*Electronics* **2020**, *9*, x FOR PEER REVIEW 4 of 16 *Electronics* **2020**, *9*, x FOR PEER REVIEW 4 of 16

identify the maximum power point of the laser PV module as well as stably control the operating point of the input source determined by the algorithm. Therefore, the small-signal characteristics of the laser PV module must be analyzed to control the operating point of the input source; the small-signal resistance (denoted as *rs*) for the increment of the measured voltage and current of the laser PV module is shown in Figure 4. As shown, the laser PV module exhibits negative small-signal resistance characteristics, and the small-signal resistance value increases as it approaches the current source region. In the next section, a method for deriving the small-signal transfer function of a boost converter that reflects the small-signal resistance characteristics of a laser PV module and a controller design method are presented. operating point of the input source determined by the algorithm. Therefore, the small-signal characteristics of the laser PV module must be analyzed to control the operating point of the input source; the small-signal resistance (denoted as *rs*) for the increment of the measured voltage and current of the laser PV module is shown in Figure 4. As shown, the laser PV module exhibits negative small-signal resistance characteristics, and the small-signal resistance value increases as it approaches the current source region. In the next section, a method for deriving the small-signal transfer function of a boost converter that reflects the small-signal resistance characteristics of a laser PV module and a controller design method are presented. operating point of the input source determined by the algorithm. Therefore, the small-signal characteristics of the laser PV module must be analyzed to control the operating point of the input source; the small-signal resistance (denoted as *rs*) for the increment of the measured voltage and current of the laser PV module is shown in Figure 4. As shown, the laser PV module exhibits negative small-signal resistance characteristics, and the small-signal resistance value increases as it approaches the current source region. In the next section, a method for deriving the small-signal transfer function of a boost converter that reflects the small-signal resistance characteristics of a laser PV module and a controller design method are presented. 

**Figure 3.** Voltage–current and voltage–power characteristics based on laser power of PV module. **Figure 3.** Voltage–current and voltage–power characteristics based on laser power of PV module. **Figure 3.** Voltage–current and voltage–power characteristics based on laser power of PV module.

#### *2.2. Modeling and Controller Design of Boost Converter 2.2. Modeling and Controller Design of Boost Converter 2.2. Modeling and Controller Design of Boost Converter*

Figure 5 shows the boost converter topology of the battery charging system to which the laser PV module was applied. In Figure 5, L is the inductor of the boost converter, C is the input capacitor, rc is the capacitor's equivalent series resistance (ESR), and Vb is the voltage source representing the battery connected to the output. The boost converter must have an operating point control function to generate the maximum power from the PV module before the battery reaches its full voltage [10– 12]. In this study, the boost converter was set to control the voltage of the PV module such that the Figure 5 shows the boost converter topology of the battery charging system to which the laser PV module was applied. In Figure 5, L is the inductor of the boost converter, C is the input capacitor, rc is the capacitor's equivalent series resistance (ESR), and Vb is the voltage source representing the battery connected to the output. The boost converter must have an operating point control function to generate the maximum power from the PV module before the battery reaches its full voltage [10– 12]. In this study, the boost converter was set to control the voltage of the PV module such that the Figure 5 shows the boost converter topology of the battery charging system to which the laser PV module was applied. In Figure 5, L is the inductor of the boost converter, C is the input capacitor, rc is the capacitor's equivalent series resistance (ESR), and Vb is the voltage source representing the battery connected to the output. The boost converter must have an operating point control function to 92

generate the maximum power from the PV module before the battery reaches its full voltage [10–12]. In this study, the boost converter was set to control the voltage of the PV module such that the laser PV module can form a stable operating point in a large signal. For the controller design and stability analysis of the boost converter, a small-signal modeling technique using the state-space averaging method was applied, which is a method of obtaining the transfer function from the input to the output of the system by linearizing the system based on the operating point in the steady state [13,14]. *Electronics* **2020**, *9*, x FOR PEER REVIEW 5 of 16 laser PV module can form a stable operating point in a large signal. For the controller design and stability analysis of the boost converter, a small-signal modeling technique using the state-space averaging method was applied, which is a method of obtaining the transfer function from the input to the output of the system by linearizing the system based on the operating point in the steady state [13,14]. *Electronics* **2020**, *9*, x FOR PEER REVIEW 5 of 16 laser PV module can form a stable operating point in a large signal. For the controller design and stability analysis of the boost converter, a small-signal modeling technique using the state-space averaging method was applied, which is a method of obtaining the transfer function from the input to the output of the system by linearizing the system based on the operating point in the steady state [13,14]. 

**Figure 5.** Boost converter for battery charging with laser PV module. **Figure 5.** Boost converter for battery charging with laser PV module. **Figure 5.** Boost converter for battery charging with laser PV module.

