Economic Optimization of Hybrid Energy Storage Capacity for Wind Power Based on Coordinated SGMD and PSO

Abstract

Under the dual carbon objectives, wind power penetration has accelerated markedly. However, the inherent volatility and insufficient peak regulation capability in energy storage allocation hamper efficient grid integration. To address these challenges, this paper presents a hybrid storage capacity configuration method that combines Symplectic Geometry Mode Decomposition (SGMD) with Particle Swarm Optimization (PSO). SGMD provides fine-grained, multi-scale decomposition of load–power curves to reduce modal aliasing, while PSO determines globally optimal ESS capacities under peak-shaving constraints. Case-study simulations showed a 25.86% reduction in the storage investment cost compared to EMD-based baselines, maintenance of the state of charge (SOC) within 0.3–0.6, and significantly enhanced overall energy management efficiency. The proposed framework thus offers a cost-effective and robust solution for energy storage at renewable energy plants.

1. Introduction

Driven by the dual carbon objectives, the global energy landscape is undergoing a rapid shift toward low-carbon generation, with the wind power penetration in modern power systems rising sharply [1]. However, the inherent randomness and volatility of wind generation impair reliable peak regulation and complicate load matching, while single-mode energy storage schemes often suffer from poor economic efficiency and limited dispatch flexibility [2,3,4].

A range of decomposition-based storage optimization techniques have been proposed. Zheng Hao et al. applied Empirical Mode Decomposition (EMD) for the joint power–capacity optimization of a lithium-ion battery ESS, which yielded moderate scheduling improvements but was still prone to modal aliasing [5,6]. Jiangbo et al. extended this to a hybrid ESS by combining Ensemble EMD (EEMD) with EMD and accounting for battery degradation. This decoupled the frequency components, but the system was still affected by aliasing errors [7,8]. Yan Qunming’s team improved CEEMD for photovoltaic signal decomposition and coordinated supercapacitor–battery smoothing across scales [9]. Zhang Ping et al. leveraged PSO-tuned Variational Mode Decomposition (VMD) alongside exponential smoothing to separate high- and low-frequency storage power. However, VMD creates high computational complexity [10]. Shi Linjun et al. introduced a battery-grouping control strategy optimized via NSGA-II to enhance economic efficiency and the life cycle but focused solely on electrochemical storage [11]. While these approaches advance individual aspects, no unified framework simultaneously reduces modal aliasing, ensures global economic optimization, and maintains computational efficiency.

To address this gap, we propose a coordinated Symplectic Geometry Mode Decomposition–Particle Swarm Optimization (SGMD–PSO) framework. SGMD performs fine-grained, multi-scale decomposition of load power signals to effectively reduce modal aliasing with low computational overhead; PSO then determines the globally optimal ESS capacities under peak-shaving constraints. Compared to EMD-based baselines, the proposed method achieves both precise power decomposition and significant economic gains. Simulation results demonstrated a 25.86% reduction in the storage investment cost and maintenance of the state of charge (SOC) within 0.3–0.6, showing a marked improvement in system stability and overall energy management performance.

The remainder of this paper is organized as follows: Section 2 introduces the SGMD-PSO methodology. Section 3 describes the case study and simulation setup. Section 4 presents and discusses the results. Finally, Section 5 summarizes the conclusions and outlines future work.

2. Energy Storage Configuration Strategy Considering Peak-Shaving Effectiveness

2.1. Peak-Shaving Power Curve of Hybrid Energy Storage System

In the peak-shaving process of a hybrid energy storage system, the original power curve is first analyzed to ensure that the power fluctuations comply with national standards [12], as shown in

Table 1. National standards for wind power output fluctuations in China.

$\mathbf{Installed} \mathbf{Capacity} \mathbf{of} \mathbf{Wind} \mathbf{Farm} ({\mathit{P}}_{\mathit{N}}/\mathit{M}\mathit{W}$)	10-Min Maximum Active Power Variation/MW	1-Min Maximum Active Power Variation/MW
$P_N<30$	$10$	$3$
$30\le P_N\le 150$	$P_N/3$	$P_N/10$
$P_N>150$	$50$	$15$

. On this basis, the load–power curve is fitted using the cubic smoothing spline (csaps) method. Compared with conventional interpolation methods, the cubic smoothing spline method introduces a smoothing factor to control the smoothness of the fitted curve, effectively avoiding overfitting and unnecessary fluctuations.

