Optimal Configuration of Hybrid Energy Storage Capacity in a Microgrid Based on Variational Mode Decomposition

Abstract

The capacity configuration of the energy storage system plays a crucial role in enhancing the reliability of the power supply, power quality, and renewable energy utilization in microgrids. Based on variational mode decomposition (VMD), a capacity optimization configuration model for a hybrid energy storage system (HESS) consisting of batteries and supercapacitors is established to achieve the optimal configuration of energy storage capacity in wind–solar complementary islanded microgrids. Firstly, based on the energy mapping relationship between the frequency domain and time domain, the decomposition mode number $K$ of VMD is determined based on the principle of minimum total mode aliasing energy. Then, considering the smoothing fluctuation characteristics of different energy storage components, the dividing point N of high frequency and low frequency in the unbalanced power between the source and load in the microgrid is selected to allocate charging and discharging power instructions for the battery and supercapacitor. Finally, taking the annual comprehensive cost of the HESS as the objective function, a hybrid energy storage capacity optimization configuration model is established, and the dividing point N is used as the optimization variable to solve the model in order to obtain the optimal configuration results. The case study results show that the proposed method is more economical and feasible than the existing energy storage configuration methods.

1. Introduction

With the growing fossil energy crisis worldwide, renewable energy generation, such as wind and photovoltaic energy, has gained traction [1]. The global trend of renewable energy generation is shown in Figure 1. Since the 20th century, the development of renewable energy, represented by wind and solar energy, has been rapid [2]. The strong growth of renewables means that their share of the global power generation mix is forecast to rise from 29% in 2022 to 35% in 2025, with the shares of coal-fired and gas-fired generation falling [3]. Renewable energy is widely used to accelerate the energy structure transition and the process of global carbon neutrality. However, the inherent randomness and fluctuations in renewable energy can lead to low energy utilization and affect grid stability and power quality [4,5]. In this scenario, microgrids have emerged as a solution, given their flexible, modular, and localized attributes [6]. Microgrids enable the consumption of renewable energy and mitigate the challenges associated with their large-scale integration into the grid. In a microgrid, unbalanced power fluctuations between the source and load will also affect the safe and stable operation of the microgrid. Energy storage systems have become crucial for maintaining the microgrid’s power balance by facilitating flexible charging and discharging to smooth power fluctuations [7]. Therefore, the optimal capacity configuration of the energy storage system is the key focus.

Kerdphol et al. [8] used the particle swarm optimization algorithm to evaluate the optimal capacity of a battery energy storage system in an islanded microgrid. Yuan et al. [9] established an optimal configuration model of a photovoltaic and battery energy storage system with the variable of energy storage capacity and solved the model using the genetic algorithm. Hesaroor et al. [10] proposed using the incremental cost approach to verify the economic feasibility of the battery unit and identify the optimal size of the battery energy storage system which minimizes the running cost. These studies only used a single material, the battery, as the carrier of the energy storage system, without considering the fluctuating characteristics of renewable energy on different time scales.

Tostado-Véliz et al. [11] applied supercapacitors and batteries to tramways, with the supercapacitors handling the peak demand consumption and the batteries primarily dedicated to providing backup long-term storage capacity. Hernández et al. [12] studied the role of a hybrid energy storage system based on batteries and ultracapacitors in increasing the income of PV home household-prosumers. Batteries are energy-type energy storage components with a high energy density which are suitable for smoothing low-frequency fluctuations, while supercapacitors are power-type energy storage components with a high power density which are suitable for smoothing high-frequency fluctuations [13]. A HESS leveraging the complementary benefits of these two systems can effectively smooth fluctuations in renewable energy and improve the stability of the microgrid. Therefore, it is important to choose an appropriate power allocation method and rationally configure the capacity of the HESS for the microgrid.

At present, scholars from across the world have studied power allocation algorithms and the capacity configuration of energy storage systems, yielding several outcomes. Wu et al. [14] and Roy et al. [15] used low-pass filtering (LPF) to decompose the unbalanced power in the microgrid at high and low frequencies, achieving better suppression of fluctuations. However, LPF has a time delay, which leads to a high HESS capacity. Li et al. [16] proposed discrete Fourier transformation (DFT) to optimize the HESS. However, DFT is vulnerable to noise, which makes spectrum decomposition complicated. Li et al. [17] used wavelet decomposition (WD) to decompose the load sequence in order to reduce the nonstationary load sequence. Santos et al. [18] used wavelet packet decomposition (WPD) to decompose net load power in the time dimension. Based on WPD, Wu et al. [19] proposed using fuzzy control to adaptively manage the SOC of power-type energy storage components, enabling the optimal allocation of power. However, WPD’s limitation is its dependence on fundamental wave selection, impacting energy storage capacity. He et al. [20] proposed empirical mode decomposition (EMD) to decompose wind power generation and establish a wind power time series prediction model. However, EMD is prone to mode aliasing. Wang et al. [21] and Sanabria-Villamizar et al. [22] used ensemble empirical mode decomposition (EEMD) to effectively resolve the modal aliasing problem. Nevertheless, boundary effects could still influence the power decomposition results. Addressing these challenges, VMD could offer a viable solution to some extent [23].

