# Distributed Model-Predictive Energy Management Strategy for Shipboard Power
Systems Considering Battery Degradation

Satish Vedula∗, Seyyed Shaho Alaviani∗∗ and
Olugbenga Moses Anubi∗

$*$ Department of Electrical and Computer Engineering, the Center for Advanced
Power Systems, Florida State University E-mail: {svedula<EMAIL_ADDRESS>$**$ Center for Advanced Power Systems (CAPS), Florida State University, and
National High Magnetic Field Laboratory (MagLab), Tallahassee, FL 32310, USA
(email<EMAIL_ADDRESS>
##  1 Abstract

With the integration of loads such as pulse power loads, a new control
challenge is presented in meeting their high ramp rate requirements. Existing
onboard generators are ramp rate limited. The inability to meet the load power
due to ramp rate limitation may lead to instability. The addition of energy
storage elements in addition to the existing generators proves a viable
solution in addressing the control challenges presented by high ramp rate
loads. A distributed energy management strategy maximizing generator
efficiency and minimizing energy storage degradation is developed that
facilitates an optimal adaptive power split between generators and energy
storage elements. The complex structure of the energy storage degradation
model makes it tough for its direct integration into the optimization problem
and is not practical for real-time implementation. A degradation heuristic to
minimize absolute power extracted from the energy storage elements is proposed
as a degradation heuristic measure. The designed strategy is tested through a
numerical case study of a consolidated shipboard power system model consisting
of a single generator, energy storage element, and load model. The results
show the impact of the designed energy management strategy in effectively
managing energy storage health.

Keywords: Battery degradation, model predictive control, energy management

##  2 Introduction

Conceptualization of Microgrids (MGs) as an amalgamation of fossil-fuel-driven
micro-turbines, renewable energy-based generating systems, and energy storage
systems (ESSs) incorporated into a distribution generation environment,
driving high demand for the power. MGs are categorized into _grid-connected_
and _islanded_ , based on the configuration and their mode of operation ([1]).
One such example of an islanded operation is Shipboard Power Systems (SPSs).
In shipboard power systems, power generation modules (PGMs) and power
conversion modules (PCMs) provide the power required by the loads through
means of the direct current (DC) distribution system. Modern SPSs are equipped
with advanced power loads such as rail guns, electromagnetic radars, and heavy
nonlinear and pulsed power loads (PPLs) ([2]).

There is a challenge to utilize the existing PGMs, which are ramp-rate
limited, to meet the power demanded by the high-ramp-rate loads. Failure to
ramp up in appropriate time to meet the load requirements can lead to system
imbalances and instability. Furthermore, adding additional generators is not a
plausible solution considering the ship hull dimension restrictions. This
imposes a need for high-ramp-rate power generation elements to be integrated
into SPSs. Energy storage systems such as batteries can support higher ramp
rates. Integrating energy storage systems (ESSs) into the existing notional
SPS framework addresses the high ramp requirements ([3]). The ESSs can store
the energy during normal periods and when needed can provide immediate ramp-up
support. Several investigators have addressed the above challenge _without_
considering battery degradation ([4, 5, 6]) where they have used dynamic
programming (DP) and model predictive control (MPC) to address the
aforementioned challenge. MPC is a renowned concept used in control literature
whose applications include industry. MPC problem solves an objective function
subject to a set of constraints that can be actuator limitations in physical
systems or rate constraints ([7]). The main _advantage_ of the MPC over DP is
that it enables incorporation of the constraints such as inequality and box
constraints capturing the upper and lower limitations and ramp rates, thus
accounting for actuator limitations.

Figure 1: A Notional Shipboard Power System with the Zonal Architecture and
Control Hierarchy.

Nevertheless, batteries degrade faster than generators. This fact imposes
another challenge to consider both _ramp rate requirements_ and _battery
degradation_ together. Some investigators have considered this challenge while
addressing the EM problem for the MGs ([8, 9, 10, 11, 12, 13, 14, 15]). In
[8], the authors use DP to solve the EM problem while considering the depth of
discharge (DoD) as a measure for capturing battery degradation. In [9], an
energy market pricing-based EM problem for an MG, considering the battery DoD
as a degradation measure, is presented. Market-energy-pricing-based approach
for MG with fast Li-ion(standing for Lithium-ion)-based batteries, considering
DoD as a measure for capturing the battery degradation, has been presented in
[10]. DoD has been also used as a battery degradation measure to solve the EM
problem in [11] where the authors proposed a neural-network-based method to
address the EM problem in the MGs. In [12], the authors have presented a
state-of-charge (SoC)-based EM scheme for MGs; the SoC is regulated around a
reference value to minimize battery degradation. Monitoring the battery
charge/discharge cycles and using them to determine the battery degradation
with an application to the MGs has been investigated in [13]. The work of [8,
9, 10, 11, 12, 13] consider battery aging in _different measures_ such as DoD,
SoC, and charge/discharge based cycle count. In [14], the measures used to
minimize the battery usage are _battery power_ and _battery SoC_ , but the
authors in [14] did _not_ consider the cost associated with the generators and
its effect on determining the battery degradation.

The main contributions of this paper are:

  * •

A model predictive energy management strategy is designed to incorporate
battery degradation in the form of minimizing its power as a heuristic
(measure) to mitigate battery deterioration as shown in Fig. 1.

  * •

We Consider both the maximizing generator efficiency and minimizing battery
degradation in the designed model predictive energy management problem along
with the component ramp rate limitations to test the performance under the
presence of pulsed power loads.

  * •

A scalable distributed plug-and-play model predictive energy management
strategy is developed and tested on a shipboard power system with the real
component ratings provided by the Office of Naval Research (ONR).

