In electromagnetic transient conditions, the power surge of linear motor loads and the high-C-rate discharge of lithium battery packs can cause a significant output voltage drop in lithium battery energy storage systems (battery energy storage system). This directly affects the stability of electromagnetic launch systems. Additionally, the peaking phenomenon caused by initial state errors in constant-gain extended state observers (ESO) deteriorates controller performance. To address these challenges, I propose a variable-gain active disturbance rejection control (VG-ADRC) chopping compensation strategy based on a conjugate pole method for configuring nonlinear extended state observer (NESO) parameters.
This paper focuses on developing a robust voltage compensation control for a high-power battery energy storage system used in pulsed power applications. The proposed method attenuates the peaking of total disturbance observation, enhances the compensation capability of the controller, and reduces the output voltage drop of the battery energy storage system. By designing a variable-gain control law and utilizing conjugate pole placement for NESO, the observer complexity is reduced while disturbance observation ability is improved. Experimental results from a pulsed high-power discharge platform validate the effectiveness of the proposed strategy.
1. Introduction
The battery energy storage system is a core component in electromagnetic launch systems, providing instantaneous high power to linear motor loads through high-C-rate discharge. However, during transient operation, the internal resistance of lithium batteries causes voltage drops that may not meet the voltage requirements of the downstream inverter. Furthermore, the increasing power demand of linear motors exacerbates this voltage drop, threatening system stability. Traditional PID control or feedforward control methods either lack disturbance rejection capability or require precise disturbance models, which are difficult to obtain due to the nonlinearity of the battery energy storage system and the complexity of the inverter system.
Active disturbance rejection control (ADRC) has emerged as a model-free approach that estimates and compensates total disturbances in real time. However, conventional linear ESO (LESO) suffers from the peaking phenomenon during initial transient, where large initial errors cause high gains that degrade control performance. To mitigate this, I develop a variable-gain ADRC (VG-ADRC) strategy that gradually increases the controller gain, smoothing the transient response. Additionally, I employ a conjugate pole method to configure NESO parameters, increasing the equivalent bandwidth of the disturbance observation transfer function and improving the response speed of the battery energy storage system.
2. Modeling of the Battery Energy Storage System
The pulsed power system consists of the battery energy storage system, parallel active neutral point clamped three-level inverter systems (ANPC-TLIS), and linear motor loads. The battery energy storage system uses two series-connected N+1 level dynamic chopping (N+1-LDC) converters. Each converter is formed by N lithium battery packs in series with one Buck converter module. The equivalent circuit of a single N+1-LDC converter is shown in the original reference. The modeling assumptions include identical battery parameters, steady-state operation, and ideal switching devices. Using state-space averaging, the small-signal transfer functions are derived as follows:
| Parameter | Symbol | Value |
|---|---|---|
| Single battery voltage | V₁ | 268 V |
| Series battery voltage | V₂ | 1072 V |
| Internal resistance of V₁ | Rr1 | 20 mΩ |
| Internal resistance of V₂ | Rr2 | 190 mΩ |
| Filter inductance | Lf | 0.3 mH |
| Bus capacitance | Cbus | 44 mF |
| Switching frequency (converter) | fs1 | 2 kHz |
| Switching frequency (inverter) | fs2 | 1.25 kHz |
From Eq. (1) and (2) of the original paper, the control-to-output transfer function of the current inner loop and voltage outer loop can be derived. The current loop is designed with a PI controller, and its closed-loop transfer function can be approximated as a first-order system with a bandwidth of 500 Hz. Therefore, the voltage outer loop plant is considered as a first-order plant suitable for a first-order ADRC design.
3. Variable-Gain ADRC Design and Analysis
3.1 NESO Parameter Design
I consider a first-order system: $$ \dot{y} = f(y,w) + b u $$ where y is the output (bus voltage), w is external disturbance, b is system gain, u is control input. Define state variables: x₁ = y, x₂ = f(y,w) = fsum (total disturbance). The state-space model becomes:
$$
\begin{cases}
\dot{x}_1 = x_2 + b_0 u, \\
\dot{x}_2 = h_f,
\end{cases}
$$
where b₀ is the estimated gain, hf is the derivative of fsum. The NESO is designed using the fal(·) function:
$$
fal(e_v, \alpha, \delta) =
\begin{cases}
|e_v|^\alpha \cdot \text{sign}(e_v), & |e_v| > \delta, \\
e_v / \delta^{1-\alpha}, & |e_v| \le \delta.
\end{cases}
$$
Observer dynamics:
$$
\begin{cases}
e_v = U_O – z_1, \\
\dot{z}_1 = z_2 + b_0 u + q_1 \, fal(e_v, \alpha_1, \delta_1), \\
\dot{z}_2 = q_2 \, fal(e_v, \alpha_2, \delta_1),
\end{cases}
$$
where UO is the sampled output voltage, z₁ tracks x₁, z₂ tracks x₂ (total disturbance). Using the bandwidth method, the nominal gains are q₁ = 2ωo, q₂ = ωo², with ωo = 1000 rad/s. The controller gain kp = ωc = 250 rad/s.
