a field investigation of the intermediate light switching by users - preprint

8/14/2019 A field investigation of the intermediate light switching by users - PREPRINT

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A field investigation of the intermediate light switching

by users

David Lindelof and Nicolas MorelSolar Energy and Building Physics Laboratory (LESO-PB)

EPFL, CH-1015 Lausanne, Switzerland

July 15, 2007

This paper describes how data collected during a continuously running dataacquisition program on the LESO building in Lausanne, Switzerland, was usedto measure the intermediate light switch probability by users as a function ofcurrent illuminance levels, i.e. the probability for a given timestep that the userwill switch on or off the electric lighting, excluding such actions that happenupon user entry to or exit from the office. We assume such a probabilityto be independent of the users history and further derive some theoreticalconsequences of this postulate. In particular, we show how a history-less userleads naturally to patterns of behaviour already observed in real buildings.

Keywords: intermediate light switching, visual comfort, Poisson process

1 Introduction

Understanding the way users interact with building services (blinds, electric lighting, cool-ing, ventilation, window opening, etc), and the impact of their use on the buildings total

energy consumption, helps us attain two goals. First, we may elaborate better models ofthe users behaviour for simulation software that will help building planners to predict andoptimize the energy use or the comfort provided to the user. Secondly, advanced controlalgorithms may use this information to increase their acceptance by users and help achieveenergy savings.

Several building simulation software packages that need a good simulation of usersbahaviour are available (1; 2) or will shortly be (3; 4). The software packages just mentionedare all based on the Lightswitch-2002 algorithm described in (2). An underlying assumption

Corresponding author. Tel.: +41 21 693.55.56; fax: +41 21 693.27.22; email: [email protected]

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behind this algorithm is that users use the manual controls at their disposal in a consciousand consistent way, which allows us to predictively model their behaviour.

The algorithm seeks also to model the intermediate light switch-on probability, i. e. theprobability that a user switches on the artificial lighting without leaving or arriving in theoffice. It uses a probability function that depends on the workplane illuminance, derivedfrom previous work by the author of the algorithm (5). For five-minute timesteps, it findsthat the intermediate switch-on probability is about 5% between 0 and 200 lux workplaneilluminance, and sharply drops to about 0.002 for higher illuminances. One purpose of thispaper is to verify this model.

More specifically, we will focus on the following themes concerning lighting actions (day-light or electric lighting):

The lapse of time between the entry of the user into the room and the use of controls,or between use and subsequent exit from the room;

The probability of a user switching on or off the electric lighting as a function ofambient illuminance levels;

The lighting conditions immediately preceding and immediately following a usersaction with the electric lighting;

The correlation between the delay before the user action and the illuminance level.

We will also discuss some theoretical consequences of the modelling of users actions and

their relationship to experimentally observed data.

2 User simulation

Some models of user behaviour assume that the time between user actions, given constantenvironmental conditions, is a random variable distributed according to an exponentialdistribution with sole parameter satisfying = 1/T where T is the average time beforethe action. In other words, its probability density function is given by

f(t) = exp(t).

This distribution is believed to hold, with different parameters of course, for most ofthe users actions, such as use of artificial lighting controls, window opening or closing, andexit or arrival.

The exponential distribution function is used for modelling the occurence of events rang-ing from earthquakes to phone calls. Similarly, the time remaining until, for instance, theusers next opening of windows can be modelled in much the same way as the time remain-ing until the users next phone call.

This postulate is justified by strong evidence that the number of user actions of a givenkind for constant or near-constant environmental conditions in a given time frame follows

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a Poisson distribution. From that fact follows that the intervals between events are dis-tributed according to an exponential distribution. In the case of user entry and exit, Wang

(6), for example, has recently verified experimentally that the duration of user absencesfrom a room indeed follows an exponential distribution.

The problem is that environmental conditions are seldom constant in an office. Temper-atures, air quality and illuminance levels change over time. How are we to compute theprobability density function for user events under varying conditions, assuming we knowhow to do so for constant conditions?

