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

Abstract

The flyback converter has been widely used in Photovoltaic microinverters, operating either in Discontinuous, Boundary, or Continuous Conduction Mode (DCM, BCM, CCM). The recently proposed hybrid DBCM operation inherits the merits of both DCM and BCM. In this work, the necessary analytical equations describing the converter operation for any given condition under DBCM are derived, and are needed due to the hybrid nature of the modulation strategy during each sinusoidal wave. Based on this analysis, a design optimization sequence used to maximize the weighted efficiency of the inverter under DBCM is then applied. The design procedure is based on a power loss analysis for each converter component and focuses on the appropriate selection of the converter parameters. To achieve this, accurate, fully parameterized loss models of the converter components are implemented. The power loss analysis is then validated by applying the optimization methodology to build an experimental prototype operating in DBCM.

1. Introduction

Distributed power generation in residential areas using solar panels has seen great progress during the last decade, to the point where the module-level microinverters are now a strong alternative solution to central inverters, due to their improved energy harvest, ease of installation, modularity, and flexibility [1,2,3,4]. Various single or multi-stage candidate topologies have been proposed [4,5,6,7,8,9,10,11], among which the flyback current source inverter stands out because of its many application-specific advantages (high voltage step-up, transformer isolation, high power density, and low cost).

The flyback inverter topology is shown in Figure 1. The circuit consists of a primary switch S_P, which operates in high frequency; a step-up transformer with two secondary windings; two diodes D₁, D₂, which ensure the flyback operation; and two secondary switches S₁, S₂, 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.

Depending on the transformer current waveform, three different operating modes exist: the Discontinuous Conduction Mode (DCM) [12,13,14,15,16], the Boundary and improved Boundary Conduction Mode (BCM and i-BCM) [12,17,18], and the Continuous Conduction Mode (CCM) [19,20,21]. CCM’s main disadvantage is the complexity of its control algorithm in order to accommodate the right half plane zero that exists [21]. DCM offers the simplest control at the expense of a reduced power density, as was experimentally proven in [22,23], whereas BCM, which has a variable switching frequency, due to increased switching losses, suffers at lower power levels (both average due to low irradiance, as well as instantaneous near the edges of the sinusoidal waveform). A hybrid discontinuous/boundary conduction mode (DBCM) modulation, which aims to combine the individual advantages of DCM and i-BCM, has been proposed and analyzed [22]. At lower power (irradiance) levels, the inverter operates in DCM, reducing the switching losses, whereas in higher power levels, it operates in i-BCM around the center of the sinusoidal wave (α < ωt < π − α) and in DCM near the edges (0 < ωt < α and π − α < ωt < π), minimizing the peak and rms currents (and therefore, the conduction losses) and maximizing the inverter power density. The transition angle α should be judiciously calculated in order to achieve a smooth transition between these two modes. The switching sequence of each semiconductor for DBCM is shown in Figure 2. However, even though this operating mode shows improved power efficiency compared to the DCM/i-BCM conduction modes, there has never been an analytical procedure to determine the converter parameters and to achieve maximum converter performance for the full operating range.

In this paper, the derivation of the average and rms current equations of all the converter key components for operation in DBCM is analytically described. Based on those equations, a design methodology for the selection of the inverter component parameters is presented, focusing on maximizing the weighted efficiency under DBCM operation. To achieve this, the power losses for each converter component are analyzed. After that, the derived formulas, as well as the inverter specifications and constraints, are integrated into an optimization algorithm, specifically created for DBCM, which determines the component design values. The application of this methodology is afterwards evaluated on a laboratory experimental prototype. Finally, the converter efficiency is compared to a flyback inverter with equivalent characteristics, designed for operation in i-BCM.

2. Power Losses in Hybrid DBCM

The converter power losses can be separated into two major categories: semiconductor losses and transformer losses. For the derivation of each component loss formulas, the rms or average values of the corresponding currents must be first analytically calculated. Equivalent mathematical formulations have been described for DCM and i-BCM in [14,18]. However, as will be shown in this section, due to the two different operating modes that exist within a sinusoidal wave half cycle, as well as because of the variable switching frequency imposed by the i-BCM operation during a segment of the sinusoidal wave, complex mathematical manipulations need to be carried out in order to derive precise equations for the DBCM operation during the full sinusoidal period.

