• Aucun résultat trouvé

Delay-Aware Control Designs of Wide-Area Power Networks: Proofs of the Results

N/A
N/A
Protected

Academic year: 2021

Partager "Delay-Aware Control Designs of Wide-Area Power Networks: Proofs of the Results"

Copied!
3
0
0

Texte intégral

(1)

Delay-Aware Control Designs of

Wide-Area Power Networks:

Proofs of the Results ?

Seyed Mehran Dibaji∗ Yildiray Yildiz∗∗ Anuradha Annaswamy∗ Aranya Chakrabortty∗∗∗

Damoon Soudbakhsh∗∗∗∗

Department of Mechanical Engineering, Massachusetts Institute of

Technology, Cambridge, MA 02139 USA, {dibaji,aannu}@mit.edu.

∗∗Department of Mechanical Engineering, Bilkent University, Ankara,

Turkey, yyildiz@bilkent.edu.tr

∗∗∗Department of Electrical and Computer Engineering, North

Carolina State University, Raleigh, NC 27695 USA, achakra2@ncsu.edu

∗∗∗∗Department of Mechanical Engineering, George Mason University,

Fairfax, VA 22030 USA, dsoudbak@gmu.edu

Abstract: The purpose of this document is to provide proofs of the results in Dibaji et al. (2017).

Keywords: Power systems stability, Wide-area measurement systems, Optimal operation and control of power systems, Cyber-Physical Systems, Systems with time-delays

1. A DELAY-AWARE CONTROL DESIGN 1.1 Main Equations

The physical network follows the ordinary continuous-time LTI model,

˙

x = Acx(t) + Bcu(t), (1)

where Bc = [B1c, . . . , Bnc]. However, if there is no delay

present in the network, the following sampled-data model is applied for the purpose of control designs.

x[k + 1] = Ax[k] + Bu[k], (2) where A = eAch and B = Z h 0 eAcsB cds. (3)

The delay-aware ANCS co-design model leads to (4), x[k + 1] = Ax[k] + B1U [k] + B2U [k − 1], (4) where Bj1i =            Z h−τij0 0 eAcsdsBi c if j = g(i), (5a) Z h−τij0 h−τ0 i(j+1) eAcsdsBi c, if j 6= g(i), (5b) Bi2i = Z h h−τ0 i1 eAcsdsBi c, (6) and Bi

j1∈ RN ×1is the coefficient of uij[k] and Bi2i ∈ RN ×1

is the coefficient of uig(i)[k − 1] in (7).

? This work was supported in part by NSF grant ECS 1054394.

Ui(t) =      uij[k] if t − kh ∈ [τij0 , τ 0 i(j+1)), j < g(i),

uig(i)[k] if t − kh ∈ [τig(i)0 , h), j = g(i),

uig(i)[k − 1] if t − kh ∈ [0, τi10 ),

(7) The extended form of (4) is given by

W4h[k + 1] = A4hW4h[k] + B4hU [k], (8) where W4h[k] =           x[k] U [k − 1] x[k − 1] U [k − 2] .. . x[k − 5] U [k − 6]           , B4h=           B1 IG 0 0 .. . 0 0           , and A4h∈ R6(N +G)×6(N +G)is A4h=         A B2 0 · · · 0 0 0 0 0 0 · · · 0 0 0 IN 0 0 · · · 0 0 0 0 IG 0 · · · 0 0 0 .. . 0 0 0 · · · IG 0 0         . (9) 1.2 Stability Analysis

Lemma 1. (Yuz and Goodwin, 2014) The matrix A in (3) is non-singular and all of its eigenvalues, provided that Ac

in (1) is Hurwitz, are inside the unit circle.

Proof. Denoting µi as the i-th eigenvalue of A, it follows

(2)

Re(λi) and βi = Im(λi), it follows that µi = ehαiejhβi. If

Ac is Hurwitz, αi is negative which in turn implies that

0 < |µi| = ehαi< 1.

Proposition 2. If the system (1) is stable, A4h is

Schur-stable with N eigenvalues coinciding with all eigenvalues of A and the remaining ones at zero.

