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6.1 Prediction Algorithms
Valid and reliable forecast information on the expected PV power production and
home consumptions play a primary role for the design of an energy management
system and to
find the best time to start an appliance.
The following approach to implement a Minimal Resource Allocating Network
(MRAN) is based on a sequential learning algorithm and an Extended Kalman Filter
(EKF) Kadirkamanathan and Niranjan (
1993
), Platt (
1991
), Sundararajan et al.
(
2002
). In particular the sequential learning algorithm adds and removes neurons on-
line to the network according to a given criterion (Platt
1991
), (Sundararajan et al.
2002
; Yingwei et al.
1998
), and an EKF is used to update the net parameters
(Kadirkamanathan and Niranjan
1993
).
6.1.1 Radial Basis Function Neural Network
m
A RBFN with input pattern x
2
R
and a scalar output
y
2
R
implements a
m
mapping f
:
R
!
R
according to
X
K
i¼1
k
i
/
^
¼
ð
Þ
¼
k
0
þ
ð
k
k
Þ
ð
7
Þ
y
f
x
x
c
i
R
þ
to
where
˕
(
·
) is a given function from
,
kk
denotes the Euclidean norm,
ʻ
i
,
R
m
, i =1,2,
i =0,1,
, K, are the radial basis
function centers (called also units or neurons) and K is the number of centers Chen
et al. (
1991
). The terms:
…
, K are the weight parameters, ci
i
2
R
…
o
i
¼ k
i
/
ð
k
x
c
i
k
Þ;
i
¼
1
; ...;
K
ð
8
Þ
are called the hidden unit outputs.
In this work the RBFN is used for the prediction of the output of a dynamical
system and the system dynamics can be taken into account through the network
input pattern x, that must be composed of a proper set of system input and output
samples acquired in a
2
finite set of past time instants Hunt et al. (
1992
), i.e. x
n
y
þ
n
u
and it is de
ned as:
R
T
x
ð
n
Þ
¼½
y
ð
n
1
Þ; ...;
y
ð
n
n
y
Þ;
u
ð
n
1
Þ; ...;
u
ð
n
n
u
Þ
ð
9
Þ
…
·
·
where n = 1,2,
) are the system output and inputs
(for a detailed description see Sect.
6.2
), respectively; n
y
, n
u
are the lags of the
output and input, respectively.
Theoretical investigation and practical results show that the choice of the non-
linearity
are the time instants, y(
) and u(
), a function of the distance di
i
between the current input x and the centre
c
i
, does not signi
˕
(
·
uence the performance of the RBFN Chen et al. (
1991
).
Therefore, the following gaussian function is considered:
cantly in
fl
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