4.1 Optimisation of LED mixtures for LER and CRI (CIE Ra)

This investigation [26] explored the optimisation of white-light mixtures of seven LEDs (four at a time) from the Lumileds Luxeon™ range [30] and it is therefore classed as constrained. The objective here was to achieve the best available combination of Ra with LER, and the fitness function for DE was defined as in Eq. 4:

$$f\_{fit} = aR\_a + bR\_b + cR\_c + d\_{rad} + eR\_{min} \tag{4}$$

It is worth noting that Ra ≥ 90 is generally considered as excellent colour performance, and LER ≥ 300 lm/W exceeds that of many other (efficient)

Relative SPD of the 4-band LED mixture (Test 1). Wavelength peaks for: blue = 460 nm, green = 530 nm, amber = 590 nm, red = 630 nm. Source parameters: Ra = 95, Rb = 86, ηrad = 339 lm/W, CCT = 3268 K,

Table 1 shows that significant step decrements in the weight e (the Rmin weighting) led to fairly small losses in the colour performance of the mixtures, but

Figures 4 and 5 show how the individual elements of ffit in Test 1 were

during the early stages of the process (G < 300). It then converges to just below 340 lm/W at G = 336. In this instance, the population has converged around G = 400, and all parameters are only fine-tuned thereafter. These curves indicate that the process could, in fact, have been terminated at G = 500, with low likelihood of loss.

4.2 Optimisation of mathematical mixtures for LER and CRI (CIE Ra)

of SPD mixtures defined by the following mathematical functions:

It can be seen that ηrad (Figure 5), and hence also ffit, undergoes significant swings

It was decided to extend the above approach to an unconstrained investigation

The optimiser was free to position four spectral bands, conforming to any one function, at any wavelength within the visible band (380–760 nm) and having any relative intensity (i.e., height) of one to another. One constraint was imposed for practical purposes, namely each band was kept to a spectral width between 25 and

to noticeable gains in LER (and hence, efficiency in use).

Test no Weighting factors Best results

Influence of different ffit weights on the optimization of LED mixtures.

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

abc d e Ra Rb Rc ηrad (lm/W) Rmin imin CCT (K)

1 1 1 1 1 10 95 96 91 339 76 12 3268 2 1 1 1 1 5 93 83 89 350 76 12 3074 3 1 1 1 0.5 0 92 79 87 384 58 9 3169

fluctuating before converging to their final values.

light sources.

R12 = 76 (blue sample).

Figure 3.

Table 1.

• Gaussian

• rectangular

• triangular

47

where Ra = CIE general colour rendering index (based on 8 medium-chroma test colours); Rb = similar figure of merit based on the 6 additional test colours in the CIE method; Rc = similar figure of merit based on all 14 test colours in the CIE method; ηrad = the value of the LER (lumen per watt figure); Rmin = the lowest value of Ri in the set of 14 individual values; a, b, c, d and e are user-selectable weights controlling the influence of Ra, Rb, Rc, ηrad and Rmin on the optimisation process.

Experimentation found that ffit converged around (or before) the 1000th generation; hence, the number of generations G was set to 1000 for fast and accurate convergence of the DE. Note that a too-small G can mean that the process has insufficient opportunity to search for effective solution vectors whereas, for excessive values of G, the optimisation process is unnecessarily slowed down without improving the optimisation result.

The other DE parameters that may influence the operation of the algorithm are P, F, and CR, all of which need to be set prior to any run. We found by experimentation that the values of these parameters had only a minor influence on the optimisation results; and the choice of suitable values for good optimisation was straightforward. We quickly settled on a value of P = 50, and we set F and CR according to the suggestions in [10], that is, F = 0.5 and CR = 0.1.

Further experimentation investigated the effects of different values for the weighting factors for ffit, and some indicative results are shown in Table 1. The relative spectral power distribution (SPD) for the result listed as Test 1 is shown in Figure 3.

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094


#### Table 1.

4. Optimisations

[23, 26–29] for full details.

mix and optimise as above.

intervals, and for lasers at 1 nm.

improving the optimisation result.

