Evaluation of Inrush Current in LED Luminaires Applied to Urban Lighting Based on Multi-Criteria Decision-Making Methods: A Case Study

ABSTRACT

In recent years, a progressive renovation of public lighting has been taking place in cities, from metal halide lamps to LED luminaires. This renovation has been carried out on a large scale by replacing some luminaires with others without taking into account other elements of the circuit. Due to the high number of electronic components, LED luminaires generate inrush currents that can exceed several times the nominal current value at ignition process. This electrical phenomenon is a basic problem to be solved when planning a street lighting renovation with LED luminaires, because it is essential to have a prior estimation of the inrush current generated by the luminaires for a correct sizing of the circuit protection elements. Currently, it is difficult to predict the value of the inrush current, for this reason, the main objective is to establish an innovative methodology based on advanced multi-criteria decision-making methods (entropy and CRITIC) to predict anomalous behaviors (inrush current peaks) in LED luminaire installations and their appropriate selection in the design phase or before are mounted in the circuit to avoid negative consequences on the components and to dimension them correctly, producing cost savings in the replacement of circuit elements such as circuit breakers and switches.

KEYWORDS:

• CRITIC
• entropy
• inrush current
• LED
• MCDM

1. Introduction

Light-emitting diode (LED) lamps are a well-established form of urban lighting due to their economic savings and energy efficiency (Uddin et al. 2013; Gil-de-castro et al. 2017; Gutierrez-Ballesteros et al. 2021). The LED luminaires are mainly composed of an electronic device called driver, which is responsible for regulating the input current and voltage of the LEDs to keep them within their optimal operating range (Unión-Sánchez et al. 2022; Jahkonen et al. 2013). This device will have a major impact on the lamp ignition process due to the design of its electronic architecture. For outdoor lighting applications, the LED drivers are based on passive elements that make up the architectures of AC–DC and DC–DC converters (Ramchandra et al. 2020; Wu et al. 2015). The key element of the driver is the DC–DC converter. An LED luminaire is considered a capacitive load due to the number of capacitors needed for energy accumulation. When a DC–DC converter starts up with a capacitive load connected it creates the inrush current problem at ignition stage (Liu et al. 2014; Hermoso-Orzaez et al. 2018).

The inrush current is the maximal instantaneous current drawn by an electrical device when first turned on (Sadeghi et al. 2018). The duration of this current is 2 ms and can exceed more than 100 times the nominal current of the system (Drgona et al. 2021; Gil-De-Castro et al. 2013). This duration and amplitude have negative influences, such as damages on switches, cables, connectors, and other parts of installation (Nan et al. 2016).

There are different studies related to the prediction and measurement of inrush current in different fields of engineering. First study shows this issue, in electrical transformers as a key problem in transformer protection (Jenner et al. 2014). They are based on the use of an optical current transducer to mathematically model the waveform to obtain exhaustive knowledge of the parameters of the inrush current (Lei et al. 2019). Second study discusses the design idea, construction of the prototype measurement set-up and experimental verification of an automated test stand for transformer inrush current identification (Bonislawski et al. 2018).

In addition to studies of inrush current measurement, there are some studies that analyze the negative effects that inrush current can have if it is not taken into account in the correct dimensioning of system components, for example in the automotive industry, specifically in the use of halogen lamps, where it has been shown that the inrush current generated when turning on a lamp is the main stress shortens the lifetime of lamp (Cho and Kim 2011; Koval et al. 2017). Also continuing in the field of electrical transformers, it has been demonstrated that these currents have undesirable effects, including potential damage to the transformer, reduced power quality on the system, and relay malfunction (Vahidi and Khorsandi 2012; Xi et al. 2018).

Due to the negative effects that inrush current can produce, as mentioned in the previous papers, it has been studied how to limit the current (Madani et al. 2012). There are different elements on which the limitation of the inrush current has been focused, firstly is the AC-DC converter, trying to limit step by step the inrush current of the capacitors used in the electronic architecture (Iuga and Tirnovan 2019; Saathoff et al. 2020) and using a conventional power factor corrector, an auxiliary circuit and additional MOSFET transistor, which is connected in series with capacitor of PFC (Lai and Lai 2013), secondly in the DC–DC converter where proposed soft start-up mechanism for limiting the input inrush current is presented (Kalenteridis et al. 2019). The idea behind the soft start-up mechanism involves skipping of the clock pulses of a Charge Pump, which is used to drive the gate of a MOSFET switch in the input path of the converter (Ballo et al. 2019).

