This work was supported by Syncrude Canada Limited and Natural. Sciences and Engineering Research Council of Canada (NSERC). A. Swarnakar, H. J. ...
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007
ThPI23.17
A Design Framework for Overlapping Controllers and its Application Adarsha Swarnakar, Horacio Jose Marquez and Tongwen Chen
Abstract— This paper presents a new practical framework for decentralized output feedback design with overlapping information structure constraints. In contrast to the earlier works, the proposed method removes restriction in the control design algorithm by utilizing congruence transformations, simplifications and the reciprocal variant of the projection lemma. This leads to a less conservative solution than the previous design methods because the choice of some design parameters by trial and error is eliminated. Moreover, in some cases, the structural constraint of having a diagonal Lyapunov function in linear matrix inequalities (LMIs) is removed. The results are then extended to capture a more general scenario of output feedback control design for nonlinear interconnected systems. The validity of the proposed approach is demonstrated through applications to an industrial utility boiler and to a multi-area power system.
problem in the expanded region. Using the inclusion principle, the expanded control laws are then transformed to the original space for implementation. One important problem that may arise while using this approach results from the fact that this method is not applicable if some subsystem (A22 ) is unstable. The eigenvalues of this subsystem represent fixed modes of the expansion space and limit the practical appeal of this algorithm. In [7], [1], an approach towards the (partial) elimination of this weakness is shown. This method solves static state feedback problem for both Type I and Type II overlapping (Fig. 1 [7]). For Type I, the input matrix and the control law has the following form h i B11 0 K12 0 B B 21 22 , K = K011 K B= , (1) K 23 24 0
I. I NTRODUCTION In many practical control systems, specific structures of controllers are often used rather than a centralized architecture. The reasons for this preference are diverse and include failure tolerance, less communication overhead and wide acceptance by operators in the industry. Moreover, robustness is another interesting feature of decentralized control, because it can lead to stability of the closed loop system in the presence of wide range of uncertainties, both within the subsystems and in their interconnections [1]. For this reason, the last few decades have seen ongoing research interest in the design of decentralized control systems [2], [3], [1], [4], [13], [14]. However, there are only few design procedures for decentralized structures due to non-convexity of the design problems. It is well known that if some state information is shared among the subsystems then the concept of overlapping control arises [2], [1]. The local controllers then use these extra information to improve the stability properties and performance characteristics of the overall closed loop system. Practical examples of this kind include vehicle platooning, power systems and web handling systems [5], [6], [7], [8]. However, in many cases, finding an explicit solution to this control design is still an open problem [7]. In the past, expansion and inclusion principles were widely used for designing overlapping control laws [9], [2], [10]. By the expansion principle, the systems are first expanded into a larger space where they appear as disjoint. As a result, the control design can be viewed as a decentralized control This work was supported by Syncrude Canada Limited and Natural Sciences and Engineering Research Council of Canada (NSERC). A. Swarnakar, H. J. Marquez, and T. Chen are with Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada.
(adarsha, marquez, tchen)@ece.ualberta.ca
1-4244-1498-9/07/$25.00 ©2007 IEEE.
B32
with Type II corresponding to the situation when B21 = B22 = 0. The proposed method combines LMIs [11] and the system expansion to eliminate the fixed mode problem. However, in the Type II overlapping case, the selection of some parameters in the optimization problem remains an open question. They are generally selected on a trial and error basis to convert the nonlinear optimization problem into LMIs. Moreover, for both overlapping (Type I and Type II), the algorithm requires a Lyapunov function with special structural constraint and, in some cases, a diagonal version of Lyapunov functions. This is very restrictive and it degrades the robustness of the closed loop system. In some cases, it may even lead to infeasibility of the optimization problem and so it may not be very user friendly to control engineers. This open and challenging problem constitutes the motivation of development in this paper. In this paper, the authors proposed two different techniques to solve the aforementioned problem. Different congruence transformations, some simplifications, change of controller variables method [15], [4] and the reciprocal variant of the projection lemma [12] are used to obtain less conservative LMI solutions. This is due to the fact that the restriction of using diagonal Lyapunov functions and choice of parameters by trial and error are ruled out in this approach. Moreover, the method is extended to capture a more general scenario of output feedback control design. The results are also generalized to include large scale nonlinear interconnected systems. Some interesting observations of the algorithm, which is a source of attraction to both theorists and practitioners are: 1) Method I: A general algorithm has been developed for linear as well as nonlinear systems, which deals with both Type I and Type II overlapping. The method can handle static state feedback, static output feedback, full order and reduced order dynamic output
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 x1
u1
x1
u1
x2
x2
u2
x3
Fig. 1.
