## 3 A Rule-of-Thumb Model of Road Lighting Illuminance Distributions
### Illuminance Distributions and Modelling in the Transverse Direction  
We simulated illuminance distributions in the transverse direction in a four-lane road with 10 kinds of luminaires including high intensity discharge (HID) lamps and LEDs. The road is 14 m in width, and the luminaires are set at a height of 8 m. The 10 luminaires are different in photometric curves, wattages, and lumen outputs as shown in Figure 6, resulting in different illuminance distributions. Figure 4: Illuminance modelling using sine functions with feature points between two luminaires of single type, double type, and staggered type. The illuminances in brackets correspond to the staggered type. LD1: Longitudinal Direction No. 1, LD2: Longitudinal Direction No. 2, and LD3: Longitudinal Direction No. 3. Figure 5: Example of applying sine model at distances from d\({}_{\text{l}}\) = 151.54 m to 267.68 m in the longitudinal direction of a road. The E\({}_{\text{min}}\) and E\({}_{\text{max}}\) corresponding to the two distances are 1 lx and 15 lx, respectively, and w = 16.14 m. All known values, E\({}_{\text{min}}\), E\({}_{\text{max}}\), w, d\({}_{\text{l}}\), and E\({}_{\text{l}}\), are used in Equation (1), and d\({}_{0}\) is calculated to be 1.37 m.  
The illuminances, generally decreasing from one side beneath luminaires to the other side, are shown in Figure 7. The maximum illuminances mostly appear beneath the luminaires, and the minimum illuminances on the other side of the road. Although the decreasing rates are different for 10 illuminance distributions, they conform to the logistic fit function in Equation (2) with high R\({}^{2}\) from 0.96 to 0.99. The E\({}_{t}\) is the illuminance in the transverse direction, E\({}_{1}\) is the first illuminance, E\({}_{2}\) is the last illuminance, d\({}_{t}\) is the distance in the transverse direction, d\({}_{0}^{\prime}\) is the distance where maximum illuminance of the feature points in the transverse direction decreases to a half, and p is the power. The p value may be varied with decreasing rates of the data. For the 10 logistical fittings in Figure 7, the p values are 3.18 \(\pm\) 0.80, namely p may be set to 3.0 in default. We can apply the least square method in the curve fitting process to obtain a precise p value by virtue of curve fitting toolbox in MATLAB (Mathworks). However, the logistic function works well in the transverse direction when many points (>3) are measured. If only three feature points are measured, the E\({}_{1}\) should be equal to E\({}_{first}\) beneath the luminaire, and the E\({}_{2}\) should be equal to E\({}_{last}\) at the opposite side in Figure 7. The R\({}^{2}\) is decreased. \[\rm E_{t}=\frac{E_{1}-E_{2}}{1+(d_{t}/d_{0}^{\prime})^{p}}+E_{2}. \tag{2}\]  
Figure 6: Photometric curves of the 10 different luminaires. The S_Roads represent the 10 simulated road lighting configurations. S_Roads 1, 2, 3, 4, 5, 9 and 10 are HID lamps, and S_Roads 6, 7 and 8 are LED luminaires. Wattages and lumen outputs: S_Road 1, 259 W & 27,000 lm; S_Road 2, 261 W & 20,500 lm; S_Road 3, 201 W & 20,500 lm; S_Road 4, 153 W & 14,500 lm; S_Road 5, 149 W & 14,500 lm; S_Road 6, 91 W & 10,280 lm; S_Road 7, 108 W & 11,000 lm; S_Road 8, 105 W & 11,690 lm; S_Road 9, 372 W & 20,000 lm; S_Road 10, 253 W & 20,500 lm. Figure 7: Illuminance distributions in the transverse direction (red-dashed line) start from the illuminance beneath the luminaire. The S_Roads represent the 10 simulated road lightings. The logistic function is complex, and the fitting precision decreases if three feature points are applied. We therefore also applied two partitioned linear fittings in the transverse direction of the 10 illuminance distributions as shown in Figure 8. The fitting also reaches a good level of R\({}^{2}\) from 0.85 to 0.99 in the first two lanes of interest, except S_Road 7. For S_Road 7, the maximum illuminance appears at the second point, and the R\({}^{2}\) is only 0.50.