Stochastic Assessment of Risk of Light Flicker in a Household

This paper introduces the concept of risk applied to light flicker. It offers a novel stochastic assessment of the risk of light flicker produced by voltage fluctuations from devices in a household. The paper has contributed with stochastic models of voltage fluctuations patterns from several household devices. The voltage fluctuations severity has been determined based on the perception of light flicker from a generalized LED lamp using an index defined as severity factor, what is a novel assessment in contrast to the incandescent lamp considered in the IEC 61000-4-15 standard. The severity factor can be equal to 1, 2 or 3 in increasing order of severity. Each of the severity factors indicates a probability of light flicker equal to or over the perception threshold in a generalized LED lamp and, therefore, a probability of light flicker perceived by an average observer. The severity factor is later used to obtain the risk of light flicker by calculating an index defined as the probabilistic number of voltage fluctuations in a 10-minute interval producing light flicker equal to or over the perception threshold ( $P_{inst}^{LM}$ = 1) in a generalized LED lamp. A $P_{st}^{LM}$ value is obtained from the risk of light flicker index calculated. The results show a low risk of annoying light flicker due to voltage fluctuations from devices in a household for an average observer.


I. INTRODUCTION
Flicker is defined in IEC 60050-161 standard [1] and can be explained as the visual perception of light fluctuation in time.The power quality issues that causes flicker can be voltage fluctuations [2], interharmonics [3] or supraharmonics [4], [5].This paper assesses the light flicker produced by voltage fluctuations from electronic devices in a household e.g., induction stoves, lighting, heaters, air conditioners, refrigerators and freezers, personal computers, entertainment equipment, UPS [6], electric vehicle (EV) and photovoltaic panel (PV).Electronic loads are always present in a household and can, depending on device, produce voltage fluctuations.Other sources of voltage fluctuations are industrial installations with e.g., arc furnaces where the voltage fluctuations propagate from the upstream grid towards a low voltage The associate editor coordinating the review of this manuscript and approving it for publication was Chandan Kumar .
installation and there produce flicker [7], [8].The light flicker inside a household would thus also depend on the area where the household is located, i.e., the proximity to large industrial installations.Severe voltage fluctuations propagating from any upstream industry such as arc furnaces are neglected here.
The standardized measure for flicker is P st and P lt [9].The instantaneous value P inst is recommended to study the flicker impact of voltage fluctuations in [10], [11] and [12], and the instantaneous value adjusted for the light flickermeter, P LM  inst , is used in [2].The IEC flickermeter [9] uses a 60 W incandescent lamp as reference while the light flickermeter [13] does not include a lamp model.To study the voltage variations severity, the IEC flickermeter [9] is sufficient.
To study the flicker perceived by the end user from nonincandescent lamps, the light flickermeter [13] is needed.The light flickermeter [13] uses the light output as input to obtain the flicker severity of the evaluated lamp.
LED lamps have a growing worldwide penetration rate that was 46.5 % in 2019 and expects to reach 75.8 % in 2025 [14].There is to date not a generic or standard model for LED lamps, each can have a different flicker response to similar voltage fluctuations [15].

A. UNCERTAINTIES
To assess the light flicker produced by voltage fluctuations from devices in a household, there are some uncertainties to consider: -The use of devices in a household.
-The voltage fluctuations from the different devices in a household.-The light flicker response of LED lamps to voltage fluctuations.These uncertainties lead to use a stochastic approach and the concept of risk to assess the light flicker.

B. CONCEPT OF RISK
There are several definitions of risk applied to different areas of knowledge.The first novelty of the paper is the application of the concept of risk defined in [16] to light flicker.The risk defined in [16] considers the probability of occurrence and the severity factor (SF) associated to a contingency.The contingencies assessing flicker in a household are the voltage fluctuations from the household devices.The other novelties are the definition of the severity factor and the stochastic models needed for the assessment of the probability of occurrence of voltage fluctuations.

