# Resonance Instability of Photovoltaic E-Bike Charging Stations: Control Parameters Analysis, Modeling and Experiment

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## Abstract

**:**

## 1. Introduction

_{p}of Canadian Solar

^{®}PV modules and 2.0 kW inverter SOLAX Power

^{®}X1 Series) and was installed by the end of 2017.

## 2. Experimental Study

#### 2.1. Experimental Set-Up

^{®}ALL-BLACK CS6K-275 (marked with red dotted squares at the upper part of Figure 1), and a single-phase grid-connected inverter with a rated power of 2 kW (at left part of Figure 1). For further details on the PV modules and the inverter, please refer to References [18,19], respectively. One can also see in Figure 1, the flexible CIGS mini modules in the front wheel PV of the solar e-bikes.

#### 2.2. Experimental Results

#### 2.3. Passivity Impedance-Based Stability Criterion

_{c}i

_{ref}) and an inverter output impedance (Z

_{inv}) connected in parallel, as shown in Figure 4, in the grey box.

_{c}(s) is the transfer function of the current control loop of the inverter; i

_{ref}is the reference current; Y

_{inv}(s) is the inverter output admittance and v is the inverter output voltage.

_{g}(s) represents the grid impedance; vg is the grid voltage, H(s) and Y(s) are the closed-loop transfer function and input admittance of the inverter-grid system, separately.

_{OL}(s) = Y

_{inv}(s)Z

_{g}(s). When there is a resonance point in the grid impedance with a small damping coefficient, the grid impedance will intersect the output impedance of the inverter near the resonance frequency ω

_{r}with a large peak. Additionally, the phase angle of grid impedance will abruptly change from π to −π at ω

_{r}. When the phase angle of the inverter output admittance meets −π ≥ arg[Yinv(jω)] ≥ π, then the phase angle of the open-loop transfer function arg[Y

_{OL}(jω)] will certainly cross over ±2π near the resonance frequency ω

_{r}. Thus, Y(s) will be unstable in this situation. Conversely, if the phase angle of the inverter output admittance meets −π ≤ arg[Y

_{inv}(jω)] ≤ π, which can also be called passivity, Y(s) can be stable.

- Y(s) is stable and;
- −π ≤ arg[Y(jω)] ≤ π and can also be expressed equivalently as;
- Re{Y(jω)} ≥ 0, ∀ω ∈ (0,fL].which means that the transfer function is passive in the interval (0,fL].

_{inv}(s) and Z

_{g}(s) need to be proven to be passive. We assume that both Y

_{inv}(s) and Z

_{g}(s) are passive, as shown in Equation (3).

_{c}(s) of a well-functioning grid-connected inverter is passive because its poles are located in the left half plane. Due to the same principle, the closed-loop transfer function of the inverter-grid system H(s) is also passive if Y(s) is already passive.

_{g}(s) is always stable and passive [14]. In this way, the stability of the inverter-grid system is determined by the passivity degree of the inverter output admittance Yinv(s). This criterion is expressed in Equation (5).

_{inv}(jω)} is equal to 0, the inverter-grid system will be in a critical stable state. The inverter output admittance can be derived from the modeling of the inverter.

## 3. Modeling the Inverter in the Frequency-Domain

- proportional resonance (PR) based current control
- synchronous rotating frame (dq-transformation) based current control

#### 3.1. PR-Based Current Control Method

^{*}with the output from the DC voltage regulator i*m. The current loop controller is a quasi-PR regulator, which may robustly track the sinusoidal waveform without a static error, its transfer function is expressed in Equation (6).

_{P-PR}is the proportional coefficient; K

_{R-PR}is the resonance coefficient; ω

_{c}is the control bandwidth; ω

_{0}is the resonant frequency.

_{0}(s) is the output admittance of the inverter hardware, G

_{id}(s) is the transfer function from the output voltage of the full bridge circuit u to output current i. T

_{PLL}(s) is the transfer function from the grid side voltage v to the normalized current reference. The controller transfer function G

_{PR}(s) is expressed in Equation (6), G

_{d}(s) is the transfer function for the time delay in the digital control. The delay time consists of the calculation time in the controller and PWM modulation time. In this case, it is 1.5 times that of the sampling period [24,25].

_{0}can be derived by assuming the inverter output is zero; G

_{id}can be derived by assuming the grid voltage is zero, as expressed in Equation (7).

_{α}and v

_{β}will be obtained. Then the q-axis component of the grid voltage v

_{q}can be calculated by performing a dq transformation on the grid voltage. Let vq become 0 through the closed loop control. When v

_{q}= 0, the output of the regulator ω

_{PLL}should be the angular velocity of the grid voltage ω

_{g}, and its integral value θ

_{PLL}should be the phase angle of the grid voltage θ

_{g}.