2.1. Semiconductor Losses

Semiconductor losses can be further divided into conduction losses and switching losses. Firstly, for the main switch S_P, since a power MOSFET is used, the conduction losses are:

$$P_{CL,SW,pri}=I_{pri,rms}^2\cdot R_{ds,pri}.$$

(1)

To calculate this value, the analytical expressions computed when the converter operates either fully in DCM or i-BCM are employed for the corresponding time intervals:

$$\begin{array}{ll}{I}_{pri,rms}^2& =\frac{1}{\mathsf{\pi }}{\displaystyle {\int }_0^{\mathsf{\pi }}{i}_{pri}^2(t)d\omega t}=\frac{1}{\mathsf{\pi }}\left({\displaystyle {\int }_0^{\alpha }{i}_{pri}^2(t)d\omega t}+{\displaystyle {\int }_{\alpha }^{\mathsf{\pi }-\alpha }{i}_{pri}^2(t)d\omega t}+{\displaystyle {\int }_{\mathsf{\pi }-\alpha }^{\mathsf{\pi }}{i}_{pri}^2(t)d\omega t}\right)\\ & =\frac{1}{\mathsf{\pi }}\left(2\cdot {\displaystyle {\int }_0^{\alpha }{i}_{pri}^2(t)d\omega t}+{\displaystyle {\int }_{\alpha }^{\mathsf{\pi }-\alpha }{i}_{pri}^2(t)d\omega t}\right)=I_{pri,rms,\mathrm{DCM}}^2+I_{pri,rms,\mathrm{BCM}}^2\end{array}.$$

(2)

Next, a similar mathematical sequence as presented in [14] is employed, but instead of calculating the value for the complete sinusoidal cycle, it is now calculated only for the DCM segments of DBCM. Thus, it is deduced that:

$$\begin{array}{ll}{I}_{pri,rms,\mathrm{DCM}}^2& =\frac{V_{dc}^2\cdot d_p^3}{3L_1^2\cdot f_s^2}\cdot \frac{2}{m}{\displaystyle \sum _{i=1}^{\frac{\alpha }{\mathsf{\pi }}m}{\sin }^3\left(\frac{\mathsf{\pi }}{m}i\right)}\\ & =\frac{V_{dc}^2\cdot d_p^3}{3L_1^2\cdot f_s^2}\cdot \frac{1}{4m}[-\csc \left(\frac{3\mathsf{\pi }}{2m}\right)\cdot \sin \left(\frac{-3\mathsf{\pi }+m\mathsf{\pi }}{2m}\right)+3\csc \left(\frac{\mathsf{\pi }}{2m}\right)\cdot \sin \left(\frac{m\mathsf{\pi }-\mathsf{\pi }}{2m}\right)\\ & +3\csc \left(\frac{\mathsf{\pi }}{2m}\right)\cdot \sin \left(\frac{2\alpha m+\mathsf{\pi }-m\mathsf{\pi }}{2m}\right)-\csc \left(\frac{3\mathsf{\pi }}{2m}\right)\cdot \sin \left(\frac{6\alpha m+3\mathsf{\pi }-m\mathsf{\pi }}{2m}\right)]\end{array}.$$

(3)

Accordingly, for the i-BCM portion of the hybrid modulation, equivalent mathematical principles as shown in [18] are employed:

$$\begin{array}{c}{I}_{pri,rms,\mathrm{BCM}}^2=\frac{1}{3}{\left(\frac{V_{dc}}{L_1}\right)}^2{\left(\frac{t_{on,p}}{1+\frac{\lambda }{n}}\right)}^2\frac{{\displaystyle {\sum }_{i=k_1}^{k_2}\left[\sin \omega t_{i-1}\cdot {\left(\sin \omega t_{i-1}+\frac{\lambda }{n}\right)}^3\right]}}{{\displaystyle {\sum }_{i=k_1}^{k_2}{\left(\frac{\lambda }{n}+\sin \omega t_{i-1}\right)}^2}}\\ =\frac{1}{3}{\left(\frac{V_{dc}}{L_1}\right)}^2{\left(\frac{t_{on,p}}{1+\frac{\lambda }{n}}\right)}^2\sum I_{pri,rms}(\alpha )\end{array},$$

(4)

where ΣIpri,rms(α) is the generalized form of the trigonometric ratio calculated in [18], in the case that i-BCM occurs during a segment of the utility grid half cycle, meaning α < ωt < π − α, and its calculation is found in Appendix A.

