# New examples of twisted Brill-Noether loci I

L. Brambila-Paz  CIMAT
Apdo. Postal 402
C.P. 36240
Guanajuato
Mexico<EMAIL_ADDRESS>and  P. E. Newstead  Department of Mathematical Sciences
University of Liverpool
Peach Street, Liverpool L69 7ZL, UK<EMAIL_ADDRESS>
###### Abstract.

Our purpose in this paper is to construct new examples of twisted Brill-
Noether loci on curves of genus $g\geq 2$. Many of these examples have
negative expected dimension. We deduce also the existence of a new region in
the Brill-Noether map, whose points support non-empty standard Brill-Noether
loci.

###### Key words and phrases:

Vector bundles on curves, Brill-Noether theory

###### 2010 Mathematics Subject Classification:

Primary: 14H60

The authors are members of the research group VBAC (Vector Bundles on
Algebraic Curves).The first author acknowledges the support of CONACYT proj.
251938.

Dedicated to Oscar García-Prada on the occasion of his 60th birthday

###### Contents

  1. 1 Introduction
  2. 2 Non-emptiness of BN loci
    1. 2.1 The BN map
    2. 2.2 Kernel and dual span bundles
    3. 2.3 Small slope
  3. 3 A non-emptiness result for twisted BN loci
  4. 4 Examples with $k=k_{1}k_{2}$
    1. 4.1 The construction
    2. 4.2 Examples
    3. 4.3 A new region in the BN map
  5. 5 Construction using kernel bundles
    1. 5.1 The construction
    2. 5.2 Examples

##  1\. Introduction

Our object in this paper and its successor [9] is to construct twisted Brill-
Noether loci different from those constructed in [15]. In particular, the
relevance of many of these examples is that they have negative Brill-Noether
numbers, whereas the examples constructed in [15, Theorem 1.1] all have
positive Brill-Noether numbers. We deduce also the existence of a new region
in the Brill-Noether map (see below for the definition), the points of which
support non-empty Brill-Noether loci, whose existence was not previously
known. This implies that our new examples also differ from those constructed
in [15, Section 9] (see Remark 4.10(iii)). In this paper, we consider
applications of the results of [6, 8, 21, 22, 23, 25] together with Butler’s
Theorem [11, Theorem 2.1]. In [9], we will, in particular, use known cases of
Butler’s Conjecture [12, Conjecture 2] (see [3, 4, 7]) to obtain further
examples. Our methods will continue to apply as further cases of Butler’s
Conjecture are proved. In what follows, we shall usually abbreviate Brill-
Noether to BN.

Let $C$ be a smooth irreducible projective curve of genus $g\geq 2$ defined
over the complex numbers. For $i=1,2$, let $M_{i}:=M(n_{i},d_{i})$ denote the
moduli space of stable bundles of rank $n_{i}$ and degree $d_{i}$. Similarly,
we write $\widetilde{M}_{i}$ for the corresponding moduli space of
S-equivalence classes of semistable bundles.

Let $E_{2}$ be any vector bundle on $C$ of rank $n_{2}$ and degree $d_{2}$.
For any integers $n_{1}$, $d_{1}$, $k$ with $n_{1}\geq 1$, one can define the
twisted BN locus

(1.1) $B(n_{1},d_{1},k)(E_{2}):=\\{E_{1}\in M_{1}\,|\,h^{0}(E_{1}\otimes
E_{2})\geq k\\}.$

The locus $B(n_{1},d_{1},k)(E_{2})$ is a closed subscheme of $M_{1}$. In fact,
it is a determinantal locus, being the pullback of a universal determinantal
variety of codimension $k(k-\chi)$, where

(1.2) $\chi:=n_{2}d_{1}+n_{1}d_{2}-n_{1}n_{2}(g-1).$

It follows that, if $\beta(n_{1},d_{1},k)(E_{2})\leq n_{1}^{2}(g-1)+1$, then
every irreducible component of $B(n_{1},d_{1},k)(E_{2})$ has dimension greater
than or equal to the BN number

(1.3) $\beta(n_{1},d_{1},k)(E_{2}):=n_{1}^{2}(g-1)+1-k(k-\chi),$

This number is often referred to as the expected dimension.

We define also

$\widetilde{B}(n_{1},d_{1},k)(E_{2}):=\\{[E_{1}]\in\widetilde{M}_{1}\,|\,h^{0}(\operatorname{gr}E_{1}\otimes
E_{2})\geq k\\},$

where $[E_{1}]$ denotes the S-equivalence class of $E_{1}$ and
$\text{gr}E_{1}$ is the corresponding graded bundle. This is a closed
subscheme of $\widetilde{M}_{1}$, but can have components of dimension less
than the BN number.

In the special case $E_{2}=\mathcal{O}_{C}$, we obtain the standard
(untwisted) higher rank BN locus, denoted by $B(n_{1},d_{1},k)$, with expected
dimension

(1.4) $\beta(n_{1},d_{1},k):=n_{1}^{2}(g-1)+1-k(k-d_{1}+n_{1}(g-1)).$

In particular, $B(1,d_{1},k)$ is the classical BN locus $W^{k-1}_{d_{1}}$.

In the case $n_{1}=1$, twisted BN loci have been studied for some time
starting with [13, 19] (see [15, Theorem 2.1] for full details and further
references). These loci include the special case of maximal line subbundles. A
systematic study for arbitrary $n_{1}$ was started in [15], where, in
particular, examples of twisted BN loci were constructed [15, Theorem 1.1]
using the technique of [22]. As we shall see in Theorem 3.1, these have the
important feature that the BN number (1.3) is positive. Many of them (maybe,
all) have the expected dimension [15, Theorem 1.2].

We can now introduce the loci which we are going to study in this paper. This
is a special case of a more general construction introduced in [15], where it
was used primarily to obtain results for $B(n_{1},d_{1},k)(E_{2})$. If
$\gcd(n_{i},d_{i})=1$ for $i=1,2$, then there exist universal bundles
$\mathcal{U}_{i}$ on $M_{i}\times C$ and, for any integer $k$, we define the
universal twisted BN locus as follows:

(1.5) $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2}):=\\{(E_{1},E_{2})\in M_{1}\times
M_{2}\,|\,h^{0}(E_{1}\otimes E_{2})\geq k\\}.$

In the non-coprime case, we need to lift to étale coverings of $M_{1}$ and
$M_{2}$ in order to construct “universal” bundles [24, Proposition 2.4].
However (1.5) still makes sense and we shall use the same notation. The locus
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ is a closed subscheme of $M_{1}\times
M_{2}$; if $k\leq 0$, then $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})=M_{1}\times
M_{2}$. The structure of $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ as a
determinantal locus allows us to define a BN number (see [15, (3.1)]

(1.6) $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2}):=\dim M_{1}+\dim
M_{2}-k(k-\chi),$

which may again be referred to as the expected dimension. Note that
$\chi=\chi(E_{1}\otimes E_{2})$ for all $(E_{1},E_{2})\in M_{1}\times M_{2}$.
If $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\leq\dim M_{1}+\dim M_{2}$, then
every irreducible component of $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ has
dimension greater than or equal to
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$. We define similarly the
corresponding semistable locus

$\widetilde{B}(\mathcal{U}_{1},\mathcal{U}_{2}):=\\{([E_{1}],[E_{2}])\in\widetilde{M}_{1}\times\widetilde{M}_{2}\,|\,h^{0}(\text{gr}E_{1}\otimes\text{gr}E_{2})\geq
k\\}.$

Note that the definition is symmetric and
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\cong
B^{k}(\mathcal{U}_{2},\mathcal{U}_{1})$. Moreover, by Serre duality,
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\cong
B^{k_{1}}(\mathcal{U}_{1}^{*},\mathcal{U}_{2}^{*}\otimes p_{C}^{*}K_{C})$,
where $k_{1}=k-\chi$ and $K_{C}$ denotes the canonical line bundle on $C$.
Note also that $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\cong
B^{k}(\mathcal{U}_{1}\otimes p_{C}^{*}L^{*},\mathcal{U}_{2}\otimes
p_{C}^{*}L)$ for any line bundle $L$. We take account of these possibilities
in Theorem 3.1(ii) and Remark 3.3(iii) (see also Remark 5.4(i)).

######  Remark 1.1.

The following conditions are equivalent:

  * •

$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$;

  * •

$B(n_{1},d_{1},k)(E_{2})\neq\emptyset$ for some $E_{2}\in M_{2}$;

  * •

$B(n_{2},d_{2},k)(E_{1})\neq\emptyset$ for some $E_{1}\in M_{1}$.

Many of our results are valid for any smooth curve. For some, however, we
require the concept of a Petri curve. The curve $C$ is said to be Petri if the
classical Petri map $H^{0}(L)\otimes H^{0}(K_{C}\otimes L^{*})\longrightarrow
H^{0}(K_{C})$ is injective for every line bundle $L$ on $C$. The general curve
is Petri in the sense that Petri curves of genus $g$ form a non-empty open
subset of the moduli space of curves of genus $g$ [14, 20]. By classical BN
theory, if $C$ is Petri, the BN locus $B(1,d_{1},k)$ has dimension precisely
$\beta(1,d_{1},k)$ whenever $0\leq\beta(1,d_{1},k)\leq g$ and is empty when
$\beta(1,d_{1},k)<0$. Sometimes we will not be able to obtain results for
arbitrary Petri curves but only for curves which are general in the sense that
they live in an unspecified open subset of the moduli space of curves of genus
$g$.

The BN loci $B(1,d_{1},k)(E_{2})$, with $E_{2}$ general, behave in a similar
way to the classical BN loci [15, Theorem 2.1]. By Remark 1.1, this implies
non-emptiness results for $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ when
$n_{1}=1$ or $n_{2}=1$. Moreover, when $n_{2}=1$, we have an isomorphism

$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\longrightarrow
B(n_{1},d_{1}+n_{1}d_{2},k)\times\mbox{Pic}^{d_{2}}(C):\
(E_{1},L)\mapsto(E_{1}\otimes L,L),$

giving a description of $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ in terms of
an untwisted BN locus. We know many cases in which such loci are non-empty;
see, in particular, Section 2. In this paper, we shall assume that $n_{i}\geq
2$ for $i=1,2$.

We can now state our first main theorem.

######  Theorem 1.2.

Suppose that $C$ is a smooth curve of genus $g\geq 2$ and that $d_{i}$,
$k_{i}$ $(i=1,2)$ are positive integers. Let $B(n_{i},d_{i},k_{i})$ be non-
empty BN loci with $n_{i}\geq 2$ and suppose that $k=k_{1}k_{2}$.

  * (i)

If $d_{1}<2n_{1}$ and $d_{2}\leq 2gn_{2}$, then
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$.

  * (ii)

If, in addition, $d_{i}/n_{i}=\mu_{i}$, $k_{i}/n_{i}=\lambda_{i}$ and
$\mu_{1}+\mu_{2}<\lambda_{1}\lambda_{2}+g-1$, then, for fixed $\mu_{i}$,
$\lambda_{i}$ and sufficiently large $n_{1}$, $n_{2}$,
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$.

Part (i) of this theorem will be stated and proved in a more detailed version
as Theorem 4.3; (ii) is Lemma 4.2. Using the results of Section 2, it is easy
to find many cases where the hypotheses of the theorem are satisfied, thus
obtaining the examples we are seeking (see Example 4.5). There exist examples
even when $n_{1}=n_{2}=2$ (see Example 4.6).

Our second main theorem (Theorem 4.8) is an important application to BN theory
which shows that this construction gives a new region in the BN map (for the
definition, see Subsection 2.1).

Returning to twisted BN loci, our third main theorem is of a rather different
nature, in that $d_{2}$ is negative and so $h^{0}(E_{2})=0$ for all $E_{2}\in
M_{2}$.

######  Theorem 1.3.

Suppose that $C$ is a smooth curve of genus $g\geq 2$, $n_{1}\geq 2$,
$k_{1}>n_{1}$, $n$ a positive integer and either $d>2ng$ or $d=2ng$ and $C$ is
non-hyperelliptic. Suppose further that $B(n_{1},d_{1},k_{1})\neq\emptyset$
and that

$k\leq(d-n(g-1))(k_{1}-n_{1})-nd_{1}.$

Let $(n_{2},d_{2})=(d-ng,-d)$. Then
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$. If, in addition,
$k=d(k_{1}-n_{1})-e$, where $e\geq n(g-1)(k_{1}-n_{1})+nd_{1}$, and

$d_{1}<k_{1}+n_{1}(g-1)-\frac{g-1}{k_{1}-n_{1}},$

then, for any fixed values of $n_{1}$, $d_{1}$, $k_{1}$, $n$ and $e$, and
$d\gg 0$, $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$.

