# An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Discrete LQR-ORT Controller Design

#### 3.1. Computation of Suboptimal LQR Controller

#### 3.2. Optimal Reference Tracking Matrix

#### 3.3. Main Grid Power Contribution Calculation Using Superposition Principle

## 4. Simulation and Experimental Results

#### 4.1. Robustness and Stability Analysis

#### 4.2. HIL Experimental Robustness Assessment

#### 4.3. Experimental Results

_{u}= {171.42, 200, 600} Ω and L = 0.4571 H. Finally, for the unbalanced load, a three-phase noncontrolled rectifier with an RLC parallel circuit with R

_{NL}= 1200 Ω, C

_{NL}= 100 μF, and L

_{NL}= 2 mH was used.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**LQR-ORT discrete-time closed-loop eigenvalues of the nominal system and the system under variations using the same controller ${\mathrm{K}}_{\mathrm{d}}$.

**Figure 7.**HIL experiment comparison between LQR-ORT and the PR-Droop controller for the nominal case and scenarios 41 and 42. (

**a**) PR-Droop controller with nominal scenario. (

**b**) PR-droop controller with scenario 41. (

**c**) PR-Droop controller with scenario 42. (

**d**) LQR-ORT controller with nominal scenario. (

**e**) LQR-ORT controller with scenario 41. (

**f**) LQR-ORT controller with scenario 42.

**Figure 8.**Photo of the experimental setup [37].

**Figure 9.**Experimental results comparing the LQR-ORT controller performance with the PR-Droop controller using the same reference signals. (

**a**) LQR-ORT response. (

**b**) PR-Droop response (

**c**) LQ-Cost values.

**Figure 10.**Experimental scheme for the evaluation of the performance of the LQR-ORT controller under unbalanced and nonlinear loads.

**Figure 11.**Experimental results evaluating the performance of the LQR-ORT controller under unbalanced and nonlinear loads. The inverter starts connected to a balanced linear load. At t = 0.7 s, event (1) occurs by connecting the unbalanced load. At t = 2.5 s, event (2) occurs by connecting the nonlinear load. (

**a**) Active power. (

**b**) Reactive power. (

**c**) Output current showing the transition between balanced and unbalanced load (LEFT), and the transition between unbalanced and nonlinear load (RIGHT). (

**d**) Load current showing the transition between balanced and unbalanced load (LEFT), and the transition between unbalanced and nonlinear load (RIGHT).

**Table 1.**Parameter Specification for the Linear Quadratic Regulator with Optimal Reference Tracking (LQR-ORT) Controller.

Parameter | Symbol | Value |
---|---|---|

Grid Voltage | $V$ | 120 $\mathrm{Vrms}$ |

DC bus Voltage | ${V}_{dc}$ | 350 $\mathrm{V}$ |

Grid Frequency | $f$ | 60 Hz |

Output Inductance | ${L}_{o}$ | 1.8 mH |

Input Inductance | ${L}_{i}$ | 1.8 mH |

Filter Capacitance | $C$ | 8.8 $\mathsf{\mu}\mathrm{F}$ |

Switching Frequency | ${f}_{s}$ | 10 $\mathrm{kHz}$ |

Sampling Period | ${T}_{s}$ | 100 $\mathsf{\mu}\mathrm{s}$ |

Error Weighting Matrix | ${Q}_{p}$ | $5000\times {I}_{2\times 2}$ |

Input Weighting Matrix | ${R}_{p}$ | $0.2\times {I}_{2\times 2}$ |

Inner Integrator Gain | ${K}_{i}$ | 1 |

Outer Integrator Gain | ${K}_{s}$ | 5 |

SOGI gain | ${K}_{SOGI}$ | 0.7 |

PLL Proportional Gain | ${K}_{pPLL}$ | 0.28307 |

PLL Integral Gain | ${K}_{iPLL}$ | 7.5102 |

ID | C $(\mathsf{\mu}\mathbf{F})$ | Li $\left(\mathbf{m}\mathbf{H}\right)$ | Lo $(\mathbf{m}\mathbf{H})$ | Max Deviation | LQR-ORT Stable | PR-Droop Stable |
---|---|---|---|---|---|---|

Nom | 8.8 | 1.8 | 1.8 | 0% | Yes | Yes |

1 | 12.6 | 2.87 | 2.57 | 42.8% | Yes | Yes |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

27 | 10.85 | 2.92 | 1.06 | −41.0% | Yes | Yes |

28 | 5.97 | 1.03 | 2.79 | −42.7% | Yes | No |

29 | 5.47 | 0.95 | 2.58 | −47.3% | No | No |

30 | 14.03 | 1.73 | 0.92 | −48.9% | Yes | Yes |

31 | 4.32 | 1.91 | 2.16 | −50.9% | Yes | Yes |

32 | 4.25 | 1.79 | 2.90 | −51.7% | Yes | Yes |

33 | 11.09 | 2.39 | 0.86 | −52.4% | Yes | No |

34 | 13.96 | 0.85 | 1.97 | −52.7% | Yes | No |

35 | 5.93 | 0.85 | 1.96 | −53.0% | No | No |

36 | 4.40 | 2.53 | 0.85 | −53.0% | Yes | Yes |

37 | 14.33 | 2.39 | 0.82 | −54.2% | No | No |

38 | 3.94 | 1.96 | 1.27 | −55.3% | Yes | No |

39 | 11.12 | 1.89 | 0.78 | −56.6% | Yes | Yes |

40 | 3.82 | 1.96 | 1.33 | −56.7% | Yes | No |

41 | 9.82 | 1.44 | 0.76 | −58.0% | Yes | Yes |

42 | 12.21 | 0.74 | 1.80 | −58.6% | Yes | No |

43 | 4.32 | 1.33 | 0.73 | −59.7% | No | No |

44 | 10.75 | 0.72 | 2.30 | −59.8% | Yes | No |

45 | 3.52 | 1.62 | 1.42 | −60.0% | Yes | No |

46 | 3.51 | 1.63 | 1.40 | −60.1% | No | No |

47 | 3.48 | 1.91 | 1.00 | −60.4% | Yes | No |

48 | 11.97 | 0.67 | 2.59 | −62.6% | Yes | No |

49 | 3.50 | 0.67 | 1.40 | −62.7% | No | No |

50 | 3.25 | 1.36 | 0.86 | −63.1% | No | No |

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**MDPI and ACS Style**

Patarroyo-Montenegro, J.F.; Vasquez-Plaza, J.D.; Andrade, F.; Fan, L.
An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame. *Appl. Sci.* **2020**, *10*, 6730.
https://doi.org/10.3390/app10196730

**AMA Style**

Patarroyo-Montenegro JF, Vasquez-Plaza JD, Andrade F, Fan L.
An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame. *Applied Sciences*. 2020; 10(19):6730.
https://doi.org/10.3390/app10196730

**Chicago/Turabian Style**

Patarroyo-Montenegro, Juan F., Jesus D. Vasquez-Plaza, Fabio Andrade, and Lingling Fan.
2020. "An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame" *Applied Sciences* 10, no. 19: 6730.
https://doi.org/10.3390/app10196730