Similarly, the conduction losses of the secondary windings’ switches S₁, S₂ are:

$$P_{CL,SW,\sec }=2\cdot I_{\sec ,rms}^2\cdot R_{ds,\sec }.$$

(5)

To calculate the rms value of each of the transformer secondary side winding currents, an equivalent sequence as shown in Equations (2)–(4) is applied:

$$\begin{array}{c}{I}_{sec,rms}^2=\frac{1}{2\mathsf{\pi }}{\displaystyle {\int }_0^{2\mathsf{\pi }}{i}_{\sec }^2(t)d\omega t}=\frac{1}{2\mathsf{\pi }}\left({\displaystyle {\int }_0^{\alpha }{i}_{\sec }^2(t)d\omega t}+{\displaystyle {\int }_{\alpha }^{\mathsf{\pi }-\alpha }{i}_{\sec }^2(t)d\omega t}+{\displaystyle {\int }_{\mathsf{\pi }-\alpha }^{\mathsf{\pi }}{i}_{\sec }^2(t)d\omega t}\right)\\ =\frac{1}{2\mathsf{\pi }}\left(2\cdot {\displaystyle {\int }_0^{\alpha }{i}_{\sec }^2(t)d\omega t}+{\displaystyle {\int }_{\alpha }^{\mathsf{\pi }-\alpha }{i}_{\sec }^2(t)d\omega t}\right)=I_{\sec ,rms,\mathrm{DCM}}^2+I_{\sec ,rms,\mathrm{BCM}}^2\end{array}$$

(6)

with:

$$\begin{array}{ll}{I}_{\sec ,rms,\mathrm{DCM}}^2& =\frac{n\cdot V_{dc}^2\cdot d_p^3}{6\cdot V_{acp}\cdot L_1^2\cdot f_s^2}\cdot \frac{2}{m}{\displaystyle \sum _{i=1}^{\frac{\alpha }{\mathsf{\pi }}m}{\sin }^2\left(\frac{\mathsf{\pi }}{m}i\right)}\\ & =\frac{n\cdot \lambda \cdot V_{dc}\cdot d_p^3}{3\cdot L_1^2\cdot f_s^2}\cdot \frac{1}{4\mathsf{\pi }}\left[2\alpha m+\mathsf{\pi }+\mathsf{\pi }\csc \left(\frac{\mathsf{\pi }}{m}\right)\cdot \sin \left(\frac{-\mathsf{\pi }-2\alpha m}{m}\right)\right]\end{array}$$

(7)

and

$$\begin{array}{l}{I}_{sec,rms,\mathrm{BCM}}^2=\frac{1}{6}\frac{\lambda }{n}{\left(\frac{n{V}_{dc}}{L_1}\right)}^2{\left(\frac{t_{on,p}}{1+\frac{\lambda }{n}}\right)}^2\frac{{\displaystyle {\sum }_{i=k_1}^{k_2}\left[{\sin }^2\omega t_{i-1}{\left(\sin \omega t_{i-1}+\frac{\lambda }{n}\right)}^3\right]}}{{\displaystyle {\sum }_{i=k_1}^{k_2}{\left(\frac{\lambda }{n}+\sin \omega t_{i-1}\right)}^2}}\\ I_{sec,rms,\mathrm{BCM}}^2=\frac{1}{6}\frac{\lambda }{n}{\left(\frac{n{V}_{dc}}{L_1}\right)}^2{\left(\frac{t_{on,p}}{1+\frac{\lambda }{n}}\right)}^2\sum I_{\sec ,rms}(\alpha )\end{array},$$

(8)

where ΣIsec,rms (α) is the generalized form of the trigonometric ratio calculated in [18], in the case that i-BCM occurs during a segment of the utility grid half cycle, meaning α < ωt < π − α, and its calculation is found in the Appendix A.

