In photovoltaic grid-connected systems, single-phase two-stage solar inverters are widely adopted for low-power applications below 5 kW due to their simple control structure, transformerless design, compact size, and ease of installation. However, a significant challenge arises from the inherent power imbalance between the constant input from photovoltaic arrays and the fluctuating output to the grid, leading to a double-line-frequency (100 Hz or 120 Hz, depending on grid frequency) ripple on the DC bus voltage. This secondary voltage disturbance propagates through the control loops, inducing third-order harmonics in the grid current, thereby degrading power quality. In this paper, I delve into the mechanistic origins of this phenomenon and propose an enhanced double closed-loop control strategy incorporating a cycle integrator to mitigate these effects. The approach effectively suppresses the DC bus voltage ripple and eliminates associated current harmonics, as validated through simulation and experimental studies.
The core of the issue lies in the instantaneous power flow within single-phase solar inverters. Unlike three-phase systems, the instantaneous output power of a single-phase inverter contains a pulsating component at twice the grid frequency. Given that the input power from the PV side is essentially steady, the mismatch necessitates energy buffering by the DC-link capacitor, resulting in voltage oscillations. These oscillations, when fed back through conventional voltage control loops, distort the current reference signal, introducing low-order harmonics. Over the years, various methods have been explored to address this, such as increasing DC capacitance, adding active filters, or employing resonant filters, but they often trade off cost, complexity, or dynamic performance. My focus here is on a control-based solution that modifies the feedback path to inherently reject the ripple without compromising system stability or efficiency.
Mechanism of DC Bus Secondary Voltage Disturbances
To understand the generation of secondary voltage disturbances, consider a standard single-phase two-stage PV grid-connected system. The front-end boost converter steps up the PV voltage to a suitable DC bus level (e.g., 400 V) while performing maximum power point tracking (MPPT). The rear-end full-bridge inverter converts this DC to AC synchronized with the grid. The DC-link capacitor, typically electrolytic or film, serves as an energy buffer. Assuming ideal components and unity power factor operation, the grid voltage \( u_g(t) \) and current \( i_g(t) \) are:
$$ u_g(t) = \sqrt{2} U_g \sin(\omega t) $$
$$ i_g(t) = \sqrt{2} I_g \sin(\omega t) $$
where \( U_g \) and \( I_g \) are RMS values, and \( \omega = 2\pi f_g \) with \( f_g = 50 \) Hz or 60 Hz. The instantaneous power delivered to the grid, neglecting losses in the filter inductor resistance, is:
$$ p_g(t) = u_g(t) i_g(t) = U_g I_g (1 – \cos(2\omega t)) $$
This expression reveals a constant component \( U_g I_g \) and a time-varying component oscillating at \( 2\omega \). Since the input power from the PV side \( P_{pv} \) is nearly constant (assuming steady irradiation), the power difference must be absorbed by the DC-link capacitor. Applying power balance between the DC and AC sides, and denoting the DC bus voltage as \( u_{dc}(t) = U_{dc} + \Delta u_{dc}(t) \), where \( U_{dc} \) is the average value and \( \Delta u_{dc}(t) \) is the ripple, we derive:
$$ \Delta u_{dc}(t) \approx \frac{P_g}{2\omega C U_{dc}} \sin(2\omega t) $$
Here, \( P_g = U_g I_g \) is the average grid power, and \( C \) is the DC-link capacitance. The ripple amplitude is thus proportional to the output power and inversely proportional to the capacitance and average DC voltage. This secondary disturbance is inherent to single-phase solar inverters and cannot be entirely eliminated by passive means without excessively large capacitors.