In summary, energy storage systems play a crucial role in microgrids. Different energy storage components have different energy storage characteristics. The HESS, combining a high energy density and high power density, has a better performance than the single energy storage system. The commonly used LPF, DFT, WPD, EMD, and EEMD power decomposition algorithms are affected by time delays, poor accuracy, boundary effects, and mode aliasing. VMD is better at processing non-stationary abrupt signals and smoothing power fluctuations in renewable energy.

In this paper, a hybrid energy storage capacity optimization configuration model is established using VMD to decompose the unbalanced power between the source and load in a wind–solar complementary islanded microgrid as the power reference signal of the HESS. The main contributions of this study are as follows:

(1)

Combining the energy mapping relationship between the frequency domain and time domain, the decomposition mode number of VMD is determined based on the principle of minimum total mode aliasing energy, which can completely decouple the original signal and effectively solve the problem of mode aliasing and the boundary effect.

(2)

Comprehensively considering the smoothing fluctuation characteristics of different energy storage components and the state of charge (SOC), the dividing point N of high frequency and low frequency in the unbalanced power between the source and load in the microgrid is selected to allocate charging and discharging power instructions for the battery and supercapacitor in order to obtain the best smoothing fluctuation effectiveness.

(3)

Taking the annual comprehensive cost of the HESS as the objective function, a hybrid energy storage capacity optimization configuration model is established, and the dividing point N is used as the optimization variable to solve the model and obtain the optimal configuration results. We analyze the advantages of VMD compared to EMD and of hybrid energy storage compared to single energy storage through the case study results.

This paper is organized as follows. Section 2 introduces the structure of the wind–solar complementary islanded microgrid. In Section 3, an energy storage capacity allocation method based on VMD is proposed. In Section 4, the optimal configuration model of the HESS is established. Section 5 presents the case study and optimization result analysis. Finally, Section 6 provides the conclusions.

2. Structure of the Microgrid

The Department of Energy in the United States defines a microgrid as “a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries that act as a single controllable entity with respect to the grid. A microgrid can connect and disconnect from the grid to enable it to operate in both grid-connected or island-mode” [24].

The wind–solar complementary islanded DC microgrid is a small-scale power generation and distribution system composed of distributed energy systems, such as wind and solar systems, a conventional load, hybrid energy storage system, a power distribution device, a central controller, etc. [25]. This microgrid can achieve self-operation, regulation, and control [26]. Its structure is shown in Figure 2.

In this system, each unit is directly connected to the DC bus through a converter [27]. To ensure power balance and improve power quality, the unbalanced power between the output of renewable energy and the load demand in the microgrid is smoothed by the HESS, which is controlled by the central controller [28]. When the load demand is higher than the renewable energy generation, the HESS works in the discharging state to meet the load demand. When the renewable energy generation is higher than the load demand, the HESS works in the charging state to absorb the excess power [29]. The power signal $P_{HESS}\left(t\right)$ of the HESS takes the unbalanced power $P_{un}\left(t\right)$ of the source and load in the microgrid as a reference, as shown in Equation (1):

$$P_{un}(t)=P_l(t)-P_w(t)-P_v(t)$$

(1)

where $P_{un}(t)$ is the unbalanced power. $P_l(t)$ is the load power. $P_w(t)$ is the output power of wind power generation. $P_v(t)$ is the output power of photovoltaic power generation.

3. VMD for Capacity Configuration

The fundamental principle of VMD is to adaptively decompose the original signal into the intrinsic mode function (IMF) with a limited bandwidth through variational constraints. Then, the alternate direction multiplier method (ADMM) is used to iteratively search for the optimal solution of the variational model [30], with the aim of minimizing the sum of the bandwidths of all $imfs$, thus allowing the original signal to be reconstructed through stacking.