The organization of the rest of the paper is as follows: Mathematical
notations used throughout the paper are presented in Section 3. Section 4
presents the physics-based mathematical model of the shipboard power system
components and their device-level control. The main results are provided in
Section 5. The trade-off between the generator and the energy storage and
deductions from the numerical simulation is presented in Section 6 followed by
the concluding remarks in Section 7.

##  3 Notations

The set of natural numbers, positive real numbers, and real numbers is
represented by $\mathbb{N}$, $\mathbb{R}_{+}$, and $\mathbb{R}$.
$\mathbb{L}_{\infty},\mathbb{L}_{2}$ denote the signal space of bounded and
square-integrable signals. A real matrix with $a$ rows and $b$ columns is
denoted as $\mathbb{R}^{a\times b}$. $\mathbb{R}_{+}$ denotes a positive real
number. Scalars are denoted by lowercase alphabets respectively (for example
$x\in\mathbb{R}$). The notation $\mathcal{X}\subset\mathbb{R}^{n}$ represents
polytopes of the form
$\\{\mathbf{x}\in\mathbb{R}^{n}|A_{i}\mathbf{x}_{i}\preceq\mathbf{b}_{i},A_{e}\mathbf{x}_{e}=\mathbf{b}_{e}\\}$,
where $A_{i},A_{e},\mathbf{b}_{i},\mathbf{b}_{e}$ are the parameters derived
from the system specific inclusion constraints. $\underline{\mathbf{0}}$ and
$\mathbf{1}$ denote the vector of zeros and ones. For any vector
$\mathbf{x}\in\mathbb{R}^{n}$,
$\|\mathbf{x}\|_{2}=\sqrt{\mathbf{x}^{\top}\mathbf{x}}$ and
$\|\mathbf{x}\|_{1}=\sum_{i=1}^{n}|\mathbf{x}_{i}|$, represents the 2-norm and
the 1-norm, respectively (where $|.|$ denotes absolute value). The symbol
$\preceq$ denotes the component-wise inequality i.e.
$\mathbf{x}\preceq\mathbf{y}$ is equivalent to
$\mathbf{x}_{i}\leq\mathbf{y}_{i}$ for $i=1,2,\ldots,n$. For a function
$f:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, $\nabla f(x)$ denotes the
gradient of the function $f$ at $x$. The energy of the signal
$x:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ is defined as:

$E(x)=\int_{0}^{\infty}|x(t)|^{2}dt.$

##  4 Model Development

In this section, we present the model development and the device level
controller structure for a Shipboard Power System components _generators/PGMs_
and _batteries/PCMs_ supplying a common load through a unified DC bus based on
Fig. 2.

The _DC Generator or the PGM_ model consists of a current-controlled DC
voltage source coupled with a series $RL$ impedance connected to the bus. The
assumption is that the bus voltage is already regulated to a set point. Thus,
it is considered to be a constant. The dynamical model PGM is given as follows

$l_{g}\frac{di_{g}}{dt}=-r_{g}i_{g}(t)+\Delta v(t)$ (1)

where $i_{g}(t)\in\mathbb{R}$ is the PGM current and $\Delta
v(t)={v}_{bus}(t)-v_{g}(t)$, where $v_{g}(t)\in\mathbb{R}$ is the controllable
voltage source and $v_{bus}(t)\in\mathbb{R}$ is the bus voltage to which the
PGM is coupled. The generator inductance $l_{g}\in\mathbb{R}_{+}$ is in Henry,
$r_{g}\in\mathbb{R}_{+}$ is the PGM resistance in Ohm. The current injected by
the PGM into the bus is dictated by controlling the voltage source. Given an
optimal power profile $p_{g_{r}}\in\mathbb{R}$ from the energy management
control layer, the reference current $i_{g_{r}}\in\mathbb{R}$ (assumed to be
bounded $\in\mathbb{L}_{\infty}$) is generated as
$i_{g_{r}}=p_{g_{r}}/{v_{bus}}$ and the local controller input $v_{g}$ is
determined via a closed-loop control scheme in which the device level state
tracks the reference current i.e. $i_{g}(t)\to i_{g_{r}}(t)$. Thus, the DLC is
designed to achieve the following objective

$\int_{0}^{\infty}\bigg{(}\underbrace{i_{g}(t)-i_{g_{r}}(t)}_{\tilde{i}_{g}}\bigg{)}^{2}dt<\infty.$

Since the design and analysis of such a DLC is not the main objective of this
paper, numerous linear (PID) and nonlinear methods can be employed to design
it ([16]).

The _Battery Energy Storage System_ (BESS) or the _PCM_ model consists of
multiple ESSs modeled as a voltage-controlled current source. The static model
consists of a resistance $r_{b}\in\mathbb{R}_{+}$ coupled to the bus
$v_{bus}\in\mathbb{R}$ in series with a controllable voltage source
$v_{b}\in\mathbb{R}$ and an open circuit voltage $v_{oc}\in\mathbb{R}$ that
make up the battery model. The algebraic equation governing the power exchange
in the battery is given as:

$\displaystyle v_{b}$
$\displaystyle=\frac{v_{bus}^{2}-p_{b}r_{b}-v_{bus}v_{oc}}{v_{bus}},$
$\displaystyle i_{b}$ $\displaystyle=\frac{v_{bus}-v_{b}-v_{oc}}{r_{b}}.$

The state of charge dynamics (SoC) of the battery is given as:

$\frac{dSoC}{dt}=-\frac{1}{Q}i_{b}(t),$ (2)

where $Q\in\mathbb{R}_{+}$ is the battery capacity in ampere-hour. The
discretized form of the SoC dynamics given in (2) is:

$SoC_{k+1}=SoC_{k}-\frac{T_{d}}{Q}i_{b_{k}},$ (3)

where $T_{d}\in\mathbb{R}_{+}$ is the discretization time-step. Here, $k,k+1$
represents the discrete time intervals.

