In modern power electronic systems, the integration of dual-active bridge (DAB) DC-DC converters with single-phase inverters is prevalent in applications such as renewable energy integration, electric vehicle charging, and grid-connected storage systems. The DAB converter offers advantages like high power density, electrical isolation, and bidirectional power flow, while the single-phase inverter facilitates AC grid interconnection. However, when these systems are cascaded, impedance mismatches can lead to oscillatory instabilities, even if each subsystem operates stably in isolation. This paper addresses the stability challenges in DAB cascade single-phase inverter systems through impedance-based analysis. We derive accurate impedance models for both the DAB output and the single-phase inverter input, considering the effects of control loops and phase-locked loops (PLL). By examining the interaction between these impedances, we analyze how key control parameters, particularly the proportional gain in the DAB voltage loop, influence system stability. Furthermore, we propose a parameter optimization method to enhance stability without additional hardware or complex control strategies. The analysis is supported by theoretical derivations, numerical simulations, and impedance frequency responses, ensuring a comprehensive understanding of the system dynamics.
The stability of power electronic systems is critical for reliable operation, especially in cascaded configurations where source and load converters interact. The Middlebrook criterion, based on impedance matching, is widely used to assess such interactions. For a DAB cascade single-phase inverter system, the DAB acts as a voltage source controlling the DC bus voltage, while the single-phase inverter regulates power flow to the grid. The single-phase inverter, due to its constant power control, exhibits negative impedance characteristics at low frequencies, which can destabilize the system when interacting with the DAB output impedance. This paper focuses on modeling these impedances accurately and analyzing their frequency-dependent behavior to identify stability boundaries. We employ harmonic linearization and dq-frame modeling techniques to capture the dynamics of the single-phase inverter, including PLL effects, which are often neglected in simplified models. The derived models are validated through frequency sweep simulations, ensuring their accuracy across a wide frequency range.
To model the DAB output impedance, we consider a dual-loop control strategy with an inner current loop and an outer voltage loop. The power transfer equation for the DAB is given by:
$$ P = \frac{n V_{in} v_{bus}}{L_o f_s} d_\phi (1 – 2d_\phi) = v_{bus} \langle i_2 \rangle $$
where \( n \) is the transformer turns ratio, \( V_{in} \) is the input voltage, \( v_{bus} \) 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 \( \langle i_2 \rangle \) is the average output current. Through small-signal analysis, the relationship between the output current and phase shift is derived as:
$$ G_{i_2 d} = \frac{\hat{i}_2}{\hat{d}_\phi} = \frac{n V_{in}}{L_o f_s} (1 – 4D_\phi) $$
The control loops are represented by transfer functions: \( G_{c1}(s) = k_{pi} + \frac{k_{ii}}{s} \) for the current controller and \( G_{c2}(s) = k_{pv} + \frac{k_{iv}}{s} \) for the voltage controller. A low-pass filter (LPF) with transfer function \( H_{LPF}(s) = \frac{1}{s/\omega_{LPF} + 1} \) is included to shape the loop gain. The output impedance of the DAB is then expressed as:
$$ Z_{out\_DAB} = \frac{\hat{v}_{bus}}{-\hat{i}_{bus}} = \frac{1}{C_{bus} s + G_{c1} G_x} $$
where \( G_x = \frac{G_{c2} G_{i_2 d}}{1 + G_{c2} G_{i_2 d} H_{LPF}} \). This model captures the resonant behavior of the DAB output impedance, which resembles an LC filter characteristic, with a resonant peak that can interact adversely with the inverter input impedance.