The small-signal transfer function required for designing the voltage controller of the laser PV module of the boost converter was derived in the following two steps. First, as shown in Figure 6, a small-signal model of the unterminated model for the boost converter modeling of the PV module as a current source was developed. Subsequently, the small-signal transfer function of the boost converter to which the laser PV module was applied was derived by including the relationship between the small-signal voltage and current of the laser PV module mentioned in Section 2.1. The small-signal transfer function required for designing the voltage controller of the laser PV module of the boost converter was derived in the following two steps. First, as shown in Figure 6, a small-signal model of the unterminated model for the boost converter modeling of the PV module as a current source was developed. Subsequently, the small-signal transfer function of the boost converter to which the laser PV module was applied was derived by including the relationship between the small-signal voltage and current of the laser PV module mentioned in Section 2.1. The small-signal transfer function required for designing the voltage controller of the laser PV module of the boost converter was derived in the following two steps. First, as shown in Figure 6, a small-signal model of the unterminated model for the boost converter modeling of the PV module as a current source was developed. Subsequently, the small-signal transfer function of the boost converter to which the laser PV module was applied was derived by including the relationship between the small-signal voltage and current of the laser PV module mentioned in Section 2.1. 

**Figure 6.** Unterminated model of boost converter with input as the current source. **Figure 6.** Unterminated model of boost converter with input as the current source. **Figure 6.** Unterminated model of boost converter with input as the current source.

The state equation obtained by averaging the differential equations of the voltage and current based on the switching states of the power switch of the boost converter in Figure 6 is shown in Equation (1), where *d* denotes the ratio of turning ON during one period of the switching frequency, and *d*' is defined as 1 − *d*. The state equation obtained by averaging the differential equations of the voltage and current based on the switching states of the power switch of the boost converter in Figure 6 is shown in Equation (1), where *d* denotes the ratio of turning ON during one period of the switching frequency, and *d*' is defined as 1 − *d*. The state equation obtained by averaging the differential equations of the voltage and current based on the switching states of the power switch of the boost converter in Figure 6 is shown in Equation (1), where *d* denotes the ratio of turning ON during one period of the switching frequency, and *d* is defined as 1 − *d*.

$$
\begin{bmatrix} i\_L \\ i\_{\bar{\upsilon}\_c} \\ \vdots \\ i\_{\bar{\upsilon}\_c} \end{bmatrix} = \begin{bmatrix} i\_{\bar{U}} \\ i\_{\bar{\upsilon}\_c} \\ \vdots \\ i\_{\bar{\upsilon}\_c} \end{bmatrix} \begin{bmatrix} \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} \\ \vdots & \vdots & \vdots \\ \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} \end{bmatrix} \begin{bmatrix} \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} \\ \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} \\ \vdots & \vdots & \vdots \\ \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} & \frac{\partial}{\partial \bar{U}} \end{bmatrix} \begin{bmatrix} i\_{\bar{s}} \\ i\_{\bar{b}} \\ i\_{\bar{b}} \end{bmatrix} \tag{1}
$$

 ¬ ¼¬ ¼ *& &* To develop the small-signal model, the state and control variables of the boost converter were defined as having small perturbations at the steady-state operating point, which is indicated by capital letters as follows: Ö *F FF Y9Y* Ö */ // L ,L* Ö *G 'G* Ö *E EE Y9Y* and Ö *V VV L ,L* . Since the small perturbation excluding the steady-state operating point yields a small-signal model, the small-signal transfer function of the boost converter is as shown in Equation (2). In addition, the small-signal block diagram of the output to the input can be represented as shown in Figure 7. To develop the small-signal model, the state and control variables of the boost converter were defined as having small perturbations at the steady-state operating point, which is indicated by capital letters as follows: Ö *F FF Y9Y* Ö */ // L ,L* Ö *G 'G* Ö *E EE Y9Y* and Ö *V VV L ,L* . Since the small perturbation excluding the steady-state operating point yields a small-signal model, the small-signal transfer function of the boost converter is as shown in Equation (2). In addition, the small-signal block diagram of the output to the input can be represented as shown in Figure 7. To develop the small-signal model, the state and control variables of the boost converter were defined as having small perturbations at the steady-state operating point, which is indicated by capital letters as follows: *vc* = *Vc* + *<sup>v</sup>*<sup>ˆ</sup>*c*, *iL* = *IL* + ˆ *iL*, *d* = *D* + ˆ *d*, *vb* = *Vb* + *v*ˆ*b* and *is* = *Is* + ˆ *is*. Since the small perturbation excluding the steady-state operating point yields a small-signal model, the small-signal

$$\begin{cases} \dot{i}\_s & \Delta\\ G\_4 = \frac{\hat{v}\_s}{\hat{v}\_b} = D' \frac{\left(1 + s r\_c C\right)}{\Delta} \\\\ G\_5 = \frac{\hat{I}\_L}{d} = \frac{s C V\_b}{\Delta} \\\\ G\_6 = \frac{\hat{v}\_s}{\Delta \text{MaxID}} = -V\_0 \frac{\left(1 + s r\_c C\right)}{\text{Coefficient (2)}} \end{cases} \tag{2}$$

transfer function of the boost converter is as shown in Equation (2). In addition, the small-signal block diagram of the output to the input can be represented as shown in Figure 7. *G* °¯ '

ˆ*iL*(<sup>1</sup>+*srcC*) The characteristic equation Δ, Q-factor, and resonance frequency of the transfer function are expressed as shown in Equation (3).

