# Modeling and Design of the Vector Control for a Three-Phase Single-Stage Grid-Connected PV System with LVRT Capability according to the Spanish Grid Code

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Control of the PV System

#### 2.1. Control of the Inverter Currents

_{id}and PI

_{iq}) in order to control its dq components. The equations include the cross-coupling terms—ωLi

_{d}and ωLi

_{q}and a decouple strategy is recommended for a better transient behavior. Thus, the outputs of PI

_{id}and PI

_{iq}plus the compensating cross terms are the dq components of the reference command inverter voltages (u

_{d}and u

_{q}). Then, u

_{d}and u

_{q}are fed to the PWM module after the application of the inverse Park’s transformation [18]. Finally, the output of the PWM module is the status of the power switches of the power semiconductor devices (e.g., Insulated Gate Bipolar Transistors (IGBTs)) of the three-phase inverter.

#### 2.2. Control of the DC Bus Link Voltage

_{VDC}regulator is fed by the error between V

^{*}

_{DC}and V

_{DC}, and its output becomes the reference of the active power (Ref_P) that is proportional to the maximum available power at the input. The d component of the reference inverter current i

^{*}

_{d}is proportional to Ref_P, but inversely proportional to the positive-sequence of the grid voltages u

_{ACd}+.

^{*}

_{q}is proportional to Ref_Q, but also inversely proportional to the positive-sequence of the grid voltages u

_{ACd}+.

#### 2.3. Synchronization Algorithm

_{ACq}+ − are fed to two PI regulators in order to reduce their values to zero producing the positive- and negative-sequences of the angular speeds ω+ − of the grid voltage vector u

_{AC}. It is worth noting that initial values of the angular speeds ω

_{offset}+ − are added to reduce the time to track the phase angles, improving the PLL stability. Finally, two integrators compute the phase angle θ+ −, although only the positive-sequence θ+ is feedback to the vector control of the inverter currents, as it is shown in Figure 1. On the other hand, the Fault Signal is activated when the normalized depth of the voltage sag (V

_{gf}) is less than 0.85 (see Equation (6)).

_{0}.

#### 2.4. Control of P and Q

_{ACdq}and i

_{dq}are the dq components of the grid voltages and the inverter currents, respectively.

#### 2.5. Limitation of the Output Currents of the Inverter during Voltage Sags

_{gf}by measuring the grid voltages, extracting its positive-sequence and normalizing with the amplitude of the line-to-line grid voltages (see Equation (6)) [9].

_{gf}bellow 0.85 in order to consider that a voltage sag has occurred. The value of S

_{max}will be calculated using the positive- and negative-sequences of the grid voltages and the nominal apparent power S

_{nom}, normalizing again with the amplitude of the line-to-line utility grid voltages (see Equation (7)) [9].

_{gf}is the normalized depth of the voltage sag.

_{nom}is around 507 kVA (see Table 2).

_{gf}≥ 0.85, no fault is detected and a normal operation is achieved with no reactive power injected into the grid and allowing the application of the Maximum Power Point Tracking (MPPT) algorithm. In this case, the PV generator delivers the maximum power to the utility grid for a given irradiance and temperature, and the output of the MPPT module is the DC voltage reference command V

^{*}

_{DC}MPP for the outer voltage control loop which corresponds to the maximum power P* (Ref_P = P*).

_{gf}< 0.85, the Fault Signal flag is activated and there are five possibilities:

- If V
_{gf}< 0.2 for more than 0.15 s or 0.2 ≤ V_{gf}< 0.5 for more than 0.58 s or 0.5 ≤ V_{gf}< 0.85 for more than 0.27 s, the inverter must be disconnected from the utility grid. - If the fault has a duration less than the previous situation, a certain amount of reactive (Q) according to Equation (8) and active (P) powers are delivered to the utility grid according to the value of V
_{gf}and S_{max}. - For depth sags, Q might be greater than S
_{max}and a limitation to the delivered reactive and active powers are imposed (Ref_Q = S_{max}, P_{max}= 0). - If P
_{max}> P*, the MPPT algorithm is activated, the maximum power is delivered by the PV generator to the grid (Ref_P = P*) and the DC link voltage reference command is V_{DC}* MPP. - If P
_{max}< P*, the non-MPPT algorithm is activated instead, and the power delivered to the grid is P_{max}(Ref_P = P_{max}) with the V_{DC}* non-MPP as the DC link voltage reference command$$\{\begin{array}{c}Q=0,{V}_{gf}\ge 0.85\\ Q=\frac{15}{7}{S}_{nom}\left(0.85-{V}_{gf}\right),\text{}0.5\le {V}_{gf}0.85\\ Q=\frac{3}{4}{S}_{nom},{V}_{gf}0.5\end{array}$$

_{DC}* MPP, P*) to (V

_{DC}* non-MPP, P

_{max}).

## 3. Simulation Studies

^{2}and temperature T = 25 °C) are shown in Table 1 and will be used in this paper.

