1. Introduction
The integration of renewable energy sources into AC microgrids introduces challenges due to their inherent volatility and randomness. In islanded operation, energy storage inverters (ESIs) with battery energy storage units (ESS) provide stable power output while regulating system voltage and frequency. Parallel operation of multiple energy storage inverters enhances system redundancy and reliability. However, traditional droop control struggles with accurate reactive power sharing due to mismatched line impedances. Additionally, variations in state-of-charge (SOC) and capacities among ESS units necessitate coordinated control strategies to prolong system lifespan and optimize energy utilization. This study focuses on improving droop control through adaptive virtual impedance and SOC-based collaborative algorithms, ensuring efficient power distribution and dynamic SOC balancing in islanded microgrids.
2. Modeling and Control Analysis of Energy Storage Inverters
2.1 ESS Modeling and SOC Estimation
The battery model simplifies ESS dynamics using a Thevenin equivalent circuit. The terminal voltage ( U{\text{bat}} ) is expressed as: [ U{\text{bat}} = n \cdot (E – I{\text{bat}} \cdot R) ] where ( E ) is the open-circuit voltage, ( R ) is the internal resistance, and ( n ) is the number of series-connected cells. SOC estimation employs ampere-hour integration: [ \text{SOC} = \text{SOC}0 – \frac{1}{C{\text{nom}}} \int{0}^{t} I{\text{bat}}(t) \, dt ] where ( C{\text{nom}} ) is the nominal capacity.
2.2 Inverter Modeling and Parameter Design
A three-phase voltage-source inverter with LC filters is modeled in the dq-frame: [ \begin{cases} L \frac{di{d}}{dt} = u{d} – R i{d} + \omega L i{q} – u{od} \ L \frac{di{q}}{dt} = u{q} – R i{q} – \omega L i{d} – u{oq} \ C \frac{du{od}}{dt} = i{d} – i{od} + \omega C u{oq} \ C \frac{du{oq}}{dt} = i{q} – i{oq} – \omega C u{od} \end{cases} ] Voltage and current control loops are designed using PI regulators. Filter parameters (( L_f ), ( C_f )) and controller gains (( k{vp} ), ( k{vi} )) are optimized for stability and dynamic response.
2.3 Control Strategies for Energy Storage Inverters
- PQ Control: Maintains fixed active/reactive power but lacks grid-support capability.
- V/f Control: Regulates voltage and frequency but is unsuitable for multi-inverter systems.
- Droop Control: Adjusts frequency and voltage based on power demand, enabling decentralized operation. Traditional droop equations are: [ \begin{cases} f = f{\text{ref}} – m \cdot P \ U = U{\text{ref}} – n \cdot Q \end{cases} ]
3. Parallel Control Strategy for Energy Storage Inverters
3.1 Challenges in Parallel Operation
Mismatched line impedances (( Z_{\text{line}} )) cause reactive power sharing errors and circulating currents. For two inverters, the circulating current ( \Delta I ) is: [ \Delta I = \frac{E_1 – E_2}{Z_1 + Z_2} ] where ( Z_1 ) and ( Z_2 ) are line impedances.
3.2 Adaptive Virtual Impedance
A virtual impedance ( Z_v = R_v + jX_v ) is introduced to decouple power flow and mitigate line impedance mismatches. The modified droop law becomes: [ \begin{cases} f_i = f{\text{ref}} – m_i \cdot P_i \ U_i = U{\text{ref}} – n_i \cdot Q_i – Z_v \cdot I_i \end{cases} ] The adaptive ( Z_v ) is dynamically adjusted to ensure: [ X{v,i} \cdot Q_i = X{v,j} \cdot Q_j \quad \forall i,j ] For inverters with unequal capacities (( C_i \neq C_j )), reactive power sharing follows: [ \frac{Q_i}{Q_j} = \frac{C_i}{C_j} ]
3.3 Simulation Results
A MATLAB/Simulink model validates the strategy under varying line impedances and load conditions:
| Case | Line Impedance (Ω) | Capacity Ratio | Reactive Power Error (%) |
|---|---|---|---|
| 1 | ( 0.24 + j1.2 ) | 1:1 | 2.1 |
| 2 | ( 0.4 + j2 ) | 1:3 | 4.8 |
Adaptive virtual impedance reduces circulating currents by 68% and improves reactive power sharing accuracy to >95%.
4. Improved Droop Control Based on SOC Collaboration
4.1 Multiplicative SOC Factor
A multiplicative SOC term modifies the droop coefficient: [ f_i = f{\text{ref}} – m \cdot P_i \cdot \left(1 – k{\text{SOC}} \cdot \text{SOC}_i\right) ] However, this introduces frequency deviations during transient states.
4.2 Exponential SOC Factor
An exponential function dynamically adjusts the droop coefficient: [ f_i = f{\text{ref}} – \frac{m}{C_i} \cdot \exp\left(-\alpha (\text{SOC}i – \text{SOC}_{\text{avg}})\right) \cdot P_i ] where ( \alpha ) tunes the convergence rate. Small-signal analysis confirms stability for ( \alpha < 60 ).
4.3 Consensus Algorithm for SOC Balancing
A multi-agent system (MAS) with distributed consensus achieves SOC synchronization without centralized control. Each agent updates its SOC estimate as: [ \text{SOC}i(k+1) = \text{SOC}i(k) + \sum{j \in \mathcal{N}i} a{ij} \left(\text{SOC}j(k) – \text{SOC}i(k)\right) ] where ( \mathcal{N}i ) denotes neighboring agents. The algorithm converges within 20 iterations (Figure 1).
4.4 Simulation and Validation
Case studies demonstrate SOC balancing under heterogeneous conditions:
| Scenario | SOC Initial Disparity | Convergence Time (s) | Frequency Deviation (Hz) |
|---|---|---|---|
| Equal Capacity | 0.8, 0.7, 0.6 | 3.2 | ≤0.1 |
| Unequal Capacity | 0.8, 0.7, 0.6 | 4.5 | ≤0.15 |
| Scalability (8 ESIs) | 0.7–0.9 | 5.8 | ≤0.2 |
The strategy maintains frequency within ±0.2 Hz and SOC imbalance below 5%.
5. Conclusion and Future Work
The proposed adaptive virtual impedance and SOC-based droop control enhance power sharing accuracy and SOC balancing in multi-inverter microgrids. Key contributions include:
- Adaptive Virtual Impedance: Mitigates line impedance mismatches, reducing reactive power errors by >95%.
- Exponential SOC Factor: Accelerates convergence while minimizing frequency deviations.
- Consensus Algorithm: Enables plug-and-play operation without centralized communication.
Future work will focus on hybrid AC/DC microgrid integration and experimental validation.
Keywords: Energy storage inverter, SOC collaboration, droop control, adaptive virtual impedance, consensus algorithm, microgrid stability.