# Grid-Connected Control Strategy of Five-level Inverter Based on Passive E-L Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topology and Mathematical Mode

#### 2.1. Topology

_{a}, R

_{b}, and R

_{c}, and the equivalent inductors are L

_{a}, L

_{b}, and L

_{c}. Further, R

_{a}= R

_{b}= R

_{c}= R, and L

_{a}= L

_{b}= L

_{c}= L

_{f}. C

_{1}= C

_{2}= C

_{3}= C

_{4}= C is the polar film capacitor between the DC stack buses. The voltage of the DC side is U

_{dc}. I

_{dc}denotes the current from the DC-link to the inverter. T

_{A1}–T

_{A8}, T

_{B1}–T

_{B8}, and T

_{C1}–T

_{C8}are the eight switches on the arms of phases A, B, and C respectively. V

_{A1}–V

_{A6}, V

_{B1}–V

_{B6}, and V

_{C1}–V

_{C6}are the six clamp diodes on the arms of phases A, B, and C respectively. S

_{A}, S

_{B}, and S

_{C}are the switch signals of each arm on the phases A, B, and C respectively. Among them, the switch signals of the arm on phase A are S

_{A1}, S

_{A2}, S

_{A3}, S

_{A4}, S

_{A5}, S

_{A6}, S

_{A7}, and S

_{A8}. Likewise, the switch signals of phase B and phase C can be obtained. U

_{eA}, U

_{eB}, and U

_{eC}denote the grid side AC voltages.

#### 2.2. Working Principle

- (1)
- T
_{A1}, T_{A2}, T_{A3}, and T_{A4}shut down at the same time; T_{A5}, T_{A6}, T_{A7}, and T_{A8}switch on at the same time; and then the output phase voltage of the inverter is +U_{dc}/2 and S_{A}= +2. - (2)
- T
_{A2}, T_{A3}, T_{A4}, and T_{A5}shut down at the same time; T_{A1}, T_{A6}, T_{A7}, and T_{A8}switch on at the same time; and then the output phase voltage of the inverter is +U_{dc}/4 and S_{A}= +1. - (3)
- T
_{A3}, T_{A4}, T_{A5}, and T_{A6}shut down at the same time, and T_{A1}, T_{A2}, T_{A7}, and T_{A8}switch on at the same time. O point and N point have the same potential at the same moment, so the output phase voltage of the inverter is 0. Meanwhile, S_{A}= 0. - (4)
- T
_{A4}, T_{A5}, T_{A6}, and T_{A7}shutdown at the same time; T_{A1}, T_{A2}, T_{A3}, and T_{A8}switch on at the same time; and the output phase voltage of the inverter is −U_{dc}/4 and S_{A}= −1. - (5)
- T
_{A5}, T_{A6}, T_{A7}, and T_{A8}shut down at the same time; T_{A1}, T_{A2}, T_{A3}, and T_{A4}switch on at the same time; and the output phase voltage of the inverter is −U_{dc}/2 and S_{A}= −2.

#### 2.3. Mathematical Model

_{k}can be decomposed into eight two-state switching functions, as in Equation (1).

_{C}

_{1}and U

_{C}

_{4}are the voltages of the capacitors C

_{1}and C

_{4}in the DC side, respectively, and i

_{A}, i

_{B}, and i

_{C}are the three-phase current of the inverter.

_{d}and i

_{q}are the three-phase current in the rotation d-q frame; U

_{ed}and U

_{eq}are the grid-connected voltage in the rotation d-q frame; S

_{d1}, S

_{q1}and S

_{d2}, S

_{q2}are the coordinate components of S

_{A1}and S

_{A2}in the rotation d-q frame; S

_{d7}, S

_{q7}and S

_{d8}, S

_{q8}are the coordinate components of S

_{A7}and S

_{A8}in the rotation d-q frame; and ω is the angular speed of rotation.

## 3. SPWM Modulation Strategy

_{a}is the amplitude of the sinusoidal modulation wave, and V

_{1}to V

_{4}are the amplitudes of the same phase triangular wave. Among them, V

_{1}controls T

_{A1}and T

_{A5}, V

_{2}controls T

_{A2}and T

_{A6}, V

_{3}controls T

_{A3}and T

_{A7}, and V

_{4}controls T

_{A4}and T

_{A8}. When the amplitude of the sine wave V

_{a}is greater than the amplitude of the triangular carrier wave V

_{1}to V

_{4}, the switching of each phase is shut down. On the contrary, when V

_{a}is smaller than V

_{1}to V

_{4}, the switching is switched on.

## 4. Design of Passive Controller

#### 4.1. Passive E-L Model of the Five-Level Inverter

**x**is the state variable,

**u**is the control variable that reflects the exchange between the system and the external energy,

**M**is a positive diagonal matrix consisting of energy storage elements,

**J**is the anti-symmetric matrix that reflects the internal interconnection structure of the system,

**J**is equal to

**−J**, and

^{T}**R**is the symmetric matrix that reflects the dissipation characteristics of the system.

