We have developed a comprehensive capacity optimization allocation method for a hybrid energy storage system (HESS) combining a solid oxide electrolysis cell (SOEC) and a lithium-ion battery energy storage system. Our approach aims to smooth wind power fluctuations while minimizing the initial investment cost of the HESS. The study is based on a 30 MW wind farm’s real output power data, and the effectiveness of the proposed method is validated through simulation experiments.
1. System Modeling and Transient Response Analysis
We first built a 20 kW SOEC module model and a 250 kW / 500 kWh lithium-ion battery energy storage unit model using MATLAB/Simulink. The electrochemical reactions in the SOEC during electrolysis are described by the following equations:
$$
\begin{aligned}
\text{Hydrogen electrode: } & \text{H}_2\text{O} + 2e^- \rightarrow \text{H}_2 + \text{O}^{2-} \\
\text{Oxygen electrode: } & \text{O}^{2-} \rightarrow 2e^- + \frac{1}{2}\text{O}_2 \\
\text{Overall: } & \text{H}_2\text{O} \rightarrow \text{H}_2 + \frac{1}{2}\text{O}_2
\end{aligned}
$$
The lithium-ion battery model employs an improved mathematical representation where the discharge and charge voltages are given by:
$$
\begin{aligned}
V_d &= E_0 – R i – K \frac{Q}{Q – i t} i^* – K \frac{Q}{Q – i t} i t + A \exp(-B \cdot i t) \\
V_c &= E_0 – R i – K \frac{Q}{0.1 Q + i t} i^* – K \frac{Q}{Q – i t} i t + A \exp(-B \cdot i t)
\end{aligned}
$$
Here, \(V_d\) and \(V_c\) are the discharge and charge voltages, \(E_0\) is the battery constant voltage, \(R\) is the internal resistance, \(i\) is the battery current, \(i^*\) is the filtered current, \(K\) is the polarization voltage constant, \(Q\) is the battery capacity, \(i t\) is the actual charge, and \(A\) and \(B\) are empirical coefficients. The battery energy storage system (battery energy storage system) operates with a charge/discharge efficiency of 97.5%.
We analyzed the transient response characteristics of both systems. The SOEC module can handle a maximum power ramp rate of 15% of rated power per minute (3 kW/min), corresponding to a reference signal frequency below 1/120 Hz. The lithium-ion battery energy storage system responds within 2 seconds under step power commands. To improve the SOEC’s power tracking under disturbances, we propose a feedforward model predictive control (FMPC) method. The control block diagram combines a PID controller with a feedforward path and a Smith predictor-like structure to compensate for system delays. The FMPC method significantly reduces tracking errors and enhances disturbance rejection compared to conventional MPC.
2. Signal Decomposition Methods
We employ two decomposition techniques to allocate power commands between the SOEC and the battery energy storage system. The variational mode decomposition (VMD) method decomposes a signal into intrinsic mode functions (IMFs) with specific center frequencies. The optimization problem is:
$$
\min \left\{ \sum_{m=1}^{M} \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_m(t) \right] e^{-j \omega_m t} \right\|_2^2 \right\} \quad \text{s.t.} \quad \sum_{m=1}^{M} u_m(t) = f(t)
$$
The Lagrangian form is solved using the alternating direction method of multipliers (ADMM), yielding updated modal functions and central frequencies:
$$
\hat{u}_{m, n+1}(\omega) = \frac{ \hat{f}(\omega) – \sum_{i \neq m} \hat{u}_i(\omega) + \frac{\hat{\lambda}(\omega)}{2} }{ 1 + 2\alpha (\omega – \omega_m)^2 }
$$
$$
\omega_{m, n+1} = \frac{ \int_0^\infty \omega |\hat{u}_m(\omega)|^2 d\omega }{ \int_0^\infty |\hat{u}_m(\omega)|^2 d\omega }
$$
The successive variational mode decomposition (SVMD) method extracts modes sequentially without predefining the number of modes, improving convergence and robustness. Its two minimization criteria are:
$$
J_1 = \left\| \partial_t \left[ \left( \delta(t) + \frac{j}{\pi t} \right) * u_L(t) \right] e^{-j \omega_L t} \right\|_2^2
$$
$$
J_2 = \left\| \beta_L(t) * f_r(t) \right\|_2^2
$$
An additional criterion \(J_3\) prevents extraction of previously obtained modes. The final Lagrangian for SVMD is:
$$
\phi(u_L, \omega_L, \lambda) = J_1 + J_2 + J_3 + \left\| f(t) – u_L(t) + f_L(t) + \sum_{i=1}^{L-1} u_i(t) \right\|_2^2
$$
3. Capacity Optimization Using Whale Optimization Algorithm
The whale optimization algorithm (WOA) mimics humpback whale hunting behavior. The mathematical model includes three phases: encircling prey, bubble-net attacking (with 50% probability for each of shrinking encircling and spiral updating), and searching for prey. The position update formulas are:
Encircling prey:
$$
X_f(t_a+1) = X_{\text{best}}(t_a) – A_w D_1, \quad D_1 = |C_w X_{\text{best}}(t_a) – X_f(t_a)|
$$
Spiral updating:
$$
X_f(t_a+1) = D_2 e^{b l} \cos(2\pi l) + X_{\text{best}}(t_a), \quad D_2 = |X_{\text{best}}(t_a) – X_f(t_a)|
$$
Search for prey (when \(|A_w| > 1\)):
$$
X_f(t_a+1) = X_{\text{rand}}(t_a) – A_w D_3, \quad D_3 = |C X_{\text{rand}}(t_a) – X_f(t_a)|
$$
Our capacity optimization method integrates SVMD, VMD, and WOA as follows:
- First, we set an initial SOEC power command as \(P_{\text{soec}}^* = P_w \times k\), where \(k\) is the power consumption ratio coefficient and \(P_w\) is the wind power.