The conduction losses of the diodes D₁, D₂ are:

$$P_{CL,d}=2\cdot I_{\sec ,avg}\cdot V_d.$$

(9)

Assuming that the output current is purely sinusoidal and that the inverter supplies power under unitary power factor, then:

$$I_{\sec ,avg}=\frac{1}{2\mathsf{\pi }}{\displaystyle {\int }_0^{\mathsf{\pi }}{i}_{\sec }(t)}d\omega t=\frac{1}{2\mathsf{\pi }}{\displaystyle {\int }_0^{\mathsf{\pi }}{I}_{grid,p}\sin (\omega t)}d\omega t=\frac{\sqrt{2}}{\mathsf{\pi }}\frac{P_{PV}}{V_{acrms}}.$$

(10)

Concerning the semiconductor switching losses, operation in DCM or i-BCM guarantees zero turn-ON losses for all the MOSFETs. The remaining turn-OFF losses of S_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:

$$P_{SL}=P_{SL,\mathrm{DCM}}+P_{SL,\mathrm{BCM}},$$

(11)

with:

$$\begin{array}{cc}{P}_{SL,\mathrm{DCM}}& =\frac{t_f\cdot V_{dc}\cdot d_p}{2L_1}\left[2\cdot V_{dc}\cdot \frac{2}{m}{\displaystyle \sum _{i=1}^{\frac{\alpha }{\mathsf{\pi }}m}\sin \left(\frac{\mathsf{\pi }}{m}i\right)}+2\cdot \frac{n\cdot V_{dc}}{\lambda }\cdot \frac{2}{m}{\displaystyle \sum _{i=1}^{\frac{\alpha }{\mathsf{\pi }}m}{\sin }^2\left(\frac{\mathsf{\pi }}{m}i\right)}\right]\\ & =\frac{t_f\cdot V_{dc}\cdot d_p}{2L_1}\{V_{dc}\cdot \frac{2}{m}\left[\csc \left(\frac{\mathsf{\pi }}{2m}\right)\sin \left(\frac{m\mathsf{\pi }-\mathsf{\pi }}{2m}\right)+\csc \left(\frac{\mathsf{\pi }}{2m}\right)\sin \left(\frac{\mathsf{\pi }-m\mathsf{\pi }+2\alpha m}{2m}\right)\right]\\ & +\frac{n\cdot V_{dc}}{\lambda }\cdot \frac{1}{\mathsf{\pi }m}\left[\mathsf{\pi }+2\alpha m+\mathsf{\pi }\csc \left(\frac{\mathsf{\pi }}{m}\right)\cdot sin\left(\frac{-\mathsf{\pi }-2\alpha m}{m}\right)\right]\}\end{array}$$

(12)

and

$$\begin{array}{ll}{P}_{SL,\mathrm{BCM}}& =\frac{t_f\cdot V_{dc}^2}{2L_1}\left[\frac{\lambda }{n}\frac{{\displaystyle {\sum }_{i=k_1}^{k_2}\sin \omega t_{i-1}}}{{\displaystyle {\sum }_{i=k_1}^{k_2}{\left(\frac{\lambda }{n}+\sin \omega t_{i-1}\right)}^2}}+2\frac{{\displaystyle {\sum }_{i=k_1}^{k_2}{\sin }^2\omega t_{i-1}}}{{\displaystyle {\sum }_{i=k_1}^{k_2}{\left(\frac{\lambda }{n}+\sin \omega t_{i-1}\right)}^2}}+\frac{n}{\lambda }\frac{{\displaystyle {\sum }_{i=k_1}^{k_2}{\sin }^3\omega t_{i-1}}}{{\displaystyle {\sum }_{i=k_1}^{k_2}{\left(\frac{\lambda }{n}+\sin \omega t_{i-1}\right)}^2}}\right]\\ & =\frac{t_f\cdot V_{dc}^2}{2L_1}\sum P_{SL}\left(\alpha \right)\end{array}$$

(13)

where ΣP_SL (α) is calculated in Appendix A.

2.2. Transformer Losses

The transformer losses can be separated into windings losses, core losses, and losses due to the energy dissipation stored on the leakage inductance (on the primary switch snubber).

Beginning with the windings, copper losses occur due the dc resistance of the windings, as well as due to the proximity and skin effect because of the high frequency current. To calculate the windings losses for the flyback inverter operating in DBCM, the semiempirical method presented in [24] is used, which calculates separately for each winding z the dc ohmic losses P_dc,z and the ac ohmic losses P_ac,z, employing the analytical equations derived in the previous section:

$$P_{dc,z}=I_{z,avg}^2\cdot R_{dc,z},$$

(14)

$$P_{ac,z}=\left(I_{z,rms}^2-I_{z,avg}^2\right)\cdot R_{dc,z}\cdot F_{r,z}.$$

(15)