Consider the example of the use of artificial lighting controls. If we could find a relation-ship between the time a user tolerates given visual conditions and the variables describingthese visual conditions (e.g., the illuminance levels), then we would be in an advantageousposition to simulate the behaviour of the user in a computer simulation of the building.

This section will describe how.Let us assume that such a relationship exists between the average time T before useraction and the illuminance E, and that we have found it. We do not specify this rela-tionship; we are in no position to do so yet. But we assume that the probability densityfunction of the time the user spends before switching on the lights follows the exponentialdistribution, and that its parameter is given by some function ofE.

We begin at time t = 0. What is now the probability P(T) that the user has not turnedon the lights yet at a time t = T? Let us discretize the time between t = 0 and t = Tinto n equal timesteps t = T/n. Let us assume that these timesteps are sufficiently smallthat the workplane illuminance can be taken as constant during each timestep, noted Ei,with i running from 0 to n 1. The corresponding parameters are noted i.

The probability that the user did not switch on his lights during the first timestep ishigh because t is small, but not quite equal to unity. It is given by

P(t) = 1

t

0

f0(t)dt

= exp(t0)

By the exponential distributions lack of memory, the probability that the user did notswitch on the lights between t and 2t knowing that he did not do so between 0 and tis similarly equal to exp(t1), and so on.

The probability that the user has not switched on his lights by the time T is thus theproduct of all these probabilities. We obtain

P(T) =n1i=0

exp(it)

= exp(tn1i=0

i)

= exp(T),

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where =

n1

i=0i/n is the average of all i.

But if p(t) denotes the probability distribution function of the time at which the user

switches on the lights for non-constant environmental conditions, then the following musthold:

T

p(t)dt = P(T),

since the left-hand term is the probability that the user switches on the lights at a timebetween T and and the right-hand term is the probability that the user did not switchthe lights on between 0 and T. They are, of course, the same thing.

Replacing with the value for P(T) found previously, and deriving with respect to T onboth sides, we obtain

p(t) = exp(t)

Note that along the way we lost any reference to t, so this formula holds for t van-ishingly small. In fact, it does not hold if t is too large for the environmental conditionsto be considered as constant.

We thus have an expression for the distribution function of the time of user action forvarying environmental conditions. It is however difficult to use this expression in practicesince it cannot readily be integrated over time, due to the non-constant parameter.

A computer running a building simulation should therefore rather compute the evolutionof environmental conditions in steps of t, and compute at each timestep P(t) withparameter = i where i indexes the timestep.

The question remains, how does one measure this parameter? One cannot put a

user in his office, bid the sun and the clouds not to move too much for the next hourand wait until the user gets up and switches on his lights, repeat N times, and derivethe average waiting time. Rather, since a process obeying an exponential distributionimplicitly assumes that the probability that the user should use his controls within thenext few minutes is independent of the users history, one should rather measure for agiven illuminance level what is the probability that the user should use his controls in agiven time window t. Then the average waiting time between two user actions, if t issmall enough, is 1/ = t/.

This probability p can be obtained over the course of a measuring campaign by countinghow many times the user finds himself in a given illuminance, and how many times that

illuminance leads to a user action.

3 Methodology

The data representing the base material for the study discussed in this paper is part of acontinuous recording program by LESO-PB (Solar Energy and Building Physics Labora-tory, EPFL, Lausanne, Switzerland), on the LESO building. The LESO building is a smalloffice building (20 office rooms of around 20 m2 floor area each, about half of them with asingle user and the other half with two), hosting the activities of LESO-PB (for a detailed

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description of the building, see (7)). Additionally, this inhabited building has been usedfor the experimentation of new passive solar systems and advanced control algorithms for

building services (heating, blinds and electric lighting). It is equipped with a commercialEIB building bus system, with the following sensors and actuators for each room:

Sensors: inside and outside air temperatures, diffuse and global solar radiation, solar illu-minance, wind speed and direction, interior illuminance1, occupancy, window open-ing;

Actuators: blinds, electric lighting (continuous dimming), heating;

User interface: blinds, electric lighting, heating temperature setpoint.