Proof. The structure of A4h in (9) implies that A4h can

be written in a block diagonal form as A4h=∆1,1

∆2,1

∆3,1 ∆4,1

 , where ∆4,1 ∈ RG×G is a zero matrix in the right lowest

corner. It can also be shown that ∆2,1 = 0. Hence, the

eigenvalues of A4h include G zeros (Demmel, 1997). We

now further partition ∆1,1 = ∆1,2

∆2,2

∆3,2 ∆4,2



, where ∆4,2 ∈

RN ×N is also a zero matrix. Therefore it follows that

A4h has an additional N zero eigenvalues. By repeating

the same process, we obtain the sub-partition ∆1,9 =



Ah 0

[IN 0] 0



. Consequently, ∆1,10 = Ah whose eigenvalues

include the N eigenvalues of A and additional G zeros. The complete process yields a total of N eigenvalues which correspond to those of A and all remaining 5N + 6G eigenvalues at zero.

1.3 Controllability Analysis

Lemma 3. (Liu and Fong, 2012) The time-delay system x[k + 1] = dx X i=0 Aix[k − i] + du X i=0 Biu[k − i],

is completely controllable if and only if Y = dx X i=0 λdx−iA i− λdx+1I du X i=0 λdu−iB i 

has full rank at all roots of | dx X i=0 λdx−iA i− λdx+1I| = 0. Proposition 4. If (Ah, Bh) is controllable, (A4h, B4h) is stabilizable.

Proof. First, we restate the sufficient condition in terms of the criterion given in Lemma 3. For (4), du = 1 and

dx = 0. Hence Y1 = A − λI λB1+ B2, where λ is the

solution to the equation |A − λI| = 0. This means that λ is an eigenvalue of A. But from Lemma 1, we know that λ 6= 0. Then, in (8), we have dx = 5 and du = 6. So

Y2 =ν5A − ν6I ν6B1+ ν5B2, where ν is the solution

to the equation |ν5A − ν6I| = ν5|A − νI| = 0. This

means that either ν = 0 or ν is an eigenvalue of A which is non-zero. If ν is zero, Y2 has not full rank. From the

sufficient condition, we know that for all ν as eigenvalues of A, A − νI νB1+ B2 has full rank. Hence, rank of

Y2 = ν5A − νI νB1+ B2 for eigenvalues of A is equal

to the rank of A − νI νB1+ B2



which is full. Thus, the extended system (8) is not completely controllable; however, its non-zero eigenvalues, corresponding to Ah,

can be altered using a proper control signal U [k] and this makes (A4h, B4h) stabilizable.

2. EMPLOYING OUTPUT FEEDBACK 2.1 Main Equations

When the states are not accessible, the control input is modified as

U [k] = K0(1)xo[k] + K (2)

0 xˆo[k] + G0U [k − 1], (10)

where the estimated observer state vector ˆxo is calculated

using (11) as ˆ

xo[k] = Axo[k − 1] + B2U [k − 2] + B1U [k − 1]

+L(y[k − 1] − Cxo[k − 1]). (11)

By using an appropriate output matrix C ∈ Rs×N, we

determine the output vector y ∈ Rsas

y[k] = Cx[k]. (12)

The observer error dynamics is obtained as ˜

x[k + 1] = (A − LC)˜x[k], (13) where ˜x = x − xois the observer error vector.

2.2 Stability Analysis

Theorem 5. The closed-loop output feedback control sys-tem consisting of (4), (10) and (11) is stable.

Proof. To obtain the closed-loop dynamics of the overall system, we first rewrite (4), and then by using (10) and (11): x[k + 1] = Ax[k] + B2U [k − 1] +B1 K (1) 0 xo[k] + K (2) 0 xˆo[k] + G0U [k − 1]  = Ax[k] + (B2+ B1G0)U [k − 1] +B1K (1) 0 xo[k] + B1K (2) 0 xˆo[k] = Ax[k] + (B2+ B1G0)U [k − 1] + B1K (1) 0 xo[k] + B1K (2) 0 Axo[k − 1] + B2U [k − 2] +B1U [k − 1] + L(y[k − 1] − Cxo[k − 1]) = Ax[k] + (B2+ B1G0+ B1K (2) 0 B1)U [k − 1] +B1K (1) 0 xo[k] + B1K (2) 0 A − LCx0[k − 1] +B1K (2) 0 B2U [k − 2] + B1K (2) 0 Ly[k − 1]. (14)