To estimate the transformer core losses, the improved Generalized Steinmetz equation (iGSE) [25] is used, which is an empirical formula derived in order to apply the Steinmetz equation [26] for non-sinusoidal current waveforms, as is usually the case for power electronics converters. The transformer core losses for the flyback inverter operating in DBCM are calculated by summing the core loss during each switching cycle:

$$P_{Core}=\frac{{\mathrm{k}}_{\mathrm{f}}}{2\mathsf{\pi }}\cdot {\displaystyle \sum _{i=0}^k{k}_f{\left({\mathsf{\Delta }\mathsf{{\rm B}}}_i\right)}^{{\beta }_{iGSE}{-\alpha }_{iGSE}}{\left|\frac{d{B}_i}{dt}\right|}^{{\alpha }_{iGSE}}\left(t_{i+1}-t_i\right)},$$

(16)

where:

$${\mathrm{k}}_{\mathrm{f}}=\frac{{\mathrm{k}}_{iGSE}}{2^{{\beta }_{iGSE}+1}{\mathsf{\pi }}^{{\alpha }_{iGSE}-1}\left(0.2761+\frac{1.7061}{{\alpha }_{iGSE}+1.354}\right)}.$$

(17)

Due to the complexity of the equations, which cannot be analytically solved because of the hybrid operation, approximated equations using curve fitting are derived using mathematical software [27] as a function of the transition angle α.

Finally, due to the leakage inductance of the transformer, an energy amount is not transferred from the primary to the secondary winding, but is dissipated in the form of heat on the primary switch snubber. The percentage of these power losses is equal to the percentage of the leakage inductance to the primary winding inductance.

The power losses described above occur under DBCM operation. As shown in [22], when the input power level drops under a critical power level (calculated based on the converter parameters), the flyback inverter operates entirely in DCM. For this case, the power losses equations are simplified, since only the DCM portion is considered for their analytical calculation.

3. Design Optimization Process for DBCM

Based on the power loss analysis for each component, it is now possible to appropriately select the flyback inverter parameters, i.e., transformer turns ratio and inductance, DCM switching frequency, and MOSFET breakdown voltage, for DBCM operation by employing a design methodology to achieve optimal converter performance. To achieve this, the numerical value that needs to be optimized (objective function) must be selected, as well as the constraints for the specific optimization problem.

3.1. Objective Function

As solar irradiance (and therefore, the PV panel output power) is not constant and varies considerably throughout the day, during the design process of a PV converter, instead of aiming for a high efficiency at nominal power, focus is given to achieve a maximum efficiency on various power levels. More specifically, the European (EU) weighted efficiency and the California Energy Commission (CEC) weighted efficiency methods were established in order to show the realistic performance of the power converter for different irradiance levels [28]. The weighting factors of the two methods for each power level are shown in

Table 1. Weighting factors of European and CEC efficiency.

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

. In this work, the European weighted efficiency was selected as the objective function, which is calculated based on the power loss formulas calculated for each converter component (1), (5), (9), (11), (14)–(16).

$$\mathsf{\eta }=\frac{P_o}{P_i}=1-\frac{P_{Losses}}{P_i},$$

(18)

where:

$$\begin{array}{ll}{P}_{Losses}& =P_{Si}+P_{xfmr}\\ & =\left(P_{CL}+P_{SW}\right)+\left(P_{dc,1}+P_{dc,2}+P_{ac,1}+P_{ac,2}+P_{Core}\right)\\ & =\left(P_{CL,pri}+P_{CL,\sec }+P_{CL,pri}+P_{SL}+P_{CL,d}\right)+\left(P_{dc,1}+P_{dc,2}+P_{ac,1}+P_{ac,2}+P_{Core}\right)\end{array}$$

(19)

3.2. Design Constraints

The limitations on the design of a flyback microinverter can be again separated as limitations imposed by the semiconductor devices and limitations imposed by the transformer.

Concerning the transformer design, as with any magnetic component, two limitations must be met [29]:

• 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).

Regarding the semiconductor devices, the voltage induced during each switching cycle must be lower than the breakdown voltage of the selected switches. However, by arbitrarily selecting a power MOSFET with a high breakdown voltage, its higher ON resistance (caused by the longer epitaxial layer) will limit the maximum converter efficiency due to the increased conduction losses. What is more, the maximum voltage induced on each switch is a function of the inverter design parameters (i.e., transformer turns ratio), and therefore, its accurate determination is essential for the selection of the design parameters through the optimization process. During this calculation, it is essential to include the effect of the output filter capacitor value, as will be shown below.