Finally, the building is also equipped with a legacy central data logger used for recordingmiscellaneous data such as room temperatures at different positions (measuring, in effect,the stratification of the temperature) or the electricity consumption for each room (countingseparately the heating and the other appliances).

The EIB was installed in 2000. Since then, two research projects focused on controlsystems have been carried out at LESO-PB that made use of mesurements on the LESObuilding, AdControl (8; 9; 10) and Ecco-Build, for which no publication is available yet.The data acquisition has been running independently of any research project and beenstored to a MySQL database (11), representing slightly more than two years of continuousmonitoring. For data due to user actions alone, we have accumulated at the time of writing

about three million datapoints.The data we consider covers the period from mid-November 2002 to mid-January 2005.Any time one of the physical variables changes by more than an adjustable threshold, thatchange is logged to the database together with the time of the event. It is trivial to usethis data to reconstruct a time series of any variable for any given constant timestep.

The data was analyzed with the open-source R data analysis environment (12).

4 Data analysis

4.1 Definition of a user action

In the following a user action is defined as a set of interventions on individual controlsavailable to the user not more than one minute apart from each other. For example, auser might come in in the morning, switch on the lights, and open the blinds. During theday, he or she might decide that the sunlight is enough to illuminate the office so he orshe switches off the light but lowers the blinds sufficiently to prevent direct sunlight from

1We use Siemens brightness sensors GE 252, which are actually roof-mounted luminance sensors shieldedfrom the windows luminance. They were calibrated by Guillemin (8) with LMT reference luxmeterswith an estimated accuracy of1.9%. The conversion from the workspaces luminance to its illuminanceis a programmable feature of the sensor.

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hitting the computer screen. These would be considered two user actions. As long as theindividual interventions are not spaced more than one minute apart, they are considered

the same user action.The data recorded on the building bus is used each night to rebuild a table with all

user actions since data acquisition began. Each entry of that table records the time ofthe beginning of the action, the time of its end, as well as the time when the user cameinto the office before the action occured and the time at which the user left the officeafterwards. We also record which control(s), among blinds, artificial lighting, or windows,were affected.

4.2 Time between user action and user entry/exit

In this study we would like to concentrate on those user actions where the user was knownto have been present in his office for a certain time before using the controls. We alsonaturally require that the user still be present in the office for some time after using thecontrols, in order to filter out those events where the user, for instance, switches off thelights before going home in the evening.

Box plots of the times in seconds between user entry and user action are given for eachoffice in Fig. 1. The middle-bar in each box is the median time. The boxs edges (notedt25 and t75) are placed at the 25% and the 75% quantiles

2. The boxes are then extendedwith so-called whiskers that extend to the most extreme data point not further awayfrom the box than 1.5 times the interquartile range t75 t25. Any datapoints beyond thewhiskers (outliers) are plotted as small circles.

For instance, office 102 has 953 actions recorded. 244 of these, or approximately 25%,happened within 3 seconds after the user had entered the room, so the 25% quantile edgeof the box is placed at 3 seconds. 479 actions, practically half of all actions, occured within31 seconds after user entry, so the median bar in the box is placed at 31 seconds. Similarly,the upper edge of the box is placed at 160 seconds.

The interquartile range is 160 3 = 157 seconds, so the upper whisker is placed at themost extreme data point not exceeding 1.5 157 + 160 = 395.5 seconds. The highest suchdata point is at 393 seconds, so the upper whisker lands there. The lower whisker ends onthe minimum of the data points, at zero seconds. The remaining data points (115 of themin total) are considered outliers.

It is apparent that for most offices, except offices 001, 104 and 201, three quarters ofall user actions occurred less than 300 seconds, or five minutes, after the user entered theoffice. In other words, users usually use the controls available to them while they are onthe move. Users do not leave their seats to adjust their settings unless the situation isclearly unconfortable.