$$\begin{cases} G\_1 = \frac{i\_l}{l\_c} = \frac{(1+sr\_cC)}{\Delta} \\ G\_2 = \frac{i\_l}{l\_b} = \frac{s\frac{\sqrt{-D'}}{\Delta}s^2}{\Delta + \Delta} \\ G\_3 = \frac{\frac{\phi\_c}{l\_c}}{\frac{1}{\Delta}} = \frac{Q\phi\_b + sr\_cD\phi\_o^2}{\Delta} \\ Q\frac{\phi\_c}{\Delta} = \frac{\frac{\phi\_c}{\Delta}}{\frac{1}{\Delta}} \cdot \frac{\frac{1}{\Delta}D'}{\frac{C\_{\phi}C\_{\phi}}{\Delta}} \\ G\_5 = \frac{\frac{\phi\_c}{\Delta}}{\frac{1}{\Delta}} \frac{1}{\Delta} \frac{C\_{\phi}V\_b}{\Delta} \\ G\_6\frac{\phi\_c - \frac{\phi\_c}{\Delta}LC}{\DeltaLC}V\_b\frac{(1+sr\_cC)}{\Delta} \end{cases} (2)$$

**Figure 7.** Small-signal block diagram of unterminated model. **Figure 7.** Small-signal block diagram of unterminated model.

As mentioned in Section 2, the laser PV module represents nonlinear voltage and current characteristics; therefore, a small-signal model can be obtained through the Taylor series for the operating point, as shown in Equation (4). The characteristic equation Δ, Q-factor, and resonance frequency of the transfer function are expressed as shown in Equation (3).

$$\begin{cases} \dot{i}\_s = f\left(\dot{V}\_s\right) \frac{1}{\omega\_s} = 1 + \frac{s}{Q\omega\_o} + \frac{s^2}{\omega\_o^2} \\ \dot{I}\_s + \hat{i}\_s = \begin{cases} \frac{1}{Q\sqrt{\epsilon}} \cdot \frac{1}{\epsilon \sqrt{\epsilon}} \sqrt{\frac{L}{\epsilon}} & f\left(V\_s\right) + f\left(V\_s\right) \cdot \hat{v}\_s \\ \omega\_s = \frac{1}{\sqrt{\epsilon}} \end{cases} \tag{3} \\ \dot{i}\_s = r\_s \hat{v}\_s^\dagger \end{cases} \tag{4}$$

In the unterminated model, the relationship between <sup>Ö</sup>*V Y* and <sup>Ö</sup>*VL* corresponds to the small-signal resistance *rs* of the laser PV module; therefore, the small-signal block diagram in Figure 7 can be reorganized as shown in Figure 8. As depicted in Figure 8, when the laser PV module is connected to the input of the boost converter, a loop gain T is formed where <sup>Ö</sup>*VY* is fed back to <sup>Ö</sup>*VL* ; therefore, the As mentioned in Section 2, the laser PV module represents nonlinear voltage and current characteristics; therefore, a small-signal model can be obtained through the Taylor series for the operating point, as shown in Equation (4). *is*=*f*(*vs*)

$$\begin{cases} \mathsf{f}\mathsf{f}\mathsf{c}\mathsf{f}\mathsf{g} \triangleq \mathsf{f}\mathsf{c}\mathsf{g}\mathsf{p} \text{ gain } \mathsf{l}' \text{ is formed where } \mathsf{v}\_{s} \text{ is fed back to } \mathsf{i}\_{s}; \text{ therefore, the} \\\ \mathsf{l}\_{s} + \mathsf{i}\_{s} = f(\mathsf{V}\_{s} + \mathsf{i}\_{s}) = f(\mathsf{V}\_{s}) + \dot{f}(\mathsf{V}\_{s}) \cdot \mathsf{d}\_{\mathsf{s}} \\\ \dot{\mathsf{i}}\_{\mathsf{s}} = \mathsf{r}\_{\mathsf{s}} \mathsf{d}\_{\mathsf{s}} \ (\mathsf{r}\_{\mathsf{s}} < 0) \end{cases} \tag{4}$$

In the unterminated model, the relationship between *v*ˆ*s* and ˆ *is* corresponds to the small-signal resistance *rs* of the laser PV module; therefore, the small-signal block diagram in Figure 7 can be reorganized as shown in Figure 8. As depicted in Figure 8, when the laser PV module is connected to the input of the boost converter, a loop gain T is formed where *v*ˆ*s* is fed back to ˆ *is*; therefore, the final final small-signal transfer function of the boost converter considering the laser PV module can be derived from Equation (5). 

*Electronics* **2020**, *9*, 1745

$$\int\_{\cdot} \frac{\hat{\boldsymbol{\hat{v}}\_{\cdot \cdot}}}{\hat{\boldsymbol{v}}\_{b}} = \frac{G\_4}{1 + T}$$

small-signal transfer function of the boost converter considering the laser PV module can be derived from Equation (5). Ö Ö*V Y \* 7*°°° 

$$\begin{cases} \frac{\partial \mathcal{G}}{\partial \tilde{x}} = \frac{1\_{\mathcal{G}\_{4}} T}{1 + T} \\ \frac{\partial \mathcal{G}}{\partial \tilde{x}} = \frac{\partial \mathcal{G}\_{6}}{1 + T} + \frac{G\_{1}}{r\_{\mathcal{S}}} \frac{G\_{4}}{1 + T} \\ \frac{\frac{\partial \mathcal{G}}{\partial \tilde{x}}}{\frac{\partial \mathcal{G}}{\partial \tilde{x}}} = G\_{2} + \frac{G\_{1}}{G\_{4}} \frac{G\_{4}}{\frac{\partial \mathcal{G}}{\partial \tilde{x}}} \\ \frac{\frac{\partial \mathcal{G}}{\partial \tilde{x}}}{\frac{\partial \mathcal{G}}{\partial \tilde{x}}} = G\_{5} + \frac{G\_{1}}{F\_{5}} \frac{G\_{6}}{1 + T} \end{cases} \tag{5}$$

**Figure 8.** Small-signal block diagram with added PV module. **Figure 8.** Small-signal block diagram with added PV module.

