*3.1.1. Analysis for resistive load-type*

When a resistive load is connected to the dc-dc converter, Figure 11 may be redrawn as per Figure 12 and equation (7) can be derived.

**Figure 12.** MPPT supplying a resistive load-type.

An Optimized Maximum Power Point Tracking Method Based on PV Surface Temperature Measurement http://dx.doi.org/10.5772/51020 99

$$V\_R = RI\_R \tag{7}$$

Taking into account a literal dc-dc converters static gain *G*, the input system variables (*V PV* and *I PV*) can strictly be associated to the output ones (*V <sup>R</sup>* and *I <sup>R</sup>*), through (8) and (9).

converter duty cycle in order to emulate a variable load from the PV terminals point of view, even when a fixed load is employed. The arrangement presented at Figure 11, com‐ posed by a PV module, a dc-dc converter and a load, defines the hardware of a maximum

It is important to emphasize that the tracking system will present distinct behaviours de‐ pending on the dc-dc converter and load-type features. Here, buck, buck-boost, boost, Cuk, SEPIC and zeta converters will be analyzed in association with resistive or constant voltage

When a resistive load is connected to the dc-dc converter, Figure 11 may be redrawn as per

power point tracking system.

98 Sustainable Energy - Recent Studies

**Figure 11.** Maximum point tracker system.

*3.1.1. Analysis for resistive load-type*

Figure 12 and equation (7) can be derived.

**Figure 12.** MPPT supplying a resistive load-type.

loads-type.

$$G = \frac{V\_R}{V\_{PV}}\tag{8}$$

$$\mathbf{G} = \frac{I\_{PV}}{I\_R} \tag{9}$$

Isolating *V <sup>R</sup>* in (8) and *I <sup>R</sup>* in (9) and substituting the found results into (7), it is possible to obtain (10).

$$\frac{V\_{PV}}{I\_{PV}} = \frac{R}{G^2} \tag{10}$$

The term *V PV /I PV* describes the effective resistance *R eff* obtained from the PV module termi‐ nals. In other words, the dc-dc converter emulates a variable resistance, whose value can be modulated in function of the converter static gain *G*. This conclusion allows redesigning Fig‐ ure 12 as Figure 13 and writes (11).

**Figure 13.** Effective resistance obtained from the PV module terminals.

$$V\_{PV} = \frac{R}{G^2} I\_{PV} \tag{11}$$

When plotted on the I-V plan, equation (11) results in a straight line whose inclination angle θ, given by 12, is modified according to the converter static gain *G*.

$$\Theta = \operatorname{atann}\left(\frac{G^2}{\mathbb{R}}\right) \tag{12}$$

Table 2 presents static gain, as a function of the duty cycle *D*, for the dc-dc converter cregard‐ ed in this chapter, for operation in continuous conduction mode (CCM). Applying the re‐ sults from Table 2 in (12), it is possible to describe the effective inclination angle θ, for each converter, as a variable dependent on the duty cycle *D*, as consequence, Table 3 is obtained.

The tracking region refers to the area on the I-V plan in which the dc-dc converter is able to emulate a proper effective load curve in order to intercept the I-V curve exactly on the MPP, ensuring the maximum power transfer. Note, when solar radiation or temperature change, the maximum power point is relocated on the I-V plan, thus, the effective load inclination angle must also be modified in order to reestablish the maximum power transfer. However, this condition is only suitable if the MPP is located inside the tracking region, otherwise, the

An Optimized Maximum Power Point Tracking Method Based on PV Surface Temperature Measurement

=atan( <sup>1</sup>

**Figure 14.** Tracking and non tracking regions for: (a) buck converter; (b) boost converter and (c) buck-boost, Cuk, SEP‐

Comparing the graphical results, it is verified that buck-boost, Cuk, SEPIC and zeta are the most appropriated converters for maximum power point tracking applications, once they may track the MPP independently on its position on the I-V plan. On the other hand, buck

<sup>θ</sup><sup>|</sup> *<sup>D</sup>*=0

**Table 4.** Minimum and maximum effective load inclination for some dc-dc converters.

*=0* <sup>θ</sup>*<sup>|</sup> D=1*

*<sup>R</sup>* ) <sup>θ</sup><sup>|</sup> *<sup>D</sup>*=1

=0 <sup>θ</sup><sup>|</sup> *<sup>D</sup>*=1

**Maximum effective load inclination angle**

> *=atan( <sup>1</sup> R )*

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101

=90°

=90°

system operating point will be set out of the MPP.

Buck <sup>θ</sup>*<sup>|</sup> D=0*

Boost <sup>θ</sup><sup>|</sup> *<sup>D</sup>*=0

Buck-boost, Cuk, SEPIC and zeta

IC and zeta converters.

**Power dc-dc converter Minimum effective load inclination angle**


**Table 2.** Static gain for some dc-dc converters in CCM.


**Table 3.** Load curve inclination angle as a function of the converter duty cycle *D*.

Theoretically, since the duty cycle is limited between 0 and 1, the effective inclination an‐ gle becomes restricted into a range whose extremes are dependent on the considered dcdc converter. For instance, when buck converter is taken into account, for null duty cycle, *D=*0, (13) is found.

$$\begin{array}{c} \Theta \mid \begin{array}{c} \\ \end{array} \end{array} \Big| \begin{array}{c} \text{atan} \left( \frac{0^2}{\mathbb{R}} \right) = 0 \end{array} \tag{13}$$

Thereby, if the duty cycle is set on its high value, *D=*1, (14) may be written.

$$\Theta \mid\_{D=1} = \operatorname{atan}\left(\frac{1}{\mathbb{R}}\right) \tag{14}$$

In other to extend the presented analysis for further converters, a similar procedure may be applied, resulting at Table 4 and Figure 14.