Based on the fitted load–power curve, the peak-shaving power curve—i.e., the reference power for the energy storage operation—is obtained by calculating the difference between the fitted curve and the original power curve:

$$P_{storage}\left(t\right)=P_{orig}\left(t\right)-P_{load}\left(t\right)$$

(1)

where $P_{storage}(t)$ is the reference power for the energy storage operation, $P_{orig}(t)$ is the original wind power output, and $P_{load}\left(t\right)$ is the fitted load power.

This adjusted peak-shaving power curve can effectively reduce power fluctuations while ensuring compliance with national fluctuation standards. It also enables flexible adjustments and rapid responses to the grid’s peak regulation demands. Ultimately, the resulting peak-shaving power curve provides a precise reference for the charge–discharge strategy of the hybrid energy storage system, contributing to stable grid load regulation and efficient coordination of the wind–solar–storage system.

Through this method, the hybrid energy storage system can perform precise scheduling across different time scales based on the peak-shaving power curve, thereby enhancing the system’s peak regulation capability and meeting the grid’s dual requirements of stability and economic efficiency.

2.2. Empirical Mode Decomposition Algorithm

Empirical Mode Decomposition (EMD) is a classical signal decomposition method proposed by Huang et al. that can effectively extract local feature signals from complex signals at different time scales. The specific procedure is shown in Figure 1.

This method first extracts the local maxima and minima from the original signal to form the upper and lower envelopes, respectively. The mean of these envelopes is then taken as a low-frequency component of the signal. This low-frequency component is subtracted from the original signal, and the above process is repeated on the resulting new signal until the convergence condition in Equation (2) is met. The resulting $h_1^k(t)$ is then taken as the first Intrinsic Mode Function (IMF) component ($c_1(t)$) in the EMD process [13].

$${\displaystyle \frac{{\displaystyle \sum {[h_1^{k-1}(t)-h_1^k(t)]}^2}}{{\displaystyle \sum {[h_1^{k-1}(t)]}^2}}}\le \epsilon$$

(2)

where $h_1^k(t)$ is the new signal after subtracting the $k$-th mean envelope and $\epsilon$ is the sifting threshold, typically set between 0.2 and 0.3.

After obtaining the first IMF component, it is subtracted from the original signal to generate a new signal with one IMF removed. The above process is then repeated until the residual becomes a monotonic function or a constant value, at which point the EMD is terminated.

2.3. Symplectic Geometry Mode Decomposition Algorithm

Symplectic Geometry Mode Decomposition (SGMD) is a signal processing method based on symplectic geometry analysis. This method applies a symplectic similarity transformation to process the Hamiltonian matrix and obtain its eigenvalues, then reconstructs individual component signals using the corresponding eigenvectors. These eigenvectors can be converted into reconstructed time series, which are referred to as symplectic geometry components (SGCs) [14,15]. By leveraging the properties of symplectic similarity transformation, SGMD can not only perform effective signal decomposition but can also better preserve the intrinsic features of the original signal and mitigate the issue of modal aliasing. The underlying principle is as follows (Figure 2):

Let $x=x_1,x_2,\cdot \cdot \cdot ,x_n$ be the original time series. Based on Takens’ theorem, phase space reconstruction is applied to generate the multi-dimensional phase space trajectory matrix ($X$):

$$X=\left[\begin{array}{cccc}{x}_1& x_{1+\tau }& \cdot \cdot \cdot & x_{1+(d-1)\tau }\\ x_2& x_{2+\tau }& \cdot \cdot \cdot & x_{2+(d-1)\tau }\\ ⋮& ⋮& \ddots & ⋮\\ x_m& x_{m+\tau }& \cdot \cdot \cdot & x_{m+(d-1)\tau }\end{array}\right]$$

(3)

where $\tau$ is the time delay and $d$ is the embedding dimension. Both are obtained by analyzing the power spectral density of the original signal.