Inrush current can be limited but not eliminated. In this situation, the approximate estimation of this current value takes great importance in the efficient control of this electrical phenomenon (Mo et al. 2022; Majithia et al. 2011). Numerous authors have used statistical methods to estimate the value of the inrush current in different fields of engineering based on high-order statistics (Zhang et al. 2017; Tajdinian et al. 2020) in electrical transformers, and in overcurrent relays, that are one of the most important devices on the electrical power system because they help to eliminate the abnormal currents. This study in relays introduces a new principle of discrimination between inrush currents and short-circuit currents based on Bayesian recursive algorithm. Bayesian rules help combine the probability information to find the most probable prediction (Nguyen et al. 2011) about the current states. Experimental results demonstrate that this method is effective and helps the relay avoid nuisance tripping of short-circuit protection (Nguyen and Nguyen 2020).

The objective of this study is to propose a statistical methodology to estimate the peak inrush current, depending on electronic architecture, of LED luminaires applied to urban lighting. To do this, we will analyze and test five LED luminaires of different years of production and different electrical and electronic characteristics. These tested luminaires were the first types of LED luminaires that were installed in the city of Jaén and are still in operation. Using a network analyzer in the laboratory, the value of the peak inrush current of each luminaire will be measured, and the data will be saved with the analyzer software. Subsequently, these laboratory data will be processed by two statistical decision-making methods, and the final results obtained will be compared. This comparison will allow an approximate prediction of the value of the peak current before the luminaires are installed on the roads, which will lead to economic savings in electrical wiring and in the correct sizing of the system protection elements.

2. Materials and methods

2.1. Instrumentation

2.1.1. Network analyzer

A network analyzer is a device for measuring the properties of electrical networks, especially those related to electrical quality (voltage, current, frequency) and power quality (power consumption, voltage harmonics, current harmonics). The model used in this research is the Fluke 437-Series II.

The main specifications of the network analyzer are shown in Table 1.

Table 1. Table of specifications of network analyzer.

Attribute	Value
Maximum input voltage	1000 Vrms
Maximum peak measurement voltage	6kV
Input impedance	1MΩ
Current measurement range	0,5–200A
Maximum inrush current	5500 A peak
Ampere Peak Factor	1–10
Frequency	42,5–69 Hz
Power Factor	0–1

Source: User’s manual of Fluke 437-Series II.

2.1.2. Current probe

The current probe used, together with the network analyzer, is the Fluke i430 Flexi-TF probe, whose technical specifications are detailed in the following Table 2.

Table 2. Table of specifications of current probe.

Attribute	Value
Current range	6000A AC RMS
Voltage output (at 1000ARMS, 50 Hz)	86,6 mV
Accuracy	±1%
Output impedance	82 Ω
Load impedance	50MΩ
Internal resistance	10,5 Ω
Bandwidth	10 Hz-7 kHz
Operating voltage	1000 V AC RMS

Source: User’s manual of Fluke i430 Flexi-TF probe.

2.2. Devices under test (DUT)

Five LED luminaires are applied to urban lighting, of different production years, and therefore different technologies and electronic components will be tested.

2.2.1. GL1A modular LED luminaire

This is a LEDUS LED luminaire, manufactured in 2014, with a rated power of 60 W and a power factor of 0.95. The LED source of this luminaire is Philips-Lumileds REBEL ES. This luminaire features an INVENTRONICS LED driver model EUC-060S105SV-HP04 of 60 W operating from 90 to 305 V and whose characteristics are presented in the following Table 3.

Table 3. Table of specifications of Inventronics 60 W LED driver.

Attribute	Minimum Value	Maximum Value
Input voltage	90 V	305 V
Frequency	47 Hz	63 Hz
Input current AC	0,36A	0,8A
Peak inrush current	-	60A
Power factor	0,95	-

Source: Inventronics catalog.

2.2.2. T1A modular LED luminaire

This is a LEDUS LED luminaire, manufactured in 2014, of 80 W nominal power, with a power factor of 0.95. As in the previous model, the LED source is Philips-Lumileds REBEL ES. This luminaire also features an INVENTRONICS brand LED driver, model EUC-085S070SV-HP04 of 85 W operating from 90 to 305 V, whose main characteristics are as follows Table 4.

Table 4. Table of specifications of Inventronics 85 W LED driver.

