# Optimal Design of a Flyback Microinverter Operating under Discontinuous-Boundary Conduction Mode (DBCM)

^{*}

## Abstract

**:**

## 1. Introduction

_{P}, which operates in high frequency; a step-up transformer with two secondary windings; two diodes D

_{1}, D

_{2}, which ensure the flyback operation; and two secondary switches S

_{1}, S

_{2}, which operate at the utility grid frequency. Based on the grid voltage polarity, the corresponding secondary winding transfers the energy from the PV module to the grid.

## 2. Power Losses in Hybrid DBCM

#### 2.1. Semiconductor Losses

_{P}, since a power MOSFET is used, the conduction losses are:

_{1}, S

_{2}are:

_{1}, D

_{2}are:

_{P}are calculated separately for the DCM and i-BCM segments during a grid half cycle, taking into consideration the operating period during each mode, and, as a result, the semiconductor switching losses are a function of the transition angle α as well:

_{SL}(α) is calculated in Appendix A.

#### 2.2. Transformer Losses

_{dc,z}and the ac ohmic losses P

_{ac,z}, employing the analytical equations derived in the previous section:

## 3. Design Optimization Process for DBCM

#### 3.1. Objective Function

#### 3.2. Design Constraints

- The core peak flux density (which for the case of the flyback microinverter operated in DBCM is for ωt = π/2 at maximum input power) must be lower than the ferrite material maximum flux density to avoid core saturation.
- The window utilization factor for the given core type, which is the ratio of the area of all the transformer windings to the transformer window, must be lower or equal to the maximum permitted value (for the windings to fit).

_{f}, shown in Figure 1, is negligible, then the peak voltage for each switch will be observed at the maximum grid voltage (ωt = π/2):

_{P}as well. To determine the maximum ΔV

_{Cf}

_{,}the charge ΔQ is first calculated for the switching cycle near ωt = π/2:

#### 3.3. Optimization Sequence

_{on,p,max}, the transformer windings current density J, and the transformer ferrite core maximum operational magnetic flux density B

_{p}. As noted in the section above, these independent variables are affected by the design constraints, e.g., a winding with a very low current density (i.e., many litz wire strands) will not fit in the given transformer utilization window. For each feasible combination of the independent variables, the expected weighted efficiency of this design is calculated using (18) with the weights of Table 1. This sequence is repeated for multiple combinations until a global maximum is found. As this is an off-line method (i.e., the optimization algorithm is applied during the design phase and not during the operation phase), the differential evolution stochastic method for constrained nonlinear global optimization is used through a software platform [27]. This optimization method is compute intensive, but provides a global maximum. For the design that achieves the maximum weighted efficiency, the rest of the parameters are calculated and presented to the designer in order to select the remaining component values [22]. Finally, the control parameters are also calculated to be input to the microcontroller program. An example of the application of this algorithm is shown in the following section, where the theoretical analysis (loss equations derived above) will be compared to experimental results for a given design.

## 4. Optimization Algorithm Verification

_{P}according to the hybrid DBCM formulas. The optimal operation values (i.e., DCM switching frequency, maximum pulse length) are calculated offline and entered as parameters, together with the converter component values. Figure 6 shows the inverter output current at maximum power.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

P_{CL,SW,pri} | Conduction losses of the primary switch (W). |

I_{pri,rms} | RMS current value of the primary transformer side (A). |

R_{ds,pri} | ON-resistance of the primary switch (Ω). |

i_{pri} | Instant current value of the primary transformer side (A). |

I_{pri,rms,DCM} | RMS current value of the primary transformer side during the DCM segment of DBCM (A). |

I_{pri,rms,BCM} | RMS current value of the primary transformer side during the i-BCM segment of DBCM (A). |

V_{dc} | Input voltage (V). |

L_{1} | Transformer primary winding inductance (H). |

d_{p} | Peak duty cycle when in DCM. |

f_{s} | DCM switching frequency (Hz). |

m | Ratio of the utility grid half cycle period to the DCM switching period. |

t_{on,p} | Peak ON-time of the primary side switch when in i-BCM (s). |

t_{on,p,max} | t_{on,p} at nominal power (s). |

k | Number of switching cycles during a utility grid half cycle for DBCM operation. |

k_{1} | Νumber of switching cycles until ωt = α. |

k_{2} | Number of switching cycles until ωt = π − α. |

λ | Ratio of the input voltage to the peak grid voltage. |

n | Transformer turns ratio. |

α | Transition angle between DCM and BCM (rad). |

P_{CL,SW,sec} | Conduction losses of the secondary switches (W). |

I_{sec,rms} | RMS current value of each of the secondary transformer side windings (A). |

R_{dc,sec} | ON-resistance of the secondary switches (Ω). |

I_{sec,rms,DCM} | RMS current value of each of the secondary transformer side windings during the DCM segment of DBCM (A). |

I_{sec,rms,BCM} | RMS current value of each of the secondary transformer side windings during the i-BCM segment of DBCM (A). |

P_{CL,d} | Conduction losses of the diodes (W). |

I_{sec,avg} | Average current value of each of the transformer secondary side windings current (A). |

