Guido De

Filip Rindler
email: f.rindler@warwick.ac.uk

Guido De Philippis
email: guido.de-philippis@ens-lyon.fr

ON THE STRUCTURE OF A -FREE MEASURES AND APPLICATIONS

Keywords: 35D30 (primary); 28B05, 42B37 (secondary) Structure of measures, Alberti rank-one theorem, functions of bounded deformation, Rademacher theorem

We establish a general structure theorem for the singular part of Afree Radon measures, where A is a linear PDE operator. By applying the theorem to suitably chosen differential operators A , we obtain a simple proof of Alberti's rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in R d is a Federer-Fleming flat chain.

Introduction

Consider a finite Radon measure µ on an open set Ω ⊂ R d with values in R m that is A -free for a k'th-order linear constant-coefficient PDE operator A (k ∈ N), i.e.

A µ := |α|≤k A α ∂ α µ = 0 in D ′ (Ω; R n ). (1.1)
Here, A α ∈ R n×m and ∂ α = ∂ α 1 1 . . . ∂ α d d for each multi-index α = (α 1 , . . . , α d ) ∈ (N ∪ {0}) d . A central question about (1.1) asks what can be said about the singular part µ s of solutions µ = gL d + µ s (µ s ⊥ L d ). Besides Alberti's celebrated rank-one theorem [START_REF] Alberti | Rank one property for derivatives of functions with bounded variation[END_REF] for A = curl, not much is known at present.

In this respect we recall that the wave cone

Λ A := |ξ|=1 ker A k (ξ) ⊂ R m with A k (ξ) := (2πi) k |α|=k A α ξ α ,
where

ξ α = ξ α 1 1 • • • ξ α d d
, plays a crucial role in the compensated compactness theory for sequences of A -free maps [START_REF] Diperna | Compensated compactness and general systems of conservation laws[END_REF][START_REF] Murat | Compacité par compensation[END_REF][START_REF]Compacité par compensation. II[END_REF][START_REF]Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms[END_REF][START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF][START_REF]The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations[END_REF]. Indeed, Λ A contains the values that an oscillating or concentrating sequence of functions is expected to attain. The corresponding characteristic ξ's determine the allowed directions of oscillations and concentrations.

Since the singular part µ s of a measure contains "condensed" oscillations and concentrations, it is natural to conjecture that for a measure µ solving (1.1), the polar dµ d|µ| , i.e. the Radon-Nikodým derivative of µ with respect to its total variation measure |µ|, must lie in the wave cone at almost all singular points. For A = curl this was conjectured by Ambrosio & De Giorgi in [START_REF] Ambrosio | Un nuovo tipo di funzionale del calcolo delle variazioni[END_REF] and proved by Alberti in [START_REF] Alberti | Rank one property for derivatives of functions with bounded variation[END_REF]. Our main result asserts the truth of this conjecture in full generality: Theorem 1.1. Let Ω ⊂ R d be an open set, let A be a k'th-order linear constantcoefficient differential operator as above, and let µ ∈ M(Ω; R m ) be an A -free Radon measure on Ω with values in R m . Then, dµ d|µ| (x) ∈ Λ A for |µ| s -a.e. x ∈ Ω.

Remark 1.2. Note that (perhaps surprisingly) we do not need to require A to satisfy Murat's constant-rank condition [START_REF]Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant[END_REF].

Remark 1.3. Let us point out that Theorem 1.1 is also valid in the situation A µ = σ for some σ ∈ M(Ω; R n ). (1.2) This can be reduced to the setting of Theorem 1.1 by defining μ = (µ, σ) ∈ M(R d ; R m+n ) and à (with an additional 0'th-order term) such that (1.2) is equivalent to à μ = 0. It is easy to check that, if k ≥ 1, Λ Ã = Λ A × R n and that for |µ|-almost every point dµ d|µ| is proportional to dµ d|μ| . Remark 1.4. Using essentially the same proof, Theorem 1.1 can be further extended to the setting of variable-coefficient linear differential operators A = α A α (x)∂ α with the coefficients satisfying suitable regularity assumptions. In this setting, the conclusion reads dµ d|µ| (x) ∈ Λ A (x) := |ξ|=1 ker A k x (ξ) for |µ| s -a.e. x,

where A k x (ξ) := |α|=k (2πi) k A α (x)ξ α .
Similar statements can be obtained if µ solves some pseudo-differential equations.

By applying Theorem 1.1 to suitably chosen differential operators, we easily obtain several remarkable consequences, which are outlined below. In particular, we provide a simple proof of Alberti's rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents in the spirit of the rank-one theorem. By relying on the results of Alberti & Marchese [START_REF] Alberti | On the differentiability of Lipschitz functions with respect to measures in the Euclidean space[END_REF] and of Schioppa [START_REF] Schioppa | Metric currents and Alberti representations[END_REF], the latter result immediately implies that the Rademacher theorem can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in R d is a Federer-Fleming flat chain (a part of the so-called "flat chain conjecture", see [START_REF] Ambrosio | Currents in metric spaces[END_REF]Section 11]).

1.1. Rank-one property of BV-derivatives. As already mentioned above, in [START_REF] Alberti | Rank one property for derivatives of functions with bounded variation[END_REF] Alberti solved a conjecture of Ambrosio & De Giorgi [START_REF] Ambrosio | Un nuovo tipo di funzionale del calcolo delle variazioni[END_REF] by showing the rank-one property for the singular part of the gradients of BV-functions (also see [START_REF] Alberti | Structure of null sets in the plane and applications[END_REF][START_REF] Lellis | A note on Alberti's rank-one theorem, Transport equations and multi-D hyperbolic conservation laws[END_REF]). Besides its theoretical interest, the rank-one theorem has many applications in the theory of functions of bounded variation, we just mention the following: lowersemicontinuity and relaxation [START_REF] Ambrosio | On the relaxation in BV(Ω; R m ) of quasi-convex integrals[END_REF][START_REF] Fonseca | Relaxation of quasiconvex functionals in BV(Ω, R p ) for integrands f (x, u, ∇u)[END_REF][START_REF]Relaxation of signed integral functionals in BV[END_REF], integral representation theorems [START_REF] Bouchitté | A global method for relaxation[END_REF], Young measure theory [START_REF] Kirchheim | On rank-one convex functions that are homogeneous of degree one[END_REF][START_REF] Kristensen | Characterization of generalized gradient Young measures generated by sequences in W 1,1 and BV[END_REF][START_REF]A local proof for the characterization of Young measures generated by sequences in BV[END_REF], approximation theory [START_REF]Piecewise affine approximations for functions of bounded variation[END_REF], and the study of continuity equations with BV-vector fields [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF] (in the latter case the use of the rank-one theorems can however be avoided, see [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF]Remark 3.7] and [START_REF]Transport Equation and Cauchy Problem for Non-Smooth Vector Fields, Calculus of Variations and Nonlinear Partial Differential Equations[END_REF]). We refer to [START_REF] Ambrosio | Functions of Bounded Variation and Free-Discontinuity Problems[END_REF]Chapter 5] for further history.

(x) ∈ R ℓ \ {0}, b(x) ∈ R d \ {0} such that dD s u d|D s u| (x) = a(x) ⊗ b(x).
Alberti's rank-one theorem easily follows by choosing A = curl in Theorem 1.1. Let us also mention that Massaccesi & Vittone have recently given a short and elegant proof of the rank-one property based on the theory of sets of finite perimeter [START_REF] Massaccesi | An elementary proof of the rank one theorem for BV functions[END_REF].

As already observed by Alberti in [1, Theorem 4.13], Theorem 1.5 implies the validity of a similar property for higher-order derivatives. A direct proof of this fact can also be obtained as a corollary of our Theorem 1.1: Theorem 1.6 (Rank-one theorem for higher-order derivatives).

Let Ω ⊂ R d be an open set and let u ∈ L 1 (Ω; R ℓ ) with D r u ∈ M(Ω; SLin r (R d ; R ℓ )) for some r ∈ N, where SLin r (R d ; R ℓ ) contains all symmetric r-linear maps from R d to R ℓ . Then, for |(D r u) s |-almost every x ∈ Ω, there exist a(x) ∈ R ℓ \ {0}, b(x) ∈ R d \ {0} such that d(D r u) s d|(D r u) s | (x) = a(x) ⊗ b(x) ⊗ • • • ⊗ b(x) r times .
1.2. Polar density theorem for BD-functions. The proofs in [START_REF] Alberti | Rank one property for derivatives of functions with bounded variation[END_REF] and in [START_REF] Massaccesi | An elementary proof of the rank one theorem for BV functions[END_REF] of Alberti's rank-one theorem strongly rely on the structure of functions of bounded variation and on their link with the theory of sets of finite perimeter. In particular, so far it has remained open whether a similar statement is valid for the larger class of functions of bounded deformation, i.e. those functions u ∈ L 1 (Ω; R d ) whose symmetric part of the (distributional) derivative is a measure,

Eu := Du + (Du) T 2 ∈ M(Ω; R d×d sym ).
We collect all these functions into the set BD(Ω); see [START_REF] Ambrosio | Fine properties of functions with bounded deformation[END_REF][START_REF] Temam | Mathematical Problems in Plasticity[END_REF][START_REF] Temam | Functions of bounded deformation[END_REF] for a detailed account of the theory of this space. The extension of Alberti's rank-one theorem to the space of functions of bounded deformation follows from our main Theorem 1.1 with the appropriate choice of the differential operator A : This theorem has consequences for the structure theory of BD-functions and lower semicontinuity theory (in the lower semicontinuity theory our structure theorem can, however, be avoided at the price of some mild restrictions on the functional, see [START_REF] Rindler | Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures[END_REF] for BD and [START_REF]Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem[END_REF] for an analogous result in BV); some of these consequences will be explored in future work.

Further, in [START_REF] Conti | Dislocation Microstructures and the Effective Behavior of Single Crystals[END_REF][START_REF] Fuchs | Variational methods for problems from plasticity theory and for generalized Newtonian fluids[END_REF][START_REF] Temam | Mathematical Problems in Plasticity[END_REF] it is motivated why the space

U(Ω) := u ∈ BD(Ω) : div u ∈ L 2 (Ω)
is the appropriate space for elasto-plasticity theory in the geometrically linear setting. For this space we immediately get the following structure result: 1.3. Normal currents, the Rademacher theorem, and metric currents. Our next application of Theorem 1.1 deals with finite families of (Euclidean) normal currents, by which we obtain some consequences concerning differentiability of Lipschitz functions and the theory of metric currents. We assume the reader to be familiar with the theory of currents and with basic multilinear algebra. We refer to [17, Chapters 1 & 4] and Section 3 below for the relevant notations and definitions.

