Page 310 - Distributed model predictive control for plant-wide systems

284 Distributed Model Predictive Control for Plant-Wide Systems

T
discrete system sampling time, x (k) = [p (k) p (k + 1) ··· p (k + n )] is the state variable,
s
d
s
s
s
A ∈ ℝ (n d +1)×(n d +1) and B ∈ ℝ (n d +1)×1 are decided by the sampling time and the time constant
of electric refrigerator. Let the sampling time be Δt, we can obtain
⎡ 0 1 ··· 0 ⎤
⎢0 0 ⋱ ⋮ ⎥
A = ⎢ ⋮ ⋱ ⋱ 1 ⎥
s
⎢ ⎥
⎣0 ··· 0 e −Δt∕ ⎦
[ ] T
B = 0 ··· 0 1 − e −Δt∕
s
[ ]
C = 1 0 ··· 0 (13.10)
s
13.3.2.2 MPC Model of Subsystems
In each subsystem, the real-time reference of supplying the cold power by an electric refrig-
erator is decided by the current MPC and other MPCs of other subsystems.
For the subsystem s (s = 1, 2, … , N), MPC coordinate strategy includes the tracking of the
upper level optimal reference r in the future P sampling time and it is not the normal MPC
s
strategy. In this chapter, we introduce a sum of square of the difference between the total
predictive power and all expected powers to track total expected power r which makes the
,
objective functions coupling from different subsystems. Above all, the MPC objective function
of the subsystem s is
P
∑ 2
min J = Q (r (k + i)− ̂p (k + i|k))
s s s
U s
i=1
2
P ⎛ N ⎞
∑ ⎜ ∑ ⎟
+ R ⎜ (k + i) − ̂p (k + i|k)− ̂ p (k + i|k) ⎟ (13.11)
r
s
j
i=1 ⎜ j=1 ⎟
⎝ j≠s ⎠
T
where U = [u(k) u(k + 1) ··· u(k + P − 1)] is the reference sequences of the electric refrig-
s
erator output power in the predictive horizon. Q, R ∈ [0, 1] are the weights used to measure
the predictive load tracking level for a single electric refrigerator and all electric refrigerators,
respectively. Each system solves the self-optimal question with not only the output in the pre-
dictive horizon based on the model itself but also the tracking level of the total expected power
based on the other subsystems. Based on the last section, we can get the predictive model as
i
⎧ ∑ i−j
i
⎪̂ x (k + i|k) = A x (k)+ (A B u (k + j − 1))
s s s s s s
⎨ j=1
⎪
⎩ ̂p (k + i|k)= C ̂ x (k + i|k)
s
s s
i = 1, 2, … , P (13.12)
Above all, for the MPC controller of the sth subsystem the optimal problem form is as
follows: