sum _{j\in N_i}{\psi }_{ij}(k+1)$$

(32)

$${\psi }_{ij}(k+1)={\psi }_{ij}\left(k\right)+SO{C}_j\left(k\right)-SO{C}_i\left(k\right)$$

(33)

where k represents the number of iterations; $Z_i$ denotes the initial state of charge (SOC) for the ith agent, with $SO{C}_i\left(0\right)=Z_i$ and ${\psi }_{ij}\left(0\right)=0$; $N_i$ refers to the ith energy storage unit agent; and ${\psi }_{ij}$ indicates the cumulative variable between the ith agent and the jth agent. If an energy storage unit leaves the system or if the communication within the storage units changes, the new system will still converge to the average value of the current number of energy storage units. This algorithm distributes the leader’s role evenly among all energy storage units, ensuring that the algorithm does not fail due to the loss of a single energy storage unit. Ultimately, the energy storage system achieves the collective average state of charge (SOC) of the storage units.

Figure 3 shows multi-fuzzy controllers in the secondary distribution of the power, which include three fuzzy controllers with three dual input single outputs. Due to SOC deviations, it is necessary to design a fuzzy control system for the hybrid energy storage system. The inputs for the fuzzy controller are $SO{C}_i$ and $SO{C}_i-SO{C}_m$, where the basic domain of $SO{C}_i$ is $[0,1]$, The fuzzy control set is $\left\{VS,S,MS,M,M1,MB,B,VB\right\}$, representing Very Small, Small, Moderately Small, Medium, Medium 1, Moderately Big, Big, Very Big; The deviation range of $SO{C}_i-SO{C}_m$ is $[-45,45]$, and after multiplying by a certain scaling factor, the basic domain becomes $[-1,1]$; The fuzzy set is $\left\{NL,NM,NS,NO,PO,PS,PM,PL\right\}$, representing Negative Large, Negative Medium, Negative Small, Negative Zero, Positive Zero, Positive Small, Positive Medium, Positive Large. The output includes power correction coefficients $K_{soc.Bat}$ and $K_{soc.SC}$. When the deviation $SO{C}_i-SO{C}_m$ is very small, the correction coefficient needs to be sufficiently large to be effective. Therefore, after multiplying by a scaling factor, the basic domain is $[2e3,2e5]$, and the fuzzy set is $\{VS,S,MS,M,M1,MB,B,VB\}$. Each controller follows the same set of fuzzy control rules, as shown in Table 1.

To illustrate, Figure 5 shows the three-dimensional surface plots of the input membership functions and the relationships between inputs and outputs for the three fuzzy controllers, using the multi-fuzzy control optimization of batteries as an example.

In the design of multi-fuzzy controllers, appropriate fuzzy controllers are selected for each interval based on the $SO{C}_i-SO{C}_m$ deviation, achieving dynamic partition control of the SOC for the hybrid energy storage system. When the slope of the membership function in the fuzzy controller is set higher, the system will make adjustments in response to slight changes in the input variables. When the slope of the membership function in the fuzzy controller is set to be more gradual, the system will adjust in response to a wider range of changes in the input variables.

