Abdelghani El Mazouni

AND Fatima Laytimi
email: fatima.laytimi@math.univ-lille1.fr

D S Nagaraj

A El Mazouni

MORPHISMS FROM P2 TO Gr(2,C4)

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

We denote by P 2 the projective plane over the field of complex numbers and Gr(2, C 4 ) be the Grassmannian of two-dimentional quotient spaces of C 4 . In this paper we investigate the possible types of nonconstant morphisms P 2 → Gr(2, C 4 ). Any non-constant morphism P 2 → Gr(2, C 4 ) determines a cohomology class in H 4 (Gr(2, C 4 ), Z). We consider the following problem: Determine the necessary and sufficient conditions that a cohomology class of H 4 (Gr(2, C 4 ), Z) has to satisfy in order to be represented by a morphism from P 2 → Gr(2, C 4 )?

It is easy to see that if a cohomology class is represented by a morphism from P 2 then it has to satisfy an obvious necessary condition (see Lemma (3.3)). We show that in general this condition is not sufficient. The following result shows that there are classes satisfying the condition but are not represented by morphisms: Theorem 1.1. Let c and a be two integers. Assume c ≥ 4 and 1 ≤ a ≤ c -2 or c 2 -c + 2 ≤ a ≤ c 2 -1. Then the cohomology class (a, c 2 -a) is not represented by a morphism P 2 → Gr(2, C 4 ).

The following theorem shows that in some cases morphisms exist: Theorem 1.2. 0) For every c ≥ 1 the cohomology classes (0, c 2 ) and (c 2 , 0) are represented by morphisms P 2 → Gr(2, C 4 ).

1) Let 1 ≤ c ≤ 3 and 0 ≤ ℓ ≤ c 2 be integers. Then the cohomology class (ℓ, c 2 -ℓ) is represented by a morphism P 2 → Gr(2, C 4 ).

2) If c = 4 and 3 ≤ ℓ ≤ 13 then there are morphisms P 2 → Gr(2, C 4 ) representing the cohomology class (ℓ, 16 -ℓ).

1991 Mathematics Subject Classification. 14F17. 3) Let c ≥ 5 be an integer. Let k be the largest integer such that k.c ≤ (c 2 -3c + 2)/2. Then for every integer ℓ in one of the following intervals [t(c -3) + 2, t.c] for 1 ≤ t ≤ k or [(c 2 -3c + 2)/2 + 1, c 2 /2] there are morphisms P 2 → Gr(2, C 4 ) representing the cohomology class (ℓ, c 2 -ℓ). Also, for every such ℓ there is a morphism whose cohomology class is (c 2 -ℓ, ℓ).

It is also shown here (see, Remark (3.12)) that for every integer n ≥ 1 there are morphisms

f n : P 2 → Gr(2, C 4 ) such that f n is one to one onto its image and f * n (O Gr(2,C 4 ) (1)) = O P 2 (n). Acknowledgments:
The third author would like to thank, IFIM, Université Lille, and Université d' Artois, France.

Morphisms to Grassmannian

Here we recall some results about morphisms from a variety to a Grassmannian variety. Definition 2.1. Let X be a variety over the field C of complex numbers and s be a positive integer. A vector bundle E over X is said to be generated by s sections if there a surjection of vector bundles

C s ⊗ O X → E.
Definition 2.2. Let X be a projective variety. Let r and k be two positive integers. Let E 1 and E 2 be two vector bundles of rank r over X. Two vector bundle surjections

φ 1 : C r+k ⊗ O X → E 1 ,
and

φ 2 : C r+k ⊗ O X → E 2
, are said to be equivalent if there exists an isomorphsim of vector bundles ψ : E 1 → E 2 over X such that the following diagram commutes:

C r+k ⊗ O X φ 1 -→ E 1 ↓ ψ C r+k ⊗ O X φ 2 -→ E 2
The following two lemmas are well known(see, for example [START_REF] Laytimi | Vector bundles generated by sections and morphism to Grassmannian[END_REF]). We include the proofs of these lemmas for the sake of completeness. Lemma 2.3. Let X be a projective variety. Let r and k be two positive integers. There is a natural bijection between the following two sets:

1) The set of equivalence classes of surjection of vector bundles

φ : C r+k ⊗ O X → E,
where E is a vector bundle of rank r on X.

2) The set of morphisms f : X → Gr(r, C r+k ), where Gr(r, C r+k ) is the Grassmannian of r dimensional quotient of C r+k .

Proof: Given a surjection φ :

C r+k ⊗ O X → E by sending x → {C r+k → E x }
defines a morphism f : X -→ Gr(r, C r+k ). This defines a map from the set in 1) to the set in 2).

To prove the existence of the map in the other direction, we first note the following: on Gr(r, C r+k ) there is a canonical vector bundle surjection C r+k ⊗ O Gr(r,C r+k ) -→ Q, where Q is the rank r bundle on Gr(r, C r+k ) whose fiber at x ∈ Gr(r, C r+k ) is the quotient vector space corresponding to x. Now given a morphism f : X -→ Gr(r, C r+k ) by pulling back by f the canonical surjection of vector bundles on the Grassmannian, we get a surjection of vector bundles

C r+k ⊗ O X = f * (C r+k ⊗ O Gr(r,C r+k ) ) -→ f * (Q).
This gives a map from the set in 2) to the set in 1). These two maps are clearly inverse to each other hence we get the required bijection.

Theorem 2.4. Let X be a projective variety. There is a natural bijection between the following two sets:

1) The set of equivalence classes of surjections of vector bundles

C r+k ⊗ O X -→ E
with rank(E) = r and det(E) is ample.

2) The set of morphisms f : X -→ Gr(r, C r+k ) with f finite (onto its image).

Proof: Given an element of the set in 1), i.e., a surjection, C r+k ⊗ O X -→ E with rk(E) = r and det(E) is ample, then as in Lemma(2.3) it determines a morphism f : X -→ Gr(r, C r+k ), also, it follows from Lemma (2.3)

, that f * (Q) ≃ E. Hence f * det(Q) ≃ det(E). Since det(E)| f -1 (f (x)) ≃ f * det(Q)| f -1 (f(x)) is trivial, the ampleness assump- tion on det(E) implies dim(f -1 (f(x))) ≤ 0. Thus, if det(E) is ample, then f is finite onto its image.
In the other direction, if f : X -→ Gr(r, C r+k ) is finite morphism onto its image, then det(f * Q) ≃ f * (detQ) is ample, as pull-back of an ample bundle remains ample under finite morphism.

Let L be an vector subspace of dimension one in C 4 . Then the projective 2 plane P 2 L = Gr(2, C 4 /L) is naturally a subvariety of Gr(2, C 4 ). Similarly, if H is an one dimensional quotient vector space of C 4 , then the projective 2 plane

P 2 H = Gr(2, (C 4 ) * /H * ) is also naturally a subvariety of Gr(2, C 4 )
, where (C 4 ) * (resp. H * ) denotes the dual vector space of C 4 (resp. of H). Note that P 2 H is the subset of Gr(2, C 4 ) consists of all those quotient vector spaces of dimension two of C 4 which has H as their quotient.

