In the context of rapid economic development, the demand for energy in modern society is increasing day by day. Solar energy, as one of the clean energy sources, has the advantages of low pollution, abundant reserves, and convenient collection, and has been widely utilized on a large scale. Distributed photovoltaic (PV) solar inverters, serving as the core equipment for energy conversion and power quality control in grid-connected photovoltaic systems, play a crucial role in the entire photovoltaic grid-connected system. Their performance indicators directly or indirectly affect the power quality. Non-isolated solar inverters are widely used due to their simple structure, small size, and light weight. However, during their operation, there is a direct electrical connection between the photovoltaic unit and the AC grid. Due to the large parasitic capacitance between the photovoltaic panel and the ground, a common-mode circuit is formed among the photovoltaic panel, parasitic capacitance, ground, and the grid. The parasitic capacitance is excited by high-frequency varying common-mode voltage (CMV), causing a significant leakage current in the common-mode loop. The presence of leakage current severely affects the output power quality of the photovoltaic grid-connected system and even endangers personal safety.
To address this issue, domestic and foreign scholars have mainly conducted research from aspects such as inverter topology, modulation methods, and control strategies. In terms of inverter topology, a new boost inverter topology with fewer switching elements, strong boost capability, and no leakage current was proposed in one study. Another study introduced a novel H8 topology that reduces leakage current and common-mode voltage change rate by separating the photovoltaic array from the grid under zero-voltage conditions. A new topology using three additional diodes in the full-bridge circuit of a traditional B6 inverter not only reduces system leakage current but also lowers grid-connected current harmonic distortion while maintaining a constant common-mode voltage. Another study proposed a new solar inverter capable of delivering standard sinusoidal current to the grid even under unbalanced or harmonic distorted grid voltages while suppressing leakage current. Changing the inverter topology to suppress leakage current is effective, but it also has obvious drawbacks: adding components to the original topology not only generates additional losses but also increases costs. In terms of modulation methods, an improved discontinuous pulse width modulation strategy for non-isolated three-phase grid-connected solar inverters was proposed to reduce the low-frequency harmonic components of common-mode voltage and thus reduce leakage current. Changing the modulation strategy directly determines the inverter switching sequence without adding extra components, thus achieving leakage current suppression. However, this method is usually specific to certain topologies and has limitations. In terms of control strategies, based on the virtual impedance concept for grid-connected solar inverters, a method of adding virtual resistance in the common-mode loop to suppress leakage current was proposed. Compared with traditional methods, this approach has a wide range of applications and can effectively control costs.
In this paper, we propose a concentrated suppression strategy for leakage current in multiple solar inverter clusters. This strategy is based on carrier phase-shift control. We initialize the carrier phase of each solar inverter in the system, and then use the simulated annealing algorithm to optimize and adjust the carrier phases in real time. By properly configuring the carrier phases of each solar inverter, we achieve centralized suppression of leakage current. The research object of this paper is the non-isolated three-phase grid-connected solar inverter.
1 Mathematical Model of Solar Inverter Leakage Current
In a non-isolated photovoltaic grid-connected system, influenced by factors such as photovoltaic panel material, temperature, and humidity, there is a large parasitic capacitance between the photovoltaic unit and the ground. This causes a common-mode voltage between the photovoltaic array and the grid. This common-mode voltage acts on both ends of the parasitic capacitance through the common-mode loop, generating common-mode leakage current. Since the grid voltage is a low-frequency signal of 50 Hz, its influence on leakage current in the common-mode loop can be neglected. Therefore, the grid side can be equivalently short-circuited. The structure of a non-isolated three-phase photovoltaic grid-connected power generation system is shown below.
For the three-phase solar inverter, the common-mode voltage is expressed as:
$$
v_{\mathrm{com}} = \frac{v_{\mathrm{AO}} + v_{\mathrm{BO}} + v_{\mathrm{CO}}}{3}
$$
The solar inverter under study adopts the space vector pulse width modulation (SVPWM) strategy. Compared with the sinusoidal pulse width modulation (SPWM) strategy, SVPWM has obvious advantages in voltage utilization, harmonic elimination, switching loss, and dynamic response speed. Moreover, the modulation results of the two are equivalent. Therefore, using the double Fourier series, we can derive the Fourier expansions of \(v_{\mathrm{AO}}, v_{\mathrm{BO}}, v_{\mathrm{CO}}\).