3.1. Mathematical Model of VMD

In this paper, the power of the HESS is taken as the original signal, and the variational model with constraints is constructed as follows:

$$\begin{array}{l}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{\displaystyle \sum _{k=1}^K{\Vert {\partial }_t\left\{\left[\delta (\mathrm{t})+\frac{j}{\pi t}\right]u_k(t)\right\}{e}^{-j{\omega }_{k}t}\Vert }_2^2}\right\}\\ s.t.{\displaystyle \sum _{k=1}^K{u}_k}=P_{hess}\left(t\right)\end{array}$$

(2)

where $\left\{u_k\right\}=\left\{u_1,u_2,\dots ,u_K\right\}$ is the $imf$ obtained via decomposition of the original signal. $\left\{{\omega }_k\right\}=\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_K\right\}$ is the corresponding central frequency of each $imf$. ${\partial }_t$ is the partial derivative with time $t$. $\delta (t)$ is the impulse function.

Then, the quadratic penalty term $\alpha$ and Lagrangian multiplier $\lambda (t)$ are introduced to render Equation (2) an unconstrained problem, as shown in Equation (3):

$$\begin{array}{l}L(\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda )=\alpha {\displaystyle \sum _{k=1}^K{\Vert {\partial }_t\left\{\left[\delta (t)+\frac{j}{\pi t}\right]u_k(t)\right\}{e}^{-j{\omega }_{k}t}\Vert }_2^2}\\ +{\Vert P_{hess}(t)-{\displaystyle \sum _{k=1}^K{u}_k(t)}\Vert }_2^2+\langle \lambda (t),P_{hess}(t)-{\displaystyle \sum _{k=1}^K{u}_k(t)}\rangle \end{array}$$

(3)

The ADMM is adopted to iteratively update $u_k^{n+1}$, ${\omega }_k^{n+1}$, and ${\lambda }^{n+1}$. Then, using Fourier transform [31], the updated mode functions and their corresponding center frequencies are obtained as follows:

$${\widehat{u}}_k^{\mathrm{n}+1}\left(\omega \right)=\frac{{\widehat{P}}_{hess}\left(\omega \right)-{\displaystyle \sum _{i\ne k}^K{\widehat{u}}_i\left(\omega \right)}+\frac{\widehat{\lambda }(\omega )}{2}}{1+2\alpha {(\omega -{\omega }_k)}^2}$$

(4)

$${\omega }_k^{n+1}=\frac{{\displaystyle {\int }_0^{\infty }\omega }{\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }{{\displaystyle {\int }_0^{\infty }{\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }}$$

(5)

where ${\widehat{P}}_{hess}\left(\omega \right)$, ${\widehat{u}}_i\left(\omega \right)$, and $\widehat{\lambda }\left(\omega \right)$ are the Fourier transforms of $P_{hess}\left(t\right)$, $u_i\left(t\right)$, and $\lambda \left(t\right)$, respectively.

The above iterative process stops when the condition Equation (6) is satisfied:

$${\displaystyle \sum _{k=1}^K\frac{{\Vert {\widehat{u}}_k^{n+1}-{\widehat{u}}_k^n\Vert }_2^2}{{\Vert {\widehat{u}}_k^n\Vert }_2^2}}<e$$

(6)

where $e$ is the convergence parameter.

3.2. Selection of the Decomposition Mode Number

Improper selection of the decomposition mode number in VMD may generate false components as a result of over-decomposition or cause a failure to separate the $imfs$ due to under-decomposition, leading to mode aliasing and affecting the accuracy of the configuration results.

Following the decomposition of unbalanced power in the microgrid using VMD, the normalized instantaneous frequency–time curve of each $imf$ is obtained by employing the Hilbert transform, as shown in Figure 3.

The Hilbert transform of $im{f}_i(t)$:

$$H\left[im{f}_i(t)\right]=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{im{f}_i(t)}{t-\tau }}d\tau$$

(7)

Next, we construct the analytic signal of $im{f}_i(t)$ in the polar coordinate form:

$$\begin{array}{l}{z}_i(t)=im{f}_i(t)+jH\left[im{f}_i(t)\right]\\ \hspace{1em}\hspace{1em}={\omega }_i(t)e^{(j{\phi }_i(t))}\end{array}$$

(8)

where ${\omega }_i(t)$ is the amplitude function of $z_i(t)$. ${\phi }_i(t)$ is the phase function of $z_i(t)$.

$${\omega }_i(t)=\sqrt{im{f}_i^2(t)+H^2\left[im{f}_i(t)\right]}$$

(9)

$${\phi }_i(t)=\arctan \frac{H\left[im{f}_i(t)\right]}{im{f}_i(t)}$$

(10)

The instantaneous frequency of the $imfs$:

$$f_i(t)=\frac{d{\phi }_i(t)}{dt}$$

(11)

As depicted in the figure above, each $imf$ exhibits a narrow frequency band, with its center frequency represented by ${\omega }_i\left(t\right)$, and the instantaneous frequency of each moment fluctuates around this center frequency. Due to the overlapping of the $imfs$ in the frequency domain, there is modal aliasing energy when they are mapped to the time domain [32]. We take three overlapping $imfs$ in the frequency domain and map them to the time domain to calculate the mode aliasing energy. The mode aliasing energy of $im{f}_2\left(t\right)$ mapped to the time domain is shown in Figure 4. The dotted line in the figure is the guiding line for mapping the frequency domain to the time domain in the same period.