A more detailed version of this theorem will be stated and proved as Theorem
5.3. In Corollary 5.5 and the related Example 5.6, we show that the
requirement that $B(n_{1},d_{1},k_{1})\neq\emptyset$ is compatible with the
condition $d_{1}<k_{1}+n_{1}(g-1)-\frac{g-1}{k_{1}-n_{1}}$. Hence Theorem 5.3
does lead to examples of twisted BN loci with negative expected dimension.

Section 2 is concerned with the non-emptiness of $B(n_{1},d_{1},k_{1})$. In
subsection 2.1, we introduce the BN map and describe some sufficient
conditions for non-emptiness. Subsection 2.2 contains a description of kernel
bundles and the dual span construction. We include also a description of BN
loci for bundles of slope $\leq 2$ (subsection 2.3). In section 3, we obtain
an extended version of the main existence theorem of [15]. Sections 4 and 5
contain our main results.

The dual span construction works also when we consider pairs $(E,V)$ where $V$
is a subspace of $H^{0}(E)$ which generates $E$. In the case where $E$ is a
line bundle, there are then further stability results, which have consequences
for the non-emptiness of $B(n_{1},d_{1},n_{1}+1)$. We shall investigate
applications of this to twisted BN loci in a future paper [9].

Throughout the paper, $C$ will denote a smooth irreducible projective curve of
genus $g\geq 2$ defined over ${\mathbb{C}}$. For a vector bundle $E$ on $C$ of
rank $n$ and degree $d$, we denote by $\mu(E):=\frac{d}{n}$ the slope of $E$.
We write also $h^{0}(E)$ for the dimension of the space of sections
$H^{0}(E)$. The canonical line bundle on $C$ is denoted by $K_{C}$. For any
real number $a$, we write $\lfloor a\rfloor$ and $\lceil a\rceil$ for the
largest integer $\leq a$ and the smallest integer $\geq a$, respectively.

We thank Lilia Alanis Lopez for help with the figures and the referee for some
useful comments.

##  2\. Non-emptiness of BN loci

In this section, we will recall and extend some of the principal known results
on the non-emptiness of the BN loci $B(n_{1},d_{1},k_{1})$. The answer is
completely known for $k_{1}\leq n_{1}$, but the problem is much more
complicated (and still not completely solved) for $k_{1}>n_{1}$ (except for
$g\leq 3$ and (almost) for hyperelliptic curves) (see [8]). For Petri curves,
the naïve conjecture is that $B(n_{1},d_{1},k_{1})$ is non-empty if and only
if $\beta(n_{1},d_{1},k_{1})\geq 0$. From (1.4), this is equivalent to

(2.1) $d_{1}\geq
k_{1}+n_{1}(g-1)-\frac{n_{1}(g-1)}{k_{1}/n_{1}}-\frac{1}{k_{1}}.$

In fact, numerous examples show that (2.1) is neither necessary nor sufficient
for non-emptiness of $B(n_{1},d_{1},k_{1})$ (see, for example,[6], [2], [21],
[18, Remark 5.5], [1]).

###  2.1. The BN map

Following [8], the non-emptiness of $B(n_{1},d_{1},k_{1})$ can be understood
in terms of the BN map. For this, we map any non-empty BN locus
$B(n_{1},d_{1},k_{1})$ (or $\widetilde{B}(n_{1},d_{1},k_{1})$) with $0\leq
d_{1}\leq n_{1}(2g-2)$ to the point in the plane

(2.2) $(\mu,\lambda)=(d_{1}/n_{1},k_{1}/n_{1}).$

We shall say that $(\mu,\lambda)$ supports $B(n_{1},d_{1},k_{1})$ (resp.,
$\widetilde{B}(n_{1},d_{1},k_{1})$) if (2.2) holds and
$B(n_{1},d_{1},k_{1})\neq\emptyset$ (resp.,
$\widetilde{B}(n_{1},d_{1},k_{1})\neq\emptyset$). We are interested here in
regions in the $(\mu,\lambda)$-plane for which it is known that all points
with rational coordinates support infinitely many BN loci.

The first examples of such regions were determined by Teixidor i Bigas [25]
and Mercat [22]. To describe these in terms of the BN map, we follow [8]. For
$s\geq 1$, write
$\widehat{\eta}^{\prime}(s):=s+g-2-\left\lfloor\frac{g-1}{s}\right\rfloor$;
note that $d\geq\widehat{\eta}^{\prime}(s)$ if and only if $\beta(1,d+1,s)\geq
1$, and that $\widehat{\eta}^{\prime}(1)=0$,
$\widehat{\eta}^{\prime}(g)=2g-2$. Now define a function $t_{g}$ for
$0\leq\mu\leq 2g-2$ as follows:

(2.3) $t_{g}(\mu):=\begin{cases}0,&\mu=0;\\\
\mu-\lceil\mu\rceil+s,&\mu\in]\widehat{\eta}^{\prime}(s),\widehat{\eta}^{\prime}(s)+1];\\\
s,&\mu\in]\widehat{\eta}^{\prime}(s)+1,\widehat{\eta}^{\prime}(s+1)].\end{cases}$

We define T to be the region in the $(\mu,\lambda)$-plane given by

(2.4) $\mu,\lambda\in{\mathbb{Q}},\ 0\leq\mu\leq 2g-2,\ 0<\lambda\leq
t_{g}(\mu)$

(see Figure 1). Note that Clifford’s Theorem can be stated in the form

$\lambda\leq\frac{\mu}{2}+1$

and the boundary line for this inequality (the Clifford line) is shown in the
figure. Moreover, the condition $\beta(n_{1},d_{1},k_{1})=1$ becomes

$\lambda(\lambda-\mu+g-1)=g-1$

(see (1.4)), which defines a branch of a hyperbola with asymptotes $\lambda=0$
and $\lambda=\mu-g+1$ (the BN curve), and is also shown in the figure.

The following lemma summarises the results of Teixidor [25] and Mercat [22].
Teixidor’s proof is for general $C$; for $\widetilde{B}(n_{1},d_{1},k_{1})$,
this is sufficient to prove it for any smooth curve. Mercat’s result [22,
Théorème A-5] is valid for any smooth curve and implies our lemma. For the
version stated below, see [8, Section 5].

######  Lemma 2.1.

Let $C$ be a smooth curve of genus $g\geq 2$ and suppose that $n_{1}\geq 2$.
If $(\mu,\lambda)\in\operatorname{T}$, then $(\mu,\lambda)$ supports
$\widetilde{B}(n_{1},d_{1},k_{1})$ for all values of $n_{1}$ for which
$n_{1}\mu$ and $n_{1}\lambda$ are both integers. The same is true for
$B(n_{1},d_{1},k_{1})$, provided that the points

(2.5) $(\mu,\lambda)=(\widehat{\eta}^{\prime}(s)+1,\lambda),\ s-1<\lambda\leq
s,\ \widehat{\eta}^{\prime}(s)+1\neq\widehat{\eta}^{\prime}(s+1).$

are excluded.

######  Remark 2.2.

Lemma 2.1 is equivalent to the statement that
$\widetilde{B}(n_{1},d_{1},k_{1})\neq\emptyset$ if

(2.6) $d_{1}\geq k_{1}+n_{1}(g-1)-n_{1}\left\lfloor\frac{g-1}{\lceil
k_{1}/n_{1}\rceil}\right\rfloor.$

This can also be stated as

(2.7)
$\mu\geq\lambda+g-1-\left\lfloor\frac{g-1}{\left\lceil\lambda\right\rceil}\right\rfloor.$

For $B(n_{1},d_{1},k_{1})$, the case

(2.8)
$d_{1}=n_{1}\left(\left\lceil\frac{k_{1}}{n_{1}}\right\rceil+g-1-\left\lfloor\frac{g-1}{\lceil
k_{1}/n_{1}\rceil}\right\rfloor\right)$

must be excluded when $\beta(1,d_{1}/n_{1}+1,\left\lceil
k_{1}/n_{1}\right\rceil+1)\leq 0$. (2.6) and (2.8) are essentially a
restatement of Teixidor’s conditions [25] and relate closely to Mercat’s
construction [22].

Another method of constructing non-empty BN loci on a smooth curve $C$ is
described in [8]. This starts from the results of [6, 21, 23], which give
precise conditions for non-emptiness when $d_{1}\leq 2n_{1}$, and proceeds by
tensoring by line bundles $L$ with $h^{0}(L)=s\geq 1$. Serre duality extends
the region further and we denote the result by BMNO. To describe BMNO in the
BN map, we write

$\hat{\eta}(s):=s+g-1-\left\lfloor\frac{g}{s}\right\rfloor$

for $s\geq 1$. The condition $d\geq\hat{\eta}(s)$ is then equivalent to
$\beta(1,d,s)\geq 0$. On a Petri curve, this is equivalent by classical BN
theory to the assertion that $B(1,d,s)\neq\emptyset$. Note that
$\hat{\eta}(1)=0$ and, on a Petri curve, $\hat{\eta}(2)$ is the gonality of
$C$. We now define a function $f_{g}(\mu)$ for $0\leq\mu\leq g-1$ as follows:

(2.9)
$f_{g}(\mu):=\begin{cases}\frac{s}{g}(\mu-\left\lceil\mu\right\rceil)+s,&\mu\in[\hat{\eta}(s),\hat{\eta}(s)+1];\\\
\frac{s}{g}(\mu-\left\lceil\mu\right\rceil+1)+s,&\mu\in]\hat{\eta}(s)+1,\hat{\eta}(s+1)-1];\\\
\frac{\hat{\eta}(s+1)-s}{g}(\mu-\left\lceil\mu\right\rceil+1)+s,&\mu\in]\hat{\eta}(s+1)-1,\hat{\eta}(s+1)[.\end{cases}$

We extend the definition of $f_{g}$, which is modified from that in [8] to
allow $\mu=0$, to the whole interval $[0,2g-2]$ by Serre duality. The region
BMNO in the $(\mu,\lambda)$-plane is then defined by the conditions

$0\leq\mu\leq 2g-2,\ 0<\lambda\leq f_{g}(\mu).$

For $g=10$, this region is illustrated in Figure 1, which also includes the
top boundary of T, the Clifford line, the BN curve and an indication of the
excluded points from Lemmas 2.1 and 2.3 and Remark 2.4(ii). This is a copy of
[8, Figure 6] with some added information.

The following lemma summarises the principal result of [8], taking account of
the results of [23] (see [8, Remark 4.4]).

######  Lemma 2.3.

Let $C$ be a smooth curve of genus $g\geq 2$ and $n_{1}\geq 2$. If
$(\mu,\lambda)\in\operatorname{BMNO}$, then $(\mu,\lambda)$ supports
$\widetilde{B}(n_{1},d_{1},k_{1})$ for infinitely many values of $n_{1}$. If
$C$ is non-hyperelliptic, the same holds for $B(n_{1},d_{1},k_{1})$, provided
that points of the form $(\mu,\lambda)$ such that $\mu=\hat{\eta}(s)$ with
$\mu\leq g-1$ and $\lambda>\frac{s-1}{g}+s-1$, together with their Serre duals
and the points $((\widehat{\eta}(s)+1,s)$, are excluded.

######  Remark 2.4.

(i) [8, Remarks 4.2 and 4.3] provide information about which values of $n_{1}$
are included in the above result for any given $(\mu,\lambda)$. In particular,
there is a dense set of points in BMNO for which all values of $n_{1}$ with
$n_{1}\mu,n_{1}\lambda\in{\mathbb{Z}}$ are allowed.

(ii) Some points excluded in Lemma 2.1 are included in Lemma 2.3. In fact, if
$g$ is a multiple of $s$, then $\widehat{\eta}^{\prime}(s)=\widehat{\eta}(s)$,
and the points $(\widehat{\eta}(s)+1,\lambda)$ for $s-1<\lambda<s$ are
excluded by (2.5), but included by Lemma 2.3.

(iii) For $g\geq 4$, there are many non-empty $B(n_{1},d_{1},k_{1})$
corresponding to points outside the union of T and BMNO; see, for example, the
figures in [16, 17, 18]. We shall construct later a new region of the BN map,
giving new points for all $g\geq 5$ (see Theorem 4.8).

(iv)[8] also contains comprehensive results for hyperelliptic curves.

