# Modeling and Stability Analysis of a Single-Phase Two-Stage Grid-Connected Photovoltaic System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description and Nonlinear Averaged Equations

_{in}is the capacitance of the input filter, L

_{b}and C

_{dc}are the inductance and the capacitance of the boost converter, respectively, and L

_{f}is the inductance of the output filter. The PV array generates electricity from solar radiation. A boost converter with an input filter connects the PV array to the DC bus in order to raise the output voltage of PV array to the voltage level of DC bus while implementing maximum power point tracking (MPPT). The boost converter is designed to operate in continuous conduction mode (CCM).

_{dc}and generates the reference current i

_{ref}for the current control loop. Then, the current control loop regulates the output current of full bridge inverter i

_{o}. In order to facilitate the analysis, the boost converter and the full bridge inverter are assumed to have same switching frequency f

_{s}.

#### 2.1. PV Array

_{pv}of the PV array is related to the output voltage u

_{pv}and also affected by other parameters of PV panel itself [25]. However, the manufacturer’s datasheet do not provide some of these parameters, such as equivalent series resistance and equivalent parallel resistance. Basic parameters provided in all datasheets of PV array are open circuit voltage U

_{OC}, short circuit current I

_{SC}, the current at maximum power point (MPP) I

_{M}, and the voltage at maximum power point U

_{M}. These parameters are all with the standard test condition (STC). To overcome the lack of detailed information of the PV array, a simplified model proposed in [26] is used in this study, which describes the terminal characteristic of the PV array with STC in the following equation:

#### 2.2. Power Stage Circuit

_{pv}, the current of L

_{b}which denoted as ${i}_{{L}_{b}}$, the DC bus voltage u

_{dc}and the output current of full bridge inverter i

_{o}. The averaged equations of the input filter and boost converter can be derived as follows [19]:

_{1}is the duty cycle of the boost converter and i

_{dc}is the input current of full bridge inverter.

_{2}is the duty cycle of S

_{1}and S

_{4}in the full bridge inverter and u

_{g}is the grid voltage.

#### 2.3. Controller

_{p}is the gain of the PI controller, and T

_{i}is the time constant of the PI controller.

_{pvref}given by the algorithm is assumed to be constant. Therefore, the state equation of PI controller 1 can be derived as

_{c}

_{1}is the output signal of PI controller 1 and K

_{p}

_{1}and T

_{i}

_{1}are the gain and the time constant of the PI controller. u

_{c}

_{1}is compared to a ramp u

_{r}in the Pulse Width Modulation (PWM) comparator to produce driving signal. To complete the model of MPPT controller, the duty cycle d

_{1}is expressed as following equation:

_{M}

_{1}is the peak-to-peak amplitude of ramp u

_{r}.

_{dcref}is the reference voltage of DC bus, u

_{e}is the output signal of PI controller 2, K

_{p}

_{2}and T

_{i}

_{2}is the gain and the time constant of the PI controller. Then the reference current i

_{ref}can be expressed as:

_{c}

_{2}is the output signal of PI controller 3 and K

_{p}

_{3}and T

_{i}

_{3}are the gain and the time constant of the PI controller. u

_{c}

_{2}is compared to a triangular wave u

_{tri}in the PWM comparator to generate driving signal. The duty cycle d

_{2}is expressed as following equation:

_{M}

_{2}is the peak-to-peak amplitude of triangular wave u

_{tri}.

_{gm}is the amplitude of u

_{g}.

## 3. Observer-Pattern Model

_{o}/dt and du

_{c}

_{2}/dt. Thus, only i

_{o}and u

_{c}

_{2}need to be processed.

_{o}and u

_{c}

_{2}as i

_{oI}and u

_{c}

_{2I}. Since i

_{o}and u

_{c}

_{2}are sinusoidal, i

_{oI}and u

_{c}

_{2I}maintain 90° phase shift with i

_{o}and u

_{c}

_{2}. The Park transformation can be expressed as

**T**is the transformation matrix given by (14).

_{o}and u

_{c}

_{2}, resulting in the following equations:

**T**gives

**i**with

_{odq}**u**satisfies this equation as well, (16) can be expressed as

_{c}_{2dq}_{c}

_{2}i

_{o}in the averaged Equation (12) can be substituted by

_{1}= cos(2ωt), g

_{2}= sin(2ωt), the following equations can be constructed:

_{dc}and output current i

_{o}, respectively. In both pictures, it can be seen obviously that observer-pattern model and simulation give almost the same results in steady state.

