2.3. Mathematical and Practical Proof of Feasibility of the Method
As explained previously, in the proposed method, UGR table numbers or their 2D interpolation or extrapolation are used as the UGR for the observer’s position on the room’s midline. Therefore, it is assumed that UGR for an observer at any point on a room’s midline is equal to UGR for an observer in a room with size equal to the space in front of the first observer when the second observer is placed in the middle of one wall looking toward the opposite-side wall in a smaller room. For proving the validity of idea, it should be proved that UGRs obtained for both observers are adequately close to each other.
To start the proof, it is necessary to refer to Equation (1), the well-known CIE 117:1995 formula for UGR calculation [
8]. The formula takes into account all but one of the variables which are directly related to the observer’s field of view. Therefore, except for one variable, all variables—
as the luminance of the luminaire i,
as the solid angle of the luminaire
i, and
as the Guth position index of the luminaire
i—are directly related to the luminaires which are in the observer’s sight. The only parameter which considers the effect of luminaires out of the observer’s sight is
Lb, the background luminance. Therefore, for two mentioned observers in two different rooms, it is enough to prove that the background luminances are close to each other such that when using them in Equation (1), the resulted UGRs are reasonably close to each other.
Consider two mentioned observers. One is in a room in a position which places some of the luminaires in front of them (in their field of view), and some of the luminaires behind them (
Figure 4a). The other observer exists in a room containing only the luminaires which are in the first observer’s field of view. The nearest wall behind the second observer is at a distance of H/2 from the first row of the luminaires in front of them (
Figure 4b). The arrangement of the luminaires is as the standard arrangement explained in CIE 117:1995 [
8].
For UGR calculation in the two rooms illustrated in
Figure 4a,b, Equation (1) can be stated as:
where the parameters with indices a and b correspond to rooms a and b as illustrated in
Figure 4a,b.
It should be proved that:
Therefore, from Equations (2)–(4):
As a result, by proving:
the equality in Equation (6) will be proved.
As can be deduced from
Figure 4a,b, since there are same luminaires in both observers’ fields of view in the mentioned figures, Equations (7)–(9) are true.
However, Lb in Equation (1) is the only variable which is not related only to the luminaires in the observer’s field of view. Therefore, the aim is to prove that the values of Lb for both of the rooms and observers’ positions are reasonably close to each other. For this aim, the range of changes in background luminance will be obtained based on observers’ position changes; we also examine whether background luminance change can be ignored.
In this research, the CIE 190:2010 [
31] method for
Lb calculation will be used:
where,
EWID is an indirect component of the illuminance on the wall produced by luminaires. In turn,
EWID is:
where
FUWID is the indirect utilization factor for walls,
N is the number of luminaires,
AW is the total area of walls (m
2) (
W stands for wall) between the reference plane and the luminaires’ installation plane, and Φ
0 equals 1000 lm (reference luminous flux) when the luminaire’s luminous intensity distribution is given in cd per 1000 lm. It should be noted that in case of giving luminous intensity distribution based on luminaire luminous flux, the luminous intensity distribution numbers should be divided by Φ
luminaire/1000 to change the luminous intensity unit to cd per 1000 lm, which makes the use of Equation (12) possible.
As the observer moves toward the opposite-side wall in the main (first) room, the size of the smaller (second) room corresponding to the observer’s position in the first room (In the second room, the observer’s distance from the opposite-side wall is the same as the first observer, but the nearest wall behind the second observer is exactly H/2 distance from the first row of the luminaires in front of them) changes as illustrated in
Figure 4a,b, and the length of the second room in the first observer’s path decreases. Meanwhile, the number of luminaires in both observers’ fields of view decreases.
If there is a room with dimensions of XH × Y
LH (the main room) and an observer moves in Y direction, it can be said when the observer in the main room is in the position (Y
0 + ΔY)H (distance from opposite-side wall), and for H = 2 m, the number of luminaires effecting background luminance would be:
and the area of wall which reflects background luminance is:
where Y
0 is the smallest distance of the observer from the opposite-side wall, ΔY is added to Y
0 for obtaining the Y corresponding to the observer’s position, and Y
L is the length of the main room (Y and X should be multiplied by H for obtaining dimensions in meters).
