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Publications

Dans le document The DART-Europe E-theses Portal (Page 17-20)

This thesis is based on the following articles, some of which have already been published or are under review.

J1 Juan E. Machado, Robert Gri˜n´o, Nikita Barabanov, Romeo Ortega, and Boris Polyak, “On Existence of Equilibria of Multi-Port Linear AC Networks With Constant-Power Loads,”, IEEE Transactions on Circuits and Systems I: Reg-ular Papers, Vol. 64, No. 10, pp. 2772–2782, 2017.

J2 Wei He, Romeo Ortega, Juan E. Machado, and Shihua Li, “An Adaptive Passivity-Based Controller of a Buck-Boost Converter With a Constant Power Load,” Asian Journal of Control, Vo. 21, No. 2, pp. 1–15, 2018.

J3 Pooya Monshizadeh, Juan E. Machado, Romeo Ortega, and Arjan van der Schaft, “Power-Controlled Hamiltonian Systems: Application to Electrical Systems with Constant Power Loads,” Automatica, 2019. (to appear)

J4 Alexey Matveev, Juan E. Machado, Romeo Ortega, Johannes Schiffer, and Anton Pyrkin, “On the Existence and Long-Term Stability of Voltage Equi-libria in Power Systems with Constant Power Loads,” IEEE Transactions on Automatic Control, 2019. (provisionally accepted)

J5 Juan E. Machado, Romeo Ortega, Alessandro Astolfi, Jos´e Arocas-P´erez, An-ton Pyrkin, Alexey Bobtsov, and Robert Gri˜n´o, “Active Damping of a DC Network with a Constant Power Load: An Adaptive Observer-based Design,”

IEEE Transactions on Control Systems Technology, 2019. (submitted)

Preliminaries and models

Synopsis This chapter presents the notation and theoretical foundations that are used throughout the thesis. The content regarding nonlinear dynamical systems reported in Section 2.2 is the base of the stability analysis of Chapters 4 and 5 and of the control design that is reported in Chapters 6 and 7. The concepts of passivity presented in section 2.2.2 and the port-Hamiltonian framework introduced in section 2.2.3 are also extensively used in the latter chapter. The brief introduction to electric power systems in Section 2.3 is helpful to better understand the developments that are presented in Chapters 3 and 4.1

2.1 Notation and mathematical foundations

Sets and Numbers: The sets of natural, real and complex numbers are denoted byN, R, and C, respectively. The notation x ∈A ⊂ R means that x is a member of A and that A is a subset of R. A complex number z ∈ C is usually written in its Cartesian form as z = a +jb, where a, b ∈ R and j = √

−1 is the imaginary unit, conj(z) denotes its complex conjugate, and the real and imaginary parts ofz are denoted by Re(z) and Im(z), respectively. For a set V, with a finite number of elements,|V| denotes its cardinality.

Vectors and Matrices: The Euclidean n-space is denoted by Rn and any x∈Rn is written as ann×1 matrix x= col(x1, x2, ..., xn); the notation col(xi) or stack(xi) is considered equivalent in the text. The positive orthant of Rn is denoted by Kn+ = {x ∈ Rn : xi > 0 ∀i}. The n-vectors of all unit and all zero entries are written as 1n and 0n, respectively; whenever clear from the context, the sub-index n may be dropped. The n×n identity matrix is In. Givenx∈Rn, diag(x) denotes a diagonal matrix with x on the diagonal; an equivalent notation is diag(xi). For x∈ Rn, kxk1 =Pn

i=1|xi|, kxk2 = (Pn

i=1x2i)1/2, and kxk = maxi|xi|. Inequalities between real,n-vectors is meant component-wise.

Anm×nmatrixAof real (complex) entries is denoted byA ∈Rm×n(A∈Cm×n).

For A ∈ Rm×n, its transpose is A>. For B ∈ Cm×n, its conjugate transpose is BH. For a square, symmetric matrixA, its smallest and largest eigenvalues are denoted by λm(A) andλM(A), respectively. The nullspace of a matrix is denoted by ker(A). For

1The structure and content of this chapter borrows heavily from the analogous chapters of the theses [96, 105, 78, 131], yet primary bibliographic sources are signaled whenever necessary.

A1, A2, ..., Ak ∈Rn×n, diag(A1, A2, ..., Ak) is a block-diagonal matrix of appropriate size.

Functions: The notation f : A ⊂ R → R means that f maps the domain A into R. A function f of argument x is denoted by x 7→ f(x) yet, for ease of notation, f(x) may be used to represent the function itself. For a vector field f :Rn →Rm, the partial derivative of fi with respect to xj is equivalently written as ∂f∂xi(x)

j and∇xjfi(x). The Jacobian matrix of f is written as∇f(x); the argument x is omitted whenever clear from the context and, for the particular case when m= 1, ∇f(x) represents the transposed gradient of f. For a map F :Rn →Rn×m and the distinguished vector ¯x ∈Rn, ¯F =F(x)|x=¯x; analogously, for f :Rn → Rk,

∇f¯=∇f(x)|x=¯x. In the text, every function is assumed to be sufficiently smooth.

Graph Theory2: A finite, undirected, graph is defined as a pair G= (V,E) where V ={1,2, ..., n}is the set of vertices (also referred as nodes, or buses), E ⊂ V × V is the set of edges (or branches, or lines). The set of edges consists of elements of the form (i, j) such thati, j = 1,2, ..., nand i6=j.3 Two vertices i, j ∈ V are said to be graphGis said to be connectedif, for every pair of (different) vertices in V, there is a path that has them as its end vertices; notice that the graphs considered in this thesis do not admit self loops,i.e., for any i∈ V, {i, i}∈ E/ .

Although this thesis deals only with simple graphs, the following discussion is pertinent for the analysis carried in Section 2.3. An orientation o of the edges set E assigns a direction to the edges in the sense that o : E → {−1,1}, with o(i, j) =−o(j, i). An edge is said to originate ini (tail) and terminate inj (head) if o(i, j) = 1, and vice versa ifo(i, j) =−1. Assume that the edges of a simple graphG have beenarbitrarilyoriented and that a unique number`∈ {1,2, ...,|E|}is assigned to each edge {i, j} ∈ E, then the node-to-edge, incidence matrix, B ∈ Rn×|E|, is Assume now that, together with the edge and vertex sets, a function w : E → R is given that associates a value to each edge, then the resulting graph, denoted as G= (V,E, w), is a weighted graph. Theweighted graph Laplacian matrix associated with the weighted graph G= (V,E, w) is defined as

L=BWB>∈Rn×n, (2.2)

whereW is a |E| × |E| diagonal matrix, with w(`),` = 1,2, ...,|E|, on the diagonal.

2The information of this section has been taken from [72, Chapter 2].

3The notation {i, j} is used in the sequel to identify the pair (i, j) ∈ E and (j, i)∈ E as the same edge.

The discussion on weighted, simple graphs is wrapped-up with the following claim, which is instrumental in this thesis; see [72, Chapter 2].

(i) If the graph is connected, then ker(B>) = ker(L) = span(1n), and all n−1 non-zero eigenvalues of L are strictly positive.

More information about graph theory can be found in [13] and [12].

Dans le document The DART-Europe E-theses Portal (Page 17-20)