_{reg}= 40.957 μs), the same as the switching frequency of the inverter in order to attain the synchronization of the sample process [35]. The magenta color block computes the MPPT and the non-MPPT algorithms shown in Figure 5 for a specific irradiance and temperature, setting the proper DC bus voltage reference (V

_{DC}*).

_{set}), the damping factor (ζ) and the amplitude of the grid voltages [36,37].

^{2}and the grid currents indicates no overcurrent during the short-time duration fault events in all cases. When the voltage sag ends, the no faulty operation mode of the PV system is established (P = 500 kW and 250 kW, respectively, and Q = 0).

#### 3.1. Three-Phase Voltage Sag

^{2}, respectively. The DC link voltage increases from 810 to 995 and 954 V, respectively, whereas the DC current delivered by the PV generator is around 50 A for both situations during the fault due to mainly the copper and the inverter losses. The delivered P is zero and Q is 50 kVAr for both irradiances.

#### 3.2. Voltage Sag in Phase 3

^{2}, respectively. The DC link voltage increases from 810 to 976 and 995 V, respectively, whereas the DC current delivered by the PV generator are 140 A and zero, respectively, during the fault. The mean values of the delivered P are 100 kW and zero, respectively; the mean values of the delivered Q are 150 kVAr for both irradiances.

^{2}, respectively. The DC link voltage increases from 810 to 937 and 878 V, respectively, whereas the DC current delivered by the PV generator reaches 330 and 230 A, respectively, during the fault. The mean values of the delivered P are 300 and 175 kW, respectively, whereas the mean values of the delivered Q are zero for both irradiances.

## 4. Experiments

_{s}) for the DRTS is set to 5.1196 μs, whereas the sample time of the controller (T

_{reg}) is set to 40.957 μs; the latter has a delay of $\frac{3}{2}{T}_{reg}=61.44\mathsf{\mu}\mathrm{s}$ in the worst case.

^{2}. The time evolution of the DC link voltage and the DC current delivered by the PV generator, the grid voltages and currents at the grid side, and P and Q powers delivered to the low-voltage utility grid are shown in (a), meanwhile the same variables are depicted in (b) during the voltage sags, where the values of the nominal period (T = 20 ms) of the voltages and currents, and P and Q powers can be observed for the normal and faulty operation modes.

_{DC}*MPP DC bus reference voltage is applied to the control algorithm to do the power balance, reaching P a value of 250 kW whereas a zero Q is attained for unity power factor operation. However, the DC link voltage increases during the voltage sag because the limitation in the amplitude of the line currents to its nominal rated value decreases the P delivered to the utility grid and a V

_{DC}*non-MPP DC bus reference voltage is set in this situation. At the same time, some amount of Q is needed according to Equation (8) so as to be able to improve the voltage profile of grid voltages during the sags.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Voltage profile for Low-Voltage-Ride-Through (LVRT) according to IEC 61400-21; (

**b**) Reactive power (Q) to be injected into the grid during voltage sags in Spain ($\Delta V\left(pu\right)=1-Vgf$).

**Figure 8.**(

**a**) Simulink model of the Solar Panel; (

**b**) Lookup Table (two-dimensional; 2-D) model of the PV generator.

**Figure 9.**Simulink model of the power subsystem: (

**a**) three-phase inverter + L Filter; (

**b**) faulty grid; (

**c**) three-phase Utility grid.

**Figure 10.**Simulink model of the control subsystem. (

**a**) Positive and negative sequence detector (PNSD) + Fault detection + PQ References Generator + dq control of the inner current loops (PI regulator) + outer DC bus control (PI regulator); (

**b**) Detailed Simulink model of the PNS Fault detection and the PQ References Generator.

**Figure 11.**Time evolution of the variables during a three-phase deep voltage sag of 90% of amplitude at an irradiance G = 1000 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 12.**Time evolution of the variables during a three-phase deep voltage sag of 90% of amplitude at an irradiance G = 500 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 13.**Time evolution of the variables during a three-phase deep voltage sag of 70% of amplitude at an irradiance G = 1000 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 14.**Time evolution of the variables during a three-phase deep voltage sag of 70% of amplitude at an irradiance G = 500 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 15.**Time evolution of the variables during a deep voltage sag of 90% of amplitude in phase 3 at an irradiance G = 1000 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 16.**Time evolution of the variables during a deep voltage sag of 90% of amplitude in phase 3 at an irradiance G = 500 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 17.**Time evolution of the variables during a moderate voltage sag of 50% of amplitude in phase 3 at an irradiance G = 1000 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive power.