The frequency $f_2\left(t\right)$ of $im{f}_2$ is lower than the frequency $f_3\left(t\right)$ of $im{f}_3$ in the period $(a,b)$ a. Mapping to the time domain indicates that $im{f}_2$ has low-frequency mode aliasing energy in this period. Direct summing of the products of power and time may cause the energies generated by positive and negative power to cancel each other out in the same period, making it challenging to determine the actual aliasing energy value. Thus, the absolute value of power is used for the calculation. Equation (12) represents the low-frequency aliasing energy of $im{f}_2$ in the period $(a,b)$.

$$E_{low,2}={\displaystyle \sum _{i=a}^b|im{f}_2(t)|\mathsf{\Delta }t}$$

(12)

Similarly, the frequency $f_2\left(t\right)$ of $im{f}_2$ is higher than the frequency $f_1\left(t\right)$ of $im{f}_1$ in the period $(c,d)$. Mapping to the time domain indicates that $im{f}_2$ has a high-frequency mode aliasing energy in this period. Equation (13) represents the high-frequency aliasing energy of $im{f}_2$ in the period $(c,d)$.

$$E_{high,2}={\displaystyle \sum _{i=c}^d|im{f}_2(t)|\mathsf{\Delta }t}$$

(13)

Therefore, the total mode aliasing energy of $im{f}_2$ in all periods is:

$$E_{s,2}=E_{low,2}+E_{high,2}$$

(14)

The total mode aliasing energy generalized to any $imf$ is shown in Equation (15):

$$E_{s,j}={\displaystyle \sum _{i=f}^g|im{f}_j(t)|}\mathsf{\Delta }t+{\displaystyle \sum _{i=m}^n|im{f}_j(t)|\mathsf{\Delta }t}$$

(15)

where $i=(f,g)$ is the period in which $im{f}_j$ has low-frequency aliasing energy. $i=(m,n)$ is the period in which $im{f}_j$ has high-frequency aliasing energy.

Assuming that the decomposition mode number of VMD is $K$, the total mode aliasing energy value of all $imfs$ under this decomposition number can be obtained through Equation (16):

$$E_s={\displaystyle \sum _{j=1}^K{E}_{s,j}}$$

(16)

The determination of the decomposition mode number of VMD can be achieved by incorporating the above equation, based on the principle of minimum total mode aliasing energy.

3.3. Rated Power and Rated Capacity of HESS

After decomposing the unbalanced power in the microgrid into $K$ $im{f}_s$ using VMD, an appropriate dividing point between high frequency and low frequency is selected to smooth the $imfs$ using the HESS. The reconstructed signal is then divided into two parts: the high-frequency part, which is the sum of the first N-order $imfs$, and the low-frequency part, which is the sum of $imfs$ larger than the N order. Based on the characteristics of supercapacitors suitable for smoothing high-frequency fluctuations and batteries suitable for smoothing low-frequency fluctuations, their charging and discharging power instructions can be obtained, as shown in Equation (17):

$$\{\begin{array}{l}{p}_{sc}(t)=x_H\left(t\right)={\displaystyle \sum _{i=1}^{N}im{f}_i}(t)\\ p_{bat}(t)=x_L\left(t\right)={\displaystyle \sum _{i=N+1}^{K}im{f}_i}(t)\end{array}$$

(17)

where $p_{sc}(t)$ and $p_{bat}(t)$ are the charging and discharging power instructions for the supercapacitors and batteries, respectively.