C. STRUCTURE OF THE PAPER
This paper offers a stochastic assessment of the risk of light flicker due to voltage fluctuations from electronic devices in a household.Fig. 1 illustrates the workflow of the methodology used in the paper.Sections II to IV address the cited uncertainties, Section V defines the risk as index of the severity of light flicker, Section VI assesses the risk of light flicker from the devices in a household, Section VII shows the results, Section VIII discuss the assessment done and Section IX states the conclusions.

II. USE OF DEVICES IN A HOUSEHOLD
Voltage fluctuations from 14 devices and background voltage fluctuations are considered for the assessment of risk of light flicker in a household.The household devices considered are described in Table 1.The use of the household devices along a day is stochastically modeled under certain restrictions.A scenario where all the household devices listed in Table 1 are used the same day is considered to assess the risk of light flicker.The devices have been considered to turn on within a time interval of 5 consecutive hours according to the period of higher power consumption in apartments and houses in Sweden as showed in [17].
The number of times that the devices are used is indicated in Table 1.The minimum and maximum working time for every device is determined by the estimation based on e.g., measurements in [15] and shown in Table 1.
For the devices that are used only once, the probability of a device to be turned on in a certain instant is defined within the 5-hour interval.For those devices that are used twice, to avoid the hypothetic overlap of the first and second use, the probability of the device to be turned on in a certain instant is defined within two intervals of similar duration along the five consecutive hours.The intervals are separated a time interval equivalent to the maximum duration of the working time defined for the specific device.
The probability that a device is turned on in a certain instant is considered as uniform, as it is the probability of a device of being working within the minimum and maximum working times defined.

III. MODELING VOLTAGE FLUCTUATIONS FROM HOUSEHOLD DEVICES
Voltage and current from household devices were measured in two different locations with different grid resistances [15]: in a laboratory (1.46 ) and in an apartment (0.55 ).The voltage fluctuations are either obtained directly from the measurements in the laboratory or indirectly calculated from the measured current in the apartment taking as reference the resistance of the laboratory grid.The light flicker produced by the calculated voltage fluctuations is approximated to the light flicker measured from voltage fluctuations with similar magnitude.The worst case of expected light flicker in stabilized LED lamps [18] is considered.

A. MODELING OCCURRENCE OF VOLTAGE FLUCTUATIONS
To stochastically address the probability of occurrence of voltage fluctuations, those from the household devices studied in [15] with the addition of other devices have been taken as a reference.Similar patterns of voltage fluctuations can also be found in e.g., [19].To model the voltage fluctuations from household devices, a resolution of 1 second is considered, so that, only one voltage fluctuation can occur per second from a certain device.The voltage fluctuation patterns have been simplified for some devices.The occurrence of voltage fluctuations is modeled for each device as follows: -Background: It has been observed that in general over 10 voltage fluctuations occur per hour at random time instants.It is modeled using a uniform probability.The voltage fluctuations are modeled with a magnitude of 0.4 % of the nominal voltage (230 V rms), according to the maximum values found in the apartment described in [15].-Dishwasher: the voltage pattern of the dishwasher is explained in [15].The probability of voltage fluctuations every 600 s-interval (P LM st interval) along the working time is calculated in [15] considering three different measurements.This probability is used to obtain in which 600 s-interval each of the 12 or 14 (randomly decided) voltage fluctuations occur.After the 600 s-interval is selected, the probability of each voltage fluctuation to occur in any of the 600 s of the time interval is uniform.
-Dryer: the patterns found in [15] have been simplified.
The dryer has been modeled with two different patterns.The pattern is randomly chosen whenever the device is used.The first pattern consists of an even number of voltage fluctuations separated 100 s of each other.The second pattern consists of an initial voltage fluctuation followed by a sequence of two voltage fluctuations separated 17 s that repeats every 600 s and by a final voltage fluctuation at the end of the working time.The patterns are illustrated in Fig. 2 -Electric vehicles (EVs): from when the state of charge (SoC) reaches 90 % until the end of the charge, two voltage fluctuations occur.-Induction stove: it is considered that during the cooking activity, a person can change temperature up to 10 times including the turning on and off of the device (2 changes as minimum).-Kettle, toaster, iron, coffee maker and vacuum cleaner: these devices produce voltage fluctuations when they are turned on and off.
-Microwave oven: the pattern of the microwave oven is explained in [15].A sequence of 20 s containing three voltage fluctuations is repeated along the working time.The three voltage fluctuations occur in different instants depending on the power selected (six different power levels considered here).The power is randomly selected with every use.-Mixer: the mixer's own operation entails repeated switching on and off producing voltage fluctuations.
The number of times it is turned on and off along the time of use is determined using a uniform distribution from one to ten.-Printer: the printer patterns according to the operation mode are explained in [15].For the model here it is considered that the printer produces voltage variations along the working time.-Washing machine: the washing machine programs described in [15] are simplified to a single pattern.The pattern is divided in three parts.The first and last part last 8 min and a voltage fluctuation occurs every 6 s.
The middle part has a duration nine times shorter than the total working time.It consists of a voltage fluctuation at the beginning and at the end, and a voltage fluctuation repeated every 16 s.