To motivate our next result, recall that any (d -1)-dimensional normal current T ∈ N d-1 (R d ) without boundary (∂T = 0) can be identified via Hodge duality with the derivative of a function u ∈ BV loc (R d ; R), that is, T = ⋆Du. Using this identification and the fact that dim Λ d-1 (V ) = 1 if and only if dim(V ) = d -1, Theorem 1.5 can be rephrased as follows.

Corollary 1.9.

Let T 1 = T 1 T 1 , . . . , T r = T r T r ∈ N d-1 (R d ) be (d-1)-dimensional boundaryless normal currents, i.e. ∂T i = 0 for i = 1, . . . , r. Let further µ ∈ M + (R d ) be a positive Radon measure such that µ ≪ T i for i = 1, . . . , r.
Then, for µ s -a.e. x ∈ R d there exists a (d -1)-dimensional subspace

V x ⊂ R d such that T 1 (x), . . . , T r (x) ∈ Λ d-1 (V x ).
As another simple application of Theorem 1.1 we can generalize the above statement to finite families of normal currents (not necessarily of the same dimension). Then, for µ s -a.e. x ∈ Ω there exists a 1-covector

Theorem 1.10. Let Ω ⊂ R d be an open set and let T 1 = T 1 T 1 ∈ N k 1 (Ω), . . . , T r = T r T r ∈ N kr (Ω)
ω x ∈ Λ 1 (R d ) \ {0} such that T 1 (x) ω x = . . . = T r (x) ω x = 0.
Equivalently, for µ s -a.e. x ∈ Ω, T 1 (x) ∈ Λ k 1 (ker ω x ), . . . , T r (x) ∈ Λ kr (ker ω x ).

Remark 1.11. Let us note in passing the following curious consequence of the above result: It is well known that, apart from the trivial cases k ∈ {1, d -1, d}, the orienting vector T of a k-dimensional normal current T need not be simple, i.e. of the form

T (x) = v 1 (x) ∧ . . . ∧ v k (x), v i (x) ∈ R d . However, if dim V = (d -1), then every w ∈ Λ d-2 (V ) is necessarily simple. Thus, we have that for T ∈ N loc d-2 (R d
) the simplicity of T holds T s -almost everywhere. Note that the current

T = (e 1 ∧ e 2 + e 3 ∧ e 4 ) H 4 {x 5 = 0} ∈ N loc 2 (R 5 ), shows that this statement is false for k-dimensional currents with 1 < k < (d -2).
A particularly relevant instance of Theorem 1.10 is obtained when r = d and k 1 = . . . = k d = 1. In view of the subsequent applications, let us state it in a slightly different (but equivalent) formulation:

Corollary 1.12. Let T 1 = T 1 T 1 , . . . , T d = T d T d ∈ N 1 (R d
) be one-dimensional normal currents such that there exists a positive Radon measure µ ∈ M + (R d ) with the following properties:

(i) µ ≪ T i for i = 1, . . . , d, (ii) for µ-almost every x, span{ T 1 (x), . . . , T d (x)} = R d . Then, µ ≪ L d .
This answers the question about a higher-dimensional analogue of [2, Proposition 8.6]. By the trivial identification of one-dimensional normal currents with vector-valued measures, Corollary 1.12 can be stated in the following equivalent formulation, which in a sense is dual to Theorem 1.5. It can be also directly inferred from Theorem 1.1.

Corollary 1.13. Let µ ∈ M(Ω; R d×d ) be a matrix-valued measure such that

div µ ∈ M(Ω; R d ).
Then,

rank dµ d|µ| (x) ≤ d -1 for |µ| s -a.e. x ∈ Ω.
It has been noted in several places that the validity of the rank-one theorem for maps u ∈ BV(R 2 ; R 2 ) has some direct implications concerning differentiability of Lipschitz functions and the structure of top-dimensional metric currents in the plane [2-4, 34, 35, 40]. Relying on [START_REF] Alberti | On the differentiability of Lipschitz functions with respect to measures in the Euclidean space[END_REF][START_REF] Schioppa | Metric currents and Alberti representations[END_REF], we use Corollary 1.12 to extend these results to every dimension. In particular, Theorem 1.15 below provides a positive answer to the case k = d of the "flat chain conjecture" stated in [START_REF] Ambrosio | Currents in metric spaces[END_REF]Section 11], see [START_REF] Schioppa | Metric currents and Alberti representations[END_REF]Theorem 1.6] for the case k = 1.

Theorem 1.14. Let µ ∈ M + (R d ) be a positive Radon measure such that every Lipschitz map f : R d → R is differentiable µ-almost everywhere. Then, µ ≪ L d . Theorem 1.15. Let T ∈ M met d (R d
) be an Ambrosio-Kirchheim metric current of dimension d, see [START_REF] Ambrosio | Currents in metric spaces[END_REF]. Then, T ≪ L d . In particular, the space of d-dimensional metric currents in R d coincides with the space of Federer-Fleming d-dimensional flat chains,

M met d (R d ) = F d (R d ).
Let us mention that the last two theorems will also follow by a stronger result announced by Csörnyei and Jones in [START_REF] Jones | Product formulas for measures and applications to analysis and geometry, talk given at the conference[END_REF], namely that for every Lebesgue null set E ⊂ R d there exists a Lipschitz map f : R d → R d which is nowhere differentiable in E, see the discussion in the introduction of [START_REF] Alberti | On the differentiability of Lipschitz functions with respect to measures in the Euclidean space[END_REF] for a detailed account of these type of results. 1.4. Sketch of the proof. We conclude this introduction with an outline of the main ideas behind the proof of Theorem 1.1. Let us assume for simplicity that A is a first-order homogeneous operator, A = ℓ A ℓ ∂ ℓ . Assume by contradiction that there is a set E of positive |µ| s -measure such that the polar vector dµ d|µ| (x) is not in the wave cone Λ A for every x ∈ E. One can then find a point x 0 ∈ E and a sequence r j ↓ 0 such that w*-lim

j→∞ (T x 0 ,r j ) ♯ µ |µ|(B r j (x 0 )) = w*-lim j→∞ (T x 0 ,r j ) ♯ µ s |µ| s (B r j (x 0 )) = P 0 ν,
where T x,r : R d → R d is the dilation map T x,r (y) = (y -x)/r, T x,r ♯ denotes the push-forward operator (that is, for any measure σ and Borel set B, [(T x,r ) ♯ σ](B) := σ(x + rB)), ν ∈ Tan(x 0 , |µ|) = Tan(x 0 , |µ| s ) is a non-zero tangent measure in the sense of Preiss [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF], and

P 0 := dµ d|µ| (x 0 ) / ∈ Λ A .
Moreover, one easily checks that

d ℓ=1 A ℓ P 0 ∂ ℓ ν = 0 in D ′ (Ω; R n ).
By taking the Fourier transform of the above equation, we get

A(ξ)P 0 ν(ξ) = 0, ξ ∈ R d .
Having assumed that P 0 / ∈ Λ A , this implies supp ν = {0} and thus ν ≪ L d . The latter fact, however, is not by itself a contradiction to ν ∈ Tan(x 0 , |µ| s ). Indeed, Preiss [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF] provided an example of a purely singular measure that has only multiples of Lebesgue measure as tangents (we also refer to [START_REF] Neil | A measure with a large set of tangent measures[END_REF] for a measure that has every measure as a tangent at almost every point).

On the other hand, P 0 / ∈ Λ A implies that A(ξ)P 0 = 0, so one can hope for some sort of "elliptic regularization" that forces not only ν ≪ L d but also |µ| s ≪ L d in a neighborhood of x 0 . In fact, this is (almost) the case: Inspired by Allard's Strong Constancy Lemma in [START_REF] Allard | An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, Geometric Measure Theory and the Calculus of Variations[END_REF] and using some basic pseudo-differential calculus, we can show that in the above situation not only

ν j := (T x 0 ,r j ) ♯ µ s |µ| s (B r j (x 0 )) * ⇀ ν ≪ L d
but that, crucially, this convergence also holds in the total variation norm,

|ν j -ν|(B 1 ) → 0.
Since ν j ⊥ L d , this latter fact easily gives a contradiction to ν ≪ L d and concludes the proof of the theorem.

Acknowledgments. The authors would like to thank A. Massaccesi and D. Vittone for useful discussions. G. D. P. is supported by the MIUR SIR-grant "Geometric Variational Problems" (RBSI14RVEZ) and F. R. acknowledges the support from an EPSRC Research Fellowship on "Singularities in Nonlinear PDEs" (EP/L018934/1). The Lebesgue-Radon-Nikodým decomposition of a Radon measure µ ∈ M(Ω; R m ) is given as

Proof of the main theorem

µ = dµ d|µ| |µ| = µ a + µ s = gL d + dµ d|µ| |µ| s ,
where dµ d|µ| ∈ L(Ω, |µ|; R m ) is the polar of µ, i.e. the Radon-Nikodým derivative of µ with respect to µ's total variation measure |µ| ∈ M + (Ω), µ a ≪ L d is the absolutely continuous part of µ with density g ∈ L 1 (Ω), and µ s ⊥ L d is the singular part of µ. Note that here and in the following the terms "singular" and "absolutely continuous" are always understood with respect to the Lebesgue measure if not otherwise specified.

We will generically denote by A a k'th-order linear partial differential operator with constant coefficients that acts on smooth functions u ∈ C ∞ (R d ; R m ) as

A u := |α|≤k A α ∂ α u ∈ C ∞ (R d ; R n ), where α = (α 1 , . . . , α d ) ∈ (N ∪ {0}) d is a multi-index, ∂ α = ∂ α 1 1 . . . ∂ α d d , and A α ∈ R n×m are matrices. A vector-valued Radon measure µ ∈ M(Ω; R m ) is said to be A -free if A µ = 0 in D ′ (Ω; R n ).
Here,

D(Ω; R n ) = C ∞ c (Ω; R n ) is the set of R n -valued test functions in Ω with the usual topology and D ′ (Ω; R n ) is the set of R n -valued distributions on Ω.
Given A as above, its symbol A : R d → R n×m is defined as

A(ξ) := |α|≤k (2πi) |α| A α ξ α , ξ ∈ R d ,
where

ξ α := ξ α 1 1 . . . ξ α d d . Note that for u in the Schwartz class S(R d ; R m ), A u(ξ) = A(ξ) u(ξ),
where for v ∈ S(R d ; R m ) we denote by v its Fourier transform,

v(ξ) = F [v](ξ) := v(x)e -2πi x•ξ dx, ξ ∈ R d .
We also recall the definition of the wave cone associated to A [START_REF] Diperna | Compensated compactness and general systems of conservation laws[END_REF][START_REF]Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant[END_REF][START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF][START_REF]The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations[END_REF]:

Λ A := |ξ|=1 ker A k (ξ) ⊂ R m with A k (ξ) := (2πi) k |α|=k A α ξ α .
2.2. First-order operators. For the sake of illustration, we first treat the case when A is a first-order homogeneous constant-coefficient differential operator, namely

A µ = d ℓ=1 A ℓ ∂ ℓ µ = 0 in D ′ (Ω; R n ). (2.1)
Proof of Theorem 1.1 assuming (2.1). We have

Λ A = |ξ|=1 ker A(ξ), A(ξ) = A 1 (ξ) = 2πi d ℓ=1 A ℓ ξ ℓ . Let E := x ∈ Ω : dµ d|µ| (x) / ∈ Λ A ,
where the existence of dµ d|µ| (x) in the sense of the Besicovitch derivation theorem, see [START_REF] Ambrosio | Functions of Bounded Variation and Free-Discontinuity Problems[END_REF]Theorem 2.22], is part of the definition of E.