To ensure the power balance in the microgrid, the rated power of the configured energy storage components should be able to smooth the maximum unbalanced power between the source and load. Considering the charging and discharging efficiency of the energy storage component, the rated power of the supercapacitor and battery can be obtained:

$$P_{SCN}\ge \max \left\{\begin{array}{l}|\min p_{sc}(t)|{\eta }_{sc,c},\\ |\max p_{sc}(t)|/{\eta }_{sc,d}\end{array}\right\}$$

(18)

$$P_{BN}\ge \max \left\{\begin{array}{l}|\min p_{bat}(t)|{\eta }_{bat,c},\\ |\max p_{bat}(t)|/{\eta }_{bat,d}\end{array}\right\}$$

(19)

where ${\eta }_{sc,c}$ and ${\eta }_{sc,d}$ are the charging and discharging efficiency of the supercapacitor, respectively. ${\eta }_{bat,c}$ and ${\eta }_{bat,d}$ are the charging and discharging efficiency of the battery, respectively.

To ensure the safe and stable operation of the HESS, the $SOC$ of the supercapacitor and battery should be considered. The $SOC$ of the supercapacitor at the time $t$ is:

$$SO{C}_{sc}(t)=SO{C}_{sc0}-{\displaystyle \sum _{t=1}^T{P}_C(t)}\mathsf{\Delta }t/E_{SCN}$$

(20)

where $SO{C}_{sc0}$ is the initial $SOC$ value of the supercapacitor. $\mathsf{\Delta }t$ is the charging and discharging power instruction time interval. $E_{SCN}$ is the rated capacity of the supercapacitor. $P_C(t)$ is the power reference value of the supercapacitor considering the charging and discharging efficiency, as shown in Equation (21):

$$p_C(t)=\{\begin{array}{ll}{p}_{sc}(t){\eta }_{sc,c}& p_{sc}(t)\le 0 \\ p_{sc}(t)/{\eta }_{sc,d}& p_{sc}(t)>0\end{array}$$

(21)

Considering the constraint of the $SOC$, we obtain:

$$\{\begin{array}{l}\min \left[SO{C}_{sc}(t)\right]\ge SO{C}_{\min }\\ \max \left[SO{C}_{sc}(t)\right]\le SO{C}_{\max }\end{array}$$

(22)

Combining Equations (20) and (22), the rated capacity of the supercapacitor can be obtained as shown in Equation (23):

$$E_{SCN}\ge max\left\{\left|\frac{\min {\displaystyle \sum _{t=1}^T{p}_C(t)\mathsf{\Delta }t}}{SO{C}_{\max }-SO{C}_{\mathrm{sc}0}}\right|,\left|\frac{\max {\displaystyle \sum _{t=1}^T{p}_C(t)\mathsf{\Delta }t}}{SO{C}_{\mathrm{sc}0}-SO{C}_{\min }}\right|\right\}$$

(23)

Similarly, the rated capacity of the battery $E_{BN}$ can be obtained.

4. Optimal Configuration Model of the HESS

This paper comprehensively considers the initial investment cost, operation and maintenance costs, and replacement cost of the HESS. Then, the optimal configuration model of the hybrid energy storage system is established by taking the annual comprehensive cost of the HESS as the objective and the dividing point of high frequency and low frequency in the unbalanced power between the source and load as the optimization variable. The flowchart of the model is shown in Figure 5.

4.1. Objective Function

The annual comprehensive cost of the HESS is shown in Equation (24):

$$C_{tol}=C_{inv}+C_{main}+C_{rep}$$

(24)

(1)

The initial investment cost of HESS is shown in Equation (25):

$$C_{inv}=(c_{Pbat}{P}_{BN}+c_{Ebat}{E}_{BN}+c_{Psc}{P}_{SCN}+c_{Esc}{E}_{SCN})\gamma$$

(25)

where $c_{Pbat}$ and $c_{Ebat}$ are the investment cost per unit power and unit capacity of the battery. $c_{Psc}$ and $c_{Esc}$ are the investment cost per unit power and unit capacity of the supercapacitor. $\gamma$ is the capital recovery factor, as shown in Equation (26):

$$\gamma =\frac{r{(1+r)}^{T_N}}{{(1+r)}^{T_N}-1}$$

(26)

where $r$ is the discount rate. $T_N$ is the rated service lifetime of the HESS.

(2)

The operation and maintenance costs of the HESS are shown in Equation (27):

$$C_{main}=c_{bat,m}{E}_{BN}+c_{sc,m}{E}_{SCN}$$

(27)

where $c_{bat,m}$ and $c_{sc,m}$ are the annual operation and maintenance costs per unit capacity of the battery and supercapacitor.

(3)

The replacement cost of the HESS is shown in Equation (28):

$$C_{rep}=\gamma (c_{Pbat}{P}_{BN}+c_{Ebat}{E}_{BN})n_b$$

(28)

where $c_{Pbat}$ and $c_{Ebat}$ are the replacement cost per unit power and unit capacity of the battery. Since the service life of the supercapacitor is long, its replacement cost is not considered. $n_b$ is the number of battery replacements, as shown in Equation (29):

$$n_b=\lceil \frac{T_N}{T_{life}}\rceil -1$$

(29)

where $\lceil \mathrm{x}\rceil$ means that one takes the smallest integer that is not less than x.