The initial wind power allocation strategy only considers frequency characteristics and does not take into account the state of charge (SOC) of supercapacitors and batteries, nor the impact of changes in their SOC on power distribution. Considering that the difference $SO{C}_i-SO{C}_m$ includes the impact of changes in the state of charge (SOC), a global variable $SO{C}_m$ is introduced. his approach ensures that all energy storage units are treated equally in the control strategy, reducing the influence of a single leader’s SOC and aligning them with the global variable $SO{C}_m$. Ultimately, this leads to a convergence towards the average SOC of all units. By taking these effects into account, the power of the hybrid energy storage system is adjusted using multi-fuzzy control. Partitioning is conducted based on the $SO{C}_i-SO{C}_m$ deviation to adjust the power of the hybrid energy storage system. The specific correction principles are as follows: When the battery and supercapacitor operate within their normal SOC ranges, inaccuracies in the initial power allocation cause deviations between each battery and supercapacitor’s SOC and their average SOC. The deviation range is $[0,45]$. When the deviation is within the range $[20,45]$, it indicates a significant SOC difference between them. To quickly align with the average SOC, the rules of Fuzzy Controller 3 are executed. When the deviation is within the range $[10,20)$, it indicates a smaller SOC difference between them, suggesting that the system has moved out of the SOC danger zone. To follow the average SOC faster, the rules of Fuzzy Controller 2 are executed. When the deviation is in the range $[0,10)$, it indicates that the SOC deviation between energy storage units is minimal, and the SOC of the storage units is nearly balanced while being out of the danger zone. In this case, the rules of Fuzzy Control 1 are applied. This approach ensures that the energy storage system continuously tracks the average SOC, minimizes the depth of charging/discharging, and reduces unnecessary charging/discharging events. It prevents any single battery from excessive charging or discharging, which could disrupt system stability and reduce the battery’s lifespan. The goal is to prevent state-of-charge (SOC) from exceeding its limits while dynamically adjusting it to keep it within a reasonable range.In Figure 5, $P_{SC}^{\prime }$ and $P_{Bat}^{\prime }$ represent the actual power consumption of the supercapacitor and battery, respectively, while the corrected powers are denoted by $P_{SC}^{″}$ and $P_{Bat}^{″}$.

Figure 5. Partition control diagram of battery and supercapacitor SOC deviations from the average SOC

Figure 5. Partition control diagram of battery and supercapacitor SOC deviations from the average SOC

4. Example Analysis

4.1. Basic Data

The wind power installed capacity used in this study is 1.5 MW, with a sampling period of 1 minute for the wind power data, resulting in a total of 1,500 data samples. Simulation analysis was conducted in Matlab. The grid connection technical specifications for wind farms selected a wind power fluctuation limit of 75 kW over 10 minutes. During the simulation, wind power data was collected every 0.014 seconds to form the wind power waveform, as shown in the figure. When optimizing the successive variational mode decomposition (SVMD) and variational mode decomposition (VMD) using the SBOA, the number of iterations was set to 20 and the population size to 25. Based on the optimized parameters, $[k,\alpha ]$ and the maximum $\alpha$ were used to decompose and reconstruct the wind power. The boundary frequency for hybrid energy storage was chosen as Hz. Modes reconstructed above the boundary frequency were used to smooth the power from the supercapacitor, while the modal combination below the boundary frequency selected the lowest fluctuation as the grid-connected power. The remaining wind power reconstruction served as the smoothing power for the lithium battery, as illustrated in the figure. The configuration of hybrid energy storage parameters is shown in the table below:

Table 2. Parameters corresponding to lithium batteries and supercapacitors

Table 2. Parameters corresponding to lithium batteries and supercapacitors

	Energy Storage Component
Parameter	LIB1	LIB2	LIB3	LIB4	SC1	SC2	SC3	SC4
$P_{max}$(kW)	112	120	120	112	100	110	110	110
$C_{Rate}$(kWh)	80	85	70	70	40	45	50	40
$SOC(\%)$	70.00	21.25	20.50	20.25	68.75	10.27	10.24	10.18
$SO{C}_{Lower}(\%)$	20	20	20	20	10	10	10	10
$SO{C}_{upper}(\%)$	80	80	80	80	90	90	90	90

In Table 2, $P_{max}\left(kW\right)$ denotes the maximum charge and discharge power of the energy storage components. $C_{rate}\left(kWh\right)$ represents the rated capacity. $SO{C}_0(\%)$ indicates the initial state of charge, while $SO{C}_L(\%)$ and $SO{C}_U(\%)$ correspond to the lower and upper safety limits of the state of charge, respectively.

The comparison between the direct grid-connected component and the initial wind power data is shown in Figure 6.; The low-frequency power fluctuations of each VMD decomposed mode are illustrated in Figure 7; The grid-connected fluctuation after smoothing versus before smoothing is depicted in Figure 8; The initial power allocation of the supercapacitor in the hybrid energy storage system is shown in Figure 9, and the initial power allocation of the lithium battery is shown in Figure 10.