- The remaining wind power \(P_w^* = P_w – P_{\text{soec,ref}}\) is decomposed by VMD into multiple IMFs. Choosing a threshold based on grid standards (1-min maximum fluctuation of 3 MW for a 30 MW wind farm), we allocate the high-frequency components to the lithium-ion battery energy storage system.
- The SOEC reference command is processed by SVMD to retain only low-frequency components (below 1/120 Hz) to meet the ramp rate constraint.
- We use WOA to optimize the two key parameters: \(k\) and the number of high-frequency IMF components assigned to the battery (denoted as \(j\)). The objective is to minimize the total initial investment cost of the HESS while ensuring that the combined output power of the wind farm and HESS meets the grid code requirements (Table 1 shows the limits for 1-min and 10-min fluctuations).
| Installed capacity \(P_N\) | 1-min limit | 10-min limit |
|---|---|---|
| < 30 | 3 | 10 |
| 30–150 | \(P_N/10\) | \(P_N/3\) |
| > 150 | 15 | 50 |
The capacity sizing of each energy storage system is determined from its reference power command. For the battery energy storage system, we calculate the required energy capacity considering the SOC limits:
$$
E_f =
\begin{cases}
\sum_{n=1}^{N} \frac{P_{x\_ref}[n]}{3600 f_s \eta_d}, & P_{x\_ref}[n] \ge 0 \\
\sum_{n=1}^{N} \frac{P_{x\_ref}[n] \eta_c}{3600 f_s}, & P_{x\_ref}[n] \le 0
\end{cases}
$$
$$
E_{\text{rated}, f} = \max\left\{ \frac{\max\{E_f[n]\}}{L_{\text{SOC,max}} – L_{\text{SOC,ref}}},\; \frac{-\min\{E_f[n]\}}{L_{\text{SOC,ref}} – L_{\text{SOC,min}}} \right\}
$$
After optimization, the actual capacities are rounded up to integer multiples of the basic module sizes (20 kW for SOEC, 250 kW/500 kWh for battery energy storage system).
4. Simulation Results
We used 24-hour wind power data from a 30 MW wind farm. Initially, with arbitrary parameters \(k = 3.5\) and \(j = 4\), the HESS consisted of 17 battery units (4.25 MW/8.5 MWh) and 37 SOEC modules (0.74 MW), costing 16.31 million CNY. After WOA optimization, we obtained \(k = 2.5\) and \(j = 3\), resulting in 13 battery units (3.25 MW/6.5 MWh) and 26 SOEC modules (0.52 MW), with a total investment cost of 11.83 million CNY – a reduction of 27.5%.
The SOEC system operates with electrolysis efficiency between 0.82 and 0.97 (above the set threshold of 0.8) and electricity consumption between 3.05 and 3.69 kWh per unit of hydrogen produced (below the limit of 3.8 kWh). The lithium-ion battery energy storage system’s SOC remains within [0.2697, 0.6421], well inside the healthy operating range [0.2, 0.8]. The combined wind-HESS output power meets the grid standard of 3 MW maximum fluctuation in 1 minute.
5. Conclusion
We have developed a capacity optimization method for a hybrid energy storage system consisting of a solid oxide electrolysis cell and a lithium-ion battery energy storage system. The method integrates feedforward model predictive control for the SOEC, variational mode decomposition techniques, and the whale optimization algorithm. Simulation results on a 30 MW wind farm demonstrate that the proposed approach effectively smooths wind power fluctuations to meet grid connection requirements while significantly reducing the initial investment cost. The SOEC achieves high electrolysis efficiency, and the battery energy storage system maintains a healthy state of charge throughout operation.