For the corresponding smaller room:
and
By putting aside Aw, N, and Φ0, the remaining part of Equation (12) is related to the indirect utilization factor of the walls, which is the next part of the formula to be considered.
Indirect utilization factor for walls is [
31]:
In Equation (17), distribution factors
FDF,
FDW, and
FDC are defined as [
31]:
FDF and FDW are derived from ΦZL, which is the sum of the cumulative CIE zonal fluxes multiplied by the geometric factor values for luminaires in the standard array. Geometric factors are related to the room’s dimensions. Therefore, they change in small rooms when the observer’s position changes. However, FDC is derived from RULO, which is only related to the luminaire’s luminous intensity specifications. Then, it is independent of the observer’s position in the room. Based on changing parts of FUWID as the observer’s position changes, its range of change will be investigated as follows.
Φ
ZL as an important part of
FUWID is expressed as:
ΦZL1, ΦZL2, ΦZL3, and ΦZL4 are only related to each luminaire’s luminous flux properties and are not related to the observer’s position or the room’s dimensions. Then, they are independent of the observer’s position in the room.
By rewriting
FUWID formula (Equation (17)) considering the change in the observer’s position:
Following that, for finding the changing range for FUWID, the changing range for the right side of Equation (22) will be investigated in detail.
For
FDF, on the right side of Equation (22), it can be written:
As previously mentioned, in Equation (23), Φ
ZLns are related to the luminaires’ specifications and do not change when the room’s specifications change. Geometric factors (
FGLns) are related to the room’s dimensions and ranges of their changing with room dimensions are illustrated in
Table 1.
From
Table 1, it can be understood that when a room’s Y dimension increases,
FGL1 and
FGL2 decrease and
FGL3 and
FGL4 increase. From each row of
Table 1 and considering Equation (21), it can be deduced that for luminaires whose Φ
ZLn values are close to each other, the increasing and decreasing of related geometric factors can cancel each other and CIE zonal flux (Φ
ZL) can change little when a room’s dimensions change. Knowing that, and also since in Equation (18),
FDF is CIE zonal flux (Φ
ZL) divided by 1000 lm,
FDF change based on changes in room dimensions is not prominent.
From Equation (19) and keeping in mind that R
DLO is not dependent on room dimensions:
Consequently, FDW change which is equal to FDF change but with a negative sign is not also prominent.
Since luminaires’ RULO value is usually negligible, the FDC also can be assumed to be a negligible value. Therefore, FDF and FDW are the only distribution factors influencing the change of the indirect utilization factor of the walls.
Table 2 illustrates a range of transfer factors’ change according to the change in a room’s dimensions.
Validity of using data illustrated in
Table 1 and
Table 2 requires maintaining a trend of change for geometric and transfer factors linearly, both inside and outside of the studied range (X = 2H, 4H, 8H, and 12H, Y = 4H, 6H, and 8H). Therefore, linear regression was applied to trends of mentioned factors for inside and outside of the studied range where data were available, and R
2-values of regression for all factors for a common range are illustrated in
Tables S3 and S4. Based on data illustrated in
Tables S2 and S3, the average of R
2-value for linear regression of geometric factors is 0.8701 and for transfer factors is 0.9651. The mentioned R
2-values show acceptable linear correlation between room dimension in the Y direction and geometric and transfer factors. It shows the possibility of generalization of the results obtained in the studied dimension range for outside the range.
From Equation (22), the average change of the indirect utilization factor for walls based on the change of distribution and transfer factors can be stated as Equation (25):
From Equations (19), (20), and (25):
Since, in many luminaires, luminous intensity in the upper hemisphere is near zero, R
ULO also equals zero. Therefore, Equation (26) would be:
By expanding Equation (27):
Multiplication of two differential values makes very small values. Therefore, the terms including multiplication of two differential values are removed from Equation (28):
By subtracting
FUWID(Y
0) obtained by Equation (17) from
FUWID(Y
0+ΔY) in Equation (29):
Previously, it was concluded that Δ
FDF does not have a prominent value, hence by removing the terms including Δ
FDF in Equation (30):
By considering
Table 2, it can be deduced that:
and
From Equations (18) and (21) and by investigating luminaires with their luminous intensity distribution in the lower hemisphere around the luminaire, F
DF can be expressed as follows:
Usually in luminaires with uniform luminous intensity distribution in bottom hemisphere:
Also, by estimation it can be assumed that Equation (35) for F
DF calculation can be expressed as:
From Equation (36) it can be concluded that:
Hence, from Equations (37) and (38):
Also, R
DLO and
FDF are numbers smaller than 1. Hence, from Equations (32)–(34), and (39):
According to Equation (12), the indirect component of the illuminance on the walls in terms of its change based on room dimension change would be:
for a sample dimension changing range (8 m to 16 m) in both directions of X and Y can be seen in
Figure 5.
