Civil Engineering Reference
In-Depth Information
ˆ
u
are the predictions of PMV index,
indoor temperature reference change, and control signal (fancoil power) change,
respectively. Note that unlike Eq.
5.8
the weights that penalise the tracking errors
and the control efforts,
In Eqs.
5.18
and
5.19
,
y
PMV
,
w
T
, and
respectively, are constant along the horizons, thus,
they can be taken out of the summations. Moreover,
˃
and
ˉ
1.
As mentioned above, two different cost functions are considered to be evalu-
ated in this control problem, Eqs.
5.18
and
5.19
.
1
In both of them, the first part
of the cost function tries to minimise the PMV index within the prediction hori-
zon
N
(
˃
value has been chosen as
˃
=
T
)byusing
the static nonlinear relation between indoor temperature and PMV index described
resented by Eq.
5.18
weighs abrupt changes in the indoor setpoint temperature
(
y
PMV
=[ˆ
ˆ
y
PMV
(
k
+
1
|
k
),
ˆ
y
PMV
(
k
+
2
|
k
),...,
ˆ
y
PMV
(
k
+
N
|
k
)
]
T
). Thus, by means of
a nonlinear optimiser, the control law based on Eq.
5.18
finds an indoor tempera-
ture setpoint which is a tradeoff between thermal comfort and abrupt changes in the
indoor temperature.
In the case of the cost function defined by Eq.
5.19
, its second part weighs the
current and future fancoil actions changes (
w
T
=[
w
T
(
k
),
w
T
(
k
+
1
), . . . ,
w
T
(
k
+
N
u
−
1
)
]
u
=[
u
(
k
),
u
(
k
+
1
), . . . ,
u
(
k
+
T
) that are necessary to reach the indoor temperature value associated to the
PMV index provided in the first part of the cost function. Notice that these fancoil
actions are related to the indoor temperature through the discrete time version of
the first order model represented by Eq.
5.16
. Moreover, the indoor temperature is
related to the decision variable of the optimisation problem (the setpoint indoor air
temperature
w
T
) as a closed loop is implemented in this lower layer implementing a
PI control aimed at tracking the desired setpoint, so that, the optimisation problemcan
be formulated in terms of
w
T
or
N
u
−
1
)
]
w
T
. Moreover, these control signals are associated
to energy costs by means of the the fancoil technical specification. Therefore, this
second cost function finds an indoor temperature setpoint as a tradeoff between
thermal comfort and energy saving.
On the other hand, the optimisation problem is subjected to several system
constraints given by Eqs.
5.20
and
5.21
. The first ones are affecting both cost func-
tions, Eqs.
5.18
and
5.19
, and gives the lower limit (
w
T
min
) and the upper limit
(
w
T
max
) of the output variable, that is, the indoor temperature reference (
w
T
). The
second constraint is only affecting Eq.
5.19
and makes reference to physical hard
constraints of the HVAC system, in other words, it takes into account the fancoil
saturation [
u
min
,
u
max
].
w
T
min
≤
w
T
(
k
+
j
−
1
)
≤
w
T
max
∀
j
=
1
,...,
N
u
(5.20)
u
min
≤
u
(
k
+
j
−
1
)
≤
u
max
∀
j
=
1
,...,
N
u
.
(5.21)
1
Notice that a combined cost function grouping
J
k
1
and
J
k
2
could also be considered.
