Sim-Geometry Modal Decomposition (SGMD)-Based Optimization Strategy for Hybrid Battery and Supercapacitor Energy Storage Frequency Regulation

Abstract

This study examines the issue of wind power smoothing in renewable-energy-grid integration scenarios. Under the “dual-carbon” policy initiative, large-scale renewable energy integration (particularly wind power) has become a global focus. However, the intermittency and uncertainty of wind power output exacerbate grid power fluctuations, posing challenges to power system stability. Consequently, smoothing strategies for wind power energy storage systems are desperately needed to improve operational economics and grid stability. According to current research, single energy storage technologies are unable to satisfy both the system-level economic operating requirements and high-frequency power fluctuation compensation at the same time, resulting in a trade-off between economic efficiency and precision of frequency regulation. Therefore, hybrid energy storage technologies have emerged as a key research focus in wind power energy storage. This study employs the SE-SGMD method, utilizing the distinct characteristics of lithium batteries and supercapacitors to decompose frequency regulation commands into low- and high-frequency components via frequency separation strategies, thereby controlling the output of supercapacitors and lithium batteries, respectively. Additionally, the GA-GWO algorithm is applied to optimize energy-storage-system configuration, with experimental validation conducted. The theoretical contributions of this study include the following: (1) introducing the SE-SGMD frequency separation strategy into hybrid energy storage systems, overcoming the performance limitations of single energy storage devices, and (2) developing a power allocation mechanism on the basis of the inherent properties of energy storage devices. In terms of methodological innovation, the designed hybrid GA-GWO algorithm achieves a balance between optimization accuracy and efficiency. Compared to PSO-DE and GWO-PSO, the GA-GWO energy storage system demonstrates improvements of 21.10% and 17.47% in revenue, along with reductions of 6.26% and 12.57% in costs, respectively.

1. Introduction

Against the backdrop of the transformation of the global energy structure, as a representative form of clean energy and renewable energy, wind energy has attracted much attention. However, the inherent unpredictable, intermittent, and stochastic features of wind energy resources have led to dramatic fluctuations in wind power output, posing a substantial threat to the stable and safe operation of power systems [1]. Energy storage technology can effectively mitigate wind power fluctuations and improve system flexibility, and its level of development has a direct effect on the overall performance of the integrated wind-storage system. The available research concerns the function of energy storage systems in the frequency regulation of wind power generation. Tewari et al. conducted a multi-objective evaluation and found that battery energy storage demonstrates significant benefits in power system frequency regulation, wind power peaking, and wind farm power control, with frequency regulation being the most effective scenario [2]. Furthermore, Tewari et al. state that a properly configured battery energy storage system not only minimizes errors in wind energy prediction, but also extends equipment lifespan by optimizing operational strategies, thus strengthening the cooperative operation of wind farms and energy storage systems. While battery storage systems are crucial for compensating for generation-consumption mismatches in resilient microgrids [3], their inherent limitations—such as short cycle life, low charging and discharging efficiency, and high costs for full-system replacement—become evident when addressing frequent wind power fluctuations [4]. To overcome the performance limitations of single energy storage technologies, research on hybrid energy storage systems (HESSs) has gained momentum. Rout et al. illustrated the basic principles of HESSs for smoothing wind power volatility via multi-timescale energy modulation and summarized the current state of the technology along with development trend of a variety of energy storage components, providing a theoretical foundation for subsequent research [5]. Among them, the composite configuration of storage batteries and supercapacitors has attracted significant attention. Supercapacitors can compensate for the shortcomings of storage batteries in high-frequency power regulation because of their safety in physical energy storage, long cycle life, high power density, and efficiency [6]. Conversely, it is possible to address the limitations of supercapacitors in satisfying long-term energy needs with the high-energy-density properties of storage batteries. Yang, F., and Ren, Y. [7] demonstrated that this hybrid energy storage scheme exhibits significant synergistic effects in smoothing wind power output and prolonging battery life. Enguang Hou and Yanliang Xu et al. presented a flywheel-battery HESS on the basis of optimal variational modal decomposition (VMD), aiming to smooth wind power fluctuations. Their findings confirm that this method effectively suppresses wind power fluctuations while ensuring stable system operation by taking advantage of the respective properties of flywheel energy storage and battery [8]. Yu Peng and Li Guang Lei et al. investigated the operation of real-time wind-power-regulated energy storage systems. Additionally, it is recommended that energy storage systems with extended cycle life, high power, and energy density be employed for real-time wind power regulation. It is determined by comparing the performance of supercapacitors and batteries that a battery–supercapacitor hybrid system has improved features that satisfy energy-storage-system performance criteria. Additionally, the advantages and disadvantages of various topologies are examined from a practical engineering perspective [9]. For thermal power units, Han Xu and Liu Zhongwen suggested a regional dynamic main-frequency-regulation concept, as well as a lithium battery–flywheel control technique. Their research confirms that hybrid energy-storage-assisted frequency regulation reduces mechanical losses in thermal power units, extends unit lifespan, improves operational stability and economic efficiency, and mitigates wind and solar curtailment, thereby promoting the integration of renewable energy [10]. In terms of system simulation, Alawasa, K.M., and Malahmeh, B.K., et al. have properly justified the time-domain simulation under reproducible variability by means of an improved VIC scheme, proving that their scheme can enhance the robustness of the system [11]. In the future, optimal configuration methods and coordinated control strategies for hybrid energy storage systems will remain a key research focus [12].

