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Analysis of the ODE of Interest

Dans le document The DART-Europe E-theses Portal (Page 52-57)

As indicated in the introduction, one of the main interests of the chapter is the analysis of voltage regularity of steady solutions of AC power systems (under the common decoupling assumption [56]), and on the study of the existence of steady states of MT-HVDC networks as well as DC microgrids—the three of them with CPLs. In Section 4.3, it is shown that these studies boil down to the analysis of solutions of the following algebraic equations in ¯x∈ Kn+3

ai11+ai22 +· · ·+ainn+ bi

¯

xi =wi, i∈ {1, . . . , n}, (4.1) whereaij ∈R,wi ∈Rand bi ∈R. These equations can be written in compact form as

Ax¯+ stack bi

¯ xi

−w= 0. (4.2)

In the chapter, the following assumption is adopted.

Assumption 4.1. The matrixA=A> is positive definite, its off-diagonal elements are non-positive andbi 6= 0 for alli.

To study the solutions of (4.2), the following ODE is considered.

˙

x=f(x) := −Ax−stack bi

xi

+w. (4.3)

The main interest of the chapter is in studying the existence, and stability, of the equilibria of (4.3). In particular, the answers to the following questions are provided.

Q1 When do equilibria exist? Is it possible to offer a simple test to establish their existence?

Q2 If there are equilibria, is there a distinguished element among them?

Q3 Is this equilibrium stable and/or attractive?

Q4 If it is attractive, can its domain of attraction be estimated?

3Recall from the preliminaries that K+n denotes the positive orthant ofRn.

Q5 Is it possible to propose a simple procedure to compute this special equilibrium using the system data (A, b, w)?

Q6 Are there other stable equilibria?

Instrumental to answer to the queries Q1-Q6 is the fact that the system (4.3) is monotone. That is, for any two solutions xa(·), xb(·) of (4.3), defined on a common interval [0, T], the inequality xa(0) ≤ xb(0) implies that xa(t) ≤ xb(t) ∀t ∈ [0, T].

This can be verified by noticing that equation (4.3) satisfies the necessary and suf-ficient condition for monotonicity [114, Proposition 1.1 and Remark 1.1, Ch. III]

∂fi(x)

∂xj ≥0, ∀i6=j, ∀x∈ Kn+. (4.4) In the sequel,x(t, x0) denotes the solution of (4.3) with initial conditionsx(0) = x0 >0, and use the following.

Definition 4.1. An equilibrium ¯x > 0 of (4.3) is said to be attractive from above if for any x0 ≥ x, the solution¯ x(t, x0) is defined on [0,∞) and converges to ¯x as t→ ∞. The equilibrium is said to be hyperbolic if the Jacobian matrix ∇f(¯x) has no eigenvalue with zero real part [43].

Remark 4.1. Along the lines of the standard Kron reduction, the findings of the chapter can be extended to the case where some of the coefficients bi are equal to zero.4

The proof of this remark is given in Appendix 4.A.

4.2.1 The simplest example

To gain an understanding of some key traits of possible results, it is instructive to start with the simplest casen = 1. Then, x∈Rand (4.3) is the scalar equation

˙

x=−ax− b

x +w, (4.5)

and a > 0, b 6= 0. Feasible behaviors of the system are exhaustively described in Figure 4.1.

The following can be inferred from this figure.

B1 The system either has no equilibria, or it has finitely many equilibria, or it has a single equilibrium.

B2 If the system has equilibria, the rightmost of them is attractive from above.

B3 Non-hyperbolic equilibria may be attractive from above but unstable; more-over, there may be no other equilibria.

B4 Hyperbolic and attractive from above equilibria are stable.

B5 If b >0, globally attractive equilibria do not exist.

It is shown next that several of the traits occurring in the scalar case are inherited by then-th order ODE (4.3).

4An anonymous reviewer is acknowledged for indicating this.

x

¯ xs

Case b < 0 f(x)

(a)

x

x(t)0 in finite time

Case b > 0

w2 ab <0

f(x)

(b)

¯ x xu

x(t)0 x(t)x¯u

0

Case b > 0

w2 ab= 0

f(x)

(c)

x

x(t)0 x(t)x¯s

0

Case b > 0

w2 ab >0

¯

xus f(x)

(d)

Figure 4.1: Feasible behaviors of the one-dimensional system (4.5): (a) A unique globally attractive equilibrium ¯xs; (b) No equilibria, all solutions converge to zero in a finite timetf; (c) Unique unstable equilibrium ¯xu, which is attractive from above, whereas any solution starting on the left diverges from ¯xu and converges to 0 in a finite time; (d) Two equilibria, the smallest of which ¯xu is unstable, whereas the larger one ¯xs is stable and attractive from above.