Figure 2: Mathematical topology of the power exchange in an SPS with a circuit
level schematic of PGM, PCM, and Load models.

An Arrhenius equation-based model which uses $Ah$-throughput as
$\displaystyle\int_{0}^{t}\left|i_{b}(\tau)\right|d\tau$ as a metric to
evaluate battery state of health (SoH) is used as a PCM degradation metric.
$i_{b}(t)\in\mathbb{R}$ is the current drawn from the individual PCM (positive
while discharging and vice-versa). The PCM capacity loss formulation is given
as follows [17].

$Q_{L}(t)=\zeta_{1}e^{\frac{-\zeta_{2}+TC_{r}}{RT}}\int_{0}^{t}\left|i_{b}(\tau)\right|d\tau,$
(4)

where $T\in\mathbb{R}_{+}$ is the PCM baseline operating temperature in
Kelvin, $C_{r}$ is the C-rate of the PCM. The capacity loss (%) is given as
follows:

$\Delta Q\%=\frac{Q-Q_{L}(t)}{Q}\times 100,$

where $Q_{L}$ is the _capacity loss_ of the BESS in ampere-hour.

The Power Load is modeled as a static resistive load $r_{L}$ model which
consumes the power generated by the PGMs and the PCMs. A variable voltage
$v_{L}\in\mathbb{R}$ enables the current flow in the load based on the active
power profile $p_{L}\in\mathbb{R}$. The assumption is that there is only one
load in the model which is consuming all the generated power. Thus, The load
model is given as follows:

$\displaystyle v_{L}=\frac{v_{bus}^{2}-p_{L}r_{L}}{v_{bus}},\hskip
5.69054pti_{L}=\frac{v_{bus}-v_{L}}{r_{L}},$ (5)

Thus, the power flow in the SPS is given by the following algebraic equation:

$\sum_{i=1}^{n_{g}}\mathbf{p}_{g_{i}}+\sum_{j=1}^{n_{b}}\mathbf{p}_{b_{j}}-p_{L}=0,$
(6)

where $n_{g},n_{b}\in\mathbb{N}$ are the number of PGMs and PCMs in SPS. Fig.
2 shows the hierarchical control structure in the SPS and the power exchange.
The power and the SoC measurements are sent to the MPC layer. The MPC computes
the optimal values based on the objective and set of constraints and the
optimal values act as a reference for DLCs.

##  5 Proposed Energy Management Method

This section presents the control development and the derivation of the
proposed EM strategy. First, we present the MPC problem formulation in the
general form. We make use of the convex analysis and the preliminaries
presented in Section 3 in MPC development. Then, we consider the objective
functions of the generator and the battery respectively, and explain the
reason behind choosing the particular objectives. Finally, the distributed
plug-and-play formulation for the centralized MPC problem is derived. Consider
the MPC problem of the form:

$\displaystyle\operatorname*{Minimize:}_{\mathbf{p}_{g_{i}},\mathbf{p}_{b_{j}}}$
$\displaystyle\sum_{{i}=1}^{n_{g}}{C}_{g_{i}}(\mathbf{p}_{{g}_{i}})+\sum_{{j}=1}^{n_{b}}{C}_{b_{j}}(\mathbf{p}_{{b}_{j}})$
(7) $\displaystyle\operatorname*{Subject\phantom{a}to:}$
$\displaystyle\sum_{{i}=1}^{n_{g}}\mathbf{p}_{{g}_{i}}+\sum_{{j}=1}^{n_{b}}\mathbf{p}_{{b}_{j}}-p_{f}\mathbf{1}=0,\hskip
2.84526pt\forall k,$
$\displaystyle\mathbf{p}_{{g}_{{i}}}\in\mathcal{X}_{g},\hskip 2.84526pt\forall
k,$ $\displaystyle\mathbf{p}_{{b}_{j}}\in\mathcal{X}_{b},\hskip
2.84526pt\forall k,$

where
$k\triangleq\left[\begin{array}[]{cccc}1&2&\ldots&h\end{array}\right]^{\top}\in\mathbb{R}^{h}$
is the length of the prediction horizon,
$\mathbf{p}_{g_{i}}\triangleq\left[\begin{array}[]{cccc}{\mathbf{p}_{g_{i}}}_{k}&{\mathbf{p}_{g_{i}}}_{k+1}&\ldots&{\mathbf{p}_{g_{i}}}_{k+h-1}\end{array}\right]^{\top}\in\mathbb{R}^{h}$
is the power profile for generator $i$ over the prediction horizon of length
$h$,
$\textbf{p}_{b_{j}}\triangleq\left[\begin{array}[]{cccc}{\mathbf{p}_{b_{j}}}_{k}&{\mathbf{p}_{b_{j}}}_{k+1}&\ldots&{\mathbf{p}_{b_{j}}}_{k+h-1}\end{array}\right]^{\top}\in\mathbb{R}^{h}$
is the power profile for battery $j$ over the prediction horizon of length $h$
and ${p}_{f}\mathbf{1}\in\mathbb{R}^{h}$ is the desired total power held fixed
over the prediction horizon. The generator cost
${C}_{g_{i}}:\mathbb{R}^{{h}}\longrightarrow\mathbb{R}_{+}$ could be
associated with an efficiency map or operating at a desired rated power. The
battery cost ${C}_{b_{j}}:\mathbb{R}^{{h}}\longrightarrow\mathbb{R}_{+}$ could
be the cost associated with battery health monitoring and degradation
management. ${h}\in\mathbb{N}$ represents the prediction horizon.
$\mathcal{X}_{g}\subset\mathbb{R}^{h}$ and
$\mathcal{X}_{b}\subset\mathbb{R}^{h}$ represents the set of inclusion
constraints each of the PGM and PCM summarizing attributes such as upper and
lower power limits, ramp rate limits, and relevant system dynamics. The
optimization variables are optimized for the entire length of the horizon and
the first value of the sequence is supplied/applied as a power reference
(control input) to the low-level (DLC). The PGM and the PCM costs are
considered so that the objective is to maintain the PGM around a rated value
($\mathbf{p}_{g}^{r}$) which is known and to minimize the PCM power. Based on
the optimization problem presented in (7), consider the following MPC problem
with the respective cost functions for the PGM and the PCM as follows:

$\displaystyle\operatorname*{Minimize:}_{\mathbf{p}_{g},\mathbf{p}_{b},\mathbf{SoC}_{b}}$
$\displaystyle\sum_{i=1}^{n_{g}}\frac{\beta_{i}}{2}\left\lVert\mathbf{p}_{g_{i}}-\mathbf{p}^{r}_{{g_{i}}}\right\rVert_{2}^{2}+\sum_{j=1}^{n_{b}}\frac{\gamma_{j}}{2}\left\lVert\mathbf{p}_{b_{j}}\right\rVert_{2}^{2}$
(8) $\displaystyle\operatorname*{Subject\phantom{a}to:}$
$\displaystyle\sum_{i=1}^{n_{g}}\mathbf{p}_{g_{i}}+\sum_{j=1}^{n_{b}}\mathbf{p}_{b_{j}}=p_{f}\mathbf{1},\forall
k,$
$\displaystyle\mathbf{SoC}_{b_{jk+1}}=\mathbf{SoC}_{b_{jk}}-\frac{T_{d}}{Qv_{bus}}\mathbf{p}_{b_{jk}},\forall
k,$
$\displaystyle\underline{\mathbf{p}}_{g}\preceq\mathbf{p}_{g_{i}}\preceq\overline{\mathbf{p}}_{g},\forall
k,$
$\displaystyle\left|\mathbf{p}_{g_{ik}}-\mathbf{p}_{g_{ik-1}}\right|\preceq{r}_{g}\mathbf{1},\forall
k$
$\displaystyle\underline{\mathbf{p}}_{b}\preceq\mathbf{p}_{b_{j}}\preceq\overline{\mathbf{p}}_{b},\forall
k,$
$\displaystyle\left|\mathbf{p}_{b_{jk}}-\mathbf{p}_{b_{jk-1}}\right|\preceq{r}_{b}\mathbf{1},\forall
k,$
$\displaystyle\underline{\mathbf{SoC}}_{b}\preceq\mathbf{SoC}_{b_{jk}}\preceq\overline{\mathbf{SoC}}_{b},\forall
k,$

where
$k\triangleq\left[\begin{array}[]{cccc}1&2&\ldots&h\end{array}\right]^{\top}\in\mathbb{R}^{h}$,
$r_{g}\in\mathbb{R}_{+}$ and $r_{b}\in\mathbb{R}_{+}$ are the ramp rates of
the generator and the battery respectively. $\underline{\mathbf{p}}_{g}$ and
$\overline{\mathbf{p}}_{g}$ are the lower and the upper power limitations on
the generator, respectively. $\underline{\mathbf{p}}_{b}$ and
$\overline{\mathbf{p}}_{b}$ are the lower and the upper power limitations on
the battery. $\underline{\mathbf{SoC}}_{b}$ and $\overline{\mathbf{SoC}}_{b}$
represent the lower and the upper limitations on the state of charge of the
battery. $\beta_{i}\in\mathbb{R}_{+}^{n_{g}}$ are the tunable parameters
penalizing the PGMs deviation from a known efficient operation point.
$\gamma_{j}\in\mathbb{R}_{+}^{n_{b}}$ are the tunable parameters penalizing
the battery power.

###  5.1 Distributed Formulation

The optimization problem in (7) is broken down into the individual
optimization problems solved at the respective PGMs and the PCMs coupled via
an aggregator.

$\displaystyle\mathcal{L}(\textbf{p}_{{g}_{i}},\textbf{p}_{{b}_{j}},\boldsymbol{\lambda})$
$\displaystyle=\sum_{{i}=1}^{n_{g}}{C}_{{g_{i}}}(\mathbf{p}_{{g}_{i}})+\sum_{{j}=1}^{n_{b}}{C}_{{b_{j}}}(\mathbf{p}_{{b}_{j}})+\mathcal{I_{X}}_{g}+\mathcal{I_{X}}_{b}$
$\displaystyle\hskip
28.45274pt+\boldsymbol{\lambda}^{\top}\bigg{(}\sum_{{i}=1}^{n_{g}}\mathbf{p}_{{g}_{i}}+\sum_{{j}=1}^{n_{b}}\mathbf{p}_{{b}_{j}}-{p}_{f}\mathbf{1}\bigg{)}$
$\displaystyle=\mathcal{I_{X}}_{g}+\mathcal{I_{X}}_{b}+\sum_{{i}=1}^{n_{g}}\bigg{(}{C}_{{g_{i}}}(\mathbf{p}_{{g}_{i}})+\boldsymbol{\lambda}^{\top}\mathbf{p}_{{g}_{i}}\bigg{)}$
$\displaystyle\hskip
14.22636pt+\sum_{{j}=1}^{n_{b}}\bigg{(}{C}_{{b_{j}}}(\mathbf{p}_{{b}_{j}})+\boldsymbol{\lambda}^{\top}\mathbf{p}_{{b}_{j}}\bigg{)}-p_{f}\boldsymbol{\lambda}^{\top}\textbf{1},$

where $\boldsymbol{\lambda}\in\mathbb{R}^{h}$ is the dual variable associated
with the main problem in (7) and $\mathcal{I_{X}}_{g}$ and
$\mathcal{I_{X}}_{b}$ are the indicator functions for the inclusion
constraints. Given $\boldsymbol{\lambda}$, let $\mathbf{p}_{g_{i}}^{*}$ and
$\mathbf{p}_{b_{j}}^{*}$ to be the solutions to the optimization problem in
(7) ([18])

