In modern power electronic systems, such as photovoltaic integration, battery energy storage, and vehicle-to-grid (V2G) applications, two-stage DC/AC converters are widely employed to interface with the AC grid while enabling bidirectional power flow. Among these, the dual active bridge (DAB) DC-DC converter is a preferred front-end stage due to its high power density, electrical isolation, and soft-switching capabilities. When cascaded with a single-phase grid-connected inverter, the system can achieve efficient power conversion. However, impedance interactions between these cascaded stages often lead to instability, manifested as oscillations in the DC bus voltage, despite each subsystem being stable in isolation. This paper addresses the stability challenges in DAB-cascaded single-phase inverter systems through impedance-based analysis and control parameter optimization.
The stability of cascaded systems is critically influenced by the impedance characteristics of the source and load converters. According to the Middlebrook criterion, the system remains stable if the output impedance of the source converter (DAB) and the input impedance of the load converter (single-phase inverter) satisfy the Nyquist stability condition. Accurate impedance modeling is therefore essential. In this work, we derive the output impedance of the DAB converter under double-loop control and the input impedance of the single-phase inverter, considering the effects of the phase-locked loop (PLL). The models are validated using frequency sweep methods, and their interactions are analyzed to identify stability boundaries.
The DAB converter typically employs a voltage outer loop and a current inner loop for control. The power transfer equation of the DAB is given by:
$$ P = \frac{n V_{\text{in}} v_{\text{bus}}}{L_o f_s} d_\phi (1 – 2d_\phi) = v_{\text{bus}} \langle i_2 \rangle $$
where \( n \) is the transformer turns ratio, \( V_{\text{in}} \) is the input voltage, \( v_{\text{bus}} \) is the output voltage, \( L_o \) is the transformer leakage inductance, \( f_s \) is the switching frequency, \( d_\phi \) is the phase shift duty ratio, and \( i_2 \) is the secondary-side current. Through small-signal analysis, the relationship between \( i_2 \) and \( d_\phi \) is derived as:
$$ G_{i_2 d} = \frac{\hat{i}_2}{\hat{d}_\phi} = \frac{n V_{\text{in}}}{L_o f_s} (1 – 4 D_\phi) $$
The control loops include a current controller \( G_{c1}(s) = k_{pi} + \frac{k_{ii}}{s} \) and a voltage controller \( G_{c2}(s) = k_{pv} + \frac{k_{iv}}{s} \), with a low-pass filter \( H_{\text{LPF}}(s) = \frac{1}{s/\omega_{\text{LPF}} + 1} \) to attenuate high-frequency noise. The output impedance of the DAB is expressed as:
$$ Z_{\text{out\_DAB}} = \frac{1}{C_{\text{bus}} s + G_{c1} G_x} $$
where \( G_x = \frac{G_{c2} G_{i_2 d}}{1 + G_{c2} G_{i_2 d} H_{\text{LPF}}} \). This model captures the resonant behavior of the DAB output impedance, which resembles an LC filter response.
For the single-phase grid-connected inverter, the input impedance on the DC side is derived considering the influence of the PLL. The inverter uses a decoupled current control strategy in the dq-frame, facilitated by a second-order generalized integrator (SOGI) based PLL. The PLL transfer function accounts for phase deviations, leading to couplings between the control and electrical quantities. The power equations in the dq-frame are:
$$ (Z_L + Z_g) \hat{i}^s_{dq} = D_{dq} \hat{v}_{\text{bus}} + \hat{d}^s_{dq} V_{\text{bus}} $$
$$ \hat{i}_{\text{bus}} = \frac{1}{2} (D^T_{dq} \hat{i}^s_{dq} + I^T_{dq} \hat{d}^s_{dq}) $$
where \( Z_L \) and \( Z_g \) represent the filter and grid impedances, respectively. The PLL introduces transformations between the control and electrical frames, resulting in the input admittance:
$$ Y_{\text{in\_INV}} = \frac{1}{2 V_{\text{bus}}} \left[ I^T_{dq} (Z_L + Z_g) + \frac{1}{2} D^T_{dq} \right] \cdot \left( \left[ Z_L + G_{ci} H_{edq} – G_{\text{PLL\_V}} Z_g \right]^{-1} D_{dq} \right) – \frac{1}{2 V_{\text{bus}}} I^T_{dq} D_{dq} $$
where \( G_{ci} \) is the current controller matrix, \( H_{edq} \) is the SOGI transfer function matrix, and \( G_{\text{PLL\_V}} \) encapsulates the PLL effects. The input impedance is then \( Z_{\text{in\_INV}} = 1 / Y_{\text{in\_INV}} \). This model reveals that the single-phase inverter exhibits negative resistance characteristics at low frequencies due to constant power load behavior, except at the fundamental frequency (50 Hz), where the impedance magnitude drops significantly.