B. MODELING CHARACTERISTICS OF VOLTAGE FLUCTUATIONS
The characteristics of the voltage fluctuations are also modeled based on the voltage patterns and types defined in [15].The types found in [15] are shown in Fig. 3.The same approach is used for the additional devices and for the background as follows: -Background: the type of voltage fluctuation is random.
-Dishwasher: the step down (SD) and step up (SU) types defined in [15]  tuations consists of a SD type followed by an inrush current (IC) type and a SU type, as shown in [15].-Mixer: SD and SU types keep repeating when turning on and off respectively.-Printer: the use of the printer is simplified to two operation modes, printing and copying, which have a drop in the first second followed by voltage fluctuations of smaller magnitude.The operation mode is randomly selected every time the device is used.-Washing machine: the pattern of the washing machine consists of a single and simplified pattern divided into three parts.In the first and last parts, the voltage fluctuations are SU type.The middle part consists of a single SD type followed by a SU type repeated every 16 seconds and a last differentiated SU type.The types of voltage fluctuations have different magnitude and rate of change depending on the household device they belong to, the working mode selected and the part of the pattern when they occur.This will then be reflected in the flicker response of the LED lamps.

IV. SEVERITY FACTOR
The voltage fluctuations severity is studied using The IEC flickermeter [9].However, the 60 W incandescent lamp that it uses as reference is no longer representative of the voltage fluctuations severity perceived by the end user.Each LED lamp can have a different light flicker response to similar voltage fluctuations [15].There is to date not a generic or standard model for LED lamps that serves as reference to measure the voltage fluctuations perception or severity.
The light flickermeter [13] measures the perception of the light flicker response by an average end observer looking to a light source using the P LM inst as index.The perception threshold (P LM inst = 1) means that in 50 % of cases an average observer can perceive flicker [13].Similarly, it is defined for the P inst [9], with the difference that a voltage fluctuation is linked to this value, since the light flicker response to a voltage fluctuation of the 60 W incandescent lamp used as reference is generalized for all the lamps with the same technology.
The severity factor is the name given here by the authors to the index that solves the issue of the link between the light flicker perception level and a certain voltage fluctuation considering LED lamps as reference, similarly to the P inst [9].The severity factor considers a generalized light flicker response of the LED lamps to a certain voltage fluctuation.The severity factor is determined as explained below.

A. LED LAMPS
A set of LED lamps representative of the market is needed to assess the different light flicker response of the LED lamps to a similar voltage fluctuation.The set of 32 LED lamps described in [15] is considered here.Not every LED lamp on the market is sensitive to voltage fluctuations, i.e., producing a measurable light flicker over a certain threshold.Thirteen out of the 32 LED lamps (40 %) are considered as sensitive to voltage fluctuations in [15] applying the methodology in [20].Eleven of the sensitive LED lamps (ranging from 3 W to 11 W) have been used to quantify the severity factor.