3.2 Variable-Gain Control Law
The peaking phenomenon arises because the initial observation error ev is large, causing a large transient in z₂. To mitigate this, I introduce a time-varying gain km that smoothly increases from a small value to the nominal kp:
$$
k_m = 2(1 + e^{-k (t-t_0)})^{-1},
$$
where k determines the convergence speed, and t₀ is determined by the linear motor trajectory. The control law becomes:
$$
\begin{cases}
e_r = V_{ref} – z_1, \\
u_0 = k_m k_p \, fal(e_r, \alpha_3, \delta_2), \\
u = (u_0 – z_2) / b_0.
\end{cases}
$$
Simulation results from the original paper confirm that the proposed VG-ADRC significantly reduces the peak of the total disturbance observation compared to conventional ADRC and adaptive ESO, without sacrificing initial observation accuracy.
3.3 Frequency Characteristics of Disturbance Observation
For NESO, the transfer function from true disturbance fsum to observed disturbance z₂ is:
$$
\frac{z_2}{f_{sum}} = \frac{q_2 \, k_{g2}(e_v)}{s^2 + q_1 k_{g1}(e_v) s + q_2 k_{g2}(e_v)},
$$
where kg(e) = fal(e,α,δ)/e. When kg1 = kg2 = 1, it reduces to the linear ESO case: $$ \frac{z_2}{f_{sum}} = \frac{q_2}{s^2 + q_1 s + q_2}.$$
To improve the disturbance tracking bandwidth, I adopt a conjugate pole parameter configuration. Instead of the standard q₁ = 2ωₒ, I set q₁ = ωₒ, leading to a higher equivalent bandwidth. The Bode plot analysis shows that this configuration increases the magnitude response at low frequencies while maintaining the same high-frequency roll-off, thus improving transient response without amplifying noise.
| Parameter | Symbol | Value |
|---|---|---|
| Controller bandwidth | ωc | 250 rad/s |
| Observer bandwidth | ωo | 1000 rad/s |
| Estimated gain | b₀ | 45.26 |
| Voltage proportional gain | kvp | 7.989 |
| Voltage integral gain | kvi | 600 |
| Current proportional gain | kip | 0.004 |
| Current integral gain | kii | 5 |
| Nonlinear factor α₁ | α₁ | 0.9 |
| Nonlinear factor α₂ | α₂ | 0.7 |
| Nonlinear factor α₃ | α₃ | 0.6 |
| Nonlinear interval δ₁ | δ₁ | 10 |
| Nonlinear interval δ₂ | δ₂ | 15 |
4. Experimental Validation and Analysis
Experiments were conducted on a pulsed high-power discharge platform. The controller was implemented using DSP for computation and FPGA for PWM generation. The parameters of the battery energy storage system and the controller are listed in Tables 1 and 2. Three-phase AC current waveforms reached a peak of 5000 A, with a ramp-up from 0.1 to 0.3 s, and then stabilized at 50 Hz for 0.3 to 0.6 s, representing the linear motor load profile.
Figure 1 shows the experimental setup of the pulsed high-power discharge platform used to verify the proposed control strategy.
The total disturbance observation waveforms are compared in Figure 1 of the original paper. The conventional LADRC with fixed gain exhibits a distinct peaking phenomenon at startup due to initial state errors. In contrast, the proposed VG-ADRC attenuates this peak substantially, confirming the simulation results. During steady-state operation, the total disturbance contains fundamental, second-order, and third-order ripple components caused by load imbalance and inverter modulation. Compared to LADRC, VG-ADRC shows a slight phase lag, consistent with the frequency analysis indicating a slower phase response but improved magnitude tracking.
Full bus voltage waveforms are presented in Figure 2 of the original paper. Under PI control, the initial voltage drop is 88 V. Under conventional LADRC with control saturation, the peaking effect causes a larger drop of 215 V, which could degrade inverter performance. The proposed VG-ADRC with conjugate pole NESO reduces the voltage drop significantly and achieves a faster recovery. Specifically, VG-ADRC compensates the bus voltage 0.12 s earlier than the control-limited LADRC. Moreover, the conjugate pole configuration enables direct transition to steady state without overshoot, demonstrating superior disturbance rejection capability of the battery energy storage system.
5. Conclusion
In this paper, I have proposed a variable-gain active disturbance rejection control (VG-ADRC) chopping compensation strategy for high-power battery energy storage systems under electromagnetic transient conditions. The key contributions are:
- Design of a variable-gain control law that attenuates the peaking phenomenon of total disturbance observation, improving the transient response of the battery energy storage system.
- Introduction of a conjugate pole method for NESO parameter configuration, which increases the equivalent bandwidth of the disturbance observation transfer function and enhances the voltage compensation speed.
- Experimental validation on a pulsed high-power discharge platform, demonstrating that the proposed VG-ADRC strategy reduces the output voltage drop of the battery energy storage system compared to conventional LADRC and PI control, ensuring stable operation of the downstream inverter.
The proposed method is effective for high-power battery energy storage systems where rapid voltage regulation and robustness to disturbances are critical. Future work will focus on extending this approach to multi-module parallel battery energy storage systems and optimizing the variable-gain function for different load profiles.