# Direct Power Control of a Single Stage Current Source Inverter Grid-Tied PV System

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## Abstract

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## 1. Introduction

_{Fmin}= 0.9 (inductive or capacitive) [10]. Beyond these international standards, additional grid codes apply in some countries, demanding higher flexibility [11] from grid-connected PV inverters. These requirements may range from mandatory galvanic isolation to low voltage ride through capabilities [3,12]. However, in recent years, efforts were made so that distributed renewable systems not only comply with international standards, but also actively contribute to minimize power quality issues in the connection to the grid, dynamically stabilizing the grid voltage and frequency, as in recent grid codes in Germany for PV power plants [4]. This can be achieved by means of active and reactive power control [13].

## 2. System Description

_{dc}(with parasitic resistance r

_{dc}). Then, the CSI is connected to the grid through a second order LC filter [27,40] designed and sized to guarantee compliance with the IEEE 1547 standard, ensuring a displacement power factor higher than 0.9 in the connection to the grid, and a total harmonic distortion of the injected AC grid currents that is lower than 5% [27,41].

#### 2.1. PV Array Model

_{ph}is the photocurrent, dependent on the irradiation and temperature of the PV cell, and i

_{D}is the diode current.

_{ph}) of the PV cell changes with the irradiance level and cell temperature (T) according to Equation (2),

_{sc}is the short circuit current of the PV cell, G is irradiation level in kW/m

^{2}, G

_{ref}is the reference irradiation, 1 kW/m

^{2}, and μ

_{T}is the temperature coefficient of i

_{sc}.

_{D}represented in Figure 2 can be obtained from Equation (3),

_{o}is the diode saturation current, q is the electric charge (1.6022 × 10

^{−19}C), k is the Boltzmann’s constant (1.3806 × 10

^{−23}J/K), T is the cell temperature (K), and m is the diode quality factor (m = 1 for an ideal diode and m > 1 for a real diode).

_{o}) changes with the cell temperature (T) and is expressed as in Equation (4):

_{PV}(Equation (5)) can then be expressed as a function of the cell’s voltage v

_{PV}, considering the diode current i

_{D}(Equation (3)), and an association of n

_{p}paralleled PV modules, and n

_{s}series-connected PV modules.

_{PV}:

#### 2.2. Current Source Inverter Model

_{mn}is defined as:

**S**:

**S**, the converter voltages (v

_{1}, v

_{2}) on the DC side and the grid side currents (i

_{a}, i

_{b}, i

_{c}) can be obtained, respectively, from Equation (11):

_{sa}, v

_{sb}, v

_{sc}) are the converter grid side phase voltages and i

_{PV}is the DC current. This equation will be further used to predict the future DC voltage of the converter (v

_{12}) and the grid side currents.

#### 2.3. Equations of PV Power Dynamics

_{PV}) dynamics (Equation (12)) is obtained by applying Kirchhoff’s laws to the circuit shown in Figure 1:

_{dc}and r

_{dc}represent the DC filtering inductance and its parasitic resistance, respectively. Using Equation (12) in Equation (8), the dynamics of PV power is obtained:

#### 2.4. Equations of the Dynamics of Active and Reactive Power in the Connection to the Grid

_{la}, i

_{lb}, i

_{lc}) and the capacitor voltages (v

_{sab}, v

_{sbc}, v

_{sca}).

_{la}, i

_{lb}, i

_{lc}) can be expressed as a function of the capacitors’ voltages (v

_{sab}, v

_{sbc}, v

_{sca}) and the grid voltages (v

_{ga}, v

_{gb}, v

_{gc}), and are obtained from Equation (14), where L

_{f}represents the filter inductances [26,43].

_{f}represents the filter capacitance.

_{ga}, i

_{gb}, i

_{gc}) are then obtained from Equation (16), and depend on the inductor currents (i

_{la}, i

_{lb}, i

_{lc}), on the filter inductance L

_{f}, and on the damping resistance R

_{f}.

**X**represents the system variables in abc coordinates. To obtain a decoupled system in αβ coordinates, the Concordia transformation (Equation (17)) is applied to Equations (14), and the variables

_{abc}**X**are transformed to αβ coordinates by computing

_{abc}**X**=

_{αβ}**C**

^{T}

**X**.

_{abc}**D**(Equation (18)) is used, where $\mathsf{\theta}=\mathsf{\omega}t$ is the phase angle of the grid voltages (Equation (19)). Then, variables

**X**can be further transformed to

_{αβ}**X**, by calculating

_{dq}**X**=

_{dq}**D**

^{T}

**X**.

_{αβ}_{gd}, v

_{gq}are expressed by Equation (20):