Assume by contradiction that |µ| s (E) > 0. We now choose a point x 0 ∈ E and a sequence r j ↓ 0 such that

(i) lim j→∞ |µ| a (B r j (x 0 )) |µ| s (B r j (x 0 )) = 0 and lim j→∞ - Br j (x 0 ) dµ d|µ| (x) - dµ d|µ| (x 0 ) d|µ| s (x) = 0;
(ii) there exists a positive Radon measure ν ∈ M + (R d ) with ν B 1/2 = 0 and such that

ν j := (T x 0 ,r j ) ♯ |µ| s |µ| s (B r j (x 0 )) * ⇀ ν;
(iii) for the polar vector it holds that

P 0 := dµ d|µ| (x 0 ) / ∈ Λ A
and there is a positive constant c > 0 such that |A(ξ

)P 0 | ≥ c|ξ| for ξ ∈ R d .
Indeed, (i) holds at |µ| s -almost every point by classical measure theory, (ii) follows by the fact that for |µ| s -almost every x ∈ Ω the space of tangent measures Tan(|µ| s , x) to |µ| s at x is non-trivial, see for instance [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF]Theorem 2.5] or [36, Lemma A.1], and finally, (iii) follows from the assumption |µ| s (E) > 0. We now claim that (i)-(iii) above imply that

0 = ν B 1/2 ≪ L d , (2.2)
lim j→∞ |ν j -ν|(B 1/2 ) = 0. (2.3)
Before proving (2.2) and (2.3), let us show how to use them to conclude the proof.

Recall that ν j ⊥ L d and take Borel sets

E j ⊂ B 1/2 with L d (E j ) = 0 = ν(E j ) and ν j (E j ) = ν j (B 1/2
). Then,

ν j (B 1/2 ) = ν j (E j ) ≤ |ν j -ν|(B 1/2 ) + ν(E j ) = |ν j -ν|(B 1/2 ) → 0,
thanks to (2.3). Hence, we infer ν(B 1/2 ) = 0, in contradiction to (2.2). Thus, |µ| s (E) = 0, concluding the proof of the theorem. We are thus left to prove (2.2) and (2.3). Let us assume that x 0 = 0 and set T r := T x 0 ,r . Clearly,

A T r ♯ µ = 0 in D ′ (B 1 ; R n ).
Therefore, with ν j defined as in (ii) above and c j := |µ| s (B r j ) -1 ,

A (P 0 ν j ) = A (P 0 ν j -c j T r j ♯ µ). (2.4)
Let now {ϕ ε } ε>0 be a compactly supported, smooth, and positive approximation of the identity. By the lower semicontinuity of the total variation,

|ν j -ν|(B 1/2 ) ≤ lim inf ε→0 |ν j * ϕ ε -ν|(B 1/2 ).
Thus, for every j we can find ε j ≤ 1/j such that

|ν j -ν|(B 1/2 ) ≤ |ν j * ϕ ε j -ν|(B 1/2 ) + 1 j . (2.5)
We now convolve (2.4) with ϕ ε j to get

A (P 0 u j ) = A (V j ), (2.6)
where we have set

u j := ν j * ϕ ε j , V j := P 0 ν j -c j T r j ♯ µ * ϕ ε j .
Note that u j , V j are smooth, u j ≥ 0, and

u j * ⇀ ν. (2.7)
Moreover, recalling that x 0 = 0 and c j = |µ| s (B r j ) -1 , by the definition of V j , ν j , P 0 and standard properties of convolutions, see [11, Theorem 2.2], for ε j ≤ 1/4 it holds that

B 3/4 |V j | dx ≤ P 0 T r j ♯ |µ| s -T r j ♯ µ (B 1 ) |µ| s (B r j ) ≤ P 0 |µ| s -µ s (B r j ) |µ| s (B r j ) + |µ| a (B r j ) |µ| s (B r j ) = - Br j dµ d|µ| (0) - dµ d|µ| (x) d|µ| s (x) + |µ| a (B r j ) |µ| s (B r j ) .
Hence, by (i) above,

lim j→∞ B 3/4 |V j | dx = 0. (2.8)
Take a cut-off function χ ∈ D(B 3/4 ) with 0 ≤ χ ≤ 1 and χ ≡ 1 on B 1/2 . Then, (2.6) implies that

A (P 0 χu j ) = χA (P 0 u j ) + A (P 0 χ)u j = A (χV j ) + R j , (2.9)
where the remainder terms R j := A (P 0 χ)u jℓ A ℓ V j ∂ ℓ χ are smooth, compactly supported in B 1 , and satisfy

sup j B 1 |R j | dx ≤ C
for some constant C thanks to (2.7) and (2.8). Taking the Fourier transform of (2.9), we obtain [A(ξ)P 0 ] χu j (ξ) = A(ξ) χV j (ξ) + R j (ξ).

Now multiply by [A(ξ)P

0 ] * = [A(ξ)P 0 ]
T and add χu j (ξ) to both sides of the above equation to obtain

(1 + |A(ξ)P 0 | 2 ) χu j (ξ) = [A(ξ)P 0 ] * A(ξ) χV j (ξ) + χu j (ξ) + [A(ξ)P 0 ] * R j (ξ),
which can be rewritten as

χu j (ξ) = [A(ξ)P 0 ] * A(ξ) χV j (ξ) 1 + |A(ξ)P 0 | 2 + 1 + 4π 2 |ξ| 2 1 + |A(ξ)P 0 | 2 • χu j (ξ) 1 + 4π 2 |ξ| 2 + (1 + 4π 2 |ξ| 2 ) 1/2 [A(ξ)P 0 ] * 1 + |A(ξ)P 0 | 2 • R j (ξ) (1 + 4π 2 |ξ| 2 ) 1/2 .
Hence,

χu j = T 0 [χV j ] + T 1 [χu j ] + T 2 [R j ] =: f j + g j + h j (2.10) with T 0 [V ] := F -1 (1 + |A(ξ)P 0 | 2 ) -1 [A(ξ)P 0 ] * A(ξ) V (ξ) , T 1 [u] := F -1 m 1 (ξ)(1 + 4π 2 |ξ| 2 ) -1 u(ξ) , T 2 [R] := F -1 m 2 (ξ)(1 + 4π 2 |ξ| 2 ) -1/2 R(ξ) ,
where we have set

m 1 (ξ) = (1 + |A(ξ)P 0 | 2 ) -1 (1 + 4π 2 |ξ| 2 ), m 2 (ξ) = (1 + |A(ξ)P 0 | 2 ) -1 (1 + 4π 2 |ξ| 2 ) 1/2 [A(ξ)P 0 ] * .
By (iii) above, T 0 is an operator associated with an Hörmander-Mihlin multiplier (meaning that it has a smooth symbol m 0 (ξ) such that |∂ β m 0 (ξ)| ≤ K|ξ| -|β| for every multi-index |β| ≤ ⌊d/2⌋ + 1 and some K > 0). The L 1 -L

λ L d {|f j | > λ} ≤ C χV j L 1 → 0.
(2.11)

Moreover, the operators T 1 and T 2 are compact from L 1 c (B 1 ) to L 1 loc (R d ), where L 1 c (B 1 ) is the set of L 1 -functions vanishing outside B 1 . Indeed, by Lemma 2.1 below, for every s > 0 the operator d,s) and by [21, Theorem 5.2.7] the symbols m 1 and m 2 are L p -multipliers. We conclude in particular that

(Id -∆) -s/2 w = F -1 (1 + 4π 2 |ξ| 2 ) -s/2 w(ξ) is compact from L 1 c (B 1 ) to L p (R d ) for 1 < p < p(
{g j + h j } j is precompact in L 1 loc (R d ). From (2.8) we further get f j , ϕ = T 0 [χV j ], ϕ = χV j , T * 0 [ϕ] → 0 for every ϕ ∈ D(R d ; R n ), (2.12)
where

T * 0 : S(R d ; R n ) → S(R d , R m
) is the adjoint of T 0 . Since χu j ≥ 0, (2.10) gives that f - j := max{0, -f j } ≤ |g j + h j |. As shown above, the family {g j +h j } j is precompact in L 1 loc (R d ) and thus the previous inequality implies the local equi-integrability of {f - j }. Together with (2.11), (2.12) and Lemma 2.2 below this yields f j → 0 in L 1 loc (R d ) and thus that the sequence

{χu j } is precompact in L 1 loc (R d ).
Since also χu j * ⇀ χν by (2.7), we deduce that χν ∈ L 1 (R d ), which implies (2.2), Moreover,

χu j → χν in L 1 (R d ),
which, taking into account (2.5), implies (2.3).

General operators.

We now treat the general situation, namely the case of a measure µ ∈ M(Ω; R n ) satisfying

A µ = |α|≤k A α ∂ α µ = 0 in D ′ (Ω; R n ). (2.13)
Proof of Theorem 1.1. As before, let us set

E := x ∈ Ω : dµ d|µ| (x) / ∈ Λ A
and assume that |µ| s (E) > 0. Arguing as in the proof for first-order operators, we may find a point x 0 ∈ E satisfying (i), (ii) above and also (iii') for the polar vector it holds that

P 0 := dµ d|µ| (x 0 ) / ∈ Λ A
and there is a positive constant c > 0 such that |A k (ξ)P 0 ≥ c|ξ| k for ξ ∈ R d . We will show that (i), (ii) and (iii') together imply (2.2) and (2.3), and thus yield the desired contradiction.

Assuming that x 0 = 0, we note that (2.13) and a simple scaling argument give

A k T r ♯ µ + k-1 h=0 A h r k-h T r ♯ µ = 0,
where A h := |α|=h A α ∂ α is the h-homogeneous part of the operator A . Hence, with ν j defined as in (ii) and

c j = |µ| s (B r j ) -1 , |α|=k A α ∂ α (P 0 ν j ) = |α|=k A α ∂ α P 0 ν j -c j T r j ♯ µ - k-1 h=1 A h r k-h j c j T r j ♯ µ .