B. QUANTIFYNG THE SEVERITY FACTOR
The maximum value of the P LM inst (P LM inst,max ) from the light flickermeter [13] is used to measure the individual light flicker response of the sensitive LED lamps to every voltage fluctuation.The set of P LM inst,max values linked to a similar voltage fluctuation determines the severity factor.The light flicker response of the sensitive LED lamps is generalized since the severity factor is determined by the behavior of the set of LED lamps and not by every LED lamp independently.
The severity factor of a voltage fluctuation is 1 (SF=1) if the P LM inst,max values from the sensitive LED lamps are all below the perception threshold, SF=2 if there are LED lamps with P LM inst,max values over, equal or below the perception threshold and SF=3 if all the LED lamps P LM inst,max values are equal to or over the perception threshold.
A total of 35 different voltage fluctuations were found during the operation of the devices listed in Table 1.Each of the voltage fluctuations found has a different magnitude and rate of change corresponding to a type shown in Fig. 3. Fig. 4 shows the P LM inst,max values of 11 of the sensitive LED lamps linked to each voltage fluctuation, as well as the severity factor obtained for each voltage fluctuation.The P LM  inst,max values are obtained from measurements of the light intensity of the LED lamps.It was not feasible to connect some household devices to the LED lamps for this measurement due to technical limitations.To solve this issue, the P LM inst,max associated to the voltage fluctuations produced by these household devices corresponds to the one obtained from the synthetic voltage profile in [15] for the same or the closest voltage step magnitude.The highest rate of change is considered, i.e., the most severe P LM inst,max for the voltage fluctuation magnitude [2].

C. QUALIFYING THE SEVERITY FACTOR
The severity factor value is determined by the generalized light flicker response of the LED lamps sensitive to voltage fluctuations.The definition linked to each value of the severity factor is generalized for the LED lamps in the market (sensitive and not sensitive LED lamps).
Voltage fluctuations with SF=1 have a probability of producing light flicker over the perception threshold near to zero according to the sample of sensitive LED lamps used.Generalizing all the lamps available in the market, it may be possible to find a LED lamp with light flicker response equal to or over the perception threshold even for these fluctuations.Therefore, voltage fluctuations with SF=1 are estimated to have 5 % of probability of producing light flicker equal to or over the perception threshold.
It is estimated that 50 % of the LED lamps in the market are sensitive to voltage fluctuations, based on the results in [15].To account for the fact that only sensitive lamps have been included in the study, the probability of the voltage fluctuations with SF=2 and SF=3 producing light flicker equal to or over the perception threshold in a generalized LED lamp is estimated to be half of the probability in the measured sensitive LED lamps.The values are shown in Table 2.

TABLE 2. Probability of voltage fluctuations producing light flicker equal to or over the perception threshold.
The severity factor is defined according to the probabilities in Table 2 of the voltage fluctuations producing light flicker equal to or over the perception threshold, defined in [13], in a generalized LED lamp: -SF=1 means that there is a probability ≥ 2.5 % that an average observer perceives light flicker.-SF=2 means that there is a probability ≥ 12.5 % or more that an average observer perceives light flicker.-SF=3 means that there is a probability ≥ 25 % that an average observer perceives light flicker.

V. RISK OF LIGHT FLICKER INDEX
The IEC flickermeter [9] statistically calculates the severity (level of annoyance for the end user) of the light flicker response to voltage fluctuations based on the occurrence and values of P inst in a 10-minute interval (P st index [9]).The severity of the light flicker response to voltage fluctuations considering LED lamps as reference is quantified based on the severity factor in a 10-minute interval here using the concept of risk.

A. DEFINITION OF RISK OF LIGHT FLICKER INDEX
The definition of risk in [16] is applied here to study light flicker as explained in Sub-section I-B.The risk of light flicker index (risk f ) is defined as the probabilistic number of voltage fluctuations in a 10-minute interval producing light flicker equal to or over the perception threshold (P LM inst = 1) in a generalized LED lamp.The risk of light flicker index is calculated as indicated in (1) for a 10-minute interval.N SF1 , N SF2 and N SF3 are the number of voltage fluctuations in a 10-minute interval with SF=1, SF=2 and SF=3 respectively.P SF1 , P SF2 and P SF3 are the probabilities per unit associated to the respective severity factors (0.05 for SF=1, 0.25 for SF=2 and 0.5 for SF=3) of the voltage fluctuation producing light flicker equal to or over the perception threshold in a generalized LED lamp.risk f = N SF1 P SF1 + N SF2 P SF2 + N SF3 P SF3 (1)