Attribute	Minimum Value	Maximum Value
Input voltage	90 V	305 V
Frequency	47 Hz	63 Hz
Input current AC	0,5A	1,1A
Peak inrush current	-	60A
Power factor	0,95	-

Source: Inventronics catalog.

2.2.3. Philips mini iridium LED luminaire

It is a Philips LED luminaire of 20 W nominal power, manufactured in 2014, from the Iridium product family, with a power factor of 0.95. This luminaire features a PHILIPS LEDSTREET’4 441137071190 driver, 40 W nominal power, which operates between 220 V and 240 V. The characteristics of the driver are summarized in the following Table 5.

Table 5. Table of specifications of LEDSTREET4 LED driver.

Attribute	Minimum Value	Maximum Value
Input voltage	220 V	240 V
Frequency	45 Hz	50 Hz
Input current AC	-	0,19A
Peak inrush current	-	20A
Power factor	0,95	-

Source: Philips LEDSTREET catalog.

2.2.4. NaviaP LED luminaire

This is a Solydi LED luminaire with a nominal power of 50 W, manufactured in 2018, from the Navia Series product family, with a power factor of 0.95. The LED source of this luminaire is CREE Xlamp XT-E. This equipment is fitted with a MEANWELL LED driver, model LPFH-60D-24. The driver characteristics are as follows Table 6.

Table 6. Table of specifications of LPFH-60D-24 LED driver.

Attribute	Minimum Value	Maximum Value
Input voltage	200 V	400 V
Frequency	47 Hz	63 Hz
Input current AC	0,32A	0,4A
Peak inrush current	-	40A
Power factor	0,95	-

Source: Meanwell catalog.

2.2.5. ATP Air 7 LED Luminaire

This is an ATP LED luminaire with a rated power of 200 W, manufactured in 2018, from the Air Series product family, with a power factor of 0.97. The LED source used in this luminaire is Osram LUW CQAR (streetwhite). The luminaire is equipped with a TRIDONIC brand driver, model LCO 200/1400 mA fixC L SNC2, whose main characteristics are shown in the following Table 7.

Table 7. Table of specifications of LCO SNC2 LED driver.

Attribute	Minimum Value	Maximum Value
Input voltage	100 V	240 V
Frequency	50 Hz	60 Hz
Input current AC	-	0,97A
Peak inrush current	-	40A
Power factor	0,97	-

Source: Tridonic catalog.

2.3. Laboratory set-up

For the acquisition of the input current data of the analyzed luminaires, a Fluke 437 Series II network analyzer was used, as shown above. The connection diagram of the luminaires and the analyzer in the laboratory is shown in the following Fig. 1:

Fig. 1. Laboratory set-up scheme for measurements. Source: own elaboration.

2.4. Methodology

2.4.1. Flowchart

The methodological structure of this study is described in this section. It is mainly divided into two blocks: Measurement process and application of multi-criteria decision-making methods for inrush current assessment, as shown in Fig. 2.

Fig. 2. Flow diagram for the analysis of inrush current. Source: own elaboration.

- Measurement process: The measurement process was carried out in the lighting and domotics laboratory of the University of Jaén. Twelve start-up processes of each of the luminaires were tested, and the data was taken with the network analyzer.

The start-up processes were performed independently and on different days of luminaire analysis in the laboratory. A cold start process was produced, which is defined as the type of start-up that occurs in the luminaires after a long period of inactivity, usually several hours.

The luminaires were connected to a wall power monophasic (220 V, 50 Hz) and inside a laboratory with computer equipment, which is the most severe test case, since this computer equipment, due to the high number of electronic components, makes the power quality of the network more unfavorable due to the high harmonic content they produce and therefore the inrush current measurements taken are more severe, since it is the most unfavorable case.

2.4.2. Multi-criteria decision-making methods

Multi-criteria decision-making methods involve selection from a set of feasible alternatives, based on a set of qualitative and/or quantitative criteria, which may be in conflict. These methods are basically divided into three types: subjective, objective and integrated.

1. The subjective methods have the following characteristics: The role of assigning the importance to the criteria is put on the shoulders of the Decision Maker, expert has to assign based on previous experience, constraints of design or designer preferences. Examples are AHP, SMART and SWARA methods.

2. In the objective methods decision maker has no role in determining the importance of the criteria. It is useful when decision maker is nonexistent. Examples are ENTROPY and CRITIC methods.