Empirical modal decomposition, first-order inertial filtering, and wavelet analysis are frequently utilized techniques in the field of signal decomposition approaches [13]. Although wavelet analysis is widely used in signal processing, its high computational complexity limits real-time performance [14]. First-order inertial filtering has the problem of insufficient accuracy in low-frequency signal processing [15]. While empirical modal decomposition is applicable to nonlinear signals, it is difficult to avoid the phenomenon of modal aliasing, and it is particularly limited in primary FM power decomposition [16]. Chen, C., and Tang, W., et al. [17] employed continuous variational modal decomposition to overcome the issue that traditional modal decomposition cannot adaptively select modal parameters. Compared with traditional methods, SGMD constructs sin geometry component signals by solving Hamilton matrix eigenvalues via sin geometry similarity transform [18], demonstrating unique advantages in nonlinear signal processing. This method overcomes the dissipation mechanism of traditional decomposition methods and can effectively avoid the over-decomposition and modal aliasing problems of EMD, VMD, and other methods, thereby completely retaining the intrinsic characteristics of the signal [19]. This provides a new technological approach for wind power decomposition and energy storage control.

For the challenges of frequency modulation (FM) power owing to the inherent instability and randomness of wind power output, this paper enhances the SGMD method to optimize the collaborative FM power distribution strategy based on hybrid energy storage characteristics. It also introduces an innovative solution utilizing SGMD for FM frequency analysis and storage FM optimization via the genetic gray wolf algorithm. First, by refining the SGMD method, the low- and high-frequency components of the FM command signal are accurately separated using the sim geometry constraint mechanism. This approach resolves the issues of modal mixing and boundary effects associated with traditional decomposition methods, providing theoretical support for energy-storage-capacity optimization. Second, a layered cooperative FM strategy is proposed, leveraging the complementary characteristics of supercapacitors and lithium batteries. The genetic gray wolf optimization algorithm optimizes the energy storage allocation between supercapacitors and lithium batteries, effectively reducing the charging and discharging frequency of lithium batteries to extend their lifespan. This achieves an optimal balance between FM performance and economic efficiency, offering a novel approach for the stable operation of smart grids and the efficient utilization of energy storage systems.

2. Decomposition of FM Signals and Optimization of Energy Storage Configurations

2.1. Singular Geometry Modal Decomposition

Rational modal decomposition approaches can effectively minimize the volatility and nonlinearity of wind power generation frequency. Empirical mode decomposition along with variational mode decomposition are two prevalent signal decomposition techniques in the field of signal processing. However, these methods exhibit modal aliasing when processing nonlinear and nonstationary signals, resulting in suboptimal signal processing outcomes. Furthermore, at least one of the following limitations is imposed on these modal decomposition methods:

• The modal decomposition method lacks adaptive selection capabilities, is extremely sensitive to user-defined parameters, and needs them.
• When noise is present, the modal decomposition method is unable to decompose the signal efficiently.
• The modal decomposition method cannot decompose a complex waveform into several precise components.

To address the modal aliasing problem commonly encountered in traditional empirical mode decomposition (EMD) and VMD when processing nonlinear and non-smooth signals, this study introduces and optimizes the singular geometric mode decomposition (SGMD) method. By constructing a Hamiltonian matrix eigenvalue-solving framework based on the singular geometric similarity transform, SGMD can accurately decompose the FM command signal into low- and high-frequency components.

The core principle of this method lies in transforming signal decomposition into a singular geometric transformation problem, i.e., decomposing the signal into a set of mutually orthogonal singular geometric components (SGCs) [20]. Compared to traditional decomposition methods, SGMD effectively suppresses the influence of boundary effects through a singular geometric constraint mechanism, significantly improving the accuracy of FM command decomposition while retaining the inherent features of the signal.

This methodological development offers a more solid theoretical basis for the optimal capacity distribution of future energy storage systems.

2.2. Optimization of Hybrid Energy Storage Characteristics for FM Based on Genetic Gray Wolf Algorithm

On the basis of the complementary nature of Li batteries and supercapacitors with respect to energy storage characteristics, to maximize the energy storage allocation strategy of Li batteries and supercapacitors, a genetic gray wolf optimization algorithm is presented, and its main concept is to gradually combine the benefits of the two algorithms in a phased manner. Although the standard GWO has few parameters and is simple to implement, its global search capability is poor, and it is prone to premature convergence, which may lead to local optima or iterative stagnation in later stages, thereby affecting the performance of the algorithm and the accuracy of the results [21]. To address this issue, a new hybrid optimization algorithm is formed by combining the improved GWO with an adapted genetic algorithm. The goal of this hybrid optimization algorithm is to combine the evolutionary process of the genetic algorithm with the search power of GWO to achieve complementary advantages, thereby effectively solving the aforementioned problems [22].

With the genetic gray wolf optimization technique, supercapacitor energy storage and Li battery power distribution are rationally adjusted based on economic objectives, which effectively reduces the frequency modulation (FM) cost of wind farms. This method can drastically lower the frequency of discharging and charging of Li battery, thereby delaying its capacity decay rate, extending its service life, and achieving an optimal balance between FM performance and economic efficiency.

3. FM System and Power Allocation Strategy for Wind Power Combined with Energy Storage

3.1. FM System for Wind Power Combined with Energy Storage

HESS may greatly strengthen the effectiveness of wind farms in primary FM by combining the capabilities of Li batteries with supercapacitors. Figure 1 illustrates the structure of the wind hybrid energy storage system using a 1 MW wind farm as an example.

As illustrated in Figure 1, the supercapacitor–Li battery HESS installed at the connection point of the wind farm plays a critical role in the primary FM of the power system. Its bidirectional regulation mechanism significantly enhances the grid’s adaptability to wind power fluctuations. With a reduction in the grid frequency, the system rapidly releases stored power through the DC/DC converter, converting it into compensating power for injection into the grid, thereby effectively mitigating the frequency decline. Conversely, when the grid frequency increases, the system promptly absorbs excess power and hinders the delivery of surplus power from the wind farm to the grid. This dynamic response strategy not only achieves precise regulation of the grid’s supply–demand balance but also substantially improves the grid’s frequency stability and reliability by combining the supercapacitors’ rapid power response with lithium batteries’ energy storage capabilities.

3.2. Hybrid Energy Storage FM Power Allocation Strategy

The conventional joint FM control technique for wind power storage systems does not adequately take into account the unique FM features of various energy storage technologies. For instance, supercapacitors exhibit high instantaneous charge/discharge power density and extended cycle life, as well as rapid response times, while Li-ion batteries offer substantial energy storage capacity and prolonged discharge duration, though their service life is significantly impacted by frequent charge/discharge cycles.

To address these characteristics, the sin geometric modal decomposition (SGMD) method is applied to decompose the wind power FM signal and reconstruct it into low- and high-frequency components, managed by lithium-ion batteries and supercapacitors, respectively. This approach optimizes the complementary advantages of both storage technologies for frequency regulation.

The power distribution strategy is illustrated in Figure 2.

3.3. FM Power Allocation Model Based on SGMD

Suppose the time series of any initial signal is denoted as $x=\left(x_1,x_2,\dots ,x_n\right)$, where n is the signal length. Based on Takens’ embedding theorem, the initial time series, x, is extended to a multidimensional trajectory signal matrix, X.

$$X=\left[\begin{array}{cccc}{x}_1& x_{1+\tau }& \dots & x_1+\left(d-1\right)\tau \\ ⋮& ⋮& & ⋮\\ x_M& x_{M+\tau }& \dots & x_M+\left(d-1\right)\tau \end{array}\right]$$

(1)

where d represents the embedding function; τ denotes the extension factor; and $M=n-\left(d-1\right)\lambda \tau$, where M is the dimensions of the matrix.

Autocorrelation analysis of the trajectory matrix, X, yields the symmetric covariance matrix, A.

$$A=X^{T}X$$

(2)

Construct the Hamiltonian matrix, M, from the symmetric matrix, A.

$$M=\left[\begin{array}{cc}A& 0\\ 0& -A^T\end{array}\right]$$

(3)

Construct the sin-orthogonal matrix, Q, using Householder transformation.

$$Q^{T}NQ=\left[\begin{array}{cc}C& R\\ 0& C^T\end{array}\right]$$

(4)

where C is an upper triangular matrix. $c_{ij}=0\left(i>j+1\right);N=M^2$.

Let the eigenvalues of the upper triangular matrix, C, be ${\lambda }_1,{\lambda }_2,\dots {\lambda }_d.$ Then, the eigenvalues of the symmetric matrix, A, are the following:

$${\sigma }_i=\sqrt{{\lambda }_i} i=1,2,\dots ,d$$

(5)

Therefore, the eigenvectors corresponding to the eigenvalues of matrix A are Q(i = 1,2,…d). In accordance with the Householder transformation theory, the trajectory matrix, Z, was obtained with the reconstruction of matrix A:

$$\begin{array}{c}{Z}_i=Q_i{Q}_i^T{X}_i\\ Z=Z_1+Z_2+\dots +Z_d\end{array}$$

(6)

where Z is the dimensional initial subcomponent matrix, and the elements in $Z_i$ are defined as $z_{ij}$ if $m<d$, ${z_{ij}}^{*}=z_{ij}$; otherwise, ${z_{ij}}^{*}=z_{ji}$.

Average diagonalize any initial subcomponent matrix A so that it is transformed into a set of time series of length n. Thus, an initial time series x is formed by the sum of d groups of time series of length n. The average diagonalization formula is the following:

$$y_k=\{\begin{array}{c}{\displaystyle \frac{1}{i}}{\displaystyle \sum ^{\underset{p=1}{i}}}{z}_{p,i-p+1}^{*} 1⩽i⩽d^{*}\\ {\displaystyle \frac{1}{d^{*}}}{\displaystyle \sum ^{\underset{p=1}{d^{*}}}}{z}_{p,i-p+1}^{*} d^{*}⩽i⩽m^{*}\\ {\displaystyle \frac{1}{n-i+1}}{\displaystyle \sum ^{\underset{p=i-m^{*}+1}{n-m^{*}+1}}}{z}_{p,i-p+1}^{*} m^{*}<i⩽n \end{array}$$

(7)

where $d^{*}=\min \left(m,d\right); m^{*}=\max \left(m,d\right);\mathrm{and} n=m+\left(d-1\right)\lambda$.

Convert $Z_i$ into a time series $Y_i\left(i=1, 2,\dots ,n\right)$ of n. From the initial signal, x is a superposition of d sets of independent components, i.e., d SGCs, denoted as follows:

$$x=Y_1+Y_2+\dots +Y_d$$

(8)

The primary FM power command is decomposed stepwise through SGMD processing. Modal components with unique features can be created when the power command reconstruction error drops below a certain threshold. The sinusoidal geometry modal decomposition process is illustrated in Figure 3.

3.4. Enhanced Sinusoidal Geometry Modal Decomposition Method (SE-SGMD)

Sample entropy (SE) was introduced by Richman and Moorman as a metric for quantifying time series complexity [23]. A low SE value indicates reduced irregularity in the time-domain signal, while a higher value reflects greater signal irregularity. Through the decomposition process, it becomes evident that fluctuating signals can be decomposed into multiple symplectic geometry component (SGC) modes using the symplectic geometry mode decomposition (SGMD) method.

In this section, the SE values of each SGC and their corresponding similarity thresholds are utilized to reorganize and reconstruct the low- and high-frequency components. This approach helps to optimize the allocation of capacity-based and power-based energy-storage leveling power. Therefore, an improved method of power allocation on the basis of sample entropy—called the SE-improved SGMD method—is proposed.

For any given time series, $S=\left\{S_1,\dots ,S_i,\dots ,S_N\right\}$, an m-dimensional embedding vector is constructed.

$$S_i^m=\left\{S_i,S_{i+1},\dots ,S_{i+m-1}\right\}1⩽i⩽N-m$$

(9)

The sample entropy of vector S can then be calculated using Equation (13).

$$SE\left(m,r\right)=\underset{v\to \infty }{lim}\left[-\mathit{ln}{\displaystyle \frac{O^{m+1}\left(r\right)}{O^m\left(r\right)}}\right]$$

(10)

where r is the similarity limit.