The expressions are given as:
$$
\begin{aligned}
v_{\mathrm{AO}} &= \frac{M_r V_{\mathrm{dc}}}{2} \sin(\omega_0 t) + \frac{\sqrt{3} M_r V_{\mathrm{dc}}}{12} \cos(\omega_0 t) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t\right) \\
v_{\mathrm{BO}} &= \frac{M_r V_{\mathrm{dc}}}{2} \sin\left(\omega_0 t – \frac{2\pi}{3}\right) + \frac{\sqrt{3} M_r V_{\mathrm{dc}}}{12} \cos\left(\omega_0 t – \frac{2\pi}{3}\right) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi}{3}\right) \\
v_{\mathrm{CO}} &= \frac{M_r V_{\mathrm{dc}}}{2} \sin\left(\omega_0 t + \frac{2\pi}{3}\right) + \frac{\sqrt{3} M_r V_{\mathrm{dc}}}{12} \cos\left(\omega_0 t + \frac{2\pi}{3}\right) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t + \frac{2\pi}{3}\right)
\end{aligned}
$$
where \(M_r\) is the modulation ratio, \(\omega_{\mathrm{sw}}\) is the angular frequency of the triangular carrier, \(\omega_0\) is the angular frequency of the modulating wave, \(V_{\mathrm{dc}}\) is the DC voltage, and \(A_{mn}\) is the harmonic amplitude given by:
$$
A_{mn} = \frac{4V_{\mathrm{dc}}}{m\pi} \sin\left(\frac{m\pi}{6}\right) \left\{ \frac{1}{2} J_n(P_2) + \frac{\sqrt{3}}{4} \left[ J_{n+1}(P_2) + J_{n-1}(P_2) \right] \cos\left(\frac{n\pi}{2}\right) + \sum_{k=1}^{\infty} \left[ \frac{\sin\left(\frac{k\pi}{3}\right)}{k\pi} J_{n+k}(P_2) + \frac{\sin\left(\frac{2k\pi}{3}\right)}{k\pi} J_{n-k}(P_2) \right] \cos\left(\frac{n\pi}{2}\right) \right\}
$$
Here \(J_n(x)\) is the Bessel function of the first kind, and parameters \(P_h\) are defined as:
$$
\begin{aligned}
P_1 &= m + n \\
P_2 &= m M_r \\
P_3 &= m – k \\
P_4 &= n – k \\
P_5 &= n + k
\end{aligned}
$$
Based on the equivalent model, using the node voltage method and superposition theorem, the leakage current of a single solar inverter can be derived as:
$$
i_{\mathrm{leakage}} = \frac{1}{3} \cdot \frac{P_6 C_{\mathrm{pv}}}{P_6 + P_7 + P_8 + P_9} \cdot (v_{\mathrm{AO}} + v_{\mathrm{BO}} + v_{\mathrm{CO}})
$$
where:
$$
\begin{aligned}
P_6 &= j\omega C_f L_g \\
P_7 &= j\omega C_{\mathrm{pv}} L_g \\
P_8 &= j\omega C_{\mathrm{pv}} L_1 \\
P_9 &= -\omega^4 C_{\mathrm{pv}} C_f L_1 L_g
\end{aligned}
$$
Denoting the passive parameter part as \(K\), we have:
$$
K = \frac{1}{3} \cdot \frac{P_6 C_{\mathrm{pv}}}{P_6 + P_7 + P_8 + P_9}
$$
Assuming that the passive parameters and DC voltage of each solar inverter are identical, the sum of leakage currents of \(a\) solar inverters can be expressed as:
$$
i_{\mathrm{lea\_sum1}} = K \left[ \frac{4a M_r V_{\mathrm{dc}}}{3} \sin(\omega_0 t) + a \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t\right) + a \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi}{3}\right) + a \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t + \frac{2\pi}{3}\right) \right]
$$
2 Leakage Current Suppression Strategy Based on Carrier Phase-Shift Control
2.1 Carrier Phase-Shift Control
For the leakage current problem of solar inverters, we propose the following carrier phase-shift control implementation: Assume the total number of solar inverters in the system is \(a\). The carrier phase of the first solar inverter remains unchanged; the carrier phase of the second solar inverter is shifted by \(2\pi (1/a)\) relative to the first; the carrier phase of the third solar inverter is shifted by \(2\pi (2/a)\) relative to the first; and so on, the carrier phase of the \(a\)-th solar inverter is shifted by \(2\pi ((a-1)/a)\) relative to the first. According to this procedure, the initial carrier phases of all solar inverters are initialized. The schematic diagram of the initial triangular carrier phases is shown conceptually.