Fig. 2 is also a box plot but of the times between the action and the departure of theuser. The distribution seems similar, but note the shift upwards of the lower 25% quantilebox edge. This is due to the intrinsic 30-seconds timeout on the occupancy sensors. Again,

2I.e., the values below which we have respectively 25% and 75% of the total number of events

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001 002 003 004 101 102 103 104 105 106 201 202 203 204

0

20

0

400

600

800

1000

Office

[s]

Figure 1: Boxplots of the times between user entry and his use of manual controls, peroffice.

about 75% of all user actions happen about five minutes before the users departure. Inother words, again, users use their controls mostly when coincidentally passing by.

This has important consequences for the choice of placement of the users controls inan office, which should be as close and as convenient as possible to the user, who willotherwise simply not use them. This observation also highlights again the obvious need forsmart building control systems since users, unless particularly energy-conscious, usuallywill not adjust their controls if the only benefit is the saving of energy. A clear discomfortis required for the user to take action.

4.3 Intermediate light switching

By intermediate light switching we mean the act of using the artificial lighting controlsin circumstances other than upon arrival to or before departure from the office.

We therefore select from the database those user actions that concerned artificial lightsonly (i.e., no blinds action) and where the user was present at least five minutes before,and at least one minute after the action.

For each such action we can query the database for the values of physical variables oneminute before the beginning of the action and one minute after the end. For each officeconsidered, a histogram of horizontal workplane illuminance before and after the actionare given in Fig. 3 and Fig. 4 respectively. The sensors accuracy is estimated to be 15%.

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001 002 003 004 101 102 103 104 105 106 201 202 203 204

0

20

0

400

600

800

1000

Office

[s]

Figure 2: Boxplots of the time between users use of manual controls and his departure,per office.

Their non-linear behaviour for illuminances above 500 lux has been corrected by the EIBmonitoring software.

That the artificial lighting should increase the amount of available workplane illuminanceis hardly surprising. Neither should one ascribe too much importance to the differencesin the distribution of illuminance after the action. Past certain hours, in particular thosehours where lighting is needed most, the lighting can only provide so many lux and we doubtthat the users in office 101 have deliberately and systematically fiddled with the dimmingcontrols to get to the shown average of about 300 lux. That value represents more likely arough estimate of the maximum workplane illuminance the lighting can provide.

More interesting are the disparities seen in the distribution of illuminance right before

the action. Some users (e.g. 104) never allow the illuminance to go below about 200 luxbefore turning the lights on. Others (such as the people in office 001) seem less botheredand tolerate even very low light levels before turning the lights on.

Only office 004 shows odd results, but the measurement of this offices illuminance valuesis known to be faulty. A new user moved in during 2003, as a result of which the main lumi-naire (a lamp projecting its light on the ceiling) has been moved right under the luminancesensor doubling as an illuminance sensor. This mistake has now have been corrected, butthe data taken on this office will be excluded from further analysis in this paper and willnot be included in plots obtained by pooling together all data. The histogram obtained bylumping together all offices except office 004 are given in Fig. 5.

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Illuminance office 001

[lux]

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Figure 3: Distribution of illuminance levels before user actions, per office. The histograms arevents where the workplane illuminance increased after the user action. On top of illuminance decreased or stayed constant are added in white.

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Figure 4: Distribution of illuminance levels after user actions, per office. See Fig. 3 for the exwhite bars.

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[lux]

Coun

ts

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50

100

150

0 500 1000 1500 2000 2500 3000 3500

(a) Before user action

[lux]

Coun

ts

0

50

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0 500 1000 1500 2000 2500 3000 3500

(b) After user action

Figure 5: Distribution of illuminance, all data. See Fig. 3 for the explanation of the blackand white bars.

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Fig. 6 shows a scatterplot for each office of the illuminance level after vs. before useraction. Data points beneath the diagonal represent events when the user found the light

too strong and decreased or turned it off, while points over the diagonal represent eventswhere the illuminance level was deemed insufficient. Fig. 7 groups together all the eventsfor all offices, with a small jitter applied to each point, in order to prevent the discreteilluminance values provided by the measurement from hiding the real data point density.