3. The integrated methods combine the subjective and objective weightings into a single component.

The two methods used in this paper are objective methods because there is no expert to assign importance to a certain criterion and because the inrush current measurements taken in the laboratory are objective, since they have been measured with a precision device such as a network analyzer. On the one hand, the CRITIC method has been used in the field of economics for the selection of contract manufacturers (Adalı and Işık 2017) or in other fields of engineering such as renewable energy for the selection of a hybrid power generation system in a home (Babatunde et al. 2019). On the other hand, the entropy method has been employed in the field of computer aided engineering (CAE) (Jee and Kang 2000) or in the field of manufacturing for the selection of an optimal material (Hussain and Mandal 2016).

3. Entropy method

Entropy method (EM) is a commonly used weighting method that measures value dispersion in decision-making (Zhu et al. 2020). The concept of entropy was introduced by American mathematician Claude Shannon in 1948 (Shannon 1948) and means how much an event can be stochastic. This method reports the randomness of an event in mathematical form (Shyu et al. 2011).

The procedure for the application of this method is as follows (Gorgij et al. 2017):

Step 1: Construction of the decision matrix

The values of the inrush current for each of the luminaires analyzed are expressed in a decision matrix as follows:

(1) $\mathit{M}=\left[\begin{array}{ccccc}{x}_{11}& \dots & x_{1\mathit{j}}& \dots & x_{1\mathit{n}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{i}1}& \dots & x_{\mathit{ij}}& \dots & x_{\mathit{in}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{m}1}& \dots & x_{\mathit{mj}}& \dots & x_{\mathit{mn}}\end{array}\right]$(1)

Where the rows of the matrix symbolize the types of urban luminaires analyzed and the columns symbolize each of the start-up processes in which the inrush current was measured.

Step 2: Normalization of the arrays of decision matrix (performance indices) to obtain the project outcomes ${\mathit{p}}_{\mathit{ij}}:$

(2) $p_{\mathit{ij}}=\frac{{\mathit{x}}_{\mathit{ij}}}{{\sum }_{i=1}^m{x}_{\mathit{ij}}}$(2)

Step 3: Computation of the entropy measure of project outcomes using the following equations:

(3) $E_j=-\mathit{k}\sum _{\mathit{i}=1}^m{p}_{\mathit{ij}}\ln p_{\mathit{ij}}$(3)

(4) $\mathit{k}=\frac{1}{ln\mathit{m}}$(4)

Step 4: Define the objective weight based on the entropy concept

(5) $w_j=\frac{1-E_j}{{\sum }_{j=1}^n\left(1-E_j\right)}$(5)

4. CRITIC method

CRITIC method (Criteria Importance Through Intercriteria Correlation) is one of the methods which determines objective weights for criteria (Adalı and Işık 2017) and was developed by Diakoulaki et al. in 1995 (Diakoulaki et al. 1995). This method uses correlation analysis to find the contrasts between criteria (Yilmaz and Harmancioglu 2010). In this method, the decision matrix is evaluated, and the standard deviation of normalized criterion values by columns and the correlation coefficients of all pairs of columns are used to determine the criteria contrast (Madic and Radovanovic 2015).

The procedure for the application of this method is as follows (Diakoulaki et al. 1995):

Step 1: Construction of the decision matrix.

The decision matrix is constructed in the same way as in the entropy method.

(6) $\mathit{M}=\left[\begin{array}{ccccc}{x}_{11}& \dots & x_{1\mathit{j}}& \dots & x_{1\mathit{n}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{i}1}& \dots & x_{\mathit{ij}}& \dots & x_{\mathit{in}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{m}1}& \dots & x_{\mathit{mj}}& \dots & x_{\mathit{mn}}\end{array}\right]$(6)

Step 2: Normalize the decision matrix.

(7) ${\bar{x}}_{\mathit{ij}}=\frac{{\mathit{x}}_{\mathit{ij}}-x_j^{w\mathrm{orst}}}{x_j^{b\mathrm{est}}-x_j^{w\mathrm{orst}}}$(7)

Step 3: Calculate standard deviation ${\mathit{\sigma }}_{\mathit{j}}$ for each j criteria.

Step 4: Determine the symmetric matrix of n × n with element ${\mathit{r}}_{\mathit{jk}}$, which is the linear correlation coefficient between vectors ${\mathit{x}}_{\mathit{j}}$ and ${\mathit{x}}_{\mathit{k}}$

Step 5: Calculate measure of the conflict created by criterion j with respect to the decision situation defined by the rest of the criteria.