Using (12) and the fact that x0 = x − ˜x, (14) can be

rewritten as x[k + 1] = Ax[k] + (B2+ B1G0+ B1K (2) 0 B1)U [k − 1] + B1K (1) 0 (x[k] − ˜x[k]) + B1K (2) 0 A − LC(x[k − 1] − ˜x[k − 1]) + B1K (2) 0 B2U [k − 2] + B1K (2) 0 LCx[k − 1] = A + B1K (1) 0 x[k] + B1K (2) 0 Ax[k − 1] + (B2+ B1G0+ B1K (2) 0 B1)U [k − 1] + B1K (2) 0 B2U [k − 2] − B1K (1) 0 x[k]˜ − B1K (1) 0 x[k − 1].˜ (15)

(3)

U [k] = K0(1) x[k] − ˜x[k] + K0(2) Axo[k − 1] + B2U [k − 2] + B1U [k − 1] + L(y[k − 1] − Cxo[k − 1]) + G0U [k − 1] = K0(1)x[k] − K0(1)x[k] + K˜ 0(2)Ax[k − 1] − K0(2)A˜x[k − 1] + K0(2)B1+ G0U [k − 1] + K0(2)B2U [k − 2] + K (2) 0 LC ˜x[k − 1] = K0(1)x[k] − K0(1)x[k] + K˜ 0(2)Ax[k − 1] − K0(2) A − LC ˜x[k − 1] + K0(2)B1+ G0U [k − 1] + K0(2)B2U [k − 2]. (16) Defining X[k] ≡x[k]T U [k − 1]T x[k]˜ TT and using (13), (15) and (16), and by using the fact that K0(1)+K0(1)= K0,

we obtain the closed-loop dynamics as  X[k + 1] X[k]  = H  X[k] X[k − 1]  , (17) where H =        A + B1K0 B2+ B1G0 −B1K0 0 0 0 K0 G0 −K0 0 0 0 0 0 A − LC 0 0 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0 0       

The characteristic equation for the system (17) can be calculated as λI − (A + B1K0) B2+ B1G0 K0 λI − G0 × |λI − (A − LC)| × λI 0 0 0 λI 0 0 0 λI = 0 (18)

As seen from (18), the closed-loop system poles of the delayed power network with output feedback are the same as the state accessible case with the additional poles coming from the observer dynamics and additional poles at the origin. It was shown in Section 1 that the state accessible case is stable. We also showed that observer dynamics are stable. Therefore, the closed-loop system with output feedback is stable.

REFERENCES

Demmel, J.W. (1997). Applied Numerical Linear Algebra. SIAM.

Dibaji, S.M., Yildiz, Y., Annaswamy, A., Chakrabortty, A., and Soudbakhsh, D. (2017). Delay-aware control designs of wide-area power networks. In Proceedings of the 20th World Congress of the International Federation of Automatic Control, to appear.

Liu, Y.M. and Fong, I.K. (2012). On the controllability and observability of discrete-time linear time-delay systems. International Journal of Systems Science, 43(4), 610– 621.

Yuz, J.I. and Goodwin, G.C. (2014). Sampled-Data Models for Linear and Nonlinear Systems. Springer.

Références

Documents relatifs

Les résultats montrent que l'isolat de Pseudomonas fluorescents est une activité antagoniste importante par confrontation directe où il est inhibé

On a vu dans les chapitres précèdent, comment écrire en CAML des fonctions qui prennent en entrée un certain nombre de variables plus ou moins complexes (ou au sens du type composé)

The root causes of the contract’s complexity and ambiguity lies not only in the contract itself, but also in the larger enterprise in the form of unclear/inconsistent

E Interactive Tuples Extraction from Semi-Structured Data 115 F Learning Multi-label Alternating Decision Trees from Texts and Data125 G Text Classification from Positive and

Il faut noter que dans le cas des applications en chimie des techniques nouvelles doivent ˆetre mises en oeuvre pour d’une part adapter la m´ethode de bases r´eduites au

Included in the description is our research in the development of provably correct behavioral transformations with applica- tions to design and implementation

Schematic diagram summarizing the principal findings of this study and showing a simplified representation of differential TRPM8 localization and function depending on the AR

One hundred and twenty-six (126) motorbike drivers were selected to evaluate the relationship between haematological (white blood cells, platelets) and inflammatory