We investigate the self-adaptive and fast state-of-charge (SOC) balancing control for high-voltage transformerless battery energy storage systems. In such systems, cascaded H-bridge converters integrate battery clusters directly to the medium-voltage grid without a transformer, offering high efficiency and scalability. However, achieving fast SOC balancing among battery clusters while respecting modulation ratio limits and battery current ratings is challenging due to the coupling between inter-phase and intra-phase balancing actions. Traditional fixed balancing coefficients are often conservative, leading to slow convergence. In this work, we derive the optimal intra-phase and inter-phase SOC balancing coefficients that fully utilize the balancing capability under varying operating conditions, and propose a decoupled adaptive control strategy to ensure the fastest possible SOC equalization without over-modulation or over-current. We validate the effectiveness through simulations on a 10 kV/5 MW/5 MWh system.
Introduction
Large-scale battery energy storage systems are essential for integrating renewable energy sources into the power grid. Among various topologies, the high-voltage transformerless battery energy storage system based on cascaded H-bridge converters has attracted significant attention due to its ability to directly connect to the medium-voltage grid without a bulky line-frequency transformer. This topology divides the large battery bank into multiple small battery clusters, each connected to a separate H-bridge module. The modular structure enables fine-grained control over each battery cluster, improving safety and capacity utilization. However, because of manufacturing tolerances and aging, battery clusters inevitably exhibit SOC differences. If left unaddressed, these differences can lead to underutilization of capacity and accelerated degradation.
To mitigate this, SOC balancing control is implemented at two levels: inter-phase balancing (between the three phases) and intra-phase balancing (among modules within a phase). For star-connected systems, inter-phase balancing is achieved by injecting a zero-sequence voltage to redistribute active power among phases. Intra-phase balancing in carrier phase-shift (CPS) modulation-based systems is achieved by superimposing a fundamental-frequency voltage proportional to the SOC error onto each module’s modulation signal. The balancing speed depends on the balancing coefficients (K0 for inter-phase, Kk for intra-phase). If these coefficients are too large, battery current may exceed its rated limit or the modulation voltage may go beyond the DC bus voltage (over-modulation). Conversely, if they are too small, SOC equalization becomes unacceptably slow.
Existing methods often set the coefficients conservatively to avoid violations, sacrificing speed. Some adaptive approaches have been proposed, but they either neglect the coupling between inter-phase and intra-phase balancing or fail to consider all constraints (battery current limits, modulation ratio limits, and operating condition variations). In this paper, we aim to bridge this gap by deriving closed-form expressions for the maximum permissible balancing coefficients under any operating condition (power factor, current amplitude, battery voltage, SOC distribution). We then propose a decoupled control that optimally allocates the balancing capability between inter-phase and intra-phase actions, ensuring the fastest convergence while staying within safe boundaries. We validate our approach using a detailed simulation model.
System Description and SOC Balancing Framework
The high-voltage transformerless battery energy storage system under study is shown in the figure below. Each phase consists of N cascaded H-bridge modules. Each module has a battery cluster connected to its DC link through a filter inductor Lb and a pre-charge circuit. The AC side of the converter is connected to the grid via a filter inductor Lf. The system can operate in all four quadrants, providing active power and reactive power support.
We define the SOC of each module, the phase-average SOC, and the system-average SOC as:
$$ S_{SOC,kj} \quad (k = a,b,c; j = 1,2,\ldots,N) $$
$$ S_{SOC,k} = \frac{1}{N}\sum_{j=1}^{N} S_{SOC,kj} $$
$$ S_{SOC,sys} = \frac{1}{3}(S_{SOC,a} + S_{SOC,b} + S_{SOC,c}) $$
Inter-phase SOC balancing: We inject a zero-sequence voltage
$$ u_0 = U_0 \cos(\omega t + \theta_0) $$
to redistribute active power among phases. The power adjustment for phase k is proportional to the SOC deviation $\Delta S_{SOC,k} = S_{SOC,sys} – S_{SOC,k}$ and the nominal balancing coefficient $K_0$:
$$ \Delta P_k = K_0 P_{nom} \Delta S_{SOC,k} $$
Solving for the zero-sequence voltage amplitude and phase:
$$ U_0 = \frac{2 K_0 P_{nom} \sqrt{\Delta S_{SOC,a}^2 + \Delta S_{SOC,b}^2 + \Delta S_{SOC,c}^2}}{3 I_s} $$
$$ \theta_0 = \theta_a + \arctan\left(\frac{\Delta S_{SOC,b} – \Delta S_{SOC,c}}{\sqrt{3}\, \Delta S_{SOC,a}}\right) $$
Intra-phase SOC balancing: For each phase, we adjust the modulation voltage of module j by a correction term:
$$ \Delta u_{kj} = \frac{K_k \Delta S_{SOC,kj}}{I_s} i_k $$
where $\Delta S_{SOC,kj} = S_{SOC,k} – S_{SOC,kj}$, $i_k$ is the phase current, and $K_k$ is the intra-phase balancing coefficient for phase k.
