Time-Step Daylighting Calculation[LINK]
A daylighting calculation is performed each time step that the sun is up for each zone that has one or two daylighting reference points specified. The exterior horizontal illuminance from the sun and sky is determined from solar irradiance data from the weather file. The interior illuminance at each reference point is found for each window by interpolating the daylight illuminance factors for the current sun position, then, for sky-related interior illuminance, multiplying by the exterior horizontal illuminance from the appropriate sky types that time step, and, for sun-related interior illuminance, multiplying by the exterior horizontal solar illuminance that time step. By summation, the net illuminance and glare due to all of the windows in a zone are found. If glare control has been specified window shading (by movable shading devices or switchable glazing) is deployed to reduce glare. Finally the illuminance at each reference point for the final window and shade configuration is used by the lighting control system simulation to determine the electric lighting power required to meet the illuminance setpoint at each reference point.
Table: Variables in Time-Step Calculations
Mathematical variable|Description|Units||FORTRAN variable ———————|———–|—–||—————- Snorm,dir|Direct normal solar irradiance|W/m2|BeamSolarRad Sh,dif|Exterior diffuse horizontal solar irradiance|W/m2|SDIFH, DifSolarRad Sh,dir|Exterior direct horizontal solar irradiance|W/m2|SDIRH Z|Solar zenith angle|radians|Zeta m|Relative optical air mass|-|AirMass Δ|Sky brightness|-|SkyBrightness ε|Sky clearness|-|SkyClearness k, k’|Sky type index|-|ISky sk,k’|Interpolation factor for skies k and k’|-|SkyWeight ψk,k’|Sky luminance distribution formed from linear interpolation of skies k and k’|cd/m2|- fk|Fraction of sky that is type k|-|- Eh,k|Horizontal illuminance from sky type k|cd/m2|HorIllSky Eh,sky|Exterior horizontal illuminance from sky|lux|HISKF Eh,sun|Exterior horizontal illuminance from sun|lux|HISUNF ηdif, ηdir|Luminous efficacy of diffuse and direct solar radiation|lm/W|DiffLumEff, DirLumEff Iwin|Interior illuminance from a window|lux|DaylIllum Swin|Window luminance|cd/m2|SourceLumFromWinAtRefPt Bwin|Window background luminance|cd/m2|BACLUM dsun, dsky,k|Interior illuminance factor for sun, for sky of type k|-|DaylIllFacSun, DFSUHR, DaylIllFacSky, DFSUHR wsun, wsky,k|Window luminance factor for sun, for sky of type k|-|DaylSourceFacSun, SFSUHR, DaylSourceFacSky, SFSKHR bsun, bsky,k|Window background luminance factor for sun, for sky of type k|-|DaylBackFacSun, BFSUHR, DaylBackFacSky, BFSKHR wj|Weighting factor for time step interpolation|-|WeightNow ||| ||| iL|Reference point index|-|IL iS|Window shade index|-|IS Itot|Total daylight illuminance at reference point|lux|DaylIllum Btot, B|Total window background luminance|cd/m2|BLUM Iset|Illuminance setpoint|lux|ZoneDaylight%IllumSetPoint fL|Fractional electric lighting output|-|FL fP|Fractional electric lighting input power|-|FP NL|Number of steps in a stepped control system|-|LightControlSteps MP|Lighting power multiplier|-|ZonePowerReductionFactor
Time-Step Sky Luminance[LINK]
The sky luminance distribution, ψ, for a particular time step is expressed as a linear interpolation of two of the four standard skies — ψcs, ψts , ψis and ψos — described above under “Sky Luminance Distributions.” The two sky types that are interpolated depend on the value of the sky clearness. The interpolation factors are a function of sky clearness and sky brightness (Perez et al., 1990). Sky clearness is given by
where Sh,dif~~is the diffuse horizontal solar irradiance, Snorm,dir is the direct normal solar irradiance, Z is the solar zenith angle and κ is a constant equal to 1.041 for Z in radians.
Sky brightness is given by
where m is the relative optical air mass and is the extraterrestrial direct normal solar irradiance.
If ε ≤ 1.2
where ψis is the intermediate sky luminance distribution, ψos is the overcast sky luminance distribution, and
If 1.2<ε ≤ 3
where ψts is the clear turbid sky luminance distribution and
If ε > 3
where ψcs is the clear sky luminance distribution and
For each time step the interior illuminance, Iwin, from a window is calculated as follows by multiplying daylight factors and exterior illuminance.
First, the sun- and sky-related daylight illuminance factors for the time step are determined by interpolation of the hourly factors:
where iL is the reference point index (1 or 2), iS is the window shade index (1 for unshaded window, 2 for shaded window), ih is the hour number, and k is the sky type index. For the j th time step in an hour, the time-step interpolation weight is given by
where Nt~~is the number of time steps per hour.
