**5. Discussion**

The dependence of the power loss (*Ploss*) and energy efficiency for a DC-DC power converter were analyzed for low and medium power applications in [38,39] and [40–42], respectively. The main findings of the aforementioned studies are as follows:


$$
\eta\_{\text{conv}ter} = \eta\_{\text{max}} - \frac{\eta\_{\text{max}} - \eta\_{\text{min}}}{4} \cdot \lg^2 \left( \frac{I\_{load}}{I\_{load(opt)}} \right) \tag{16}
$$

where η*max* is the maximum of the energy efficiency (obtained at the optimal load current *Iload(opt)*) and η*min* is the minimum of the energy efficiency (obtained in the considered load range).

3) For low-power applications using the control mode based on the pulse frequency modulation technique, the energy efficiency can be approximated by (17) [38,39]:

$$
\eta\_{\text{convvertex}} = \eta\_{\text{max}} - \frac{\zeta\_{\eta}}{I\_{\text{load}}},\tag{17}
$$

where η*max* is the maximum energy efficiency (obtained at the maximum load current *Iload(max)*) and ζη is a parameter (that must be determined using the experimental values in the considered range of load). Relationship (17) highlights the nonlinear increase in energy efficiency in the range of light loads and the saturation that appears in the rest of the load range;

4) For most types of control used in medium and high-power applications, the energy efficiency in the normal load range (therefore, except for light loads), where the converter operates in continuous current mode [43,44], can be considered as constant or linearly increasing (18):

$$
\eta\_{\text{counter}} = \eta\_{\text{min}} + \chi\_{\eta} \cdot \frac{I\_{\text{load}}}{I\_{\text{load}(\text{max})}},\tag{18}
$$

where η*min* is the energy efficiency obtained at the load current *Iload(min)*, which is the upper limit of light loads, and χη is a parameter to be determined using the experimental values in the considered load range (except the light loads).

The assumption that the energy efficiency linearly increases is valid for the medium-power FC HPS analyzed in this paper, because the load range was higher than 1 kW (so the case of light load was not considered). For different values of the load current, the LF control and optimization loops set the values of the FC current, *IFC*0, and *IFC*1, using the sFF strategy and the fuel economy strategy analyzed in this paper. So, (18) can be rewritten using as a variable the FC current as (19):

$$
\eta\_{\text{boost}} = \eta\_{\text{min}} + K\_{\eta} \cdot \frac{I\_{\text{FC}}}{I\_{\text{FC}(\text{max})}},\tag{19}
$$

where η*min* - 88.5% is the energy efficiency obtained at the nominal FC current, *IFC(min)* - 30 A, and *K*η = 4 is a parameter which has been determined using the experimental values of the energy efficiency η*max* - 92% obtained at the maximum FC current, *IFC(max)* - 240 A. Note that the energy efficiency obtained at the nominal FC current, *IFC(nom)* - 130 A, was η*nom* - 90.17% (which is very close to the constant value considered in simulation).

The power loss of the boost converter (*Ploss*) was estimated using (20):

$$P\_{\rm loss} = P\_{\rm load} \cdot \left(\frac{1}{\eta\_{\rm boost}} - 1\right). \tag{20}$$

The FC current, *IFC*0, and *IFC*1, using the sFF strategy and the fuel economy strategy, are registered in the second and third columns of Table 11 for different load levels and *fd* =100 Hz. The energy efficiency (η*boost*<sup>0</sup> and η*boost*1) and the power loss of the boost converter (*Ploss*<sup>0</sup> and *Ploss*1) were estimated for the sFF strategy and the fuel economy strategy with (19) and (20), and are registered in Table 11. The difference in power loss of the boost converter (Δ*Ploss*) is registered in the eighth column of Table 11. The influence of Δ*Ploss* on Δ*PFCnet* for *fd* =100 Hz was estimated as Δ*Ploss*/Δ*PFCnet* (%) and is registered in the last column of Table 11. As expected, the biggest error of 0.10485% was obtained at the maximum load.


**Table 11.** Influence of variable energy efficiency on the FC net power estimated for *fd* = 100 Hz.

It is worth mentioning that the biggest differences mentioned in Table 4, Table 6, and Table 8 were less than 0.2% for variable energy efficiency compared to the constant efficiency and also, this value was obtained at maximum load. So, the conclusions of this study are valid for both constant and variable energy efficiency.