Theoretical Derivation of Optimal Balancing Coefficients
Intra-phase Coefficient Constrained by Battery Current Limits
Without balancing, the average battery current of a cluster is $I_{avg} = P_{nom} / (3 N E_{bat})$. With intra-phase balancing, the current of the module with the lowest SOC (during charging) or highest SOC (during discharging) may hit the rated current limit $I_{rated}$. Considering the power factor angle $\varphi$, the phase current amplitude $I_m$, the battery cluster voltage $E_{bat}$, and the modulation voltage amplitude $U_m$, we derive two constraints. For the lowest-SOC module (which increases its charging current), we require:
$$ \frac{P_{nom}}{3N E_{bat}} + \frac{K_k I_m \cos\varphi}{2 E_{bat}} \cdot (S_{SOC,k} – S_{SOC,k,\min}) \le I_{rated} $$
$$ \Rightarrow K_{k1} = \frac{2 E_{bat} I_{rated} – \frac{2 P_{nom}}{3N}}{I_m \cos\varphi \, (S_{SOC,k} – S_{SOC,k,\min})} $$
Similarly, for the highest-SOC module (which reduces its charging current or increases discharging), we obtain:
$$ K_{k2} = \frac{2 E_{bat} I_{rated} + \frac{2 P_{nom}}{3N}}{I_m \cos\varphi \, (S_{SOC,k,\max} – S_{SOC,k})} $$
Intra-phase Coefficient Constrained by Modulation Ratio Limits
When the balancing voltage is superimposed, the modulation index of a module must not exceed 1. Let the average modulation voltage amplitude be $U_{avg} = U_s / N$. The module with the lowest SOC (which gets a positive in-phase boost during charging) or highest SOC (which gets a negative in-phase boost) may reach over-modulation. The condition for the lowest-SOC module is:
$$ \sqrt{\left( U_{avg} \cos\varphi + \frac{K_k I_m \cos\varphi}{2} (S_{SOC,k} – S_{SOC,k,\min}) \right)^2 + \left( U_{avg} \sin\varphi \right)^2} \le U_C $$
leading to:
$$ K_{k3} = \frac{2\left( \sqrt{U_C^2 – (U_{avg} \sin\varphi)^2} – U_{avg} \cos\varphi \right)}{I_m \cos\varphi \, (S_{SOC,k} – S_{SOC,k,\min})} $$
Similarly for the highest-SOC module:
$$ K_{k4} = \frac{2\left( \sqrt{U_C^2 – (U_{avg} \sin\varphi)^2} + U_{avg} \cos\varphi \right)}{I_m \cos\varphi \, (S_{SOC,k,\max} – S_{SOC,k})} $$
The intra-phase balancing coefficient for phase k is then:
$$ K_k = \min (K_{k1}, K_{k2}, K_{k3}, K_{k4}) $$
Table 1 summarizes the four limiting cases and the factors influencing each.
