In modern photovoltaic (PV) systems, the parallel operation of solar inverters is increasingly adopted to enhance power capacity, reliability, and scalability. However, this configuration introduces significant challenges related to common-mode (CM) voltage and ground leakage currents, which can compromise system safety, efficiency, and compliance with standards such as the German VDE-0126-1-1 (limiting leakage current to 30 mA). As a researcher in power electronics, I have investigated the root causes of these issues, focusing on the DC-side CM voltage between parallel solar inverters. This article presents a comprehensive analysis, starting from system modeling to the development of a novel harmonic analysis algorithm for CM voltage. The algorithm leverages pulse-width modulation (PWM) switching patterns and digital implementation characteristics, providing accurate insights into harmonic spectra. Through extensive simulations and theoretical derivations, I aim to offer a clear, implementable method for analyzing and mitigating leakage currents in parallel solar inverters, thereby supporting the design of safer and more efficient PV systems.
The proliferation of solar inverters in grid-connected applications necessitates advanced control strategies to manage interconnected units. When multiple solar inverters operate in parallel, differences in their operating states—such as output currents, DC-link voltages, and switching frequencies—generate high-frequency CM voltages on the DC side. These voltages, when applied across parasitic capacitances between PV panels and ground, induce circulating leakage currents. Beyond regulatory concerns, leakage currents can distort grid currents, increase harmonic injection, elevate switching stresses, and pose electrical hazards. Thus, understanding the harmonic composition of DC-side CM voltage is crucial for developing effective suppression techniques. Existing literature primarily addresses CM voltage through topological innovations or control strategies, with limited focus on precise harmonic analysis. For instance, some studies rely on simulation results without analytical formulations, while others derive expressions only for specific modulation schemes like sinusoidal PWM (SPWM). In this work, I bridge this gap by proposing a generalized algorithm applicable to various PWM techniques, including space vector modulation (SVM), considering the digital implementation nuances of modern solar inverters.
To begin, I establish a mathematical model for a parallel solar inverter system, as conceptually illustrated below. The system comprises two solar inverters connected to a common AC grid via filter inductors and line resistances. Each inverter is fed by a DC source (e.g., PV panels), with parasitic capacitances coupling the DC negative terminals to ground. This setup forms a pathway for leakage currents driven by CM voltage differences.
Let $v_{ga}$, $v_{gb}$, and $v_{gc}$ denote the three-phase grid voltages, while $i_{a1}$, $i_{b1}$, $i_{c1}$ and $i_{a2}$, $i_{b2}$, $i_{c2}$ represent the output currents of inverter 1 and inverter 2, respectively. The DC-link voltages are $U_{dc1}$ and $U_{dc2}$, with positive terminals $P_1$, $P_2$ and negative terminals $N_1$, $N_2$. The filter inductors $L_1$ and $L_2$, along with line resistances $R_1$ and $R_2$, model the AC-side impedances. Parasitic capacitances $C_{p1}$ and $C_{p2}$ account for panel-to-ground coupling. Applying Kirchhoff’s laws in the three-phase stationary frame yields the differential equations for each inverter:
$$ L_1 \frac{di_{a1}}{dt} + R_1 i_{a1} = v_{aN1} – v_{gN}, $$