Therefore, the final transfer function of the boost converter considering the PV module is expressed as shown in Equations (6)–(9), and the small-signal block diagram including the inductor current of the boost converter and the input voltage controller of the laser PV module is as shown in Figure 9. To control the input voltage with the voltage command generated by the MPPT algorithm that tracks the maximum power point of the laser PV module, a two-loop controller structure was adopted, in which a voltage controller was applied to the outer loop while an internal current controller was applied. Since the converter is operated by a digital controller through the micro controller unit (MCU), the transfer function of the block diagram is expressed as a discrete-time transfer function. Therefore, the final transfer function of the boost converter considering the PV module is expressed as shown in Equations (6)–(9), and the small-signal block diagram including the inductor current of the boost converter and the input voltage controller of the laser PV module is as shown in Figure 9. To control the input voltage with the voltage command generated by the MPPT algorithm that tracks the maximum power point of the laser PV module, a two-loop controller structure was adopted, in which a voltage controller was applied to the outer loop while an internal current controller was applied. Since the converter is operated by a digital controller through the micro controller unit (MCU), the transfer function of the block diagram is expressed as a discrete-time transfer function. *Electronics* **2020**, *9*, x FOR PEER REVIEW 8 of 16 Ö Ö */ E V LG V F F V V L 9 VU & \* G / U U /& V U & V /& U U* § ·§ · ¨ ¸¨ ¸ © ¹© ¹ (9) ½ ° ° ® ¾ ° ° ¯ ¿ *V V7 S S H \*] = \*V V* (10) 

The open-loop current gain (Ti) for controlling the inductor current of the boost converter is shown in Figure 9. At this time, for digital control, the continuous-time small-signal transfer function obtained previously was converted into a discrete-time transfer function using the zero-order hold (ZOH) method, and the conversion equation is as shown in Equation (10). Here, *Z*{·} represents the z-transform, *V V7 H V* represents the transfer function of the ZOH, and *Gp*(*s*) represents the transfer function of Equations (6)–(9). \*YJ] \*LG] \*LJ] \*YG] 9EDW G L/ 9V Ʒ Ʒ Ʒ Ʒ

$$\begin{bmatrix} \frac{\hat{\mathbf{v}}\_{s}}{\hat{\mathbf{v}}\_{b}} \mathbf{G}\_{\text{vg}} = \mathbf{D} \frac{\mathbf{I}\_{s} \mathbf{I}\_{s} \mathbf{I}\_{s}}{\mathbf{T}\_{\text{b}} + s \left( \mathbf{r}\_{\text{c}} \mathbf{C} \underbrace{\frac{\mathbf{I}\_{\text{c}} \mathbf{I}\_{\text{d}}}{\mathbf{H}(\mathbf{x})} \right) \left( \mathbf{X} - \frac{\mathbf{r}\_{\text{c}} \mathbf{L} \mathbf{f}}{\mathbf{r}\_{\text{c}}} \right)} \mathbf{T}\_{\text{s}} \\\\ \frac{\hat{\mathbf{i}}\_{L}}{\hat{\mathbf{v}}\_{b}} - \mathbf{G}\_{\text{g}} - \frac{\mathbf{D}^{\*}}{\mathbf{r}\_{\text{s}}} \frac{\mathbf{L}^{\*} \mathbf{L} \mathbf{f}}{\mathbf{1} + s \left( \mathbf{r}\_{\text{c}} \mathbf{C} - \frac{\mathbf{L}}{\mathbf{L}} \right) + s^{2} \left( \frac{\mathbf{L} \mathbf{H}(\mathbf{x})}{\mathbf{L} \mathbf{C} - \frac{\mathbf{x}^{\*}}{\mathbf{x}^{\*}}} \right) \mathbf{C}^{\*}} \mathbf{M}^{\*} \end{bmatrix} \tag{6}$$

 *V V U U* © ¹© ¹ **Figure 9.** Small-signal block diagram for two-loop control. **Figure 9.** Small-signal block diagram for two-loop control.

 Ö Ö *V F YG E F F V V Y VU & \* 9 G / U /& V U & V /& U U* § ·§ · ¨ ¸¨ ¸ © ¹© ¹ (8) In this study, a proportional and integral (PI) controller was used for current control, as shown in Equation (11), and the Ti applied with the PI current controller is shown in Equation (12). At this time, *<sup>z</sup>*−1 represents the time delay until the calculated duty is reflected, and it was considered as a one-sampling delay in this study. Figure 10 shows the Bode diagram of the Ti of the designed current controller. CSR denotes the current-source region of the laser PV module, MPP denotes the region near the maximum power point, and VSR denotes the voltage-source region. Since the laser PV module has nonlinear characteristics, it must be designed to ensure stability in the CSR, VSR, and MPP of the laser PV module, as shown in the Bode diagram. The open-loop current gain (Ti) for controlling the inductor current of the boost converter is shown in Figure 9. At this time, for digital control, the continuous-time small-signal transfer function obtained previously was converted into a discrete-time transfer function using the zero-order hold (ZOH) method, and the conversion equation is as shown in Equation (10). Here, *Z*{·} represents the

$$H\_l\left(z\right) = 0.078 \frac{\left(z - 0.93\right)}{\left(z - 1\right)}\tag{11}$$

$$T\_\cdot = G\_{\cdot \cdot \cdot} (\underline{z}) \cdot H\_\cdot (\underline{z}) \cdot \underline{z}^{-1} \tag{1\mathcal{D}}$$