0 2 00 4 00 6 00 8 00 1 00 0

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Figure 6: Illuminances recorded right after (Y-axis) vs right before (X-axis), per office.

There is a marked tendency for all offices to prefer switching on the lights to switchingthem off, something coherent with our personal experience. People are, in general, moreconcerned with their visual comfort than with unnecessary energy expenditures3. Of all

offices, only the occupant of office 104 seems to switch off somewhat regularly the lightswhen not necessary.

Note also that for most offices, but most notably in offices 001, 002, 101, 104, 106, 203and 204, there is a clear clustering of the points along two diagonals running parallel to themain diagonal, and equally distanced from that diagonal. This reflects the fact that usersusually dont bother fiddling with the light dimming commands and content themselveswith switching the lights on or off, thus resulting in a constant increase or decrease ofavailable illuminance of roughly 300 lux, the maximum the current light installation can

3or even forget they have left the lights on.

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0 200 400 600 800 1000

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Illuminance before action [lux]

Illumi

nanceafteraction[lux]

Figure 7: Illuminances recorded right after vs right before, all offices.

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provide. This shows that even when dimming commands are available, few users will makeuse of them in an energetically optimal way when they are placed close to the offices

entrance rather than close to the user4

.The occupants of offices 001, 104 and 201 obviously use their electric lighting controls

much more than the other users. Since the placement of the electric lighting controls issimilar in all offices (close to the offices entrance), one can only conclude that these usersare much more concerned with a rational use of the artificial lighting than the others.This leads credence to the notion ofactive vs. passive users found in the Lightswitch-2002model, which distinguishes users based on their willingness to use the controls at theirdisposal.

From now on we shall assume that all user actions on the artificial lights are switchon/off events and neglect the extremely rare dimming events.

4.4 Intermediate switch probability

We now turn to the determination of the intermediate switch probability, i.e. the probabilitythat the user will switch on or off the lights in a given time window as a function of theilluminance.

We choose a time window of five minutes, and slice up the periods of user presence intoperiods of five minutes each, always beginning five minutes after the initial user entry intothe room. Two periods of presence separated by no more than two minutes absence areconsidered as an uninterrupted presence.

We query the database for the value of the illuminance at each such time step, and checktwo things: whether an interaction with the artificial lighting (alone) occured within thenext five minutes, and whether any interaction with blinds or artificial lighting occured.Remember that all interactions with the artificial lighting are assumed to be switchingevents.

If an interaction with the artificial lighting alone occurred, we count it as an intermediateswitch event. If no interaction with the artificial lighting nor with the blinds occurred, wecount it as a situation where the user was satisfied with his visual environment. If only aninteraction with the blinds occurred we exclude the timestep.

For a given range of illuminance values we can thus compute the ratio between thenumber of times the user acted on the aritificial lighting at that illuminance, and the total

number of times the user spent at that illuminance without altering the visual environmentby means of the blinds.

We obtain thus respectively the switch-on and switch-off probabilities for a time windowof five minutes for different ranges of illuminances. These probabilities are given for eachindividual office in Fig. 8 and 9 respectively, and again for the combined data from alloffices in Fig. 10.

4A user even told one of the authors that she actually did not even know she could dim her light.

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Office 001

[lux]

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0.00

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.000

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.000

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Office 201

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Office 204

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0.000

0.01

5

0.030

Figure 8: Intermediate switch-on probability, per office.

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Office 001

[lux]

Switchoffprobability

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.000

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0.004

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.000

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0

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Office 101

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0.0010

Office 102

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0.0010

Office 103

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0.0010


0.000

0.006

0.012

Office 105

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0.000

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0.0000

0.0015

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0.000

0.004

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0.000

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Figure 9: Intermediate switch-off probability, per office.