B. QUALIFYING THE RISK OF LIGHT FLICKER INDEX FOR THE END USER
The risk of light flicker index is qualified in terms of severity of light flicker to the end user using the P LM st [13].Several P LM st values can be obtained depending on the P LM inst,max values assigned to the number of voltage fluctuations calculated from risk f .The light flickermeter [13] was modified to make the P LM inst the input to obtain the P LM st .The P LM inst profile is obtained considering that the voltage fluctuations occur consecutively and there is no overlap of the P LM inst response shown in Fig. 5.The shape of the response shown in Fig. 5 corresponds to a SU type, but it has also been observed in an IC type.Under the cited considerations, the 10-minute interval P LM inst profile is built scaling the P LM inst response of P LM inst,max = 1 shown in Fig. 5 to the selected P LM inst,max values to convert from the discrete P LM inst,max value to a continuous P LM inst .The duration of P LM inst response limits the calculation of the P LM st to a maximum risk f of 294 voltage fluctuations.The definition of the risk of light flicker index implies that it has a minimum P LM st value associated when the P LM inst,max of all the quantified voltage fluctuations is equal to the perception threshold.It is represented in Fig. 6.To obtain a P LM st characteristic of voltage fluctuations in a household, it has been considered the probability distribution of P LM inst,max values over 1 obtained from the voltage fluctuations produced by each of the household devices considered in this study.The probability distribution of P LM inst,max , shown in Fig. 7, is calculated with a resolution of 0.1.The average value ( PLM st ) and the standard deviation (std) resulting of the stochastic calculation of 200 P LM st values are considered to measure the severity of the light flicker associated to a certain value of risk f according to (1).
Integer values of risk f up to 294 with the respective values of the PLM st and standard deviation calculated are shown in Table 3.For simplification when showing the results, only the values that make a change in the first decimal of the average P LM st are shown in Table 3.
Attending to the criteria stablished for P LM st in IEC TR 61547-1 [13], the risk of light flicker is qualified as high if the resulting P LM st value is higher or equal to 1 and low if it is below 1.From the stochastic calculation of the P LM st , the

average P LM
st value determines the level of risk of light flicker since the standard deviation values obtained are between 0.0491 and 0.0960.
The threshold of the risk of light flicker index in a household is obtained for a value of 84 voltage fluctuations (low risk if risk f < 84, cf.Table 3 ).
The definition of high or low risk of light flicker is stated: -High risk of light flicker means that in 50 % or more of the cases an average observer may perceive annoying light flicker.-Low risk of light flicker means that in less than 50 % of the cases an average observer may perceive annoying light flicker.

VI. STOCHASTIC QUANTIFICATION OF THE RISK OF LIGHT FLICKER IN A HOUSEHOLD
The risk of light flicker index in a household is calculated along 1-day interval, i.e., 144 10-minute intervals.Two thousand iterations of this calculation are executed to obtain a stochastic approach of the risk of light flicker considering the following two uncertainties: -The use of devices during a day in a household (stochastically modeled in Section II).-The voltage fluctuations produced during the operation time of the household devices (stochastically modeled in Section III).
In each iteration, a new distribution of voltage fluctuations along 1-day interval with 1 s resolution is obtained for each household device.The severity factor of every voltage fluctuation is determined as explained in Section IV.The risk of light flicker index is quantified according to (1) without overlapping the 10-minute intervals along the 1-day interval.The 95 th percentile value is used as result of the stochastic approach of the risk of light flicker index in a household.The results obtained from each of the 2000 iterations and from the 95 th percentile are organized as indicated in Fig. 9.The flow of the calculations to assess the light flicker in a household is represented in Fig. 10.