Ö

*L*

*Electronics* **2020**, *9*,1745

$$\frac{\dot{G}\_L}{\dot{d}} = G\_{\text{idf}} = -\frac{V\_b}{r\_s} \frac{\left(1 - s r\_s C\right)}{1 + s\left(r\_c C - \frac{L}{r\_s}\right) + s^2 \left(LC - \frac{r\_c L C}{r\_s}\right)}\tag{9}$$

z-transform, (1 − *e*<sup>−</sup>*sTs*)/*s* represents the transfer function of the ZOH, and *Gp*(*s*) represents the transfer function of Equations (6)–(9). ½ ° ° ® ¾ ° ° ¯ ¿ *V V7 S S H \*] = \*V V* (10) 

$$\frac{\frac{\hat{\sigma}\_{\text{y}}}{\text{V}\hat{\mathbf{p}}\_{\text{S}}} = G\_{\text{vy}} \equiv \frac{D'}{\sum\_{s \in \text{(\%)}} (1 + sr\_{c}\mathbf{C})} \frac{(1 + sr\_{c}\mathbf{C})}{s^{2} \left(\mathbf{r}\_{c}\mathbf{C} \cdot \mathbf{C} \mathbf{\hat{p}}\_{\text{s}}\right) + s^{2} \left(\mathbf{r}\_{c}\mathbf{C} \cdot \frac{r\_{c}\mathbf{C} \mathbf{\hat{c}}}{\|\mathbf{r}\_{c}\|}\right)}} \tag{6}$$

$$\begin{array}{c|c} \cline{2-4} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \begin{array}{c} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \begin{array}{c} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \end{array} \end{array} G\_{\mathsf{f}\boldsymbol{\xi}} = \ \begin{array}{c} \cline{2-4} D' \\ \cline{2-4} \text{Tr}\begin{array}{c} \cline{2-4} \text{Tr}\begin{array}{c} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \end{array} \end{array} \end{array} G\_{\mathsf{f}\boldsymbol{\xi}} \begin{array}{c} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \end{array} G\_{\mathsf{f}\boldsymbol{\xi}} \begin{array}{c} \cline{2-4} G\_{\mathsf{f}\boldsymbol{\xi}} \end{array} \end{array} \tag{7}$$

*v* ˆ *s* ˆ *d* = *Gvd* = −*Vb* (1 + *srcC*) 1 + *srcC* − *Lrs* + *s*<sup>2</sup>*LC* − *rcLCrs* (8) \*LG] L/ = **7Y** Ʒ 

$$\left| \frac{\hat{\mathbf{i}}\_{\text{L}}}{\hat{d}} \right| = \underbrace{\mathbf{Q}\_{\text{M}}}\_{\mathbf{I}} = \underbrace{-\frac{\mathbf{W}\_{\text{b}}}{r\_{\text{s}}} \underbrace{1 + s\left(r\_{\text{c}} \underbrace{\frac{\mathbf{I}\_{\text{H}}}{\mathbf{I}\_{\text{H}}} \left(\mathbf{x}\right)}\_{\mathbf{I}\_{\text{L}}} + s^{2} \underbrace{\mathbf{L}\mathbf{C}}\_{\mathbf{I}\_{\text{H}}} - \underbrace{s^{1.0}\mathbf{C}}\_{\mathbf{I}\_{\text{H}}}}\_{\mathbf{I}\_{\text{L}}}}\_{\mathbf{I}\_{\text{L}}} \right) \tag{9}$$

*Gp*(*z*) = *Z* ⎧⎪⎪⎨⎪⎪⎩1 − *e*<sup>−</sup>*sTss Gp*(*s*)⎫⎪⎪⎬⎪⎪⎭ (10) **Figure 9.** Small-signal block diagram for two-loop control. 

In this study, a proportional and integral (PI) controller was used for current control, as shown in Equation (11), and the Ti applied with the PI current controller is shown in Equation (12). At this time, *z*<sup>−</sup><sup>1</sup> represents the time delay until the calculated duty is reflected, and it was considered as a one-sampling delay in this study. Figure 10 shows the Bode diagram of the Ti of the designed current controller. CSR denotes the current-source region of the laser PV module, MPP denotes the region near the maximum power point, and VSR denotes the voltage-source region. Since the laser PV module has nonlinear characteristics, it must be designed to ensure stability in the CSR, VSR, and MPP of the laser PV module, as shown in the Bode diagram. In this study, a proportional and integral (PI) controller was used for current control, as shown in Equation (11), and the Ti applied with the PI current controller is shown in Equation (12). At this time, *<sup>z</sup>*−1 represents the time delay until the calculated duty is reflected, and it was considered as a one-sampling delay in this study. Figure 10 shows the Bode diagram of the Ti of the designed current controller. CSR denotes the current-source region of the laser PV module, MPP denotes the region near the maximum power point, and VSR denotes the voltage-source region. Since the laser PV module has nonlinear characteristics, it must be designed to ensure stability in the CSR, VSR, and MPP of the laser PV module, as shown in the Bode diagram. 