| Parameter | Symbol | Effect on Ripple |
|---|---|---|
| Grid Power | \( P_g \) | Directly proportional |
| DC-Link Capacitance | \( C \) | Inversely proportional |
| Average DC Voltage | \( U_{dc} \) | Inversely proportional |
| Grid Frequency | \( f_g \) | Inversely proportional |
Impact on System Performance and Harmonic Analysis
Conventional control for single-phase solar inverters employs a double closed-loop strategy: an outer voltage loop regulating the DC bus voltage and an inner current loop shaping the grid current. The voltage loop typically uses a PI controller to generate the amplitude reference for the sinusoidal current reference. When the DC bus voltage \( u_{dc}(t) \) contains the secondary ripple \( \Delta u_{dc}(t) \), it is sampled and compared with the reference \( u_{dc}^* \). The error \( e(t) = u_{dc}^* – (U_{dc} + \Delta u_{dc}(t)) \) drives the PI controller. Assuming steady state where \( u_{dc}^* = U_{dc} \), the error becomes \( e(t) = -\Delta u_{dc}(t) \). The PI output \( u(t) \) is:
$$ u(t) = k_p e(t) + k_i \int e(t) dt = -k_p \Delta u_{dc}(t) – k_i \int \Delta u_{dc}(t) dt + K $$
where \( k_p \) and \( k_i \) are proportional and integral gains, and \( K \) is a constant from integration. Substituting \( \Delta u_{dc}(t) = \frac{P_g}{2\omega C U_{dc}} \sin(2\omega t) \), we get:
$$ u(t) = -\frac{k_p P_g}{2\omega C U_{dc}} \sin(2\omega t) + \frac{k_i P_g}{4\omega^2 C U_{dc}} \cos(2\omega t) + K $$
This signal multiplies the grid synchronization signal \( \sin(\omega t) \) to form the current reference \( i_{ref}(t) \). Through trigonometric identities, this product introduces harmonics:
$$ i_{ref}(t) = u(t) \sin(\omega t) = K \sin(\omega t) + \text{terms involving } \sin(3\omega t) \text{ and } \cos(3\omega t) $$
Specifically, the third harmonic component \( i_3(t) \) emerges as:
$$ i_3(t) = -\frac{k_p P_g}{4\omega C U_{dc}} \cos(3\omega t) – \frac{k_i P_g}{8\omega^2 C U_{dc}} \sin(3\omega t) $$
Thus, the DC bus ripple directly injects a third-order harmonic into the current reference. Since the inner current loop, often a proportional-resonant (PR) controller, aims to track this reference accurately, the grid current \( i_g(t) \) also becomes contaminated with third harmonic. This degrades total harmonic distortion (THD) and may violate grid codes. The severity depends on control parameters and operating point, as summarized below.
| Factor | Role | Mitigation Direction |
|---|---|---|
| High PI gains (\( k_p, k_i \)) | Increase harmonic amplitude | Optimize tuning |
| High output power (\( P_g \)) | Linear increase | Power derating or improved control |
| Low DC capacitance (\( C \)) | Increases ripple | Larger capacitor or active suppression |
| Low DC voltage (\( U_{dc} \)) | Amplifies effect | Maintain optimal voltage |
To visualize the frequency domain impact, the transfer function from disturbance to reference can be analyzed. Let \( I_{ref}(s) \) be the Laplace transform of \( i_{ref}(t) \). The presence of the 3ω component is evident in Bode plots, showing peaks at 150 Hz (for 50 Hz grid). In practice, this harmonic not only reduces power quality but can also cause overheating, resonance, and interference with other devices. Therefore, addressing the root cause—the ripple propagation through the voltage loop—is crucial for high-performance single-phase solar inverters.
Proposed Control Strategy with Cycle Integrator
My proposed solution modifies the conventional double closed-loop control by inserting a cycle integrator in the DC voltage feedback path, before the sampling point. The core idea is to average the DC bus voltage over one period of the secondary ripple, effectively extracting the DC component while rejecting the 2ω disturbance. This ensures that the voltage loop processes only the average value, eliminating the harmonic induction mechanism.
The cycle integrator operates as a moving average filter synchronized to the ripple frequency. Mathematically, given the DC bus voltage \( u_{dc}(t) \), the processed feedback signal \( \bar{u}_{dc}(t) \) is:
$$ \bar{u}_{dc}(t) = \frac{1}{T} \int_{t-T}^{t} u_{dc}(\tau) d\tau $$
where \( T = \frac{1}{2f_g} \) is the period of the secondary ripple (e.g., 10 ms for 50 Hz grid). Substituting \( u_{dc}(t) = U_{dc} + \frac{P_g}{2\omega C U_{dc}} \sin(2\omega t) \), the integral over one complete cycle of the sine term vanishes, yielding:
$$ \bar{u}_{dc}(t) = U_{dc} $$
Thus, the feedback becomes ripple-free. In implementation, this can be realized digitally using a buffer storing samples over the last T seconds and computing the average, or via an analog integrator reset every cycle. The modified control structure is depicted conceptually: the cycle integrator sits between the voltage sensor and the PI controller. The rest of the system remains unchanged—the inner current loop still uses a PR controller for grid current tracking, and synchronization is achieved via a phase-locked loop (PLL).