4.2. Constraint

(1)

Power balance constraint. Ensure that the HESS can always smooth out the unbalanced power in the microgrid.

$$P_{hess}(t)=P_l(t)-P_w(t)-P_v(t)$$

(30)

(2)

Charging and discharging power constraints for batteries and supercapacitors. Ensure that the power and capacity of the HESS are within the rated range.

$$\{\begin{array}{l}-P_{SCN}\le P_C\left(t\right)\le P_{SCN}\\ -P_{BN}\le P_B\left(t\right)\le P_{BN}\end{array}$$

(31)

(3)

$SOC$ constraints for batteries and supercapacitors. Ensure the safe and stable operation of the HESS.

$$\{\begin{array}{l}SO{C}_{sc}^{\min }\le SO{C}_{sc}\left(t\right)\le SO{C}_{sc}^{\max }\\ SO{C}_{bat}^{\min }\le SO{C}_{bat}\left(t\right)\le SO{C}_{bat}^{\max }\end{array}$$

(32)

where $SO{C}_{sc}(t)$ is the $SOC$ of the supercapacitor at the time $t$. $SO{C}_{sc}^{\max }$ and $SO{C}_{sc}^{\min }$ are the upper and lower limits of the supercapacitor $SOC$. $SO{C}_{bat}(t)$ is the $SOC$ of the battery at the time $t$. $SO{C}_{bat}^{\max }$ and $SO{C}_{bat}^{\min }$ are the upper and lower limits of the battery $SOC$.

5. Case Study

In order to verify the feasibility and cost-effectiveness of the model proposed in this paper, the historical data of a microgrid in a given region on a typical day are analyzed as an example. The sampling interval is 5 min, the sampling time is 24 h, and the number of sampling points is 288. The wind power, photovoltaic power, and load data of the microgrid are shown in Figure 6.

The unbalanced power at each sampling point is the difference between the load, wind, and photovoltaic power generation in a given moment. According to Equation (1), $P_{un}(t)=P_l(t)-P_w(t)-P_v(t)$, a polynomial model is established to obtain the unbalanced power curve, as shown in Figure 7.

Before solving the capacity optimization configuration model, the relevant parameters of the HESS must be set as shown in

Table 1. Relevant parameters of the HESS.

Component	Parameter	Value
Battery	$SO{C}_{bat}^{\min }$ (%)	20
	$SO{C}_{bat}^{\max }$ (%)	80
	${\eta }_{bat}$ (%)	90
	$c_{bat,m}$ (¥/kWh)	0.05
	$c_{Pbat}$ (¥/kW)	2700
	$c_{Ebat}$ (¥/kWh)	640
	$T_N$ (year)	5
	$r$ (%)	6
Supercapacitor	$SO{C}_{sc}^{\min }$ (%)	10
	$SO{C}_{sc}^{\max }$ (%)	90
	${\eta }_{sc}$ (%)	95
	$c_{sc,m}$ (¥/kWh)	0.05
	$c_{Psc}$ (¥/kW)	1500
	$c_{Esc}$ (¥/kWh)	27,000
	$T_N$ (year)	20
	$r$ (%)	6

.

5.1. Decomposition Mode Number

According to the mapping relationship between the frequency domain and time domain, the decomposition mode number $K$ is determined based on the principle of minimum total mode aliasing energy. The values of mode aliasing energy under different decomposition mode numbers are shown in

Table 2. Mode aliasing energy under different decomposition mode numbers.

K	2	3	4	5	6	7	8
Total	10,571.69	10,980.57	13,326.62	5083.28	4142.25	8196.95	10,829.16
$im{f}_1$	4128.84	385.94	343.63	71.03	68.87	556.01	482.24
$im{f}_2$	6442.85	3221.96	875.91	512.65	483.74	972.09	762.07
$im{f}_3$		7372.67	3050.49	861.57	655.75	1090.66	775.68
$im{f}_4$			9056.59	947.28	713.41	1164.48	1660.26
$im{f}_5$				2690.75	618.63	1177.05	1477.69
$im{f}_6$					1601.85	1777.64	1103.81
$im{f}_7$						1459.02	1986.94
$im{f}_8$							2580.47

.

It can be seen from the table that the mode aliasing energy is at the minimum when $K$ = 6. Therefore, $K$ = 6 is used as the reference value to normalize the total energy of mode aliasing under different decomposition mode numbers. Figure 8 shows the normalized mode aliasing total energy curve with decomposition mode numbers ranging from 2 to 35.

