As the penetration of intermittent renewable energy sources and stochastic loads increases in microgrids, maintaining power balance becomes increasingly challenging. The conventional approach of installing a single homogeneous battery energy storage system (BESS) is often insufficient, leading to the construction of heterogeneous battery energy storage systems (HBESS). These systems consist of multiple BESS units with distinct electrical parameters—such as rated capacity, rated power, and efficiency—and differing operational states, including state of charge (SOC) and state of health (SOH). This heterogeneity, combined with long-term operation, results in uneven battery unit (BU) characteristics and accelerates lifespan degradation. In this paper, we present a novel fixed-time proportional power control strategy for HBESS based on an improved grouping model designed specifically for time-delay communication environments. Our approach addresses three critical challenges: high lifespan degradation, poor balance among battery units, and insufficient control precision.
The core of our methodology is a three-stage process. First, we evaluate the regulation capability of each BU by considering its electrical parameters (rated power, efficiency, capacity) and operational states (SOC, SOH), alongside the characteristics of the power command (magnitude and ramp rate). Using a competition-cooperation mechanism, we partition the HBESS into several battery groups (BGs). Second, we design a hierarchical regulation method that determines which BGs participate in power control at each time step, ensuring that power is allocated primarily to BGs with higher regulation capability. Third, we propose a time-delay-based fixed-time proportional consensus algorithm (TFPCA) that enables distributed iterative computation among BUs. We prove that TFPCA guarantees fixed-time convergence and proportional consensus, and we eliminate iterative errors through a reverse calculation step. Finally, we integrate TFPCA with the hierarchical regulation method to achieve precise power tracking while promoting SOC balance.
Ⅰ. Assessment of Battery Unit Regulation Capability
To effectively group BUs, we first quantify each BU’s capability to regulate power. The regulation capability at time \(t\) is defined by two components: the capability to undertake power commands and the capability to provide reserve capacity. The formulas are given below.
$$
A_i^{(1)}(t) = \frac{\eta_i^c P_{N,i}^c u(P(t)) + \eta_i^d P_{N,i}^d u(-P(t))}{\alpha P(t) + \varepsilon_1}, \quad \alpha = \frac{\|\mathbf{P}\|_\infty}{\max(\|\mathbf{P}\|_\infty, \varepsilon_1)}
$$
$$
A_i^{(2)}(t) = \frac{\bigg[S_{i}^{\mathrm{c}}(t) – S_{\min}^{\mathrm{c}} \cdot u(P(t)) + (S_{\max}^{\mathrm{c}} – S_{i}^{\mathrm{c}}(t)) \cdot u(-P(t))\bigg] \cdot E_{N,i} \cdot S_{i}^{\mathrm{h}}(t)}{\beta \big| P'(t) \big| \Delta t + \varepsilon_2}, \quad \beta = \frac{\max_{k} |P'(t_k)|}{\max(\max_{k} |P'(t_k)|, \varepsilon_2)}
$$
Where:
| Symbol | Meaning |
|---|---|
| \(\eta_i^c, \eta_i^d\) | Charging/discharging efficiency of BU \(i\) |
| \(P_{N,i}\) | Rated power of BU \(i\) |
| \(P(t)\) | Power command at time \(t\) |
| \(P'(t)\) | Ramp rate of power command |
| \(S_i^{\mathrm{c}}(t)\) | State of charge of BU \(i\) |
| \(S_i^{\mathrm{h}}(t)\) | State of health of BU \(i\) |
| \(E_{N,i}\) | Rated capacity of BU \(i\) |
| \(\Delta t\) | Control interval |
| \(\varepsilon_1, \varepsilon_2\) | Small positive constants to avoid division by zero |
| \(u(\cdot)\) | Unit step function |
The overall regulation capability is \(A_i(t) = A_i^{(1)}(t) + A_i^{(2)}(t)\). This evaluation is updated at each control interval.
Ⅱ. Improved Grouping Model Based on Competition-Cooperation Mechanism
Using the assessment results, we apply a competition-cooperation mechanism to partition the HBESS into battery groups. The objective is twofold: (1) maximize the difference in average regulation capability between groups (competition), and (2) minimize within-group variation of regulation capability and balance the total capacity among groups (cooperation). The optimization problem is:
$$
\min \left\{ \sum_{j=1}^{n_{\mathrm{bg}}} \sum_{i \in G_j} \left( A_i^*(t) – \overline{A}_j^*(t) \right)^2 + \left( E_j^*(t) – \overline{E}^*(t) \right)^2 \right\}
$$
where \(n_{\mathrm{bg}}\) is the number of groups, \(G_j\) is the set of BUs in group \(j\), \(\overline{A}_j^*(t)\) is the per-unit average regulation capability of group \(j\), \(E_j^*(t)\) is the per-unit total capacity of group \(j\), and \(\overline{E}^*(t)\) is the overall per-unit average capacity. The number of groups \(n_{\mathrm{bg}}\) is dynamically adjusted based on the dispersion of regulation capabilities, which in turn depends on \(\alpha\) and \(\beta\).