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4.4.1 Switch-on probability

The behaviour of users with respect to swich-on probability shows remarkable consistency.Most users seem to have an illuminance threshold under which the switch-on probabilitysharply rises to a level of between 1% and 10%. As long as the illuminance is above thatthreshold, the switch-on probability is negligible. That threshold varies from user to userbut lies between 100 and 200 lux.

Fig. 10, obtained by lumping together the data from all office rooms except rooms 003and 004 (which both seem to display an abnormal behaviour) shows that our average userhas a switch-on probability of about 3.3% between 0 and 100 lux, which drops to about1.4% between 100 and 200 lux, then to about 0.6% between 200 and 300 lux, and whichthen becomes more or less negligible.

Should the switch-on process be considered as a Poisson process, the above figures would

then correspond to an average switch-on time of about 150 minutes, 360 minutes, and 830minutes respectively. In terms of half-lives, it means that if left in constant conditions,half the users will have switched their lights on after 104 minutes, 250 minutes and 575minutes respectively (assuming anyone is still left in the office).

[lux]

Sw

itc

h

onpro

ba

bility

0 200 400 600 800 1000

0.0

0

0.0

1

0.0

2

0.0

3

0.0

4

Figure 11: Intermediate switch-on probability for low illuminances, all data.

Fig. 11 shows the combined switch-on probability for low illuminance values, detail-ing what happens below 100 lux. We see that below 50 lux the intermediate switch-onprobability continues to rise up to about 4%. It is difficult, however, to see whether theprobability should rise up to 1 for vanishing illuminance.

The figures obtained are very comparable to the ones proposed in the Lightswitch-2002model (2), where the intermediate switch-off probability almost constant and equal to 2%between 0 and 200 lux, and drops to 0.002 for higher illuminances. This model, and theSuntool program based on it (3), further set this probability equal to 1 for zero illuminance.

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This makes arguably sense but cannot be confirmed nor ruled out from our data alone, fora practical reason: it is simply extremely unlikely that a user would allow the workplane

illuminance to drop to zero without switching on the lights before, and hence, we do nothave in our data these events.

4.4.2 Switch-off probability

The switch-off probability poses more problems. The statistics are poor; for instance thereare roughly ten times less intermediate switch-off events in office 001 than switch-on events,which explains the huge error bars on the graphics. This indicates that the users switch offtheir lights mostly on the way out of the office, rather than as an intermediate switch-offevent. Fig. 10 shows that the intermediate switch-off probability for the data gatheredfrom all users (except offices 003 and 004) is rather flat and lies at roughly 0.1%, ratherindependently of the illuminance value.

4.5 Correlation between action delay and illuminance

Some models of user behaviour postulate that the time users tolerate a visual discomfortbefore deciding to use the manual controls should be correlated to the level of their dis-comfort. In other words, a user in a very dark or very bright room will act on the controlsearlier than a user in a room whose visual environment is just at the discomfort threshold.

The exact relationship between this delay and the level of discomfort is a question leftunanswered for the moment. Furthermore, it is difficult to measure such a delay until the

user acts on the controls since, usually, visual conditions in the room vary over time andthe user acts only when some discomfort threshold has been crossed. The best we cando, since we have non-constant environmental conditions, is to see if there is at least acorrelation between the illuminance at the time of the users action and the time since theusers entry in the room, in the hope that environmental conditions remain more or lessconstant during the users presence.

Unfortunately, as can be seen on Fig. 12 on a per-office basis or on Fig. 13 for all datagrouped together, there is no such readily discernible pattern. However, this lack of apattern could be entirely due to two statistical reasons. First, users are unlikely to allowlighting conditions at the end of the day to degrade far beyond the discomfort limit and

will thus deny us data points for low illuminance levels. In other words, data points forhigh discomfort levels will not exist simply because the users will have adjusted theircontrols before. Secondly, users can tolerate lighting conditions just at the lower limit ofthe discomfort zone (roughly 200300 lux) indefinitely and will switch the lights on onlywhen moving close to the controls, again depriving us of data points for higher illuminancelevels closer to the comfort zone.