Mollification and localization now yield

|α|=k A α ∂ α (P 0 χu j ) = |α|=k A α ∂ α (χV j ) + R j . (2.14)
Here, as before,

u j := ν j * ϕ ε j , V j = [P 0 ν j -c j T r j ♯ µ] * ϕ ε j , where χ ∈ D(B 3/4 ) with 0 ≤ χ ≤ 1, χ ≡ 1 on B 1/2 ,
and ϕ ε j is a sequence of mollifier such that (2.5) is satisfied. In particular, by (i), χV j L 1 → 0. Moreover, the remainder term R j can be written as a finite sum of smooth-coefficient partial differential operators of order at most k-1 applied to smooth functions with bounded L 1 -norm and compact support:

R j = k-1 h=0 |α|=h b α (x)∂ α z α j ,
where b α (x) ∈ D(B 3/4 ), the functions z α j are smooth and compactly supported, and

sup j z α j L 1 ≤ C for some constant C. Namely, R j = R 1 j + R 2 j + R 2 j where R 1 j = |α|=k β+γ=α |γ|≥1 c βγ ∂ γ χ ∂ β (A α P 0 χu j ), R 2 j = |α|=k β+γ=α |γ|≥1 c βγ ∂ γ χ ∂ β (A α χV j ), R 3 j = |α|≤k-1 |α|=h χ ∂ α χA α (r k-h j c j T r j ♯ µ) * ϕ ε j
with c βγ ∈ R, and χ ∈ D(B 1 ) is identically equal to 1 on the support of χ. By taking the Fourier transform of (2.14) and performing the same computations as in the first part, but now multiplying with [A k (ξ)P 0 ] * instead of [A(ξ)P 0 ] * , we obtain

χu j = S 0 [χV j ] + S 1 [χu j ] + R j , (2.15
) where S 0 and S 1 are given by

S 0 [V ] = F -1 [A k (ξ)P 0 ] * A k (ξ) V (ξ) 1 + |A k (ξ)P 0 | 2 , S 1 [u] = F -1 (1 + 4π 2 |ξ| 2 ) k 1 + |A k (ξ)P 0 | 2 • u(ξ) (1 + 4π 2 |ξ| 2 ) k .
Applying the Hörmander-Mihlin multiplier theorem and arguing as for first-order operators, we deduce that

sup λ≥0 λ L d {|S 0 [χV j ]| > λ} ≤ C χV j L 1 (B 1 ) → 0 and S 0 [χV j ] * ⇀ 0.
Moreover, the family {S 1 [χu j ]} is precompact in L 1 loc (R d ). To conclude the proof it is enough to show that { R j } is precompact in L 1 loc (R d ), since then the application of Lemma 2.2 as in the first part will imply the validity of (2.2) and (2.3). The generic term of R j can be written as

f α j = Q • (Id -∆) -k 2 • P α • (Id -∆) |α|-k 2 [z α j ] where 0 ≤ |α| ≤ (k -1), sup j z α j L 1 ≤ C, Q[z] = F -1 (1 + |A k (ξ)P 0 | 2 ) -1 (1 + 4π 2 |ξ| 2 ) k/2 A k (ξ) z(ξ) ,
and P α is the k'th-order pseudo-differential operator given by

P α [z](x) = b α (x) (2πi) |α| ξ α (1 + 4π 2 |ξ| 2 ) |α|-k 2 z(ξ) e 2πix•ξ dξ, x ∈ R d .
The composition (Id-∆) -k/2 2.4. Auxiliary results. Finally, we prove the two simple technical lemmas that have been used in the proofs above. The first is an L 1 -compactness result in the spirit of the Sobolev embedding theorems. Since we have not been able to find a reference we provide its simple proof.

(s, d) = b 1,ε + b 2,ε with b 1,ε ∈ C 1 c (R d ) and b 2,ε L 1 < ε,
see [START_REF]Modern Fourier Analysis[END_REF]Proposition 6.1.6]. Thus,

(Id -∆) -s/2 u = b 1,ε * u + b 2,ε * u := T 1,ε [u] + T 2,ε [u]. Because b 1,ε ∈ C 1 c (R d ), T 1,ε is compact from L 1 c (B 1 ) to L 1 (R d ). Moreover, (Id -∆) -s/2 -T 1,ε L 1 →L 1 ≤ T 2,ε L 1 →L 1 ≤ ε,
so that (Id -∆) -s/2 is the limit in the uniform topology of compact operators and thus compact as well. The conclusion of the lemma now follows by Hölder's inequality.

The second lemma is an easy consequence of the Vitali convergence theorem: (c) the sequence of negative parts {f - j } is equi-integrable, lim

L d (E)→0 sup j∈N B 1 f - j dx = 0. Then, f j → 0 in L 1 loc (B 1 ). Proof. Let ϕ ∈ D(B 1 ), 0 ≤ ϕ ≤ 1. It is enough to show that lim j→∞ ϕ|f j | dx = 0.
(2.16)

We write

ϕ|f j | dx = ϕf j dx + 2 ϕf - j dx ≤ ϕf j dx + 2 f - j dx.
The first term on the right-hand side goes to 0 as j → ∞ by assumption (a). Thanks to the Vitali convergence theorem, assumptions (b) and (c) further give that also the third term vanishes in the limit. Hence, (2.16) follows.

Applications

Theorems 1.5, 1.6 and 1.7 follow from Theorem 1.1 simply by applying it to the differential constraints that gradients, higher gradients, or symmetrized gradients, respectively, have to satisfy.

Proof of Theorem 1.5. Let µ = (µ k j ) ∈ M(Ω; R ℓ×d ) be the (distributional) gradient of a function u ∈ BV(Ω; R ℓ ), µ = Du. Then,

0 = ∂ i µ k j -∂ j µ k i i, j = 1, . . . , d; k = 1, . . . , ℓ. Setting A µ := ∂ j µ k i -∂ i µ k j i,j=1,...,d; k=1,...,ℓ , it is a simple algebraic exercise, carried out for instance in [19, Remark 3.5(iii)], to compute that Λ A = a ⊗ ξ ∈ R ℓ×d : a ∈ R ℓ , ξ ∈ R d \ {0} .
Corollary 1.5 then follows directly from Theorem 1.1.

Proof of Theorem 1.6. For the operator

A µ := ∂ j µ k α 1 •••α h iα h+2 •••αr -∂ i µ k α 1 •••α h jα h+2 •••αr i,j,
Λ A = a ⊗ ξ ⊗ • • • ⊗ ξ ∈ SLin r (R d ; R ℓ ) : a ∈ R ℓ , ξ ∈ R d \ {0} ;
see [START_REF]A-quasiconvexity, lower semicontinuity, and Young measures[END_REF]Example 3.10(d)] for the details.

Proof of Theorem 1.7. Let µ = (µ k j ) ∈ M(Ω, R d×d sym ) be the (distributional) symmetrized gradient of u ∈ BD(Ω), µ = Eu. Then, by direct computation, see [19, Example 3.10(e)],

0 = A µ := d i=1 ∂ ik µ j i + ∂ ij µ k i -∂ jk µ i i -∂ ii µ k j j,k=1,...,d .
These equations are often called the Saint-Venant compatibility conditions in applications. Hence, for Before proving Theorem 1.10, let us recall some simple facts concerning (Euclidean) currents and multi-linear algebra. We refer to [START_REF] Federer | Geometric measure theory[END_REF] for more details.

M ∈ R d×d sym , -(4π) -2 A(ξ)M = (M ξ) ⊗ ξ + ξ ⊗ (M ξ) -(trM ) ξ ⊗ ξ -|ξ| 2 M, which gives ker A 2 (ξ) = ker A(ξ) = a ⊗ ξ + ξ ⊗ a : a ∈ R d , ξ ∈ R d .
Given a finite dimensional vector space V we let Λ k (V ) be the set of k-vectors and Λ k (V ) ≃ (Λ k (V )) * be the set of k-covectors. If v ∈ Λ k (V ) and η ∈ Λ 1 (V ), then the interior product of η with v is the (k -1)-vector v η ∈ Λ k-1 (V ) defined by duality as v η, ω := v, η ∧ ω for every ω ∈ Λ k-1 (V ), see [START_REF] Federer | Geometric measure theory[END_REF]Section 1.5].

Following [17,

Theorem 1 . 5 (

 15 Alberti's rank-one theorem). Let Ω ⊂ R d be an open set and let u ∈ BV(Ω; R ℓ ). Then, for |D s u|-almost every x ∈ Ω, there exist a

Theorem 1 . 7 .

 17 Let Ω ⊂ R d be an open set and let u ∈ BD(Ω). Then, for |E s u|almost every x ∈ Ω, there exist a(x), b(x) ∈ R d \ {0} such that dE s u d|E s u| (x) = a(x) ⊙ b(x), where we define the symmetrized tensor product as a ⊙ b := (a ⊗ b + b ⊗ a)/2 for a, b ∈ R d .

Corollary 1 . 8 .

 18 Let Ω ⊂ R d be an open set and let u ∈ U(Ω). Then, for |E s u|-almost every x ∈ Ω, there exist a(x), b(x) ∈ R d \ {0} with a(x) ⊥ b(x) such that dE s u d|E s u| (x) = a(x) ⊙ b(x).

  be normal currents, where k 1 , . . . , k r ∈ {1, . . . , d}, r ∈ N. Let further µ ∈ M + (Ω) be a positive Radon measure such that µ ≪ T i for i = 1, . . . , r.

2. 1 .

 1 Notation. We denote by M(Ω; R m ) the space of all finite Radon measures on an open set Ω ⊂ R d with values in R m and by M + (Ω) the space of positive Radon measures on Ω. We write µ = w*-lim j→∞ µ j or µ j * ⇀ µ for the local weak*convergence of µ j to µ, that is ϕ dµ j → ϕ dµ for all ϕ ∈ C 0 c (Ω), the set of all continuous functions with compact support in Ω. The d-dimensional Lebesgue measure is L d . Given a Borel set B ⊂ Ω and a measure µ ∈ M(Ω; R m ) (or µ ∈ M + (Ω)), we denote by µ B the restriction of µ to B.

Lemma 2 . 1 .