After constructing the trajectory matrix, autocorrelation analysis is performed to compute the covariance matrix ($A$), which is then used to construct the Hamiltonian matrix ($M$):

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

(4)

Using the Hamiltonian matrix ($M$) and the Householder matrix ($H=\left[\begin{array}{cc}Q& 0\\ 0& Q\end{array}\right]$), the upper triangular Hessenberg matrix ($B$) is constructed, as shown in Equation (5).

$$HM{H}^T=\left[\begin{array}{cc}B& 0\\ 0& -B^T\end{array}\right]$$

(5)

According to the relationship between the matrix eigenvalues, $\lambda (A)=\lambda (B)={\lambda }^2(X)$, the eigenvalues of matrix $X$ can be expressed as follows:

$${\sigma }_i=\sqrt{{\lambda }_i}(i=1,2,\cdots ,d)$$

(6)

We arrange the eigenvectors of $X$ in descending order to form the eigenvector matrix ($Q_i$) and define $N_i=Q_i^T{X}^T$ and $M_i=Q_i{N}_i$. Then, the initial reconstructed component matrix ($Z_{i(m\times d)}$) is given by the following equation:

$$Z_i=M_i^T$$

(7)

Thus, the reconstructed phase space time matrix is as follows:

$$Z=Z_1+Z_2+\cdots +Z_d$$

(8)

After reconstruction, diagonal averaging is performed to convert it into $\mathrm{d}$ sets of one-dimensional time series ($Y_i$), as shown in Equation (9). The original time series can then be obtained by summing these $\mathrm{d}$ one-dimensional sequences:

$$Y=Y_1+Y_2+\cdots +Y_d$$

(9)

At this point, the $\mathrm{d}$ sets of one-dimensional time series have not yet achieved complete independence, and further reconstruction is required.

Because the dominant components of the reconstructed original time series are mainly located in the leading part of the matrix, $Y_1$ is compared with other components based on certain features. The components with high similarity are summed to obtain the first decomposed component ($SG{C}_1$). The same procedure is then repeated after removing this component until the remaining components satisfy the normalized mean square error criterion. At this point, the iteration stops, yielding the final set of symplectic geometry components:

$$x(n)={\displaystyle \sum _{h=1}^{N}SG{C}^h(n)}+g^{(N+1)}(n)$$

(10)

where $g$ denotes the final residual signal.

In this study, we set the normalized mean square error (NMSE) threshold parameter, threshold_nmse, to 0.3, which was revised from the literature-recommended value of 0.95. A lower NMSE threshold enables the extraction of more mode components for higher decomposition accuracy at the expense of slightly increased computation costs. Through preliminary experiments, we determined that 0.3 struck a good balance between decomposition precision and computational efficiency.

2.4. Particle Swarm Optimization

Particle Swarm Optimization (PSO) is an intelligent optimization technique that searches from a spatial perspective and gradually converges toward the global optimal solution. It is well suited to dynamic and multi-objective optimization problems [16,17]. This study employed Particle Swarm Optimization (PSO) for energy storage capacity configuration, primarily because of its fast convergence, simple parameter tuning, and strong performance on continuous high-dimensional problems (Figure 3). In contrast, Genetic Algorithms (GAs) require careful design of crossover and mutation operators and are prone to premature convergence [18,19], while Ant Colony Optimization (ACO) is less efficient for continuous-variable optimization [20]. Due to limitations related to experimental resources and manuscript length, simulation experiments using other algorithms were not included and will be pursued in future work.

PSO is initialized with a swarm of randomly distributed particles that iteratively search for the global optimum. In each iteration, the particles update their positions and velocities by considering both their own best-known position and the best-known position of the entire swarm.

3. Peak Regulation Control and Economic Optimization Strategy for Hybrid Energy Storage Systems

3.1. Hybrid Energy Storage System Configuration

With the increasing penetration of renewable energy, energy storage technologies have also developed rapidly. Based on the different methods of storing electrical energy within energy storage systems, these technologies can be broadly categorized into four types: mechanical energy storage (including flywheel energy storage, pumped hydro storage, and compressed air energy storage), electrochemical energy storage, electromagnetic energy storage, and thermal energy storage [21,22,23].

A properly configured hybrid energy storage system should include energy-type storage capable of delivering large output power with relatively smooth fluctuations, as well as power-type storage capable of responding rapidly to power variations. Such a configuration enables a system to meet energy storage demands across multiple time scales, achieving good performance in both fast power regulation and long-term energy balancing. Moreover, selecting energy storage technologies with high stability and long service lives allows for sustained stable operation over extended periods, reducing the costs and operational risks associated with frequent equipment replacement. This approach leads to a lower total lifecycle cost and a higher return on investment.

To this end, in this study flywheel energy storage (FESS)—with a fast response and high stability—was selected as the power-type storage system to smooth out high-frequency fluctuations. Meanwhile, liquefied compressed air energy storage (LAES), characterized by low energy costs and a high energy density, was chosen as the energy-type storage system to mitigate low-frequency fluctuations. Their economic indicators are shown in

Table 2. Comparison of technical and economic characteristics of various energy storage systems.

Storage Type	Discharge Duration	Main Advantages	Power Cost (CNY/kW)	Energy Cost (CNY/kWh)	Operation and Maintenance Cost (CNY/kWh)
Flywheel Energy Storage	Seconds to Minutes	Fast response, long service life, excellent stability, and high power density.	3000–6000	3000–8000	0.008
Pumped Hydro Storage	4–10 h	Large storage capacity, long service life, and rapid load response, but significantly constrained by geographical conditions.	5200–6480	200–1200	0.006
Liquefied Compressed Air Energy Storage	Minutes–28 h	High energy density, relatively fast response, long service life, and low-cost storage using atmospheric pressure in liquefied tanks.	1500–3000	400–1000	0.033
Lithium-Ion Battery	1–4 h	Fast charging and discharging capability and high energy density but a relatively limited service life.	2000–4000	1000–2500	0.010
Lead–Acid Battery Energy Storage	6–10 h	Short service life, but offers the advantage of low costs.	800–1500	600–1000	0.020
Superconducting Magnetic Energy Storage	Virtually unlimited	Fast response time and high operational efficiency.	4000–8000	2000–4000	0.006

.

In this study, only the capacity configuration of the flywheel energy storage (FESS) and liquefied compressed air energy storage (LAES) was optimized, assuming all upstream components (the wind farm output, grid conditions, generators, and loads) were known and fixed, to focus on storage capacity and power allocation. “Reconstruction” refers solely to the optimal power allocation between the FESS and the LAES, excluding other storage types, generator configurations, and locations. Type and location optimization will be addressed in future work.

The investment cost of the energy storage system is denoted as $F$.

$$F=C_{inv,P}{P}_{rated}+C_{inv,E}{E}_{rated}+C_{OM}$$

(11)

where $C_{inv,P}$ and $C_{inv,E}$ are the unit power cost and unit energy cost, respectively; $P_{rated}$ and $E_{rated}$ represent the rated power and rated energy capacity of the configured energy storage system, respectively; and $C_{OM}$ denotes the operation and maintenance (O&M) cost of the energy storage system.

In this study, $C_{OM}$ was calculated as the product of the total energy discharged during the simulation period and the benchmark O&M price (CNY/kWh), thereby capturing both routine maintenance and long-term degradation expenses. Because we focused on the one-time upfront investment for the capacity configuration, market dynamics such as electricity price volatility were not considered and will be addressed in future work.

3.2. Construction of an Optimization Model for Hybrid Energy Storage System Configuration

The capacity configuration of a hybrid energy storage system must be optimized under constraints such as the state of charge (SOC) and system power balance. To address this, a configuration strategy is proposed in this paper, which integrates SGMD and PSO for coordinated optimization. The specific process is illustrated in Figure 4.

Based on the obtained reference power of the hybrid energy storage system, power decomposition is performed to extract characteristic components ($SG{C}_n$). These components are then used to allocate the reference power for energy storage. The extracted $SGC$ s are assigned as reference power inputs to the FESS and the LAES. If the first $i$ components are designated as the reference power for the FES, the corresponding reference power for the LAES is given by ${\displaystyle \sum _{\mathrm{n}=i+1}^{n}SG{C}_n}$. Thus, by reconstructing the combinations of different components, $n$ power allocation schemes can be obtained. Therefore, in this study the Particle Swarm Optimization (PSO) algorithm was employed to identify the most economical configuration that met the power-smoothing requirements for each allocation scenario. To enhance PSO’s convergence efficiency and solution stability in the hybrid energy storage capacity configuration problem, we followed empirical guidelines from [16,17] and evaluated population sizes of 30, 50, and 80, as well as iteration counts of 100, 200, and 300. The results show that increasing the population size from 30 to 50 yielded a significant reduction in the total configuration cost, while further increasing it to 80 provided only marginal cost improvements at the expense of substantially longer computation times. Likewise, raising the number of iterations from 100 to 200 led to a notable cost decrease, whereas extending to 300 iterations offered minimal additional benefit but greatly increased the computational effort. Based on these findings, a population size of 50 and 200 iterations were adopted for all subsequent PSO simulations in this study.