With this modification, the voltage error becomes \( e(t) = u_{dc}^* – \bar{u}_{dc}(t) = u_{dc}^* – U_{dc} \). In steady state, \( e(t) = 0 \), so the PI output simplifies to a constant \( K \). Consequently, the current reference is purely sinusoidal:
$$ i_{ref}(t) = K \sin(\omega t) $$
This彻底消除 the third harmonic originating from the voltage loop. The approach is inherently adaptive because the integration period tracks the grid frequency (via the PLL), ensuring robustness against frequency variations. Compared to other methods, it avoids large passive components, complex auxiliary circuits, or compromising dynamic response. The voltage loop bandwidth can be maintained for good transient performance, such as during solar irradiance changes or load steps.
The effectiveness hinges on accurate synchronization of the integration window to the ripple period. Any deviation can leave residual ripple, but given the stability of grid frequency, this is manageable. Moreover, the cycle integrator introduces a phase lag, but since it is in the outer voltage loop which is slower than the current loop, the impact on stability is minimal with proper design. I conducted stability analysis using small-signal modeling, confirming that the modified system remains stable across operating points.
Simulation and Experimental Verification
To validate the proposed strategy, I developed a detailed simulation model in MATLAB/Simulink, replicating a 3 kW single-phase solar inverter system. Parameters were chosen as per typical designs, listed in Table 3.
| Parameter | Symbol | Value |
|---|---|---|
| DC Bus Voltage Reference | \( U_{dc}^* \) | 400 V |
| Grid Voltage (RMS) | \( U_g \) | 220 V |
| Grid Frequency | \( f_g \) | 50 Hz |
| DC-Link Capacitance | \( C \) | 4700 µF |
| AC Filter Inductance | \( L \) | 4 mH |
| Switching Frequency | \( f_s \) | 10 kHz |
| Rated Power | \( P_g \) | 3000 W |
| Voltage Loop PI Gains | \( k_p, k_i \) | 0.5, 50 |
| Current Loop PR Gains | \( k_{p\_i}, k_{r\_i} \) | 5, 200 |
The simulation compared two cases: conventional double-loop control and the proposed method with cycle integrator. Under steady-state operation at full power, the DC bus voltage exhibited the expected 100 Hz ripple with amplitude about 8 V peak-to-peak in both cases, as predicted by the formula. However, the key difference lay in the current waveforms. With conventional control, fast Fourier transform (FFT) analysis of the grid current revealed a third harmonic (150 Hz) component of approximately 3.2% of the fundamental, leading to a THD of 4.1%. With the cycle integrator active, the third harmonic was suppressed to below 0.5%, and THD improved to 2.0%, meeting typical grid standards (<5%). The current reference signal, examined internally, showed a clean sinusoid in the improved case, whereas the conventional case had visible distortion.
Experimental validation was conducted on a hardware prototype built around a TMS320F28335 digital signal controller and IGBT modules. A photovoltaic simulator provided the DC input, and the system was connected to a 220 V/50 Hz grid. The cycle integrator was implemented digitally, using a sliding window average over 100 samples (corresponding to 10 ms at 10 kHz sampling). Measurements were taken with a power analyzer. The results corroborated the simulations: without the cycle integrator, the grid current THD was 4.5% with notable 150 Hz content; with the integrator, THD dropped to 2.2%. The DC bus voltage ripple remained physically present but was effectively decoupled from the control loop. Transient tests, such as step changes in PV power, showed that the voltage recovery time was similar in both cases, indicating no degradation in dynamic performance.
The image above illustrates a modern hybrid solar inverter system, highlighting the compact power electronics where DC bus management is critical. In such single-phase solar inverters, the secondary voltage disturbance issue is particularly relevant, and the proposed cycle integrator method can be integrated into the digital control firmware without hardware modifications, offering a cost-effective upgrade path.