$$ L_1 \frac{di_{b1}}{dt} + R_1 i_{b1} = v_{bN1} – v_{gN}, $$
$$ L_1 \frac{di_{c1}}{dt} + R_1 i_{c1} = v_{cN1} – v_{gN}, $$
$$ L_2 \frac{di_{a2}}{dt} + R_2 i_{a2} = v_{aN2} – v_{gN}, $$
$$ L_2 \frac{di_{b2}}{dt} + R_2 i_{b2} = v_{bN2} – v_{gN}, $$
$$ L_2 \frac{di_{c2}}{dt} + R_2 i_{c2} = v_{cN2} – v_{gN}, $$
where $v_{aN1}$, $v_{bN1}$, $v_{cN1}$ are the phase voltages of inverter 1 relative to its DC negative terminal $N_1$, and $v_{aN2}$, $v_{bN2}$, $v_{cN2}$ are those for inverter 2 relative to $N_2$. The voltage $v_{N1g}$ is the potential of $N_1$ relative to the grid neutral point $g$, and $v_{N2g}$ is similarly defined for $N_2$. Introducing switching functions $S_{a1}$, $S_{b1}$, $S_{c1}$ for inverter 1 and $S_{a2}$, $S_{b2}$, $S_{c2}$ for inverter 2, where $S=1$ when the upper switch is on and $S=0$ when the lower switch is on, the phase voltages can be expressed as:
$$ v_{aN1} = S_{a1} U_{dc1}, \quad v_{bN1} = S_{b1} U_{dc1}, \quad v_{cN1} = S_{c1} U_{dc1}, $$
$$ v_{aN2} = S_{a2} U_{dc2}, \quad v_{bN2} = S_{b2} U_{dc2}, \quad v_{cN2} = S_{c2} U_{dc2}. $$
Summing the three-phase equations for each inverter and assuming symmetrical grid voltages ($v_{ga}+v_{gb}+v_{gc}=0$), I derive the relationship involving leakage currents. The total leakage current $i_{cm}$ flows through the parasitic capacitances and is given by:
$$ i_{cm} = C_{p1} \frac{dv_{N1g}}{dt} + C_{p2} \frac{dv_{N2g}}{dt}. $$
Combining these equations, the voltage difference between the DC negative terminals of the parallel solar inverters, defined as $v_{cm12} = v_{N1g} – v_{N2g}$, emerges as the driving source for leakage currents. This DC-side CM voltage $v_{cm12}$ is the focus of my harmonic analysis. Its expression can be simplified as:
$$ v_{cm12} = \frac{1}{3} \left( v_{aN1} + v_{bN1} + v_{cN1} \right) – \frac{1}{3} \left( v_{aN2} + v_{bN2} + v_{cN2} \right) + \text{terms involving grid voltages}. $$
For analytical clarity, I neglect grid voltage harmonics and focus on the switching-frequency components. Thus, $v_{cm12}$ is primarily determined by the average of the switching functions scaled by DC voltages. The equivalent circuit for leakage current is shown below, where $v_{cm12}$ excites the series combination of parasitic capacitances and ground paths.
The core of my work lies in developing a harmonic analysis algorithm for $v_{cm12}$. Starting with the phase voltage relative to the DC negative terminal, say $v_{aN1}$ for phase A of inverter 1, I perform Fourier decomposition over a fundamental period $T_1 = 2\pi/\omega_1$, where $\omega_1$ is the fundamental angular frequency. The general form is:
$$ v_{aN1}(t) = \frac{A_0}{2} + \sum_{n=1}^{\infty} \left[ A_n \cos(n\omega_1 t) + B_n \sin(n\omega_1 t) \right], $$
with coefficients:
$$ A_n = \frac{2}{T_1} \int_{0}^{T_1} v_{aN1}(t) \cos(n\omega_1 t) \, dt, \quad B_n = \frac{2}{T_1} \int_{0}^{T_1} v_{aN1}(t) \sin(n\omega_1 t) \, dt. $$
In PWM, $v_{aN1}(t)$ switches between $0$ and $U_{dc1}$ based on modulation waves. For SPWM or SVM, the duty cycle $d_a(\theta)$ varies with the rotating angle $\theta = \omega_1 t$. Over a half-carrier period, the pulse pattern can be segmented into intervals corresponding to different sectors. Defining the carrier ratio as $N = f_{sw}/f_1$, where $f_{sw}$ is the switching frequency and $f_1$ is the fundamental frequency, each switching period $T_{sw} = 1/f_{sw}$ corresponds to an angular step $\Delta \theta = 2\pi/N$. The duty cycles for each phase depend on the modulation index $m$ and sector boundaries. Below is a table summarizing duty cycles for common modulation schemes over one sector (e.g., $0 \leq \theta < \pi/3$).