The following is for when v takes finite values:

$$SE\left(m,r,v\right)=-\mathit{ln}{\displaystyle \frac{O^{m+1}\left(r\right)}{O^m\left(r\right)}}$$

(11)

where $O^m\left(r\right)$ and $O^{m+1}\left(r\right)$ represent the probabilities that the SGC matches m points and m + 1 points under r, respectively. The reconstruction of low- and high-frequency fluctuation components is achieved by decomposing a signal into multiple SGCs through SGMD and calculating the SE value of each component to compare with the corresponding similarity value. The expression for the similarity threshold calculation is as follows:

$$\omega ={\displaystyle \frac{m}{\sqrt{r}}}\sum ^{\underset{n=1}{N}}{S}_i$$

(12)

The threshold, ω, should lie between the mean SE values of the low- and high-frequency components, which can be identified via calculating the SE distribution of all SGCs.

The threshold, ω, is a critical parameter for balancing the performance and cost of hybrid energy storage systems:

An excessively small ω results in an overestimation of the high-frequency component, leading to supercapacitor capacity redundancy and increased costs.

Conversely, an excessively large ω leads to an overestimation of the low-frequency component, resulting in lithium-ion battery capacity redundancy and reduced lifespan.

Reasonable ω: adaptive partitioning based on sample entropy can avoid modal mixing and achieve the optimal configuration through economic evaluation. Practical applications require comprehensive consideration of signal characteristics, energy-storage device parameters, and operational constraints.

The total reference power in Figure 4 is decomposed using the improved SGMD method, and the modal distribution of each SGC sub-mode obtained after decomposition is shown in Figure 5c. A total of 14 SGC sub-mode components are obtained through the SGMD decomposition. Furthermore, these 14 SGCs are reconstructed using the aforementioned sample entropy similarity to obtain high-frequency and low-frequency fluctuation components, as presented in Figure 6.

The calculations of the SE and the similarity threshold, ω, required for reconstruction are closely related to the values of parameters ‘m’ and ‘r’. According to Reference [24], the statistical characterization of the sample entropy is more reasonable when $m = 1$ or 2 and $r=0.1~0.25 std$ (where std denotes the standard deviation of the time series under analysis). Therefore, $m=2$ and $r=0.19 std$ are selected for the calculations in this section. The similarity threshold, ω, is determined to be 0.921 using Equation (12).

To verify the accuracy and stability of SE-SGMD, power decomposition was performed using ICEEMDAN, SGMD, and SE-SGMD. The objects of the study are the measurement data of an 80 MW wind farm in Guangxi Province, China. Figure 4 shows the amount of electricity generated by this wind farm.

The ICEEMDAN method is employed in Figure 5a, the SGMD method in Figure 5b, and the SE-SGMD method in Figure 5c. As can be seen from the modal components derived from the decomposition, the high-frequency components obtained from ICEEMDAN produce satisfactory results, whereas its low-frequency performance is inferior compared to SE-SGMD. The low-frequency modal fluctuations exhibit minimal variation across different time intervals, rendering them unsuitable as a reference for the actual FM output of the Li-ion battery energy storage. On the contrary, the low-frequency modal component acquired by SE-SGMD demonstrates greater prominence and reduced volatility compared to ICEEMDAN, making SE-SGMD more appropriate as an actual reference value for the FM of Li battery energy storage. Compared with SGMD, SE-SGMD effectively eliminates signal decomposition noise, reorganizes the signal decomposition frequency into high and low frequencies through an appropriate ω and efficiently leverages the distinct characteristics of supercapacitors and lithium batteries.

4. Energy Storage System Modeling

4.1. Constraint Function

• Power limitations in hybrid energy storage system charging and discharging are calculated as follows:

$$\begin{array}{c}\{\begin{array}{c}{P}_{lb}^{rate }=\max \left\{\left|\max \left[\mathsf{\Delta }{P}_{lb}(t)\right]\right|{\eta }_{lb}^{cha },{\displaystyle \frac{\left|\min \left[\mathsf{\Delta }{P}_{lb}(t)\right]\right|}{{\eta }_{lb}^{dis }}}\right\}\\ -P_{lb}^{rate }<\mathsf{\Delta }{P}_{lb}(t)<P_{lb}^{rate }\end{array}\\ \{\begin{array}{c}{P}_{sc}^{rate }=\max \left\{\left|\max \left[\mathsf{\Delta }{P}_{sc}(t)\right]\right|{\eta }_{sc}^{cha },{\displaystyle \frac{\left|\min \left[\mathsf{\Delta }{P}_{sc}(t)\right]\right|}{{\eta }_{sc}^{dis }}}\right\}\\ -P_{sc}^{rate }<\mathsf{\Delta }{P}_{sc}(t)<P_{sc}^{rate }\end{array}\end{array}$$

(13)

where $\Delta P_{lb}\left(t\right)$ and $\Delta P_{sc}\left(t\right)$ represent the discharge and charge power required by the Li battery and supercapacitor at time t, respectively; ${\eta }_{lb}^{cha }$ and ${\eta }_{lb}^{cha }$ denote the charging and discharging efficiencies of Li batteries, respectively, while ${\eta }_{sc}^{cha }$ and ${\eta }_{sc}^{dis }$ are those of the supercapacitor.

2.

