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Fig. 2. Binary variables of MILP approach
constraints. Finally, it is common to define an upper limit to the number of
turns on and off during the scheduling period N on = N off = N change .
N I
y ( i )
N change
i =0
(7)
N I
z ( i )
N change
i =0
For instance, for a one-day scheduling period with the CHP in one day, and
N on = N off = 1, this means that CHP can be turned on and off just once.
4
Immune Scheduling Approach
The second approach is based on the opt-aiNet version [6] of the clonal selec-
tion algorithm. The optimization procedure (AIS-LP) is divided into two nested
stages: the inner one is the LP problem derived in Section 2 which defines the
optimal production levels at each time interval once the on/off profiles are de-
fined. The outer stage is responsible defining the on/off status of the generation
units.
It is useful to use as degrees of freedom of the optimization the time ampli-
tudes of the on and off intervals τ j of the CHP (Fig. 3). These values are treated
as integer variables representing the number of on and off intervals of each con-
trol period. The variables are then decoded in terms of 0-1 strings representing,
for each utility, its on/off status. This assumption drastically simplify the op-
timization search. The number of available solutions is in fact equal to M N ,
where N is the number of degrees of freedom and M the number of possible val-
ues assumed by each variable. A fine discretization does not affect the number
of variables but only their range of values M , thus the overall complexity of the
problem is polynomial. With a MILP approach, M is always equal to 2, because
the problem is modeled by binary variables. The time discretization affects the
value of N , giving rise to an exponential complexity of the problem. Moreover,
in AIS-LP approach, the value of M is restricted when including MOT/MST
 
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