Definition 2.5. A morphism f : P 2 → Gr(2, C 4 ) is said to be trivial if it is one of the following: • a) f is a constant morphism. • b) Image of f is P 2 L for some one dimensional subspace L of C 4 . • c) Image of f is P 2
H for some one dimensional quotient H of C 4 . Example 2.6. It is known that (see, for example [START_REF] Inamdar | Cycle class map and restriction of subvarieties[END_REF]) the Veronese surface (2-uple embedding of P 2 in P 5 ) is contained in a smooth quadric in P 5 . As any two smooth quadrics are isomorphic via a projective automorphism we see that there is an embedding of P 2 in Gr(2, C 4 ). In fact, the global sections

s 1 = (X, Z), s 2 = (Y, X), s 3 = (Z, Y ) and s 4 = (X, 0) of O P 2 (1) ⊕ O P 2 (1)
gives a vector bundle surjection

C 4 ⊗ O P 2 → O P 2 (1) ⊕ O P 2 (1).
Then by Lemma (2.3) we get morphism φ : P 2 → Gr(2, C 4 ). It is easy to see that φ is an embedding and φ composed with the natural embedding of Gr(2, C 4 ) into P 5 is given by the quadrics

X 2 -Y Z, XY -Z 2 , Y 2 -XZ, XZ, X 2 , XY.
As these quadrics form a basis for the space of quadrics on P 2 we get that the veronese embedding of P 2 in P 5 factors through Gr(2, C 4 ).

Example 2.7. More generally, for any positive integers a and b the global sections

s 1 = (X a , Z b ), s 2 = (Y a , X b ), s 3 = (Z a , Y b ) and s 4 = (X a , 0) of O P 2 (a) ⊕ O P 2 (b) gives a vector bundle surjection C 4 ⊗ O P 2 → O P 2 (a) ⊕ O P 2 (b).
Then by Lemma (2.3) we get a morphism φ : P 2 → Gr(2, C 4 ).

Example 2.8. Let T P 2 denotes the tangent bundle of P 2 . Then one has an exact sequence (See, page 409 of [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF])

0 → O P 2 → O P 2 (1) 3 → T P 2 → 0.
¿From this we see that the space of all global sections H 0 (T P 2 ) of the tangent bundle is a vector space of dimension eight. Now it is easy to see that if we choose four linearly independent general sections of the tangent bundle we get a surjective morphism

O 4 P 2 → T P 2 .
This surjection give rise to a morphism from P 2 to Gr(2, C 4 ). Since the tangent bundle is not direct sum of line bundles, this morphism is different from the morphisms given by example above.

3. Non-trivial morphisms from P 2 to Gr(2, C 4 ) On the Grassmaian Gr(2, C 4 ) one has universal exact sequence:

(1) 0 → S → C 4 ⊗ O Gr(2,C 4 ) → Q → 0,
where S and Q are respectively the universal sub bundle and quotient bundle of rank two on Gr(2, C 4 ). The fiber of Q (resp. of S) at a point p ∈ Gr(2, C 4 ) is the two dimensional quotient space (resp. subspace, which is the kernal of this quotient map) of C 4 corresponding to the point p. It is known (see, page 197 and 411 of [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF]) that the cohomology group H 4 (Gr(2, C 4 ), Z) is equal to

(2) Z[c 2 (Q)] ⊕ Z[c 2 (S)],
where

c 2 (Q) (resp. c 2 (S)
(3) 0 → Ω 1 (1) → C 4 ⊗ O P 2 → O P 2 (1) ⊕ O P 2 → 0.
The dual of the exact sequence Question 2) Let Q be the vector bundle on Gr(2, C 4 ) as in the equation [START_REF] Almeida | Une involution sur un espace de modules de fibrés instantons Bull[END_REF]. For which cohomology classes (a, b) does there exists a morphism φ : P 2 → Gr(2, C 4 ) the bundle φ * (Q) is indecomposable? Remark 3.1. Let φ : P 2 → Gr(2, C 4 ) be a morphism. By pulling back the universal exact sequence (1) on Gr(2, C 4 ) we get an exact sequence

(4) 0 → φ * (S) → C 4 ⊗ O P 2 → φ * (Q) → 0,
of vector bundles on P 2 . By dualizing the exact sequence (4) we get another exact sequence

(5) 0 → φ * (Q) ∨ → C 4 ⊗ O P 2 → φ * (S) ∨ → 0,
of vector bundles on P 2 . By Lemma (2.3) the surjection

(6) C 4 ⊗ O P 2 → φ * (S) ∨ → 0,
gives a morphism from P 2 to Gr(2, C 4 ) which we denote by φ ∨ and call the dual morphism. If (a, b) is the cohomology class of a morphism φ then it is clear that the cohomology class of the dual morphism φ ∨ is (b, a).

Remark 3.2. Let φ : P 2 → Gr(2, C 4 ) be a closed immersion. Assume that φ is a non trivial imbedding as in Definition (2.5). It follows from [START_REF] Tango | On (n-1)-dimensional projective spaces contained in the Grassmann variety Gr(n, 1)[END_REF] the morphism φ or φ ∨ , is given by a surjection

(7) C 4 ⊗ O P 2 → O P 2 (1) ⊕ O P 2 (1).
If the morphism φ is given by [START_REF] Kausz | A Gieseker type degeneration of moduli stack of vector bundles on curves Trans[END_REF] Proof: By pulling back the universal exact sequence (1) by φ we get the following exact sequence of vector bundles on P 2 :

(8) 0 → φ * (S) → C 4 ⊗ O P 2 → φ * (Q) → 0.
Since, the bundle φ * (Q) is non-trivial and is a quotient of trivial bundle, we see that the Chern classes c 1 (φ * (Q)) and c 2 (φ * (Q)) are both non-negative integers (we identify H 4 (P

= c 2 (φ * (Q)) and c = c 1 (φ * (Q)). Note that c 2 -a = c 2 (φ * (S)) = c 2 ((φ * (S)) ∨ )
, where (φ * (S)) ∨ is the dual of the bundle φ * (S). Since (φ * (S)) ∨ is generated by sections we must have c 2 -a ≥ 0. This completes the proof of the lemma.

Remark 3.4. The following question arises naturally. Given (a, c 2 -a), with c > 0 and 0 ≤ a ≤ c 2 does there exists a morphism φ : P 2 → Gr(2, C 4 ) such that the cohomolgy class given by the morphism φ is of the form (a, c 2 -a)? In general this question has negative answer (see Lemma (3.6) below). Also, we give below partial answer to the above question (see Theorem (3.7) and Theorem (3.8)). We believe that there are no morphisms from P 2 which represent the remaining cohomology classes.

We need the following theorem:

Theorem 3.5. (Cayley-Bacharach theorem) Let S be a non-singular surface and Z be a sub scheme of S of dimension zero. Then there exists a rank two vector bundle E on S with a section s such that the zero sub scheme of s is Z if and only if for every point p ∈ Z the linear system

|I Z K S ⊗ det(E)| = |I Z-{p} K S ⊗ det(E)|,
where I Z (respectively, I Z-{p} ) denotes the ideal sheaf of Z (respectively, of Z -{p}).