From
Figure 5 it can be deduced that the average change of
in the considered dimension change range is 0.125. Consequently, Equation (41) would be:
For finding the change of the indirect component of the illuminance on the walls according to the change in the room’s dimensions:
The right side of Equation (44) can be rewritten as:
Therefore, average change in
EWID for a ±4 m change in observer position would be:
From Equation (17), transfer factors
FT,FW,
FT,WW-1, and
FT,CW and distribution factors
FDF,
FDW, and
FDC are used for obtaining the indirect utilization factor for walls (
FUWID). The average of transfer factors for usual calculations as mentioned in Table 5 in CIE 190:2010 are 0.135, 0.251, and 0.371, for
FT,FW, FT,WW-1, and
FT,CW, respectively.
FDF and
FDW are smaller than one and
FDC for luminaires with their luminous intensity distribution in lower hemisphere is approximately zero. Therefore:
or
Since, as shown in Equation (11),
is proportional to background luminance average, from Equation (46) it is known that for finding average change in background luminance,
and
are needed. Also, as mentioned before, R
DLO is smaller than one and an average value of 0.5 can be considered for it. Therefore, from Equation (39) and average value of R
DLO, in Equation (39)
would be equal to 0.225. Hence,
in Equation (48) would be equal to 0.099. Then,
, based on average values of
N/AW and
FUWID in the studied range, would be:
Ultimately, the ratio of background luminance change to its absolute value is:
Therefore, it is concluded that for an observer position change (or room dimension change) from Y = 16 m to 8 m, background luminance change average is 3.67% for average values of R
DLO = 0.5 and
FDF = 0.225. This value for background luminance change can lead to UGR change as:
By applying Equations (7)–(9) to Equation (51):
Therefore, on average, while an observer moves from a distance of 16 m from the opposite-side wall toward it, at distance of 8 m, only an approximate change in UGR of 0.13 units occurs for them. This change is far from the 3 UGR units of change which is perceptible according to the Hopkinson criterion. Remembering that previously it was shown that geometric and transfer factors have linear correlation with dimension change, the obtained result can be generalized and extrapolated for dimensions greater or smaller than values studied in this proof.
For evaluating the difference between the background luminance in the main room and the corresponding smaller room in another manner, the effect of a change in observer position on the background luminance for rooms with widths of 24 m, 16 m, and 4 m, and lengths of 24 m, 4 m, and 2 m () is investigated both by manual calculation and simulation. The simulation has been performed by DIALux software. The mentioned maximum and minimum dimensions for X and Y are approximately the minimum and maximum values which are usually used in issued tables for UGR calculation. X changing is related to the width of the observer’s path of movement. Small and large values of X correspond to narrow and wide pathways, respectively. This can give a view of the effect of the distance between side walls on the indirect light reflectance of the wall surfaces. Y value is the distance between the observer and the opposite-side wall.
Therefore, this comprehensive study on usual dimensions in UGR tables can provide a good view of the range of difference between background luminance in the main room and the smaller rooms corresponding to the observer’s position.
For mentioned Xs and Ys, distribution factors (
FDF,
FDW, and
FDC), and consequently the indirect utilization factor for walls from Equations (17)–(21), can be obtained. The obtained distribution factors are illustrated in
Supplementary Table S4 for the luminaire used for comparing background luminance in main and corresponding smaller rooms.