(8) $\sum _{\mathit{k}=1}^m\left(1-r_{\mathit{jk}}\right)$(8)

Step 6: Determining the quantity of the information in relation to each criterion.

(9) $C_j={\sigma }_j\cdot \sum _{\mathit{k}=1}^m\left(1-r_{\mathit{jk}}\right)$(9)

Step 7: Determining the objective weights.

(10) $w_j=\frac{C_j}{{\sum }_{k=1}^m{C}_j}$(10)

5. Results

The matrix of inrush currents in amperes, measured in the laboratory for each of the luminaires and therefore the decision matrix is as follows: Table 8.

Table 8. Decision matrix.

Luminaire	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
GL1A	3,7	2	3,4	2,1	3,9	4	2,1	2,1	3,7	2,1	3,9	2,6
T1A	4,2	2,2	2,2	2,5	3,5	3,8	2,2	3,1	2,2	2,3	4,2	2,9
Min.Irid	0,3	0,2	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3	0,3
NaviaP	2,4	2,8	4,3	3,6	1,9	2,8	2,3	4,6	3,1	2,5	4,5	4,6
Air 7	5,9	4,6	6,5	6,1	6,4	8	7,7	5,2	7,8	8	7	5,7

5.1. Entropy method

When applying ENTROPY method, the first step is to normalize the decision matrix using equation (2). The results are presented in Table 9.

Table 9. Normalized decision matrix.

Luminaire	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
GL1A	0,22	0,17	0,20	0,14	0,24	0,21	0,14	0,14	0,22	0,14	0,20	0,16
T1A	0,25	0,19	0,13	0,17	0,22	0,20	0,15	0,20	0,13	0,15	0,21	0,18
Min.Irid	0,02	0,02	0,02	0,02	0,02	0,02	0,02	0,02	0,02	0,02	0,02	0,02
NaviaP	0,15	0,24	0,26	0,25	0,12	0,15	0,16	0,30	0,18	0,16	0,23	0,29
Air 7	0,36	0,39	0,39	0,42	0,40	0,42	0,53	0,34	0,46	0,53	0,35	0,35

Once the decision matrix has been normalized, the entropy value ${\mathit{E}}_{\mathit{j}}$ is given by equations (3)(3) $E_j=-\mathit{k}\sum _{\mathit{i}=1}^m{p}_{\mathit{ij}}\ln p_{\mathit{ij}}$(3) and (4(4) $\mathit{k}=\frac{1}{ln\mathit{m}}$(4) ), where m in this case represents the number of luminaires tested. The following Table 10 shows the entropy value.

Table 10. Entropy value.

${\mathit{p}}_{\mathit{ij}}\cdot \mathbf{Ln}{\mathit{p}}_{\mathit{ij}}$
Lum	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
GL1A	−0,34	−0,30	−0,32	−0,28	−0,34	−0,33	−0,28	−0,27	−0,33	−0,27	−0,32	−0,29
T1A	−0,35	−0,31	−0,27	−0,30	−0,33	−0,32	−0,29	−0,32	−0,26	−0,29	−0,33	−0,31
M.I	−0,07	−0,07	−0,07	−0,08	−0,07	−0,07	−0,08	−0,08	−0,07	−0,08	−0,06	−0,07
Nav	−0,28	−0,34	−0,35	−0,35	−0,25	−0,28	−0,29	−0,36	−0,31	−0,30	−0,34	−0,36
Air 7	−0,37	−0,37	−0,37	−0,36	−0,37	−0,36	−0,34	−0,37	−0,36	−0,34	−0,37	−0,37
$\sum _{\mathit{i}=1}^m{\mathit{p}}_{\mathit{ij}}\cdot \mathbf{Ln}{\mathit{p}}_{\mathit{ij}}$
	−1,40	−1,39	−1,38	−1,37	−1,37	−1,36	−1,27	−1,40	−1,33	−1,27	−1,41	−1,40
${\mathit{E}}_{\mathit{j}}$
	0,87	0,86	0,86	0,85	0,85	0,85	0,79	0,87	0,83	0,79	0,88	0,87

Finally, the objective weight is calculated based on the concept of entropy by means of equation (5)(5) $w_j=\frac{1-E_j}{{\sum }_{j=1}^n\left(1-E_j\right)}$(5) . The following Table 11 shows the results:

Table 11. Objective weight-based entropy concept.