Substituting the initial carrier phases into the leakage current expression of each solar inverter, the leakage current of the \(a\)-th solar inverter can be expressed as:
$$
i_{\mathrm{lea\_a}} = K \left[ \frac{4 M_r V_{\mathrm{dc}}}{3} \sin(\omega_0 t) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi (a-1)}{a}\right) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi (a-1)}{a} – \frac{2\pi}{3}\right) + \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi (a-1)}{a} + \frac{2\pi}{3}\right) \right]
$$
Let the first harmonic part be \(G_{a1}\). Then summing \(G_{11}, G_{21}, \dots, G_{a1}\), we obtain:
$$
\sum_{a=1}^{a} G_{a1} = \sum_{m=1}^{\infty}\sum_{n=-\infty}^{\infty} A_{mn} \left[ \cos(\theta) + \cos\left(\theta – \frac{2\pi}{a}\right) + \dots + \cos\left(\theta – \frac{2\pi(a-1)}{a}\right) \right]
$$
where \(\theta = m\omega_{\mathrm{sw}} t + n\omega_0 t\). Considering the parity of \(a\) and the values of \(m\) and \(n\), the terms cancel except when \(m = ka (k=1,2,\dots)\). Thus we get:
$$
\sum_{a=1}^{a} G_{a1} = a \sum_{m=a,2a,\dots}\sum_{n=-\infty}^{\infty} A_{mn} \cos(\theta)
$$
Similarly, the other two harmonic components also cancel except for \(m = ka\). Therefore, the total leakage current after carrier phase-shift control becomes:
$$
i_{\mathrm{lea\_sum2}} = K \left[ \frac{4a M_r V_{\mathrm{dc}}}{3} \sin(\omega_0 t) + a \sum_{m=a,2a,\dots}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t\right) + a \sum_{m=a,2a,\dots}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t – \frac{2\pi}{3}\right) + a \sum_{m=a,2a,\dots}\sum_{n=-\infty}^{\infty} A_{mn} \cos\left(m\omega_{\mathrm{sw}} t + n\omega_0 t + \frac{2\pi}{3}\right) \right]
$$
Comparing \(i_{\mathrm{lea\_sum1}}\) and \(i_{\mathrm{lea\_sum2}}\), we can clearly see that after using the carrier phase-shift control strategy, the harmonic components in the total leakage current expression partially cancel each other, reducing the total leakage current amplitude. This theoretically proves the effectiveness of the carrier phase-shift control strategy in achieving concentrated suppression of leakage current in the solar inverter system.
2.2 Simulated Annealing Algorithm
In practical systems, solar inverters may have different DC voltages and power levels, leading to differences in leakage current amplitudes. Phase shifting by a fixed angle may not achieve precise suppression of the total leakage current. Therefore, considering the differences in leakage current amplitudes caused by different DC voltages and power levels, we adopt the simulated annealing algorithm to optimize the carrier phases of each solar inverter. According to the system’s total leakage current standard, a preset maximum total leakage current \(\eta\) is defined. When the sum of leakage currents of all solar inverters exceeds this value, based on the initial phases of each solar inverter, the carrier phase compensation values are taken as variables. The simulated annealing algorithm is used to obtain the optimal solution. The system then outputs the results for real-time optimization and compensation of the leakage current phases of each solar inverter.
The flow of the simulated annealing algorithm is shown conceptually. The specific steps are as follows:
- Initialize algorithm parameters and system passive parameters, including initial temperature, termination temperature, annealing rate, and the number of iterations at each temperature.
- Randomly generate carrier phase data sets for the solar inverters within \([0, 2\pi]\). Substitute each phase data into the leakage current expression and calculate the total leakage current sum. Determine whether to accept the phase data according to the Metropolis criterion:
$$
p = \begin{cases}
1, & \text{if } i(S_i) < i(S_{i-1}) \\
\exp\left(-\frac{\Delta i}{T}\right), & \text{if } i(S_i) \geq i(S_{i-1})
\end{cases}
$$
where \(p\) is the probability of accepting the phase data.