$$\text{M}\_{\text{M}}(\text{z}) = 0.028 \frac{(\text{z} - \text{z})}{(\text{z} - \text{z} \text{M})} \tag{101}$$

$$\Psi\_i \equiv G \otimes\_i (\not\models \varepsilon) \cdot \not\models \Psi(\not\models \varepsilon) \cdot \not\models \varepsilon^{-1} \tag{182}$$

**Figure 10. Figure 10.**Bode diagram of open-loop current gain. Bode diagram of open-loop current gain.

The voltage controller of the laser PV module was designed to achieve dynamic performances (bandwidth) and stability (phase margin) for a system with a closed current loop, as shown in Equation (13). The designed PI voltage controller is expressed as shown in Equation (14); Equation (15) shows

*Electronics* **2020**, *9*, 1745 the equation of the open-loop voltage gain when the designed voltage controller is applied. As shown from the voltage loop gain of Figure 11, even when the operating point of the laser PV module has changed, the voltage controller maintained a bandwidth of 400–700 Hz, and the phase margin is designed to exceed 70◦. The voltage controller of the laser PV module was designed to achieve dynamic performances (bandwidth) and stability (phase margin) for a system with a closed current loop, as shown in Equation (13). The designed PI voltage controller is expressed as shown in Equation (14); Equation (15) shows the equation of the open-loop voltage gain when the designed voltage controller is applied. As shown from the voltage loop gain of Figure 11, even when the operating point of the laser PV module has changed, the voltage controller maintained a bandwidth of 400–700 Hz, and the phase margin is designed to exceed 70°. 

$$\frac{V\_{sa}V\_{su}}{V\_c} \overset{V\_{su}}{\underset{V\_c}{\mathbf{V}\_c}} \mathbf{G}\_{\mathbf{v}\mathbf{M}\xi}(\mathbf{z}) \equiv \frac{G\_{\mathbf{G}}(\mathbf{z})(\mathbf{z}\mathbf{f})\mathbf{f}(\mathbf{z})(\mathbf{z})}{^1\mathbf{1}^T + ^tT\_i} \tag{159}$$

$$W^v(\mathbb{R})^{(\pm)\text{-}0\text{-}9\text{-}9\frac{\text{J}\cdot\text{J}\cdot\text{J}}{(\mp - 0.0\text{-}9\text{-}9\text{-}9)}}\tag{144}$$

$$T\_{\mathbb{Z}}\mathbb{T} \equiv \mathbb{G}\_{\mathbb{H}}(\mathfrak{f}(\mathfrak{z})\mathbb{Y}\mathfrak{h}(\mathfrak{z})\,\tag{135}$$

**Figure 11.** Board diagram of voltage loop gain of two-loop control. **Figure 11.** Board diagram of voltage loop gain of two-loop control.

#### *2.3. MPPT Algorithm Design 2.3. MPPT Algorithm Design*

In this study, the incremental conductance method with variable amplitude was applied to track the maximum power point based on the output power of the laser light source [15]. This method compares the load impedance with the impedance of the laser PV module and controls the voltage of the PV module to correspond to the maximum power point. As shown in Figure 12, when the output of the PV module is located to the left of the maximum power point, the power increases with the voltage. Conversely, when it is located to the right of the maximum power point, the power decreases with the increase in the voltage. This relationship is expressed as shown in Equations (16)–(19). In this study, the incremental conductance method with variable amplitude was applied to track the maximum power point based on the output power of the laser light source [15]. This method compares the load impedance with the impedance of the laser PV module and controls the voltage of the PV module to correspond to the maximum power point. As shown in Figure 12, when the output of the PV module is located to the left of the maximum power point, the power increases with the voltage. Conversely, when it is located to the right of the maximum power point, the power decreases with the increase in the voltage. This relationship is expressed as shown in Equations (16)–(19).

Therefore, the voltage command of the laser PV module can be positioned as the maximum power point by measuring and comparing the increment for the conductance of the laser PV module and the instantaneous resistance value. At this time, the fluctuation value of the command value was set as a variable voltage such that when the operating point was far from the maximum power point, it rapidly converged to the maximum power point, and in the vicinity of the maximum power point, the periodic vibration width reduced compared with using a fixed value. As described above, Figure 13 shows the flow chart of the MPPT algorithm with variable amplitude based on the incremental conductance method. Therefore, the voltage command of the laser PV module can be positioned as the maximum power point by measuring and comparing the increment for the conductance of the laser PV module and the instantaneous resistance value. At this time, the fluctuation value of the command value was set as a variable voltage such that when the operating point was far from the maximum power point, it rapidly converged to the maximum power point, and in the vicinity of the maximum power point, the periodic vibration width reduced compared with using a fixed value. As described above, Figure 13 shows the flow chart of the MPPT algorithm with variable amplitude based on the incremental conductance method.