_{ld}, i

_{lq}dynamics (Equation (21)), the grid side currents are obtained in dq coordinates (Equation (22)), where ω is the grid angular frequency:

_{d}and i

_{q}. Consequently, to design the DPPC, it is necessary to calculate the first derivative of active and reactive power, considering that in dq coordinates the grid voltages v

_{gd}, v

_{gq}are constant, and their derivatives are zero:

_{gd}, v

_{gq}, the capacitor voltages v

_{sd}, v

_{sq}, and the CSI currents, i

_{d}, i

_{q}:

_{gq}= 0 (Equation (20)), the previous equation can be further simplified to:

## 3. Design of the Direct Power Predictive Controller

_{s}is much lower than the period of the electric variables. Then the equations for the Euler backward (Equation (28)) integration method can be obtained.

#### 3.1. Prediction of the PV Power

_{PV}(k + 1) can be obtained by discretizing Equation (12) at T

_{s}, where ${v}_{12}\left(k+1\right)$ represents the future DC converter voltage and can be computed from Equation (11).

_{s}, with the same yield as Equation (29).

#### 3.2. Prediction of Active and Reactive Power in the Connection to the Grid

_{sd}(k + 1), v

_{sq}, (k + 1), which can be calculated by discretizing Equation (21):

_{d}(k + 1) and i

_{q}(k + 1), as expected. All the other variables, as the capacitor voltages (v

_{sd}, v

_{sq}), are for the present sample t

_{s}= k and can be measured.

#### 3.3. Cost Function and Selection of Swtiching Vector

_{1}, w

_{2}are the weights for the active and reactive power, respectively.

## 4. Results

#### 4.1. Response to a Step in Power

_{PV}= 400 W to P

_{PV}= 280 W at Q = 0 var. The main values used in the simulations and in the experiments are presented in Table 4. The system is tested at 110 V phase grid voltage and around 70 V at the DC side (PV panel emulator).

_{gan}and the corresponding grid current i

_{ga}, the power measured in the DC side P

_{dc}, and the reactive power Q in the connection to the grid. When the PV power is reduced from P

_{PV}= 400 W to P

_{PV}= 280 W, the AC current reduces from i

_{ga}= 1.4 A to i

_{ga}= 1.07 A. At the PV power change, the DC current i

_{dc}reduces from i

_{dc}= 6 A to i

_{dc}= 4 A, as shown in Figure 7c,d.

_{PV}= 540 W) is presented in Figure 8, calculated using a fast Fourier tool analysis (FFT) in PowerGUI block. It can be seen that the measured value, obtained with a Fluke 1735 (compliant with IEC 61000-4-7 Class II), is 4.1%, thus less than the 5% set as the maximum by international standards.

_{pv}= 90 V and 110 V grid phase voltage, at maximum PV power (P

_{PV}= 540 W). From Figure 9a,b, it can be seen that the current and voltage are out of phase, thus guaranteeing a nearly unitary power factor. The power supplied to the grid is slightly lower than 540 W and the reactive power is Q = 0 var.

#### 4.2. Active Power and Reactive Power Control Using DPPC

#### 4.2.1. Step in the Reactive Power Q (Leading and Lagging)

_{gan}and the current i

_{ga}, in the same phase, for reactive power changing from Q = −40 var to Q = 40 var at P = 350 W operation. Figure 10b shows the three-phase grid currents i

_{gabc}, (i

_{gmax}= 2.6 A) and the current in the DC link, i

_{dc}.

#### 4.2.2. PV Power Profile Control

_{ga}and the corresponding grid phase current i

_{ga}for a step change in the reactive power Q, from Q = 0 var to Q = 100 var (Figure 11b), and then from Q = 100 var back to Q = 0 var (Figure 11c). The results obtained show that the grid current i

_{ga}is leading the grid voltage v

_{ga}at maximum Q and out of phase at Q = 0.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the single stage current source inverter (CSI) based three-phase grid-connected photovoltaic (PV) system.

**Figure 3.**P–I characteristic of PV array at different irradiance levels and at constant temperature 25 °C.

**Figure 6.**Experimental setup of the proposed system: (

**a**) auto transformer; (

**b**) grid side filter; (

**c**) matrix converter; (

**d**) field-programmable gate array (FPGA); (

**e**) DSPACE hardware; (

**f**) DSPACE control desk; (

**g**) controlled DC power supply; and (

**h**) DC inductance coil.

**Figure 7.**Simulation results (

**a**,

**c**) and experimental results (

**b**,

**d**) for a step change in PV power; (

**a**,

**b**) grid voltage v

_{gan}

_{,}(yellow), grid current i

_{ga}

_{,}(blue), power P

_{dc}in the DC side (pink), reactive power Q in the connection to the grid (green), for a time scale (10 ms/div); (

**c**,

**d**) grid current i

_{ga}

_{,}(yellow), current in the DC link i

_{dc}(blue), for a time scale (50 ms/div).