$\displaystyle\textbf{p}^{*}_{g_{i}}(\boldsymbol{\lambda})=\operatorname*{arg\,min}_{{\mathbf{p}_{{g}_{i}}}\in\mathcal{X}_{g}}\bigg{\\{}\text{C}_{{g_{i}}}(\mathbf{p}_{{g}_{i}})+\boldsymbol{\lambda}^{\top}\textbf{p}_{{g}_{i}}\bigg{\\}},$
(9a)
$\displaystyle\textbf{p}^{*}_{b_{j}}(\boldsymbol{\lambda})=\operatorname*{arg\,min}_{{\mathbf{p}_{{b}_{j}}}\in\mathcal{X}_{b}}\bigg{\\{}\text{C}_{{b_{j}}}(\mathbf{p}_{{b}_{j}})+\boldsymbol{\lambda}^{\top}\textbf{p}_{{b}_{i}}\bigg{\\}}.$
(9b)

Thus, the dual problem is given as:

$\operatorname*{Maximize:}\quad\mathcal{L}(\textbf{p}^{*}_{g_{i}},\textbf{p}^{*}_{b_{j}},\boldsymbol{\lambda})\triangleq
d(\boldsymbol{\lambda}),$

which is solved using the gradient ascent algorithm

$\displaystyle\boldsymbol{\lambda}_{t+1}$
$\displaystyle=\boldsymbol{\lambda}_{t}+\alpha\nabla
d(\boldsymbol{\lambda}_{t})$
$\displaystyle=\boldsymbol{\lambda}_{t}+\alpha\left(\sum\limits_{i=1}^{n_{g}}\mathbf{p}_{g_{i}}^{*}+\sum\limits_{j=1}^{n_{b}}\mathbf{p}_{b_{j}}^{*}-p_{f}\mathbf{1}\right).$
(10)

for a step size $\alpha>0$ and $t$ is the _iteration counter_. Since the
minimizing power profiles $\textbf{p}^{*}_{g_{i}}$ and
$\textbf{p}^{*}_{b_{j}}$ depends on $\boldsymbol{\lambda}$, the dual ascent
step in (10) becomes a recursion and challenging to evaluate. Consequently, we
incorporate the primal problems in (11) within a sequential minimization-
maximization scheme [19] as follows:

$\displaystyle\mathbf{p}_{g_{i}}^{t+1}$
$\displaystyle=\operatorname*{arg\,min}_{{\mathbf{p}_{{g}_{i}}}\in\mathcal{X}_{g}}\bigg{\\{}{C}_{{i}}(\mathbf{p}_{{g}_{i}})+\boldsymbol{\lambda}^{\top}_{t}\mathbf{p}_{{g}_{i}}\bigg{\\}},$
(11a) $\displaystyle\mathbf{p}_{b_{j}}^{t+1}$
$\displaystyle=\operatorname*{arg\,min}_{{\mathbf{p}_{{b}_{j}}}\in\mathcal{X}_{b}}\bigg{\\{}{C}_{{j}}(\mathbf{p}_{{b}_{j}})+\boldsymbol{\lambda}^{\top}_{t}\mathbf{p}_{{b}_{j}}\bigg{\\}},$
(11b) $\displaystyle\boldsymbol{\lambda}_{t+1}$
$\displaystyle=\boldsymbol{\lambda}_{t}+\alpha\bigg{(}\sum_{{i}=1}^{n_{g}}\mathbf{p}^{t+1}_{{g}_{i}}+\sum_{{j}=1}^{n_{b}}\mathbf{p}^{t+1}_{{b}_{j}}-p_{f}\mathbf{1}\bigg{)}.$
(11c)

Next, more details on the resulting nodal problems in the above scheme with
the choice of the cost functions and the inclusion constraints are given.

###  5.2 PGM Node Optimization Problem

From (11a) the optimization problem at the $i^{th}$ PGM node assuming the cost
function $C_{i}(\mathbf{p}_{g_{i}})$ to be a quadratic function is given in
(12). The objective of the considered cost is to maintain the PGM power at
around a given rated value or a set point value at every instant of the
horizon given by $\mathbf{p}^{r}_{g_{i}}$. This rated value is the operating
point for that generator as prescribed by the manufacturer. The optimization
problem is given as follows:

$\displaystyle\operatorname*{Minimize:}_{\mathbf{p}_{{g}_{i}}}$
$\displaystyle\frac{\beta_{i}}{2}\left\lVert\mathbf{p}_{{g}_{i}}-\mathbf{p}^{r}_{{g_{i}}}\right\rVert_{2}^{2}+\boldsymbol{\lambda}^{\top}_{t}\mathbf{p}_{{g}_{{i}}}$
(12) $\displaystyle\operatorname*{Subject\phantom{a}to:}$
$\displaystyle\underline{\mathbf{p}}_{g_{i}}\preceq\mathbf{p}_{{g_{i}}_{k}}\preceq\overline{\mathbf{p}}_{g_{i}},\hskip
2.84526pt\forall k$
$\displaystyle\left|\mathbf{p}_{{g_{i}}_{k}}-\mathbf{p}_{{{g}_{{i}}}_{k-1}}\right|\preceq{r}_{g}\mathbf{1},\hskip
2.84526pt\forall k,$

where ${r}_{g}\in\mathbb{R}_{+}$ is the ramp-rate limitation associated with
generator. $\overline{\mathbf{p}}_{{{{g}_{i}}}}\in\mathbb{R}_{+}$ and
$\underline{\mathbf{p}}_{{{{g}_{i}}}}\in\mathbb{R}_{+}$ are the upper and
lower limits on generator power respectively.