 21 For u ∈ S(R d ) and s > 0 define(Id -∆) -s/2 u := F -1 (1 + 4π 2 |ξ| 2 ) -s/2 u(ξ) .Then, (Id -∆) -s/2 extends to a compact map from L 1 c (B 1 ) to L p (R d ) for 1 ≤ p < p(d, s), wherep(d, s) c (B 1 ) ⊂ L 1 (R d ) is the set of L 1 -functions supported in B 1 .Proof. For u in the Schwartz class we can write(Id -∆) -s/2 u = b(s, d) * u where b(s, d) = F -1 [(1 + 4π|ξ| 2 ) -s/2 ]is the Bessel potential of order s, see [22, Section 6.1.2]. By classical estimates [22, Proposition 6.1.5], b(s, d) ∈ L p for 1 ≤ p < p(d, s) so that by Young's inequality for convolutions, (Id -∆) -s/2 u ∈ L p for 1 ≤ p < p(d, s) (actually also for p = p(d, s) if s = d). For every ε > 0 we can write b

Lemma 2 . 2 .

 22 Let {f j } ⊂ L 1 c (B 1 ) be a family of functions such that (a) f j * ⇀ 0 in D ′ (B 1 ); (b) The negative parts of f j tends to zero in measure in, i.e. lim j→∞ L d {f - j > λ} = 0 for every λ > 0;

Theorem 1 .

 1 1 now implies the conclusion. Proof of Corollary 1.8. The only fact to show in addition to the assertion of Corollary 1.7 is that a(x) • b(x) = 0. For Eu we have the Lebesgue-Radon-Nikodým decomposition Eu = Eu L d + E s u and thus div u = tr(Eu) L d + a(x) • b(x) |E s u| Since div u ∈ L 2 (Ω), we must have a(x) • b(x) = 0 for |E s u|-almost every x ∈ Ω.

  Section 4.1.7], we let D k (Ω) := D(Ω, Λ k (R d )) and D k (Ω) := D ′ (Ω, Λ k (R d ))be the sets of compactly supported k-differential forms with smooth coefficients and the set of k-dimensional currents, respectively. For T ∈ D k (Ω) the boundary ∂T ∈ D k-1 (Ω) is defined by duality with the exterior differential via ∂T, ω := T, dω , where ω ∈ D k-1 (Ω). One easily checks that∂T = -d i=1 ∂ i T dx i ,(3.1)see[17, p. 356]. Here, forT ∈ D k (Ω) and η ∈ C ∞ (Ω; Λ 1 (R d )), T η ∈ D k-1 (Ω) is defined as T η, ω := T, η ∧ ω , ω ∈ D k-1 (Ω) and ∂ i T ∈ D k (Ω) is defined by duality via ∂ i T, φ dx j 1 ∧ • • • ∧ dx j k = -T, ∂ i φ dx j 1 ∧ • • • ∧ dx j k . We endow Λ k (R d )with the mass norm, see [17, Section 1.8]. A k-current is said to have finite mass if it can be extended to a Λ k (R d )-valued (finite) Radon measure and we let T be the total variation of T and T := dT d T , see [17, Section 4.1.7]. In this context, the Radon-Nikodým theorem reads as T = T T . We denote by N k (Ω) the set of k-dimensional normal currents, i.e. those currents such that T and ∂T both have finite mass. Note that the boundary of a k-dimensional normal current T can be seen as a Λ k-1 (R d )-valued Radon measure, ∂T ∈ M(Ω; Λ k-1 (R d )). (ii) There are d normal one-dimensional currents T 1 = T 1 T 1 , . . . , T d = T d T d ∈ N 1 (R d ) such that µ ≪ T i for i = 1, . . . , d, and span T 1 (x), . . . , T d (x) = R d for µ-a.e. x ∈ R d . Proof. The implication (i) ⇒ (ii) is obtained by choosing (in a measurable way) for µ-a.e. x ∈ R d a basis {e 1 (x), . . . , e d (x)} of V (µ, x) and by applying to each e i the implication (i) ⇒ (ii) of [4, Corollary 6.5]. For the other implication, simply notice that, by the implication (ii) ⇒ (i) of [4, Corollary 6.5], T i (x) ∈ V (µ, x) for µ-a.e. x ∈ R d . Proof of Theorem 1.14. By [4, Theorem 1.1] the assumptions in the statement of the theorem are equivalent to V (µ, x) = R d for µ-a.e. x ∈ R d . This implies that µ ≪ L d by Lemma 3.1 and Corollary 1.12. Proof of Theorem 1.15. By [40, Theorem 1.3] the mass measure T associated with a d-dimensional metric current T ∈ M met d (R d ) admits d independent Alberti representations, which, by the very definition of decomposability bundle, see [4, Section 2.6], implies that V ( T , x) = R d for T -a.e. x ∈ R d . Theorem 1.15 hence follows from Theorem 1.14, see also the discussion after Theorem 1.3 in [40].

  1,∞ estimates [21, Theorem 5.2.7] in conjunction with (2.8) give

	sup
	λ≥0

  •P α is a pseudo-differential operator of order 0, see [41, Theorem 2, Chapter VI], and thus bounded from L p (R d ) to itself, see [41, Proposition 4, Chapter VI]. By (iii') and the Hörmander-Mihlin multiplier theorem, also Q is a bounded operator from L p(R d ) to itself. Since |α| ≤ k -1, Lemma 2.1 below implies that (Id -∆) (|α|-k)/2 is compact from L 1 c (B 1 ) to L p (R d ) for 1 < p < p(d, |α| -k). We conclude that {f α j } is precompact in L 1 loc (R d ).The validity of (2.2) and (2.3) now follows from (2.15) by arguing as before.

  α 1 ,...,αn=1,...,d; k=1,...,ℓ; h=1,...,r one can see that A µ = 0 if and only if µ is an r'th-order derivative, and furthermore compute that

Proof of Theorem 1.10. Let us set

and note that the assumption of Theorem 1.10 can be rewritten as

By applying Theorem 1.1 in conjunction with Remark 1.3 we deduce that for |T | salmost every x ∈ Ω there exists ξ x = 0 such that

Thanks to (3.1), one easily checks that for

where

Let µ ∈ M + (Ω) be as in the statement of the theorem and note that, since µ ≪ T i for every i = 1, . . . , r, the Radon-Nikodým derivatives d|T | d T i and dT i d|T | exist µ-almost everywhere. Then, Proof of Corollary 1.12. By Theorem 1.10, assumption (i) implies that for µ s -almost every x ∈ R d there exists a (d -1)-dimensional subspace V x such that

Assumption (ii) hence gives that µ s = 0, which is the desired conclusion.

Proof of Corollary 1.13. Let µ = (µ k j ) ∈ M(Ω; R d×d ) and let

Then, for M ∈ R d×d , A(ξ)M = (2πi)M ξ, so that

The conclusion follows from Theorem 1.1 and Remark 1.3.

We will now show how to obtain Theorems 1.14 and 1.15 from Corollary 1.12. In order to do so, we assume the reader to be familiar with the work of Alberti & Marchese [START_REF] Alberti | On the differentiability of Lipschitz functions with respect to measures in the Euclidean space[END_REF] concerning differentiability of Lipschitz functions, with the definition of metric currents given in [START_REF] Ambrosio | Currents in metric spaces[END_REF], as well as with the work of Schioppa in [START_REF] Schioppa | Metric currents and Alberti representations[END_REF]. We refer to these papers also for notations and definitions.

Let us start with the following lemma, which is essentially [4, Corollary 6.5].


Ilenia Tinnirello

Giovanni Neglia

Laura Giarré

Giuseppe Di Bella

Alain Jean-Marie

Mariano Ippolito

Zigbee Plc

Hv

Large Scale Control of Deferrable Domestic Loads in Smart Grids

Keywords: PLC, GPRS Data Concentrator Data Concentrator ADSL, FIBER, 3G … … ADSL

In this paper, we investigate a realistic and lowcost deployment of large scale direct control of inelastic home appliances whose energy demand cannot be shaped, but simply deferred. The idea is to exploit 1) some simple actuators to be placed on the electric plugs for connecting or disconnecting appliances with heterogeneous control interfaces, including nonsmart appliances, and 2) the Internet connections of customers for transporting the activation requests from the actuators to a centralized controller. Our solution requires no interaction with home users: in particular, it does not require them to express their energy demand in advance. A queuing theory model is derived to quantify how many users should adopt this solution in order to control a significant aggregated power load without significantly impairing their quality of service.

I. INTRODUCTION

Load control in modern power grids is becoming increasingly essential for maintaining a balance between energy supply and demand. Traditionally, demand was much more variable and less controllable than supply, so energy balance was achieved by adapting dynamically generation levels to match consumption. Now, the increasing penetration of unpredictable renewable energy has radically changed the scenario. The possibility of controlling power demand is becoming more appealing both for the energy utilities (who can better plan production as well as control the grid reliability) and the end customers (who can actively participate in the energy market). Two envisaged main approaches are: Demand-Response and Direct Load Control (DLC). The former refers to the possibility of the end users changing their normal consumption patterns in response to a dynamic price signal, the latter to the possibility of the energy utility (or third-party entities) switching some specific users appliances on and off during peak demand periods.

In this paper, we deal with DLC mechanisms for deferrable domestic loads, i.e. loads of residential users whose power demand can be postponed but not interrupted. Examples of deferrable domestic loads are dishwashers and laundry machines. Indeed, most of the current DLC programs work on large industrial loads or domestic thermostatic loads. Large industrial loads have a significant and well characterized power demand and can be disconnected (according to prearranged agreements between the energy utility and the customer) with very simple decisions and control networks (often based on voice dispatch) [START_REF] Callaway | Achieving controllability of electric loads[END_REF]. Domestic thermostatic loads, such as air conditioners and heating systems, allow a fine-tuning regulation of power demand [START_REF] Mathieu | State estimation and control of electric loads to manage real-time energy imbalance[END_REF], [START_REF] Hao | Ancillary service to the grid through control of fans in commercial building HVAC systems[END_REF]. However, a larger number of customers has to be involved, thus making the control algorithms more complex [START_REF] Hsu | Dispatch of direct load control using dynamic programming[END_REF], [START_REF] Sheble | Direct load control-a profit-based load management using linear programming[END_REF]. Only recently, deferrable loads have been considered for DLC, by assuming that each appliance is able to provide its energy profile, as well as an operation deadline, to the controller [START_REF] O'brien | Scheduling non-preemptive deferrable loads[END_REF].

In this work, we deal with a large number of domestic deferrable loads whose location affects a substantial power demand in a given geographic area. This choice of load allows the energy utility to potentially reach all the residential customers with limited investments. Indeed, rather than considering appliances with smart models supporting advanced control interfaces [START_REF] O'brien | Scheduling non-preemptive deferrable loads[END_REF], it is sufficient to deploy simple controlled switches between the electric plugs and the loads and to rely on a stochastic characterization of the appliances' profile. Differently from previous works, this study assumes that the appliances can be disconnected only once (at their activation request), while the control period is longer than the few minutes characterizing the on/off cycle of the thermostatic loads in [START_REF] Keep | Using load switches to control aggregated electricity demand for load following and regulation[END_REF], but shorter than the several hours characterizing the complete charging cycle of electric vehicles in [START_REF] Alizadeh | A scalable stochastic model for the electricity demand of electric and plug-in hybrid vehicles[END_REF].