VII. RESULTS
The daily total risk obtained is shown in Fig. 11.The results of the quantification of the risk of light flicker index in a household are concentrated in 47 10-minute intervals (7 h 50 min) shown in Fig. 11.It is due to the 5-hour interval within which the household devices are considered to be turned on and the working time of the household devices (cf.Table 1).The maximum total risk and maximum partial risk calculated have been rounded to the nearest integer number to qualify the severity of the ligth flicker experienced by the end 104112 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.user as described in Sub-section V-B.The severity obtained as well as the level of risk of light flicker is stated in Table 4.The average P LM st determines a low risk of light flicker for the maximum total risk and maximum partial risk.The dispersion (standard deviation) of the P LM st for the total risk does not point to any case of P LM st equal to or over 1.The dispersion of the P LM st for the partial risk could indicate the possibility of P LM st slightly over 1 on few occasions.The washing machine has the most significant contribution with up to 100 voltage fluctuations per 10-minute interval.The results obtained without considering the washing machine are significantly lower as shown in Table 5 and Fig. 12.

A. STOCHASTIC APPROACH
The results shown are obtained considering a stochastic approach with 2000 iterations.Up to 5000 iterations were run without any significant change in the results.
To calculate the PLM st , up to 2000 iterations were run without any significant change in the results compared to the 200 iterations used.

B. ACCURACY OF THE MODELED UNCERTAINTIES
The risk of light flicker is assessed considering a scenario where all the household devices considered are used the same day within a certain range of hours based on the hours of higher consumption in apartments and houses in Sweden [17].A more accurate model of the number of devices as well as time of use of devices in a household would improve the accuracy of the assessment of risk of light flicker.
A simplification of the voltage fluctuation patterns is done considering the most characteristic voltage fluctuations found for each device and that may have the most significant impact on the light flicker response.The accuracy of the voltage fluctuation patterns lies on the different models of household devices that there are in the market, the programs (working modes) of the device and the complexity of the voltage pattern of the individual devices.The simplification of the voltage fluctuation patterns from an individual device may result in a generic model of such device.
The accuracy of the severity factor relies in the amount of LED lamps measured and how well they represent the market.The set of 32 LED lamps used on this study is chosen according to these criteria.

C. LIGHT FLICKER
The values of light flicker are obtained for voltage fluctuations considering a supply impedance of 1.46 .This type of grid can be found in rural areas.In urban areas, the grid usually has lower resistance [15], [21].The voltage fluctuations produced by the same device in a low resistance grid have lower magnitude than in a higher resistance grid resulting in a lower risk of light flicker.
The P LM inst,max was measured after the LED lamps reached the stabilization [18].To know how the stabilization of the LED lamps influences on the light flicker obtained, the LED lamps were also measured when they were cold, i.e. in the first seconds after switching on.The synthetic voltage profile in [15] has been used to compare the resulting light flicker in stable and cold LED lamps using the methodology in [20].For most of the cases, the LED lamps considered as sensitive are more sensitive after the stabilization time.For three of the eleven LED lamps measured, it depends on the type of voltage fluctuation or if they are subjected to overvoltage or undervoltage.The assessment of risk of light flicker presented is assumed to consider the worst case of light flicker for most of the LED lamps measured.
The P LM inst response duration can vary between different voltage fluctuations.The duration of the P LM inst response shown in Fig. 5 is assumed as fixed for all the P LM inst,max values in the calculation of the P LM st .The P LM inst response is more abrupt at the beginning, so a representative contribution of the P LM inst response to the P LM st is considered, independently whether the duration of the flicker response is shorter or longer.
The risk of light flicker index is qualified according to the PLM st obtained assuming that there is no overlap in the P LM inst response to voltage fluctuations.IEC TR 61000-3-14 [22] states that the probability of several voltage fluctuations to occur at the same instant is very low.It is uncertain if it holds for the probability of overlap of the P LM inst response, which has a duration of approximately 2 s in Fig. 5.A study on how to model the overlap of the P LM inst response is needed in order to address this uncertainty since the P LM inst response is different depending on the instant of the overlap as shown in [2] for two voltage fluctuations using the P inst [9].
The P st is not suitable for assessing sudden and short voltage fluctuations [2], [23].The instantaneous value P inst is recommended instead [10], [11], [12].The instantaneous value adjusted for the light flickermeter P LM inst is used in [2].Therefore, the severity factor of a voltage fluctuation is defined according to the P LM inst , and the risk of light flicker index considers in its definition the severity factor of the voltage fluctuations to quantify the severity of light flicker.However, the P LM inst assess the perception of the light flicker and not the severity it does the P LM st .For this reason, the P LM st was necessary to qualify the severity of light flicker.More research is needed to find an alternative to the P LM st for qualifying the risk of light flicker index.