Finding a correlation between the time a user spends before deciding to use manualcontrols (placed within arms length) at given environmental conditions is probably aproject best suited for laboratory conditions, not a real-life building.

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0 100 200 300 400 500

0

4000

8

000

Office 001

[lux]

[s]

0 100 200 300 400 500

1000

3000

5000

Office 002

[lux]

[s]

0 100 200 300 400 500

1000

3

000

Office 003

[lux]

[s]

0 100 200 300 400 500

1000

30

00

Office 004

[lux]

[s]

0 100 200 300 400 500

1000

3000

5000

Office 101

[lux]

[s]

0 100 200 300 400 500

1000

3000

5000

Office 102

[lux]

[s]

0 100 200 300 400 500

500

1000

1500

Office 103

[lux]

[s]

0 100 200 300 400 500

2000

6000

Office 104

[lux]

[s]

0 100 200 300 400 500

500

1000

2000

Office 105

[lux]

[s]

0 100 200 300 400 500

1000

3000

500

0

Office 106

[lux]

[s]

0 100 200 300 400 500

1000

3000

5000

Office 201

[lux]

[s]

0 100 200 300 400 500

0

4000

8000

Office 202

[lux]

[s]

0 100 200 300 400 500

500

1500

2500

Office 203

[lux]

[s]

0 100 200 300 400 500

1000

3000

5000

Office 204

[lux]

[s]

Figure 12: Time between user entry and use of controls vs illuminance levels, per office.

0 500 1000 1500 2000

0

2000

4000

6000

8000

10000

Illuminance before action [lux]

De

lay

be

foreac

tion

[s]

Figure 13: Time between user entry and use of controls vs illuminance levels, all data.

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5 Conclusion

Three-quarters of the LESO users use of manual controls occur less than five minutes aftertheir arrival in their office or before their departure, most likely a direct consequence ofthe traditional placement of their controls close to the door rather than close to their desk.

We have measured intermediate switch-on probabilities for the users of the LESO build-ing as a function of horizontal workplane illuminance for five-minute time intervals. Theincrease in probability for lower illuminances is consistent between users and can be takenas being equal to about 3.3% between 0 and 100 lux, to about 1.4% between 100 and 200lux, to about 0.6% between 200 and 300 lux, and negligible for higher illuminances.

When it comes to actual use of electric lighting controls, we have observed that the users(about 30 people) behave quite differently from each other. It is remarkable that some users

give very frequent commands, while others seem more passive and do not bother using thesystem. This lends credence to the classification in the Lightswitch-2002 algorithm of usersinto different types according to their dynamic or static use of manual controls.

We have also realized that in the LESO building, where lighting controls are placed closeto the offices entrances, the users very seldomly use the dimmable feature of their electriclighting. They almost always switch it completely on or completely off.

Appendix

Link with Nicols probit function

For this final section we will try to establish a link between user actions described by anexponential distribution and the results given by Nicol in his 2001 paper (13).

It will be recalled that Nicol found that the fraction of offices within a given buildingexhibiting a certain user behaviour (such as having the windows open, or having the fansrunning) was a function of an external stimulus; in his case, he considers only the externaltemperature, a hypothesis born out by the study by Fritsch (14). That function was givenby p(x) = exp(a + bx)/

1 + exp(a + bx)

where x was the external temperature and a

and b were parameters to be fitted from experimental data. We will see if we can derivethis result from our assumption about an exponential distribution of delays between useractions.

Let us consider the case of window opening by the user. We choose this example because,although the theory is similar, an open window will usually be closed after a short whileand reopened again during the day, whereas a lamp that has been turned on might remainon until the end of the day. Therefore, we expect the average time between windowopenings and closings to be shorter than the average time between the turning on or off ofartificial lighting, and it should be easier to verify this theory with windows during fieldmeasurements. For this reason, we are going to derive an expression for the fraction ofwindows open after a long enough time, but which will hold only for reversible useractions, i.e. actions that the user effectively undoes during a day. The user will usually

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close a window he or she has opened, but will seldomly switch off a light he or she hasturned on before leaving the office.