The main goal of this work is to evaluate if DLC applied to deferrable domestic loads may allow the energy utility to control a significant percentage of power at the geographical scale and time scale of interest, while maintaining an acceptable quality of service for the customers. Since loads can only be deferred, the customers experience the effect of the direct control in terms of a possible service delay. Obviously, the more significant the change required to the power demand profile is (either in terms of instantaneous power reduction or duration of the control time interval), the longer the user delay will be. At the same time, the larger the number of appliances under control is, the smaller the probability that the plug of a given customer will be disconnected. Therefore, we aim to answer the fundamental question of how many customers should be controlled in order to have a desired power reduction with an acceptable delay experienced by each customer.

To this purpose, we apply queuing theory to study the relation between the load availability and the temporal constraints on the acceptable service delay. Although queuing theory has been already used for scheduling the energy suppliers working on renewable sources [START_REF] Neely | Efficent Algorithms for Renewable Energy Allocation to Delay Tolerant Consumers[END_REF] or characterizing the aggregated power demand of a number of users [START_REF] Alizadeh | A scalable stochastic model for the electricity demand of electric and plug-in hybrid vehicles[END_REF], our achievements are more general since we study the service delays experienced when the number of electric loads is limited by admission control. This framework also offers some theoretical challenges, since it requires studying the transient period of a queue with The paper is organized as follows. In Section II we present our reference scenario and the architectural choices that allow DLC on deferrable domestic loads to be implemented with limited investments. In Section III we present the model used for characterizing the aggregated power demand of a large number of deferrable appliances, while in Section IV we model the effect of the admission control on the number of active appliances and quantify the service delay distribution. Section V presents some numerical results enlightening the tradeoffs between power reduction, service delay and number of controlled users. Finally, some concluding remarks are drawn in Section VI.

II. SYSTEM DESCRIPTION We consider the problem of power shaving for an energy supplier which wants to reduce its customers consumption during a given time interval. To achieve this goal, the energy supplier may interact with the distribution system operators (DSO), and/or with novel intermediate figures called load aggregators [START_REF] Ruiz | A Direct Load Control Model for Virtual Power Plant Management[END_REF] by means of a communication infrastructure transporting the control messages. Both possibilities are illustrated in Fig. 1, where dashed lines indicate the flows of control messages.

The solution based on the DSO can take advantage of the advanced metering infrastructure (AMI), mainly based on PLCs (Power Line Communications) and GPRS (General Packet Radio Services) technologies, for reaching a largescale number of domestic users in the low-voltage (LV) distribution grid [START_REF]An overview to new plc applications and new challenges -maxium integrated products[END_REF], [START_REF]Demand-response technology roadmap[END_REF]. Conversely, the load aggregator can exploit standard Internet connectivity to interact with appliances at customers premises. The presence of the load aggregator allows to have an intermediary for interacting with large scale numbers of customers [START_REF] Ancillotti | The role of communication systems in smart grids: Architectures, technical solutions and research challenges[END_REF]. The aggregator can collect information on user profiles and willingness to respond to control actions for dispatching the controlled loads as a response of the energy supplier commands. Since the two approaches can coexist, we will talk generically about the load controller to denote the entity which drives the appliances.

The energy supplier specifies an high-level command for the load controller in terms of maximum tolerable probability to exceed a given power demand in the controlled area [START_REF] Di Bella | Modeling Energy Demand Aggregator for Residential Users[END_REF], [START_REF] Neglia | Unidirectional Probabilistic Direct Control for Deferrable Loads[END_REF]. The load controller is responsible to meet this requirement by deciding about the activation of deferrable loads in the controlled area. To this purpose, each household can be equipped with a gateway which transmits and receivers the admission control messages and interacts with the domestic appliances by means of local area technologies, such as ZigBee and WiFi.

While smart appliances can be natively able to communicate with the gateway, the choice of working with deferrable loads allows to easily control also dumb appliances by means of programmable switches. Examples of these switches are the smart plugs, which can act as a physical interface between the appliances and the control application by exposing simple control primitives such as the activation of disactivation of the electric plug [START_REF] Keep | Using load switches to control aggregated electricity demand for load following and regulation[END_REF] or the setting of an operation timer [START_REF] Galioto | Smart plugs: A low cost solution for programmable control of domestic loads[END_REF].

A. Use Cases

According to the employed admission control logic, to the actors (energy utilities, DSO, load aggregators) involved in the control and to the geographical scale of the controlled area, DLC can be used for different goals, such as improving the grid reliability or operation savings [START_REF] Chen | Scheduling direct load control to minimize system operation cost[END_REF]. to the previous case, the control actions enforced by the energy utilities on the demand side can help to ease the regulation process of the voltage profile in the network, since the set-point in the automatic regulator of the station affects the voltage profile across the entire MV network. For advanced systems, with a management mode inspired by the smart-grid model, this second level of control may also participate in the control of the frequency, especially in emergency conditions in which the grid may operate in stand-alone configuration. • Power System Zone: While the control actions at the previous levels are implemented by the electric utilities, the third level admission control can be performed at the community of customers level to respond to price signals, with the goal to make the power demand elastic in the electricity market. For this type of applications, the control of the energy consumption can be realized on a quite long time scale (from several hours to several days). Regardless of the control level and involved actors, we assume that a centralized controller implements an admission control logic devised to enforce a probabilistic cap on the maximum power absorbed during a specific time interval, in the sense that the probability that the instantaneous power exceeds a desired limit is bounded (including a deterministic bound in case such a probability is set to zero). Such a bound, as detailed in the Section IV under the assumption of homogeneous appliances, can be equivalently given in terms of maximum number of appliances that can be active at the same time.

III. MODEL OF APPLIANCES In this section, we characterize the power consumption originated by the aggregation of one type only of deferrable appliances, during activation time can be flexible for the users (e.g. dishwashers or laundry machines). Since each user consumption coincides with its appliance consumption, we indifferently refer to the total number of appliances or users.

Recent studies have characterized the percentage of users activating a specific residential appliance along different intervals of the day [START_REF] Miceli | Sustainable development and energy saving laboratory[END_REF] and estimated their statistical properties. Moreover, it has been shown [START_REF] Pagliuca | Capacity assessment of residential demand response mechanisms[END_REF] that the aggregation of even a few thousands of customers makes residential demand uniform enough in a given period. As reported in Table I, in these studies the day is divided into equal duration intervals and the percentage of active users over a given population is averaged in each interval. Assuming that the user population U is large and considering an observation time of one day, we can model the activation of a new appliance as a nonhomogeneous Poisson process with arrival rate λ(t). We also assume that the arrival rate λ(t) is perfectly known.

The working interval of the reference appliance is usually deterministic or a deterministic function of the machine working program. Let D be the time interval during which the appliance keeps working after its activation. We assume every appliance has the same deterministic operation time D and its instantaneous power consumption is a random variable X(t) with known time-invariant probability density function.

Let u(t) be then the total number of users whose appliance is on at time t. Since, in absence of critical problems, the energy production is able to follow exactly the energy demand, we can model the u(t) random process as the number of jobs in a M (t)/D/∞ queue. 1 The instantaneous power consumption P (t) at time t can then be calculated as

P (t) = u(t) i=1 X i (t). (1)
We can easily characterize the probability distribution p i (t) = P r{u(t) = i}, that is the probability to find i active users at time t. Since all the appliances activated before t -D are deterministically switched off in t, we find i users at time t if exactly i new appliances have been switched on during the last D interval:

p j (t) = [ λt-D D D] j j! e -λt-D D D (2)
where λt-D We observe that there is no correlation between the number of active users in t and t + D. Conversely, we can derive the probability distribution p j (t + ∆t|u) of j active users at t + ∆t for ∆t < D given that there are u active users at t, by considering the joint probability to have k departures and j -u + k arrivals in [t, t + ∆t]. The probability a(t, ∆t) that an appliance active in t is switched off at time t + ∆t can be expressed as the probability that its arrival has occurred at time t -D + ∆t:

a(t, ∆t) = t-D+∆t t-D λ(x)dx t t-D λ(x)dx . (3)
Eq. 3 follows from the definition of conditional probability. Therefore, the probability d k (t, ∆t|u) that k appliances among the u active at time t are switched off during [t, t + ∆t] is expressed by the binomial:

d k (t, ∆t|u) = u k a(t, ∆t) k (1 -a(t, ∆t)) u-k ,
for k = 0, 1, . . . u. Being l = max{0, u -j}, the conditional probability to have j active users can be expressed as:

p j (t + ∆t|u) = e -λt ∆t ∆t u k=l d k (t, ∆t|u) ( λt ∆t ∆t) j-u+k (j -u + k)! . (4)
Under the assumption that the load controller can monitor the actual number of active users at the current time instant t, Eq. 4 allows us to evaluate the distribution of the number of active users at t + ∆t. Numerical Example. For the laundry machine data in Table I, assuming D equal to 90 min, the thin dotted curves in Fig. 2 shows a realization of the random process u(t) with a total population of 10000 users over three days. In the figure we can clearly identify the peak hours and the effect of the timevarying activation rates. The maximum number of users into the system is about 3100, which corresponds to a load of 4.65 MW if we consider a power consumption equal to 1.5 KW for each laundry machine. At noon, being λ10:30 ([18], [START_REF] Pagliuca | Capacity assessment of residential demand response mechanisms[END_REF]) be simultaneously active during a given time interval. The dashed and continuous curves in Fig. 2 show the number of active appliances and the total number of appliances (active and disconnected appliances), respectively, when c is set to 2500 (first day) or 2200 (second and third day). The figure also shows the effect of different resumption policies after load control: in the first and second day, all the disconnected appliances are immediately activated at the end of the control interval, thus leading to a potentially sudden increase of the power demand (second day); in the third day, the waiting users are admitted gradually by considering a maximum increment rate for the limit c. From the figure, it is evident that load control performs similarly to load-shifting: for example, in the second day the peak hour is shifted from noon to 2 p.m. Load shifting capacity is constrained by the number of users involved into the admission control and by the temporal delay that can be tolerated. To design the control mechanism, we need to quantify these two constraints.