IX. CONCLUSION
This paper offers a stochastic assessment of the risk of light flicker produced by voltage fluctuations from devices in a household.The results show a low risk of annoying light flicker for an average observer.
The risk of light flicker index (risk f ) is defined as the probabilistic number of voltage fluctuations in a 10-minute interval producing light flicker equal to or over the perception threshold (P LM inst = 1) in a generalized LED lamp.It is evaluated according to the PLM st associated as low risk if PLM st < 1 and high risk otherwise.This threshold is equivalent to a risk f equal to 84 (low risk if risk f < 84).
The risk of light flicker index is calculated stochastically, i.e., several risk f values are obtained for the same 10-minute interval.Different terms are given to the risk of light flicker index calculated for a single iteration during the stochastic approach (''partial risk'') and to the risk obtained from the 95 th percentile of all iterations (''total risk'').This terminology facilitates the understanding of the results.
The severity of the voltage fluctuations is determined based on the perception of light flicker from a generalized LED lamp using the index defined as severity factor.The severity factor can be equal to 1, 2 or 3 in increasing order of severity.Each of the severity factors indicates a probability of light flicker equal to or over the perception threshold in a generalized LED lamp and, therefore, a probability of light flicker perceived by an average observer.
The voltage fluctuations pattern from the household devices are simplified and stochastically modeled pursuing a generic model based on the most characteristic voltage fluctuations produced by the household devices and that may have the most significant impact on the light flicker response.
The use of the household devices along a day is stochastically modeled under certain restrictions and according to the period of higher power consumption in apartments and houses in Sweden showed in [17].Further research on this matter would improve the accuracy of the assessment presented of risk of light flicker.

FIGURE 1 .
FIGURE 1. Workflow of the methodology used in the paper.

FIGURE 2 .
FIGURE 2. Illustration of the patterns of the dryer.
are repeated for the six or seven pairs of voltage fluctuations.-Dryer: the SD and SU types follow each other for the two patterns described in Figure 2. -Electric vehicles (EVs): all the voltage fluctuations considered for the EVs are SU type, since the measured voltage drops are considered too slow to produce flicker in the LED lamps based on the results in [15].-Induction stove: the type associated to each voltage fluctuation is obtained from the probability of the types stated in [15] associated with a change in temperature.-Kettle, toaster, iron, coffee maker and vacuum cleaner: these devices produce a SD type volage fluctuation when turning on the device and a SU type when turning off the device.-Microwave oven: the sequence of three voltage fluc-

FIGURE 4 .
FIGURE 4. P LM inst ,max from sensitive LED lamps set obtained for each voltage fluctuation.The vertical black dot-dashed lines separate the voltage fluctuations according to the severity factor.The horizontal red dashed line indicates P LM inst ,max = 1.

FIGURE 6 .
FIGURE 6. Minimum P LMst associated to the risk of light flicker index.

FIGURE 7 .
FIGURE 7. Distribution of P LMinst ,max for all the voltage fluctuation obtained from all the household devices with a resolution of 0.1.

Fig. 8
exemplifies the quantification of the risk of light flicker index in each iteration considering 4 household devices.The seconds not shown in Fig. 8 are free of voltage fluctuations.

FIGURE 8 .
FIGURE 8. Example of the calculation of the risk of light flicker index in a household in 1 iteration.

FIGURE 9 .
FIGURE 9. Organization of the results from the stochastic approach of risk of light flicker index in a household.

FIGURE 10 .
FIGURE 10.Flow of the calculations to assess the light flicker in a household.

TABLE 4 .
Maximum values of the results of the stochastic assessment of the risk of light flicker in a household.

FIGURE 12 .
FIGURE 12. Daily total risk results without washing machine.

TABLE 1 .
Use of household devices in 5-hours interval.

TABLE 5 .
Maximum values of the results of the stochastic assessment of the risk of light flicker in a household without washing machine.