**Figure 9.**Simulation results (

**a**,

**c**) and experimental results (

**b**,

**d**) for a constant PV power P

_{dc}= 540 W; (

**a**,

**b**) grid voltage v

_{ga}(yellow), grid current i

_{ga}, power P

_{dc}in the DC side (pink), reactive power Q in the connection to the grid (green); (

**c**,

**d**) grid currents i

_{ga}, i

_{gb}, i

_{gc}.

**Figure 10.**Experimental results for a step in the reactive power. (

**a**) Grid voltage v

_{gan}(yellow), grid current i

_{ga}(blue), power P

_{dc}in the DC side (pink); reactive power Q in the connection to the grid (green); (

**b**) three-phase grid currents i

_{gabc}, and current i

_{dc}(green).

**Figure 11.**Experimental results for a daily PV power profile. (

**a**) Power P

_{dc}produced by the PV (blue) and reactive power Q (pink); (

**b**) reactive power step from Q = 0 var to Q = 100 var; (

**c**) reactive power step from Q = 100 var to Q = 0 var; (

**b**,

**c**) grid voltage v

_{gan}

_{,}(yellow), grid current i

_{ga}

_{,}(green), reactive power Q (pink).

Item | Value |
---|---|

Maximum Power, P_{max} | 305 W |

Open circuit voltage, v_{oc} | 64.2 V |

Short circuit current, i_{sc} | 5.96 A |

Voltage at P_{max}, v_{mpp} | 54.7 V |

Current at P_{max}, i_{mpp} | 5.58 A |

Parallel strings, n_{p} | 5 |

Series-connected modules per string, n_{s} | 1 |

**Table 2.**Switching states of CSI and corresponding PV side voltage and grid side currents of each state.

Vector | S_{1a} | S_{1b} | S_{1c} | S_{2a} | S_{2b} | S_{2c} | v_{12} | i_{sa} | i_{sb} | i_{sc} |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 0 | 0 | 1 | 0 | v_{ab} | i_{PV} | −i_{PV} | 0 |

2 | 0 | 1 | 0 | 1 | 0 | 0 | −v_{ab} | −i_{PV} | i_{PV} | 0 |

3 | 0 | 1 | 0 | 0 | 0 | 1 | v_{bc} | 0 | i_{PV} | −i_{PV} |

4 | 0 | 0 | 1 | 0 | 1 | 0 | −v_{bc} | 0 | −i_{PV} | i_{PV} |

5 | 0 | 0 | 1 | 1 | 0 | 0 | v_{ca} | −i_{PV} | 0 | i_{PV} |

6 | 1 | 0 | 0 | 0 | 0 | 1 | −v_{ca} | i_{PV} | 0 | −i_{PV} |

7 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

9 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

Item | Value |
---|---|

Grid frequency (Hz) | 50 |

Grid phase voltage, rms value (V) | 110 |

Grid filter inductance (mH) | 4 |

Grid side filter damping resistance (Ω) | 33 |

Grid filter capacitance delta (μF) | 6 |

DC link inductance (mH) | 12.5 |

Sampling time (μs) | 20 |

PV Power (W) | 600 |

Grid Phase Voltage (V) | DC Voltage (V) | P_{dc, ref} (W) | Q_{ref} (var) |
---|---|---|---|

110 | 70 | Step: 400 to 280 | 0 |

Grid Phase Voltage (V) | DC Voltage (V) | P_{dc, ref} (W) | Q_{ref} (var) |
---|---|---|---|

110 | 90 | 0 to 500 | Step: 0 to 100 |

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## Share and Cite

**MDPI and ACS Style**

Youssef, E.; Costa, P.B.C.; Pinto, S.F.; Amin, A.; El Samahy, A.A.
Direct Power Control of a Single Stage Current Source Inverter Grid-Tied PV System. *Energies* **2020**, *13*, 3165.
https://doi.org/10.3390/en13123165

**AMA Style**

Youssef E, Costa PBC, Pinto SF, Amin A, El Samahy AA.
Direct Power Control of a Single Stage Current Source Inverter Grid-Tied PV System. *Energies*. 2020; 13(12):3165.
https://doi.org/10.3390/en13123165

**Chicago/Turabian Style**

Youssef, Erhab, Pedro B. C. Costa, Sonia F. Pinto, Amr Amin, and Adel A. El Samahy.
2020. "Direct Power Control of a Single Stage Current Source Inverter Grid-Tied PV System" *Energies* 13, no. 12: 3165.
https://doi.org/10.3390/en13123165