More specifically, assuming that the high frequency voltage fluctuation across the filter capacitor C_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):

$$V_{pri}=V_{dc}+n\cdot V_{C_f}\stackrel{\Delta V_{C_f}\approx 0}{\Rightarrow }{V}_{pri,p}=V_{dc}+n\cdot V_{acp},$$

(20)

$$V_{\sec }=2\cdot V_{C_f}\stackrel{\Delta V_{C_f}\approx 0}{\Rightarrow }{V}_{\sec ,p}=2\cdot V_{acp}.$$

(21)

However, this implies that a large capacitor value would be used for the output filter, which would lower the converter power density and, more importantly, increase the phase difference between the grid voltage and the converter output current, lowering the inverter power factor. This is not acceptable for a grid-connected PV inverter and, thus, the capacitance value must be appropriately selected. In Figure 3, where the current and voltage waveforms on the transformer secondary side are shown, the increase of the maximum voltage on the transformer secondary side switch due to the capacitor voltage fluctuation can be observed. This voltage fluctuation is transferred to S_P as well. To determine the maximum ΔV_Cf, the charge ΔQ is first calculated for the switching cycle near ωt = π/2:

$$\Delta Q={\displaystyle \underset{0}{\overset{t_q}{\int }}{i}_{C_f}(t)} dt={\displaystyle \underset{0}{\overset{t_q}{\int }}\left(i_{\sec }(t)-I_{grid,p}\right) }dt=\frac{1}{2}\left(I_{\sec ,p}-I_{grid,p}\right)t_q.$$

(22)

Using Thales’ theorem for similar triangles, it is deduced that:

$$\frac{t_q}{t_{off,p}}=\frac{I_{\sec ,p}-I_{grid,p}}{I_{\sec ,p}}\Rightarrow t_q=\frac{I_{\sec ,p}-I_{grid,p}}{I_{\sec ,p}}{t}_{off,p}.$$

(23)

Thus, the capacitor fluctuation is calculated using (22) and (23):

$$\Delta V_{C_f}=\frac{1}{C_f}\Delta Q=\frac{1}{C_f}\frac{1}{2}\frac{{\left(I_{\sec ,p}-I_{grid,p}\right)}^2}{I_{\sec ,p}}{t}_{off,p}.$$

(24)

The maximum capacitor fluctuation will occur under nominal converter output power at ωt = π/2. Therefore, as shown in Figure 2, the converter will operate in the i-BCM segment, and the i-BCM equations should be employed [18], which, after some mathematical manipulations, leads to:

$$\begin{array}{c}\Delta V_{C_f}=\frac{1}{C_f}\frac{1}{2}\frac{{\left(n\frac{V_{dc}}{L_1}{t}_{on,p}-\frac{2P_{PV,nom}}{V_{acp}}\right)}^2}{n\frac{V_{dc}}{L_1}{t}_{on,p}}\frac{\frac{\lambda }{n}}{1+\frac{\lambda }{n}}{t}_{on,p}\left(1+\frac{\lambda }{n}\right)\\ =\frac{1}{C_f}\frac{{(-4f_{s,avg}{L}_1{P}_{PV,nom}\mathsf{\pi }+V_{dc}(\mathsf{\pi }{V}_{dc}+2n{V}_{acp}))}^2}{2f_{s,avg}^2{L}_1{n}^2{\mathsf{\pi }}^2{V}_{acp}^3}\end{array}.$$

(25)

Finally, taking into account the voltage fluctuation of the capacitor, the constraints for the selection of the semiconductor switches are:

$$V_{pri,p}=V_{dc}+n{V}_{C_f,p}=V_{dc}+n\cdot \left(V_{acp}+\frac{\Delta V_{C_f}}{2}\right),$$

(26)

$$V_{sec,p}=2V_{C_f,p}=2\cdot \left(V_{acp}+\frac{\Delta V_{C_f}}{2}\right)=2V_{acp}+\Delta V_{C_f}.$$

(27)

3.3. Optimization Sequence

The flowchart of the optimization process is shown in Figure 4. Having analytically obtained the efficiency equation for the microinverter operating in hybrid DBCM (objective function), a global maximum can be determined by recursively selecting different values for the design independent variables. Those variables are the design values that determine the characteristics of each converter instance implementation and are: the transformer turns ratio n, the switching frequency when the converter operates in DCM f, the peak ON time at nominal power t_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

To verify the optimization process as well as the power loss analysis, a laboratory experimental prototype was built (shown in Figure 5), with the component values, outlined in

Table 2. Inverter design parameters.