Proof: This is a consequence of [Theorem 7. [START_REF] Tan | A note on Caylay-Bacharach property for vector bundles Complex analysis and Algebraic geometry[END_REF]]. (See, [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF] page 731, for the case Z is reduced). Lemma 3.6. There is no morphism φ : P 2 → Gr(2, C 4 ) such that the cohomolgy class given by the morphism φ is of the form [START_REF] Almeida | Une involution sur un espace de modules de fibrés instantons Bull[END_REF]15).

Proof: Assume there exists a morphism φ : P 2 → Gr(2, C 4 ) such that the cohomolgy class given by the morphism φ is of the form [START_REF] Almeida | Une involution sur un espace de modules de fibrés instantons Bull[END_REF]15). By pulling back the universal exact sequence (1) by φ gives the following exact sequence of vector bundles on P 2 : Proof: If φ : P 2 → Gr(2, C 4 ) is a morphism such that the cohomology class of the morphism is φ is (a, c 2 -a) then the cohomology class of the dual morphism (see 3.6) φ ∨ : P 2 → Gr(2, C 4 ) is (c 2 -a, a). So it is enough to show that there are no morphisms from P 2 to Gr(2, C 4 ) such that the cohomolgy class is given by (a, c 2 -a) for 1 ≤ a ≤ c -2. But note that for any zero dimensional sub scheme Z of P 2 the natural morphism

(9) 0 → φ * (S) → C 4 ⊗ O P 2 → φ * (Q) → 0. with c 2 (φ * (Q)) = [H]
H 0 (O P 2 (c -3)) → H 0 (O Z (c -3) is surjective, if length of O Z is
less than or equal to c -2. This will imply the following: if Z is a zero dimensional sub scheme of P 2 such that the length of O Z is less than or equal to c -2, then for any p ∈ Z (10)

H 0 (I Z (c -3)) = H 0 (I Z-p (c -3)),
where I T denotes the ideal sheaf of the sub scheme T. On the other hand if there exists a morphism φ : P 2 → Gr(2, C 4 ) such that the cohomology class of the morphism is φ is (a, c 2 -a) then the pull back E of the universal quotient bundle on Gr(2,

C 4 ) has c 1 (E) = c, c 2 (E) = a.
Since E is generated by sections we can find section s such that the scheme Z of zeros of this section is zero dimensional and length of O Z is equal to a ≤ c -2. Now by Cayley-Bacharach theorem (3.5) we must have

H 0 (I Z (c -3)) = H 0 (I Z-{p} (c - 3
)), for every p ∈ Z. But this leads to a contradiction to [START_REF] Tango | On (n-1)-dimensional projective spaces contained in the Grassmann variety Gr(n, 1)[END_REF]. This proves the reqired result. Theorem 3.8. 0) For every c ≥ 1 the cohomology classes (0, c 2 ) and (c 2 , 0) are represented by morphisms P 2 → Gr(2, C 4 ).

1) Let 1 ≤ c ≤ 3 and 0 ≤ ℓ ≤ c 2 be integers. Then the cohomology class (ℓ, c 2 -ℓ) is represented by a morphism P 2 → Gr(2, C 4 ).

2) If c = 4 and 3 ≤ ℓ ≤ 13 then there are morphisms P 2 → Gr(2, C 4 ) representing the cohomology class (ℓ, 16 -ℓ).

3) Let c ≥ 5 be an integer. Let k be the largest integer such that k.c ≤ (c 2 -3c + 2)/2. Then for every integer ℓ in one of the following intervals

[t(c -3) + 2, t.c] for 1 ≤ t ≤ k or [(c 2 -3c + 2)/2 + 1, c 2 /2]
there are morphisms P 2 → Gr(2, C 4 ) representing the cohomology class (ℓ, c 2 -ℓ). Also, for every such ℓ there is a morphism whose cohomology class is (c 2 -ℓ, ℓ).

Proof: 0) If f : P 2 → P 2 is a finite morphism then f * (O P 2 (1)) = O P 2 (
n) for some integer n > 0. Then the degree of the morphism is equal to n 2 . Thus we see that if a cohomology class of the form (0, a) or (a, 0) is represented by a non constant morphism f : P 2 → P 2 if and only if a = n 2 for a positive integer n.

1) If 1 ≤ c ≤ 3 then for any zero dimensional sub scheme Z the vector space H 0 (I Z (c-3)) is 0. Hence by Cayley-Bacharach theorem(3.5) there exists a vector bundle E and a section s such that c 1 (E) = c and zero scheme of s is Z. It is enough to consider the case ℓ ≤ c 2 /2, for if φ : P 2 → Gr(2, C 4 ) such that the cohomology class of the morphism given by φ is (ℓ, c 2 -ℓ) then the cohomology class of dual morphism (see 3.1) φ ∨ is (c 2 -ℓ, ℓ). For ℓ ≤ c 2 /2, by considering any reduced sub scheme Z consisting of ℓ points we get an exact sequence

0 → O P 2 → E → I Z (c) → 0,
where I Z denotes the ideal sheaf of Z, where E is the vector bundle obtained by Cayley Bacharach theorem. Then we see that E is generated by sections and hence by 4 sections. This gives the required morphism φ.

2) For ℓ = 3 choose Z a reduced scheme consisting of three points lying on a line. Then we see that Z satisfies Cayley -Bacharach conditions with respect to line bundle O P 2 (1). Thus there exists vector bundle E with c 1 (det(E)) = 4[H] and a section s such that (s) 0 = Z. Also, We get an exact sequence

0 → O P 2 → E → I Z (4) → 0,
where I Z denotes the ideal sheaf of Z. Then we see that E is generated by sections and hence by 4 sections. This gives the required morphism φ.

For 4 ≤ ℓ ≤ 8 Let Z = {P 1 , . . . , P ℓ } be a reduced closed scheme consisting of ℓ points such that no three points lie on a line. Then we see that Z satisfies Cayley -Bacharach conditions with respect to line bundle O P 2 (1). Thus there exists vector bundle E with c 1 (det(E)) = 4[H] and a section s such that (s) 0 = Z. Also, We get an exact sequence 0 → O P 2 → E → I Z (4) → 0, where I Z denotes the ideal sheaf of Z. Then we see that E is generated by sections and hence by 4 sections. This gives the required morphism φ. For 8 ≤ ℓ ≤ 13 the dual morphism φ ∨ of appropriate φ above gives the reqired morphism.