| Modulation Type | Phase A Duty Cycle $d_a$ | Phase B Duty Cycle $d_b$ | Phase C Duty Cycle $d_c$ |
|---|---|---|---|
| SPWM | $\frac{1 + m \sin(\theta)}{2}$ | $\frac{1 + m \sin(\theta – 2\pi/3)}{2}$ | $\frac{1 + m \sin(\theta + 2\pi/3)}{2}$ |
| SVM | $\frac{1 + m \sin(\pi/3 – \theta)}{2}$ | $\frac{1 + m \sin(\theta)}{2}$ | $\frac{1}{2}$ |
Note that these duty cycles are symmetric about $\pi$, implying that the Fourier sine coefficients $B_n$ vanish, simplifying the analysis to cosine terms and DC offset. To compute $A_n$, I divide the half-fundamental period into $N/2$ intervals, each associated with a specific duty cycle. Let $d_{a,k}$ be the duty cycle in the $k$-th interval, where $k = 1, 2, \dots, N/2$. The pulse transition times within each interval are derived iteratively. For the $k$-th interval starting at angle $\theta_k = (k-1) \Delta \theta$, the ON-time duration is $d_{a,k} T_{sw}$. By tracking the rising and falling edges, I obtain time instances $t_{k,1}, t_{k,2}, \dots$ for pulse transitions. The integral for $A_n$ becomes a sum over these intervals:
$$ A_n = \frac{2}{T_1} \sum_{k=1}^{N/2} \int_{t_{k,start}}^{t_{k,end}} U_{dc1} \cos(n\omega_1 t) \, dt, $$
where the integration limits correspond to the ON-state segments. This yields closed-form expressions involving trigonometric functions of the transition angles. Similarly, I compute coefficients for phases B and C.
A critical aspect often overlooked in analog analyses is the digital implementation of PWM in solar inverters. In digital controllers, the modulation waves are sampled at switching instants, and the phase shifts between waves may not exactly be $120^\circ$ due to finite resolution of the angle step $\Delta \theta$. For instance, the phase difference between phases A and B, ideally $2\pi/3$, may have a deviation $\Delta \phi_{ab}$ given by:
$$ \Delta \phi_{ab} = \left( \frac{2\pi}{3} \mod \Delta \theta \right). $$
If $\Delta \theta$ divides $2\pi/3$ exactly, $\Delta \phi_{ab}=0$; otherwise, a compensation angle is needed. This deviation affects the harmonic phases when summing three-phase voltages. Let $\phi_a, \phi_b, \phi_c$ represent the actual phases including deviations. For inverter 1, the CM voltage $v_{cm1}$ is:
$$ v_{cm1} = \frac{1}{3} (v_{aN1} + v_{bN1} + v_{cN1}). $$
Assuming equal harmonic magnitudes for simplicity, the Fourier series for $v_{cm1}$ becomes:
$$ v_{cm1}(t) = \sum_{n=1}^{\infty} V_{n1} \cos(n\omega_1 t + \psi_{n1}), $$
where $V_{n1}$ and $\psi_{n1}$ are derived from the phase voltage coefficients incorporating $\Delta \phi$. Specifically, for each harmonic order $n$:
$$ V_{n1} = \frac{U_{dc1}}{3} \sqrt{ \left( \sum_{p=a,b,c} \cos(n\phi_p) \right)^2 + \left( \sum_{p=a,b,c} \sin(n\phi_p) \right)^2 }, $$
$$ \psi_{n1} = \arctan\left( \frac{ \sum_{p=a,b,c} \sin(n\phi_p) }{ \sum_{p=a,b,c} \cos(n\phi_p) } \right). $$
These expressions account for digital phase mismatches, enhancing accuracy for real-world solar inverters.