${\mathit{w}}_{\mathit{j}}\left(\mathbf{\%}\right)$
S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
6,98	7,42	7,82	8,13	8,13	8,37	11,48	7,09	9,40	11,52	6,64	7,03

The evolution of the weights for each of the luminaire start-up processes can be seen in Fig. 3.

Fig. 3. Entropy weight for start-up processes.

As can be seen in the graph above, the highest weight value is 11.52, which corresponds to start-up process number 10. Therefore, the currents assigned to this weight and to this ignition process are as follows Table 12.

Table 12. Inrush current entropy method.

Inrush Current Entropy (A)
Luminaire	S10
GL1A	2,1
T1A	2,3
Mini-Iridium	0,3
NaviaP	2,5
Air 7	8

5.2. CRITIC method

As well as entropy method, when applying CRITIC method, the first step is to normalize the decision matrix using equation (6)(6) $\mathit{M}=\left[\begin{array}{ccccc}{x}_{11}& \dots & x_{1\mathit{j}}& \dots & x_{1\mathit{n}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{i}1}& \dots & x_{\mathit{ij}}& \dots & x_{\mathit{in}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{\mathit{m}1}& \dots & x_{\mathit{mj}}& \dots & x_{\mathit{mn}}\end{array}\right]$(6) . The results are presented in Table 13.

Table 13. Normalized decision matrix.

Luminaire	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
GL1A	0,39	0,59	0,50	0,69	0,41	0,52	0,76	0,63	0,55	0,77	0,46	0,57
T1A	0,30	0,55	0,69	0,62	0,48	0,55	0,74	0,43	0,75	0,74	0,42	0,52
Min.Irid	1,00	1,00	1,00	1,00	1,00	1,00	1,00	1,00	1,00	1,00	1,00	1,00
NaviaP	0,63	0,41	0,35	0,43	0,74	0,68	0,73	0,12	0,63	0,71	0,37	0,20
Air 7	0,00	0,00	0,00	0,00	0,00	0,00	0,00	0,00	0,00	0,00	0,00	0,00

Calculating the standard deviation for each start-up process, the results are as follows: Table 14.

Table 14. Standard deviation for each start-up process.

	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
${\mathit{\sigma }}_{\mathit{j}}$	0,37	0,36	0,37	0,37	0,37	0,36	0,38	0,40	0,37	0,38	0,36	0,38

Once the decision matrix and the value of the standard deviation have been calculated, the symmetric matrix n × n is calculated by means of the linear correlation coefficient. The results are presented in the following Table 15.

Table 15. Coefficient correlation matrix r_jk.

Process	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
S1	1	0,88	0,80	0,83	0,98	0,96	0,85	0,73	0,87	0,84	0,93	0,78
S2	0,88	1	0,97	0,99	0,87	0,93	0,93	0,94	0,94	0,94	0,98	0,97
S3	0,80	0,97	1	0,96	0,82	0,89	0,90	0,91	0,95	0,90	0,94	0,96
S4	0,83	0,99	0,96	1	0,83	0,91	0,95	0,94	0,94	0,95	0,95	0,97
S5	0,98	0,87	0,82	0,83	1	0,98	0,90	0,68	0,92	0,89	0,90	0,75
S6	0,96	0,93	0,89	0,91	0,98	1	0,95	0,77	0,97	0,95	0,94	0,83
S7	0,85	0,93	0,90	0,95	0,90	0,95	1	0,78	0,97	1,00	0,88	0,84
S8	0,73	0,94	0,91	0,94	0,68	0,77	0,78	1	0,79	0,80	0,93	0,99
S9	0,87	0,94	0,95	0,94	0,92	0,97	0,97	0,79	1	0,97	0,91	0,86
S10	0,84	0,94	0,90	0,95	0,89	0,95	1,00	0,80	0,97	1	0,89	0,85
S11	0,93	0,98	0,94	0,95	0,90	0,94	0,88	0,93	0,91	0,89	1	0,96
S12	0,78	0,97	0,96	0,97	0,75	0,83	0,84	0,99	0,86	0,85	0,96	1

Performing the operation $1-{\mathit{r}}_{\mathit{jk}}$, the resulting confliction matrix will be: Table 16

Table 16. Confliction matrix.