A. Power constraints

In order to achieve a reduction of the power demand expected during a time interval [T s , T e ], we assume that the controller would like to enforce a maximum power consumption level P g for the set of appliances under control with probabilistic guarantees: the instantaneous power consumption (P (t)) can exceed P g with probability at most , i.e., Prob{P (t) > P g } ≤ . The problem that we consider is finding the maximum number c of appliances active during the control period which guarantees that Prob{P (t) = u(t) i=1 X i > P g } ≤ , when u(t) ≤ c. For simplicity, we develop the calculations for the case when c is large enough for the aggregated power consumption to be approximated by a normal distribution. This case is also likely the most relevant from a practical point of view, given that we are interested in controlling a large number of appliances and we do not want to excessively limit the number of appliances active at a given time instant (otherwise users would experience too large delays). It is clear that

Prob{P (t) = u(t) i=1 X i > P g } is maximized for u(t) = c.
In such a case, the average of the instantaneous power in Eq. 1, is given by E [P (t)] = cE [X] and Var (P (t)) = cVar (X). Under the normal approximation, Prob{P (t) > P g } is lower than if and only if (P g -cE [X])/ cVar (X) ≥ z 1-, where z 1-is the percentile of the standard normal distribution. The power consumption profile is then satisfied if: cE [X] + z 1-cVar (X) ≤ P g . Then c = n * , where n * is the smallest solution of the quadratic equation

(n * E [X] -P g ) 2 = z 2 1-n * Var (X) . (5)

B. Active and disconnected appliances

We assume that the load controller can only work on the new activation requests, because disconnecting appliances whose working cycle is in progress can be inefficient and uncomfortable for the users. In order to guarantee that less than c appliances are active at time T s , the load controller anticipates the action at T sc = T s -D, because the appliances active at time T sc (whose number can be larger than c) will have necessarily terminated their operation at time T s .

During the control period [T sc , T e ], each appliance activation is conditioned to an admission control: the appliance can be switched on only if the number of appliances already active is lower than c. When the new appliance cannot be activated, the controller disconnects the relative plug until some power resources become available. We assume that there is no limit to the maximum number of disconnected appliances and that they can be orderly reconnected into the system (when possible) according to the arrival time of their activation request. Under these assumptions, we can model the controlled load process as an M (t)/D/c process.

Recall that u(t) is the number of appliances at time t that are working or are waiting for some available power in order to start. The probability distribution p j (T sc ) of u(T sc ) can be evaluated on the basis of the previous analysis without admission control. Consider now a generic time duration ∆t < D. The number of users u(T sc +∆t) in the system can be obtained from u(T sc ) considering the difference between the number of new activation requests and the number of appliances that have been switched off in the interval [t, t + ∆t]. While the new requests depend on the average arrival rate λTsc ∆t , the number of appliances switched off depends on the number of arrivals in the previous interval [T sc -D, T sc -D + ∆t], being the total number of arrivals in the interval [T sc -D, T sc ] equal to u(T sc ). In other words, p j (T sc + ∆t|u(T sc )) can be expressed using Eq. 4. By weighting each conditional probability with the probability of the conditioning event to have u(T sc ) users in T sc , we can find the probability to have j users in the system at time T sc + ∆t, given by

p j (T sc + ∆t) = ∞ u=0 p j (T sc + ∆t|u)p u (T sc ). (6)
Consider now a generic instant t > T sc + D. The number of appliances u(t) is now depending on u(t -D), because the appliances disconnected at t -D are still into the system at time t. Indeed, u(t) is given by the sum of u(t -D) -c (if positive) and the new arrivals, i.e. we can find j appliances in t if we have j + min{0, c -u(t -D)} new arrivals, with a probability given by:

p j (t) = j+c i=0 p i (t -D) [ λt-D D D] j+min{0,c-i} (j + min{0, c -i})! e -λt-D D D . (7)
Since Eq. 6 yields p i (t) for t ∈ [T sc , T sc + D], for t ∈ [T sc + D, T e ] it is possible to apply Eq. 7 in k = (t -T sc )/D consecutive time intervals starting from the distribution p i (T sc + (t -T sc )%D). Therefore, the behavior of the user population can be characterized during the whole control period [T sc , T e ]. The probability distributions q j (t) to have j disconnected appliances is obviously p j+c (t) for j > 0 and c i=0 p i (t) for j = 0.

C. Delay analysis

During the control period, some appliances cannot be activated exactly when the user makes the activation request. In this case, they experience a service delay until some power resources become available. Consider a generic appliance whose activation request is originated at time t ∈ [T sc , T e ].

Let E[W (t)] be its average service delay. Since the arrival rate of activation requests at time t is given by λ(t), the average delay experienced by a random user when the load control is applied can be expressed as

W Tsc Te-Tsc = Te Tsc λ(t)E[W (t)]dt Te Tsc λ(t)dt . ( 8
)
The proof of Eq. 8 is in the appendix.

To derive E[W (t)] we evaluate the cumulative distribution of the delay W (t) experienced by a user arriving at time t following the methods proposed in [START_REF] Franx | A simple solution for the M/D/c waiting time distribution[END_REF], [START_REF]The transient m/d/c queueing system[END_REF]. We generalize the approach considering non-homogeneous arrival rates and an unknown initial state u(T sc ).

Let t be the arrival instant of an appliance and u(t) = kc+i with i ∈ [1, c]. We assume that the u(t) appliances are ranked according to their arrival order. The appliance arrived at t has to wait that (k-1)c+i appliances complete their service before being reconnected (i.e. before being in the first c positions). Since only c appliances can complete their work in an interval equal to D, the new appliance is going to be reconnected in the interval [t + (k -1)D, t + kD]. Consider for example the scenario depicted in Fig. 3. The new arrival at time t (the yellow one in the figure) is in position 7 in a system in which c = 2. Since it has to wait that 5 users complete their service, it will be activated after t+2D and before t+3D. Specifically, its delay will be lower than 3D -x with x ∈ [0, D], if the target appliance occupies one of the first c positions of the queue by t + 3D -x. If we go backward in the past, this condition implies that the target appliance has to be in one of the first 2c positions by t + 2D -x, and in one of the first 3c positions by t + D -x. To satisfy this last condition, the number of users in the queue at t -x plus the number of new arrivals before t has to be strictly lower than 3c (note that the number of active appliances at t -x are switched off by t -x + D and therefore are not in the system anymore). We can generalize the previous considerations and evaluate the cumulative probability P {W (t) ≤ kD -x} as the probability that the number of users N t (t+D-x) arrived before t that are still into the system at t+D -x is strictly lower than kc. For a given value of the queue p in t-x, N t (t+D-x) is lower than kc if the number of new arrivals is strictly lower than kc -p. Therefore P {W (t) ≤ kD -x} is 1 and for kD -x ≥ T e -t is given by

∞ p=0 q p (t -x) kc-p-1 j=0 ( λt-x x x) j j! e λt-x x x (9)
for kD -x < T e -t. Obviously, for t -x < T sc (i.e. before the starting of the admission control), q 0 (t -x) is equal to 1 (there is no disconnected user in the system).

V. NUMERICAL RESULTS

In order to answer our initial question about the scale of the energy demand controller, we quantify the service delay experienced by the users for different control parameters (P g , [T s , T e ]) and for different user populations.

We assume here that the controller manages a group of U users served by the same primary substation. These users accept a delay in the appliances activation. In particular, we focus on the control of a given appliance type, which in our experiments is the laundry machine, whose uncontrolled activation rate generates the profile summarized in Table I. For simplicity we assume a constant power absorption per appliance equal to 1.5kW.

The energy supplier has historical knowledge about the power demand in similar days and can then characterize statistically the power demand in absence of any control. We assume that the energy supplier asks a power reduction K to be applied starting from 10.00 a.m. (during the peak hours) to a reference level P that can be the average power demand or the a given percentile of the power demand. The maximum aggregated power P g is then P -K and the maximum number c of appliances simultaneously active in [T s , T e ] can be determined. 3Consider first the case of the maximum size of user population, i.e. all the primary substation users (typically, tens of thousands) join the energy controller controller. Fig. 4 shows the average service delay experienced when the power reduction K varies in the range 100 KW-5MW for a total population of 30000 users. To guarantee the desired power reduction K at 10 a.m., the limitation on the number of active appliances is started at 8.30 a.m., i.e. at T s -D. On the basis of historical consumption data, the reference level P is evaluated as the average power demand at 10 a.m. in the left plot of the figure, and as the 90% percentile of the power demand in the right plot. The first case corresponds to a smaller value of c. From the figure, we can see that in the longest considered control period (namely, 180 min), asking for a power reduction of 0.5 MW from 10.00 a.m. to 13.00 a.m. leads to a service delay lower than 20 min. For higher power reduction values, the service delay can be still acceptable if the control time interval is limited. By comparing Fig. 4-a and 4-b, we can also observe that there is not a significant difference between applying the power reduction to the 90% percentile of the power demand or to the average value. This is due to the fact that we are considering a large population of users which corresponds to a small variability of the aggregated power demand.

Fig. 5 shows the curves (analogous to the ones plotted in Fig. 4) obtained for different user populations. As the number of users increases, for a given power reduction, the system is obviously able to provide a lower service delay. For example, for a power reduction of 100 KW and a control interval of 3 hours, the average service delay is lower than 20 min when U = 10000 and about 70 min when U = 3000. For small user populations, the service delay experienced when the power reduction refers to the 90% percentile of the power demand is significantly smaller than when it refers to the average value (e.g. about 10 min of reduction for U = 3000, K = 100KW and T e -T s =3 hours).

The projection of the curves in Fig. 5-a is plotted in Fig. 6-a, for the cases U = 10000 and U = 3000, while a wide range of population sizes vs. the mean queue delay are considered in Fig. 6-b control period. For example, in Fig. 6-b, an average delay lower than 50 min can be guaranteed with U = 3000 for a power reduction of 500 KW for 60 min, or with U = 8000 for a power reduction of 1 MW for 180 min. By considering the maximum delay that users can tolerate and the desired power reduction, it is then possible to have an estimate of the percentage of residential users in a given area that should adopt our solution.

A. System uncertainty

Our model requires some knowledge about the user population, in particular about the appliances' operation time, their power consumption and finally the activation rate.

Even if one of our model's assumptions is a homogeneous deterministic operation time D, we carried out some numerical Monte-Carlo simulations to evaluate how its variability affects the experienced mean queue delay. In particular we have considered the case when the operation time is uniformly distributed in the range [D-∆D, D+∆D], where D = 90 min and ∆D varies from 5 to 30 min with steps of 5 minutes. As shown in Table II for the case U = 3000 and K = 500KW , the mean queue delay, expressed in terms of percentage variation compared to the case of constant operation time, is quite small: it ranges from less than 1% up to about 4% when ∆D = 30 min. Fig. 7 shows a random realization of the service delays experienced in case of deterministic operation time and random operation time with ∆D=30 min, which intuitively justifies the model robustness to the variability of the operation time 4 . This leads us to conclude that using a model with constant operation times provides provide good estimates of the results obtained also in case of heterogeneous operation times.

