Wireless Sensor and Actuator Networks for Lighting Energy ...

E sub
=
e pq
e rs
e xy
1
= L sub
d =
1
1
l pq
l rs
l xy
2
l pq
2
l rs
2
l xy
K
l pq
K
l rs
K
l xy
d 1
d 2
d K
(6.4)
This problem is formulated into a linear programming problem as shown in (6.5)
. By minimizing the 1-norm of vector d, the summation of the output levels from each
luminaire is minimized, which translates into minimizing the energy usage of the
resulting settings. The lighting preferences of the occupants are posed as the constraints
to the linear programming problem. The physical dimming capabilities of each
luminaire, DimLevel min and DimLevel max , are also considered as constraints.
min d
1
, subject to
L sub
d = E sub
,
(6.5)
DimLevel min
d DimLevel max
However, this optimization problem may not have a solution, depending on how
each of the points of interest is specified in the equality constraint. Infeasible problems
are most likely caused by the occupants’ conflicting lighting preferences. In case of
infeasibility, the equality constraints are relieved to inequality constraints, thus allowing
some tolerances. The relief of the equality constraints makes sense in that the physical
meaning of this is to allow the lights at the points of interest to be regulated in certain
tolerable ranges ( tol ) instead of demanding the exact illuminances. Studies have shown
that people are insensitive to 20% of illuminance changes and are willing to accept an
illuminance change up to 30% [100]. In summary, the algorithm starts with the original
linear programming problem with equality constraints and gradually expands the
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