u2
x3
Type I and Type II overlapping [7]
feedback control design. There is no need of selecting any parameters by trial and error or imposing any structural constraint on the Lyapunov function. The overlapping control design problem is converted into an optimization problem which involve LMIs and only one equality constraint of the form: Q = M T M . This constraint is then relaxed as h i Q MT ≥ 0, (2) ⋆ I
and an iterative algorithm is used to compute the controller parameters. The proof of convergence of this algorithm guarantees that the objective function value strictly decreases in each step. It should be noted that Q = M T M corresponds to the boundary of the convex sets in (2). As the optimization involves only one equality constraint, few iterations are required for the control design. Moreover, each step involves solving LMIs and there is no requirement of any initial guess. The generalization of the results to N nonlinear interconnected systems are also quite straightforward. The algorithm can also accommodate many other structures of the controllers, namely, decentralized design and control design when overlapping states are shared by multiple subsystems. This makes the result very general and increases its applicability to a wide range of practical systems. 2) Method II: The control design problem for Type II is converted into a sequential two-part optimization problem using different congruence transformations and change of controller variables method. Then, a two step method (similar to [13]) is used for computing the controller parameters. The advantage of this approach is that no iteration is required and the control law can be obtained only in two steps. However, this method requires block diagonal Lyapunov functions and cannot handle static output feedback control designs. To show that the approach is also practically relevant, two engineering problems are considered. In the first case, an overlapping load frequency control law for a two area power system is designed. The areas represent the subsystems and tie-lines are the overlapping parts. It is shown that the scenario corresponds to Type II overlapping case and the designed controller keeps the system frequency and the interarea tie line power to scheduled values in the presence of load variations. The stability is also studied in the presence of generating rate constraint (GRC), i.e., the practical limit on the response speed of the turbine, and the results are
ThPI23.17 compared with the decentralized design. In the second case, the authors focused their attention on the control problem of Syncrude Canada Ltd. integrated energy facility. This utility plant consists of a boiler system, an electricity generating system and a header system at four different pressure levels (6.306, 4.24, 1.068 and 0.372 MPa). At present, decentralized PI controllers for most part work well, however, the 6.306 MPa header pressure shows oscillatory behavior under load fluctuations which the controllers are unable to damp out quickly. This happens due to the nonlinear characteristics of the plant and the interactions arising from overlooking the multivariable nature of the plant. To overcome this problem and to bring overall stability under load variations, the authors identified the nonlinear interconnected model of utility boilers and the 6.306 MPa header. The pressure equation is defined by an identified data fit and the drum water level based on first principles. Inputs to the model are feedwater flow rate, firing rate (to control air and fuel flow), attemperator spray flow rate and the outputs are: drum level, header pressure and steam temperature. Different overlapping controllers are designed and their performance under load variations are compared with the existing PI controllers in a nonlinear simulation package of Syncrude boilers called SYNSIM. All controllers are linear, so their implementation is straightforward and cost effective. II. A S OLUTION TO OVERLAPPING C ONTROL D ESIGN Consider a nonlinear process of the form [3], [7], [4] x˙ = Ax + Bu + h(x), y = Cx,
(3)
where x(t) ∈ ℜn is the state, u(t) ∈ ℜm is the control input, y(t) ∈ ℜp is the output, C = diag (C1 , C2 , C3 ) and B11 0 A11 A12 A13 B B A A A , , B= A= 21 22 21 22 23 A31
A32
A33
0
B32
with B21 = B22 = 0 for Type II. The function h(x) is assumed to be uncertain, but bounded by [3], [7], [4] hT (x)h(x) ≤ α2 xT H T Hx, where H is a constant matrix and α is a scalar parameter that reflects the degree of robustness. Consider a static output feedback overlapping controller of form u = Ky (K in (1)), and a dynamic output feedback overlapping control law x˙ k1 = Ak1 xk1 + Bk11 C1 x1 + Bk12 C2 x2 , u1 = Ck1 xk1 +Dk11 C1 x1 + Dk12 C2 x2 , x˙ k2 = Ak2 xk2 + Bk21 C2 x2 +Bk22 C3 x3 , u2 = Ck2 xk2 + Dk21 C2 x2 + Dk22 C3 x3 . (4) In both cases, the closed loop system is given by ¯ d C)x ¯ cl + hr (xcl ) = AˆD xcl + hr (xcl ), (5) x˙ cl = (A0 + BK ¯ = B, Kd = K, C¯ = C, where, for static case, A0 = A, B xcl = x and hr (xcl ) = h(x) is bounded by hTr (xcl )hr (xcl ) ≤ α2 xTcl H T Hxcl = α2 xTcl HlT Hl xcl .