3) Let t and ℓ be as in the Theorem. Let Z = {P 1 , . . . , P ℓ } be a reduced closed scheme consisting of ℓ points such that no r.c + 1 points lie on a curve of degree r for 1 ≤ r ≤ t. Note that Z satisfies Cayley -Bacharach conditions with respect to line bundle O P 2 (c -3). Thus there exists vector bundle E with c 1 (det(E)) = d[H] and a section s such that (s) 0 = Z. Also, We get an exact sequence

0 → O P 2 → E → I Z (d) → 0,
where I Z denotes the ideal sheaf of Z. Then we see that E is generated by sections and hence by 4 sections. This morphism φ corresponds to the cohomology class (ℓ, c 2 -ℓ). The dual morphism φ ∨ corresponds to the cohomology class (c 2 -ℓ, ℓ). For ℓ in the interval [(c 2 -3c + 2)/2 + 1, c 2 /2] let Z = {P 1 , . . . , P ℓ } be a reduced closed scheme consisting of ℓ points such that no r.c + 1 points lie on a curve of degree r for 1 ≤ r ≤ c -3. Then the rest of the proof is as before. Let Z 1 ⊂ Y × S be the sub variety on which the sheaf F supported. Let q : Z 1 → S be the natural projection and let U = {s ∈ S|dimq -1 (s) < dim(X)} then we see that by semi continuity (see, page 95, Exercise (3.22) [START_REF] Hartshorne | Algebraic Geometry[END_REF]), U is an open subset and is non-empty as s 0 ∈ U. For s ∈ U the morphism F s is an isomorphism on X ×s-G -1 (q -1 (s). Since G is finite G -1 (q -1 (s) is proper closed sub set of X × s and hence the morphism F s is birational onto its image. This proves the Proposition. The result follows from Proposition(3.10), if we show the existence of one such f φ . If φ is given by the matrix

    X a Z b Y a X b Z a Y b X a 0    
Since a, b are coprime, it is easy to see that the morphism f φ : P 2 → Gr(2, C 4 ) corresponding to φ is birational onto its image. Remark 3.12. By choosing a = 1 and b = n and using the fact that Gr(2, C 4 ) is imbedded in P 5 as a smooth quadric we get the following:

for every integer n ≥ 1 there are morphisms

f n : P 2 → Gr(2, C 4 )
such that f n is one to one onto its image and f * n (O Gr(2,C 4 ) (1)) = O P 2 (n).

1

 1

  (3) correspondence to the cohomology class (1, 0). Question 1) a) Given the cohomology class (a, b) of the decomposition in (2) of the cohomology group does there exists a morphism φ : P 2 → Gr(2, C 4 ) such that the cohomology class of this morphism is (a, b)? b) For which cohomology class (a, b) does there exists a generically injective morphism φ : P 2 → Gr(2, C 4 ) such the cohomology class of this morphism is (a, b)?

Lemma 3 . 3 .

 33 then φ * ([P 2 ]) =[START_REF] Almeida | Une involution sur un espace de modules de fibrés instantons Bull[END_REF][START_REF] Inamdar | Cycle class map and restriction of subvarieties[END_REF] and the class of (φ ∨ ) * ([P 2 ]) =[START_REF] Inamdar | Cycle class map and restriction of subvarieties[END_REF][START_REF] Almeida | Une involution sur un espace de modules de fibrés instantons Bull[END_REF]. Thus the only classes (a, b) represented by regularly imbedded P 2 in Gr(2, C 4 ) are (1, 0), (0, 1), (1, 3) and (3, 1). Let φ : P 2 → Gr(2, C 4 ) be a non constant morphism. Then the cohomolgy class given by the morphism φ is of the form (a, c 2 -a), for some integers c > 0 and 0 ≤ a ≤ c 2 .

2 Theorem 3 . 7 .

 237 and c 2 (φ * (S)) = 15[H]2 , where [H] is the hyper plane class. More over it can be easily seen that c 1 (φ * (Q)) = 4[H]. Since E = φ * (Q) is generated by sections there exists a section of E which vanishes at exactly one point p of P 2 with multiplicity one. But then by Cayley-Bacharach theorem(3.5) the point must be a base point for the complete linear system of the line bundle det(E)⊗K P 2 = O P 2 (1). But this is a contradiction. This proves the required result.More generally we have the following: Let c and a be two integers. Assume c ≥ 4 and1 ≤ a ≤ c -2 or c 2 -c + 2 ≤ a ≤ c 2 -1. Then the cohomology class (a, c 2 -a)is not represented by a morphism P 2 → Gr(2, C 4 ).

Lemma 3 . 9 .

 39 Let φ : P 2 → Gr(2, C 4 ) be a non constant morphism. Assume that φ * (Q) is decomposable. Then the cohomolgy class given by the morphism φ is of the form (a.b, (a + b) 2 -a.b) for some non negative integers a, b with a + b > 0. Moreover, for any such tuple (a.b, (a + b) 2 -a.b), there are morphisms φ : P 2 → Gr(2, C 4 ) whose corresponding cohomology class is (a.b, (a + b) 2 -a.b).Proof: By pulling back the universal exact sequence ((1)) by φ gives the following exact sequence of vector bundles on P 2 :(11) 0 → φ * (S) → C 4 ⊗ O P 2 → φ * (Q) → 0. By assumption φ * (Q) ≃ O P 2 (a) ⊕ O P 2 (b). Since the bundle φ * (Q) is quotient oftrivial bundle implies we must have a, b ≥ 0. On the other hand φ is non constant morphism implies a + b > 0. Now, it is easy to see that c 2 (φ * (Q)) = a.b and c 2 (φ * (S)) = (a + b) 2 -a.b Last assertion of the Lemma follows from Example(2.7). Proposition 3.10. Let X and Y be two irreducible projective varieties. Let S be any irreducible quasi-projective variety and s 0 ∈ S be a point. Let F : X × S → Y be a morphism. Assume that F s := F | X×s : X → Y is finite for all s ∈ S and F s 0 is a birational onto its image. Then there is an open subvariety U of S such that s 0 ∈ U and for s ∈ U the morphism F s is a birational onto its image. Proof: Consider the morphism G = F × Id S : X × S → Y × S. Then the assumption F s is finite implies the morphism G is finite and proper. Hence G = G * (O X×S ) is coherent sheaf of O Y ×S modules. Let Z ⊂ Y × S be the sub variety on which the sheaf G * (O X×S ) is supported. Then clearly the map p : Z → S, restriction of the natural projection, is surjective. The section 1 ∈ O X×S gives an inclusion of O Z in G. Let F = G/O Z .

Proposition 3 . 11 .

 311 Let a, b be two coprime positive integers. Letf φ : P 2 → Gr(2, C 4 )be the morphism given by a surjectionφ : C 4 ⊗ O P 2 → O P 2 (a) ⊗ O P 2 (b).Then for a generic choice of φ the morphism is birational onto its image. Proof: Since the set of surjections φ is an open subset of Hom(C 4 , H 0 (O P 2 (a) ⊕ O P 2 (b)))


domain: info math

text:
Alix Dupont
email: alix.dupont@edf.fr

Jean-Baptiste Breal

Yezekael Hayel
email: yezekael.hayel@univ-avignon.fr

Panagiotis Andrianesis

Optimal Price Menu Design of Electric Vehicle Charging Stations Providing Demand Response

Keywords: Electric vehicle, charging station operator, optimal price menu design, demand response

The growing adoption of Electric Vehicles (EVs) creates new challenges due to the increasing charging demand, but also opportunities for EV Charging Station Operators (CSOs) to leverage EV user flexibility. In this work, we consider an EV CSO who offers several power rates at different prices, and explore the problem of optimal price menu design. In this setting, EV users are inflexible in their parking duration but flexible in terms of their energy demand, and choose the option that maximizes their welfare, i.e., their utility minus their cost of charging. We formulate the optimal price menu design problem for a profit maximizing CSO as a Mixed Integer Linear Programming problem, which we compare against the outcome of social welfare maximization problem. We further account for the provision of demand response by the CSO, i.e., lowering its power consumption for a certain period given a certain price for remuneration, by adjusting the price menu (in real time), so as to discourage EV users with a lower need for energy from choosing higher power rates. Our numerical demonstrations provide useful insights on the construction of the optimal price menu.

I. INTRODUCTION

Considering the growing adoption of Electric Vehicles (EVs) [START_REF]Global EV Outlook[END_REF] public charging stations are attracting significant attention. Their aggregate power consumption may stress the grid and its assets, hence, leveraging the flexibility of the EV users is key to avoid costly and potentially unnecessary upgrades. An EV Charging Station Operator (CSO), with a capability to modulate its aggregate power consumption for a given period, can provide some type of demand response to the grid operator. For instance, the French distribution system operator offers remuneration in exchange for this type of services [START_REF]Enedis' website: call for tenders[END_REF]. To this end, the CSO can intelligently design the options/services provided to EV users, by forming a dynamic charging price menu, with a variety of charging options/services, which can adapt to heterogeneous requirements of EV users using information that may be either provided by the users or obtained from the CSO statistics/data collected about its EV user characteristics/behavior.

There are a few works [START_REF] Bitar | Deadline differentiated pricing of deferrable electric loads[END_REF], [START_REF] Ghosh | Control of charging of electric vehicles through menu-based pricing[END_REF], [START_REF] Owen | Smart charging of electric vehicles: An innovative business model for utility firms[END_REF], [START_REF] Lu | Deadline differentiated dynamic ev charging price menu design[END_REF] that have addressed the problem of designing a charging price menu for an EV Submitted to the 23rd Power Systems Computation Conference (PSCC 2024).

CSO. In a rather general setting for electric power loads [START_REF] Bitar | Deadline differentiated pricing of deferrable electric loads[END_REF], the consumers can choose an energy for intermediary times before their true deadline, with a price that depends on the aggregate power consumption. It is found that marginal cost pricing, combined with "earliest-deadline-first" charging maximizes the social welfare. In [START_REF] Ghosh | Control of charging of electric vehicles through menu-based pricing[END_REF], the CSO offers a menubased pricing, considering the EV user flexibility in the amount of energy and the deadline for charging. The charging station electricity is provided by either a local renewable energy source with zero cost, or by buying from the grid, therefore the cost depends on the charging station aggregate power consumption. EV users are assumed to provide information on both their energy demand and deadline. In [START_REF] Owen | Smart charging of electric vehicles: An innovative business model for utility firms[END_REF], apart from modulating the price of each deadline in the service menu, the focus shifts to identifying optimal deadline values. The charging station scheduling problem is formulated as an optimal control problem, aiming at smoothing the aggregate power consumption and assuming a convex electricity cost for the CSO. In [START_REF] Lu | Deadline differentiated dynamic ev charging price menu design[END_REF], the CSO offers different charging options to EV users that are flexible with their deadline for charging but have a pre-determined energy demand. A flexible deadline allows the CSO to charge EVs when the cost of electricity is low, but a long occupancy of the charging point may deprive a future arrival from charging. This deadline-differentiated pricing considers either full information on the EV user utility or a clustering to EV classes with similar energy demand and utilities. However, [START_REF] Owen | Smart charging of electric vehicles: An innovative business model for utility firms[END_REF], [START_REF] Lu | Deadline differentiated dynamic ev charging price menu design[END_REF] do not account for the EV user flexibility in their energy demand. In another relevant work [START_REF] Zhang | Optimal charging scheduling by pricing for ev charging station with dual charging modes[END_REF], the EV users can choose between two charging levels, and the CSO objective is to set the prices so as to minimize the "dropping rate," i.e., the proportion of EV users declining the entrance to the chosen charging station because of a long queue.

The aforementioned works require EV users to provide information on their parking duration (deadline) and/or their energy demand. Apart from privacy concerns, it is not unreasonable to assume that many users may not have a specific need in energy, and/or there may be some uncertainty in their parking duration. Furthermore, in the deadline-differentiated pricing, there may be no guarantee on the amount of energy charged if the EV user leaves sooner than expected (also not unreasonable to assume). For these reasons, we suggest a price menu that offers a guaranteed power for a certain 23rd Power Systems Computation Conference price, so that EVs can choose among different power rates, thus guaranteeing a certain quality of service.

In this paper, our aim is to investigate the problem of the CSO price menu design and its capability to provide demand response. We consider a CSO equipped with a large enough number of charging points, i.e., the station's capacity (in terms of charging points) does not become a limiting factor for serving EVs. We also consider EV users that view charging more as an opportunity rather than an indispensable need. Hence, the parking duration depends only on their on-site activities -see, e.g., [START_REF] Pierce | Getting the prices right[END_REF] on the in-elasticity of the parking demand. A profit maximizing CSO, using the information (or estimate) on the EV user utilities, would offer a price menu with different prices per power rate. Assuming a certain price for offering demand response in real-time, the CSO could then adjust the price menu to reduce its aggregate power consumption.

Our main contribution is three-fold. First, we formulate the price menu design problem of an EV CSO that differentiates the options in the power rate, instead of a deadline, as a Mixed Integer Linear Programming (MILP) problem. Second, we propose a setting for the provision of demand response by the CSO, given a price for reducing its aggregate power consumption over a certain time period, by adjusting its price menu in real-time. Third, we provide insights through numerical experimentation on the construction of the optimal price menu, and the trade-offs considered in terms of the CSO profits and the social welfare under different prices for electricity and demand response.

The remainder of the paper is organized as follows. Section II presents the preliminaries for the CSO and EV models. Section III formulates the price menu design problem, and Section (IV) describes the provision of demand response by the CSO. Section V discusses a numerical experimentation, and Section VI concludes and provides further research directions.

II. MODEL PRELIMINARIES

In this section, we introduce the EV CSO offered charging options (in Subsection II-A), the utility function of EV users, (in Subsection II-B), and we define the CSO profit (in Subsection II-C) and the social welfare (in Subsection II-D).

A. EV CSO Offered Charging Options

The CSO offers a discrete set of K options, denoted by K = {1, ..., K}, where option k corresponds to a power rate P k and a price per energy unit (kWh) π k . For notational simplicity, we denote the set of available options, which includes option k = 0 corresponding to "not charging," i.e., P 0 = 0 and π 0 = 0, by K + := K ∪ {0}.

The CSO options are ordered from lowest power rate to the highest, i.e., P k-1 < P k , ∀k ∈ K, with prices that are non-decreasing with the power rate, i.e.,

π k-1 ≤ π k , ∀k ∈ K. ( 1
)
In most of the charging stations in reality, a higher charging price incurs a higher cost, see for example [START_REF]Charge place scotland tariffs[END_REF].

B. EV Utility

Let EV class i ∈ I, where I = {1, ..., I} is the set of EV classes, have an initial state of charge (upon arrival) denoted by e 0 i , and parking duration denoted by d i . Let e max i denote the EV battery capacity, and e i the state of charge at the time of departure. For brevity, we shall refer to EV i instead of EV class i.

If EV i charges at a power rate P , its state of charge at the time of departure is given by:

e i = e 0 i + P d i . (2)
Given any option k, EV i can charge at most min{e max i e 0 i , P k d i }, so that the battery capacity is not exceeded. We can therefore denote the options available to EV i using a subset K i = {1, ..., K i } ⊆ K, where K i is defined as the highest available power rate such that the battery capacity is not exceeded given the parking duration d i .

Let U i denote the utility of EV i, which depends on the state of charge e i at the time of departure, and is increasing and concave in e i -standard in the EV literature [START_REF] Samadi | Optimal real-time pricing algorithm based on utility maximization for smart grid[END_REF]. Essentially, increasing the state of charge increases the EV utility but with diminishing returns (due to the concavity). From (2), we can express the dependency of the EV utility on the power rate P using U i (P ).

Let u i (P k ) denote the marginal utility of EV i choosing option k related to power P k , i.e.,

u i (P k ) = ∂U i (P ) ∂P P =P k . (3)
Given the increasing and concave utility in e i , which also, from (2), implies that the utility is increasing and concave in P , the marginal utility is non-negative and decreasing in k, i.e.,

u i (P k-1 ) ≥ u i (P k ) ≥ 0, ∀k ∈ K + i . (4)
Eq. ( 4) simply states that the marginal utility is positive but decreasing in the power rate, hence, there are diminishing returns when choosing an option with a higher power rate.

C. CSO Profit

The CSO buys electricity from the grid and sells electricity to EVs. Let λ t be the price (per kWh) at which the CSO buys electricity at time t. Let c i denote the average cost (per kWh) over time that the CSO incurs when charging EV i, which is given by:

c i = 1 d i τ 0 i +di τ 0 i λ t dt, (5)
where τ 0 i and τ 0 i + d i are the arrival and departure times, respectively of EV i.

When EV i chooses option k, the CSO charges π k (per kWh), whereas incurs a cost c i (per kWh). Hence, the CSO profit for charging EV i choosing option k, denoted by Π i (π k ) to indicate the dependence on π k , is then given by: Let W i (π k ) denote the welfare of EV i choosing option k, which is the difference between their utility U i (P k ), and the cost of charging at the charging station, i.e.,

Π i (π k ) = (π k -c i ) P k d i . (6
W i (π k ) := U i (P k ) -π k P k d i . (7)
Note that we express the dependence of the welfare on the price π k , whereas the dependence of the utility is on the associated power rate P k . EV i aims at maximizing its welfare, hence, the optimal choice, k * i , is given by:

k * i = arg max k∈K + i W i (π k ). (8)
Using the concavity of the utility function and the price ordering (1), one can show that for any set of options the welfare increases (in k) until the option that corresponds to the optimal choice for the EV user, and then decreases. Hence, one can find the optimal choice, with a simple search among ordered options (from the lowest power rate to the highest power rate) until the welfare begins to decrease. The social welfare, W S i (π k ), i.e., the sum of the CSO's profit and the welfare of EV i choosing option k, is given by:

W S i (π k ) := Π i (π k ) + W i (π k ) = U i (P k ) -c i P k d i , (9)
where we added the dependence on π k , which naturally associates to the power rate P k .

III. PRICE MENU DESIGN PROBLEM

In this section, we consider the price menu design problem of a CSO who has information on the distribution of EV users among a finite set of classes, I. These classes and their weight over the population, denoted by θ i for EV (class) i, can be obtained using data on the EV user behavior -see e.g., [START_REF] Lu | Deadline differentiated dynamic ev charging price menu design[END_REF].

In what follows, we consider the price menu design problem under two objectives: (i) maximization of the expected CSO profit (in Subsection III-A), and (ii) maximization of the expected social welfare (in Subsection III-B). Notably, the two problems can be solved for each time period separately (no time coupling constraints), i.e., the CSO can update the price menu at each time period, assuming without loss of generality an hourly granularity, (although the offered price when an EV arrives will be binding for that EV for its entire parking duration). Hence, in the formulation we will not include a time index for brevity.

A. CSO Profit Maximization

The CSO expected profit is given by:

E θ [Π i (π k * i )] = i∈I θ i (π k * i -c i )P k * i d i , (10)
where k * i is the optimal choice of EV (class) i, which is given by [START_REF] Pierce | Getting the prices right[END_REF]. Hence, the CSO profit maximization problem is given by:

max π,k * E θ [Π i (π k * i )]
given by (10), s.t. ( 1) and ( 8).

(

) 11
The optimal choice, k * i , for EV user i can be modeled using a binary variable, a i,k , ∀i ∈ I, k ∈ K + i , denoting whether EV i chooses option k (value 1, otherwise 0). The expected profit is thus given by:

E θ [Π i ] = i∈I k∈K + i θ i a i,k (π k -c i )P k d i , (12)
where we require that only one option is chosen, i.e.,

k∈K + i a i,k = 1, ∀i ∈ I. (13)
Constraint ( 8) can then be reformulated using the big M method as follows:

W i (π k ) + M W m̸ =k a i,m ≥ W i (π ℓ ), ∀i ∈ I, k, ℓ ∈ K + i , ℓ ̸ = k., (14)
where M W is a positive and sufficiently large number. Together with (13), Constraints (14) ensure that a i,k * i = 1. The idea of this constraint is that for k such that a i,k = 0, the inequality is verified because the lhs sum take a very large value with M W . For the unique k such that a i,k = 1, the lhs sum is equal to zeros, which enforce option k to be the optimal option in terms of EV welfare maximization.

The objective function in [START_REF] Wolbertus | Fast charging systems for passenger electric vehicles[END_REF] contains a product of a binary variable a i,k and a non-negative continuous variable π k . Using standard big M linearization techniques [START_REF] Fortet | L'algebre de boole et ses applications en recherche operationnelle[END_REF], we can replace the product by a new, non-negative, continuous variable, say b i,k , ∀i ∈ I, k ∈ K i , so that [START_REF] Wolbertus | Fast charging systems for passenger electric vehicles[END_REF] becomes:

E θ [Π i ] = i∈I k∈Ki θ i (b i,k -a i,k c i )P k d i , (15)
and add the following constraints:

b i,k ≤ a i,k M π , ∀i ∈ I, k ∈ K i , (16a) b i,k ≥ 0, ∀i ∈ I, k ∈ K i , (16b) b i,k ≤ π k , ∀i ∈ I, k ∈ K i , (16c) b i,k ≥ π k -(1 -a i,k )M π , ∀i ∈ I, k ∈ K i . (16d)
Constraints ( 16) ensure that the variables b i,k take the value π k when a i,k = 1 and 0 otherwise. Specifically, constraints (16a) and (16b) enforce b i,k = 0 when a i,k = 0, whereas (16a) becomes redundant when a i,k = 1. Constraints (16c) and (16d) enforce b i,k = π k when a i,k = 1, and they become redundant otherwise. Summarizing, the CSO expected profit maximization problem, Π CSO max is given by:

Π CSO max : max π,a,b E θ [Π i ]
given by (15), s.t. ( 1), ( 7), ( 13), ( 14), and ( 16).

B. Social Welfare Maximization

Using the aforementioned binary variable a i,k , the expected social welfare is given by:

E θ [W S i ] = i∈I k∈Ki θ i a i,k [U i (P k ) -c i P k d i ]. (18)
For practical purposes, in order to ensure a non-negative profit for the CSO (who sets the prices), we require from (15) that:

i∈I k∈Ki θ i (b i,k -a i,k c i )P k d i ≥ 0, (19)
where b i,k are described by constraints (16). Summarizing, the expected social welfare maximization problem, W S max , is as follows:

W S max : max π,a E θ [W S i ]
given by (18), s.t. ( 1), ( 7), ( 13), ( 14), ( 16) and ( 19). ( 20)

Problem ( 20) is also a MILP problem.