_{PLL}is equal to θ

_{g}and the dq components of the voltage ${\mathrm{v}}_{\mathrm{dq}}^{\mathrm{PLL}}$ calculated by the PLL is equal to the dq components of the grid voltage vdq. In the case of non-steady state behavior (by small-signal perturbation), due to the influence of the control loop in PLL, the phase angle obtained by the PLL θ

_{PLL}is no longer equal to the phase angle of the grid voltage θ

_{g}, so the calculated dq component of the grid voltage ${\mathrm{v}}_{\mathrm{dq}}^{\mathrm{PLL}}$ in the controller system does not represent the exact grid situation. We then set the difference between θ

_{PLL}and θg as δ

_{PLL}. The transfer function from the grid voltage disturbance to the voltage disturbance in the controller is expressed in Equations (9) and (10).

#### 3.2. Synchronous Rotating Frame Based Current Control Method

_{dq}are obtained by dq transformation. Then the dq components of the current are controlled separately. The reference value of the dq components of the current i*

_{dq}is obtained by a DC voltage regulator or/and calculated by the active/reactive power requirements. The output of the current regulator is converted into a reference of the inverter output voltage u*

_{dq}after decoupling and inverse dq transformation.

_{PLL}affects the results of the dq transformation and inverse dq transformation. With a small-signal perturbation, due to the PLL output difference δ

_{PLL}, there is an influence on the value associated with the dq transformation. In order to present this influence, a small signal model is shown in Figure 9.

_{iPL}L(s) is the transfer function matrix from the grid voltage disturbance to the inverter current disturbance in the controller, as expressed in Equation (19). G

_{uPLL}(s) is the transfer function matrix from the grid voltage disturbance to the inverter output reference voltage disturbance in the controller, as expressed in Equation (20). G

_{deco}(s) is the decoupling control matrix and G

_{C}(s) is the current regulator matrix, as expressed in Equations (21) and (22), separately [27].

_{0}and G

_{id}can be derived, as expressed in Equation (23).

_{inv}and the current loop control transfer function H

_{C}can be derived, as shown in Equations (30)–(32), respectively.

## 4. Simulation Verification

#### Verification

_{inv}(jω)} of the inverter with the dq-based control method is near 0 in a large frequency range (1000 Hz~5000 Hz), as shown in Figure 11.

_{r}= 1530 Hz is less than the critical resonant frequency (1700 Hz), the inverter-grid system with the PR-based control method is stable after 0.3 s. As shown in Figure 12f, when f

_{r}= 1730 is larger than the critical resonant frequency (1700 Hz), the system is unstable after 0.3 s.

## 5. Analysis of the Influence of the Control Parameter and Improvement Methods

#### 5.1. Analysis of the Influence of the Control Parameter

_{inv}(jω)}, the ordinate represents the PLL bandwidth and the color in the figure represents the amplitude of the Re{Y

_{inv}(jω)}, with warm colors representing larger amplitudes and cool colors representing smaller amplitudes. Dark blue represents the negative amplitude of the Re{Y

_{inv}(jω)}. The blue vertical dotted line in Figure 13a and red vertical dotted line in Figure 13b represent the 1700 Hz line.

_{inv}(jω)} of the dq-based control method. With the increasing PLL bandwidth, the negative area of the Re{Y

_{inv}(jω)} of the PR-based control method expands at a low frequency (Figure 13b). The positive area shrinks even though the amplitude increases. In general, the stability frequency area of the system becomes smaller.

_{inv}(jω)} of the PR-based control method to enter a stable area at 1730 Hz (Figure 13b). Thus, when the resonant frequency of the grid impedance is around 1700 Hz, the inverter-grid system can still be stable.

_{inv}(jω)}, the ordinate represents the PI bandwidth for the current control loop of the dq-based control method (Figure 14a), and the proportion coefficient KP-PR of the PR-based control method (Figure 14b). The red vertical dotted line in Figure 14a and the blue vertical dotted line in Figure 14b represents the 1700 Hz line.

_{inv(}jω)} expands at a low frequency, the positive area moves to a higher frequency area, and the amplitude peak of Re{Y

_{inv}(jω)} also decreases, as shown in Figure 14a. For the PR-based control method, by increasing the proportion coefficient K

_{P−PR}, the positive area (Figure 14b, in the red dotted box) shrinks. Thus, increasing the PI control bandwidth or proportion coefficient will result in a reduction in the system’s stable frequency area.