The interaction between the DAB output impedance and the inverter input impedance determines the system stability. The minor loop gain \( T_m = Z_{\text{out\_DAB}} / Z_{\text{in\_INV}} \) must satisfy the Nyquist criterion for stability. The Bode plots of these impedances show that the DAB output impedance has a resonant peak, while the inverter input impedance is negative at low frequencies. If the resonant peak intersects with the inverter impedance at a phase margin less than 180°, instability occurs. The following table summarizes the key parameters used in the analysis:
| Parameter | Value |
|---|---|
| DAB Input Voltage \( V_{\text{in}} \) | 400 V |
| DC Bus Voltage \( V_{\text{bus}} \) | 400 V |
| DC Bus Capacitance \( C_{\text{bus}} \) | 1500 μF |
| Transformer Leakage Inductance \( L_o \) | 30 μH |
| Transformer Turns Ratio \( n \) | 1:1 |
| Switching Frequency \( f_s \) | 20 kHz |
| Low-Pass Filter Cutoff \( \omega_{\text{LPF}} \) | 4000π rad/s |
| Grid Voltage \( V_g \) | 220 V (rms) |
| Output Power \( P \) | 10 kW |
| Filter Inductance \( L_f \) | 10 mH |
| Filter Resistance \( R_f \) | 50 mΩ |
| Grid Inductance \( L_g \) | 1 mH |
| Grid Resistance \( R_g \) | 15 mΩ |
To simplify the analysis, the DAB output impedance is approximated under the assumption that the current controller proportional gain \( k_{pi} = 0 \) and the inner loop bandwidth matches the LPF cutoff. The simplified impedance is:
$$ Z’_{\text{out\_DAB}} \approx \frac{1}{G_{i_2 d} k_{ii}} \cdot \frac{s(s + G_{i_2 d} k_{ii})}{C_{\text{bus}} s^2 + k_{pv} s + k_{iv}} $$
This model accurately represents the low-frequency behavior (1–200 Hz), where stability issues arise. The resonant frequency and peak of the DAB output impedance are derived as:
$$ \omega_r = \sqrt{\frac{G_{i_2 d} k_{ii} k_{iv}}{C_{\text{bus}} G_{i_2 d} k_{ii} – k_{pv}}} $$
and the magnitude at resonance is proportional to the controller parameters. The effects of the DAB voltage loop PI parameters on the resonant characteristics are summarized below:
| Parameter | Effect on Resonant Frequency | Effect on Resonant Peak |
|---|---|---|
| Voltage Proportional Gain \( k_{pv} \) | Negligible change | Significant reduction |
| Voltage Integral Gain \( k_{iv} \) | Increase | Negligible change |
| Current Integral Gain \( k_{ii} \) | Negligible change | Negligible change |
Thus, \( k_{pv} \) is the key parameter for flattening the resonant peak, while \( k_{iv} \) adjusts the resonant frequency. Bode plots of the DAB output impedance for varying \( k_{pv} \) and \( k_{iv} \) confirm these trends: increasing \( k_{pv} \) from 0.02 to 0.4 drastically reduces the peak without shifting the frequency, whereas increasing \( k_{iv} \) from 10 to 120 raises the frequency without affecting the peak.
The stability of the cascaded system is analyzed using the root locus of the characteristic equation \( 1 + T_m = 0 \). The equivalent open-loop transfer function is:
$$ D(s) = \frac{k_{pv} s G_x Z_{\text{in\_INV}}}{s + Z_{\text{in\_INV}} (C s^2 + k_{iv} G_x)} $$
The root locus plot with respect to \( k_{pv} \) shows that for \( k_{pv} < 0.0634 \), the system has right-half-plane poles, indicating instability. At \( k_{pv} = 0.0634 \), poles lie on the imaginary axis at \( \pm j251 \) rad/s (40 Hz), corresponding to critical stability. For \( k_{pv} > 0.0634 \), the system becomes stable. The Nyquist plot of \( T_m \) at the critical condition crosses the (-1, j0) point, validating the stability boundary.
Time-domain simulations in MATLAB/Simulink corroborate these findings. For \( k_{pv} = 0.03 \), the DC bus voltage and grid current exhibit sustained oscillations, indicating instability. At \( k_{pv} = 0.0634 \), the system is marginally stable with 40 Hz oscillations, and for \( k_{pv} = 0.258 \) or 2, the system operates stably. The dynamic response is optimized at \( k_{pv} = 0.258 \), as evidenced by faster settling times in active power transitions. Further increasing \( k_{pv} \) beyond this point slows down the response, highlighting a trade-off between stability and dynamic performance.
To enhance system stability, we propose an optimization method for the DAB voltage proportional gain \( k_{pv} \). This approach does not require additional compensators or control loops, simplifying implementation. The design criterion is to ensure that the DAB output impedance peak remains below the magnitude of the inverter input impedance at low frequencies, which is approximately \( -V_{\text{bus}}^2 / P \) for a single-phase inverter under constant power control. The optimal \( k_{pv} \) is selected based on the root locus and Bode plots to achieve sufficient phase margin while maintaining acceptable dynamic performance.
For instance, when the power level increases from 5 kW to 10 kW, the inverter input impedance magnitude decreases, potentially causing instability. By adjusting \( k_{pv} \) from 0.0634 to 0.452, the resonant peak is suppressed, and stability is restored. The root locus for 10 kW operation indicates that \( k_{pv} > 0.0645 \) ensures stability, with \( k_{pv} = 0.452 \) providing the best dynamic response. Simulation results confirm that after the adjustment, the DC bus voltage stabilizes with minimal oscillation, and the single-phase inverter maintains smooth grid current injection.
The impedance models and stability analysis provide a framework for designing robust DAB-cascaded single-phase inverter systems. The single-phase inverter’s behavior, particularly its negative input impedance, necessitates careful tuning of the DAB controller to avoid interactions. The proposed method of optimizing \( k_{pv} \) offers a straightforward solution to improve stability without compromising efficiency or adding hardware complexity. Future work could explore adaptive control strategies to automatically adjust parameters under varying operating conditions.
In conclusion, this paper presents a comprehensive impedance-based stability analysis for DAB-cascaded single-phase inverter systems. The derived impedance models accurately capture the interactions between the DAB and single-phase inverter, enabling the identification of stability boundaries. The analysis reveals that the DAB voltage proportional gain \( k_{pv} \) plays a critical role in shaping the output impedance and determining system stability. By optimizing \( k_{pv} \), stability can be enhanced while preserving dynamic performance. Simulation results validate the theoretical findings, demonstrating the effectiveness of the proposed approach. This work provides valuable insights for the design and control of cascaded power electronic systems in renewable energy and storage applications.