Hybrid energy storage SOC constraints are as follows:

$$\begin{array}{c}SO{C}_{lb\min }\le SO{C}_{lb}(t)\le SO{C}_{lb\max }\\ SO{C}_{sc\min }\le SO{C}_{sc}(t)\le SO{C}_{sc\max }\end{array}$$

(14)

where $SO{C}_{lb}(t)$ represents the state of charge of the Li battery at time t; $SO{C}_{lb\max }$ and $SO{C}_{lb\min }$ denote the upper and lower SOC limits of the Li battery, respectively, while $SO{C}_{sc\max }$ and $SO{C}_{sc\min }$ denote those of the supercapacitor; and $SO{C}_{sc}(t)$ stands for the SOC of the supercapacitor at time t.

3.

Power balance constraints are as follows:

$$\mathsf{\Delta }{P}_{te}(t)=\mathsf{\Delta }{P}_{lb}(t)+\mathsf{\Delta }{P}_{sc}(t)$$

(15)

4.2. Lithium Battery Life Function: Hybrid Energy-Storage-Capacity Optimization Function

Since supercapacitors store energy through a double electric layer at the electrode–electrolyte interface without undergoing internal chemical reactions like batteries, their charge–discharge processes are highly reversible. Consequently, supercapacitors exhibit exceptionally long cycle lives, capable of enduring millions of long charge–discharge cycles. Therefore, this study treats the supercapacitor’s cycle life as a fixed parameter. On the other hand, the cycle life of Li-ion batteries deteriorates irreversibly with each charge–discharge cycle. As discussed in the previous section, multiple factors influence battery cycle life, including operating temperature, discharge rate, number of charge–discharge cycles, and depth of discharge (DOD). This research mainly examines the effects of cycle count and discharge depth on Li-ion battery longevity in order to streamline the analysis. The method for calculating the discharge depth of lithium-ion batteries is expressed as follows:

$$D_{cir}(t)=1-SOC(t)$$

(16)

where $D_{cir}(t)$ stands for the actual discharge depth of the lithium battery during operation.

According to the method described by Kaisheng Sun [25], the correspondence between DOD and the number of cycles of lithium iron phosphate batteries is displayed in

Table 1. Relationship between lithium battery cycle times and battery DOD.

Depth of Discharge	Number of Cycles	Depth of Discharge	Number of Cycles
0.1	150,000	0.6	8000
0.2	50,000	0.7	7500
0.3	30,000	0.8	6000
0.4	14,000	0.9	5000
0.5	10,000	1	4000

.

Based on the data in Table 1, a Gaussian function was fitted to the battery DOD and the number of cycles. Figure 7 displays the fitted connection between cycle life and battery DOD.

The expression for the fitted relationship is derived as follows:

$$N=5.27\times {10}^5{e}^{-15.16D_{od}}+4.45\times {10}^4{e}^{-2.68D_{od}}$$

(17)

Here, N and $D_{od}$ are the number of cycles and DOD of the Li battery. The equivalent number of cycles of a Li-ion battery in operation can be denoted as follows:

$$N\left(D_{od,i}\right)={\displaystyle \frac{N_c\left(D_{od,1}\right)}{N_c\left(D_{od,i}\right)}}$$

(18)

Here, $N\left(D_{od,i}\right)$ represents the equivalent number of cycles of the Li battery at the i-th cycle discharge depth of $D_{od,i}$; $N_c\left(D_{od,1}\right)$ is the number of cycles of the Li battery at a discharge depth of 1; and $N_c\left(D_{od,i}\right)$ is the number of cycles with a discharge depth of $D_{od,i}$.

Let the lithium battery undergo m cycles of charging and discharging each time, with discharge depths of $D_{od,1}$, $D_{od,2}$, and $D_{od,m}$, respectively. The equivalent number of cycles for the battery in one day is the following:

$$N_1=\sum _m^{i=1}N\left(D_{od,i}\right)={\displaystyle \frac{N_c\left(D_{od,1}\right)}{N_c\left(D_{od,i}\right)}}$$

(19)

where $N_1$ stands for the equivalent number of cycles in a day for a Li battery. The life of the lithium battery in the operating cycle can be expressed as the following:

$$Y={\displaystyle \frac{N_c\left(D_{od,1}\right)}{365N_1}}$$

(20)

where Y is the number of years of life of the Li-battery over the entire operating cycle.

From the above analysis, it can be observed that the lifespan of the battery and the functioning of the energy storage system are directly correlated with the service life of Li-batteries and the number of cycles. The capacity configuration of the energy storage system is affected by the battery lifespan, making it meaningful to consider its impact on capacity configuration.

The utilization of Li-ion batteries under typical operating conditions can be divided into four classes [26]: The first tier is employed in electric devices such as new-energy vehicles, requiring a depth-of-discharge (DoD) range of 80–100%; the second tier (DoD range of 50–80%) is commonly used in new-energy storage devices and power grids; the third tier (DoD range of 40–50%) is utilized by low-end users; and the fourth tier (DoD below 40%) is designated for battery dismantling and recycling.

Based on this classification, it can be concluded that energy storage batteries in the new-energy sector require replacement when the depth of discharge reaches 60%, with the battery’s cycle life being approximately 17,000 cycles.

4.3. Modeling the Life-Cycle Cost of HESSs

The original investment cost and yearly operational and maintenance (O&M) costs, as well as differential generation costs, represent a major portion of the overall cost of an energy storage system. The following formula can be used to express its cost model:

$$C=M_{\mathrm{i}}+M_{\mathrm{d}}+M_{\mathrm{a}}$$

(21)

where C represents the mean annual cost; $M_{\mathrm{i}}$ is the annual cost of completing the major investment in the project; $M_{\mathrm{d}}$ is the annual cost of the O&M of the wind, light, and storage system; and $M_{\mathrm{a}}$ is the annual cost of the differential generation.