Figure 3: Distributed Dual Ascent Algorithm Implementation Schematic for
Numerical Simulation.

###  5.3 PCM Node Optimization problem

From (11b) the optimization problem at each battery/PCM node is given in (13).
The discretized SoC dynamics in (3) is introduced as an equality constraint.
Since capturing the PCM degradation is a long process, the heuristic
considered in this work is to minimize the PCM power, which translates into
minimizing the PCM degradation. Thus, the cost is introduced as the weighted
2-norm of the PCM power minimization. The goal is to minimize the usage of
batteries as much as possible. Thus, the optimization problem is as follows:

$\displaystyle\operatorname*{Minimize:}_{\mathbf{p}_{{b_{j}}},\mathbf{SoC}_{b_{j}}}$
$\displaystyle\frac{\gamma_{j}}{2}\left\lVert\mathbf{p}_{b_{j}}\right\rVert_{2}^{2}+\boldsymbol{\lambda}^{\top}_{t}\mathbf{p}_{b_{j}}$
(13) $\displaystyle\operatorname*{Subject\phantom{a}to:}$
$\displaystyle\mathbf{SoC}_{b_{jk+1}}=\mathbf{SoC}_{b_{jk}}-\kappa\mathbf{p}_{b_{jk}},\hskip
2.84526pt\forall k,$
$\displaystyle\underline{\mathbf{p}}_{b_{j}}\preceq\mathbf{p}_{{b_{j}}_{k}}\preceq\overline{\mathbf{p}}_{b_{j}},\hskip
2.84526pt\forall k,$
$\displaystyle\underline{\mathbf{0}}\preceq\mathbf{SoC}_{{b_{j}}_{k}}\preceq\mathbf{1},\hskip
2.84526pt\forall k,$
$\displaystyle\left|\mathbf{p}_{{b_{j}}_{k}}-\mathbf{p}_{{b_{j}}_{k-1}}\right|\preceq{r}_{b}\mathbf{1},\hskip
2.84526pt,\forall k,$

where, $\displaystyle\kappa={T_{d}}/{Qv_{bus}}$, $r_{b}\in\mathbb{R}_{+}$ is
the ramp-rate limitation associated with battery,
$\overline{\mathbf{p}}_{b_{j}}\in\mathbb{R}_{+}$ and
$\underline{\mathbf{p}}_{b_{j}}\in\mathbb{R}_{-}$ are the upper and lower
limits on battery power. This indicates the discharging and the charging mode
of the PCM. The solution to the optimization problems in (12) and (13) are
used in calculating the dual variable update. The schematic for this
aggregation and distribution setup is provided in Fig. 3. All the nodes
present in the system send their current iterate value to the dual updater
node and the information of the previous iterate of the dual variable
$\boldsymbol{\lambda}_{t}$ is sent to all the nodes. Thus, the dual updater,
which acts as a common scheduler to all nodes implements the update law given
in (11c). The convergence of the proposed algorithm under the assumed
objectives and the constraint sets which are polytopes is well established in
the literature ([18]; [19]).

##  6 Numerical Simulation

The designed model predictive energy management strategy is tested on a single
PGM, PCM, and Load scenario to fully study and analyze the effect of the
weights $\beta$ and $\gamma$ on the total energy delivered by the PCM, PGM,
and its effect on the PCM capacity loss %. The simulation environment used in
this setup is based on MATLAB-Simulink on a real-time simulator. The
simulation is run on a desktop with the following configurations Digital Storm
Intel Core i9-13900K (5.7GHz Turbo) with 64GB RAM acting as a Host PC and a
real-time target with Intel Core 3.6GHz 8-core and 32GB RAM (shown in Fig. 3).
The optimization problem presented in (8) with $n_{g}=1$ and $n_{b}=1$ is
implemented to study the trade-off characteristics. The parameters used are
based on the documentation and the notional SPS component sizing provided by
the Office of Naval Research (ONRs) ESRDC ([3]). For all the results
presented, the prediction horizon is fixed at ($h=5$seconds). The simulation
was run at the fixed step of $10^{-3}$seconds. The rate transition or the
communication (measurement exchange) delay between the model and the MPC EM
layer was set at $1$second to facilitate optimization convergence.

Figure 4: The effect of different weighting of $\gamma$ and $\beta$ versus the
battery energy.

Since the core focus of the work is on determining the effect of the weights
$\beta$ and $\gamma$ on the generator and the battery usage and finding a
trade-off between them, the results focus on the trade-off part. First, the
developed EM algorithm is tested on a _single PGM, PCM, and Load_ model. Fig.
4 shows the total energy the battery utilized in MWHr on the y-axis versus the
weight on the x-axis. The _solid lines_ in the figure present the effect of
the battery penalty parameter $\gamma$ tuned from 0-10 against the generator
penalty parameter $\beta$. As the $\gamma$ increases the total energy utilized
by the battery decreases and can be seen around 30MWHr. This trend shifts
upward for the penalty tuning of $\gamma$ from 0-10 but for the increased
value of $\beta$. This shift is attributed to the power balance constraint.
Similarly _dotted lines_ depict the effect of the generator penalty parameter
$\beta$ tuned from 0-10 against the battery penalty $\gamma$. It can be seen
that as the value of the $\beta$ increases, the total energy utilized by the
battery increases since the emphasis is on maintaining the generator around
the rated operating point.

Figure 5: The effect of different weighting $\gamma$ and $\beta$ versus the
generator energy. Figure 6: Battery Capacity Loss $\%$ vs the Weight $\gamma$.