When $K$ ranges from 2 to 5, it indicates an under-decomposition state, and the $imfs$ are aliased with each other. With the increase in $K$, the number of decomposed $imfs$ increases, and the separation of modes is more apparent. The total energy of modal aliasing generally decreases. When $K$ is 6, the $imfs$ are clearly separated, and the total energy of mode aliasing reaches the minimum. As $K$ continues to increase, this indicates an over-decomposition state, resulting in false components. The mode aliasing energy fluctuates up and down irregularly.

Therefore, $K$ = 6 is selected as the optimal decomposition mode number. Figure 9 is the decomposition result of VMD.

From $im{f}_1$ to $im{f}_6$, the frequency of each mode function gradually decreases, while the amplitude gradually increases. Each $imf$ can fully reflect the fluctuation characteristics of the original unbalanced power.

5.2. Frequency-Dividing Point

The HESS takes the unbalanced power in the microgrid as the reference signal. Batteries are suitable for low-frequency fluctuations, and supercapacitors for high-frequency fluctuations. If the dividing point between high frequency and low frequency in the unbalanced power is not selected reasonably, this will affect the charging and discharging power instructions of the battery and the supercapacitor and thus affect the configuration results.

With the optimal annual comprehensive cost of the HESS as the objective and the dividing point as the optimization variable, the cost curve under different dividing points is obtained, as shown in Figure 10.

From the above figure, it can be seen that the annual comprehensive cost of the HESS is the lowest when N = 4. When the dividing point is too small, the charging and discharging power instruction of the battery contains a high degree of high-frequency fluctuation, which will increase the initial investment cost of the battery and affect its service life. When the dividing point is too large, the charging and discharging power instruction of the supercapacitor contains a high degree of low-frequency fluctuation, which will increase the initial investment cost of the supercapacitor.

According to Section 3.3, the rated power and rated capacity of the HESS under different dividing points can be obtained as shown in Figure 11.

When N is less than 4, the battery contains high-frequency fluctuations, resulting in a higher rated power and capacity of the battery. As N increases, the high-frequency fluctuations are smoothed by the supercapacitor, and the capacity of the battery gradually decreases. When N is greater than 4, the supercapacitor contains more low-frequency fluctuations, resulting in a high rated power and rated capacity of the supercapacitor. Taking N = 4, the optimal configuration results are shown in

Table 3. Optimal configuration results.

Component	Rated Power/kW	Rated Capacity/kWh
Battery	76	164
Supercapacitor	32	41

.

5.3. Configuration Result Analysis

(1)

Analysis of the Cost

To compare the advantages of VMD as compared to EMD and hybrid energy storage as compared to single energy storage, four configuration schemes are analyzed. Scheme I is a single supercapacitor energy storage system. Scheme II is a single battery energy storage system. Scheme III is a hybrid energy storage system based on EMD. Scheme IV is a hybrid energy storage system based on VMD.

Table 4. Configuration results of the four schemes.

Parameter	Scheme I	Scheme II	Scheme III	Scheme IV
$P_{SCN}$ (kW)	110	-	38	32
$E_{SCN}$ (kWh)	256	-	46	41
$C_{SC,inv}$ (¥)	424,620	-	270,192	228,700
$P_{BN}$ (kW)	-	192	87	76
$E_{BN}$ (kWh)	-	592	196	164
$T_{BN}$ (year)	-	2.68	5.84	6.20
$C_{B,inv}$ (¥)	-	372,371	74,951	61,412
$C_{main}$ (¥)	12.8	29.6	12.1	10.25
$C_{tol}$ (¥)	424,632.8	372,400.6	345,155.1	290,122.3

shows the configuration results of the four schemes.

As shown in Table 4, the capacity of the battery in Scheme II is higher than the capacity of the supercapacitor in Scheme I in terms of the energy storage capacity configuration. However, the initial investment cost per unit capacity of the supercapacitor is much higher than the initial investment cost per unit capacity of the battery, resulting in the poor economics of Scheme I. Scheme II reduces the cost by 12.3% compared with Scheme I. Compared with Scheme III and Scheme IV, the configuration result based on VMD is significantly lower than that based on EMD. This is because VMD effectively avoids mode aliasing and the inability to separate modes of similar frequency. As a result, the rated capacity and rated power of the HESS are reduced, further reducing the annual comprehensive cost. VMD reduces the cost by 15.9% compared with EMD. In general, hybrid energy storage is better than single energy storage, and VMD is better than the EMD. It not only improves the economic aspect but also increases the battery life.