**Figure 18.**Time evolution of the variables during a moderate voltage sag of 50% of amplitude in phase 3 at an irradiance G = 500 W/m

^{2}. (

**a**) Normalized amplitude of the voltage sag, DC bus voltage, inverter voltage and DC current at the ouput of the PV generator; (

**b**) Three-phase grid voltages, three-phase grid currents, active power and reactive powers.

**Figure 20.**(

**a**) PLECS model of the power subsystem; (

**b**) Simulation Parameters (fixed-step discrete solver).

**Figure 22.**Detailed Simulink model of the control subsystem; (

**a**) PNSD + Fault detection + PQ References Generator + dq control of the inner current loops (PI regulator) + outer DC bus control (PI regulator); (

**b**) Detailed Simulink model of the duty cycle generator for the TMS320F28379D.

**Figure 23.**Time evolution of the variables during a three-phase deep voltage sag of 90% of amplitude at an irradiance G = 500 W/m

^{2}. (

**a**) DC bus voltage, DC current at the ouput of the PV generator, three-phase grid voltages, three-phase grid currents, active power and reactive power; (

**b**) the same variables during the voltage sag.

**Figure 24.**Time evolution of the variables during a three-phase deep voltage sag of 70% of amplitude at an irradiance G = 500 W/m

^{2}, (

**a**) DC bus voltage, DC current at the ouput of the PV generator, three-phase grid voltages, three-phase grid currents, active power and reactive power; (

**b**) the same variables during the voltage sag.

Parameters | Value |
---|---|

Maximum power | P_{mp} = 320.024 W |

Maximum power voltage | V_{mp} = 36.7 V |

Maximum power current | I_{mp} = 8.72 A |

Open circuit voltage | V_{oc} = 45.6 V |

Short circuit current | I_{sc} = 9.07 A |

Cell number per module | N_{cell} = 72 |

Temperature coefficient of I_{sc} | α_{i} = 0.0867%/°C |

Temperature coefficient of V_{oc} | α_{v} = −0.4278%/°C |

Parallel resistance | R_{p} = 4.24074 Ω |

Series resistance | R_{s} = 0.3437 Ω |

Ideally factor of the diode | m = 1.1238 |

Parameters | Value |
---|---|

Number of the strings | N_{pvst} = 72 |

Number of the series modules | N_{pvs} = 22 |

Maximum current | I_{mpv} = N_{pvst} × I_{mp} = 627.84 A |

Maximum voltage | V_{mpv} = N_{pvs} × V_{mp} = 807.4 V |

DC output power | P_{dc} = V_{mpv} × I_{mpv} = 506.91 kW |

Open circuit voltage | V_{PV-oc} = N_{pvs} × V_{oc} = 1003.2 V |

Short circuit current | I_{PV-sc} = N_{pvst} × I_{sc} = 653.04 A |

Maximum DC link voltage (V_{DC}) | 807.4 V |

Maximum DC current (I_{DC}) | 627.84 A |

Link Capacitor (C_{link}) | 65,000 μF |

Converter Gain (K_{PWM}) | $\frac{2}{3}{\mathrm{V}}_{\mathrm{DC}}^{}$ |

Switching frequency (F_{PWM}) | 24.416 kHz |

Line Inductance (L) | 0.15 mH |

AC system (v_{RST}) | - Voltage amplitude: 230 V(rms) phase-to-neutral |

- Nominal frequency: 50 Hz |

Crossover frequency of the open current loop (f_{cI}) | 610.4 Hz |

Phase Margin of the open current loop (PM_{I}) | 63.5° |

Proportional constant of the current regulators in dq axes (k_{p,Idq}) | 0.0011 |

Integral constant of the current regulators in dq axes (k_{i,Idq}) | 0.942 |

Crossover frequency of the open voltage loop (f_{cV}) | 12.208 Hz |

Phase Margin of the open voltage loop (PM_{V}) | 63.5° |

Proportional constant of the PI voltage regulator (k_{p,VDC}) | 3977.5 |

Integral constant of the PI voltage regulator (k_{i,VDC}) | 152,110 |

Settling time of the PLL (T_{set}) | 20 ms |

Damping factor of the PLL (ζ) | $\frac{\sqrt{2}}{2}$ |

Proportional constant of the PI PLL regulator (k_{p,PLL}) | $\sqrt{2}$ |

Integral constant of the PI PLL regulator (k_{i,PLL}) | 325.2691 |

Sample time of the power subsystem (T_{S}) | 5.1196 μs |

Sample time of the control subsystem (T_{reg}) | 40.957 μs |

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## Share and Cite

**MDPI and ACS Style**

Rey-Boué, A.B.; Guerrero-Rodríguez, N.F.; Stöckl, J.; Strasser, T.I.
Modeling and Design of the Vector Control for a Three-Phase Single-Stage Grid-Connected PV System with LVRT Capability according to the Spanish Grid Code. *Energies* **2019**, *12*, 2899.
https://doi.org/10.3390/en12152899

**AMA Style**

Rey-Boué AB, Guerrero-Rodríguez NF, Stöckl J, Strasser TI.
Modeling and Design of the Vector Control for a Three-Phase Single-Stage Grid-Connected PV System with LVRT Capability according to the Spanish Grid Code. *Energies*. 2019; 12(15):2899.
https://doi.org/10.3390/en12152899

**Chicago/Turabian Style**

Rey-Boué, Alexis B., N. F. Guerrero-Rodríguez, Johannes Stöckl, and Thomas I. Strasser.
2019. "Modeling and Design of the Vector Control for a Three-Phase Single-Stage Grid-Connected PV System with LVRT Capability according to the Spanish Grid Code" *Energies* 12, no. 15: 2899.
https://doi.org/10.3390/en12152899