###  2.2. Kernel and dual span bundles

Let $E$ be a generated bundle of rank $n$ and degree $d$. Consider the exact
sequence

(2.10) $0\longrightarrow D_{E}^{*}\longrightarrow
H^{0}(E)\otimes\mathcal{O}\longrightarrow E\longrightarrow 0.$

The bundle $D_{E}^{*}$ may be referred to as a kernel bundle (or syzygy
bundle) and has rank $h^{0}(E)-ng$ and degree $-d$. Dualising (2.10), we have

(2.11) $0\longrightarrow E^{*}\longrightarrow
H^{0}(E)^{*}\otimes\mathcal{O}\longrightarrow D_{E}\longrightarrow 0.$

Following [12], we call $D_{E}$ a dual span bundle.

The following theorem addresses the stability of $D_{E}$ and summarises
results of Butler and Mercat (see also [7, Theorem 3.11]). We write

(2.12) $S:=\\{E\in M(n,d)|D_{E}\text{ is stable}\\}.$

######  Theorem 2.5.

Let $C$ be a smooth curve of genus $g\geq 2$. Suppose that either $d>2ng$ and
$E\in M(n,d)$ or $d=2ng$, $C$ is non-hyperelliptic and $E$ is a general
element of $M(n,d)$. Then $E$ is generated and $D_{E}$ is stable of slope
$\leq 2$. In particular, $S$ is a non-empty open subset of $M(n,d)$ and
$S=M(n,d)$ when $d>2ng$. Moreover, the morphism

(2.13) $S\longrightarrow B(d-ng,d,d-n(g-1)):E\mapsto D_{E}$

is an isomorphism.

###### Proof.

Any stable bundle of slope $>2g-1$ is generated with $h^{1}(E)=0$. Hence
$h^{0}(E)=d-n(g-1)$. It follows from (2.11) that $h^{0}(D_{E})\geq d-n(g-1)$.
Moreover, under the given hypotheses, $D_{E}$ is stable by [11, Theorem 2.1].
It is easy to check that $\mu(D_{E})\leq 2$. The fact that (2.13) is an
isomorphism follows from [21, Théorème 2-B-1] and [23, Proposition 2]. ∎

###  2.3. Small slope

When $d_{1}<2n_{1}$, Lemma 2.3 includes a complete answer to the question of
the non-emptiness of $B(n_{1},d_{1},k_{1})$. The references [6, 21] (see also
[5]) contain much further information about these BN loci, in particular that
there exists $E_{1}\in B(n_{1},d_{1},k_{1})$, taking one of a number of
special forms.

######  Theorem 2.6.

Let $C$ be a curve of genus $g\geq 2$. If $d_{1}<2n_{1}$ and
$B(n_{1},d_{1},k_{1})\neq\emptyset$, then there exists $E_{1}\in
B(n_{1},d_{1},k_{1})$ taking one of the following forms.

  * I

$d_{1}<n_{1}+g$. $E_{1}$ fits into an exact sequence

(2.14) $0\longrightarrow\mathcal{O}^{k_{1}}\longrightarrow
E_{1}\longrightarrow F\longrightarrow 0$

for some sheaf $F$.

  * II

$d_{1}=n_{1}+g\ell<2n_{1}$, $\ell\geq 1$. $E_{1}$ is the dual span of a stable
bundle $E$ of slope $>2g$ and is therefore given by an exact sequence

(2.15) $0\longrightarrow E^{*}\longrightarrow
H^{0}(E)^{*}\otimes\mathcal{O}\longrightarrow E_{1}\longrightarrow 0.$

Moreover, $h^{0}(E)=h^{0}(E_{1})=n_{1}+\ell$.

  * III

$d_{1}=n_{1}+g\ell+\ell^{\prime}<2n_{1}$ with $\ell>0$ and
$0<\ell^{\prime}<g$. There exist exact sequences

(2.16) $0\longrightarrow D_{E^{\prime}}^{*}\longrightarrow
H^{0}(E^{\prime})\otimes\mathcal{O}\longrightarrow E^{\prime}\longrightarrow
0$

with $E^{\prime}$ stable of rank $n_{1}+\ell^{\prime}$, degree $d_{1}$ and
$h^{0}(E^{\prime})=n_{1}+\ell+\ell^{\prime}$, and

(2.17) $0\longrightarrow\mathcal{O}^{\ell^{\prime}}\longrightarrow
E^{\prime}\longrightarrow E_{1}\longrightarrow 0.$

###### Proof.

I. By [6] (for $d_{1}\leq n_{1}$) and [21, Lemme 3-A-2] (for
$n_{1}<d_{1}<n_{1}+g$), the sections of $E_{1}$ are all carried by a subsheaf
$\mathcal{O}^{k_{1}}$. This gives (2.14)

II. We know that $B(n_{1},d_{1},k_{1})\neq\emptyset$ if and only if $d_{1}\geq
n_{1}+g(k_{1}-n_{1})$. We can therefore choose $E_{1}\in
B(n_{1},d_{1},n_{1}+\ell)$ and then $h^{0}(E_{1})=n_{1}+\ell$. By [21,
Théorème 2-B-1], $E_{1}=D_{E}$ for some stable bundle $E$ of rank $\ell$ and
degree $d_{1}$, hence of slope $>2g$. So $h^{0}(E)=n_{1}+\ell$ and we have the
exact sequence (2.15).

III. By [21, Théorème 2-B-1] (see Case II), there exists a stable bundle
$E^{\prime}$ of rank $n_{1}+\ell^{\prime}$ and degree $d_{1}$ with
$h^{0}(E^{\prime})=n_{1}+\ell+\ell^{\prime}$, fitting in an exact sequence
2.16. Now, as in the proof of [21, Théorème 3-B-1], we have an exact sequence
(2.17), with $E_{1}\in B(n_{1},d_{1},n_{1}+\ell)$. Since
$B(n_{1},d_{1},k_{1})\neq\emptyset$, we have (see [21])

$n_{1}+g\ell+\ell^{\prime}=d_{1}\geq n_{1}+g(k_{1}-n_{1}),$

hence $\ell\geq k_{1}-n_{1}$ and $E_{1}\in B(n_{1},d_{1},k_{1})$. ∎

######  Remark 2.7.

Theorem 2.6 can be extended to the case $d_{1}=2n_{1}$ (see [23]). We have two
cases.

  * IV

$C$ non-hyperelliptic. By the results of [23], we are in one of Cases I-III,
except that in Case II, we now have $\deg E=2n_{1}g$. There is one further
case, namely when $E_{1}=D_{K_{C}}\in B(g-1,2g-2,g)$.

  * V

$C$ hyperelliptic. In this case $B(n_{1},2n_{1},k_{1})\neq\emptyset$ if and
only if $k_{1}\leq n_{1}$ (see [23]). When this holds, there exists $E_{1}\in
B(n_{1},2n_{1},k_{1})$ which admits $\mathcal{O}^{k_{1}}$ as a subsheaf.

##  3\. A non-emptiness result for twisted BN loci

In this section we give an extended version of [15, Theorem 1.1], which we
need for comparison purposes.

######  Theorem 3.1.

Let $C$ be a smooth curve of genus $g\geq 2$ and $E_{2}$ any vector bundle of
rank $n_{2}$ and degree $d_{2}$ on $C$. Let $d_{0}$ and $k_{0}$ be integers
satisfying $\beta(1,d_{0},k_{0})(E_{2})\geq 1$, and suppose that $n_{1}\geq
2$. Then,

  * (i)

if $k\leq n_{1}k_{0}$ and $d_{1}\geq n_{1}d_{0}+1\ (\text{resp.},d_{1}\geq
n_{1}d_{0})$, the twisted BN locus $B(n_{1},d_{1},k)(E_{2})\
(\text{resp.},\widetilde{B}(n_{1},d_{1},k)(E_{2}))$ is non-empty and
$\beta(n_{1},d_{1},k)(E_{2})>(\text{resp.},\geq)\,1$;

  * (ii)

if $k_{1}:=k-n_{2}d_{1}+n_{1}d_{2}-n_{1}n_{2}(g-1)\leq n_{1}k_{0}$ and
$-d_{1}\geq n_{1}d_{0}+1\ (\text{resp.},-d_{1}\geq n_{1}d_{0})$, the twisted
BN locus $B(n_{1},d_{1},k)(K_{C}\otimes E_{2}^{*})\
(\text{resp.},\widetilde{B}(n_{1},d_{1},k)(K_{C}\otimes E_{2}^{*}))$ is non-
empty and

$\beta(n_{1},d_{1},k)(K_{C}\otimes E_{2}^{*})>(\text{resp.},\geq)\,1.$

###### Proof.

(i) If $k\leq 0$, then $B(n_{1},d_{1},k)(E_{2})=M_{1}$ and the result holda.
Otherwise, the non-emptiness of $B(n_{1},d_{1},k)(E_{2})$ (resp.,
$\widetilde{B}(n_{1},d_{1},k)(E_{2})$) is [15, Theorem 1.1]. The inequality
$\beta(n_{1},d_{1},k)(E_{2})>(\geq)1$ follows from the observation that

(3.1) $\beta(n_{1},n_{1}d_{0},n_{1}k_{0})(E_{2})\geq
1\Longleftrightarrow\beta(1,d_{0},k_{0})(E_{2})\geq 1$

and the fact that $\beta(n_{1},d_{1},k)(E_{2})$ is a strictly increasing
function of $d_{1}$ and a decreasing function of $k$.

(ii) Under the stated hypotheses, in the case $-d_{1}\geq n_{1}d_{0}+1$, (i)
implies that $B(n_{1},-d_{1},k_{1})(E_{2})\neq\emptyset$. By Serre duality,

$B(n_{1},d_{1},k)(K_{C}\otimes E_{2}^{*})\cong B(n_{1},-d_{1},k_{1})(E_{2}).$

So $B(n_{1},d_{1},k)(K_{C}\otimes E_{2}^{*})\neq\emptyset$. Moreover, as in
the proof of (i),

(3.2) $\beta(n_{1},-d_{1},k_{1})(E_{2})>1$

and

$\beta(n_{1},d_{1},k)(K_{C}\otimes
E_{2}^{*})=n_{1}^{2}(g-1)+1-kk_{1}=\beta(n_{1},-d_{1},k_{1})(E_{2}).$

The same proof applies in the case $-d_{1}\geq n_{1}d_{0}$ if we replace $>$
in (3.2) by $\geq$ and $B$ with $\widetilde{B}$ throughout. ∎

######  Corollary 3.2.

Under the hypotheses of Theorem 3.1, the twisted BN locus
$B^{k}(\mathcal{U}_{!},\mathcal{U}_{2})\
(\text{resp.},\widetilde{B}(\mathcal{U}_{1},\mathcal{U}_{2}))$ is non-empty.
Moreover

$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})>n_{2}^{2}(g-1)+2\ >0\
(\mbox{resp.,}\geq n_{2}^{2}(g-1)+2>0).$

###### Proof.

In the theorem, we can take $E_{2}$ to be stable; the required non-emptiness
then follows (see Remark 1.1). The inequality for
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ follows from the fact that

$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})=\dim
M_{2}+\beta(n_{1},d_{1},k)(E_{2})$

(see (1.3) and (1.6)). ∎

######  Remark 3.3.

(i) As indicated in the introduction, the fact that the BN numbers are
negative will be crucial for us in establishing that many of the examples
constructed in this paper are new.

(ii) (3.1) makes Theorem 3.1(i) the precise analogue of the theorem of
Teixidor i Bigas [25] for twisted BN loci $B(n_{1},d_{1},k)(E_{2})$, at least
on a general curve; the proof in [15] depends on Mercat’s construction [22]
and holds for any curve.

(iii) If we replace $E_{2}$ by $E_{2}\otimes L$, where $L$ is a line bundle of
degree $\ell$, and simultaneously replace $d_{0}$ by $d_{0}-\ell$ and $d_{1}$
by $d_{1}-n_{1}\ell$ in Theorem 3.1(i), the BN numbers and the isomorphism
classes of the BN loci are unchanged. This applies also to Theorem 3.1(ii)
except that now $d_{1}$ must be replaced by $d_{1}+n_{1}\ell$.

##  4\. Examples with $k=k_{1}k_{2}$

In this section, we consider the following situation. Suppose that
$B(n_{i},d_{i},k_{i})\neq\emptyset$ for $i=1,2$ and that there exist $E_{i}\in
B(n_{i},d_{i},k_{i})$ such that $h^{0}(E_{1}\otimes E_{2})\geq k_{1}k_{2}$.
When $n_{2}=1$, this is exactly the situation considered in [8]. Here,
however, we assume $n_{i}\geq 2$. One scenario to which this applies is when
$E_{1}=D_{E}$.

###  4.1. The construction

######  Proposition 4.1.

Let $C$ be a smooth curve of genus $g$. Suppose that $E$ is a generated stable
(resp., semistable) bundle of rank $n$ and degree $d_{1}$ and that
$h^{0}(E)=k_{1}=n+n_{1}$. Suppose further that $D_{E}$ is stable (resp.,
semistable) and that $E_{2}\in B(n_{2},d_{2},k_{2})\
(\text{resp.},[E_{2}]\in\widetilde{B}(n_{2},d_{2},k_{2}))$ with
$\mu(E_{2})<\mu(E)$. Then $h^{0}(D_{E}\otimes E_{2})\geq k_{1}k_{2}\
(\text{resp.},h^{0}(D_{E}\otimes\text{gr}E_{2})\geq k_{1}k_{2})$. In
particular, $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset\
(\text{resp.},\widetilde{B}^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset)$.

###### Proof.

Consider the sequence (2.11) and suppose first that we are in the stable case.
Note that $D_{E}\in B(n_{1},d_{1},k_{1})$, since $D_{E}$ is stable by
hypothesis and $h^{0}(E^{*})=0$. Note also that $h^{0}(E^{*}\otimes E_{2})=0$
since $E^{*}\otimes E_{2}$ is semistable of negative slope. Tensoring (2.11)
by $E_{2}$ and taking global sections, we obtain

$h^{0}(D_{E}\otimes E_{2})\geq h^{0}(E)\cdot h^{0}(E_{2})\geq k_{1}k_{2}.$

The semistable case proceeds similarly. ∎

Note that, by (1.6), we have

(4.1)
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})=(n_{1}^{2}+n_{2}^{2})(g-1)+2-k(k-\chi).$

We have also

(4.2)
$\beta(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)=n_{1}^{2}n_{2}^{2}(g-1)+1-k(k-\chi).$

######  Lemma 4.2.

Let $\mu_{i}$, $\lambda_{i}$ be positive rational numbers for $i=1,2$ and
suppose that $d_{i}=n_{i}\mu_{i}$, $k_{i}=n_{i}\lambda_{i}$, $k=k_{1}k_{2}$.
If

(4.3) $\mu_{1}+\mu_{2}<\lambda_{1}\lambda_{2}+g-1,$

then, for fixed $\mu_{i}$, $\lambda_{i}$ and sufficiently large $n_{1}$,
$n_{2}$, $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$.

###### Proof.

Since $k=k_{1}k_{2}$, (4.1) is equivalent to

(4.4)
$\frac{\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})}{n_{1}^{2}n_{2}^{2}}=\left(\frac{1}{n_{2}^{2}}+\frac{1}{n_{1}^{2}}\right)(g-1)+\frac{2}{n_{1}^{2}n_{2}^{2}}-\lambda_{1}\lambda_{2}(\lambda_{1}\lambda_{2}-(\mu_{1}+\mu_{2})+g-1).$

The result follows. ∎

######  Theorem 4.3.

Suppose that $C$ is a smooth curve of genus $g\geq 2$ and that $\mu_{i}$,
$\lambda_{i}$ ($i=1,2$) are positive rational numbers, $\mu_{1}<2$ and
$\mu_{2}\leq 2g$.

  * (i)

If $B(n_{i},d_{i},k_{i})$ are non-empty BN loci with $n_{i}\geq 2$ supported
by $(\mu_{i},\lambda_{i})$ and $k=k_{1}k_{2}$, then
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$.

  * (ii)

If $\widetilde{B}(n_{i},d_{i},k_{i})$ are non-empty BN loci with $n_{i}\geq 2$
supported by $(\mu_{i},\lambda_{i})$ and $k=k_{1}k_{2}$, then
$\widetilde{B}^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$ and

$\widetilde{B}(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)\neq\emptyset.$

###### Proof.

(i) Let $E_{2}\in B(n_{2},d_{2},k_{2})$ with $d_{2}\leq 2n_{2}g$. We prove
that there exists $E_{1}\in B(n_{1},d_{1},k_{1})$ such that
$h^{0}(E_{1}\otimes E_{2})\geq k_{1}k_{2}$. The proof is the same as in [8]
with $L$ replaced by $E_{2}$. We need to consider cases I-III of Theorem 2.6.

I. $\mathcal{O}^{k_{1}}\otimes E_{2}$ is a subsheaf of $E_{1}\otimes E_{2}$.
It follows at once that $k_{1}k_{2}\leq h^{0}(E_{1}\otimes E_{2}$).

II. in (2.15), $E_{1}\cong D_{E}$ and $\mu(E_{2})\leq 2g<\mu(E)$. The result
follows from Proposition 4.1.

III. Tensoring (2.16) and (2.17) by $E_{2}$ and noting that
$D_{E^{\prime}}^{*}\otimes E_{2}$ is semistable of negative slope, we obtain
$h^{0}(E^{\prime}\otimes E_{2})\geq h^{0}(E^{\prime})h^{0}(E_{2})$ and hence

$h^{0}(E_{1}\otimes
E_{2})\geq(h^{0}(E^{\prime})-\ell^{\prime})h^{0}(E_{2})=(n_{1}+\ell)h^{0}(E_{2})\geq
k_{1}k_{2}.$

(ii) This follows in the same way as (i), together with the fact that the
tensor product of semistable bundles is semistable. ∎

######  Remark 4.4.

If $C$ is non-hyperelliptic, the theorem holds also under the hypotheses
$\mu_{1}\leq 2$, $\mu_{2}<2g$ (see Case IV in Remark 2.7).

###  4.2. Examples

Using Lemmas 2.1 and 2.3, it is easy to find many cases as in Theorem 4.3
where (4.3) is also satisfied, thus yielding examples of non-empty BN loci
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ with negative BN number. These cannot
be obtained from Theorem 3.1.

######  Example 4.5.

Suppose that $\mu_{i}\leq\frac{1}{2}$ for $i=1,2$. From [6] (see Lemma 2.3 and
Figure 1), we can take $\lambda_{i}$ to be any positive rational numbers with
$\lambda_{i}\leq 1-\frac{1}{g}(1-\mu_{i})$. Then, Lemma 4.2 and Theorem 4.3
apply, giving $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$ and
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$ for sufficiently large $n_{1}$,
$n_{2}$, even when $g=2$. More generally, we can take $\mu_{1}\leq 1$ and
$\mu_{2}\leq g-2$ for $g\geq 3$, or $\mu_{1}<2$ and $\mu_{2}\leq g-3$ for
$g\geq 4$, with $\lambda_{i}$ any positive rational numbers for which there
exist non-empty BN loci $B(n_{i},d_{i},k_{i})$ supported by
$(\mu_{i},\lambda_{i})$.

It is even possible to find examples with $n_{1}=n_{2}=2$.

######  Example 4.6.

Suppose that $n_{1}=n_{2}=2$. Then, under the hypotheses of Theorem 4.3,
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$ if and only if

(4.5)
$\left(\frac{1}{2}-\lambda_{1}\lambda_{2}\right)(g-1)+\frac{1}{8}-\lambda_{1}\lambda_{2}(\lambda_{1}\lambda_{2}-(\mu_{1}+\mu_{2}))<0$

(see (4.4)). We can take $B(n_{i},d_{i},k_{i})=B(2,3,2)$ for $i=1,2$; this is
non-empty by either Lemma 2.1 or Lemma 2.3. (4.5) now becomes

$-\frac{1}{2}(g-1)+\frac{1}{8}+2<0,$

which is true for $g\geq 6$. Here, $B(2,3,2)$ is the only candidate for
$B(2,d_{1},k_{1})$ for applying Theorem 4.3, but, for $i=2$, we could take any
non-empty $B(2,d_{2},k_{2})$ instead of $B(2,3,2)$; once we have fixed $d_{2}$
with $d_{2}\leq 4g$ and $k_{2}\geq 2$, (4.5) will hold for sufficiently large
$g$.

###  4.3. A new region in the BN map

Substituting $k=k_{1}k_{2}$ in (4.2) and dividing by $n_{1}^{2}n_{2}^{2}$, we
have

(4.6)
$\beta:=\frac{\beta(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)}{n_{1}^{2}n_{2}^{2}}=\\\
=g-1+\frac{1}{n_{1}^{2}n_{2}^{2}}-\lambda_{1}\lambda_{2}(\lambda_{1}\lambda_{2}-(\mu_{1}+\mu_{2})+g-1).$

It is an interesting question as to whether $\beta$ can ever be negative under
the hypotheses of Theorem 4.3 and the assumption that
$\beta(n_{2},d_{2},k_{2})\geq 1$. One can see easily that $\beta>0$ if
$\lambda_{1}\leq 1$, but other cases are not so straightforward. Independently
of this, one can ask whether the theorem can give rise to new examples of non-
empty BN loci in $M(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k_{1}k_{2})$. Our next
result shows that this is indeed the case.

We define a new region of the BN map as follows. Let

$R:=\left\\{(\mu_{1},\mu_{2},\lambda_{1},\lambda_{2})\in{\mathbb{Q}}^{4}\,\left|\begin{array}[]{c}\,0<\mu_{1}<2,0<\mu_{2}\leq
2g-2-\mu_{1},\\\
\lambda_{i}\leq\max\\{f_{g}(\mu_{i}),t_{g}(\mu_{i})\\}\end{array}\right.\right\\}$

and let $h:R\to{\mathbb{Q}}^{2}$ be defined by

$h(\mu_{1},\mu_{2},\lambda_{1},\lambda_{2})=(\mu_{1}+\mu_{2},\lambda_{1}\lambda_{2}).$

######  Definition 4.7.

The region BPN in the BN map is defined to be the union of
$\operatorname{Im}h$ and its Serre dual.

######  Theorem 4.8.

Let $C$ be a smooth curve of genus $g\geq 2$ and $n_{i}\geq 2$.

  * (i)

If
$(\mu,\lambda)=h(\mu_{1},\mu_{2},\lambda_{1},\lambda_{2})\in\operatorname{BPN}$,
then $(\mu_{1}+\mu_{2},\lambda_{1}\lambda_{2})$ supports a BN locus
$\widetilde{B}(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)$ with $k=k_{1}k_{2}$ for
infinitely many values of $n_{1}$, $n_{2}$;

  * (ii)

If $g\geq 5$, $\operatorname{BPN}$ is not contained in $\operatorname{T\cup
BMNO}$.

###### Proof.

(i) By Lemmas 2.1 and 2.3, $(\mu_{i},\lambda_{i})$ supports the BN locus
$\widetilde{B}(n_{i},d_{i},k_{i})$ ($i=1,2$) for infinitely many values of
$n_{i}$. The result follows from Theorem 4.3(ii).

(ii) Suppose that $2<\mu<4$. We can write $\mu=\mu_{1}+\mu_{2}$ with
$1<\mu_{i}<2$ for $i=1,2$. Now, if $\lambda_{i}=1+\frac{1}{g}(\mu_{i}-1)$,
then, by [21] (see also Lemma 2.3), $(\mu_{i},\lambda_{i}$) is in the region
BMNO and $(\mu_{i},\lambda_{i})$ supports $B(n_{i},d_{i},k_{i})$ for any
allowable $n_{i}$. So, by Theorem 4.3(ii),
$\widetilde{B}(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)\neq\emptyset$ when
$k=k_{1}k_{2}$. On the other hand,

(4.7) $\lambda_{1}\lambda_{2}>1+\frac{1}{g}(\mu_{1}+\mu_{2}-2)\geq
1+\frac{1}{g}(\mu_{1}+\mu_{2}-\left\lceil\mu_{1}+\mu_{2}\right\rceil+1).$

It follows from Lemma 2.3 and (2.9) that
$(\mu_{1}+\mu_{2},\lambda_{1}\lambda_{2})$ is not in the region BMNO if
$\widehat{\eta}(2)\geq 5$, in other words, if $g\geq 7$. It is clear also that
$(\mu_{1}+\mu_{2},\lambda_{1}\lambda_{2})$ is not in the region T, whose upper
boundary is given by $\lambda=1$ in this range. In fact, our construction also
gives new points for $g=5$ and $g=6$. ∎

######  Remark 4.9.

The construction in the proof of Theorem 4.8 shows that, for $g\geq 7$, the
top boundary of BPN in the range $2<\mu<4$ is given by
$\lambda=1+\frac{1}{g}(\mu-2)+\frac{1}{g^{2}}\left(\frac{\mu-2}{2}\right)^{2}$,
whereas the top boundary of BMNO is given by $\lambda=1+\frac{1}{g}(\mu-2)$
for $2<\mu<3$ and by $\lambda=1+\frac{1}{g}(\mu-3)$ for $3<\mu<4$. At the
point $(2,1)$, the line $\lambda=1+\frac{1}{g}(\mu-2)$ is the tangent to the
parabola
$\lambda=1+\frac{1}{g}(\mu-2)+\frac{1}{g^{2}}\left(\frac{\mu-2}{2}\right)^{2}$.
For $g=10$, BPN is illustrated in Figure 2, which extends to the range
$4<\mu<5$. Figure 2 also shows an expanded version of the BN map close to
$\mu=3$.

######  Remark 4.10.

(i) BPN is a new region in the BN map (see Figure 2), whereas the papers [17,
18] give examples for isolated values of $\mu$, some of which extend to higher
values of $g$. When $g=4$, on the other hand, there is an additional grey area
of the BN map (see [16, section 7]), which contains the new part of the region
BPN in the range $2<\mu<3$.

(ii) At least for Petri curves, some of the non-empty BN loci

$\widetilde{B}(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)\subset\widetilde{M}(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2})$

constructed in Theorem 4.8 are new. It is possible to choose the ranks and
degrees so that $\gcd(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2})=1$; for example, if
$g$ is odd, one can take

$(n_{1},d_{1},k_{1})=(g+1,2g+1,g+2),\ (n_{2},d_{2},k_{2})=(g+2,2g+2,g+3).$

Then $B(n_{1}n_{2},n_{2}d_{1}+n_{1}d_{2},k)\neq\emptyset$.

(iii) One can check that, in the framework of Theorem 4.8, the examples of
twisted BN loci with negative expected dimension constructed in [15, Section
9] give rise to points of T. It follows that our examples are different.

##  5\. Construction using kernel bundles

###  5.1. The construction

The essential ingredient in proving Theorem 3.1 is Mercat’s construction [22],
while Theorem 4.3 depends on the constructions of [6, 21]. Another way of
constructing non-empty twisted BN loci is described in [10] and uses kernel
bundles. The following proposition is contained in [10, Remark 5.5] and plays
an essential rôle in the proof of [10, Theorems 5.6 and 5.7]. We include a
proof for the convenience of the reader.

######  Proposition 5.1.

Suppose that $E_{1}$ has rank $n_{1}$ and degree $d_{1}$ with
$h^{0}(E_{1})\geq k_{1}$ and that $E$ is as in (2.10). If $h^{1}(E_{1}\otimes
E)=0$ and

(5.1) $k\leq(d-n(g-1))(k_{1}-n_{1})-nd_{1},$

then $h^{0}(E_{1}\otimes D_{E}^{*})\geq k$.

###### Proof.

We have $h^{0}(E_{1}\otimes E)=nd_{1}+n_{1}d-nn_{1}(g-1)$ and $h^{0}(E)\geq
d-n(g-1)$. So, tensoring (2.10) by $E_{1}$ and taking global sections,

$h^{0}(E_{1}\otimes D_{E}^{*})\geq(d-n(g-1))k_{1}-nd_{1}-n_{1}d+nn_{1}(g-1).$

Simplifying and comparing with (5.1), this gives the result. ∎

We have the following immediate corollary (see Remark 1.1).

######  Corollary 5.2.

Suppose that the hypotheses of Proposition 5.1 are satisfied.

  * (i)

If $E_{1}$ is stable $(\text{resp., semistable})$, then
$B(n_{1},d_{1},k)(D_{E}^{*})\neq\emptyset$
$(\text{resp.},\widetilde{B}(n_{1},d_{1},k)(D_{E}^{*})\neq\emptyset)$.

  * (ii)

If $D_{E}^{*}$ is stable $(\text{resp., semistable})$, then
$B(h^{0}(E)-n,-d,k)(E_{1})\neq\emptyset\
(\text{resp.},\widetilde{B}(h^{0}(E)-n,-d,k)(E_{1})\neq\emptyset)$.

  * (iii)

If both $E_{1}$ and $D_{E}^{*}$ are stable $(\text{resp., semistable})$ and
$(n_{2},d_{2})=(h^{0}(E)-n,-d)$, then
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset\
(\text{resp.},\widetilde{B}^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset)$.

Our object here is to use Corollary 5.2(iii) to obtain new non-empty examples
of twisted BN loci. We assume that $n_{1}\geq 2$. In order for (5.1) to yield
positive values of $k$ when $d>n(g-1)$, which will always be the case, we
clearly require $k_{1}>n_{1}$.

The key point we require in order to apply Corollary 5.2(iii) is that
$D_{E}^{*}$ and $E_{1}$ are stable. We have already discussed the existence of
stable $E_{1}$ in subsection 2.1 and the stability of $D_{E}$ in subsection
2.2.

We can now state and prove a general theorem, which includes Theorem 1.3.

######  Theorem 5.3.

Suppose that $C$ is a smooth curve of genus $g\geq 2$, $n_{1}\geq 2$,
$k_{1}>n_{1}$ and either $d>2ng$ or $d=2ng$ and $C$ is non-hyperelliptic.
Suppose further that $B(n_{1},d_{1},k_{1})\neq\emptyset$ and that (5.1) holds.
Let $(n_{2},d_{2})=(d-ng,-d)$ and suppose that $S$ is defined as in (2.12).
Then $S\neq\emptyset$ and the morphism

(5.2) $B(n_{1},d_{1},k_{1})\times S\longrightarrow M_{1}\times
M_{2}:(E_{1},E)\mapsto(E_{1},D_{E}^{*})$

is injective and has image contained in
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$; in particular,

$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset.$

If, in addition, $k=d(k_{1}-n_{1})-e$, where

(5.3) $e\geq n(g-1)(k_{1}-n_{1})+nd_{1},$

and

(5.4) $d_{1}<k_{1}+n_{1}(g-1)-\frac{g-1}{k_{1}-n_{1}},$

then, for any fixed values of $n_{1}$, $d_{1}$, $k_{1}$, $n$ and $e$
satisfying (5.3), and $d\gg 0$,

(5.5) $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0.$

###### Proof.

The first part follows from Theorem 2.5, Proposition 5.1 and Corollary
5.2(iii). We need to note that $h^{1}(E_{1}\otimes E)=0$ since $E_{1}\otimes
E$ is semistable of slope greater than $2g$.

For the second part, we have

(5.6) $\displaystyle\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$
$\displaystyle=$ $\displaystyle n_{1}^{2}(g-1)+1+(d-ng)^{2}(g-1)+1$
$\displaystyle-(d(k_{1}-n_{1})-e)(d(k_{1}-n_{1})-e-\chi),$

where, by (1.2),

(5.7) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle(d-ng)d_{1}-n_{1}d-n_{1}(d-ng)(g-1)$ $\displaystyle=$
$\displaystyle(d-ng)(d_{1}-n_{1}g)-nn_{1}g.$

The right-hand side of (5.6) is a quadratic in $d$, with leading coefficient

(5.8) $g-1-(k_{1}-n_{1})(k_{1}-n_{1}-d_{1}+n_{1}g).$

So $\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$ for $d\gg 0$ provided that

$d_{1}<k_{1}+n_{1}(g-1)-\frac{g-1}{k_{1}-n_{1}}.$

∎

######  Remark 5.4.

(i) Under the hypotheses of Theorem 5.3, including the inequalities (5.3) and
(5.4), we deduce that the non-emptiness of
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ cannot be obtained from Theorem 3.1.
In fact, there are several choices we can make in attempting to apply Theorem
3.1 to this situation. We can take $E_{2}$ to be any of $E_{1}\otimes L$,
$D_{E}^{*}\otimes L$, $K_{C}\otimes E_{1}^{*}\otimes L$ and $K_{C}\otimes
D_{E}\otimes L$, where $L$ is any line bundle, with the appropriate choice of
$(n_{1},d_{1})$ in each case. In all cases, we obtain
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\geq 1$. In the absence of (5.5),
we would have to work through all these cases separately.

(ii) If we assume only that $\widetilde{B}(n_{1},d_{1},k_{1})\neq\emptyset$
and $d\geq 2ng$ with $C$ any smooth curve of genus $g\geq 2$, we obtain
$\widetilde{B}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$. Also, if (5.4)
holds and $d\gg 0$, then the non-emptiness of
$\widetilde{B}(\mathcal{U}_{1},\mathcal{U}_{2})$ cannot be obtained from
Theorem 3.1.

###  5.2. Examples

To show that Theorem 5.3 provides the examples we are seeking, we need to show
that the conditions $B(n_{1},d_{1},k_{1})\neq\emptyset$ and (5.4) are
compatible. For this, we need more precise statements concerning the possible
values of $d_{1}$.

######  Corollary 5.5.

Suppose that $C$ is a smooth curve of genus $g\geq 3$, $n_{1}\geq 2$,
$k_{1}>n_{1}$ and $(n_{2},d_{2})=(d-ng,-d)$ with either $d>2ng$ or $d=2ng$ and
$C$ non-hyperelliptic. Suppose further that

(5.9) $k_{1}+n_{1}(g-1)-n_{1}\left\lfloor\frac{g-1}{\lceil
k_{1}/n_{1}\rceil}\right\rfloor\leq
d_{1}<k_{1}+n_{1}(g-1)-\frac{g-1}{k_{1}-n_{1}}$

and that $d_{1}$ is not divisible by $n_{1}$. Then,
$B(n_{1},d_{1},k_{1})\neq\emptyset$. Moreover, for any fixed value of $e$
satisfying (5.3) and $d\gg 0$,
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$, but
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$. In particular,
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ is of negative expected dimension and
the non-emptiness of $B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})$ cannot be
obtained from Theorem 3.1.

###### Proof.

The fact that $B(n_{1},d_{1},k_{1})\neq\emptyset$ follows from Remark 2.2.
Theorem 5.3 then implies that
$B^{k}(\mathcal{U}_{1},\mathcal{U}_{2})\neq\emptyset$ and that
$\beta^{k}(\mathcal{U}_{1},\mathcal{U}_{2})<0$ when (5.3) holds and $d\gg 0$.
Hence, the non-emptiness cannot be obtained from Theorem 3.1 (see Remark
3.3(i)). ∎

To obtain examples from this corollary, we need, in particular, to prove the
existence of $d_{1}$ satisfying (5.9). This is equivalent to showing that

(5.10) $n_{1}\left\lfloor\frac{g-1}{\lceil
k_{1}/n_{1}\rceil}\right\rfloor>\frac{g-1}{k_{1}-n_{1}}$

In fact (5.10) cannot hold if $g=2$ (which is why we have assumed $g\geq 3$ in
Corollary 5.5) or, more generally, if $k_{1}\geq n_{1}(g-1)+1$, since, in that
case, the left-hand side of (5.10) is $0$. We therefore assume that

(5.11) $n_{1}<k_{1}\leq n_{1}(g-1)$

######  Example 5.6.

Suppose that $g=3$ and (5.11) holds. Then (5.10) holds if and only
$n_{1}(k_{1}-n_{1})>2$, in other words, either $n_{1}\geq 3$ or $n_{1}=2$ and
$k_{1}=4$. In the latter case, the only possible value for $d_{1}$ is
$d_{1}=6$. This is not permitted by the hypotheses of Corollary 5.5 and, in
fact $B(2,6,4)=\emptyset$. For $n_{1}\geq 3$, there do exist solutions of
(5.9) for which $d_{1}$ is not divisible by $n_{1}$. In fact, if
$k_{1}=n_{1}+2$, (5.9) gives $2n_{1}+2\leq d_{1}<3n_{1}+1$, giving $n_{1}-2$
values of $d_{1}$ not divisible by $n_{1}$. If $n_{1}+3\leq k_{1}\leq 2n_{1}$,
there are $n_{1}-1$ permitted values of $d_{1}$ in the range $k_{1}+n_{1}\leq
d_{1}<k_{1}+2n_{1}$. A similar analysis can be carried out for $g\geq 4$; in
this case, in particular, $B(2,4g-5,2g-2)\neq\emptyset$ and (5.9) is
satisfied, so examples exist for $n_{1}=2$, as well as for larger $n_{1}$.

######  Remark 5.7.

When $k_{1}=n_{1}+1$, there are additional cases in which the non-emptiness of
$B(n_{1},d_{1},k_{1})$ is known. This is related to the extended dual span
construction, which involves the stability of bundles $D_{L,V}$, where $L$ is
a line bundle and $V$ is a subspace of $H^{0}(L)$ which generates $L$.
Applications of this will be considered in a paper currently in preparation
[9].

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POT), we train a support vector machine with the radial basis kernel
(parameter $\sigma^{2}=1$) using the transformed source domain. In the Figure
8 we report the accuracy on the target domain using 20,000 test examples. All
hyperparameters are set to be equal.

Here we implement and compare Domain Adaptation with OT to Domain Adaptation
with POT, which is calculated with two different algorithms Sinkhorn
(infeasible rounding) and APDAGD (with our novel Round-POT). In (Courty et al.
2017a), the authors presented an OT-based method to transform a source domain
to a reference target domain such that the their densities are approximately
equal. We present a binary classification setting involving the ”moons”
dataset. The source dataset is displayed as colored scatter points. The target
dataset comes from the same distribution, but the points go to a rotation of
60 degrees (black scatter points), representing a domain covariate shift.
Additional experimental details are further described in Further Experiments
Setup Details section in Appendix. To summarize the results, POT offers much
more practicality as it does not require normalizing both marginals to have a
sum of 1 in contrast to OT. More importantly, POT avoids bad matchings to
outliers due to the flexibility in choosing the amount of mass transported,
resulting in a higher accuracy than OT. Furthermore, linking back to Remark 4,
the infeasibility Sinkhorn degrades the practical performance of POT for
Domain Adaption and consequently leads to a worse accuracy compared APDAGD.
This is because the optimal mapping between the source and target domains
requires the exact fraction of mass to be transported which APDAGD fulfills
practically in contrast to Sinkhorn.

Figure 8: Domain adaptation with OT and with POT. POT offers a flexibility in
choosing the mass transported and helps avoid the need to normalize both
marginals to have a sum of $1$, resulting in better complexity than OT. APDAGD
demonstrates better accuracy in the novel domain than Sinkhorn, which suffers
from infeasibility.

### Synthetic Data

Figure 9: Convergence of APDAGD on a synthetic setting.

In this section we design a POT setting and evaluate the convergence of
APDAGD. The two marginals are unnormalized Gaussian mixtures. We then
discretize them into histograms of $n=100$ bins each and perform normalization
so that $\left\lVert\mathbf{r}\right\rVert_{1}=5$,
$\left\lVert\mathbf{c}\right\rVert_{1}=3$. The total transported mass is set
to
$s=0.9\min\\{\left\lVert\mathbf{r}\right\rVert_{1},\left\lVert\mathbf{c}\right\rVert_{1}\\}=2.7$,
and the cost matrix represents squared Euclidean distances between every pair
of bins in the source and target histograms. In other words,
$C_{i,j}=(i-j)^{2}$ for all $i,j=1,\ldots,n$. Finally, we divide $\mathbf{C}$
by its maximum entry to have
$\left\lVert\mathbf{C}\right\rVert_{\text{max}}=1$.

We use cvxpy with the GUROBI backend to solve the linear problem (2), and
denote by $f^{*}$ the optimal objective value. This will be used to calculate
the primal optimality gap.

The approximate POT solution is solved using APDAGD, where we set the additive
suboptimality error $\varepsilon$ to $10^{-3}$. We run APDAGD for 10,000
iterations and keep track of two errors. The first error is constraint
violation, equal to
$\left\lVert\mathbf{X}\mathbf{1}+\mathbf{p}-\mathbf{r}\right\rVert_{1}+\left\lVert\mathbf{X}^{\top}\mathbf{1}+\mathbf{q}-\mathbf{c}\right\rVert_{1}+\lvert\mathbf{1}^{\top}\mathbf{X}\mathbf{1}-s\rvert$
where $\mathbf{X}$, $\mathbf{p}$ and $\mathbf{q}$ are part of the solution
$\mathbf{x}_{k}$ found after each iteration. The second error is the
optimality gap for the primal objective. To calculate this, we need to round
the solution so that it becomes primal feasible. We use our rounding algorithm
described in Rounding Algorithm Section to round $\mathbf{X}$, $\mathbf{p}$,
and $\mathbf{q}$ after each iteration, giving us $\bar{\mathbf{X}}$,
$\bar{\mathbf{p}}$ and $\bar{\mathbf{q}}$. The primal gap then is
$\langle\mathbf{C},\bar{\mathbf{X}}\rangle-f^{*}$.

### Scalability of APDAGD

Given the complexity claims in the paper, it is important to show how running
time scales with $n$, the number of supports in each marginal. In the below
Figure 10, we show the wall-clock time of running APDAGD to solve the POT
problem between two Gaussian mixtures (shown in the end of the appendix). We
fix the error tolerance to $\varepsilon=10^{-1}$ and vary $n$ in the set
$\\{10,30,100,300,1000,3000,10000\\}$. The figure plots wall-clock time in
seconds against $n$. The expected slope of the best-fit line between
$\log(\text{time})$ and $\log(n)$ should be at most 2.5, as the complexity is
$\widetilde{O}(n^{2.5}/\varepsilon)$. We find that the slope is $2.19$, which
is consistent with the upper bound. Furthermore, the correlation is
statistically significant.

Figure 10: Wall-clock time of solving POT with APDAGD against $n$


# An ADMM-based Coordination and Control Strategy for PV and Storage to
Dispatch Stochastic Prosumers: Theory and Experimental Validation

Rahul Gupta1, Fabrizio Sossan1, Enrica Scolari1, Emil Namor1, Luca Fabietti2,
Colin Jones2, Mario Paolone1
1Distributed Electrical Systems Laboratory, 2Automatic Control Laboratory
EPFL, Lausanne, Switzerland

###### Abstract

This paper describes a two-layer control and coordination framework for
distributed energy resources. The lower layer is a real-time model predictive
control (MPC) executed at 10 s resolution to achieve fine tuning of a given
energy set-point. The upper layer is a slower MPC coordination mechanism based
on distributed optimization, and solved with the alternating direction method
of multipliers (ADMM) at 5 minutes resolution. It is needed to coordinate the
power flow among the controllable resources such that enough power is
available in real-time to achieve a pre-established energy trajectory in the
long term. Although the formulation is generic, it is developed for the case
of a battery system and a curtailable PV facility to dispatch stochastic
prosumption according to a trajectory at 5 minutes resolution established the
day before the operation. The proposed method is experimentally validated in a
real-life setup to dispatch the operation of a building with rooftop PV
generation (i.e., 101 kW average load, 350 kW peak demand, 82 kW peak PV
generation) by controlling a 560 kWh/720 kVA battery and a 13 kW peak
curtailable PV facility.

###### Index Terms:

Distributed control, storage, Photovoltaic (PV)

##  I Introduction

Due to the displacement of conventional generation in favor of production from
renewables, decentralized control schemes for distributed energy resources
(DERs) have gained considerable attention in the recent literature to support
grid operation. Control frameworks for DERs have been focusing on achieving
targets at both the local level, e.g., voltage control and congestion
managements[1, 2, 3], and system level, e.g. primary and secondary frequency
regulation[4, 5, 6, 7]. Other reported methods proposed open-loop control of
DERs by broadcasting electricity price signals, as in [8], and virtual power
plant strategies to enable control by explicit power set-points[9, 10]. A
concept which has emerged in the recent literature is dispatching the active
power flows of stochastic resources according to a trajectory established
before the operation thanks to controlling energy storage systems, as in [11,
12], in combination with flexible demand[13, 14, 15, 16], and also accounting
for grid constraints [17].

The advantage of dispatching traditional stochastic resources is that it
inherently reduces the amount of regulating power required to operate the grid
without the necessity of coordination between the local system and grid
operators. With respect to control strategies for virtual power plants, which
normally consists in myopically tracking a power set-point, the dispatch
problem as defined in [11] implicitly includes an energy management policy
that does represent a key feature in the context of managing the flexibility
of energy storage systems.

In this paper, we describe the formulation and experimental validation of a
control framework to coordinate the operation of heterogeneous DERs to
dispatch the active power flow at the grid connection point (GCP) of an active
distribution network (ADN) according to a trajectory, called dispatch plan,
determined the day before the operation. The framework consists in two
algorithmic layers running at different paces. The lower layer, executed at 10
s resolution, has the objective of pursuing an accurate tracking of the
dispatch plan at the GCP. The upper layer, executed once each 5 minutes and
implemented through distributed optimization, has the objective of
coordinating available neighboring DERs such that enough power capacity is
available in real-time to achieve a successful dispatch. In the existing
literature, distributed optimization has been applied to meet manifold
objectives such as solving optimal power flow problems [18], optimal dispatch
of photovoltaic (PV) inverters in residential distribution networks [19] and
demand response strategy with electric vehicles (EVs) [20]. Unlike centralized
methods, it is highly scalable, an appealing feature in the context of rapidly
growing distributed energy networks, and it exhibits better privacy and
security properties compared to centralized schemes. Controlling and
coordinating multiple units in real-time can be effectively achieved through
distributed methods, thanks to the Douglas-Rachford splitting [21] which
divides the problem into small objectives and solve it through an aggregator.
For distributed methods, on the contrary to dual decomposition that employs
the subgradient projection technique with quite slow convergence, the Douglas-
Rachford splitting technique is faster [22] and efficient even when applied to
non-linear objectives. In this paper, the alternating direction method of
multipliers (ADMM)[23], a special case of Douglas-Rachford splitting, is
applied to solve the distributed optimization problem. The proposed
algorithmic toolchain is tested in a real-life experimental setup to dispatch
the operation of a group of stochastic prosumers by controlling a battery
energy storage system (BESS) and a curtailable rooftop PV facility.

A similar approach was proposed in [13] to control the operation of a BESS and
a building with flexible demand (electric space heating). With respect to
[13], in this paper we apply ADMM, which represents more agnostic formulation
since all DERs share the same information.

The paper is organized as follows: Section II states the problem we intend to
solve, Section III describes the formulation of the proposed framework,
Section V presents the experimental results and Section VI summarizes the
contributions of this paper and states the conclusions.

##  II Problem Statement

The objective of the proposed control strategy is to dispatch the active power
flow of a distribution network according to a deterministic profile, called
_dispatch plan_ , established the day before the operation. The dispatch plan
is a sequence of average power flow values at 5 minute resolution. It is
denoted by $P_{disp}$ and, in this work, it is assumed known and computed with
the procedure described in [11]. In brief, the dispatch plan consists in the
sum of two terms: the forecast of the prosumption underneath the distribution
system (i.e., intuitively, it is the best guess one could do to dispatch
stochastic prosumption), and the offset profile which is with the specific
objective of restoring a suitable level of flexibility in the available
storage resources111For example, during the current day of operation, flexible
resources might be close to their state-of-charge bounds. Therefore, it is
necessary to charge/discharge them to restore an adequate level of SOC such
that enough flexibility is available to compensate for the tracking error
during the incoming day of operation.. It is noteworthy that in the current
formulation, we do not include grid constraints (e.g., see [17, 9]), which
will be considered in future works. In other words, we assume that the target
grid, as the one considered in the real-life experimental setup, is robust and
not subject to voltage and line ampacity constraints violations.

In real-time, the problem is to compensate the average power mismatch between
the dispatch plan set-point and the measured stochastic realization, on a 5
minute basis. This is achieved by the coordinated control of multiple
resources thanks to applying distributed optimization. Although the proposed
formulation is generic enough to accommodate the use of different kinds of
flexible resources, in this paper it is formulated for the case of a
curtailable PV facility and battery energy storage system (BESS). The choice
of these kinds of units is motivated by the fact that they are both well-
established technologies, characterized by a high level of maturity and
foreseen to undergo a massive growth in the near future. Fig. 1 shows the
real-life experimental setup used to test and validate the proposed control:
it consists in a radial distribution feeder interfacing a heterogeneous mix of
demand and generation (office buildings with uncontrollable rooftop PV
installations) and two controllable resources as explained in Section IV.

EPFL sub-transmission gridGrid connection point (GCP) 50/21 kV 20 MVA $P$
(composite power flow) $L$ (aggregated feeder _prosumption_) 350 kWp
Prosumption (office buildings) $G$ (PV power) 13 kWp Curtailable PV facility
21/0.3 kV 0.75 MVA $B$ (BESS charging demand) Lithium Titanate BESS 720
kVA/500 kWh Dispatchable feeder Measured power flows Figure 1: The EPFL’s
experimental setup used for validation: a MV feeder interfacing office
buildings equipped with uncontrollable rooftop PV plants, a grid-connected
battery system, and a curtailable PV plant.

##  III Methods

The objective is to coordinate the operation of and control multiple resources
to compensate the mismatch between the dispatch plan set-point and the
measured average power consumption on a 5 minute basis. In our case, the
formulation is carried out by considering a curtailable PV facility and a BESS
because they are the unit available in the experimental setup used for the
real-life validation. Nonetheless, the setup is scalable and can be augmented
to consider other controllable resources. The control framework consists in
two real-time algorithms executed at different paces:

  * •

a lower level tracking problem, executed at 10 s resolution, to achieve a fine
tracking of the dispatch plan by controlling the BESS’s active power set-
point. This is achieved with MPC with the formulation proposed in [11], and it
is not a contribution of this work.

  * •

an upper level coordination mechanism, running at 5 minutes resolution, to
coordinate the operation between elements. Its role is essentially
implementing an energy management strategy to, _i)_ , make sure that enough
power capacity is available for the faster control loop to compensate for the
power mismatch, and, _ii)_ , longer-term managing the BESS state-of-charge
(SOC) such that enough flexibility is available to compensate for the energy
error during the remaining part of the day.

The latter control loop is the main contribution of this paper and consists in
an optimization problem formulated and solved for each controllable device
with a coupling constraint on the energy error. In the following formulation,
the upper level problem is responsible for computing set-points for both the
PV plant and battery system.

It is worth noting that decoupling the control actions into two discerned time
scales is practical for two reasons: _i)_ , the distributed optimization
coordination mechanism is slower to be solved than the faster pace power set-
point tracking (especially because of the communication latency when agents
are implemented on different machines); _ii)_ , due to technological
limitations or inherent process uncertainties, certain devices (like PV plants
and fuel cells) are not curtailable/controllable with high accuracy, see
e.g.[24]. Therefore, their use is more likely for energy management rather
than precise power point tracking. On the other hand, resources like grid-
connected battery systems are able to achieve accurate power control and are
suitable for fine power point tracking. Overall, the proposed solution (i.e.,
fast pace control coupled with slower distributed optimization problem) allows
accommodating both kinds of resources in a cooperative and thoughtful manner.

In the following of this section, the model predictive control problem is
first formulated in a centralized way. Later in this section, it is shown how
it is decomposed until having one problem per each controllable unit, and how
it is solved by applying the ADMM algorithm.

###  III-A Centralized model predictive control

Let the index $i$ denote the rolling current 5 minute interval, $N$ the number
of 5 minutes interval in 24 hours, $j=i,\dots,N$ a 5 minute time index
spanning from the current time until the end of the day, the sequence
$\widehat{e}_{j}$ the forecasted deviation between the dispatch plan set-point
and forecasted realization for the remaining part of the day, $B^{o}_{j}$ and
$G^{o}_{j}$ the battery and PV plant set-point trajectories, respectively, and
$\widehat{G}_{j}$ the maximum theoretical PV power plant output. The sequences
$\widehat{e}_{j}$ and $\widehat{G}_{j}$ are calculated by applying forecasting
tools and are assumed given in the following formulation. For our specific
application, the latter is determined with the physical-based modelling tool-
chain proposed in [25], as a function of global horizontal irradiance
forecast. In the following, we focus on the formulation of the optimization
problem in a centralized manner. It is aimed at determining trajectories
$B^{o}_{j}$ and $G^{o}_{j}$ for the remaining part of the day $j=i\dots,N$ in
order to compensate for the dispatch plan mismatch $e_{j}$ while performing
minimal amount of curtailment subject to PV and battery system constraints.
The PV system constraint is that the active power should be within 0 and the
theoretical maximum PV power production $\widehat{G}_{j}$. For the battery,
the constraints are that the power should be within the four-quadrant apparent
power converter capability, and the battery SOC within the bounds
($\text{SOC}_{j}^{\text{min}},\text{SOC}_{j}^{\text{max}}$) which can be time
variant to account for dynamic constraints. In this formulation and
experiments, the battery converter is operated at the unitary power factor, so
the apparent power constraint reduces to the active power being between
plus/minus the converter apparent power rating222Power converter capability
depend on grid frequency and voltage, in this case we choose a conservative
$B^{\text{nom}}$ bound so to satisfy any operating condition., denoted by
$B^{\text{nom}}$. Battery SOC is normally modelled considering the
charging/discharging efficiency of the system, nevertheless, in the current
formulation we neglect it because the considered experimental unit has a
round-trip efficiency on the AC side close the unit (in the range 97-99%
according to the power)333We note that one might model battery efficiency
while still preserving convexity, see, e.g., [11]. Another strategy to account
for the efficiency is to account for efficiency losses in the next
rescheduling interval.. The dynamic model of the battery’s SOC is:

$\displaystyle\text{SOC}_{j+1}=\text{SOC}_{j}+\alpha
B_{j},~{}~{}~{}\alpha=\frac{300}{3600}\frac{1}{E}$ (1)

where the ratio 300/3600 is the share of 5 minutes period in an hour, and $E$
the battery nominal energy capacity in kWh. The SOC value at time interval
$i+1$ is calculated by applying (1) recursively to propagate the current
battery state-of-charge $\text{SOC}_{i}$:

$\displaystyle\text{SOC}_{i+1}$ $\displaystyle=\text{SOC}_{i}+\alpha
B_{i}+\dots+\alpha B_{j}=$ (2)
$\displaystyle=f(\text{SOC}_{i},\mathbf{B}_{i,j}),$ (3)

where the bold typeface notation $\mathbf{B}_{i,j}$ denotes the sequence
$[B_{i},\dots,B_{j}]$.

Formally, the centralized optimization problem is:

$\displaystyle\underset{\begin{matrix}\scriptstyle G_{i},\dots,G_{N}\\\
\scriptstyle
B_{i},\dots,B_{N}\end{matrix}}{\operatorname{arg}\,\operatorname{min}}\;\sum_{j=i}^{N}\left(G_{j}-\widehat{G}_{j}\right)^{2}$
(4)

subject to:

$\displaystyle B_{j}+G_{j}=e_{j}$ $\displaystyle j=i,\dots,N$ (5)
$\displaystyle 0\leq G_{j}\leq\widehat{G}_{j}$ $\displaystyle j=i,\dots,N$ (6)
$\displaystyle B^{\text{min}}\leq B_{j}\leq B^{\text{max}}$ $\displaystyle
j=i,\dots,N$ (7) $\displaystyle\text{SOC}^{\text{min}}_{j}\leq
f(\text{SOC}_{i},\mathbf{B}_{i-1,j-1})\leq\text{SOC}^{\text{max}}_{j}$
$\displaystyle j=i,\dots,N.$ (8)

In words, it determines the PV trajectory at minimum curtailment and the BESS
power trajectory respectful of battery’s constraints to satisfy the tracking
error on the shrinking horizon from $i$ to $N$. The optimization problem
(4)-(8) is centralized because, to be solved, it requires all the information
from the controllable resources. By exploiting well-known results from
distributed optimization theory, it is possible to decompose the centralized
problem into smaller ones, which can be solved iteratively until reaching a
consensus on a coupling constraint – in this case, the dispatch constraint
(5).

###  III-B From the centralized to distributed optimization problem and ADMM

Let $g_{j}(\cdot)$ be a barrier function with zero cost when the tracking
error constraint (5) is respected and infinity otherwise:

$\displaystyle g_{j}(G_{j},B_{j})=\begin{cases}0&B_{j}+G_{j}=e_{j}\\\
\infty&\text{otherwise.}\end{cases}$ (9)

Let $\mathcal{G}_{j}$, $\mathcal{B}_{j}$ be the variables that mimic the
behavior of the original variables ${G}_{j}$, ${B}_{j}$, the so-called copied
variables. The centralized objective can be re-written by moving the system
constraints into the objective function by using the barrier function
$g_{j}(\cdot)$. It is:

$\displaystyle\begin{aligned} \underset{\begin{matrix}\scriptstyle
G_{i},\dots,G_{N}\\\ \scriptstyle
B_{i},\dots,B_{N}\end{matrix}}{\operatorname{arg}\,\operatorname{min}}\;\sum_{j=i}^{N}\left(G_{j}-\widehat{G}_{j}\right)^{2}&+\sum_{j=i}^{N}g_{j}(\mathcal{G}_{j},\mathcal{B}_{j})\end{aligned}$
(10)

subject to:

$\displaystyle B_{j}-\mathcal{B}_{j}=0$ $\displaystyle j=i,\dots,N$ (11)
$\displaystyle G_{j}-\mathcal{G}_{j}=0$ $\displaystyle j=i,\dots,N.$ (12)

Constraints (11) and (12) can be moved in the cost function by using two
sequences of Lagrangian multipliers, denoted by $\boldsymbol{y_{G}}_{i,N}$ and
$\boldsymbol{y_{B}}_{i,N}$, referred to as dual variables in the following.
The augmented Lagrangian cost function is obtained by moving the equality
constraints (11)-(12) in the cost function (10). It is:

$\displaystyle\begin{aligned}
L_{\rho}=&\sum_{j=i}^{N}\left(G_{j}-\widehat{G}_{j}\right)^{2}+\sum_{j=i}^{N}g_{j}(\mathcal{G}_{j}+\mathcal{B}_{j})+\\\
+&\frac{\rho}{2}\left(\left|\left|\boldsymbol{G}_{i,N}-\boldsymbol{\mathcal{G}}_{i,N}\right|\right|_{2}^{2}+\left|\left|\boldsymbol{B}_{i,N}-\boldsymbol{\mathcal{B}}_{i,N}\right|\right|_{2}^{2}\right)+\\\
+&\boldsymbol{y_{G}}^{T}_{i,N}(\boldsymbol{G}_{i,N}-\boldsymbol{\mathcal{G}}_{i,N})+\boldsymbol{y_{B}}^{T}_{i,N}(\boldsymbol{B}_{i,N}-\boldsymbol{\mathcal{B}}_{i,N}).\end{aligned}$
(13)

The ADMM consensus and sharing problem [23] consists in the following three
steps.

####  III-B1 Original variables updates

Let $\boldsymbol{u_{G}}_{i,N}$ = $\boldsymbol{y_{G}}_{i,N}/\rho$ and
$\boldsymbol{u_{B}}_{i,N}$ = $\boldsymbol{y_{B}}_{i,N}/\rho$ be the scaled
dual variable for the PV and battery problems. The update of the decision
variable of the PV power plant is:

$\displaystyle\begin{aligned}
\boldsymbol{G}_{i,N}^{k+1}=\underset{G_{i},\dots,G_{N}}{\operatorname{arg}\,\operatorname{min}}\;\Bigg{\\{}\sum_{j=i}^{N}\left(G_{j}-\widehat{G}_{j}\right)^{2}+\\\
+\frac{{\rho}^{k}}{2}\left|\left|\boldsymbol{G}_{i,N}-\boldsymbol{\mathcal{G}}^{k}_{i,N}+\boldsymbol{u_{G}}_{i,N}^{k}\right|\right|_{2}^{2}\Bigg{\\}},\end{aligned}$
(14)

subject to (6). For the battery, it is:

$\displaystyle\boldsymbol{B}^{k+1}_{i,N}=\underset{B_{i},\dots,B_{N}}{\operatorname{arg}\,\operatorname{min}}\;\Bigg{\\{}\frac{{\rho}^{k}}{2}\left|\left|\boldsymbol{B}_{i,N}-\boldsymbol{\mathcal{B}}^{k}_{i,N}+\boldsymbol{u_{B}}_{i,N}^{k}\right|\right|_{2}^{2}\Bigg{\\}},$
(15)

subject to (7) and (8).

####  III-B2 Copied variable update

$\displaystyle\begin{aligned}
=\underset{\mathcal{G}_{i},\dots,\mathcal{G}_{N},\mathcal{B}_{i},\dots,\mathcal{B}_{N}}{\operatorname{arg}\,\operatorname{min}}\;\Bigg{\\{}\sum_{j=i}^{N}g_{j}(\mathcal{G}_{j}+\mathcal{B}_{j})\\\
+\frac{{\rho}^{k}}{2}\left|\left|\boldsymbol{G}_{i,N}^{k+1}-\boldsymbol{\mathcal{G}}_{i,N}+\boldsymbol{u_{G}}_{i,N}^{k}\right|\right|_{2}^{2}+\\\
\frac{{\rho}^{k}}{2}\left|\left|\boldsymbol{B}_{i,N}^{k+1}-\boldsymbol{\mathcal{B}}_{i,N}+\boldsymbol{u_{B}}_{i,N}^{k}\right|\right|_{2}^{2}\Bigg{\\}}.\end{aligned}$
(16)

####  III-B3 Dual variable update

$\displaystyle\boldsymbol{u_{G}}_{i,N}^{k+1}=\boldsymbol{G}_{i,N}^{k+1}-\boldsymbol{\mathcal{G}}^{k+1}_{i,N}+\boldsymbol{u_{G}}_{i,N}^{k}$
(17)
$\displaystyle\boldsymbol{u_{B}}_{i,N}^{k+1}=\boldsymbol{B}_{i,N}^{k+1}-\boldsymbol{\mathcal{B}}^{k+1}_{i,N}+\boldsymbol{u_{B}}_{i,N}^{k}.$
(18)

###  III-C Implementation

The updates of the original variable $G_{i}$ and $B_{i}$ in (14) and (15) are
computed in parallel. The updates of the copied variables $\mathcal{G}_{i}$
and $\mathcal{B}_{i}$ in (16) require gathering the local solutions, therefore
it is solved in a centralized manner by the aggregator. Once the dual
variables $u_{G_{i}}$ and $u_{B_{i}}$ are computed with (17) and (18), they
are disseminated to the subsystems together with the copied variables. The
iterations are repeated till convergence. The convergence criterion is met
when the primal residual norm ${r}^{k}$

$\displaystyle{r}^{k+1}_{i}=||\boldsymbol{G}_{i,N}^{k+1}-\boldsymbol{\mathcal{G}}^{k+1}_{i,N}||_{2}+||\boldsymbol{B}_{i,N}^{k+1}-\boldsymbol{\mathcal{B}}^{k+1}_{i,N}||_{2}$
(19)

and dual residual norm ${s}^{k}$

$\displaystyle{s}^{k+1}_{i}=\left|\left|-\rho^{k}\Bigg{[}\begin{pmatrix}\boldsymbol{\mathcal{B}}^{k+1}_{i,N}\\\
\boldsymbol{\mathcal{G}}^{k+1}_{i,N}\end{pmatrix}-\begin{pmatrix}\boldsymbol{\mathcal{B}}^{k}_{i,N}\\\
\boldsymbol{\mathcal{G}}^{k}_{i,N}\end{pmatrix}\Bigg{]}\right|\right|_{2}$
(20)

are both smaller than a dynamic feasibility tolerances calculated ad described
in [23].

For the penalty parameter $\rho^{k}$ in (21), we follow a self-adaptive scheme
[26], where $\tau_{incr}$ and $\tau_{decr}$ effectuate an adjustment scheme
via multiplying $\rho^{k}$ by $\tau_{incr}$ when the primal residual norm is
larger than the dual residual norm, and dividing $\rho^{k}$ by $\tau_{decr}$
in the opposite case[23]. It is to retain the primal and dual residual norms
within a factor of $\mu$ times one another as they both converge to zero. We
fix $\mu$ = 10 and $\tau_{incr}$ = 2 and $\tau_{decr}$ = 2 as reported in
[26].

$\displaystyle{\rho}^{k+1}:=\begin{cases}{\tau}^{incr}{\rho}^{k}&{r}_{i}^{k+1}>\mu{s}_{i}^{k+1}\\\
\frac{{\rho}^{k}}{{\tau}^{decr}}&{s}_{i}^{k+1}<\mu{r}_{i}^{k+1}\\\
{\rho}^{k}&\text{otherwise.}\end{cases}$ (21)

At each computation $i$, prosumption forecast and PV generation forecasts are
necessary to update the sequence $\boldsymbol{e}_{i,N}$. The former are
generated with a persistent predictor, the latter are computed by applying a
physical-based modelling tool-chain (transposition + PV plant models) as a
function of global horizontal irradiance and air temperature
predictions444provided by MeteoTest.ch.. The procedure describe in [27] is
used to transpose horizontal irradiance forecasts into tilted irradiance
values, while the five-parameter cell model proposed in [28], extended to the
whole PV array, is used to model the PV plant. Also, the current battery SOC
is necessary to update the initial condition of the state-of-charge integral
model. The flow chart depicting the real-time operation procedure is sketched
in Fig. 2. In the flow diagram, controllers running at two time samples: _i)_
faster MPC controller executed each 10 s and _ii)_ slower ADMM based
coordination mechanism computing PV set-points each 5 minute.

Figure 2: Flow chart showing real-time operation during 24 hours

The algorithm is implemented on a computer equipped with an Intel i7
processor. Statistics on the computational burden of the algorithm are
discussed in the Results section.

##  IV Experimental Setup

The proposed control framework is validated experimentally in the real-life
setup sketched in Fig. 1, which includes office buildings with rooftop PV
installations (i.e. uncontrollable prosumption), a curtailable PV facility,
and a grid-connected BESS. Characteristics of these units are summarized in
Table I. The active power flow of the distribution network is monitored by
sensing the consumption at the GCP with a PMU-based metering system. The BESS
is controllable by sending active/reactive power set-points to the power
converter over a Modbus interface, and the PV plant is controllable by sending
active power set-points to the converter over CAN communication.

TABLE I: Description of connected elements at GCP Component |  Parameter |  Value
---|---|---
Grid-connected BESS |  Nominal power |  720 kVA
Energy capacity |  560 kWh
Ramping rate |  $\pm$20 MW/s
Prosumption (office building + rooftop PV) |  Peak active power demand |  350 kW
Average demand |  101 kW
Generation capacity |  82 kWp
Curtailable PV Plant |  Generation capacity |  13 kWp

##  V Experimental Results and Discussion

The experimental results for one day of operation are shown in Fig. 3. In
particular, Fig. 3a shows the dispatch plan (in black), the prosumption
realization (dashed red) and the realization at the GCP (gray shaded area):
the prosumption realization differs from the dispatch plan due to forecasting
errors, whereas the realization at the GCP, which is corrected by controlling
the contribution of the battery and curtailable PV facility, achieves a good
tracking of the dispatch plan.

Fig. 3b shows the battery injection (in the upper panel), and the SOC
evolution and its bounds (bottom panel, full and dashed line, respectively),
which are an input of the decision problem. In this case, SOC bounds are
deliberately constrained at time 20h to reproduce a situation where the
flexibility is constrained (due to, e.g., the need of providing additional
service or a foreseen contingency situation, like maintenance to battery
modules, or simply to emulate a situation where the battery capacity is
saturated). It is noteworthy that SOC evolution is always within the allowed
bounds.

Fig. 3c shows the PV curtailment action, in particular the theoretical PV
maximum power point (in red), the PV converter set-point (black) and measured
PV active power measured at the converter grid connection point. As visible,
during the first part of the day, the PV set-point corresponds to the maximum
available power, indeed the PV plant produces close to the theoretical limit:
the small differences between the two profiles are due to the fact that
maximum power point operation is achieved by MPPT algorithms (maximum power
point tracking), which normally rely on perturb-and-observe strategies and not
on a physical model-based toolchain, as we do. As visible in Fig. 3c, the PV
power is curtailed around midday. This is due to the distributed optimization
policy, which – as explained in the next paragraph – decides to implement
curtailment in order to satisfy the constraints of the optimization problem.

(a) Dispatch plan (black), measured prosumption realization (shaded area),
measured active power flow at the GCP (dashed red).

(b) Battery power injection (upper panel), and battery SOC evolution and
respective limits (bottom panel).

(c) Curtailed PV (black), theoretical maximum power point (MPP) PV (dashed
red) and measured PV after implementation (shaded area)

Figure 3: Operation of the dispatchable feeder with ADMM strategy on 17
September 2017.

To evaluate whether the curtailment action determined by the distributed
optimization problem was necessary, we playback the experimental measurements
of the prosumption realization in a ad-hoc simulation framework555The reason
why we rely on a simulation framework is the impossibility of replicating the
same stochastic realization of the prosumption in two different experiments.
where the feeder dispatch is enforced by controlling the battery only. In
other words, we want to verify if battery SOC constraints are still respected
in the same stochastic conditions but without leveraging the controllability
of the curtailable PV facility. The evolution of the SOC is modelled as in
(1). Fig. 4 compares the experimental SOC (dashed red, from the battery
management system, which includes ADMM action), the simulated SOC with ADMM
(shaded gray band), the simulated SOC without ADMM action, and the SOC upper
and lower bounds (dashed blue and black lines). The close matching between the
trajectories of the experimental and simulated SOC with ADMM denotes that the
SOC estimation model (3) are in agreement and validates the reliability of the
used simulation playback approach. As visible in Fig. 4, the experimental SOC
and simulated SOC without ADMM matches until approximately midday, the time
when the PV curtailment action begins (from Fig. 3c). After midday, the two
trajectories diverge because in the latter case the battery has to charge more
in order to track the dispatch plan. Nevertheless, in the former case, the
battery SOC respect the SOC upper bound constraint, whereas the latter
strategy (without ADMM) fails at time 20:00 UTC. We can therefore conclude
that the curtailment action determined by the formulation with distributed
optimization was necessary and finally accomplished the target of respecting
BESS constraints.

Figure 4: SOC evolution with and without ADMM. The former is experimental,
whereas the latter is obtained by playing back into simulations experimental
data. With ADMM, SOC constraints are respected, whereas they are not without
ADMM.

Numerical results comparing the control performance with and without ADMM are
shown in Table II. They denote that the distributed optimization control, even
if it sacrifices PV generation, it achieves to respect BESS capacity
constraints and that it is able to harmonize the operation of the controllable
resources to achieve to dispatch stochastic prosumption.

TABLE II: PV generation, PV curtailments and SOC constraint violation  Quantity | no ADMM | with ADMM
---|---|---
Max distance from SOC upper bound constraint ($>0$ violation) | 3.09 % | -0.47 %
PV Generation | 33.8 kWh | 21.60 kWh
PV Curtailement | – | 12.2 kWh

Table III shows the statistics on the tracking error for the case where there
is no control at all (no dispatch), dispatch strategy without ADMM upper layer
coordination strategy, and dispatch with ADMM. In this case, the dispatch
strategy with achieves the best tracking performance with a RMS error less
than 0.5 kW over 24 hours.

TABLE III: Tracking error statistics without dispatch, with dispatch and no ADMM, and dispatch + ADMM (kW). Scenario | RMSE | Mean | Max
---|---|---|---
No dispatch | 11.1 | -4.1 | 36.0
Dispatch without ADMM | 1.60 | 0.53 | 7.8
Dispatch + ADMM | 0.32 |  $\leq$ 0.01 | 2.27

Finally, Table IV shows mean, standard deviation and maximum value of the
computation time necessary to finalize a single iteration, the number of total
iteration required to convergence to a solution and the reached
accuracy666Accuracy here is defined as $B_{j}+G_{j}-e_{i}$, i.e. the coupling
constraint to achieve in (5).. The mean time required to reach a solution is
$1.01\times 12.69=12.82~{}$seconds, denoting that the ADMM is more suitable to
perform short-term energy management, as proposed in this paper, rather than
real-time power control.

TABLE IV: Computation performance Quantities | Mean | Standard Deviation | Max
---|---|---|---
Iteration time (second) | 1.01 | 0.91 | 4.50
Iteration count | 12.69 | 2.11 | 16
Accuracy (kW) | 0.03 | 0.14 | 1.11

##  VI Conclusions and future work

In this paper, we proposed and experimentally validated a control and
coordination framework for distributed energy resources. It consisted in two
layers. The lower layer is a real-time MPC executed at 10 s resolution to
achieve fine tuning of a given energy set-point. The upper layer is a slower
MPC coordination mechanism based on distributed optimization and solved with
ADMM. It runs each 5 minutes and it has the objective of coordinating the
power flow among the resources such that enough power and energy is available
in real-time to achieve a pre-established energy trajectory.

The setup was tested in a real-life setup to dispatch the operation of
stochastic prosumption (e.g. office buildings with 101 kW average load, 350 kW
peak demand and 82 kW peak PV generation facility) according to a dispatch
plan at 5 minutes resolution, established the day before the operation. The
experimental controllable elements are a 720 kVA/560 kWh battery energy
storage system and a 13 kWp curtailable PV facility.

The experimental results showed that:

  * •

the inclusion of the ADMM-based coordination mechanism successfully achieves
to curtail PV generation in contingency situation (e.g., loss of battery
capacity) and respect battery state-of-charge constrains;

  * •

the mean time to convergence to a solution in the proposed setup was 12
seconds, therefore suitable for short-term energy management;

  * •

imagining to cluster controllable resources into two classes (accurate and
with fast actuation, like grid-connected BESS, and inaccurate and slow
reaction, like PV, due to be subject to solar irradiance and MPPT dynamics),
the proposed framework allows to cope with both of them and coordinate them in
a cooperative and fruitful manner to track a shared control objective.

The future work is in the direction of integrating more controllable elements
in the setup.

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