46

confined solely to results for 4-band mixing.

We here present, in outline form, the results of optimised spectral designs achieved using DE. Interested readers are referred to previous publications

Computer Architecture in Industrial, Biomechanical and Biomedical Engineering

It is noted at the outset that it is, in principle, possible to create white-light mixtures by the use of as few as two or three wavelengths. Our early experiments showed that these practically always lead to suboptimal mixtures in the sense that either one or both of LER and Xc are unacceptably low (Xc is here used to denote any one of the colour metrics mentioned in Section 3). Further experiments indicated that mixtures of five, six or seven wavelengths gave little or no practical advantage over 4-band mixtures, while adding to the complexity (and possible unreliability) of the light source. For the sake of brevity, therefore, the following descriptions are

Two types of optimisation conditions are described: (i) constrained, in which the

In the optimisations described below, all SPDs for LEDs were computed at 5-nm

This investigation [26] explored the optimisation of white-light mixtures of seven LEDs (four at a time) from the Lumileds Luxeon™ range [30] and it is therefore classed as constrained. The objective here was to achieve the best available combination of Ra with LER, and the fitness function for DE was defined as in Eq. 4:

where Ra = CIE general colour rendering index (based on 8 medium-chroma test colours); Rb = similar figure of merit based on the 6 additional test colours in the CIE method; Rc = similar figure of merit based on all 14 test colours in the CIE method; ηrad = the value of the LER (lumen per watt figure); Rmin = the lowest value of Ri in the set of 14 individual values; a, b, c, d and e are user-selectable weights controlling the influence of Ra, Rb, Rc, ηrad and Rmin on the optimisation process. Experimentation found that ffit converged around (or before) the 1000th generation; hence, the number of generations G was set to 1000 for fast and accurate convergence of the DE. Note that a too-small G can mean that the process has insufficient opportunity to search for effective solution vectors whereas, for excessive values of G, the optimisation process is unnecessarily slowed down without

The other DE parameters that may influence the operation of the algorithm are P, F, and CR, all of which need to be set prior to any run. We found by experimentation that the values of these parameters had only a minor influence on the optimisation results; and the choice of suitable values for good optimisation was straightforward. We quickly settled on a value of P = 50, and we set F and CR

Further experimentation investigated the effects of different values for the weighting factors for ffit, and some indicative results are shown in Table 1. The relative spectral power distribution (SPD) for the result listed as Test 1 is shown in Figure 3.

according to the suggestions in [10], that is, F = 0.5 and CR = 0.1.

ffit ¼ aRa þ bRb þ cRc þ drad þ eRmin (4)

optimiser is presented with a set of known (e.g., commercial) monochromatic spectra, and is required to find the best available mixtures in terms of defined criteria and (ii) unconstrained, when the program is given mathematical descriptions for the shapes of potential monochromatic spectra, which it then proceeds to

4.1 Optimisation of LED mixtures for LER and CRI (CIE Ra)

Influence of different ffit weights on the optimization of LED mixtures.

#### Figure 3.

Relative SPD of the 4-band LED mixture (Test 1). Wavelength peaks for: blue = 460 nm, green = 530 nm, amber = 590 nm, red = 630 nm. Source parameters: Ra = 95, Rb = 86, ηrad = 339 lm/W, CCT = 3268 K, R12 = 76 (blue sample).

It is worth noting that Ra ≥ 90 is generally considered as excellent colour performance, and LER ≥ 300 lm/W exceeds that of many other (efficient) light sources.

Table 1 shows that significant step decrements in the weight e (the Rmin weighting) led to fairly small losses in the colour performance of the mixtures, but to noticeable gains in LER (and hence, efficiency in use).

Figures 4 and 5 show how the individual elements of ffit in Test 1 were fluctuating before converging to their final values.

It can be seen that ηrad (Figure 5), and hence also ffit, undergoes significant swings during the early stages of the process (G < 300). It then converges to just below 340 lm/W at G = 336. In this instance, the population has converged around G = 400, and all parameters are only fine-tuned thereafter. These curves indicate that the process could, in fact, have been terminated at G = 500, with low likelihood of loss.