$$\frac{dP}{dV} \underset{\overline{dP}}{\equiv} \frac{d(IV)}{d\hbar\{IV\}} \equiv \frac{I+V}{I+V} \frac{dI}{d\hbar V} \cong I + \frac{V}{\Delta I} \frac{\Delta I}{\Delta V} \tag{16}$$

$$
\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\cdots}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
}}}}}}}}}
$$
}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

$$\begin{array}{c c c} \stackrel{\cdots}{\Delta V} = -\frac{1}{V} & \text{at } \text{MPP} \\ \stackrel{\Delta V}{\Delta V} = \frac{1}{V} & \text{at } \text{MPP} \end{array} \tag{17}$$

$$\frac{\Delta I}{\Delta V} > -\frac{I}{V} \stackrel{\text{def}}{\longrightarrow} \begin{cases} V & \text{un} \\ \end{cases} \tag{18}$$

$$\text{1\textsuperscript{\Delta I}} \bigd\_{\Delta V} \bigd\_{\Delta V} \supset \bigd\_{\widehat{V}} \bigd\_{V} \bigd\_{V} \text{ 1\textsuperscript{\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft}}}}}}}} \text{\tiny}\_{V} \text{ 1\textsuperscript{\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\tinyleft\text{\textPi}}}}}}}}} \dots} \dots \}} \dots \} $$

Δ*I* Δ*V* < − *I V right o f MPP* (19) ' '*, , ULJKW RI 033 9 9* (19) ' '*, , ULJKW RI 033 9 9* (19) 

**/DVHU390RGXOHYROWDJH /DVHU390RGXOHYROWDJH**

**Figure 12.** Characteristic curve of the laser PV module. **Figure 12.** Characteristic curve of the laser PV module. **Figure 12.** Characteristic curve of the laser PV module. 

**Figure 13.** Algorithm flow chart of an incremental conductance method with variable amplitude. **Figure 13.** Algorithm flow chart of an incremental conductance method with variable amplitude. **Figure 13.** Algorithm flow chart of an incremental conductance method with variable amplitude.

#### **3. Simulation and Experimental Results 3. Simulation and Experimental Results 3. Simulation and Experimental Results**

#### *3.1. Simulation Result 3.1. Simulation Result 3.1. Simulation Result*

The method proposed herein was first verified using the simulation model shown in Figure 14. In the simulation model, the laser module was modeled with a diode model-based equation [7,16]. The laser PV module output current can be described as shown in Equations (20)–(22) in Figure 15a, where *Isc* is the short circuit current of the laser PV module, *Io* is the reverse saturation current of the diode, *q* is the electronic charge, *vd* is the diode voltage, *K* is the Boltzmann constant, *T* is the temperature in Kelvin, *n* is the ideality factor, *Rsh* is the shunt resistance, *Rs* is the series resistance, *voc* is the open circuit voltage of the cell, and *Ns* is the serial number of cells constituting the module. Figure 15b shows the modeling of the laser PV module implemented in Matlab/Simulink using Equations (20)–(22). Illumination is a variable representing the laser power intensity irradiated to the laser PV module and has a value of 0 to 1 normalized to 2.874 W/cm2. The method proposed herein was first verified using the simulation model shown in Figure 14. In the simulation model, the laser module was modeled with a diode model-based equation [7,16]. The laser PV module output current can be described as shown in Equations (20)–(22) in Figure 15a, where *Isc* is the short circuit current of the laser PV module, *Io* is the reverse saturation current of the diode, *q* is the electronic charge, *vd* is the diode voltage, *K* is the Boltzmann constant, *T* is the temperature in Kelvin, *n* is the ideality factor, *Rsh* is the shunt resistance, *Rs* is the series resistance, *voc* is the open circuit voltage of the cell, and *Ns* is the serial number of cells constituting the module. Figure 15b shows the modeling of the laser PV module implemented in Matlab/Simulink using Equations (20)–(22). Illumination is a variable representing the laser power intensity irradiated to the laser PV module and has a value of 0 to 1 normalized to 2.874 W/cm2. The method proposed herein was first verified using the simulation model shown in Figure 14. In the simulation model, the laser module was modeled with a diode model-based equation [7,16]. The laser PV module output current can be described as shown in Equations (20)–(22) in Figure 15a, where *Isc* is the short circuit current of the laser PV module, *Io* is the reverse saturation current of the diode, *q* is the electronic charge, *vd* is the diode voltage, *K* is the Boltzmann constant, *T* is the temperature in Kelvin, *n* is the ideality factor, *Rsh* is the shunt resistance, *Rs* is the series resistance, *voc* is the open circuit voltage of the cell, and *Ns* is the serial number of cells constituting the module. Figure 15b shows the modeling of the laser PV module implemented in Matlab/Simulink using Equations (20)–(22). Illumination is a variable representing the laser power intensity irradiated to the laser PV module and has a value of 0 to 1 normalized to 2.874 <sup>W</sup>/cm2.

$$I\_s = I\_{s\varepsilon} - I\_o \left( \begin{matrix} \frac{\partial \mathcal{V}\_{\text{eff}}}{\partial \mathcal{V}\_{\text{eff}}} \frac{\partial \mathcal{V}\_{\text{eff}}}{\partial \mathcal{V}\_{\text{eff}}} \end{matrix} \right) + \frac{\mathcal{V}\_d}{\mathcal{V}\_{\text{eff}}^{\text{V}\_{\text{eff}}}} \frac{\mathcal{V}\_d}{\mathcal{V}\_{\text{eff}}^{\text{V}\_{\text{eff}}}} \tag{20}$$

$$I\_o = \left( I\_{s\mathcal{L}} - \frac{\bigvee\_{s\mathcal{L}}}{R\_{sl}} \right) / \left( e^{\frac{qv\_{s\mathcal{L}}}{nkT}} - 1 \right) \tag{21}$$

**Table 1.** Parameters of laser PV module in simulation model. **Table 1.** Parameters of laser PV module in simulation model.