(6)
However, for the case of dynamic output feedback with xTcl = xT1 xT2 xT3 xTk1 xTk2 ,
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 ¯ d C¯ = AˆD = A0 + BK
"
"
Ak 1 0 Ck 1 0
×
0 0 0 I 0
0 0 0 0 I
"
0 0 C1 0 0
#"
0 B22 B32 0 0
B11 B21 0 0 0 0 0 0 C2 0
0 0 0 0 C3
I 0 0 0 0
and hTr (xcl )hr (xcl )
0 I 0 0 0
A11 A21 A31 0 0
A12 A22 A32 0 0 0 Ak 2 0 Ck 2
#
Bk 11 0 Dk 11 0
0 0 0 0 0
Bk 12 Bk 21 Dk 12 Dk 21
#
+ #
0 Bk 22 0 Dk 22
,
(7)
HT H 0 0 0 0 0 2 T T α xcl Hl Hl xcl .
≤ α2 xTcl =
0 0 0 0 0
A13 A23 A33 0 0
0 0 0
(8)
x˙ = Ax + Bu + hint (x), y = Cx, (9) T T where hint (x) = h1 (x) hT2 (x) hT3 (x) . The functions hi (x) contains the nonlinearities in the subsystems itself and in the interconnections. They are assumed to be bounded by a quadratic inequality [3], [4], [14] hTi (x)hi (x) ≤ α2i xT HiT Hi x, i = 1, 2, 3
(10)
where αi ’s are the interconnection parameters. Here, with static output feedback control law 3 X α2i HiTl Hil )xcl , hTint (xcl )hint (xcl ) ≤ xTcl (
(11)
i=1
where Hil = Hi . With dynamic output feedback control law: hTi (xcl )hi (xcl ) ≤
HiT Hi 0 α2i xTcl 0 0 0 0 | {z HiT Hil l
0 0 xcl , 0 }
i = 1, 2, 3.
In the following theorem, sufficient conditions are provided for the existence of a stabilizing overlapping control law for the nonlinear system in (5). It is assumed that (A, B) is stabilizable, (C, A) is detectable and the system has no unstable fixed modes for the control structure in (1). Theorem 2.1: If there exists a controller Kd such that the following optimization is feasible
−Q ⋆ ⋆ ⋆ ⋆
ˆT + M T A D −I ⋆ ⋆ ⋆
XP 0 −I ⋆ ⋆
HlT 0 0 −γI ⋆
min γ s.t XP > T
XP − M 0 0 0 −I
0 such that P > 0 and h ˆT i ˆD + τ α2 H T Hl AD P + P A P l < 0. (14) ⋆ −τ I
This LMI cannot be used to compute the controller parameters because it is not affine in Kd . Therefore, preand post multiplying by diag(τ P −1 , I) and diag(τ P −1 , I), respectively and using the Schur’s complement method [11] 1 AˆD Y + Y AˆTD + Y HlT Hl Y + I γ
xcl
Next, consider an interconnected system of the form:
ThPI23.17
0,
(12)
0,
(13)
M T M,
then the system in (5) is asymptotically stable for nonlinearities satisfying the quadratic constraint in (6) or (8). Proof: Let us consider a quadratic Lyapunov function v = xTcl P xcl . The sufficient condition for stability of the closed loop system is the derivative of v to be negative along the solutions of (5). This can be expressed as P > 0, and xTcl AˆTD P xcl + hTr P xcl + xTcl P AˆD xcl + xTcl P hr < 0.