#### 4.2. Design of Passive E-L Model Controller

**x**,

**u**, and

**y**are the state variables, input variables, and output variables of the system, respectively; the spaces are

**x**∈ R

^{n},

**u**∈ R

^{m}, and

**y**∈ R

^{m}, respectively; and f is the local Lipschitz about (

**x**,

**u**).

**x**) and a positive definite function Q(

**x**) for $\forall $t > 0, the dissipation inequality is satisfied:

**u**, the output

**y**and the energy supply rate

**u**

^{T}

**y**are established.

**y**=

**x**,

**Q**(

**x**) =

**x**

^{T}

**Rx**, it can be concluded that the system satisfies the strict passive inequality.

**x**

_{e}=

**x**−

**x***, it can be deduced from Equation (7):

**x*** is the desired equilibrium point in the system. It can be expressed as:

_{dref}and i

_{qref}are the expected components in the d and q axis of the three-phase current i

_{A}, i

_{B}, and i

_{C}. U

_{C}

_{1ref}and U

_{C}

_{4ref}are the expected voltages of the capacitors C

_{1}and C

_{4}in the DC side, respectively.

**x**

_{e}quickly becoming zero. The injected damping dissipation term is:

**R**

_{d}

**x**

_{e}= (

**R**+

**R**

_{a})

**x**

_{e}

**R**

_{a}is a semi-definite diagonal matrix that is similar to the form of matrix

**R**and

**R**

_{a}= [R

_{a1}R

_{a2}R

_{a3}R

_{a4}]. Equation (12) can be rewritten as:

_{d}and u

_{q}, the input signal from the SPWM algorithm can be obtained from Equation (4):

_{qref}is equal to 0 in order to ensure that the unit power factor is connected to the grid. The d-axis desired current i

_{dref}is obtained by the difference between the actual capacitor voltage and the expected capacitor voltage through PI control.

## 5. Software Simulation

#### 5.1. Simulation of Passivity Control Mentioned in This Paper

_{dc}/2, +U

_{dc}/4, 0, −U

_{dc}/4, and −U

_{dc}/2. Similarly, Figure 4b shows that the line voltage consists of +600 V, +450 V, +300 V, +150 V, 0 V, −150 V, −300 V, −450 V, and −600 V, which is respectively close to the nine levels of the inverter output in theory, which are +U

_{dc}, +U

_{dc}3/4, +U

_{dc}/2, +U

_{dc}/4, 0, −U

_{dc}/4, −U

_{dc}/2, −U

_{dc}3/4, and −U

_{dc}.

_{d}and u

_{q}from the passive controller, when the injection damping R

_{a1}= R

_{a2}= R

_{a}= 5 Ω, 15 Ω, 25 Ω, 50 Ω, and 100 Ω, are shown in Figure 4c,d, respectively. In Figure 4c, when R

_{a}≤ 25 Ω, u

_{d}tends to be stable at about 0.005 s, and, when R

_{a}> 25 Ω, the stability time of u

_{d}is significantly greater than 0.01 s. Among them, when R

_{a}= 50 Ω, u

_{d}tends to be stable at about 0.02 s, and, when R

_{a}= 100 Ω, the stability time is about 0.04 s.

_{a}≤ 25 Ω, u

_{q}tends to be stable about 0.01 s, and the stability time of the output function u

_{q}is longer than 0.01 s, when R

_{a}> 25 Ω. Especially, when R

_{a}= 50 Ω, u

_{q}tends to be stable at about 0.02 s, and, when R

_{a}= 100 Ω, u

_{q}does not reach stability at 0.04 s.

_{a1}= R

_{a2}= R

_{a}= 25 Ω.

#### 5.2. The Passivity-Based Control of This Paper Is Compared with the Traditional PI Control

_{p}is relatively small and the value of K

_{i}is relatively large, which ensures the accuracy of voltage tracking. The function of the PI controller of the current loop is to track the three-phase grid current and voltage. If K

_{p}is too large, the stability of the system is poor. In summary, we select K

_{p}= 0.8 and K

_{i}= 5.

_{d}is getting smaller and smaller. Especially, when R

_{a}≥ 25 Ω, the stability time is close to 0.003 s. Meanwhile, the fluctuation range of i

_{d}is getting smaller, and the dynamic stability is getting higher.

_{d}is under different resistances in the traditional PI control system. It can be seen from Figure 5d that, when R

_{a}≤ 25 Ω, i

_{d}tends to be stable at 0.01 s. When R

_{a}= 50 Ω, i

_{d}tends to be stable at about 0.02 s, and, when R

_{a}= 100 Ω, i

_{d}does not reach stability at 0.04 s.

_{d}is significantly longer in Figure 5d. Meanwhile, the magnitude of the current amplitude fluctuates considerably in Figure 5c. In summary, the dynamic stability of passivity-based control is better than that of traditional PI control.