For the single-phase inverter, we derive the DC-side input impedance considering a decoupled current control strategy and PLL effects. The inverter model is developed in the dq-frame to facilitate analysis. The power equations are:
$$ (Z_L + Z_g) \hat{i}_{dq}^s = D_{dq} \hat{v}_{bus} + \hat{d}_{dq}^s V_{bus} $$
$$ \hat{i}_{bus} = \frac{1}{2} (D_{dq}^T \hat{i}_{dq}^s + I_{dq}^T \hat{d}_{dq}^s) $$
where \( Z_L \) and \( Z_g \) are the filter and grid impedances, respectively. The PLL, based on a second-order generalized integrator (SOGI), introduces phase deviations that affect the control variables. The relationship between electrical and control quantities is given by:
$$ \hat{v}_{dq}^c = G_v^{PLL} \hat{v}_{dq}^s $$
$$ \hat{d}_{dq}^s = \hat{d}_{dq}^c + G_d^{PLL} \hat{v}_{dq}^s $$
$$ \hat{i}_{dq}^c = H_{edq} \hat{i}_{dq}^s + G_i^{PLL} \hat{v}_{dq}^s $$
where \( G_v^{PLL} \), \( G_d^{PLL} \), and \( G_i^{PLL} \) are transfer matrices capturing PLL dynamics. The current control law with PCC voltage feedforward yields the duty ratio modulation. The DC-side input admittance is derived as:
$$ Y_{in\_INV} = \frac{\hat{i}_{bus}}{\hat{v}_{bus}} = \frac{1}{2 V_{bus}} \left[ I_{dq}^T (Z_L + Z_g) + \frac{1}{2} D_{dq}^T \right] \cdot \left[ Z_L + G_{ci} H_{edq} – G_{PLL\_V} Z_g \right]^{-1} D_{dq} – \frac{1}{2 V_{bus}} I_{dq}^T D_{dq} $$
where \( G_{ci} \) is the current controller matrix and \( G_{PLL\_V} = G_v^{PLL} – G_{ci} G_i^{PLL} + V_{bus} G_d^{PLL} – E \). The input impedance is \( Z_{in\_INV} = 1 / Y_{in\_INV} \). This model accurately represents the negative resistance behavior of the single-phase inverter at low frequencies, which is crucial for stability analysis.
The system’s stability is assessed using the minor loop gain \( T_m = Z_{out\_DAB} / Z_{in\_INV} \). According to the Middlebrook criterion, the system is stable if \( T_m \) satisfies the Nyquist stability criterion. The Bode plots of the impedances reveal that the DAB output impedance has a resonant peak, while the single-phase inverter input impedance shows a negative resistance region. If these impedances intersect with insufficient phase margin, oscillations occur. The following table summarizes key parameters used in the analysis:
| Parameter | Symbol | Value |
|---|---|---|
| DAB Input Voltage | \( V_{in} \) | 400 V |
| DC Bus Voltage | \( V_{bus} \) | 400 V |
| DC Bus Capacitance | \( C_{bus} \) | 1500 μF |
| Transformer Leakage Inductance | \( L_o \) | 30 μH |
| Switching Frequency | \( f_s \) | 20 kHz |
| Grid Voltage (RMS) | \( V_g \) | 220 V |
| Output Power | \( P \) | 10 kW |
| Filter Inductance | \( L_f \) | 10 mH |
| Grid Inductance | \( L_g \) | 1 mH |
The impedance models are verified through frequency sweep simulations, showing close agreement with theoretical predictions. The DAB output impedance exhibits a resonant frequency \( f_r \) that depends on control parameters, while the single-phase inverter input impedance has a constant power load characteristic, leading to a negative resistance \( -V_{bus}^2 / P \) at frequencies below the current controller bandwidth. The interaction between these impedances is analyzed using Nyquist plots and root locus techniques.
To understand the impact of DAB control parameters on stability, we simplify the output impedance model by setting the current controller proportional gain \( k_{pi} = 0 \) and ignoring higher-order terms. The simplified impedance is:
$$ Z_{out\_DAB}’ \approx \frac{1}{G_{i_2 d} k_{ii}} \frac{s(s + G_{i_2 d} k_{ii})}{C_{bus} s^2 + k_{pv} s + k_{iv}} $$
The magnitude and phase of this impedance are analyzed to determine the resonant peak. The resonant frequency \( \omega_r \) is given by:
$$ \omega_r = \sqrt{\frac{G_{i_2 d} k_{ii} k_{iv}}{C_{bus} G_{i_2 d} k_{ii} – k_{pv}}} $$
and the peak magnitude is influenced by \( k_{pv} \) and \( k_{iv} \). Variations in these parameters show that \( k_{pv} \) primarily affects the peak magnitude, while \( k_{iv} \) shifts the resonant frequency. The current integral gain \( k_{ii} \) has minimal impact. This is summarized in the table below:
| Control Parameter | Effect on Resonant Peak | Effect on Resonant Frequency |
|---|---|---|
| \( k_{pv} \) (Voltage Proportional Gain) | Significant reduction in peak magnitude | Negligible change |
| \( k_{iv} \) (Voltage Integral Gain) | Minimal change | Increase in frequency |
| \( k_{ii} \) (Current Integral Gain) | Negligible change | Negligible change |
Bode plots of the DAB output impedance for different \( k_{pv} \) values confirm that increasing \( k_{pv} \) suppresses the resonant peak without altering the resonant frequency significantly. This behavior is crucial for stability enhancement, as it reduces the likelihood of impedance crossover with the single-phase inverter input impedance.