The next step in obtaining background luminance based on Equations (11) and (12) is to find
N/AW. According to Equations (13) and (14) for the main room:
For X and Y = 1, 2, 12, based on Equations (15) and (16) for the corresponding smaller room:
Therefore, for X = 12, and Y
0 + ΔY = 1, 2, 12:
And, for X = 8, and Y
0 + ΔY = 1, 2, 12:
Also, for X = 2, and Y
0 + ΔY = 1, 2, 12:
Then, having distribution and transfer factors,
N/AW, and Φ
0 = 1000 lm, values for
EWID and in turn,
Lb (background luminance) for
can be obtained. The background luminances for mentioned Xs and Ys obtained by manual calculation and simulation for different types of luminaires are illustrated in
Table 3 and
Table 4, respectively. In the mentioned tables, for each X as the observer’s path width, deviation of background luminance for Y = 1 and 2 from background luminance at the start point of path (Y = 12) is also illustrated as a percentage of background luminance at the starting point.
Due to what is mentioned in CIE 117:1995 about insensitivity of the UGR to the errors produced in
Lb calculation (in which +33% error in
Lb leads to only 1-unit error in UGR), by using data in
Table 3 for manual calculation and
Table 4 for simulation of
Lb, it can be deduced that starting from a distance of 24 m from the opposite-side wall, for the manual calculation example ending at a distance of at least 4 m and for the simulation ending at a distance of less than 2 m from the opposite-side wall, the UGR calculation error is limited to only 1 unit, which is far from the 3 units of UGR introduced in the Hopkins criterion for perceptible UGR difference.
Also, for the previously obtained range of UGR change for and for an average value of RDLO equal to 0.5, the UGR change was obtained as 3.67%, which corresponds to 0.13 units of UGR error.
Hence, in addition to
Li,
ωi, and
pi, which are totally independent of the luminaires and space out of an observer’s sight, background luminance also has relatively small changes according to the number of luminaires and change in dimensions of the space both inside and outside the observer’s sight. Therefore, in addition to Equations (7)–(9), it was proved that Equation (10) is also true with approximation and it can be assumed that UGR obtained for the main room and corresponding smaller room illustrated in
Figure 4a,b are close enough and UGR table can be used for points on the room’s midline.
More practical samples for evaluating the feasibility of the method will be examined in
Section 2.4.
The tabular method was originally designed for UGR calculation in the middle of one wall in a room. When using it at a point inside a room on the room’s midline other than wall midpoints, it should be shown that for a smaller room size corresponding to the space in front of the observer in a larger room when observer’s location is other than the room’s margins, by using Equation (1), the obtained UGR for smaller room is close enough to the real UGR for the observer in the larger room. Therefore, the only difference between the parameters of the main and corresponding rooms exists in background luminance. Then, by showing acceptable difference between the two background luminances, it was shown that the proposed tabular method is applicable for an observer’s location on the room’s midline other than the wall midpoints.
All of these lead to the valid use of UGR table for changes in an observer’s position in a room instead of its conventional use for observers in the middle of one wall of the room. Also, since the proposed method has been proved mathematically, repeatability of the method’s results is obvious.
Since the proposed method uses the UGR table introduced to CIE standards [
8], the limitations of the UGR calculation method in CIE 117:1995 such as limitations for indirect lighting, luminaires’ dimensions, luminous intensity distribution uniformity [
4], etc. are applied to the proposed method. Iacomussi et al. suggested that initiatives in formulations regarding investigation and evaluation of solid-state light source (SSL) discomfort glare is necessary. They also concluded that existing criteria for discomfort glare evaluation can be applied to LED luminaires provided that they are similar to conventional fluorescent luminaires in luminous intensity and luminance distribution. They recommended equipping SSLs with diffusers [
7]. Additionally, position index is used in UGR calculation and was developed by examining light sources with uniform luminous intensity; it appears that it is not suitable for luminaires without a set circular geometry and a constant spectrum. In addition, the results of studies on position index of luminaires with non-uniform luminous intensity distribution were contradictory. Furthermore, the form and structure of modern luminaires make it problematic to calculate the luminance needed for UGR calculation from luminous intensity distribution and apparent area of the luminaire. It is not usually easy to specify an apparent area for LED luminaires, particularly for those with complex retrofits [
4]. The aforementioned limitations make applicability of the UGR method for evaluation of discomfort glare for LED luminaires vague.
Here, by performing case studies, the proposed method will be examined.