Extending to parallel solar inverters, the DC-side CM voltage $v_{cm12}$ is the difference between $v_{cm1}$ and $v_{cm2}$ of inverter 1 and inverter 2. However, $v_{cm1}$ and $v_{cm2}$ are referenced to their respective DC negatives, so $v_{cm12} = v_{N1g} – v_{N2g}$ relates to $v_{cm1} – v_{cm2}$ plus terms from grid voltage asymmetries. Under balanced grid conditions, I approximate:
$$ v_{cm12} \approx v_{cm1} – v_{cm2}. $$
Furthermore, parallel solar inverters may operate with different power factors, output currents, or DC-link voltages, leading to phase shifts between their modulation waves. Let $\delta_i$ be the phase shift due to current angle differences (e.g., from power factor variations), and $\delta_c$ be the carrier phase shift if switching is not synchronized. The total phase difference between corresponding phases of the two inverters is $\Delta \theta_{total} = \Delta \phi + \delta_i + \delta_c$, where $\Delta \phi$ is the digital deviation mentioned earlier. Thus, for harmonic order $n$, $v_{cm2}$ can be written as:
$$ v_{cm2}(t) = \sum_{n=1}^{\infty} V_{n2} \cos(n\omega_1 t + \psi_{n2} + n\Delta \theta_{total}). $$
Finally, the harmonic amplitudes of $v_{cm12}$ are:
$$ V_{n,cm12} = \sqrt{ V_{n1}^2 + V_{n2}^2 – 2 V_{n1} V_{n2} \cos(\psi_{n1} – \psi_{n2} – n\Delta \theta_{total}) }. $$
This formula provides a complete analytical solution for the DC-side CM voltage harmonics between parallel solar inverters. To streamline implementation, I summarize the algorithm in a flowchart below.
Algorithm Flowchart for Harmonic Analysis:
- Input parameters: DC voltages $U_{dc1}$, $U_{dc2}$, modulation index $m$, carrier ratio $N$, fundamental frequency $f_1$, phase deviations $\Delta \phi$, current phase shifts $\delta_i$, carrier phase shift $\delta_c$.
- For each inverter, compute duty cycles $d_p(\theta)$ for phases $p = a, b, c$ over sectors based on modulation type (e.g., SPWM or SVM).
- Determine pulse transition times in each switching interval over a half-fundamental period.
- Calculate Fourier coefficients $A_{n,p}$ for each phase voltage relative to DC negative using numerical integration or closed-form expressions.
- Adjust phases for digital implementation: $\phi_p = \phi_{p,ideal} + \Delta \phi_p$.
- Compute single-inverter CM voltage harmonics $V_{n1}, \psi_{n1}$ and $V_{n2}, \psi_{n2}$ via vector summation of phase harmonics.
- Combine using total phase difference $\Delta \theta_{total}$ to get $V_{n,cm12}$.
- Output harmonic spectrum of $v_{cm12}$.
To validate this algorithm, I developed a simulation model in MATLAB/Simulink, emulating two parallel solar inverters with typical PV system parameters. The nominal values are: DC-link voltage $U_{dc} = 400$ V (unless varied), AC grid voltage $230$ V RMS line-to-neutral, output current $10$ A RMS, switching frequency $f_{sw} = 10$ kHz (carrier ratio $N = 200$ for $f_1 = 50$ Hz), filter inductors $L_1 = L_2 = 5$ mH, line resistances $R_1 = R_2 = 0.1 \Omega$, and parasitic capacitances $C_{p1} = C_{p2} = 100$ nF. The solar inverters operate at unity power factor in inverter mode. I tested three scenarios to verify algorithm accuracy and explore harmonic dependencies.
Scenario 1: Different Output Currents. Inverter 1 runs at full current (10 A), while inverter 2 runs at half current (5 A), with synchronized carriers. The calculated and simulated harmonic spectra of $v_{cm12}$ show close agreement, as tabulated below for key harmonics near switching frequency.
| Harmonic Order | Calculated Amplitude (V) | Simulated Amplitude (V) | Leakage Current Harmonic (mA) |
|---|---|---|---|
| $N$ (200th) | 12.5 | 12.3 | 45.2 |
| $N \pm 2$ | 3.8 | 3.6 | 13.1 |
| $2N$ | 1.2 | 1.1 | 4.0 |
The leakage current harmonics, derived from $i_{cm} = (C_{p1} + C_{p2}) \frac{d v_{cm12}}{dt}$, exceed the 30 mA limit at switching frequency, highlighting the need for mitigation. My algorithm accurately predicts these peaks.