Process	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
S1	0	0,12	0,20	0,17	0,02	0,04	0,15	0,27	0,13	0,16	0,07	0,22
S2	0,12	0	0,03	0,01	0,13	0,07	0,07	0,06	0,06	0,06	0,02	0,03
S3	0,20	0,03	0	0,04	0,18	0,11	0,10	0,09	0,05	0,10	0,06	0,04
S4	0,17	0,01	0,04	0	0,17	0,09	0,05	0,06	0,06	0,05	0,05	0,03
S5	0,02	0,13	0,18	0,17	0	0,02	0,10	0,32	0,08	0,11	0,10	0,25
S6	0,04	0,07	0,11	0,09	0,02	0	0,05	0,23	0,03	0,05	0,06	0,17
S7	0,15	0,07	0,10	0,05	0,10	0,05	0	0,22	0,03	0,00	0,12	0,16
S8	0,27	0,06	0,09	0,06	0,32	0,23	0,22	0	0,21	0,20	0,07	0,01
S9	0,13	0,06	0,05	0,06	0,08	0,03	0,03	0,21	0	0,03	0,09	0,14
S10	0,16	0,06	0,10	0,05	0,11	0,05	0,00	0,20	0,03	0	0,11	0,15
S11	0,07	0,02	0,06	0,05	0,10	0,06	0,12	0,07	0,09	0,11	0	0,04
S12	0,22	0,03	0,04	0,03	0,25	0,17	0,16	0,01	0,14	0,15	0,04	0

Once the matrix is obtained, the measure of conflict, the quantity of information in relation to each criterion and the objective weights are calculated, using equations (7)(7) ${\bar{x}}_{\mathit{ij}}=\frac{{\mathit{x}}_{\mathit{ij}}-x_j^{w\mathrm{orst}}}{x_j^{b\mathrm{est}}-x_j^{w\mathrm{orst}}}$(7) , (8(8) $\sum _{\mathit{k}=1}^m\left(1-r_{\mathit{jk}}\right)$(8) ) and (9(9) $C_j={\sigma }_j\cdot \sum _{\mathit{k}=1}^m\left(1-r_{\mathit{jk}}\right)$(9) ). The results can be seen in the following Table 17.

Table 17. Objective weights.

Process	$\sum _{\mathit{k}=1}^{\mathit{m}}\left(1-{\mathit{r}}_{\mathit{jk}}\right)$	Cj	wj(%)
S1	1,57	0,59	11,83
S2	0,65	0,23	4,72
S3	0,99	0,37	7,51
S4	0,78	0,29	5,79
S5	1,50	0,56	11,35
S6	0,90	0,33	6,58
S7	1,05	0,40	8,05
S8	1,74	0,70	14,11
S9	0,91	0,34	6,80
S10	1,03	0,39	7,88
S11	0,79	0,28	5,72
S12	1,25	0,48	9,65

The evolution of the weights for each of the luminaire start-up processes can be seen in Fig. 4.

Fig. 4. CRITIC weight for start-up processes.

As can be seen in the graph above, the highest weight value is 14,11 which corresponds to start-up process number 8. Therefore, the currents assigned to this weight and to this ignition process are as follows Table 18.

Table 18. Inrush current CRITIC method.

Inrush Current CRITIC (A)
Luminaire	S8
GL1A	2,1
T1A	3,1
Mini-Iridium	0,3
NaviaP	4,6
Air 7	5,2

6. Discussion

In this section, we will analyze the results obtained for each luminaire and start-up process, comparing them with the two methods.

The first comparison to be made will be the objective weights obtained from the application of the ENTROPY and CRITIC methods for each of the 12 start-up processes tested on the LED luminaires. As can be seen in Fig. 5, the start-up processes where there is a greater difference in weights are S1 and S8 and on the other hand where there is a smaller difference in weights are in processes S3 and S11.

Fig. 5. Weight comparative for start-up processes.

It can therefore be deduced that in processes S3 and S11 the two methods of analysis have the greatest convergence and in processes S1 and S8 the greatest divergence. The similarity between the weights of both methods is due to the fact that the start-up processes S3 and S11 have very close values to each other in the inrush current, as can be seen in the decision matrix.

From this graph, it can also be deduced that the higher the weight value indicates that the parameter shows a lower degree of variability in the inrush current. For this reason, the entropy method gives S10 and the CRITIC method gives S8 as the best starting processes.

The current comparison between the two methods is shown in Table 19 and in Fig. 6.