| Constraint | Critical module | Expression for maximum $K_k$ | Key factors |
|---|---|---|---|
| Charging current limit | Lowest SOC (charging) | $K_{k1}$ | $E_{bat}, I_{rated}, I_m, \cos\varphi, \Delta S_{low}$ |
| Discharging current limit | Highest SOC (discharging) | $K_{k2}$ | $E_{bat}, I_{rated}, I_m, \cos\varphi, \Delta S_{high}$ |
| Over-modulation on low-SOC | Lowest SOC | $K_{k3}$ | $U_C, U_{avg}, \varphi, \Delta S_{low}$ |
| Over-modulation on high-SOC | Highest SOC | $K_{k4}$ | $U_C, U_{avg}, \varphi, \Delta S_{high}$ |
Inter-phase Coefficient Constrained by Battery Current Limits
When zero-sequence voltage is injected, the average battery current of a phase changes. For the phase with the lowest SOC (which receives extra charging power), we require:
$$ \frac{P_{nom}}{3N E_{bat}} + \frac{K_0 P_{nom} \Delta S_{SOC,k,\min}}{3N E_{bat}} \le I_{rated} $$
$$ \Rightarrow K_{01} = \frac{3N E_{bat} I_{rated} – P_{nom}}{P_{nom} \Delta S_{SOC,k,\min}} $$
For the phase with the highest SOC (which reduces its charging power):
$$ \frac{P_{nom}}{3N E_{bat}} + \frac{K_0 P_{nom} \Delta S_{SOC,k,\max}}{3N E_{bat}} \ge -I_{rated} $$
$$ \Rightarrow K_{02} = \frac{3N E_{bat} I_{rated} + P_{nom}}{P_{nom} |\Delta S_{SOC,k,\max}|} $$
Inter-phase Coefficient Constrained by Modulation Ratio Limits
After injecting zero-sequence voltage, the peak voltage of each phase arm becomes $U_k$ as per the vector diagram. The maximum among the three phases must not exceed $N U_C$. Using the law of cosines:
$$ U_k = \sqrt{U_s^2 + U_0^2 – 2 U_s U_0 \cos(\theta_k)} $$
where $\theta_k$ are the phase angles of the zero-sequence voltage relative to the grid voltages. The constraint $\max(U_a, U_b, U_c) \le N U_C$ yields a maximum $U_0$, and consequently a maximum $K_{03}$. The inter-phase coefficient is then:
$$ K_0 = \min(K_{01}, K_{02}, K_{03}) $$
Adaptive Decoupled SOC Balancing Control
The inter-phase and intra-phase balancing actions are coupled because injecting zero-sequence voltage changes the modulation ratio margins and the active power distribution, which in turn affect the available intra-phase balancing capability. To resolve this coupling, we introduce a quantitative metric for the imbalance:
$$ \xi = \frac{\Theta_{SOC}}{ \Delta S_{SOC} } $$
where $\Theta_{SOC} = \Theta_{SOC,a} + \Theta_{SOC,b} + \Theta_{SOC,c}$ and $\Theta_{SOC,k} = \sqrt{\frac{1}{N}\sum_{j=1}^{N} (S_{SOC,kj} – S_{SOC,k})^2}$ is the intra-phase imbalance of phase k, and $\Delta S_{SOC} = \sqrt{\Delta S_{SOC,a}^2 + \Delta S_{SOC,b}^2 + \Delta S_{SOC,c}^2}$ is the inter-phase imbalance.
We define an inter-phase effort factor:
$$ K_\xi = \begin{cases}
0.5 \cdot (2 – \xi), & \xi < 1 \\
0.5 / \xi, & \xi \ge 1
\end{cases} $$
This factor reduces the inter-phase balancing strength when intra-phase imbalances dominate, and vice versa. The actual zero-sequence voltage amplitude is then:
$$ U_0 = \frac{2 K_\xi K_0 P_{nom} \sqrt{\Delta S_{SOC,a}^2 + \Delta S_{SOC,b}^2 + \Delta S_{SOC,c}^2}}{3 I_s} $$
After applying this zero-sequence voltage, the new operating point (phase voltage amplitudes, power angles) is computed, and the intra-phase coefficients $K_k$ are recalculated using the formulas derived earlier (including the effect of the revised $U_{avg}$). The procedure is executed in real-time at each control cycle.