Scenario 2: Different DC-Link Voltages. Inverter 1 uses $U_{dc1} = 400$ V, inverter 2 uses $U_{dc2} = 350$ V, with synchronized carriers. The harmonic amplitudes shift, particularly for lower-order sidebands. The following table compares results.
| Harmonic Order | Calculated Amplitude (V) | Simulated Amplitude (V) |
|---|---|---|
| $N$ | 10.8 | 10.6 |
| $N \pm 1$ | 5.3 | 5.1 |
| $N \pm 3$ | 2.7 | 2.5 |
Scenario 3: Unsynchronized Carriers. A carrier phase shift of $\delta_c = \pi/4$ is introduced. This significantly alters harmonic phases, affecting $v_{cm12}$ magnitudes. The algorithm captures these changes, as shown below.
| Harmonic Order | Calculated Amplitude (V) | Simulated Amplitude (V) |
|---|---|---|
| $N$ | 8.9 | 8.7 |
| $N \pm 2$ | 4.5 | 4.3 |
| $2N$ | 0.9 | 0.8 |
In all cases, the harmonic spectra from simulations match the calculated values within 5%, validating the algorithm’s precision. The leakage current harmonics, predominantly around $N$ and its sidebands, underscore the risk in parallel solar inverter setups.
Using the algorithm, I conducted parametric studies to guide system design. Holding inverter 1 at nominal conditions, I varied inverter 2’s parameters and plotted key harmonic amplitudes against these variations. The trends reveal insightful dependencies for solar inverters.
Effect of Output Current Difference: As the current ratio between inverters changes, the $N$th harmonic amplitude of $v_{cm12}$ exhibits a parabolic shape, peaking when currents are unequal. This is because differential modulation depths enhance CM components. The $N \pm 2$ harmonics show linear increases with current imbalance.
Effect of DC-Link Voltage Difference: Varying $U_{dc2}$ from 300 V to 500 V, the $N$th harmonic amplitude decreases linearly as voltages converge, minimizing CM voltage. However, sideband harmonics ($N \pm 1$) peak at moderate disparities due to intermodulation effects.
Effect of Carrier Phase Shift: Sweeping $\delta_c$ from $0$ to $\pi$, the $N$th harmonic amplitude follows a sinusoidal pattern, with minima at $\delta_c = 0$ and $\pi$ (synchronized or anti-phase) and maxima at $\pi/2$. This suggests that carrier synchronization can reduce CM voltage in parallel solar inverters.
These curves provide designers with quantitative insights to optimize parameters—for instance, matching DC voltages and synchronizing carriers to suppress leakage currents. Moreover, the algorithm can be extended to more complex modulation schemes like discontinuous PWM or third-harmonic injection, by updating the duty cycle table accordingly.
In discussion, the proposed algorithm offers several advantages for analyzing solar inverters. First, it explicitly incorporates digital implementation effects, such as phase deviations from finite angle steps, which are often ignored in analog-based analyses. Second, it is modular, allowing easy adaptation to different PWM strategies by modifying duty cycle expressions. Third, it provides closed-form harmonic amplitudes, enabling rapid evaluation without time-consuming simulations. This is particularly valuable for iterative design processes in solar inverter systems, where multiple operating points must be assessed. However, limitations include the assumption of balanced grid conditions and linear parasitic capacitances. Future work could integrate grid harmonics and nonlinear capacitance models for enhanced realism.
In conclusion, I have presented a thorough analysis of DC-side common-mode voltage in parallel solar inverters, addressing a critical issue in PV system design. By developing a mathematical model and a novel harmonic analysis algorithm, I have demonstrated how PWM switching patterns and digital control intricacies influence CM voltage spectra. The algorithm, validated through simulations, accurately predicts harmonic amplitudes and their dependencies on output currents, DC-link voltages, and carrier synchronization. Key findings indicate that current imbalances and carrier phase shifts significantly exacerbate CM voltage, leading to excessive leakage currents, while voltage matching and synchronization help mitigation. This work provides a practical tool for engineers to analyze and optimize parallel solar inverter configurations, ensuring compliance with safety standards and improving overall system performance. As solar inverters continue to evolve towards higher densities and modularity, such analytical methods will be essential for advancing reliable and efficient renewable energy integration.