**Figure 15.** Modeling of laser PV module: (**a**) equivalent circuit model (**b**) simulation model in Matlab. **Figure 15. Figure 15.** Modeling of laser PV module: ( Modeling of laser PV module: (**<sup>a</sup>a**) equivalent circuit model ( ) equivalent circuit model (**bb**) simulation model in Matlab. ) simulation model in Matlab.

(**a**) (**b**)

The parameters used for modeling the laser PV module are shown in Table 1. In the parameters of the laser PV module, *Isc* and *Voc* were set as the short-circuit current and open-circuit voltage values measured in the experiment, and the *Rs*, *Rsh*, and *n* values were extracted by the trial and error method through the simulation model in Figure 15b. In order to extract more accurate parameters of the laser PV module using an optimization algorithm, the method mentioned in Sheng and Anani's articles can be applied [17,18]. The accuracy of the simulation model compared with the experimental data of the module mentioned in Section 2 is shown in Figure 16. As can be seen in Figure 16a, the root mean square error (RMSE) of the simulation and experimental results is shown as 0.0027 or less, and the simulation model reflects the experimental results well.


**Table 1.** Parameters of laser PV module in simulation model.

**Figure 16.** Characteristics of laser PV module using simulation model: (**a**) V–I characteristics comparison between the simulation model and experimental results; (**b**) V–P characteristics of the simulation model. **Figure 16.** Characteristics of laser PV module using simulation model: (**a**) V–I characteristics comparison between the simulation model and experimental results; (**b**) V–P characteristics of the simulation model.

The power circuit of the boost converter for charging a 24 V battery was modeled using PLECS software [19] and implemented in the PLECS circuit block, as shown in Figure 14. In the boost converter, the battery was modeled with a capacitor and an ESR, and the value was set to 2 F, which afforded a smaller capacity than the actual battery to reduce the simulation time. The parameter values of the components applied to the simulation are summarized in Table 2. The power circuit of the boost converter for charging a 24 V battery was modeled using PLECS software [19] and implemented in the PLECS circuit block, as shown in Figure 14. In the boost converter, the battery was modeled with a capacitor and an ESR, and the value was set to 2 F, which afforded a smaller capacity than the actual battery to reduce the simulation time. The parameter values of the components applied to the simulation are summarized in Table 2.


**Table 2.** Experimental setup of the laser charging system.

Figure 17 shows the battery charging control result through the MPPT operation when the laser power was 2.874 W/cm2. The converter operated in the MPPT mode because the battery voltage has not reached the 24.5 V set as the full-charge voltage. The sampling period of the MPPT algorithm was set to 0.1 s to reduce the simulation time, and it was observed that the input voltage set point was generate to track the 7.5 V point in the MPP. Since the converter was well controlled by the voltage command, it can be confirmed that the battery was charged with the maximum power generated not reached the 24.5 V set as the full-charge voltage. The sampling period of the MPPT algorithm was set to 0.1 s to reduce the simulation time, and it was observed that the input voltage set point was generate to track the 7.5 V point in the MPP. Since the converter was well controlled by the voltage command, it can be confirmed that the battery was charged with the maximum power generated from the laser PV module. *3.2.Results*

from the laser PV module.  *Experimental* 

Figure 18 shows the experimental configuration of a battery charging system using a laser beam. The laser light source was configured to be generated by passing a multilens array at the rear end of the MFSC-200 device with a wavelength of 1080 nm. The power generated from the laser PV module was supplied to the battery after passing through the prototype boost converter, and the electronic load was connected in parallel to the battery for the load test.

Figure 18 shows the experimental configuration of a battery charging system using a laser beam. The laser light source was configured to be generated by passing a multilens array at the rear end of the MFSC-200 device with a wavelength of 1080 nm. The power generated from the laser PV module was supplied to the battery after passing through the prototype boost converter, and the electronic load was connected in parallel to the battery for the load test. **Figure 17.** Simulation result of charging battery to the maximum power point. *3.2. Experimental Results* 

**Figure 17.** Simulation result of charging battery to the maximum power point. **Figure 17.** Simulation result of charging battery to the maximum power point. Figure 19 shows the main configuration of the prototype boost converter system, where a TMS320f28069 MCU was applied to communicate with the UAV's host controller, perform MPPT functions, and control the converter. Table 2 shows the experimental configuration of the laser wireless power transmission system and the main specifications of the manufactured converter. Figure 18 shows the experimental configuration of a battery charging system using a laser beam. The laser light source was configured to be generated by passing a multilens array at the rear end of the MFSC-200 device with a wavelength of 1080 nm. The power generated from the laser PV module was supplied to the battery after passing through the prototype boost converter, and the electronic load was connected in parallel to the battery for the load test. 

the PV module, and the battery charging device was operated. The activated converter performed a self-diagnostic verification and operated after 5 s ("BCR On") when no faults were encountered.