V_{d} | Diode forward voltage (V). |

i_{sec} | Instant current value of the secondary transformer side (A). |

P_{PV} | Inverter input power (W). |

P_{PV,nom} | Maximum inverter input power (W). |

u_{ac} | Instant voltage value of the utility grid (V). |

V_{acrms} | RMS voltage value of the utility grid (V). |

P_{SL} | Semiconductor switching losses (W). |

P_{SL,DCM} | Semiconductor switching losses for the DCM segment of DBCM (W). |

P_{SL,BCM} | Semiconductor switching losses for the i-BCM segment of DBCM (W). |

t_{f} | Current fall time of the primary switch (s). |

P_{dc,z} | DC copper losses of the winding z (W). |

P_{ac,z} | AC copper losses of the winding z (W). |

I_{z,avg} | Average current value of the winding z (A). |

I_{z,rms} | RMS current value of the winding z (A). |

F_{r,z} | Resistance factor of the winding z. |

P_{Core} | Transformer core losses (W). |

ΔB_{i} | Flux swing during each switching cycle i (T). |

α_{iGSE},β_{iGSE}, k_{iGSE} | Parameters of the Steinmetz equation loss formula. |

B_{i} | Transformer core magnetic flux density at time t_{i} (Τ). |

η | Converter efficiency at a given power level. |

P_{i} | Converter input power at a given power level (W). |

P_{o} | Converter output power at a given power level (W). |

P_{Losses} | Converter losses at a given power level (W). |

P_{Si} | Converter semiconductor losses at a given power level (W). |

P_{xfmr} | Converter transformer losses at a givern power level (W). |

P_{CL} | Converter semiconductor conduction losses at a given power level (W) |

P_{SW} | Converter semiconductor switching losses at a given power level (W). |

V_{pri} | Voltage of the primary switch when in OFF state (V). |

V_{pri,p} | Maximum value of V_{pri} (V). |

V_{sec} | Voltage of each secondary switch when in OFF state (V). |

V_{sec,p} | Maximum value of V_{sec} (V). |

t_{off,p} | OFF-time of the primary switch at ωt = π/2 (s). |

V_{acp} | Utility grid voltage value at ωt = π/2 (V). |

I_{pri,p} | Peak current value of the primary transformer side at ωt = π/2 (A). |

I_{sec,p} | Peak current value of the secondary transformer side at ωt = π/2 (A). |

ΔV_{Cf} | Voltage fluctuation of filter capacitor (V). |

t_{q} | Time segment during which the filter capacitor is charged at ωt = π/2 (s). |

i_{grid} | Instant value of the converter current supplied to the grid (A). |

I_{grid,p} | Value of the converter current supplied to the grid at ωt = π/2 (A). |

i_{Cf} | Instant current value of the filter capacitor (A). |

f_{s,avg} | Average converter switching frequency (Hz). |

J | Current density of the transformer windings (A/mm^{2}). |

B_{p} | Maximum operational flux density (T). |

## Appendix A

_{0}[30]:

_{0}[30]:

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**Figure 6.**Grid voltage and converter output current (100 V/div, 0.5 A/div, 10 ms/div, V

_{dc}= 40 V).

**Figure 7.**Calculated and measured efficiency versus power ratio for 0 °C (P

_{PV,max}= 205 W, V

_{dc}= 40 V).

**Figure 8.**Calculated and measured efficiency versus power ratio for STC (P

_{PV,max}= 180 W, V

_{dc}= 36 V).

**Figure 9.**Calculated and measured efficiency versus power ratio for 60 °C (P

_{PV,max}= 140 W, V

_{dc}= 31 V).

Power Level | 5% | 10% | 20% | 30% | 50% | 75% | 100% |
---|---|---|---|---|---|---|---|

EU | 0.03 | 0.06 | 0.13 | 0.1 | 0.48 | 0 | 0.2 |

CEC | 0 | 0.04 | 0.05 | 0.12 | 0.21 | 0.53 | 0.05 |

Specifications | Design Constraints | Optimization Independent Variables | Design Values |
---|---|---|---|

STC: P_{PV,max} = 180 W/V_{dc} = 36 V | Peak Flux Density: 280 mT | n = 0.129 | S_{P}: IXFX180N15 |

0 °C: P_{PV,max} = 205 W/V_{dc} = 40 V | Maximum. Transformer Fill Factor: 35% | f = 29 kHz | S_{1}, S_{2}: IXFX26N120 |

60 °C: P_{PV,max} = 140 W/V_{dc} = 31 V | Maximum MOSFET breakdown voltage: 1200 V | t_{onp,max} = 37.2 us | D_{1}, D_{2}: STTH1512G |

Grid: 230 V/50 Hz | J = 4.1 A/mm^{2} | Turns = 19:147 | |

C_{f} = 440 nF | B_{p} = 280 mT | L_{1} = 41.2 μH | |

Core Type: ETD54 | Leakage inductance ratio: 2.2% | ||

Material: 3F3 | Core gap = 3.41 mm |

Maximum Power | Input Voltage | Measured Efficiency | Calculated Efficiency | Calculated Efficiency of [18] (i-BCM) |
---|---|---|---|---|

205 W | 40 V | 92.15% | 92.27% | 91.59% |

180 W | 36 V | 92.24% | 92.43% | 91.38% |

140 W | 31 V | 92.35% | 92.35% | 91.51% |

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**MDPI and ACS Style**

Christidis, G.; Nanakos, A.; Tatakis, E.
Optimal Design of a Flyback Microinverter Operating under Discontinuous-Boundary Conduction Mode (DBCM). *Energies* **2021**, *14*, 7480.
https://doi.org/10.3390/en14227480

**AMA Style**

Christidis G, Nanakos A, Tatakis E.
Optimal Design of a Flyback Microinverter Operating under Discontinuous-Boundary Conduction Mode (DBCM). *Energies*. 2021; 14(22):7480.
https://doi.org/10.3390/en14227480

**Chicago/Turabian Style**

Christidis, Georgios, Anastasios Nanakos, and Emmanuel Tatakis.
2021. "Optimal Design of a Flyback Microinverter Operating under Discontinuous-Boundary Conduction Mode (DBCM)" *Energies* 14, no. 22: 7480.
https://doi.org/10.3390/en14227480