4. Results

In this study, actual power data from a wind farm in Shanghai with an installed capacity of 45.2 MW and a duration of 81 h were selected for analysis (with a sampling interval of 1 min), as shown in Figure 5. By comparing the configuration schemes obtained using two different decomposition algorithms, the effectiveness of the proposed strategy was validated.

According to the Chinese national standards for wind power grid-connection fluctuation limits, for a wind farm with an installed capacity of 45.15 MW, the maximum allowable active power fluctuations are 4.52 MW within 1 min and 15.05 MW within 10 min. Using the csaps algorithm, the fitted load demand–power curve was obtained, as shown in Figure 6. The fitted curve has a 1 min maximum power fluctuation of 1.5 MW and a 10 min fluctuation of 14.75 MW, both of which comply with the national wind power grid-connection standards. Based on this load–power curve, the reference power required for the energy storage operation was calculated, and it is shown in Figure 7.

Based on the obtained reference power for the hybrid energy storage operation, two decomposition methods—EMD and SGMD—were used to perform power decomposition, yielding IMF components and SGCs, respectively. The resulting modal decomposition diagrams are shown in Figure 8.

As shown in Figure 8, the EMD method yielded 10 IMF components, while the SGMD method produced six SGCs. Based on the strategy proposed in Section 3.2 of this paper, the high- and low-frequency power components were reconstructed from the decomposed results of both methods. All possible reconstruction schemes are enumerated and illustrated in Figure 9 and Figure 10.

Among the reconstructed schemes, the PSO algorithm was employed to search for the economically optimal configuration that met the power-smoothing requirements. In the optimization process, the high-frequency components in each scheme were used as the reference power for the FESS, while the low-frequency components served as the reference power for the LAES. The constraints of the power and energy capacities, as well as the overall system power balance, were fully considered. The optimal economic configurations obtained from both decomposition methods are summarized in

Table 3. Optimized configuration results obtained via PSO.

Energy Storage Configuration	EMD-Based Scheme	SGMD-Based Scheme
FESS Power (MW)	12.17	6.70
FESS Capacity (MWh)	8.94	0.61
LAES Power (MW)	2.23	9.99
LAES Capacity (MWh)	10.69	10.00
Total Configuration Cost (CNY·104)	12,371.85	9172.17

.

For the EMD-based optimization, the optimal reconstruction scheme was obtained at k = 4, meaning that IMF components 1 to 4 were considered high-frequency components and assigned to the FESS system, while the remaining components were handled by the LAES system. The total configuration cost in this case was CNY 123.71 million. For the SGMD-based optimization, the optimal scheme was found at k = 5, where SGCs 1 to 5 were treated as low-frequency components and allocated to the LAES system. The resulting total configuration cost was CNY 91.72 million.

To further verify the effectiveness of the proposed strategy, a comparative analysis of the SOC performance of the energy storage systems was conducted, as illustrated in Figure 11.

As shown in Figure 11a, the hybrid energy storage system based on EMD exhibited pronounced overcharging and overdischarging over extended periods, with large SOC fluctuations that could compromise system stability and shorten equipment lifespans. These fluctuations also substantially increased the required storage capacity and the overall configuration cost. In contrast, as shown in Figure 11b, under the SGMD scheme, the SOC remained predominantly within the 0.3–0.6 range, indicating smoother energy-state management that supports system stability and prolongs equipment service lives.

5. Conclusions

To address the common issues of insufficient peak-shaving capability and poor economic efficiency in energy storage configurations for renewable energy plants, this paper proposes an economic optimization strategy for hybrid energy storage capacity based on the coordinated application of Symplectic Geometry Mode Decomposition (SGMD) and Particle Swarm Optimization (PSO). The simulation results demonstrate that the proposed method achieved a 25.86% reduction in the total system investment cost compared to conventional approaches while maintaining the state of charge (SOC) within the optimal range of 0.3–0.6. This framework enables refined multi-time-scale power decomposition and effectively mitigates modal aliasing among components, thereby enhancing both peak-shaving performance and overall system stability. Future work will explore the integration of additional intelligent optimization algorithms and their validation in practical engineering applications to provide a more comprehensive solution for energy storage configuration at renewable energy stations.

To demonstrate full viability, future work will include hardware-in-the-loop (HIL) simulations and a pilot-scale demonstration in a microgrid testbed. We will detail the required ESS hardware specifications, real-time controller implementation of the SGMD–PSO algorithm, communication and SCADA integration, and performance evaluation under field conditions. These empirical studies will provide concrete evidence of this framework’s applicability in practical renewable energy installations.