The hybrid storage system and wind farm construction costs are included in the one-time investment cost. It is possible to convert the one-time investment cost of a wind-storage system into an annual one. The following equation displays the one-time investment cost’s calculation formula:

$$M_{\mathrm{i}}={\displaystyle \frac{nr{(1+r)}^n}{{(1+r)}^n-1}}\left(a_{\mathrm{A}}{S}_{\mathrm{A}}+k{a}_{\mathrm{B}}{S}_{\mathrm{B}}\right)$$

(22)

where $a_{\mathrm{A}}$ and $a_{\mathrm{B}}$ are the intrinsic cost factors for the initial investment in the supercapacitor energy storage and battery energy storage; $S_{\mathrm{A}}$ and $S_{\mathrm{B}}$ are the capacities of the supercapacitor energy storage and battery energy storage; r is the annual interest rate of the loan, which is about 0.07; and n is the loan repayment period, set at 10 years.

The equipment maintenance costs, which are computed as a proportion of the initial investment cost, represent a major portion of the O&M costs of a wind-storage system. Thus, the O&M cost can be presented as follows:

$$M_{\mathrm{d}}=\delta M_{\mathrm{i}}$$

(23)

These include miscellaneous costs such as construction and commissioning costs (about 5% of the total equipment cost) and labor O&M (about 10% of the total equipment cost).

Shortage and wind curtailment costs are the two main components of differential generation costs. The shortfall cost is the power reduction necessary when the charging capacity of the energy storage array is insufficient to meet its discharging capacity, necessitating charging to an appropriate power state. The wind curtailment cost occurs due to light fluctuations when the wind farm power increases excessively within a short period, leading to wasted wind energy. By converting these two components into economic costs, we obtain the penalty cost.

$$M_a=M_{a1}+M_{a2}$$

(24)

where $M_{a1}$ and $M_{a2}$ stand for the shortage and wind abandonment costs of the energy storage array, respectively

$$M_{a1}={\displaystyle \frac{\rho }{24}}{C}_{e}E\left(SO{C}_1-SO{C}_2\right)$$

(25)

$$M_{a2}={\displaystyle \frac{\rho }{24}}{C}_k\sum _N^{t=1}\left\{g\left[P_{\mathrm{A}}(t)-P_{WB}(t)\right]\mathsf{\Delta }t\right\}$$

(26)

where $\rho$ is the number of operating hours over 1 year of the wind energy storage system, generally 2400–3000 h; $E$ is the rated power of the energy storage array; $P_{WB}(t)$ represents the wind-storage system’s output power at time t; $P_{\mathrm{A}}(t)$ denotes the raw power of the wind farm; and $C_k$ is the penalty factor.

4.4. Revenue Accounting Method for Energy Storage Systems

The primary revenue source of the wind-storage system is electricity sales, with additional revenue streams including supplementary power generation income, peak–valley tariff differentials, energy-storage policy incentives, and energy-storage recovery benefits. Among these, electricity sale revenue primarily depends on local electricity prices and the quantity of electricity sold, as expressed in the following formulas:

$$P_{WB}(t)=P_{\mathrm{W}}(t)+P_{\mathrm{B}}(t)$$

(27)

$$R_{\mathrm{e}}={\displaystyle \frac{\rho }{24}}\sum _N^{t=1}{c}_{\mathrm{e}}{P}_{WB}(t)\mathsf{\Delta }t$$

(28)

where $P_{WB}(t)$ and $P_{\mathrm{B}}(t)$ represent the output powers of the wind farm and HESS, respectively; $P_{\mathrm{W}}(t)$ is the actual output power of the wind farm at time t; $C_{\mathrm{e}}$ is the feed-in tariff, with $C_{\mathrm{e}}$ = CNY 0.42; $\rho$ denotes the number of operating hours for 1 year of the wind farm; $\mathsf{\Delta }t$ is the sampling interval, with $\mathsf{\Delta }t=34s$; N is the number of samples of the data; and R is the gain from the sale of electricity. The additional power generation revenue, also known as the reduced retained power revenue, is the revenue obtained from selling a portion of the power generation that is withheld from renewable energy farms to make sure that grid-connected generation is stabilized. It can be calculated by the following formula:

$$P_{rd}(d)=\{\begin{array}{l}{\displaystyle \sum _{N_{\mathrm{day}?}}^{i=1}}(1-x)P_w(t)\hfill \\ {\displaystyle \sum _{N_{\mathrm{day}?}}^{t=1}}{P}_{\mathrm{rate}?}\hfill \end{array}$$

(29)

$$B_r=\sum _D^{d=1}{e}_r{P}_{rd}(d)$$

(30)

where $P_{rd}(d)$ is the wind farm’s spare capacity, and $B_r$ is the spare capacity’s revenue.

Peak and valley tariff revenue refers to the economic benefits derived from time-of-use pricing (higher on-peak tariffs and lower off-peak tariffs) when the HESS utilizes a “low-storage, high-generation” strategy [27]. This relationship is expressed by the following equation:

$$f_W\left(P_{BESS},E_{BESS}\right)=C_{peak}-C_{val}$$

(31)

where $C_{peak}$ is the grid’s peak-hour electricity price, and $C_{val}$ is the grid’s valley-hour electricity price.

The revenue from energy storage recycling primarily comes from recycled batteries. Based on the gradient utilization of lithium-ion batteries, these batteries can either be resold or their internal metal components can be recycled. The specific returns range from 5% to 10% of the original cost [28].

4.5. Wind-Storage Economic Model

An economic model of a wind-storage system is derived by combining the cost and revenue accounting methods of the wind-storage system. Specifically, the following formula is presented:

$$f=R_e-M_i-M_d-M_a$$

(32)

where f is the annual net income of the wind-storage system for the annual operating income.

4.6. Parameters for Energy-Storage Capacity Configuration

The charging and discharging efficiency of Li-ion battery is set at 80% on the basis of comprehensive charging and discharging efficiency numbers while that of the supercapacitor is set at 95%. Both the supercapacitor and the Li-ion battery have their initial state of charge (SOC) set at 0.5. In order to avoid over charging and discharging of the Li-ion batteries, which could damage their cycle life, the lower SOC limit during discharge is set to 0.15, and the upper SOC limit during charging is set to 0.85.

