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or, in scalar form:
or
∆F ∆k
=
∆F ∆k
Thermoeconomic Diagnosis
( ) t ∗
( K
P ) t ( + ⋅ ∆ 〈 KP〉
) ⋅ P
=
∆ K ext
n
∑
i = 1
⎛ n
⎞
⎜
K
∗ ⎟
⎜∑P, j ⋅ ∆kji⎟
⋅
⎜ ⎟
⎝j= 0 ⎠
Thermoeconomic diagnosis of an urban heating system based on cogenerative steam and gas turbines 140
(4.42)
. (4.43)
The total fuel impact associated to the variation of the overall production is so:
∆F ∆P
∆F ∆P
=
K
∗
P, i ∆Pexti . (4.45)
If the free and reference conditions are considered, the total fuel impact is zero, so:
. (4.46)
4.7 A procedure for the multiple malfunction detection
In a general operation condition a plant can be characterized by more than one anomaly.
The procedure shown in the previous paragraph allows to locate only the anomaly associated
to the maximum element of the ∆K matrix. In this way the location of all the anomalies passes
through the location and the complete removal of every anomaly separately. This means that
this procedure must be repeated as many times as the number of malfunctions.
Supposing that all the induced effects could be eliminated, the contemporary location of all
the anomalies would be possible. In this paragraph a procedure to eliminate the induced
effects is proposed.
The plant components are each characterized by its own behaviour, so they act differently
when the working conditions vary. The effects must be analysed for each one separately. If
the productive structure is considered, the characteristic behaviour of a component can be
modeled by varying all its resources and determining the corresponding product.
The dependence of the product of every component from its fuels can be found if three
hypotheses are complied:
1) the known working conditions are linearly independent;
2) the number of known conditions is higher than the fuels of every components;
3) the anomalies are sufficiently low.
The procedure for the location of the anomalies, proposed in the previous paragraphes, is
based on the use of known working conditions corresponding to different regulations. The
same knowledge can be used to eliminate the induced effects from the matrix ∆K. The values
of the fuels of a component are assumed equal to the known values in free condition. If the
third condition is complied, the product can assumed linearly dependent on its fuels:
0
Pi ∆F ∆P
∗
( K
P ) , (4.44)
t = ⋅ ∆Pext n
∑
i = 1
∆F∆k =
⋅
– ∆F∆P