0,
0) instead of I may lead to a fast convergence speed. This is because, the inequality in (15) is equivalent to I + γ1 Y HlT Hl Y + rI − (W + W T )+ r1 (Y AˆTD + W T )(AˆD Y + W ) < 0, which can always be satisfied by selecting r large enough. The constraints in Theorem 2.1 should also be augmented with additional pole placement constraints [15], [4] to avoid fast controller dynamics. In Theorem 2.1, the matrix XP is decoupled from the controller Kd . This is an important advantage, because any other structure of Kd (decentralized design or control design when x2 in Fig. 1 is reachable from u1 or u2 only) can be easily assigned. Moreover, reduced order dynamic controllers can be easily designed, because different orders of Ak1 and Ak2 in (7) can be imposed. For large scale nonlinear interconnected systems, where the nonlinearities satisfies the constraint [3], [4], [14]: hTi (t, x)hi (t, x) ≤ α2i xT HiT Hi x, i = 1, 2, 3, . . . , N (17) the following optimization problem should be solved
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
⋆ ⋆ ⋆
ˆT + M T A D −I ⋆ ⋆
XP 0 −I ⋆
. . . ⋆ ⋆
. . . ⋆ ⋆
. . . ⋆ ⋆
−Q
T H1 l 0 0 −γ1 I . . . ⋆ ⋆
ThPI23.17
min γ1 + γ2 + . . . + γN s.t XP > 0, Q = M T M, ... HT X − MT .
.
. ⋆ ⋆
The idea behind this approach is straightforward. It is well h i known that any matrix F12 F = F⋆11 F < 0 (19) 22
P
Nl 0 0 0
... ... ...
III. S EQUENTIAL T WO - PART O PTIMIZATION P ROCEDURE
0 0 0
. . . −γN I ⋆
. . . 0 −I
< 0.
Therefore, if F22 and F11 are affine in controller parameters and the bilinear terms appear in F12 , then 1) Solve the feasibility problem F22 < 0. 2) Define tuning parameter ξ = diag(ξ1 I, ξ2 I, . . .) > 0. Substitute the variables from step h 1 and solve i [13] F12 min ξ1 + ξ2 + . . . , s.t Fξ = F⋆11 ξF < 0. 22
Hence, unlike [7], the generalization of the result here is very straightforward. The linear system is a special case of (3) with h(x) = 0 and (10) is a special case of (17). Therefore, the derivation follows along the same lines. Remark 2.1: For solving the optimization problem using the available numerical software, a key idea is to relax the equality constraint as: h i Q MT ≥ 0, (18) ⋆ I
and to apply a cone complementary linearization (CCL) algorithm [16] for computing the controller parameters. A modified version of CCL algorithm is as follows. Computational method: As Q = M T M corresponds to the boundary of the convex set in (18), let Hβ , {XP , Kd , Q, M ; (12), (13) and (18) are satisfied}. Here, Hβ is a closed and convex set. Algorithm OC (overlapping control): 1) Find the feasible set (XP0 , Kd0 , Q0 , M 0 ) ∈ Hβ ; k := 0. 2) Solve the convex optimization problem, namely, min trace Q − (M k )T M − M T M k subject to (12), Hβ
(13) and (18). 3) Substitute the value of (XP , Kd , M ) in (16). If the condition is satisfied then output the feasible solution (XP , Kd , Q, M ), else set k = k + 1, (XPk , Kdk , Qk , M k ) = (XP , Kd , Q, M ), go to step 2. Remark 2.2: It is important to note that the optimization ⋆ problem at the k’th step, J˜k = min J˜k = min trace (Qk + T Q − M k M − M T M k ), subject to (12), (13), (18) and the step 2 in algorithm OC are equivalent. This is because Qk is a constant matrix; therefore, both optimization problem has the same solution. Using the ideas from [17], it can be ⋆ ⋆ ⋆ easily derived that J˜k ≥ 0 and the sequence {J˜1 , J˜2 , . . .} is monotonically decreasing and convergent. Moreover, in the algorithm, the set can be expanded to include the equality constraint by substituting −Q with −Q + αI (α > 0, say), and the iterative algorithm can be stopped to output the T feasible solution if trace(Qk − M k M k ) < α. Remark 2.3: It is interesting to note that in [7], the overlapping is limited to pairs of subsystems only. However, large power systems have their overlapping states shared by multiple number of subsystems. In this situation, the algorithms presented in this paper can be easily used. This is because the optimization problem involves AˆD which ¯ ¯ d C. is affine in controller parameters AˆD = A0 + BK ¯ Therefore, different structures for Kd and B can be easily assigned (different control laws and different overlapping). For a three area power system, Kd has the structure h K i K12 0 K14 0 0 11 0 K22 K23 0 K24 0 Kd = . 0 0 0 K K K 34
35
36
T F22 < 0, and F11 − F12 (F22 )−1 F12 < 0.