Supercapacitors exhibit superior charging and discharging characteristics compared with storage batteries, allowing for longer charge and discharge cycles during operation. Therefore, the lower SOC limit for supercapacitor discharge is set to 0.1, while the upper SOC limit for charging is set to 0.9.

Table 2. Parameters of hybrid energy storage configuration.

Parameter	Li Battery	SC
Unit power cost CNY/kW	2700	1500
Unit cost of capacity CNY/kWh	640	27,000
O&M costs CNY/kWh	0.05	0.05
Processing costs CNY/kWh	0.04	0.05
Charge and discharge efficiency %	80	95
SOC range	(0.15, 0.85)	(0.1, 0.9)
Cycle life/year	3	15
Funds discount rate	6	9

and

Table 3. Other parameters.

Other		Other	SC
Wind farm size	80 MW	Scale of energy storage	3 MW
Daily power generation	12 h	Recycling income	5–10%
Annual running time	3000 h	Life cycle	20 years

displays the capacity configuration parameters of HESS [29].

5. Optimized Solution Method Based on Genetic Gray Wolf Algorithm

5.1. Genetic Algorithm

The genetic algorithm (GA) was first proposed by American scholar J. Holland in 1975. It simulates the principles of “natural selection and survival of the fittest” in nature, obtaining optimal solutions through information exchange among individuals and group-based search methods. This approach represents a multi-combination and multi-parameter simultaneous optimization technique, which is particularly suitable for solving nonlinear or optimization problems. Figure 8 depicts a flow chart of the GA, with its main components including the following:

(1)

Algorithm Encoding

The GA cannot directly process the parameters of the problem space; these parameters must be transformed into the genetic space to form an individual or chromosome. This transformation process is referred to as encoding. Commonly used encoding methods include character encoding, integer encoding, and binary encoding.

(2)

Fitness Function

The fitness function, also called the objective function in the GA, stands for the capacity of an individual to adapt to its environment and to reproduce its offspring, which quantifies the degree of an individual’s superiority or inferiority.

(3)

Genetic Operations

Genetic operations consist of the following three fundamental genetic operators: selection, crossover, and mutation.

(4)

Algorithm Termination Conditions

When the predetermined number of iterations is reached or the fitness function value stabilizes, the algorithm terminates.

5.2. Gray Wolf Optimizer

The Grey Wolf Optimizer (GWO) is an intelligent swarm optimization algorithm based on the cooperative hunting behavior and social hierarchy of grey wolf packs. Gray wolves are pack animals, with packs typically consisting of 5 to 10 individuals, exhibiting a well-defined hierarchical structure, as illustrated in Figure 9.

The first tier of the pyramid is the leader, designated as α. The α possesses managerial capabilities and is primarily responsible for decision-making processes such as hunting and resource allocation. The second tier consists of β wolves, which assist the α in decision making. In the event of a vacancy in the α position, a β wolf will assume leadership. The δ wolves follow the directives of the β wolves, while underperforming α and β wolves may be demoted to δ status. The base of the hierarchy comprises η wolves, which receive orders from the three higher ranks. Although η wolves are the least-dominant individuals, their absence would destabilize the population’s structure, as they serve as emotional outlets for other ranks and contribute to maintaining group stability.

The gray wolf algorithm involves three principal phases, as follows: prey encirclement, hunting, and prey attack. The mathematical model representing the collective prey encirclement can be represented as follows:

$$\begin{array}{c}D=\left|C\cdot X_p(t)-X(t)\right|\\ X(t+1)=X_p(t)-A\cdot D\end{array}$$

(33)

where t is the current iteration number; $X_p(t)$ and $X(t)$ denote the positions of the prey and the gray wolf, respectively; and A and C are the coefficients. The computational expression is as follows:

$$\{\begin{array}{c}A=2a\cdot r_1-a\\ C=2r_2\end{array}$$

(34)

Here, $r_1$ and $r_2$ are random values from [0, 1]; A is the proximity of the simulation to the prey and is a random value from [−a, a], which diminishes from 2 to 0 as the iteration proceeds. The mathematical model for the gray wolf’s tracking of prey can be denoted as follows:

$$\{\begin{array}{l}{D}_{\mu }=\left|C_1\cdot X_a-X\right|\hfill \\ D_{\beta }=\left|C_2\cdot X_{\beta }-X\right|\hfill \\ D_{\delta }=\left|C_3\cdot X_{\delta }-X\right|\hfill \end{array}$$

(35)

where $D_a$, $D_{\beta }$, and $D_{\delta }$ denote the positions of the α, β, and δ wolves, as well as other individuals, respectively; $X_a$, $X_{\beta }$, and $X_{\delta }$ are their current positions, respectively; $C_1$, $C_2$, and $C_3$ are random numbers; and $X$ represents the current position of the gray wolf.

$$\{\begin{array}{l}{X}_1=X_a-A_1\cdot \left(D_a\right)\hfill \\ X_2=X_{\beta }-A_2\cdot \left(D_{\beta }\right)\hfill \\ X_3=X_{\delta }-A_3\cdot \left(D_{\delta }\right)\hfill \end{array}$$

(36)

where $X_1$, $X_2$, and $X_3$, adjusted by the influence of the α wolf, β wolf, and δ wolf, respectively, are averaged as follows:

$$X(t+1)={\displaystyle \frac{X_1+X_2+X_3}{3}}$$

(37)

A gray wolf attacks its target when the prey stops moving, thus completing the hunting process. The GWO algorithm initially constructs a population of gray wolves, then defines the α, β, and δ wolves. The remaining individuals locate their prey by referring to the positions of the α, β, and δ wolves and subsequently updating their positions until the prey is captured. The GWO algorithm’s workflow is displayed in Figure 10.

5.3. GA-GWO

The specific steps of the enhanced grey wolf optimization algorithm are illustrated in Figure 11.