- Compare \(p\) with a random number generated in \([0,1]\). If \(p\) is less than the random number, accept the current carrier phase data and check whether the total leakage current is less than the standard maximum \(\eta\). If satisfied, exit the loop and designate this carrier phase data as the compensation value. Otherwise, reject the current phase data and continue the iteration until an ideal carrier phase compensation value is found.
3 Simulation Results and Analysis
3.1 Simulation Analysis of Carrier Phase-Shift Control Strategy
To further verify the effectiveness of the carrier phase-shift control strategy, we take three grid-connected solar inverters with identical parameters as an example and build a simulation model using MATLAB/Simulink. The simulation analysis is performed for the leakage current of the parallel system. The main system parameters are listed in the table below.
| Parameter | Value |
|---|---|
| Grid voltage \(e_g\) / V | 380 |
| Inverter-side inductor \(L_1\) / mH | 1.5 |
| Grid-side inductor \(L_g\) / mH | 1.2 |
| Filter capacitor \(C_f\) / μF | 9.4 |
| Switching frequency \(f_s\) / kHz | 5 |
| Parasitic capacitance \(C_{\mathrm{pv}}\) / nF | 1 |
According to the definition of initial carrier phase shift angle, when the number of solar inverters in the system is 3, the adjacent carrier phase difference is 120°. Specifically, the carrier phase of the first solar inverter remains unchanged, the initial phase of the second solar inverter is 120°, and the initial phase of the third solar inverter is 240°. Figure 5 (omitted for brevity) shows a comparison of system waveforms before and after using the carrier phase-shift control strategy for three solar inverters with a DC voltage of 1000 V. From the plots, we observe that compared to before using carrier phase-shift control, the total leakage current amplitude of the system decreased by 0.9 A after adopting the strategy, a significant reduction, and it remained within an appropriate range. The RMS value of the total leakage current decreased by about 0.4 A, reaching approximately 0.28 A after the control. Moreover, the three-phase grid-connected current THD decreased from 2.61% to 1.03%, with smoother waveforms and significantly improved power quality. The simulation results confirm the effectiveness of the proposed system-level solar inverter leakage current suppression method.
3.2 Simulation Analysis of Simulated Annealing Algorithm
In practical applications, the DC voltages of solar inverters in a parallel system may differ, causing differences in leakage current amplitudes. Phase shifting by a fixed angle cannot achieve precise suppression of the total leakage current. To improve the accuracy of leakage current suppression, we use the simulated annealing algorithm to optimize the carrier phase shift angles. To verify the effectiveness of this phase optimization strategy, we take three solar inverters with DC voltages of 1000 V, 800 V, and 600 V as an example and build a simulation model in MATLAB/Simulink. After running the simulated annealing algorithm (written in MATLAB), the obtained carrier shift compensation angles are 173.07°, 183.78°, and 171.72° respectively, corresponding to the optimized carrier phases of 173.07°, 303.78°, and 51.72°. Figure 6 (omitted for brevity) shows a comparison of system waveforms before and after using the simulated annealing algorithm optimization. From the plots, it is evident that after shifting the carrier phases by the new angles, the total leakage current amplitude is slightly reduced. The RMS value of the total leakage current decreased by about 0.02 A, reaching approximately 0.19 A. The three-phase grid-connected current THD decreased from 1.49% to 1.37%, and the grid-connected current waveforms are good. The simulation results validate the effectiveness of the carrier phase optimization strategy based on the simulated annealing algorithm.
4 Conclusion
For the leakage current problem caused by common-mode voltage acting on parasitic capacitance in systems composed of non-isolated three-phase grid-connected solar inverters, we have proposed a concentrated suppression strategy based on carrier phase-shift control. Taking a single solar inverter as the research object, we first determine the initial carrier phases of each solar inverter according to the total number of inverters in the system. Then, we use the simulated annealing algorithm to optimize and compensate the initial carrier phases. Finally, theoretical analysis and simulation verify the effectiveness of the proposed leakage current centralized suppression strategy. This strategy does not require additional hardware, is simple to implement, achieves significant results, reduces costs, has a wide range of applications, and can effectively improve the overall safety level of the system.