After the initial allocation, it can be observed that the application of SBOA-VMD for processing the original signal effectively smooths wind power fluctuations and enhances grid-connected performance. Secondly, the battery energy storage system primarily handles the relatively stable low-frequency power components, while the supercapacitor is responsible for the high-frequency power components that involve frequent charging and discharging with larger instantaneous amplitudes. This allocation aligns well with the characteristics of each component in the hybrid energy storage system.

4.2. Secondary Power Allocation

Due to the inaccuracies in the initial power allocation and the differences among energy storage units, there is a risk of individual storage units having their SOC exceed acceptable limits and exiting operation, which increases the burden on other storage units. This can ultimately lead to the entire energy storage system ceasing to function. Therefore, while storage units work to smooth the power fluctuations from wind energy, maintaining balanced SOC among storage units is also critically important. The risk of SOC exceeding acceptable limits varies across different ranges of $SO{C}_i-SO{C}_m$ for energy storage units. By applying different fuzzy control rules, the lithium battery and supercapacitor can operate within safe limits while also achieving SOC balance.

Figure 11 and Figure 12 illustrate the initial and secondary power allocation for supercapacitor groups 2, 3, and 4, as well as battery groups 2, 3, and 4. Figure 13 and 14 show the correction of the SOC of energy storage units after initial and secondary power allocation. Inaccuracies in the initial power allocation can easily lead to group SOC exceeding the specified limits. In Figure 14 and Figure 15, under the control of the multi-fuzzy strategy, the SOC of the energy storage system is in the most critical operating range during the time intervals [1, 6] and [1, 8]. During these periods, a significant adjustment of the hybrid energy storage system’s power is required, and the rules of fuzzy control 3 are implemented to quickly bring the system into the optimal operating range. Subsequently, during the time intervals [6, 8.8] and [8, 12.5], the SOC of the energy storage system has moved away from the danger zone. However, since the SOC has not yet reached equilibrium, power adjustments to the hybrid energy storage system are reduced, and the rules of fuzzy control 2 are applied. During the time intervals [8.8, 21] and [12.5, 21], the SOC of the energy storage units gradually approaches equilibrium. The power adjustment is further reduced, eventually bringing the SOC of the storage units closer to the global average SOC, thereby achieving SOC balance.

Figure 11. Comparison of initial and secondary power allocation for supercapacitor.

Figure 11. Comparison of initial and secondary power allocation for supercapacitor.

Figure 12. Comparison of initial and secondary power allocation for lithium battery.

Figure 12. Comparison of initial and secondary power allocation for lithium battery.

Figure 13. Comparison of SOC under initial power allocation and SOC under secondary power allocation for supercapacitors.

Figure 13. Comparison of SOC under initial power allocation and SOC under secondary power allocation for supercapacitors.

Figure 14. Comparison of SOC under initial power allocation and SOC under secondary power allocation for lithium batteries.

Figure 14. Comparison of SOC under initial power allocation and SOC under secondary power allocation for lithium batteries.

Figure 15 and Figure 16 show the SOC balancing control curves for the supercapacitors and batteries. Figure 17 compares the reference power for grid connection with the actual tracking power. Figure 18 indicates that the fluctuation in tracking power is below 75 kW, meeting the grid connection power requirements.

Secondary power allocation effectively corrects the issue of SOC exceeding limits in hybrid energy storage, ensuring stable operation within a safe SOC range. Additionally, it significantly reduces power fluctuations in the grid-connected wind power hybrid energy storage system, enhancing equipment lifespan and power quality.

Figure 15. Balance curves of SOC for four groups of supercapacitors.

Figure 15. Balance curves of SOC for four groups of supercapacitors.

Figure 16. Balance curves of SOC for four groups of lithium batteries.

Figure 16. Balance curves of SOC for four groups of lithium batteries.

Figure 17. Comparison of wind power grid reference power and actual grid power.

Figure 17. Comparison of wind power grid reference power and actual grid power.

Figure 18. Actual grid power fluctuation amplitude.

Figure 18. Actual grid power fluctuation amplitude.

5. Conclusion

Abbreviations

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