IV. CSO DEMAND RESPONSE

In this section, we consider the case of a CSO providing demand response, by reducing the aggregate power consumption by a certain amount, during specific time periods. This reduction can be viewed as the provision of a reserve product (to be used interchangeably with demand response) -or flexibility as is often mentioned in the literature -w.r.t. its "usual" power consumption. Indeed, there is a long debate on how to determine such a "base case," or whether this is even possible. However, for the purposes of this paper, we can employ the outcome of the CSO profit maximization problem as the "usual consumption," upon which any reduction can be calculated. Alternatively, one can think of a day-ahead commitment on a certain power consumption (considering an expected profit maximization problem and the resulting power consumption given the "optimal price menu") and a "real-time adjustment" of the price menu to reduce the actual consumption by a certain amount. In what follows, we consider the problem for the provision of demand response through a certain reduction in the aggregate consumption during a certain time period.

For clarity, we introduce the notation in the multi-period problem, whose horizon is represented by the set T = {1, . . . , T }, assuming without loss of generality that t refers to an hourly period.

Let N t denote the number of EV arrivals at time t, of expectation Nt , and with weights per class i at hour t, θ i,t . For brevity, we define parameter n i,t = Nt θ i,t representing the expected number of vehicles of class i arriving at time t.

The price menu is denoted by π k,t , i.e., the prices can change every hour, assuming without loss of generality that the power rates, P k , do not change every hour. Still, for each time period, prices should be ordered as in (1), i.e.:

π k-1,t ≤ π k,t , ∀k ∈ K, t ∈ T . (21)
The binary variables a i,k,t denote whether EV i, arriving at time t, chooses option k, with:

k∈K + i a i,k,t = 1, ∀i ∈ I, t ∈ T . (22)
The optimal choice of EV i, that is k * i such that a i,k * i ,t = 1, arriving at time t is given by the following constraints:

U i (P k ) -π k,t P k d i + M W m̸ =k a i,m,t ≥ U i (P ℓ ) -π ℓ,t P ℓ d i , ∀i ∈ I, k, ℓ ∈ K + i , ℓ ̸ = k, t ∈ T , (23)
where we replaced the EV welfare by its definition -see [START_REF] Zhang | Optimal charging scheduling by pricing for ev charging station with dual charging modes[END_REF].

Similarly to (16), the continuous variables b i,k,t verify the following conditions:

∀i ∈ I, k ∈ K i , t ∈ T : (24) 0 ≤ b i,k,t ≤ a i,k,t M π , π k,t -(1 -a i,k,t )M π ≤ b i,k,t ≤ π k,t .
The aggregate expected profit over the time horizon, using (15) and parameter n i,t , is given by:

ΠT = t∈T i∈I k∈Ki n i,t (b i,k,t -a i,k,t c i )P k d i . ( 25
)
Let T R = {τ 1 , . . . , τ 2 } be the set of time periods during which the CSO reduces the aggregate power consumption by a certain amount denoted by P R t , from the aggregate power consumption, denoted by P Π t that pertains to the solution of problem ( 17), which can be solved separately for each time period.

Let indicator (parameter) δ i,t ′ ,t denote whether an EV of class i arriving at time t ′ is connected at time t, where t ≥ t ′ -i.e., for as long as its parking duration d i . Let variable Pt denote the expected aggregate power consumption, given by:

Pt = i∈I k∈Ki t t ′ =1 n i,t ′ a i,k,t ′ P k δ i,t ′ ,t , ∀t ∈ T . ( 26
)
The reduction of the power consumption for the provision of demand response is then enforced through:

Pt ≤ P Π t -P R t , ∀t ∈ T R . ( 27
)
Summarizing, the CSO demand response problem, given a price λ R t for reducing the power consumption during hour t, is now as follows: 20), Problem (28) is a MILP problem. They all can be solved using commercials solvers such as CPLEX.

DR CSO λ R : max π,
So far, we have considered the expected number of arrivals per class, n i,t , and calculated expected profits and power consumption. One may think that the provision of demand response, i.e., the reduction in power, should be enforced with 23rd Power Systems Computation Conference PSCC 2024 Paris, France -June 4 -7, 2024 a higher probability. A simple way of enforcing a more conservative would be to replace the expected number of arrivals n i,t in constraint ( 26) by an appropriately-defined "worstcase" number of arrivals, during the periods T R (potentially even earlier). Indeed, in a practical setting, the CSO can resolve the problem, after each hour, and define the price menu dynamically every hour, with the information available up to that hour. However, due to space considerations, we will refrain from such settings in this first work. Future work will be directed to more elaborate robust optimization formulations and appropriate uncertainty sets as well as more dynamic settings that employ the information that becomes available during the day.

V. NUMERICAL RESULTS

In this section, we provide a numerical experimentation of the optimal price menu design problem. In Subsection V-A, we describe the test case (CSO and EVs). In Subsection V-B, we present the results from the profit and social welfare maximization problems, whereas in Subsection V-C, we discuss the results from the CSO demand response provision.

A. Test Case 1) CSO: We consider the provision of 4 power rates for charging: P 1 = 2.5 kW, P 2 = 5 kW, P 3 = 7.5 kW, and P 4 = 10 kW, i.e., K = 4. We note that we consider "low" charging rates [START_REF] Wolbertus | Fast charging systems for passenger electric vehicles[END_REF] in our numerical illustrations to emphasize the fact that EVs view charging more as an opportunity than as an absolute need (which would make them less flexible or direct them to fast charging stations). However, our analysis remains relevant for higher charging rates.

2) EVs: We consider that all EVs have a battery capacity of 50 kWh, and charge between 20% and 80% of their state of charge. EV classes are determined by their initial state of charge and parking duration. We consider 3 different initial states of charge: 10 kWh, 20 kWh, and 30 kWh. We also consider 4 different parking durations: 1h, 2h, 3h, and 4h. Hence, in total, we have 12 EV classes, assumed to be equally distributed. EVs arrive at the charging station with a constant rate of 120 vehicles per hour, from 7am to 8pm. They share a quadratic utility function given by

U i (P ) = α i [ P d i -1 2 β i (P d i ) 2 ]
, where α i , β i , are positive scalars, which parameterize EV i willingness to charge while ensuring an increasing utility for practical power rates, and is assumed, without loss of generality, to depend only on the state of charge at arrival. Table I summarizes the 12 EV classes, the highest (in power rate) option available (K i ) for EV class i, and utility parameters. Big M values are set at 15 for M W and 0.5 for M π .

All problems were solved using CPLEX 22.1.1.0 solver.