4.2.2 An extra assumption

The situation described in item B3 above may happen in the general case —e.g., considering a diagonal matrixA. However, some evidence suggesting that this situ-ation “almost never” occurs is now given. This forms a ground to treat the opposite situation as “typical”, “regular”, and “prevalent” and to pay special attention to it.

To be specific, a case where item B3 is impossible is identified.

Assumption 4.2. There are no non-hyperbolic equilibria of the system (4.3). In other words, the following set identity holds

The lemma below proves that Assumption 4.2 is almost surely true.

Lemma 4.1. For any givenA and bi 6= 0, the set of all w∈Rn for which Assump-tion 4.2 does not hold has zero Lebesgue measure and is nowhere dense.

Proof. See Appendix 4.A at the end of the chapter.

4.2.3 Main results on the system (4.3)

The first claim contains a qualitative analysis of the system.

Proposition 4.1. Consider the system (4.3) verifying Assumption 4.1. One and only one of the following two mutually exclusive statements holds.

S1 There are no equilibria ¯x, either stable or unstable, and any solution x(·) is defined only on a finite time interval [0, tf) ⊂ [0,∞), since there exists at least one coordinate xi such that xi(t) → 0,x˙i(t) → −∞ as t → tf. Such a coordinate is necessarily associated with bi >0.5

S2 Equilibria ¯xk do exist. One of them ¯xmax > 0 verifies ¯xmax ≥ x¯k, ∀k, and is attractive from above.

Suppose that Assumption 4.2 is also met. Then in the case S2 there are finitely many equilibria and ¯xmax is locally asymptotically stable. If in addition all bi’s are of the same sign, then there are no other stable equilibria apart from ¯xmax.

Proof. See Appendix 4.D at the end of the chapter.

Lemma 4.1 allows to claim that the traits described in the last two sentences of Proposition 4.1 occur almost surely.

A constructive test to identify which of the cases S1 or S2 holds, as well as a method to find ¯xmaxin the case S2, are presented. To this end, the following concept is introduced.

5This implies that the case in claim S1 does not occur ifbj<0,j.

Definition 4.2. A solution x(·) of the ODE (4.3) is said to be distinguished if its initial condition lives in the set

E :=

x∈ Kn+|Ax >stack

hwii+h−bii xi

.6 (4.7)

Ifbi >0 for alli, then the set (4.7) reduces to the (convex open polyhedral) cone x∈ Kn+|Ax >stack (hwii) .

Proposition 4.2. Consider the system (4.3) verifying Assumption 4.1.

I The set E is non-empty, consequently there are distinguished solutions.

II All such solutions strictly decay, in the sense that ˙x(t) < 0, for all t in the domain of definition of x(·).

III One and only one of the following two mutually exclusive statements holds simultaneously for all distinguished solutions x(·):

i For a finite time tf ∈(0,∞), some coordinatexi(·) approaches zero, that is,

tlimtf xi(t) = 0, (4.8) and the solution x(·) is defined only on a finite time interval [0, tf).

ii There is no coordinate approaching zero, the solution is defined on [0,∞), and the following limit exists and verifies

t→∞lim x(t)>0. (4.9)

This limit is the same for all distinguished solutions.

IV If the case III.i holds for a distinguished solution, the situation S1 from Propo-sition 4.1 occurs.

V If the case III.ii holds for a distinguished solution, the situation S2 from Propo-sition 4.1 occurs, and the dominant equilibrium ¯xmaxis equal to the limit (4.9).

Proof. See Appendix 4.D at the end of the chapter.

Remark 4.2 (Additional properties of (4.3)).

P1 In III.i, there may be several components xi with the described property, but not all of them necessarily possess it.

P2 The claim S1 in Proposition 4.1 and IV in Proposition 4.2 yield that (4.8) is necessarily associated with bi >0 and ˙xi(t)→ −∞as t→tf.

P3 Regarding the claim S2 in Proposition 4.1 the basin of attraction of the equi-librium ¯xmax contains all statesx≥x¯max. Under Assumption 4.2, this basin is open.

6The operatorh·idenotes the clipping functionhai:= max{a,0}.

P4 The linear programming problem of finding elements in the set x∈ Kn+|Ax >0

has been widely studied in the literature [28, 100, 11]. There is a whole variety of computationally efficient methods to solve this problem, including the Fourier-Motzkin elimination, the simplex method, interior-point/barrier-like approaches, and many others; for a recent survey, the reader is referred to [35].

4.3 A Numerical Procedure and a Robustness

Dans le document The DART-Europe E-theses Portal (Page 52-57)