1.1 Problem formulation

The goal in this chapter is to study the use of the feed-water flow-rate as a manipulated variable to maintain the SG water level within allowable limits, in the face of the changing steam demand resulting from a change in the electrical power demand. The design goal of an FMPC is to minimize the predictive error between an output and a given reference trajectory in the next Ny steps through the selection of Nu step optimal control policies.

The optimization problem can be formulated as

min J(n) (6)

Au(n),Au(n+1,…,Au(n+Nu))

Ny Nu

J(n) = (y(n + i) — yr (n + i ))2 + ‘^uiAu(n + i )2

І=1 І=1

where:
Mi and 4	are the weighting factors for
y(n +i)	ith step output prediction;
yr(n +i)	ith step reference trajectory;
Au(n + i)	ith step control action.

The weighted sum of the local control policies gives the overall control policy:

P

Au(n + i) = ^®;AMj (n + i) (8)

j=1

Substituting (2) and (8) into (7) yields (9)

‘ p у

^ffljAuj (n + i)

U1 )

To simplify the computation, an alternative objective function is proposed as a satisfactory approximation of (9) (Huang et al., 2000).

min J(n) =

Au(n),Au(n+1),…,Au(n+Nu-1)

P 2

min L(yi) +J (n)

Au(n),Au(n+1),…,Au(n+Nu-1) і=1 V ‘

~ У / ‘ 2

V (n) = Ltt (y і (n+г)- у (n+г))

i=1

Nu-1 2

+ L yi (Aui (n+г))

Using the alternative objective function (12), we can derive a controller by a hierarchical control design approach.

1.2 Controller design

1. Lower Layer Design: For the jth subsystem, the optimization problem is defined as follows:

min Ji (n) (13)

Au(n),Au(n+1),…/Au(n+Nu-1)

IF у(n + k -1) is A0/•••, у(n + k- m) is A, m_1 Rj: . . t (14)

THEN у. (n + k) = у. (n + k -1) +L h. Au (n + k — i) +6і (n + k -1)

i=1

where є1 (n + k -1) serves for system coordination and it is determined at the upper layer.

2. Upper Layer Design: The upper layer coordination targets the identification of globally optimal control policies through coordinating 6і (n + k -1) for each local subsystem.

3. System Coordination: The controller is designed through a hierarchical control design (Figure 1). From the lower layer, the local information of output and control is transmitted to the upper layer. The whole design is decomposed into the derivation of p local controllers. The subsystems regulated by those local controllers will be coordinated to derive a globally optimal control policy.

The objective function defined in (11) can be rewritten in a matrix form:

J. (n) = (Y+ (n) — Yr (n))T W. (Y+ (n) — Yr (n)) +

+ (A U + (n))T W2 (A U + (n))

y* (n / k) = y* (n / k — 1) / 2 hj Au(n / k — i) / S (n / k — l) i=1 to obtain: (aU / (n)) = —Kj (y (n) — Yr (n) / P (n)/P (n)) Y/ (n) = AJ AU/ (n)/ Y(n)/ PJ (n)/ EJ (n) Subsystem j

where:

Y/ (n) = (y 1 (n /1) y 1 (n / 2) — y 1A / Ny)) (16) Yr (n) = (yr (n /1) yr (n / 2) • ••y A/ Ny )f (17) AU/ (n) = (и1 (n) Au1 (n A1) — Au1 A / Nu — 1)) (18) Wj = diag | Y, Yi, — ‘,Mk} (19) W2j = diag |vj, v2′, — ‘j } (20)

1. STEP 1: y(n / k-1) = 2P yj (n / k -1), Є (n / k—1) = y(n / k — 1) — yl (n / k—1) 2. STEP 2: P Ny I, I etot = 2 2 eJ (n / k — 1) — s(n / k — 1) j=1k=1 1 3. STEP 3: If etot < g, then an optimal control action is found; else, let s’ (n / k — 1) = e (n / k — 1) and send it down to each local controller for recalculation.

Y/(n/k) AU/(n / k)

S (n / k — 1)

Jj (n) = 2Mi ((n /’)— yr (n / i) / 2v/ (amj (n /1)) i=1 i=1 for:

J

Fig. 1. Hierarchical controller design

(21) (22) (23) (24) (25) (26) (27) (28) (29) (30)
a[ 0 0 — 0 4 a1 0 0 a3 a2 a1 0 aN У a UNy-1 a1 — UNy — 2 a uNy — Nu+1 al J? —W2 II
Y(n) = (УпаІ (n) УпаІ (n) — У (n)Dd)
P1 (n )=( Pi (n) P’ (n) — Pjy (n))
yT
E+ (n ) =
i T РІ (n) = ^ ^ hlAu(n + k -1) k=1 l=k+1
The resulting control policy for the fh subsystem can be derived as
J1 (n) = (a U+ (n) ) T (a 1 W1A + W2′) A U+ (n) + + (a U+ (n)) T A1 W/Z1 (n) + + ((n)) T W1A1AU+ (n) + ((n)) T WjZ1 (n) where:
Z1 (n) = Y(n) — Yr (n) + P1 (n) + E+ (n) Minimizing (26) yields
S~J (n) = 2(A1T W11A1+ W2) AU+ (n) SAU+ (n) 1 2) +У ’ + 2A1 W1Z1 (n) = 0

The control law by the jth FI can be identified as

(A U + (и ))=- Kj Zj (n) (31)

Kj = (a ’T W1Aj + W2) 1A’T W/ (32)

The optimal global control policies can be derived at the upper layer.

AU+ (n) = (u(и) Am(n +1) ••• Au(n + Nu -1)T (33)