## 4. Stability Analysis

**X**= [u

_{pv}, ${i}_{{L}_{b}}$, u

_{dc}, i

_{od}, i

_{oq}, u

_{c}

_{1}, u

_{e}, u

_{c}

_{2d}, u

_{c}

_{2q}, g

_{1}, g

_{2}]

^{T}. By setting all the differential items in (21) to zero, the equilibrium point

**X**

^{e}= [u

_{pv}

^{e}, ${i}_{{L}_{b}}$

^{e}, u

_{dc}

^{e}, i

_{od}

^{e}, i

_{oq}

^{e}, u

_{c}

_{1}

^{e}, u

_{e}

^{e}, u

_{c}

_{2d}

^{e}, u

_{c}

_{2q}

^{e}, g

_{1}

^{e}, g

_{2}

^{e}]

^{T}is obtained. Then. the Jacobian

**A**of the observer-pattern model at the equilibrium point can be derived as

**A**is presented in Appendix A.

**I**is unit matrix.

_{i}with respect to an uncertain parameter μ can be calculated as follow:

_{i}and υ

_{i}are the left and right eigenvectors corresponding to the eigenvalue λ

_{i}, respectively. If the real part of ∂λ

_{i}/∂μ is positive, an increase in the parameter μ causes the eigenvalue λ

_{i}to move towards right in horizontal direction. The size of the horizontal movement is decided by the magnitude of the real part of ∂λ

_{i}/∂μ. Similarly, the imaginary part of ∂λ

_{i}/∂μ is associated with the movement in vertical direction.

_{1,2}, the most sensitive parameter is K

_{p}

_{3}as the real part of the sensitivity of λ

_{1,2}with respect to K

_{p}

_{3}is the largest. The decrease in K

_{p}

_{3}makes λ

_{1,2}move toward right in the s-plane. For λ

_{3,4}, the most sensitive parameters are T

_{i}

_{1}and K

_{p}

_{1}. A negative perturbation in T

_{i}

_{1}makes λ

_{3,4}move towards right in the s-plane. In contrast, the decrease in K

_{p}

_{3}makes λ

_{3,4}move to left. For λ

_{5}, the most critical parameter is T

_{i}

_{1}. The increase in T

_{i}

_{1}leads to λ

_{5}moving towards right-half plane. For λ

_{6,7}, the most sensitive parameter is K

_{p}

_{2}. When K

_{p}

_{2}decreases, λ

_{6,7}moves towards right in the s-plane. For λ

_{8,9}, the most sensitive parameters is T

_{i}

_{3}. The increase in T

_{i}

_{3}leads to λ

_{8,9}moving towards right in the s-plane. λ

_{10,11}are insensitive to all these parameters listed in Table 3.

_{3,4}move across the imaginary axis from the left-half plane to the right-half plane when T

_{i}

_{1}decreases to 0.01, which means the system becomes unstable.

_{i}

_{1}equals to 0.01 and 0.03 are listed in Table 4 as a comparison. It can be seen clearly that λ

_{10,11}, which originated from Equation (20), always equals to ±j628. Therefore, λ

_{10,11}is only associated with the angular frequency of power grid and has no influence on the stability of the system. Actually, λ

_{1,2}and λ

_{8,9}are different when T

_{i}

_{1}varies. However, the slight differences are disregarded in this paper. Neglecting λ

_{10,11}, all eigenvalues are in the left half of the s-plane when T

_{i}

_{1}= 0.03, and the system operates in a stable state. However, when T

_{i}

_{1}decreases to 0.01, λ

_{3,4}becomes 26.8 ± j1453, which indicates the system is unstable and a low-frequency oscillation occurs. The frequency of the oscillation can be calculated according to the magnitude of the imaginary part of λ

_{3,4}as follows:

_{dc}obtained by PSIM when T

_{i}

_{1}equals to 0.03 and 0.01. In Figure 5a, the waveform of u

_{dc}is sinusoidal and fluctuates around the nominal value. It can be seen clearly that u

_{dc}contains a DC component and a ripple at 100 Hz. The ripple at 100 Hz is due to pulsating output power of the single-phase inverter [30]. In Figure 5c, the waveform of u

_{dc}is non-sinusoidal. It can be seen from Figure 5d that the waveform contains a ripple at 100 Hz and a component at 230.5 Hz, which matches the theoretical analysis given by the observer-pattern model.

## 5. Conclusions

_{i}

_{1}is closely related to λ

_{3,4}. The decrease in T

_{i}

_{1}makes λ

_{3,4}move to left in the s-plane and is disadvantageous to the stability of the system. The theoretical results have been validated by PSIM simulations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**A**is given as:

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**Figure 1.**Diagram of a single-phase two-stage grid-connected photovoltaic system: (

**a**) Power stage circuit; (

**b**) MPPT controller; (

**c**) double loop controller.

**Figure 4.**Loci of the eigenvalues with respect to various PI controller parameters: (

**a**) λ

_{1,2}when K

_{p}

_{3}is varied within the range (0.2, 1.8); (

**b**) λ

_{3,4}when T

_{i}

_{1}is varied within the range (0.01, 0.19); (

**c**) λ

_{3,4}when K

_{p}

_{1}is varied within the range (0.005, 0.095); (

**d**) λ

_{5}when T

_{i}

_{1}is varied within the range (0.01, 0.19); (

**e**) λ

_{6,7}when K

_{p}

_{2}is varied within the range (0.002, 0.038); (

**f**) λ

_{8,9}when T

_{i}

_{3}is varied within the range (0.02, 0.38).

**Figure 5.**Simulation results of u

_{dc}: (

**a**) Time domain waveforms when T

_{i}

_{1}= 0.03; (

**b**) Fast Fourier Transformation (FFT) analysis when T

_{i}

_{1}= 0.03; (

**c**) Time domain waveforms when T

_{i}

_{1}= 0.01; (

**d**) FFT analysis when T

_{i}

_{1}= 0.01.

Parameter | Symbol | Quantity |
---|---|---|

Voltage at MPP | U_{M} | 119.6 V |

Current at MPP | I_{M} | 8.36 A |

Open circuit voltage | U_{OC} | 149.2 V |

Short circuit current | I_{SC} | 8.81 A |

Parameter | Symbol | Quantity |
---|---|---|

Capacitance of input filter | C_{in} | 1000 μF |

Inductance of boost converter | L_{b} | 10 mH |

Capacitance of boost converter | C_{dc} | 1500 μF |

Inductance of output filter | Lf | 25 mH |

Amplitude of grid voltage | U_{gm} | 220√2 V |

Switching frequency | f_{s} | 10 kHz |

Gain of PI controller 1 | K_{p1} | 0.05 |

Time constant of PI controller 1 | T_{i}_{1} | 0.1 |

Gain of PI controller 2 | K_{p}_{2} | 0.02 |

Time constant of PI controller 2 | T_{i}_{2} | 0.01 |

Gain of PI controller 3 | K_{p}_{3} | 1 |

Time constant of PI controller 3 | T_{i}_{3} | 0.2 |

λ_{1,2} | λ_{3,4} | λ_{5} | λ_{6,7} | λ_{8,9} | λ_{10,11} | |
---|---|---|---|---|---|---|

K_{p1} | −8.93 × 10^{-4} ± j7.47 × 10^{-5} | 5.57 ± j1.38 × 10^{4} | −9.31 | −0.91 ± j0.145 | −2.45 × 10^{-5} ± j2.42 × 10^{-4} | 0 |

T_{i1} | −2.78 × 10^{-7} ± j2.88 × 10^{-8} | −47.5 ± j0.21 | 94.9 | 0.0286 ± j0.193 | 3.85 × 10^{-6} ± j3.28 × 10^{-7} | 0 |

K_{p2} | −937 ± j35.6 | −0.605 ± j0.188 | −0.664 | −134 ± j553 | 0.00347 ± j0.00126 | 0 |

T_{i2} | −11.8 ± j0.68 | −0.0211 ± j0.0846 | 1.47 | 11 ± j1144 | 1.22 × 10^{-4} ± j0.00224 | 0 |

K_{p3} | −1.6 × 10^{4} ± j0.977 | 7.45 × 10^{-4} ± j0.00145 | −1.35 × 10^{-4} | 0.236 ± j0.0336 | 0.00154 ± j0.00409 | 0 |

T_{i3} | −25 ± j7.68 × 10^{-4} | 2.62 × 10^{-5} ± j1.36 × 10^{-5} | 2.21 × 10^{-5} | 0.00141 ± j0.0192 | 25 ± j0.0208 | 0 |

T_{i}_{1} | λ_{1,2} | λ_{3,4} | λ_{5} | λ_{6,7} | λ_{8,9} | λ_{10,11} |
---|---|---|---|---|---|---|

0.01 | −16016 ± j314 | 26.8 ± j1453 | −94.7 | −2.947 ± j22.55 | −5 ± j314 | ±j628 |

0.03 | −16016 ± j314 | −4.743 ± j1451 | −31.6 | −2.927 ± j22.56 | −5 ± j314 | ±j628 |

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## Share and Cite

**MDPI and ACS Style**

Huang, L.; Qiu, D.; Xie, F.; Chen, Y.; Zhang, B.
Modeling and Stability Analysis of a Single-Phase Two-Stage Grid-Connected Photovoltaic System. *Energies* **2017**, *10*, 2176.
https://doi.org/10.3390/en10122176

**AMA Style**

Huang L, Qiu D, Xie F, Chen Y, Zhang B.
Modeling and Stability Analysis of a Single-Phase Two-Stage Grid-Connected Photovoltaic System. *Energies*. 2017; 10(12):2176.
https://doi.org/10.3390/en10122176

**Chicago/Turabian Style**

Huang, Liying, Dongyuan Qiu, Fan Xie, Yanfeng Chen, and Bo Zhang.
2017. "Modeling and Stability Analysis of a Single-Phase Two-Stage Grid-Connected Photovoltaic System" *Energies* 10, no. 12: 2176.
https://doi.org/10.3390/en10122176