Table 2 summarizes the adaptive control algorithm steps.
| Step | Action |
|---|---|
| 1 | Measure all battery cluster SOCs and compute $S_{SOC,k}$, $S_{SOC,sys}$, $\Theta_{SOC,k}$, $\Delta S_{SOC,k}$. |
| 2 | Compute $\xi$ and $K_\xi$ according to the formulas. |
| 3 | Compute the maximum inter-phase coefficient $K_0$ using the current operating conditions (considering both current and modulation limits). |
| 4 | Calculate the zero-sequence voltage $U_0$ using $K_\xi K_0$ and $\Delta S_{SOC,k}$. |
| 5 | Update the three-phase voltages and power angles after injecting $U_0$. |
| 6 | For each phase, compute the intra-phase coefficient $K_k$ using the updated modulation voltage and power factor. |
| 7 | Apply the balancing voltages to each module; verify that no module exceeds constraints. |
Simulation Verification
We built a 10 kV/5 MW/5 MWh high-voltage transformerless battery energy storage system model in MATLAB/Simulink using the parameters given in Table 3.
| Parameter | Value |
|---|---|
| Grid line voltage (rms) | 10 kV |
| Rated active power | 5 MW |
| Rated energy capacity | 5 MWh |
| Battery cluster rated voltage | 768 V |
| Number of modules per phase | 16 |
| Grid-side filter inductor | 6 mH |
| DC-side filter inductor | 2.5 mH |
| DC-link capacitor | 7.2 mF |
| Control frequency | 10 kHz |
We performed several test cases to validate the adaptive balancing control.
Case 1: Even intra-phase SOC distribution, varying power and power factor. Initially, phase A modules have SOCs uniformly distributed between 29.98% and 30.08%. The system charges at 5 MW from t=0 to 2 s, then switches to 2.5 MW charging (or to pure reactive power 5 MVA) from t=2 s onward. The adaptive intra-phase coefficient $K_a$ automatically adjusts: during full-power charging, the constraint is the charging current limit ($K_a = K_{a1}$); when power reduces, modulation ratio limit ($K_a = K_{a3}$) takes over; under pure reactive power, the limit shifts to $K_a = K_{a4}$ due to the highest-SOC module reaching over-modulation. The SOC differences converge quickly without violating any boundary.
Case 2: Single high-SOC cluster in phase B. One cluster in phase B starts at 30.2% while others are at 30%. During 5 MW charging, the high-SOC cluster actually discharges (negative current) while others charge. The balancing coefficient $K_b = K_{b2}$ (discharge current limit). When power reduces to 2.5 MW, the constraint becomes modulation ratio ($K_{b4}$). Under pure reactive power, the high-SOC cluster continues discharging, and balancing accelerates. All limits are respected.
Case 3: Single low-SOC cluster in phase C. One cluster at 29.98% vs others at 30%. During charging, the low-SOC cluster has higher current ($K_c = K_{c1}$). After power reduction, modulation ratio limit ($K_{c3}$) dominates. Again, no violations occur.
Case 4: Combined inter-phase and intra-phase balancing. We start with both inter-phase and intra-phase imbalances. The algorithm automatically allocates the balancing capability. Early in the simulation, intra-phase imbalance dominates ($\xi$ large), so inter-phase effort is reduced; the intra-phase errors shrink first. After 2 s, power decreases, intra-phase convergence speeds up, and $\xi$ becomes small. Then inter-phase balancing intensifies, with $K_0$ limited by modulation ratio ($K_{03}$). Eventually, all SOCs become equal within 5 s. Throughout the process, the module with the most critical condition always operates at the boundary (current or modulation limit), proving that the system fully utilizes its balancing capacity.
These simulation results confirm that the proposed adaptive balancing control achieves the fastest possible SOC equalization while ensuring safe operation for all battery clusters and modules.
Conclusion
We have presented a self-adaptive and fast SOC balancing control strategy for high-voltage transformerless battery energy storage systems. The key contributions are:
- Closed-form expressions for the maximum intra-phase and inter-phase balancing coefficients, considering battery current limits and modulation ratio limits under varying power factor, current amplitude, battery voltage, and SOC distribution.
- An adaptive decoupling mechanism that quantifies the relative imbalance and allocates the balancing capability between inter-phase and intra-phase actions optimally.
- The proposed control ensures that at least one battery cluster or module operates at the boundary of its safe operating area at all times, enabling the fastest possible convergence without over-modulation or over-current.
- The algorithm is computationally efficient and can be implemented in real-time digital controllers.
Simulation results on a 10 kV/5 MW/5 MWh system validate the theoretical analysis and demonstrate the effectiveness of the proposed approach. This work provides a practical solution for achieving fast SOC balancing in large-scale high-voltage transformerless battery energy storage systems, thereby improving capacity utilization and system reliability.