## 6. Prototype Experiment

_{a}≤ 25 Ω, the magnitude of the current amplitude fluctuates considerably. However, the ring of the fluctuation is significantly smaller under the PBC strategy, which proves that a passivity-based control strategy has good dynamic stability.

_{C}

_{1}, U

_{C}

_{2}, U

_{C}

_{3}, and U

_{C}

_{4}are the voltages of the capacitors C

_{1}, C

_{2}, C

_{3}, and C

_{4}in the DC side, respectively. The capacitor voltage in the figure is close to 150 V, which proves that the problem of the balance of the neutral point is relatively small.

_{dc}/2, +U

_{dc}/4, 0, −U

_{dc}/4, and −U

_{dc}/2. Similarly, Figure 9b shows that the output line voltage consists of +600 V, +450 V, +300 V, +150 V, 0 V, −150 V, −300 V, −450 V, and −600 V. Meanwhile, the experimental results are approximately consistent with the results obtained in the software simulation.

## 7. Conclusions

- (1)
- The optimum stability of the system can be achieved by selecting the appropriate energy function and injection damping.
- (2)
- Compared with the traditional PI control method, the passivity-based control strategy has better static and dynamic stability and makes the system achieve a higher power factor.
- (3)
- In terms of economy, the harmonic loss very small under passivity-based control.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Simulation of passivity-based control. (

**a**) Phase voltage of the inverter; (

**b**) Line voltage of the inverter; (

**c**) The output function u

_{d}; and (

**d**) The output function u

_{q}.

**Figure 5.**Comparison of static and dynamic stability under two strategies. (

**a**) The output active power of the inverter under a passivity-based control (PBC) strategy; (

**b**) The output active power of the inverter under the traditional PI control strategy; (

**c**) The d-axis current under a PBC strategy; and (

**d**) The d-axis current under the traditional PI control strategy.

**Figure 6.**Comparison of phase and harmonics in two strategies; (

**a**) A-phase grid voltage and current under a passivity-based control strategy; (

**b**) A-phase grid voltage and current under the traditional PI control strategy; (

**c**) A-phase current harmonics of the inverter under a passivity-based control strategy; and (

**d**) A-phase current harmonics of the inverter under the traditional PI control strategy.

**Figure 8.**Comparison of hardware experiments under two strategies; (

**a**) A-phase grid-connected voltage and current under a PBC strategy; (

**b**) A-phase grid-connected voltage and current under the traditional PI control strategy; (

**c**) The d-axis current of different injection dampings under the two control strategies; and (

**d**) The polar film capacitor voltage under a PBC strategy.

**Figure 9.**Phase voltage and line voltage of the inverter; (

**a**) Phase voltage of the inverter; and (

**b**) Line voltage of the inverter.

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

U_{dc} | 600 V | L_{f} | 1 mH |

C_{1} (C_{2} C_{3} C_{4}) | 220 μF | R | 1 Ω |

C | 50 μF | U_{eA} U_{eB} U_{eC} | 311 V |

L | 500 mH | frequency (f) | 50 HZ |

Injection Damping | Stability Time of u_{d} | Stability Time of u_{q} |
---|---|---|

R_{a} = 5 Ω | 0.006 s | 0.005 s |

R_{a} = 15 Ω | 0.005 s | 0.005 s |

R_{a} = 25 Ω | 0.005 s | 0.006 s |

R_{a} = 50 Ω | 0.02 s | 0.02 s |

R_{a} = 100 Ω | 0.04 s | 0.045 s |

Parameter | PBC | Traditional PI | ||
---|---|---|---|---|

Stability Time | Amplitude of Fluctation | Stability Time | Amplitude of Fluctation | |

p | 0.01 s | smaller | 0.01 s | larger |

R_{a} = 5 Ω | 0.02 s | smaller | 0.01 s | larger |

R_{a} = 15 Ω | 0.003 s | smaller | 0.01 s | larger |

R_{a} = 25 Ω | 0.003 s | smaller | 0.01 s | larger |

R_{a} = 50 Ω | 0.004 s | smaller | 0.02 s | smaller |

R_{a} = 100 Ω | 0.004 s | smaller | 0.04 s | smaller |

THD (%) | 0.26 | 3.39 |

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## Share and Cite

**MDPI and ACS Style**

Li, T.; Cheng, Q.; Sun, W.; Chen, L.
Grid-Connected Control Strategy of Five-level Inverter Based on Passive E-L Model. *Energies* **2017**, *10*, 1657.
https://doi.org/10.3390/en10101657

**AMA Style**

Li T, Cheng Q, Sun W, Chen L.
Grid-Connected Control Strategy of Five-level Inverter Based on Passive E-L Model. *Energies*. 2017; 10(10):1657.
https://doi.org/10.3390/en10101657

**Chicago/Turabian Style**

Li, Tao, Qiming Cheng, Weisha Sun, and Lu Chen.
2017. "Grid-Connected Control Strategy of Five-level Inverter Based on Passive E-L Model" *Energies* 10, no. 10: 1657.
https://doi.org/10.3390/en10101657