## 2 Materials and Methods
### Temperature Compensation and Luminous Flux Regulation  
The luminous flux produced by the LED MP-5050-240E is affected by the operating temperature. Thus, a current compensation is required to ensure a constant luminous flux during the LED operation. The design of this temperature-based current compensation requires to model four relations inside the LED:  
* The voltage vs. current relation. * The luminous flux vs. current relation. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Battery** & **Voltage (V)** & **Capacity (Ah)** & **Cost (USD)** & **Service Time (h)** \\ \hline CA1233 & 12 & 3.3 & \$11.56 & 2 \\ BW 1280 F1 & 12 & 8 & \$24.99 & 5 \\ PC36-12 NB & 10.5 & 38.6 & \$81.75 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example of battery service times per group. Figure 4: **(a)** Comparison of the exact and approximate models, and **(b)** controller performance with both exact and approximate models. * The luminous flux vs. temperature relation. * The temperature variation vs. the operating time. The datasheet of the LED MP-5050-240E [19] provides data for the first three relations. The voltage vs. current relation is modeled using the expression given in (35), where increments in the LED current \(i_{LED}\) (mA) also produce increments in the LED voltage \(v_{LED}\) (V). The luminous flux vs. current relation is modeled using expression (36), where increments in the LED current (mA) also increase the luminous flux \(Fx_{LED}\) (Im) produced by the LED. Figure 5 shows the data extracted from the manufacturer's datasheet and the modeling expressions reported in (35) and (36), where a satisfactory agreement is achieved. \[v_{LED}=0.0013\cdot i_{LED}+5.2936 \tag{35}\] \[Fx_{LED}=1.1685\cdot i_{LED}+23.5714 \tag{36}\]  
The data provided by the manufacturer's datasheet about the reduction in luminous flux caused by the temperature of the LED is normalized. Hence, it is given as a relative luminous flux (\(RFX_{LED}\)) vs. temperature (\(T_{LED}\)) relation. Equation (37) provides a polynomial expression that models such a relative luminous flux (%) for different temperatures (\({}^{\circ}\)C). Finally, the effective luminous flux (\(EFx_{LED}\)) produced by the LED, for different temperatures, is calculated by multiplying the theoretical luminous flux (36) by the relative luminous flux (37), thus resulting in Equation (38). \[RFx_{LED}=-5.5592\times 10^{-6}\cdot T_{LED}^{2}-5.0571\times 10^{- 4}\cdot T_{LED}+1.0154 \tag{37}\] \[EFx_{LED}=Fx_{LED}\cdot RFx_{LED} \tag{38}\]  
Figure 6 shows the \(RFx_{LED}\) data extracted from the Manufacturer' datasheet, moreover the satisfactory agreement of the modeling expression (37) is observed. Then, Equation (38) is used to calculate the effective luminous flux \(EFx_{LED}\) produced by the LED for different current and temperature conditions. Figure 5: LED current, voltage, and luminous flux relations. The temperature variation vs. operation time relation is not reported in the manufacturer's datasheet. However, the work published in [20] reports experimental measurements of changes in thermal resistance during the stabilization time of the LED, for three different commercial LEDs. That work shows the same dynamic behavior for the three LEDs, and the main difference corresponds to the steady-state (final) value, which is equal to the thermal resistance \(R_{th}\) reported in the manufacturer's datasheet. Those experimental data were normalized with the \(R_{th}\) value of each LED to produce a relative thermal resistance behavior \(RR_{th}\) (%), which is modeled by expression (39). This expression is a polynomial calculation based on the logarithm of the operation time, and the maximum value of \(RR_{th}\) is 100%. Finally, the dynamic behavior of the thermal resistance \(DR_{th}\) in the LED MP-5050-240E is obtained by multiplying \(RR_{th}/100\) by the static thermal resistance \(R_{th}=2\ ^{\circ}C/W\) reported in [19], as given in (40).