The interior illuminance from a window is calculated as
where Eh,sun and Eh,sky are the exterior horizontal illuminance from the sun and sky, respectively, and fk and fk’ are the fraction of the exterior horizontal illuminance from the sky that is due to sky type k and k’, respectively.
The horizontal illuminance from sun and sky are given by
where Z is the solar zenith angle, ηdif is the luminous efficacy (in lumens/Watt) of diffuse solar radiation from the sky and ηdir is the luminous efficacy of direct radiation from the sun. The efficacies are calculated from direct and global solar irradiance using a method described in (Perez et al, 1990).
The fractions fk and fk’ are given by
where Eh,k and Eh,k’ are the horizontal illuminances from skies k and k’, respectively (see “Exterior Horizontal Luminance,” above), and sk,k’ is the interpolation factor for skies k and k’ (see “Time-Step Sky Luminance,” above). For example, if ε > 3, k = cs (clear sky), k’ = ts (clear turbid sky) and
Similarly, the window source luminance, Swin, and window background luminance, Bwin, for a window are calculated from
The total illuminance at a reference point from all of the exterior windows in a zone is
where iS = 1 if the window is unshaded and iS = 2 if the window is shaded that time step. (Before the illuminance calculation is done the window shading control will have been simulated to determine whether or not the window is shaded.)
Similarly, the total background luminance is calculated:
The net glare index at each reference point is calculated as
In the last relationship, the background luminance is approximated as the larger of the background luminance from daylight and the average background luminance that would be produced by the electric lighting at full power if the illuminance on the room surfaces were equal to the setpoint illuminance. In a more detailed calculation, where the luminance of each room surface is separately determined, B(iL) would be better approximated as an area-weighted luminance of the surfaces surrounding a window, taking into account the luminance contribution from the electric lights.
Glare Control Logic[LINK]
If glare control has been specified and the glare index at either reference point exceeds a user-specified maximum value, GI,max, then the windows in the zone are shaded one by one in attempt to bring the glare at both points below GI,max. (Each time a window is shaded the glare and illuminance at each reference point is recalculated.) The following logic is used:
If there is only one reference point, shade a window if it is unshaded and shading it decreases the glare, even if it does not decrease the glare below GI,max. Note that if a window has already been shaded, say to control solar gain, it will be left in the shaded state.
If there are two reference points, then:
If glare is too high at both points, shade the window if it decreases glare at both points.
If glare is too high only at the first point, shade the window if the glare at the first point decreases, and the glare at the second point stays below GI,max.
If glare is too high only at the second point, shade the window if the glare at the second point decreases, and the glare at the first point stays below GI,max.
Shades are closed in the order of window input until glare at both points is below GI,max, or until there are no more windows left to shade.
Lighting Control System Simulation[LINK]
Once the final daylight illuminance value at each reference point has been determined, the electric lighting control is simulated. The fractional electric lighting output, fL, required to meet the setpoint at reference point iL~~is given by
Here, Iset is the illuminance setpoint and Itot
is the daylight illuminance at the reference point. This relationship assumes that the electric lights at full power produce an illuminance equal to Isetat the reference point.
The fractional electric lighting input power, fP, corresponding to fL is then calculated. The relationship between fP and fL depends on the lighting control type.
Continuous Dimming Control[LINK]
For a continuously-dimmable control system, it is assumed that fP is constant and equal to fP,min for fL<fL,min and that fP increases linearly from fP,min to 1.0 as fL increases from fL,min to 1.0 (Figure 59). This gives
Control action for a continuous dimming system.
Continuous/Off Dimming Control[LINK]
A “continuous/off” dimming system has the same behavior as a continuous dimming system except that the lights switch off for fL< fL,min rather than staying at fP,min.
For a stepped control system, fP takes on discrete values depending on the range of fL and the number of steps, NL~~(Figure 60). This gives
If a lighting control probability, pL, is specified, fP is set one level higher a fraction of the time equal to 1-pL. Specifically, if fP < 1, fP fP + 1/NL if a random number between 0 and 1 exceeds pL. This can be used to simulate the uncertainty associated with manual switching of lights.
Stepped lighting control with three steps.
Lighting Power Reduction[LINK]
Using the value of fP at each reference point and the fraction fZ of the zone controlled by the reference point, the net lighting power multiplier, MP, for the entire zone is calculated; this value multiplies the lighting power output without daylighting.
In this expression, the term to the right in the parentheses corresponds to the fraction of the zone not controlled by either reference point. For this fraction the electric lighting is unaffected and the power multiplier is 1.0.
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Hopkinson, R.G., P. Petherbridge and J. Longmore. 1966. Daylighting. Heinnemann, London, p. 322.
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