In Fig. 5 an experiment similar to the previous one is performed, and the
energy of the generator is captured versus the weights $\gamma$ and $\beta$.
The trends observed for the generator are opposite to the ones observed for
the battery. Mathematically this is what is expected to maintain the power
supply and demand equality constraint. The _solid lines_ depicts the effect of
the weight $\gamma$ tuned from 0-10 while keeping $\beta=1$ and then at
$\beta=2$ and $3$. When $\gamma=0$, the objective functions goal is minimizing
$\frac{\beta}{2}\left\lVert\mathbf{p}_{g}-\mathbf{p}_{g}^{r}\right\rVert_{2}^{2}$.
It can be seen that in this case, the energy of the generator is around
6830MWHr. But as the value of the weight $\gamma$ goes to 10, the objective
shifts to minimizing the battery usage, and thus, the generator energy goes up
to maintain the power balance. Similarly, the _dotted lines_ show the effect
of weight $\beta$ swept from 0-10 on the generator energy when the weight
$\gamma=1$, $\gamma=2$, and $\gamma=3$. Fig. 6 shows the effect of the weight
$\gamma$ incremented from 0-10 while keeping the weight $\beta=1$ on the
battery capacity loss %. It can be seen that, as the emphasis on the weight
$\gamma$ increases the capacity loss percent decreases. In summary, the total
capacity loss decreases with $\gamma$ but increases with $\beta$. From the
simulated results, we can conclude that a designer can tune the weights
$\beta$ and $\gamma$ to obtain the optimal performance of the generator and
the battery.

##  7 Conclusion

A distributed MPC-based energy management strategy considering energy storage
degradation is presented for a shipboard power system. The generator, battery
ramp rate limitations, generator rated conditions, the pulsed power load
conditions are considered. Absolute power extracted from, the battery as a
model-based battery degradation heuristic capturing the battery usage is
proposed and used in the optimization. Finally, a numerical case study shows
the trade-off in the power-sharing and the energy exchange for different
optimization weights is presented.

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Hyperbolic geometry of the ample cone
of a hyperkähler manifold

Ekaterina Amerik111Partially supported by RScF grant, project 14-21-00053,
11.08.14., Misha Verbitsky222Partially supported by RScF grant, project
14-21-00053, 11.08.14. Keywords: hyperkähler manifold, Kähler cone, hyperbolic
geometry, cusp points 2010 Mathematics Subject Classification: 53C26, 32G13

Abstract

Let $M$ be a compact hyperkähler manifold with maximal holonomy (IHS). The
group $H^{2}(M,{\mathbb{R}})$ is equipped with a quadratic form of signature
$(3,b_{2}-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form
restricted to the rational Hodge lattice $H^{1,1}(M,{\mathbb{Q}})$, has
signature $(1,k)$. This gives a hyperbolic Riemannian metric on the
projectivisation of the positive cone in $H^{1,1}(M,{\mathbb{Q}})$, denoted by
$H$. Torelli theorem implies that the Hodge monodromy group $\Gamma$ acts on
$H$ with finite covolume, giving a hyperbolic orbifold $X=H/\Gamma$. We show
that there are finitely many geodesic hypersurfaces which cut $X$ into
finitely many polyhedral pieces in such a way that each of these pieces is
isometric to a quotient $P(M^{\prime})/\operatorname{Aut}(M^{\prime})$, where
$P(M^{\prime})$ is the projectivization of the ample cone of a birational
model $M^{\prime}$ of $M$, and $\operatorname{Aut}(M^{\prime})$ the group of
its holomorphic automorphisms. This is used to prove the existence of nef
isotropic line bundles on a hyperkähler birational model of a simple
hyperkähler manifold of Picard number at least 5, and also illustrates the
fact that an IHS manifold has only finitely many birational models up to
isomorphism (cf. [MY]).

###### Contents

  1. 1 Introduction
  2. 2 Hyperkähler manifolds: basic results
    1. 2.1 Hyperkähler manifolds
    2. 2.2 MBM classes
    3. 2.3 Morrison-Kawamata cone conjecture
  3. 3 Hyperbolic geometry and the Kähler cone
    1. 3.1 Kleinian groups and hyperbolic manifolds
    2. 3.2 The cone conjecture and hyperbolic geometry
  4. 4 Cusps and nef parabolic classes

##  1 Introduction

Let $M$ be an irreducible holomorphically symplectic manifold, that is, a
simply-connected compact Kähler manifold with $H^{2,0}(M)={\mathbb{C}}\Omega$
where $\Omega$ is nowhere degenerate. In dimension two, such manifolds are K3
surfaces; in higher dimension $2n,\ n>1,$ one knows, up to deformation, two
infinite series of such manifolds, namely the punctual Hilbert schemes of K3
surfaces and the generalized Kummer varieties, and two sporadic examples
constructed by O’Grady. Though considerable effort has been made to construct
other examples, none is known at present, and the classification problem for
irreducible holomorphic symplectic manifolds (IHSM) looks equally out of
reach.

One of the main features of an IHSM $M$ is the existence of an integral
quadratic form $q$ on the second cohomology $H^{2}(M,{\mathbb{Z}})$, the
Beauville-Bogomolov-Fujiki form (BBF) form. It generalizes the intersection
form on a surface; in particular its signature is $(3,b_{2}-3)$, and the
signature of its restriction to $H^{1,1}_{\mathbb{R}}(M)$ is $(1,b_{2}-3)$.
The cone $\\{x\in H^{1,1}_{\mathbb{R}}(M)|x^{2}>0\\}$ thus has two connected
components; we call the positive cone $\operatorname{Pos}(X)$ the one which
contains the Kähler classes. The BBF form is in fact of topological origin: by
a formula due to Fujiki, $q(\alpha)^{n}$ is proportional to $\alpha^{2n}$ with
a positive coefficient depending only on $M$.