B. Profit and Social Welfare Maximization Results

For convenience, we shall refer to the outcome of the CSO expected profit maximization problem (17), Π CSO max , as the "Base Case," and to the expected social welfare maximization problem (18), W S max , as the "Benchmark." Fig. 1 presents the optimal price menu for the Base Case and the Benchmark, as well as the electricity price (parameter), λ t , that the CSO has to pay for charging the EVs (for comparison purposes). We observe that the Base Case optimal price menu remains always higher than the electricity price (maximizing CSO profit). For the most part of the day (from 7am to 5pm), when the electricity price is low (around 20 cents per kWh), the Base Case price menu remains stable (spanning around 2 cents from option 1 to option 4), whereas when the electricity price increases (from 6pm to 8pm), the price menu increases and remains practically stable for all options. The Base Case shape of the menu indicates that the optimal price menu when the CSO maximizes its profits is not very sensitive to small changes in the electricity price. This is attributed to the fact that the price menu considers also the utility of the EVs, whose optimal choices are discrete; hence, a small increase in the price might result in the EVs choosing a lower power rate or even not charging. Consider for example the case of EV class 5, which chooses at 5pm option 2. Note that this class could choose any option, however, it chooses option 2 and not the higher power rate option 3, even though the price difference (between options 2 and 3) is only 0.625 cents; hence, this small difference in price is enough to discourage EV class 5 from choosing option 3.

The Benchmark optimal price menu in Fig. 1 exhibits a different shape with a larger span during the day, and with some values that may be well below the electricity price. We elaborate further on this shape by illustrating in Fig. 2, the hourly profit and social welfare, and in Fig. 3 the aggregate power consumption and the number of EVs deciding to charge, for the Base Case and the Benchmark. We observe in Fig. 2 that the maximum social welfare is achieved with zero profits for the CSO, and maximum welfare for the EV users. The trend of the Benchmark price menu follows the trend of a "weighted moving average" of the electricity cost of the charging station. Low prices attract EV users with a low marginal utility (i.e., those classes that need less to charge), and the price span is such that the CSO profit is zero, whereas the welfare of the EVs is maximized. Clearly, as shown in Fig. 3, the Benchmark maximizes social welfare by charging more EVs at lower prices (e.g., all arriving EVs decide to charge from 7am to 4pm), and consuming significantly higher power, peaking at 1,925 kW, whereas the Base Case maximizes profit by charging less EVs at higher prices, and peaking at 875 kW. The total social welfare over the entire day for the Benchmark is 2519.22 C (821.58 C higher compared to the social welfare

C. Demand Response Results

Using the solution of the Base Case as a "baseline" for the CSO aggregate consumption, we solve the demand response problem (28), with a 5-hour reserve period between 4pm and 9pm. We consider two cases for the reserve price (constant for the entire 5-hour period): λ R t = 5 cC/kW and λ R t = 15 cC/kW.

In Fig. 4, we present the price menu for the Base Case, and for the two different values of the reserve price. For reasons that will soon become apparent, we present the 5 hours affected (4pm, 5pm, 6pm, 7pm, 8pm) as well as the previous hour (3pm). In Fig. 5, we illustrate the aggregate power consumption, for the Base Case and the two reserve prices.

Considering first the lower reserve price (Fig. 4, middle), we observe that the main change w.r.t. the Base Case refers to hours 4pm and 5pm, when the prices increase (they span between 32 and 34 cents at 4pm, and reach 37 cents at 5pm). The reduction in the aggregate power consumption (Fig. 5, green line) is 75 kW at 4pm and 275 kW at 5pm. Note that the price menu at 6pm is the same with the Base Case, however, we still see a reduction by 200 kW, which is due to the lower amount of EVs charging in the previous hours (we remind that EVs may charge for 4 hours). The reduction reduces to 175 kW at 7pm and 75 kW at 8pm. Considering next the higher reserve price (Fig. 4, right), we observe that the prices increase earlier (slightly at 3pm, and then reach around 37 cents) achieving a higher reduction in the aggregate power consumption (Fig. 5, red line) starting with 50 kW at 3pm, to 275 kW at 4 pm, to 450 kW at 5pm, to 350 kW at 6pm to 200 kW at 7pm, and 100 kW at 8pm.

In Fig. 6, we illustrate the CSO profit for the Base Case and the two reserve prices, which includes the revenue for providing demand response. We observe that at 3pm, the Base Case and the lower reserve price yield the same profit, whereas the higher reserve price yields lower profit (compare with the higher prices in Fig. 4, right). However, the profit of the higher reserve price (red column) is the highest in all other hours. The profit of the lower reserve price (green column) becomes higher than the Base Case at 5pm and remains higher until 8pm. The total profit over the entire day for the Base Case is 1,323.45 C. The total profit when providing demand response increases by 16.04 C (1.21 %) for the low reserve price, and by 128.41 C (9.70%) for the high reserve price. The total social welfare over the entire day for the Base Case is 1697.64 C. Accounting for the revenues from demand response, it decreases by 40.66 C for the low reserve price, and by 3.28C for the high reserve price. Considering the total welfare of the EVs over the entire day, which is 374.19 C for the Base Case, it decreases to 301.46 C for the low reserve price, and to 114.08 C for the high reserve price.

VI. CONCLUSIONS AND FURTHER RESEARCH

In this work, we considered a CSO who serves EVs that are not flexible in their parking duration but they are flexible in their energy demand, offering different charging rates at different prices. Arguably, this CSO model can fit well public charging stations, considering the future EV adoption, where charging will be viewed as an opportunity (e.g., in a public parking station), and charging point availability is unlikely to be a limiting factor. In this context, we formulated the optimal price menu design problem as a MILP problem, considering both a profit maximizing CSO and a social welfare maximization (as a benchmark). We further accounted for the provision of demand response by considering an adjustment of the price menu (in real time) to reduce the aggregate power consumption and benefit from a certain price for remuneration (as a type of reserve deployment). Our numerical experimentation illustrated the construction of the optimal price menu, the trade-offs considered for the CSO profit and the social welfare, and the price menu adjustment to provide demand response under different (reserve) prices.

Future research is directed to account for uncertainty in the parking duration and the EV user utility through robust optimization approaches and the construction of appropriate datadriven uncertainty sets. More elaborate price menus which might also relate to the EV state of charge, as well as CSO-EV contracts for lowering the charging rate in case of demand response provision, are also interesting directions for further research.

  (17) Problem (17) is a MILP problem. 23rd Power Systems Computation Conference PSCC 2024 Paris, France -June 4 -7, 2024

Fig. 2 .

 2 Fig. 2. Hourly profit and social welfare, Base Case and Benchmark.

Fig. 3 .

 3 Fig. 3. Aggregate power consumption (left axis, line) and number of EVs deciding to charge (right axis, columns, lower part: EVs arriving at the hour; upper part EVs remaining from previous hours).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Optimal price menu design during demand response period and for different price reserve

Fig. 6 .

 6 Fig. 6. CSO profit (lower part of column: profit for energy; upper part: revenue from demand response).

TABLE I PARAMETERS

 I OF EV CLASSES

			Class i: K i		Utility Parameters
	e i / d i	1 h	2 h	3 h	4 h	α i	β i
	10 kWh 1: 4	2: 4	3: 4	4: 3	0.425	0.017
	20 kWh 5: 4	6: 4	7: 2	8: 2	0.35	0.021
	30 kWh 9: 4 10: 2	11: 1 12: 1	0.275	0.027

Fig. 1. Optimal price menu, Base Case and Benchmark.

Paris, France -June 4 -7, 2024

ACKNOWLEDGMENT

This research paper benefited from the support of the FMJH Program PGMO and from the EDF support to this program.