6. Simulation Verification

6.1. Frequency Stability Analysis Under Step-Load Disturbance

Under step perturbation, the effectiveness of the presented strategy is demonstrated by simulating and comparing the three control strategies under various wind speeds. To facilitate the analysis, the time to reach the maximum frequency deviation (|t_s|) together with the absolute value of the maximum frequency deviation (|Δf_max|) are used as evaluation indices for the frequency regulation effect. Smaller values of these indices indicate a better frequency regulation performance. Figure 12 presents the frequency stability analysis during a sudden load increase at low wind speed.

When $t=2 s$, the load changes abruptly, causing the system frequency to change, and the evaluation metrics are presented in

Table 4. Evaluation indicators in different models.

Frequency Modulation	$\left|\mathbf{\Delta }{\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}\right|$	$\left|\mathit{t}\mathit{s}\right|$
Wind power does not participate in frequency modulation	0.1936	5.28
Single lithium FM battery	0.1072	2.37
Lithium-ion battery with supercapacitor frequency modulation	0.0535	1.76

.

Figure 12 and Table 4 reveal that the hybrid energy storage mode demonstrates superior frequency modulation (FM) performance compared to both wind power without FM participation and single Li-ion battery FM systems. Specifically, the hybrid system reduces f_max by 72.37% and 50.10% and decreases ts by 66.67% and 25.74%, respectively. These results clearly indicate that the HESS provides more effective suppression of frequency deviation and improved frequency regulation performance.

6.2. Results of Capacity Optimization Allocation

With the objective of minimizing the cost of hybrid energy storage, the benefits of hybrid energy storage systems participating in wind power smoothing were comparatively optimized using the GA-GWO algorithm, PSO-DE algorithm, and GWO-PSO algorithm under the given constraints. Figure 13 illustrates the comparative performance data for these various optimization algorithms.

As shown in Figure 13, GA-GWO requires seven fewer iterations than PSO-DE while achieving a 6.26% reduction in energy storage costs. In comparison with GWO-PSO, GA-GWO exhibits four additional iterations but demonstrates a 12.57% reduction in energy storage costs. These results indicate that GA-GWO offers superior economic performance.

Table 5. Configuration results for various optimization algorithms.

Optimization Algorithm	GA-GWO	PSO-DE	GWO-PSO
Number of iterations	38	45	34
Running time/s	6	10	9
Cost/CNY	5,853,772	6,244,595	6,695,633
Net benefit/CNY	3,755,140	3,100,866	3,196,547
Li battery/MW	0.62	0.51	0.57
Supercapacitor/MW	0.37	0.45	0.39
Li battery/MWh	0.59	0.26	0.31
Supercapacitor/MWh	0.42	1.21	1.18

shows that optimizing the capacity of the HESS using GA-GWO can effectively lower the cost of the energy storage system while increasing its net benefits and maintaining the effectiveness of FM. The optimization time is shortened by 33.33% and 40%, and the benefits of the energy storage system are raised by 17.47% and 21.10%, respectively, in contrast to GWO-PSO and PSO-DE, showing significant economic advantages.

6.3. MATLAB System Simulation

The wind power simulation system is developed to smooth the wind-generated output power. Utilizing MATLAB R2022b for simulation, this system employs the SGMD method to process frequency modulation signals and obtain the threshold value, w. The low- and high-frequency bounds of the filter are set to w, with the Li battery and supercapacitor handling the low- and high-FM components, respectively. The simulation system model is illustrated in Figure 14.

The simulation utilized wind speed data obtained from an 80 MW wind farm. Due to computational limitations, only the portion of data illustrated in Figure 15 was employed for the simulation. The simulation results are presented in Figure 16 and Figure 17.

6.4. Analysis of Simulation Results

Figure 16 demonstrates that after processing by the HESS, the high-frequency components were handled by the supercapacitor, while the low-frequency components were managed by the battery. This strategy extends the life of the battery, thus enhancing the economic efficiency of the energy storage system. Figure 17 indicates that the HESS also exhibits significant effectiveness in smoothing wind power output. Stable wind power output is eventually attained by the supercapacitor and battery working together. The experimental results confirm that an HESS combining supercapacitors with lithium-ion batteries not only enhances the stability of wind power systems and reduces generation costs but also effectively mitigates wind power fluctuations, demonstrating the practical utility of this hybrid energy storage approach.

7. Conclusions

Considering the dual constraints of power fluctuation and energy storage life-cycle costs in wind-power grid-connected scenarios, an HESS architecture based on SGC modal decomposition is proposed as a hybrid supercapacitor–lithium battery energy-storage-control strategy. This architecture integrates the complementary characteristics of Li-ion batteries and supercapacitors, employing a genetic gray wolf optimization algorithm to optimize energy storage allocation. From the simulation analysis, the following conclusions can be drawn:

• The primary frequency modulation power command is decomposed using SGC modal decomposition, which decouples the original frequency modulation command into low- and high-frequency power commands. The problem of frequent output during the FM process is successfully mitigated by the lithium battery energy storage, which distributes the low-frequency power commands. The benefits of the rapid charging and the discharge properties of the flywheel energy storage are completely realized when the supercapacitor energy storage reacts to the high-frequency part of the FM command.
• The genetic gray wolf algorithm is applied to determine the ideal configuration on the basis of typical intraday FM power characteristics. The principal frequency modulation performance of the wind power production system could effectively be improved by the hybrid energy storage. The HESS outperformed the single storage configurations in terms of frequency stability under the step-load disruption test scenario.

Future research could integrate frequency-modulation signal optimization methods to further improve energy storage frequency-modulation mode decomposition and prevent energy storage deviation accumulation. Additionally, it is important to examine the effect of capacity on energy storage systems in various environments and optimize the coordination between supercapacitors and lithium-ion batteries to maximize energy storage utilization.