Ⅲ. Hierarchical Regulation Method
After grouping, we assign a hierarchy to each BG: BG1 has the highest regulation capability, BG2 the second highest, and so on. The power command \(P(t)\) is then allocated hierarchically as shown in the table below.
| Condition | Participating BGs | Non-Participating BGs |
|---|---|---|
| \(|P(t)| \leq P_{\mathrm{BG1}}\) | BG1 | All others |
| \(P_{\mathrm{BG1}} < |P(t)| \leq P_{\mathrm{BG1}}+P_{\mathrm{BG2}}\) | BG1, BG2 | All others |
| \(\vdots\) | \(\vdots\) | \(\vdots\) |
| \(\sum_{i=1}^{n_{\mathrm{bg}}-1} P_{\mathrm{BG}_i} < |P(t)| \leq \sum_{i=1}^{n_{\mathrm{bg}}} P_{\mathrm{BG}_i}\) | All BGs | — |
Here \(P_{\mathrm{BG}_i}\) is the maximum charging/discharging power of BG \(i\). This ensures that power is supplied by the most capable groups first, thereby reducing stress on weaker BUs and extending overall battery life.
Ⅳ. Time-Delay Based Fixed-Time Proportional Consensus Algorithm (TFPCA)
For the distributed execution of power allocation, we propose TFPCA. The discrete-time state update for BU \(i\) is:
$$
x_i(k+1) = x_i(k) – \varepsilon \cdot \mathrm{sgn}\left( Q_i \right) |Q_i|^{\varphi} – \varepsilon \cdot \mathrm{sgn}\left( Q_i \right) |Q_i|^{\phi}
$$
$$
Q_i = \sum_{j=1}^{n} l_{ij} w_j x_j(k – \tau_{ij})
$$
where:
- \(x_i(k)\) is the state variable (power allocated to BU \(i\)) at iteration \(k\).
- \(\varepsilon\) is the step size.
- \(\tau_{ij}\) is the communication delay between BU \(i\) and BU \(j\).
- \(l_{ij}\) are elements of the Laplacian matrix \(\mathbf{L}\) of the communication graph.
- \(w_j > 0\) is the weight for BU \(j\), designed as a function of regulation capability and balancing factor.
- \(\varphi > 1\) and \(0 < \phi < 1\) are parameters controlling convergence speed.
- \(\mathrm{sgn}(\cdot)\) is the sign function.
We prove that TFPCA ensures fixed-time convergence: for any initial condition, the states \(x_i(k)\) reach consensus to a common value within a time \(T \leq T_{\max}\), where
$$
T_{\max} = \frac{1}{\rho_1 (m_1-1)} + \frac{1}{\rho_2 (1-m_2)}
$$
with parameters derived from the Lyapunov analysis. Furthermore, the consensus value is proportional to \(1/w_i\), i.e., \(x_i^* \propto 1/w_i\). This proportionality is preserved even under time delays, enabling us to eliminate iterative errors by scaling the converged values to match the total power command.
Ⅴ. Weight Design for State-of-Charge Balancing
To promote SOC balance among BUs, especially those within the same BESS, we design the weight \(w_i(t)\) as:
$$
w_i(t) = \frac{ \left( S_i^{\mathrm{c}}(t) – S_{\min}^{\mathrm{c}} \right) u(P(t)) + \left( S_{\max}^{\mathrm{c}} – S_i^{\mathrm{c}}(t) \right) u(-P(t)) }{ \eta_i^{\mathrm{c}} P_{N,i} u(P(t)) + \eta_i^{\mathrm{d}} P_{N,i} u(-P(t)) } \cdot S_i^{\mathrm{h}}(t) \cdot A_i(t)
$$
This weight is larger for BUs with higher SOC (when discharging) or lower SOC (when charging), thereby making them contribute less power, which naturally balances SOC. The inclusion of \(A_i(t)\) ensures that the weight also respects regulation capability, aligning with the hierarchical regulation logic.