⇐⇒
In the following, dynamic output feedback overlapping control design problem is converted into the form of (19), using different transformations, simplifications and new variable definitions. It helps to utilize the two-step approach. Dynamic output feedback overlapping control design: Let us start with the nonlinear interconnected system in (9), where a dynamic output feedback overlapping controller has to be designed. With the control law in (4), the closed loop system is given by x˙ 1 I 0 0 h1 (x) x˙ k1 0 0 0 x˙ 2 = x˙ cl = 0 I 0 h2 (x) + AˆD xcl , 0 0 I x˙ 3 h3 (x) 0 0 0 x˙ k2 | {z }| {z }
where Aˆ = " A11 + BD 11 Dk Bk11 C1 A21r 0
11
hint (x)
G
C1
B11 Ck1 A12r + B11 Dk12r C2r 0 Ak1 Bk12r C2r 0 0 A22r + B32r Dk2r Cr B32r Ck2 0 Bk2r Cr Ak2 0 ], A12 A13 ] , Bk12r = [ Bk12 h i h A23 21 22 A21r = A , A22r = A A31 A32 A33
#
.
Here, A12r = [ i C2r = diag(C2 , 0), , h i B32r = B032 , Dk2r = [ Dk21 Dk22 ] , Cr = diag(C2 , C3 ), Dk12r = [ Dk12 0 ] , Bk2r = [ Bk21 Bk22 ] . At this point, it is clear that the conditions of asymptotic stability of the closed loop system are given by ˆ T T T ˆT Y > 0,
AD Y + Y AD ⋆ ⋆ ⋆ ⋆
GD −I ⋆ ⋆ ⋆
Y H1 l 0 −γ1 I ⋆ ⋆
Y H2 l 0 0 −γ2 I ⋆
Y H3 l 0 0 0 −γ3 I
< 0.
By using different congruence transformations and change of controller variables method, the inequalities boil down to h i h i X1 I diag , X⋆2 YI2 > 0, ⋆ Y1 T1 T1 T1
F11 F12 F22 {z } | T ˆ Y +Y A ˆT )Π Π (A 2 D 2 D ⋆ ⋆ ⋆ ⋆
0 G2 {z ΠT G 2
G1 0
|
−I ⋆ ⋆ ⋆
}
1l 2 T1 l {z } T Π Y HT 2 1l 0 −γ1 I ⋆ ⋆
|
2l 2 T2 l {z } T Π Y HT 2 2l 0 0 −γ2 I ⋆
|
3l 2 T3 l {z } T Π Y HT 2 3l 0 0 0 −γ3 I
|
< 0, i i h h I Y2 I Y1 and Y Π , where Π2 = diag T T 2 = 0 N2 h i h 0 N1 i X1 I X2 I diag , . Hence, F12 in (19) is T T M 1 0 M21 0 1 G1 0 F12 T1l T2l T31l . This method requires rank constraints [15] for reduced order control design.