From Table 4 it is noticed that effective load inclination angle defines an area on the I-V plan where the maximum power can be tracked. For a better understanding, Table 4 is graphical‐ ly explained through Figure 14, from where two distinct regions are identified: tracking and non tracking regions.

The tracking region refers to the area on the I-V plan in which the dc-dc converter is able to emulate a proper effective load curve in order to intercept the I-V curve exactly on the MPP, ensuring the maximum power transfer. Note, when solar radiation or temperature change, the maximum power point is relocated on the I-V plan, thus, the effective load inclination angle must also be modified in order to reestablish the maximum power transfer. However, this condition is only suitable if the MPP is located inside the tracking region, otherwise, the system operating point will be set out of the MPP.


**Table 4.** Minimum and maximum effective load inclination for some dc-dc converters.

Table 2 presents static gain, as a function of the duty cycle *D*, for the dc-dc converter cregard‐ ed in this chapter, for operation in continuous conduction mode (CCM). Applying the re‐ sults from Table 2 in (12), it is possible to describe the effective inclination angle θ, for each converter, as a variable dependent on the duty cycle *D*, as consequence, Table 3 is obtained.

**Power dc-dc converter Static Gain**

Buck-boost, Cuk, SEPIC and zeta *G* = *<sup>D</sup>*

**Table 2.** Static gain for some dc-dc converters in CCM.

100 Sustainable Energy - Recent Studies

applied, resulting at Table 4 and Figure 14.

*D=*0, (13) is found.

non tracking regions.

Buck *G = D* Boost *G* = <sup>1</sup>

**Power dc-dc converter Effective load inclination angle θ**

Buck θ=atan*( <sup>D</sup> <sup>2</sup>*

Boost θ=atan <sup>1</sup>

Theoretically, since the duty cycle is limited between 0 and 1, the effective inclination an‐ gle becomes restricted into a range whose extremes are dependent on the considered dcdc converter. For instance, when buck converter is taken into account, for null duty cycle,

=atan( <sup>0</sup><sup>2</sup>

=atan( <sup>1</sup>

In other to extend the presented analysis for further converters, a similar procedure may be

From Table 4 it is noticed that effective load inclination angle defines an area on the I-V plan where the maximum power can be tracked. For a better understanding, Table 4 is graphical‐ ly explained through Figure 14, from where two distinct regions are identified: tracking and

Buck-boost, Cuk, SEPIC and zeta θ=atan *<sup>D</sup>* <sup>2</sup>

**Table 3.** Load curve inclination angle as a function of the converter duty cycle *D*.

<sup>θ</sup><sup>|</sup> *<sup>D</sup>*=0

Thereby, if the duty cycle is set on its high value, *D=*1, (14) may be written.

<sup>θ</sup><sup>|</sup> *<sup>D</sup>*=1

1 − *D*

1 − *D*

*R )*

(1 − *D*)2*R*

(1 − *D*)2*R*

*<sup>R</sup>* ) =0 (13)

*<sup>R</sup>* ) (14)

**Figure 14.** Tracking and non tracking regions for: (a) buck converter; (b) boost converter and (c) buck-boost, Cuk, SEP‐ IC and zeta converters.

Comparing the graphical results, it is verified that buck-boost, Cuk, SEPIC and zeta are the most appropriated converters for maximum power point tracking applications, once they may track the MPP independently on its position on the I-V plan. On the other hand, buck and boost converters are not indicate for this proposal, since their tracking area is only a part of the whole I-V plan. In order to validate the proposed theory, buck, boost and buckboost converters were designed and assembled in laboratory, according to Figure 15.

*3.1.2. Analysis for constant voltage load-type*

plied by a PV module through a literal dc-dc converter.

**Figure 16.** Experimental results for (a) buck, (b) boost and (c) buck-boost converters.

**Figure 17.** MPPT supplying a constant voltage load-type.

The analysis concerning to dc-dc converters operating as MPPT when a constant voltage load-type is considered follows the same procedures presented for resistive loads. For be‐ ginning, consider the MPPT system shown in Figure 17, in which a dc voltage source is sup‐

An Optimized Maximum Power Point Tracking Method Based on PV Surface Temperature Measurement

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103

**Figure 15.** a) buck, (b) boost and (c) buck-boost power stage converters.

For achieving the experimental tests, the converters duty cycle was linearly varied from 0 to 1, while PV voltage and current were measured. By the use of a scope on XY mode, the I-V curve was traced, and the found results are shown at Figure 16.

Notice that I-V curve is partially plotted on the I-V plan when buck and boost converters are regarded, and on the whole I-V plan, when buck-boost converter is considered. Additional‐ ly, the area in which the I-V curves were traced is in accordance with the tracking region, theoretically defined for each converter, validating the analysis.

On the next subsection, the resistive load will be replaced by a constant voltage load-type. This analysis is relevant and mandatory, since in many applications, PV systems are em‐ ployed in battery charges, or even, delivering power to a regulated dc bus.