Ⅵ. Precise Hierarchical Power Control Strategy
At each control interval, the process is as follows:
- Evaluate regulation capability for all BUs.
- Apply the improved grouping model to form BGs.
- Use the hierarchical regulation method to determine which BGs participate.
- Initialize states \(x_i(0)\) by randomly distributing the power command among participating BUs.
- Run TFPCA iterations until convergence (within the fixed time bound).
- Obtain converged values \(x_i^{\mathrm{end}}\). Since the ratio \(x_i^{\mathrm{end}} : x_j^{\mathrm{end}} = 1/w_i : 1/w_j\) is known, we compute the corrected power command for each BU as:
$$
P_i^{\mathrm{bu}}(t) = \frac{ x_i^{\mathrm{end}} / w_i }{ \sum_{j \in \Omega} x_j^{\mathrm{end}} / w_j } \cdot P(t)
$$
where \(\Omega\) is the set of participating BUs. This step eliminates any residual error from the iterative process, achieving precise power control.
Ⅶ. Simulation and Experimental Validation
We simulate a typical microgrid with an HBESS consisting of three BESS units (each with multiple BUs) under realistic parameters. The communication delays range from 0 to 33 ms. We compare our strategy with three existing methods from the literature. Key performance metrics are summarized in the tables below.
7.1 Algorithm Comparison (TFPCA vs. Others)
| Algorithm | Fixed-Time Consensus | Proportional Consensus | Iteration Error Elimination | Convergence Time (s) |
|---|---|---|---|---|
| Proposed TFPCA | Yes | Yes | Yes | 1.59 |
| Algorithm 1 [24] | Yes | No | No | 1.56 |
| Algorithm 2 [26] | Yes | Yes | No | 2.07 |
| Algorithm 3 [29] | Yes | No | No | 1.85 |
Only TFPCA achieves both fixed-time and proportional consensus while eliminating errors, making it the most suitable for HBESS power control.
7.2 Grouping Model Comparison
| Grouping Method | Lifespan Degradation Reduction (vs. No Grouping) | Dynamic BG Count | Maintains Hierarchical Regulation Capability |
|---|---|---|---|
| Proposed (Method 0) | ~8% further reduction | Yes | Yes |
| Method 1 [9] | 0 (no grouping) | No | — |
| Method 2 [14] | Moderate | No | Partially |
| Method 3 [15] | Moderate | No | Partially |
Our grouping model reduces lifespan degradation by approximately 8% compared to other group-based methods, and dynamically adjusts the number of groups.
7.3 Overall Power Control Strategy Performance
| Strategy | SOC Balance (std reduction) | Power Tracking Accuracy (%) | Hierarchical Regulation Maintained |
|---|---|---|---|
| Proposed (Strategy 0) | >50% improvement | 100% | Yes |
| Strategy 1 [15] | Moderate | 97.6% | No (overuse of one group) |
| Strategy 2 [16] | Good | 99.6% | Partially |
| Strategy 3 [31] | Poor | 100% | Yes |
Our strategy achieves perfect power tracking, significantly improves SOC balance, and preserves the hierarchical regulation order throughout the simulation.
7.4 Experimental Results
We built a hardware platform comprising two BESS units (each with six BUs) connected via wireless communication with delays. The experiment lasted 20 minutes with a 5‑second sampling interval. The results confirm that:
- The grouping model maintains hierarchical regulation capability over time.
- BU SOC std deviation decreases by more than 50%.
- Power command is tracked with zero steady-state error.
These experimental findings align with the simulation outcomes.
Ⅷ. Conclusion
We have presented a comprehensive fixed-time proportional power control strategy for heterogeneous battery energy storage systems operating under communication delays. The key innovations are:
- An improved grouping model that considers both electrical parameters and operational states, dynamically adjusting the number of groups to reduce lifespan degradation by approximately 8%.
- The time-delay-based fixed-time proportional consensus algorithm (TFPCA) that guarantees fixed-time convergence and proportional consensus while eliminating iterative errors, achieving higher precision than existing methods.
- A hierarchical regulation method integrated with TFPCA that promotes state-of-charge balance across BUs by over 50% while ensuring 100% power tracking accuracy.
The proposed method is validated through extensive simulations and hardware experiments, demonstrating its effectiveness for real-world heterogeneous battery energy storage systems in microgrid applications.
Future work will extend the approach to consider more complex battery degradation models and investigate event-triggered communication to further reduce network load.