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 IV. P OWER S YSTEM AND I NDUSTRIAL U TILITY B OILER Let us first consider a two area power system shown in Fig. 2, where the parameters are obtained from [18]. The areas represent the subsystems and the tie lines are the overlapping parts. Therefore, the controllers can be designed to share the overlapping state (∆Ptie ) for improving the performance of the overall system. The overall system is of ninth order and the output measurements (frequency deviations) as well as the system input matrix B is given by y = [ ∆f1 ∆f2 ]T , h iT 0 0 0 0 0 0 0 0 T . B = 0 0 g01 0 0 0 0 Tg2 0
In Fig. 3, the frequency deviations in the two areas due to disturbance of ∆PD1 = 0.01 pu in area 1 are shown (by dotted lines, without controller). This and the Nyquist array with the column Gershgorin circles (Fig. 4, first row) on the diagonal element show that the system is highly interacting. Only Gershgorin circles for the first subsystem (g11 and g12 ) is drawn, because transfer functions of both subsystems are same. The response in Fig. 3 also show that local controllers should be designed to minimize the oscillations. By using the Algorithm OC for linear system in section II, output feedback (static and dynamic) decentralized and overlapping controllers are designed. They have the following form: −0.4638 −0.4098 0.2778 0 0 , , 0 −0.4638 0 −0.2778 −0.4098 −5.1443 0 1.9535 −2.8866 0 0 −5.1443 0 2.8866 1.9535 . 1 0 −0.7943 0.3099 0 0
1
0
−0.3099
Second row of Fig. 4 shows the Gershgorin circles of the closed loop system with static o/p feedback decentralized controller. It is clear that only at low frequencies, some circles are enclosing the origin and at medium and high frequencies, the system is diagonal dominant. With the first order dynamic output feedback decentralized controller in Fig. 4 (third row), the radius of circles at medium frequencies are very small compared to the second row. Hence, this controller is capable of minimizing the effect of interactions between different loops and has better performance. It has been found that if all the local states are available for measurement, then the oscillations are completely removed. Industrial Utility Boiler The final control oriented nonlinear drum-boiler-header model of Syncrude utility plant is governed by 10−4 x2 + 0.00157 x˙ 1 = −0.0157x1 + 1.866 × (x1 + x4 ) 10−4 x2 , ×x5 − 0.0000395u1 − 3.545 × x1 + x4 x˙ 2 = x3 + 0.009151u1 + 0.02988x5 + 0.2239u3, x˙ 3
=
−0.001864x2 − 0.1533x3 − 0.001987u1 + 0.03634x5 − 0.03288u3,
x˙ 4 x˙ 6
= =
x˙ 7
=
ylevel
=
−0.08x4 + 0.006u3, x˙ 5 = x6 + 0.1758u2, −0.001833x5 − 0.1731x6 − 0.0177u2, r u1 − x2 x2 , yheader = (x1 + x4 )2 − 2 , 155.1411 900 0.01028x7 − 13.644(0.000219x2 + 0.00439x5
−0.7943
From Fig. 3, it can be seen that the controllers are capable of attenuating most of the oscillations. The performance of the first order dynamic controller is better than the static overlapping control law, which is in turn shows better response than the static decentralized control law. For performance evaluation,Z an ITAE criteria is also used [18]: 20
Jf re =
ThPI23.17
t|∆f1 (t)|dt.
0
The table below shows the values of Jf re for different controllers, which verifies that dynamic controller is better.
+0.000469u1) + 0.0149(0.5x2 + 6.35x5 + 0.09u1) + 0.005 × (u1 − 40.68) − 7.0707. The nonlinear model here and the fourth order linear model for steam temperature dynamics shows good fitness at all operating points. Linearization of the overall model has one pole at the origin, one RHP zero at 0.0619 and the following overlapping controllers are designed # " −11.98s−12.92 5.318s+3.154 Kf ull
0
1 R1
B1
PD1
'
-
+
K1 s
+
'
Pc1
+
-
1 1 sTG1
PG1
'
+
Integral part
u1
1 1 sTT 1
'
PT 1
+
Turbine
Governor
f1
'
Generator
Ptie
T12 s
+ -
a12
u2 '
+
+
Kpartial
K P1 1 sTP1
P12
'
'
a12
K2 s
'
+
Pc 2
+
1 1 sTG 2
-
'
PG 2
1 1 sTT 2
'
P21
-
PT 2
+
-
K P2 1 sTP 2
'
Governor
Integral part
Turbine
f2
Generator
PD 2
'
B2
1 R2
Fig. 2.