The uncertainty about the instantaneous power consumption X is taken into account by the procedure described in Sec. IV and, in particular, by the term Var (X) in Eq. ( 5). A too conservative choice for Var (X) leads to a too small threshold n * and then to uselessly longer activation delays. On the contrary, underestimating Var (X) leads to violate the probabilistic bound.

About the activation rate, energy utilities are currently able to forecast the aggregate consumption of a large population (millions of users) with very high accuracy (a few percents of error) one day-ahead. DLC solutions, like the one proposed in this paper, would not reach such deployment on a short timehorizon, but, at the same time, they need consumption forecasts on shorter timescales. An underestimate of the appliance activation rates, obviously, results in an increased service delay that, according to the simulations summarized in Table II, is (in percentage) comparable to the error on the activation rate. We also considered additional experiments with a non-Poisson exogenous activation request process. The results suggest that our findings are quite robust to deviation from the Poisson assumption.

VI. FINAL REMARKS

In this paper, we propose a large scale control mechanism for deferrable domestic loads, whose deployment requires minimal local communication overhead to allow a prompt user penetration, and to quantify the aggregation scale that should be reached in order to have appreciable power reduction with a minimal user discomfort.

Unlike other load control proposals published so far, which mainly work on thermostatic loads, the proposed scheme acts on deferrable loads (such as dishwashers or laundry machines), whose activation requests have to be accepted or postponed by a central server. A queuing theory model is derived for quantifying the user population to achieve a significant aggregated power load. Although the model has been derived for the case of homogeneous appliances, our numerical results are to be quite robust to predict the system performance in the case of heterogeneous operation times.

We are currently working on different model extensions for taking into account heterogeneous appliances, periodic sampling of instantaneous power consumption and alternative admission mechanisms, including distributed decisions. Moreover, in [START_REF] Neglia | Scalable and privacy-preserving admission control for smart grids[END_REF] we introduce the concept of energy-bucket to take into account power consumption correlation.

APPENDIX

We consider a non-homogeneous Poisson arrival process N (t), t ≥ 0 with a continuous intensity rate λ(t) and expectation function Λ(t) = t 0 λ(x)dx. {τ1, τ2, ...τn, . . . } denotes the sequence of arrivals for t ≥ 0. Let Wτ i be a stochastic process defined on the points of N (t).

We prove that:

E i|τ i <T Wτ i N (
where s (1) , . . . s (n) are the order statistics of s1, . . . sn. Let s and s-i denote the vectors (s1, . . . si, . . . sn) and (s1, . . . si-1, si+1, . . . , sn) respectively. We observe that both density functions fN and f N are invariant under permutations of s. We define gi(s1, . . . sn) = E Ws i |τ1 = s (1) , . . . τn = s (n) and g(s1, . . . sn) = n i=1 gi(s1, . . . sn). Then, gi() is invariant under permutations of s-i, while g() is invariant under permutations of s. We are now ready to prove Eq. 11. Eq. 14 follows from the invariance of g(s)fN (s|N (T ) = n) to permutations of s. The equality in Eq. 15 relies on the factorization of f () in 13. The invariance of gi(s)fN (s-i|N (T ) = n) to permutations of s-i is used in Eq. 16.

Fig. 1 .

 1 Fig. 1. Reference scenario: communication infrastructure and control system for DLC managed by a DSO and/or a load aggregator. a dynamic number of servers.The paper is organized as follows. In Section II we present our reference scenario and the architectural choices that allow DLC on deferrable domestic loads to be implemented with limited investments. In Section III we present the model used for characterizing the aggregated power demand of a large number of deferrable appliances, while in Section IV we model the effect of the admission control on the number of active appliances and quantify the service delay distribution. Section V presents some numerical results enlightening the tradeoffs between power reduction, service delay and number of controlled users. Finally, some concluding remarks are drawn in Section VI.II. SYSTEM DESCRIPTION We consider the problem of power shaving for an energy supplier which wants to reduce its customers consumption during a given time interval. To achieve this goal, the energy supplier may interact with the distribution system operators (DSO), and/or with novel intermediate figures called load aggregators[START_REF] Ruiz | A Direct Load Control Model for Virtual Power Plant Management[END_REF] by means of a communication infrastructure transporting the control messages. Both possibilities are illustrated in Fig.1, where dashed lines indicate the flows of control messages.The solution based on the DSO can take advantage of the advanced metering infrastructure (AMI), mainly based on PLCs (Power Line Communications) and GPRS (General Packet Radio Services) technologies, for reaching a largescale number of domestic users in the low-voltage (LV) distribution grid[START_REF]An overview to new plc applications and new challenges -maxium integrated products[END_REF],[START_REF]Demand-response technology roadmap[END_REF]. Conversely, the load aggregator can exploit standard Internet connectivity to interact with appliances at customers premises. The presence of the load aggregator allows to have an intermediary for interacting with large scale numbers of customers[START_REF] Ancillotti | The role of communication systems in smart grids: Architectures, technical solutions and research challenges[END_REF]. The aggregator can collect information on user profiles and willingness to respond to control actions for dispatching the controlled loads as a response of the energy supplier commands. Since the two approaches can coexist, we will talk generically about the load controller to denote the entity which drives the appliances.The energy supplier specifies an high-level command for the load controller in terms of maximum tolerable probability to

  λ(x)dx is the average arrival rate in an interval of length D starting at t -D.

2

 2

Fig. 2 .

 2 Fig.2. Random realizations of the number of active appliances for a population of 10000 total users, with and without control.

Fig. 3 .

 3 Fig. 3. A possible arrival and queue scenario (the yellow is the new arrival)

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Average service delay for a population of 30000 laundry machines.

Fig. 7 .

 7 Fig. 7. Random realization of queuing delays of the activation requests arrived during the control interval for constant D values (blue) and uniform D values in [60min, 120min] (red), U = 3000, K = 400KW .

  where E [Wt] indicates the expected value of the random variable Wt and it is then conditioned on the fact that there is an arrival in t. In particular, we are going to prove that:E n i=1 Wτ i N (T ) = n = n T 0 λ(t)E [Wt|N (t) = n, τj = t for some j = 1, . . . n] dt Λ(T ) ,(11)from which our result follows immediately by deconditioning. Let fN (τ1 = s1, τ2 = s2, . . . τn = sn|N (T ) = n) be the joint probability density function of the ordered arrival times of N (t) in the interval [0, T ]. Let us define τi = Λ(τi). The point process {τi} is a homogeneous Poisson process ( N (t)) with rate 1[START_REF] ¸inlar | Introduction to Stochastic Processes[END_REF], thenf N (τ1 = u1, τ2 = u2, . . . τn = un| N (t) = n) = n! t n .Clearly, the event N (T ) = n corresponds to the event N (Λ(T )) = n. For a given number of arrivals n, let (λ(si))ii be the (diagonal) Jacobian of the transformation ui = Λ(si). We can express fN (s1, s2, . . . sn|N (T ) = n) as:f N Λ(s1), Λ(s2), . . . Λ(sn)| N (Λ(T )) = n |(λ(si)ii)|(12)and combining the previous equations we obtainfN (τ1 = s1, τ2 = s2, . . . τn = sn|N (T ) = n)Similarly we can calculate the joint probability density function fN (s1, . . . sn|N (T ) = n) of the arrival times as: 1 n! fN τ1 = s (1) , . . . τn = s (n) |N (T ) = n = n i=1 λ(si) Λ(T ) ,

E 1 •E

 1 [Wτ i |τ1 = s1, . . . τn = sn] • fN (s|N (T ) = n) ds = 1 n! [0,T ] n g(s)fN (s|N (T ) = n) ds (14) = i [0,T ] n gi(s) fN (s|N (T ) = n) ds = i [0,T ] [0,T ] n-1 gi(s) fN (s-i|N (T ) = n -1) ds-i λ(si) T ] 0≤s 1 •••≤s i-1 ≤s i+1 ≤...sn≤T gi(s) • fN (s-i|N (T ) = n -1) ds-i λ(si) ••≤s i-1 ≤s i+1 ≤...sn≤T E Ws i |τ1 = s (1) , . . . τn = s (n) • fN (s-i|N (T ) = n -1) ds-i λ[Ws i |N (T ) = n, τj = si for some j] • λ(si) Λ(T ) dsi = n [0,T ]E [Ws|N (T ) = n, τj = s for some j] • λ(s) Λ(T ) ds.

TABLE II PERCENTAGE

 II . Both the figures clearly show the tradeoff between user population, power reduction and OF DELAY VARIATIONS IN CASE OF DATA UNCERTAINTY FOR U = 3000, K = 500KW , AND Te -Ts=180 MIN.

	∆λ(t)	1%	2%	5%	10%	15%	20%
	∆E[W ]	1.3%	2.6%	6.1%	11.2%	16.2%	21.3%
	∆D	5 min 10 min	15 min 20 min 25 min 30 min
	∆E[W ]	0.1%	0.5%	0.7%	1.8%	2.9%	4.2%

Here "M(t)" (for "Markovian") denotes the Poisson time-varying appliance activation process, "D" indicate that the operation time is deterministic and ∞ the presence of an infinite number of servers, so that new requests can immediately be served. The reader can refer to[START_REF] Kleinrock | [END_REF] for a basic introduction to queuing theory.

Note that for large values of j, the probabilities p j in Eq. 2 (as well as in Eq. 9) need to be evaluated approximating the Poisson distribution by the Gaussian one.

In this simple case where we consider appliances with constant power demand, we can calculate c simply dividing Pg by the power demand of a single appliance.

Note that, for an activation request arrived at t, the waiting time is the minimum between the queuing time of the finite servers model and Te -t, because at Te all the appliances are reactivated; therefore, when equal to Te -t it does not depend on the operation times of the active appliances.

Ilenia Tinnirello received her Ph.D. degree in telecommunications engineering from the University of Palermo in 2004. She is currently an associate professor at the University of Palermo. She has also been Visiting Researcher at the Seoul National University, Korea, in 2004, and at the Nanyang Technological University of Singapore in 2006. Her research activities have been focused on wireless networks, and in particular on the design and prototyping of protocols and architectures for emerging reconfigurable wireless networks. Recently, she is also working on the definition of novel services (smart grid, smart metering, indoor localization, etc.) enabled by the pervasive availability of ICT technologies. I. Tinnirello has been involved in several European research projects, among which the FP7 FLAVIA project, with the role of technical coordinator, and the H2020 WiSHFUL and Flex5Gware projects. In his areas of interest, he has author or co-author of more than 170 technical papers published in international journals or presented at various national and international conferences. His main research interests include power systems analysis, optimal planning, and design and control of electrical distribution systems. In recent years, his research activities have included the integration of distributed generation and storage systems into electrical networks and smart grids.