Let us assume that the mean time before a user opens a window is To, and once thewindow is open, the mean time before the user closes it again is Tc. The distributionfunction of the time remaining until the next user action is therefore an exponential functionwith parameter o = 1/To or c = 1/Tc respectively.

For simplicitys sake, let us assume all windows in the building start closed. We choose atimestep t sufficiently small so that the probability of the user both opening and closingthe window during that timestep is vanishingly small. If the window begins a timestepclosed, the probability that it should be open at the next timestep is exp(ct). Similarly,if the window begins the timestep open, the probability that it should be closed at thenext timestep is exp(ot).

At time t = 0, the probability that the window is closed is Pc(0) = 1. At t = t wehave:Pc(t) = exp(ot)

At t = 2t the probability that the window is closed is given by the probability thatthe window was closed at t = t and that it remained so at t = 2t, plus the probabilitythat it was open at t = t but that it closed again at t = 2t:

Pc(2t) = Pc(t) exp(ot) +

1 Pc(t)

1 exp(ct)

By recursion, we see that for arbitrary n,

Pc(nt) = Pc

(n 1)t

exp(ot) +

1 Pc

(n 1)t

1 exp(ct)

Expanding the right-hand side all the way down to Pc(0), we finally obtain:

Pc(nt) =

exp(ot) + exp(ct) 1n

+

1 exp(ct) n1

i=0

exp(ot) + exp(ct) 1

i

For large n, and writing T = nt , the first term in that sum vanishes and the

second one is the sum of a geometric serie. We obtain thus

Pc(T = ) =1 exp(ct)

1 exp(ot) + 1 exp(ct)5

5We obtain exactly the same result if we consider the state of the window over time as a Markov processwith two states. The transition probability from closed to open is Tco = 1 exp(ot), and the onefrom open to closed is Toc = 1 exp(ct). The Markov process asymptotically tends to a state inwhich the probability of having the window closed is Toc

Tco+Toc, which is none other than the equation

above.

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Taking the limit t 0, we obtain

limt0

Pc(T = ) = co + c

In his paper, Nicol observed that the fraction of open windows obeyed a relationship withthe outside temperature. Reordering a little bit, he found essentially that the fraction ofclosed windows could be written in the form

Pc(T = ) =1

1 + exp(a + bt),

where t is the outside temperature (not the time). Identifying his finding with ours, wesee immediately that

oc

= exp(a + bt)

We have just found a theoretical relationship between Nicols a and b probit parametersand the average times between users opening and closing of windows.

Notice also that the preceding equation can be rewritten by taking c = 1/Tc, where Tcis the average time before window closure, and similarly for average time before windowopening, and taking the logarithm on both sides:

log Tc log To = a + bt

But since a was an arbitrary constant, and assuming independence between To and t ifthe users need to open the window is taken as independent of the outside temperature,we can redefine it as a = log To + a, and we see thus that

log Tc = a + bt

In other words, we find a theoretical affine relationship between the logarithm of themean time before window closure, and the outside temperature.

Acknowledgements

We wish to thank Jessen Page (LESO-PB/EPFL) for kindly reviewing this paper andproviding helpful and insightful comments on the more mathematical aspects.

We thank Lee-Ann Nicol (LESO-PB/EPFL) for going through this paper and leavingno page without comments for improvements of style or english grammar.

We thank Denis Bourgeois (Ecole dArchitecture, Universite Laval, Canada) for havingbeen the catalyst behind some of the more theoretical developments in this work.

We thank Christoph Reinhart (National Research Concil, Canada) for going throughthis paper and offering his helpful comments for improvement.

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References

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[11] MySQL Development Team. MySQL Reference Manual.

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[13] J. F. Nicol. Characterising Occupant Behaviour in Buildings: Towards a StochasticModel of Occupant Use of Windows, Lights, Blinds, Heaters and Fans. In Proceedingsof the 7th IBPSA conference, pages 10731077, 2001.

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