_{inv}(jω)} expands at a low frequency, the positive area moves to a higher frequency area, and the amplitude peak of Re{Y

_{inv}(jω)} also decreases, as shown in Figure 14a. For the PR-based control method, by increasing the proportion coefficient K

_{P-PR}, the positive area (Figure 14b, in the red dotted box) shrinks. Thus, increasing the PI control bandwidth or the proportion coefficient will result in a reduction in the system’s stable frequency area.

_{inv}(jω)}, as shown in Figure 14a (the red line). The increase of the proportion coefficient K

_{P−PR}for the current control loop of the PR-based control method will let Re{Y

_{inv}(jω)} < 0 if K

_{P−PR}> 34. In other words, when K

_{P−PR}< 34, if the resonant frequency of the grid impedance is near 1700 Hz, the inverter will be stable.

_{inv}(jω)}. However, these parameters are limited by the inverter’s hardware and controller performance, it is not convenient for them to be changed in practical operations.

#### 5.2. Improvement Methods

_{inv}(jω)} expand its positive area to increase the stability margin of the inverter. In the case that the resonant frequency of the grid impedance is unknown, the method of reducing the control bandwidth can maximize the robustness of the inverter.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Disclaimer

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**Figure 1.**The PV (photovoltaic)charging station with 6 PV modules (top), a grid-connected inverter (at left column), and 4 solar e-bikes that are being charged.

**Figure 3.**The measurement data for the Photovoltaic Charging Station: (

**a**) grid admittance amplitude spectrogram; (

**b**) partial enlargement of (

**a**); (

**c**) THD (total harmonic distortion) of inverter output current.

**Figure 5.**The block diagram of a typical PR (proportional resonance) based current control inverter.

**Figure 7.**The block diagram of a typical SRF (synchronous rotation frame) PLL. (

**a**) Block diagram of a typical single-phase SRF PLL; (

**b**) Small signal model of SRF PLL.

**Figure 10.**The Bode diagram of the inverter output impedance. (

**a**) Impedance amplitude (

**b**) Impedance phase angle; the solid line for theoretical calculation; * for simulation result; red for dq-based control and blue for PR-based control.

**Figure 11.**The real part of the inverter output admittance; blue for the dq-based control method and red for the PR-based control method.

**Figure 12.**The simulation waveform result of (

**a**) dq-based control method with a 1530 Hz resonant frequency; (

**b**) PR-based control method with a 1530 Hz resonant frequency; (

**c**) dq-based control method with a 1730 Hz resonant frequency; (

**d**) PR-based control method with a 1730 Hz resonant frequency.

**Figure 13.**The influence from the PLL bandwidth to Re{Yinv(jω)}, full band. (

**a**) dq-based control method; (

**b**) PR-based control method.

**Figure 14.**The influence from the current control parameters to Re{Yinv(jω)}, full band. (

**a**) dq-based control method; (

**b**) PR-based control method.

**Figure 15.**The simulation waveform result of (

**a**) the dq-based control method with a low pass filter; (

**b**) the PR-based control method with a low pass filter; (

**c**) the dq-based control method with a reduced control bandwidth; (

**d**) the PR-based control method with a reduced control bandwidth.

Parameter | DQ | PR |
---|---|---|

Rated power | 3.3 kW | |

Rated voltage | 230 V | |

Filter inductor | 3.5 mH | |

PLL bandwidth | 25 Hz | |

Sampling frequency | 10 kHz | |

KP-C | 2.46 | / |

KI-C | 548.81 | / |

KP-PR | / | 34.99 |

KR-PR | / | 400 |

ωc | / | π |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Z.; Gercek, C.; Renner, H.; Reinders, A.; Fickert, L.
Resonance Instability of Photovoltaic E-Bike Charging Stations: Control Parameters Analysis, Modeling and Experiment. *Appl. Sci.* **2019**, *9*, 252.
https://doi.org/10.3390/app9020252

**AMA Style**

Zhang Z, Gercek C, Renner H, Reinders A, Fickert L.
Resonance Instability of Photovoltaic E-Bike Charging Stations: Control Parameters Analysis, Modeling and Experiment. *Applied Sciences*. 2019; 9(2):252.
https://doi.org/10.3390/app9020252

**Chicago/Turabian Style**

Zhang, Ziqian, Cihan Gercek, Herwig Renner, Angèle Reinders, and Lothar Fickert.
2019. "Resonance Instability of Photovoltaic E-Bike Charging Stations: Control Parameters Analysis, Modeling and Experiment" *Applied Sciences* 9, no. 2: 252.
https://doi.org/10.3390/app9020252