In this section, the derivation of the BCM reference current and the parameter that interact with the MPPT are elaborated. According to the volt-second balance of the inductance, the turn-on and turn-off time can be deducted as in (2) and (3), while the primary current and secondary current (i_s) relation is realized as in (4) and (5).

$T_{ON}=L_p\times I_p\times \frac{1}{v_{dc}}$ (2)

$T_{OFF}=L_s\times I_s\times \frac{1}{v_g}$ (3)

$I_p=i_{ref}$ (4)

$I_s=n\times I_p=I_p\sqrt{\frac{L_p}{L_s}}$ (5)

where $\sqrt{L_s/L_p}=n$, the transformer’s turn ratio. By considering that the efficiency of the converter is 100%, the amount of power at the output is equal to the input. Therefore, based on Figure 2 , i_in can be calculated according to the assumption that the area of A₁ and A₂ are approximately the equal. i_in can be expressed as (6). Likewise, since the i_out is yielded from filtering the i_s, the value of the i_out is approximately equal to the average value of the secondary current (I_s). Hence, area B₁ and B₂ are considered equal. i_out can be expressed as in (7).

$i_{in}=\frac{1}{2}\times I_p\times \frac{T_{ON}}{T_{ON}+T_{OFF}}$ (6)

$i_{out}=\frac{1}{2}\times I_s\times \frac{T_{OFF}}{T_{ON}+T_{OFF}}$ (7)

Equation (8) is realized to show the relationship between i_in and i_ref by substituting (2)-(3) into (6). At the steady state condition, the dc voltage v_dc and the grid voltage v_g can be considered stable at line-frequency cycle, in which the input current can be controlled directly by the reference current.

$i_{in}=\frac{1}{2}\times i_{ref}\times \frac{v_g}{v_g+\left(v_{dc}\times \sqrt{L_s/L_p}\right)}$(8)

Meanwhile, the relationship between i_ref and i_out in the line-frequency can be expressed in (9) by substituting (2)-(3) into (7). The relationship is deducted by v_g, the transformer’s turn ratio and the steady v_in due to the selection of large value of input capacitance, C_in.

$i_{ref}=2i_{out}\times \left(\frac{v_g}{v_{dc}}+\sqrt{\frac{L_s}{L_p}}\right)$(9)

Subsequently, by solving (8) and (9), the relationship of i_in and i_out can be expressed as in (10), which proved the instantaneous input power is equivalent to the output power in the ideal condition. According to the standard sinusoidal output current equation, $i_{out}=I_A\sin \left(\omega t\right)$, and output voltage $v_g=V_m\sin \left(\omega t\right)$, equation (10) can be further expended to (11).

$i_{in}=i_{out}\times \frac{v_g}{v_{dc}}$(10)

$i_{in}=\frac{V_m}{v_{dc}}\times I_A\times {\sin }^2\left(\omega t\right)$(11)

The expression shows that the i_in can be controlled by I_A. This parameter will be interacted and adjusted by MPPT control to track the MPP under BCM operation. I_A is closely related to the output power as shown in (12).

$I_A=\frac{P_{out}}{V_m}$(12)

In this work, the flyback-based inverter is a current-source that operates under peak current control and BCM. The duty cycle of the converter will vary throughout line-frequency cycle following the shape of given reference cycle. Therefore, unlike the conventional IncCond that usually alter terminal voltage of the PV panel in accordance with MPP, the proposed modification is altering the terminal current that controlled by the parameter I_A as mentioned in (12). Figure 3 shows the flowchart of the proposed algorithm. The controller initiates the process by sampling V_pv and I_pv, subsequently computing P_pv and determining the parameter I_A. In the event that the sequence begins at zero, the initial values for V and I are established at zero, whereas I_ref is assigned the calculated value of I_A. Subsequently, the incremental changes in voltage, current, and power are computed at the sampling instant Q.

The adjustable reference current is governed by a discrete current law as detailed in (13), with the sign determined by the incremental conductance test. S[k] in Equation (13) represents a step-size multiplier coefficient that scales with the local gradient magnitude, as derived using the Pythagorean theorem in Equation (14). This implements a variable step in incremental conductance, allowing for larger steps when operating point is distant from the MPP to accommodate rapid transients, while minimizing the step size when near to the MPP to mitigate steady-state oscillations. The concept of the Pythagorean theorem was initially applied in P&O by [9] and [10] to address uniform conditions and partial shading conditions, respectively. The adapted I_ref is then tracked by an inner current loop as expressed in (15), which includes a PI compensator C(z).

$I_{ref}\left[k+1\right]=I_{ref}\left[k\right]\pm d{I}_{ref}\ast S\left[k\right]$(13)

$S=\left|\sqrt{d{P}^2+d{I}^2}\right|$(14)

$I_{re{f}_a}=\frac{I_A\left(z\right)+G_{nx}C\left(z\right)\left(I_{pv}\left(z\right)-I_{ref}\left(z\right)\right)}{G_n}$(15)

To maintain uniform dynamic across the non-linear I-V curve, the power zoning with empirical tuning scheduled gains is used, which are applied in (15). The PV operating range is clustered into a mid-power zone (30% - 90% of rated power) and two edge zones (<30% and >90%). The selection of these cluster values is based on the analysis of the MPPT response and characteristics observed across the extensive range of PV power. The nominal loop gain G_n and cross gain G_nx are scheduled as explicit functions of the instantaneous PV power P_pv, which expressed as a percentage of P_rated.

$G_n\left(P_{pv}\right)=\{\begin{array}{l}1.5,P_{pv}<30\%\text{or}{P}_{pv}>90\%\\ 0.6,30\%\le P_{pv}\le 90\%\end{array}$(16)

$G_{nx}\left(P_{pv}\right)=\{\begin{array}{l}6.0,P_{pv}<30\%\text{or}{P}_{pv}>90\%\\ 1.0,30\%\le P_{pv}\le 90\%\end{array}$(17)

Aggressive gains in the edge zones enhance recovery when the array functions far from the maximum power point, particularly under conditions of weak irradiance, rapid ramps, or proximity to power clipping, thereby improving transient tracking. In contrast, conservative gains within the 30% - 90% range reduce steady-state ripple and electromagnetic interference (EMI), mitigate current overshoot at rated power, and improve resilience to measurement noise.