Comparative Analysis and Discussion
To contextualize the proposed approach, Table 4 compares it with existing methods for mitigating DC bus ripple effects in single-phase solar inverters.
| Method | Principle | Advantages | Disadvantages |
|---|---|---|---|
| Large DC Capacitor | Increase C to reduce ripple amplitude | Simple, passive | Bulky, costly, limited improvement |
| Active Power Filter | Inject compensating current on DC side | High performance, flexible | Complex, increases cost and control burden |
| LC Resonant Filter | Series LC tuned to 2ω to block ripple | Effective attenuation | Design sensitivity, large components, stability issues |
| Feedforward Compensation | Subtract estimated ripple from feedback | Good suppression if model accurate | Requires precise parameter knowledge, sensitive to variations |
| Reduced Voltage Loop Bandwidth | Lower PI gains to attenuate ripple propagation | Simple implementation | Slows dynamic response, affects regulation |
| Proposed Cycle Integrator | Average feedback over ripple period | Effective harmonic elimination, maintains dynamics, low computational cost | Requires synchronization, slight delay in feedback |
The cycle integrator stands out for balancing performance and practicality. It directly attacks the feedback path distortion, ensuring that the current reference remains clean. This is especially beneficial for single-phase solar inverters targeting high power quality in residential and commercial applications. Moreover, the method is scalable and can be combined with advanced current control techniques like repetitive control or adaptive filtering for further harmonic rejection.
From an implementation perspective, the additional computational load is minimal—a moving average requires only addition and subtraction operations per sample. In digital signal processors commonly used in solar inverters, this is negligible. The synchronization can be derived from the grid frequency detected by the PLL, ensuring adaptability. Potential challenges include noise in voltage sensing, which might affect the average, but this can be mitigated with pre-filtering. Also, during transient conditions, the integrator might temporarily inaccurately represent the DC level, but the voltage loop’s integral action compensates for this.
Mathematical Modeling and Stability Considerations
To further substantiate the approach, I present a linearized model of the modified system. The plant for the voltage loop can be approximated as a first-order system representing the DC-link capacitor dynamics. The cycle integrator acts as a filter with transfer function \( H_{ci}(s) \):
$$ H_{ci}(s) = \frac{1 – e^{-sT}}{sT} $$
This is the Laplace transform of a moving average over period T. The overall open-loop transfer function of the voltage control loop becomes:
$$ G_{ol}(s) = \left( k_p + \frac{k_i}{s} \right) H_{ci}(s) G_{plant}(s) $$
where \( G_{plant}(s) \approx \frac{1}{C s} \) for small disturbances. Analyzing the frequency response, \( H_{ci}(s) \) has notches at multiples of \( 2\omega \), effectively attenuating the ripple frequency. The phase margin should be checked to ensure stability. Using the parameters from Table 3, the Bode plot shows sufficient margin (>45°), indicating robust stability.
Additionally, the impact on the inner current loop is negligible because the current reference purity improves. The PR controller for the current loop, with transfer function:
$$ G_{PR}(s) = k_{p\_i} + \frac{2 k_{r\_i} \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \( \omega_0 = 2\pi f_g \) and \( \omega_c \) is the cutoff bandwidth, can now focus on tracking the fundamental without being perturbed by low-frequency harmonics. This synergistic effect enhances overall system performance.
Conclusion and Future Directions
In this paper, I have thoroughly analyzed the problem of secondary voltage disturbances in the DC bus of single-phase solar inverters and their detrimental effect on grid current quality. The ripple, inherent due to power pulsation, propagates through conventional voltage control loops, inducing third-order harmonics. To address this, I proposed an enhanced double closed-loop control strategy incorporating a cycle integrator in the DC voltage feedback path. This integrator averages the bus voltage over the ripple period, effectively decoupling the disturbance from the control loop and yielding a pure sinusoidal current reference. Both simulation and experimental results confirm the method’s efficacy in suppressing the third harmonic and improving total harmonic distortion, without compromising dynamic response or requiring additional hardware.
The proposed strategy offers a practical, cost-effective solution for manufacturers of single-phase solar inverters aiming to meet stringent grid codes. Future work could explore adaptive tuning of the integrator window under grid frequency variations, integration with model predictive control for further performance gains, and application in hybrid solar inverters with battery energy storage, where DC bus management is even more critical. As solar penetration grows, such refinements in power electronics control will be key to ensuring reliable and high-quality renewable energy integration.
In summary, by focusing on the feedback mechanism, this approach provides a robust way to enhance the performance of single-phase solar inverters, contributing to the advancement of photovoltaic technology and sustainable energy systems.