Specifications	Design Constraints	Optimization Independent Variables	Design Values
STC: PPV,max = 180 W/Vdc = 36 V	Peak Flux Density: 280 mT	n = 0.129	SP: IXFX180N15
0 °C: PPV,max = 205 W/Vdc = 40 V	Maximum. Transformer Fill Factor: 35%	f = 29 kHz	S1, S2: IXFX26N120
60 °C: PPV,max = 140 W/Vdc = 31 V	Maximum MOSFET breakdown voltage: 1200 V	tonp,max = 37.2 us	D1, D2: STTH1512G
Grid: 230 V/50 Hz		J = 4.1 A/mm2	Turns = 19:147
Cf = 440 nF		Bp = 280 mT	L1 = 41.2 μH
Core Type: ETD54			Leakage inductance ratio: 2.2%
Material: 3F3			Core gap = 3.41 mm

, selected by the algorithm. The flyback inverter was designed to be powered by a monocrystalline 72-cell panel with a nominal power of 180 W at 36 V (at STC). The panel operating temperature limits were selected to be 0 °C and 60 °C. Therefore, the optimization process was carried out for the worst case, which, for a PV system, is the minimum temperature (resulting to the highest input power).

The flyback inverter is controlled by the STM32F746 microcontroller, which samples the converter input and output voltage and determines the modulation of S_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.

Efficiency measurements for different input voltages (corresponding to different PV panel temperature) and power levels were carried out using the precision power analyzer LMG500 of ZES Zimmer manufacturer in order to determine the weighted efficiency of the prototype as well as to validate the power loss analysis for the DBCM modulation. The measured efficiency as a function of the input power compared to the calculated one, based on the power loss analysis, is shown in Figure 7, Figure 8 and Figure 9, for different input voltages.

The convergence between the theoretical and experimental results validates the power loss analysis as well as the optimization process. As can be observed, the component values are appropriately selected by the algorithm in order to have the maximum efficiency between 20% and 50% of the nominal input power, where the weights of the European efficiency are more significant. This is true for all three input voltage values. Since the optimization algorithm was carried out for the worst case (minimum temperature and maximum input voltage and power), the converter efficiency for the other cases (STC/36 V and 60 °C/31 V) is slightly improved.

The calculated and measured European efficiencies are shown in

Table 3. European Weighted Efficiency.

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%

. As a comparison, the calculated European efficiency for a converter with the same specifications optimally designed for i-BCM operation [18] is also shown in this table. The achieved weighted efficiency is always higher than the one of the converter operating in pure i-BCM, underlining the performance benefit of the flyback inverter designed for DBCM operation.

Having validated the power loss analysis, it is possible to break down the power losses of each of the components as a function of the input power level (shown in Figure 10). As the input power increases, the percentage of the semiconductor switching losses decreases, whereas the rest of the component losses increase. As a result, the optimization algorithm appropriately selects the component parameters to achieve the global minimum of the total losses near the 50% mark of the nominal power, achieving the highest weighted efficiency.

5. Conclusions

The characteristics of the flyback current source inverter qualify it to be used as a single-stage converter in photovoltaic modules. Operation in DBCM combines the advantages of both DCM and i-BCM (lower switching and conduction losses and higher energy density). In this work, a fully parameterized power loss model of the converter for the DBCM modulation was analyzed and implemented. This was made possible by generating the equations that describe the average and rms current values of all the converter key components for operation in DBCM. Moreover, it then led to the formulation of an optimization algorithm in order to determine the design values based on the converter specifications, so as to achieve the maximum weighted efficiency. Experimental results validated this analysis and also showed the increased performance compared to a flyback inverter with the same specifications optimized for operation exclusively in i-BCM.

The generation of the analytical equations that describe the operation of each component of the flyback micro-inverter allows the designer to appropriately select the design values based on the application specifications (e.g., power and voltage range), as well as other technical or non-technical constraints (e.g., component cost, availability, packaging). Through the power loss analysis, the effect of each component on the total performance of the converter can also be predicted, leading to a better cooling design. The mathematical manipulations presented in this paper can be applied to other inverters with similar hybrid switching strategies.