The configuration result of Scheme IV is optimal and can be used as the recommended scheme for a wind–solar complementary islanded microgrid energy storage configuration.

(2)

Analysis of VMD over EMD

When using the HESS to smooth the power fluctuations in renewable energy in microgrids, it is crucial to select an appropriate algorithm. The Introduction presented the common algorithms for smoothing power fluctuations, and this section mainly analyzes the advantages of VMD as compared to EMD. EMD is an adaptive time–frequency analysis algorithm that overcomes the limitation of wavelet packet decomposition, which depends on the selection of the fundamental wavefunction. However, EMD is still affected by mode aliasing. VMD has advantages in terms of frequency extraction and mode separation, which means that it can better suppress mode aliasing [33].

Figure 12 shows the charging and discharging power curves of the battery and supercapacitor based on EMD.

The charging and discharging power curve of the supercapacitor $P_C(t)$ is floating in a narrow range during the period shown in the dotted box A in the figure, indicating that the supercapacitor is continuously charging and discharging during this period. This is due to mode aliasing in the EMD, which leads to the aliasing of low-frequency fluctuations in period A.

Figure 13 shows the charging and discharging power curves of the battery and supercapacitor based on VMD.

During the time period shown in the dotted box B in the figure, the supercapacitor is in a fast charging and discharging state. This indicates that VMD suppresses mode aliasing and enables the supercapacitor to smooth the high-frequency fluctuations effectively.

The charging and discharging power curves of the battery $P_B(t)$ are rather similar between the two algorithms, reflecting the overall power deficit and power excess in the microgrid, with both smoothing the low-frequency fluctuations.

The SOC is an essential indicator of the safe and stable operation of the battery and supercapacitor. By analyzing and comparing the SOC under EMD and VMD, respectively, the feasibility of the proposed model can be demonstrated. Figure 14 shows the SOC based on EMD. Figure 15 shows the SOC based on VMD.

The trends of $SO{C}_{bat}(t)$ for the two algorithms are basically consistent and relatively flat. The battery is in a continuous charging and discharging state, indicating that as an energy-type storage component, the battery is mainly used to smooth low-frequency fluctuations.

The trend of $SO{C}_{sc}(t)$ is evident. The supercapacitor is in a fast charging and discharging state, indicating that as a power-type storage component, the supercapacitor is mainly used to smooth high-frequency fluctuations. The EMD-based $SO{C}_{sc}(t)$ is flatter than the VMD-based one. This is due to the mode aliasing of EMD, which leads to the mixing of low-frequency $imfs$ with high-frequency $imfs$, causing the supercapacitor to be in a continuous charging and discharging state during some periods. In addition, it can be seen from Figure 14 that $SO{C}_{sc}(t)$ crosses the SOC limit when using EMD, while VMD can effectively be avoided.

6. Conclusions

Based on VMD, this paper established a capacity optimization configuration model for a HESS consisting of batteries and supercapacitors to achieve the optimal configuration of energy storage capacity in a wind–solar complementary islanded DC microgrid. The following conclusions were obtained:

(1)

In order to take full advantage of the energy storage of the battery and supercapacitor, VMD is used to decompose the unbalanced power in the microgrid. The proposed decomposition mode number selection method, which is based on the principle of minimizing the total energy of modal aliasing, can prevent the configuration results from being influenced by subjective settings and significantly improve the applicability of VMD.

(2)

With the annual comprehensive cost of the HESS as the objective function and the high-frequency and low-frequency dividing points of the unbalanced power as the optimization variables, an optimal configuration model of the HESS capacity can be established. The unbalanced power is decoupled into two parts, low-frequency and high-frequency, as the charging and discharging power instructions for the battery and supercapacitor to obtain reasonable capacity configuration results. The service life of the battery in the HESS is 124.63% higher than that of the single energy storage battery. The annual comprehensive cost of the HESS is 31.68% lower than that of the single energy storage supercapacitor and 22.1% lower than that of the single energy storage battery. This indicates that the VMD-based hybrid energy storage capacity optimization configuration model proposed in this paper can not only realize the complementary advantages of power-type energy storage and energy-type energy storage but also improve the lifetime of the battery and the cost-effectiveness of the HESS.

(3)

By comparing the configuration costs, charging and discharging power curves, and SOC under both the EMD and VMD algorithms, it can be concluded that VMD is superior to EMD. The annual comprehensive cost of VMD is 15.9% lower than that of EMD. VMD can solve the problems of mode aliasing and SOC limit crossing that exist in EMD.