#### 4.2 Optimisation of mathematical mixtures for LER and CRI (CIE Ra)

It was decided to extend the above approach to an unconstrained investigation of SPD mixtures defined by the following mathematical functions:


The optimiser was free to position four spectral bands, conforming to any one function, at any wavelength within the visible band (380–760 nm) and having any relative intensity (i.e., height) of one to another. One constraint was imposed for practical purposes, namely each band was kept to a spectral width between 25 and

#### Computer Architecture in Industrial, Biomechanical and Biomedical Engineering

Figure 4. Behaviour of Ra, Rb, Rc and Rmin in Test 1.

4.3 Optimisation of LED mixtures for LER and CQS (NIST Qa)

Results for the mixtures of mathematical functions shown in Figure 4.

The following fitness function (Eq. 5) was employed:

of the LED mixtures using various weightings is shown in Table 3.

Weights Optimised spectra

Optimizations of five tetrachromatic LED mixtures (CQS domain).

for every candidate SPD.

Table 2.

Table 3.

49

Figure 6.

triangular ffit = Ra.

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

This was again a constrained optimisation process. It was based on the same set of Luxeon™ LEDs as used in Section 4.1, and also used tetrachromatic mixtures. The optimisation process [23] was also broadly similar to that in Section 4.1, and it used a similar interface, but the software was rewritten to compute the performance of candidate SPDs in terms of CQS parameters, specifically the average colour quality index, Q <sup>a</sup>, and the minimum, Q min (the lowest individual index, Q <sup>i</sup>)

Expt Ra Rb Rc η<sup>R</sup> (lm/W) Rmin imin TC (K) λ<sup>1</sup> λ<sup>2</sup> λ<sup>3</sup> λ<sup>4</sup> G1 98 93 96 228 80 12 7870 456 524 591 671 R1 97 90 94 317 72 12 3542 445 510 570 625 T1 98 95 97 303 84 12 4133 460 525 585 645

Relative SPDs of the mathematical 4-band mixtures. (a) G1, Gaussian; (b) R1, rectangular; (c) T1,

where a, b, c are the weights controlling the influence of Q a, LER and Qmin on the optimisation of the LED mixtures. A selection of the results of the optimisation

a bcQ <sup>a</sup> ηrad (lm/W) Q min CCT (K) 0 0 96 306 93 3638 0 1 96 305 94 3386 0 1 96 298 94 3379 1 0 95 322 91 3452 1 2 84 367 75 5041

ffit ¼ aQ <sup>a</sup> þ bηrad þ cQmin (5)

Figure 5. Behaviour of ηrad in Test 1.

45 nm (full width at half maximum, or FWHM) [27]. Apart from these changes, the procedure was essentially the same as that used in Section 4.1, and the same fitness function was used.

A total of 36 optimisations were performed, using 12 different combinations of ffit weightings for each of the three functions. The following extract from this set of results shows the SPDs obtained with the weightings: a = 1; b = c = d = e = 0. See Figure 6.

It was intended that the unconstrained approach will help to identify those wavelength combinations that will lead to optimum SPDs. The findings relating to the spectra in Figure 6 are summarised in Table 2.

This is too small a sample from which to generalise; however, it is possible to make the following observations. The colour performance of these three mixtures is exceptionally good (other than the high CCT for G1) possibly at the expense of relatively moderate LER values. It is probable that the Gaussian and triangular shapes are more realistic simulations of real physical spectra (whether originating from LEDs or phosphors); hence their corresponding centre wavelengths (λ1–λ4) may be useful indicators for use in light-source design. See [27] for a more complete discussion.

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

Figure 6.

Relative SPDs of the mathematical 4-band mixtures. (a) G1, Gaussian; (b) R1, rectangular; (c) T1, triangular ffit = Ra.


Table 2.

Results for the mixtures of mathematical functions shown in Figure 4.