no control 28.3728
s+1.122 0
=
Two area power system [18]
static decentralized 6.2166
static overlapping 4.3148
dynamic overlapping 3.0808
=
"
−11.97s−13.05 s+1.134 0 0
s+1.122 27.61s+33.76 s+1.227 0
0 −113.4s−139.2 s+1.227 −3.797s−6.5 s+1.908
0 61.08s+74.81 s+1.227 0
0 −113.5s−139.3 s+1.227 −3.797s−6.506 s+1.91
#
.
The feedwater controller (in full overlapping) is using the measurement of header pressure and the firing rate controller (in both cases) is utilizing the extra measurement of steam temperature. Figs. 5-6 shows the responses of different process variables due to a load change of 30 kpph on 6.306 MPa header. The overlapping controllers are capable of attenuating oscillations and have smoother response than the existing PI controllers. More importantly, the 50# header steam pressure shows a sharp improvement which can lead to an improvement in power generation since 50# header pressure is the back pressure affecting the turbines. The slight offset in Fig. 6 is due to lack of integrators in the controllers, however, the response of partial overlapping controller is within the range and is acceptable in the present plant.
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
ThPI23.17
50# header pressure
900# steam temp.[C] Steam flow [kg/sec]
V. C ONCLUSION In this paper, two different approaches for solving the overlapping control design problem is introduced. In the first case, an iterative algorithm is used to obtain the controller parameters. This method helps attacking a vast panorama of control problems including static state feedback, static output feedback, full order and reduced order dynamic output feedback control designs. In the second case, a two step approach has been used, which requires no iteration. Simulation results in SYNSIM shows that when only the firing rate controller (to control the header pressure) is utilizing the extra measurement of steam temperature, then it is capable of attenuating the oscillations in the 6.306 MPa header (during load changes). This makes it practically implementable. The presented algorithm has also the capability of capturing many other overlapping cases (not only Type I and Type II).
1
without controller with static output feedback overlapping controller with static output feedback decentralized controller first order dynamic output feedback overlapping controller 0
5
10
15
Steam temp.[C]
∆ f [Hertz]
0
−0.03
20
Area 2 0.02
275
partial overlapping 1500 full overlapping PI controller
0
500
1000
0
500
1000
1500
2000
0
500
1000 Time [sec]
1500
2000
505
2000
500 495
51.94 51.92
Stabilizing effect of overlapping controllers
load change of 30 kpph
1.05
partial overlapping full overlapping PI controller
1 0.95
0
500
1000
1500
2000
2500
0
500
1000
1500
2000
2500
0
500
1000 1500 Time [sec]
2000
2500
6320 6300 6280 6260 500.5 500 499.5
0.01
Fig. 6.
0
Header pressure response during sudden load change
2
∆ f [Hertz]
280
51.96
Drum level [m] Header pressure [kPa]
0.01
−0.02
285
Fig. 5.
Area 1
−0.01
load change of 30 kpph 290
−0.01 −0.02
0
5
10 Time[sec]
15
20
Fig. 3. Frequency deviations of the areas with output feedback controllers 10
10 g12 IMAG
IMAG
g11 0
−10 −20 2
0
10
5
−2
0
2
0
1
REAL
0.5 IMAG
IMAG
−2 −4 1
0
0 −2 −4 −2
0 REAL
0
−4 −2 REAL
−5
2
IMAG
IMAG
0
−6 2
0 −5 −10 4
20
REAL
−2
5
0
−0.5 0
2
4
−1 −2
−1
Fig. 4. Nyquist array with column Gershgorin circles of the first area: without controller (first row), with static output feedback controller (second row), with dynamic output feedback controller (third row).
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