Table 19. Inrush current entropy vs CRITIC method.

	Inrush Current Entropy (A)	Inrush Current CRITIC (A)
Luminaire	S10	S8
GL1A	2,1	2,1
T1A	2,3	3,1
Mini-Iridium	0,3	0,3
NaviaP	2,5	4,6
Air 7	8	5,2

Fig. 6. Inrush current comparison.

A very important fact that we can deduct from the above graph is that the CRITIC method is more restrictive and, therefore, more severe when evaluating the inrush current because in all luminaires except Air 7 the current is higher than using the entropy method.

This result is very important because both methods can be combined for a more accurate evaluation, choosing to take the most severe currents obtained in each luminaire as the final result of the study.

Table 20 and Fig. 7 show the current prediction of the combination of both methods.

Table 20. Final inrush current predicted.

Inrush Current Predicted
Luminaire	Current (A)
GL1A	2,1
T1A	3,1
Mini-Iridium	0,3
NaviaP	4,6
Air 7	8

Fig. 7. Inrush current predicted.

Both multicriteria decision-making methods (entropy and CRITIC) are approaches used in multicriteria analysis to make decisions in complex situations. These methods differ in their approach and application, which implies different implications in their use.

1. Measurement approach: The Entropy method is based on information theory and uses entropy measures to quantify the uncertainty or dispersion of criteria values. The CRITIC method focuses on the correlation between criteria and seeks to determine the relative importance of each criterion based on its relationship with other criteria.

2. Application: The Entropy method is suitable for situations where the aim is to select the most informative criteria and reduce uncertainty in decision making. The CRITIC method is useful when it is desired to evaluate the relative importance of criteria in terms of their relationship to each other, which may be relevant in contexts where criteria are interconnected or may influence each other.

For these reasons, the combination of using the more severe method in each case leads to a more realistic approximation of the prediction of the inrush current, since the combination of both methods seeks to reduce the uncertainty in decision-making and to evaluate the relative importance of the start-up processes in terms of their relationship with each other.

7. Conclusions

A novel technique has been developed in the present work that provides an accurate conjoint determination of the inrush current in LED luminaires applied to urban lighting. The most important innovation is that the process includes estimation of the inrush current through a combination of Multi-Criteria Decision-Making methods.

The results obtained from the application of the two methods make it possible to predict the inrush current of the luminaires before they are installed on the roads and streets of the cities. The potential of multi-criteria decision methods has been demonstrated, as the prediction of the current allows significant economic savings to be made in terms of the correct sizing of operating elements such as cables or protection elements such as switches or earth leakage protection.

Although LED technology is the most economically and energy-efficient form of lighting, it has been shown throughout this study that there are problems related to power quality such as inrush current which is a key parameter in the proper functioning of both the lamps and the electrical circuit. Therefore, a correct evaluation of this current will allow a better knowledge of the system and an important anticipation of future electrical problems.

This study has shown that the use of the entropy method and CRITIC method are useful techniques for improving the understanding of power quality issues regarding LED luminaires applied to urban lighting.

Author contributions

Conceptualization, M.J.H.-O., J.T-C, and J.d.D.U.-S.; methodology M.J.H.-O, J.T-C and J.d.D.U.-S.; validation, M.J.H.-O., J.T-C and B.O.-F.; formal analysis M.J.H.-O., J.T-C, and J.d.D.U.-S.; research, M.J. H.-O., J-T-C. and J.d.D.U.-S.; resources, writing M.J.H.-O., J.T-C and J.d.D.U.-S.; writing --review and editing, M.J.H.-O., J.T-C, and J.d.D.U.-S.; visualization M.J.H.-O., J.T-C and J.d.D.U.-S. supervision, M.J.H.-O., J.T-C and B.O.-F.; project administration, M.J.H-O and J.T-C.

Acknowledgments

University of Jaén for materials used in this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research received external funding. This work is funded by The Spanish Government’s state-wide research project entitled: Sustainability and resilience of medium-sized cities and their contribution to the energy transition: circular urban metabolism, energy scenarios and indicator proposals. Acronym: METURBAN2030. Project type: Competitive R+D+I projects. Start date: 12/01/2022. End date: 11/30/2024. Funding entity: MINISTRY OF SCIENCE AND INNOVATION. GOVERNMENT OF SPAIN. Management centre: Research Management Service. Code: 2022/00407/001. Internal reference: MICIN_Transition_ECO_DIG_2021.