The stability of the cascade system is further analyzed using the root locus method with respect to \( k_{pv} \). The characteristic equation is \( 1 + T_m = 0 \), which can be rearranged to:
$$ D(s) = \frac{k_{pv} s G_x Z_{in\_INV}}{s + Z_{in\_INV} (C_{bus} s^2 + k_{iv} G_x)} $$
The root locus plot 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, corresponding to critical stability with an oscillation frequency of approximately 40 Hz. For \( k_{pv} > 0.0634 \), the system becomes stable. However, beyond \( k_{pv} = 0.258 \), the dynamic response slows down, indicating an optimal range for \( k_{pv} \). Time-domain simulations validate these findings, showing stable operation for \( k_{pv} = 0.258 \) and \( k_{pv} = 2 \), critical stability at \( k_{pv} = 0.0634 \), and instability at \( k_{pv} = 0.03 \). The FFT analysis of DC bus voltage waveforms reveals harmonic components consistent with the root locus predictions.
Based on this analysis, we propose a stability improvement method by optimizing the DAB voltage proportional gain \( k_{pv} \). This approach does not require additional compensators or hardware, making it efficient and cost-effective. The design procedure involves:
- Deriving the impedance models for the DAB and single-phase inverter.
- Identifying the resonant peak of the DAB output impedance and the negative resistance region of the single-phase inverter input impedance.
- Using root locus and Nyquist plots to determine the critical \( k_{pv} \) value for stability.
- Selecting \( k_{pv} \) within the optimal range to ensure stability while maintaining acceptable dynamic performance.
For example, when the power level increases from 5 kW to 10 kW, the single-phase inverter input impedance magnitude decreases, leading to instability. By adjusting \( k_{pv} \) from 0.102 to 0.452, the system regains stability, as confirmed by simulations. The table below compares system behavior for different \( k_{pv} \) values at 10 kW power:
| \( k_{pv} \) Value | Stability Status | Oscillation Frequency | Remarks |
|---|---|---|---|
| 0.03 | Unstable | Multiple frequencies | System collapses |
| 0.0634 | Critically Stable | 40 Hz | Borderline operation |
| 0.258 | Stable | None | Good dynamic response |
| 0.452 | Stable | None | Optimal for high power |
This method ensures that the DAB output impedance peak remains below the magnitude of the single-phase inverter input impedance in the critical frequency range, avoiding adverse interactions. The single-phase inverter’s role in this cascade system is highlighted repeatedly, as its constant power behavior is a primary source of instability. By carefully tuning the DAB controller, we can achieve robust performance across various operating conditions.
In conclusion, the impedance-based stability analysis of DAB cascade single-phase inverter systems provides valuable insights into the interaction dynamics. The derived impedance models accurately capture the system behavior, and the analysis of control parameters reveals that the DAB voltage proportional gain \( k_{pv} \) is a key factor in stabilizing the system. The proposed optimization method enhances stability without compromising dynamic performance, offering a practical solution for real-world applications. Future work could explore adaptive control strategies to automatically adjust parameters in response to changing operating conditions, further improving the reliability of single-phase inverter based systems.
The implications of this study extend to various fields, including renewable energy integration and electric vehicle charging, where single-phase inverters are commonly used. By ensuring stability in these cascaded systems, we can enhance the overall efficiency and reliability of power electronic interfaces. The methods presented here can be adapted to other converter topologies, providing a general framework for impedance-based stability analysis and control design.