To understand better the geometry of an IHSM, it can be useful to fiber it,
whenever possible, over a lower-dimensional variety. Note that by a result of
Matsushita, the fibers are always Lagrangian (in particular, $n$-dimensional,
where $2n=\dim_{\mathbb{C}}M$), and the general fiber is a torus. Such
fibrations are important for the classification-related problems, and one can
also hope to get some interesting geometry from their degenerate fibers (for
instance, use them to construct rational curves on $M$).

Note that a fibration of $M$ is necessarily given by a linear system $|L|$
where $|L|$ is a nef line bundle with $q(L)=0$. Conjecturally, any such bundle
is semiample, that is, for large $m$ the linear system $|L^{\otimes m}|$ is
base-point-free and thus gives a desired fibration.

It is therefore important to understand which irreducible holomorphic
symplectic varieties carry nef line bundles of square zero. By Meyer’s theorem
(see for example [Se]), $M$ has an integral $(1,1)$-class of square zero as
soon as the Picard number $\rho(M)$ is at least five. By definition, such a
class is nef when it is in the closure of the Kähler cone
$\operatorname{Kah}(M)\subset\operatorname{Pos}(M)$. The question is thus to
understand whether one can find an isotropic integral $(1,1)$-class in the
closure of the Kähler cone.

For projective K3 surfaces, this is easy and has been done in [PSh-Sh]. Indeed
$\operatorname{Kah}(M)\subset\operatorname{Pos}(M)$ is cut out by the
orthogonal hyperplanes to $(-2)$-classes, since a positive $(1,1)$-class is
Kähler if and only if it restricts positively on all $(-2)$-curves, and
$(-2)$-classes on a K3 surface are $\pm$-effective by Riemann-Roch. Let $x$ be
an isotropic integral $(1,1)$-class and suppose that
$x\not\in\overline{\operatorname{Kah}(M)}$, that is, there is a $(-2)$-curve
$p$ with $\langle x,p\rangle<0$. Fix an ample integral $(1,1)$-class $h$. Then
the image of $x$ under the reflection in $p^{\bot}$, $x^{\prime}=x+\langle
x,p\rangle p$, satisfies $\langle x^{\prime},h\rangle<\langle x,h\rangle$.
Therefore the image of $x$ under successive reflexions in such $p$’s becomes
nef after finitely many steps. The non-projective case is even easier, since
an isotropic line bundle must then be in the kernel of the Neron-Severi
lattice and so has zero intersection with every curve, in particular, it is
nef.

Trying to apply the same argument to higher-dimensional IHSM we see that the
existence of an isotropic line bundle yields and isotropic element in the
closure of the birational Kähler cone $\operatorname{BK}(M)$. By definition,
$\operatorname{BK}(M)$ is a union of inverse images of the Kähler cone on all
IHSM birational models of $M$, and its closure is cut out in
$\operatorname{Pos}(X)$ by the Beauville-Bogomolov orthogonals to the classes
of the prime uniruled exceptional divisors ([Bou1]). One knows that the
reflections in those hyperplanes are integral ([M1]); in particular the
divisors have bounded squares and the “reflections argument” above applies
with obvious modifications.

A priori, the closure of $\operatorname{BK}(M)$ may strictly contain the union
of the closures of the inverse images of the Kähler cones of all birational
models, so an additional argument is required to conclude that there is an
isotropic nef class on some birational model of $M$. One way to deal with this
is explained in the paper [MZ]: the termination of flops on an IHSM implies
that any element of the closure of $\operatorname{BK}(M)$ does indeed become
nef on some birational model. These observations, though, require the use of
rather heavy machinery of the Minimal Model Program (MMP) which are in
principle valid on all varieties (though the termination of flops itself
remains unproven in general).

The purpose of the present note is to give another proof of the existence of
nef isotropic classes, which does not rely on the MMP. Instead it relies on
the “cone conjecture” which was established in [AV2] using completely
different methods, namely ergodic theory and hyperbolic geometry. We find the
hyperbolic geometry picture which appears in our proof particularly appealing,
and believe that it might provide an alternative, perhaps sometimes more
efficient, approach to birational geometry in the particular case of the
irreducible holomorphic symplectic manifolds.

The main advantage of the present construction is its geometric
interpretation. The BBF quadratic form, restricted to the rational Hodge
lattice $H^{1,1}(M,{\mathbb{Q}})$, has signature $(1,k)$ (unless $M$ is non-
algebraic, in which case our results are tautologies). This gives a hyperbolic
Riemannian metric on the projectivisation of the positive cone in
$H^{1,1}(M,{\mathbb{Q}})$, denoted by $H$. Torelli theorem implies that the
group $\Gamma^{\operatorname{\sf Hdg}}$ of Hodge monodromy acts on $H$ with
finite covolume, giving a hyperbolic orbifold $X=H/\Gamma^{\operatorname{\sf
Hdg}}$. Using Selberg lemma, one easily reduces to the case when $X$ is a
manifold. We prove that $X$ is cut into finitely many polyhedral pieces by
finitely many geodesic hypersurfaces in such a way that each of these pieces
is isometric to a quotient
$\operatorname{Amp}(M^{\prime})/\operatorname{Aut}(M^{\prime})$, where
$\operatorname{Amp}(M^{\prime})$ is the projectivization of the ample cone of
a birational model of $M$, and $\operatorname{Aut}(M^{\prime})$ the group of
holomorphic automorphisms.

In this interpretation, equivalence classes of birational models are in
bijective correspondence with these polyhedral pieces $H_{i}$, and the
isotropic nef line bundles correspond to the cusp points of these $H_{i}$.
Existence of cusp points is implied by Meyer’s theorem, and finiteness of
$H_{i}$ by our results on the cone conjecture from [AV2] (Section 3). Finally,
the geometric finiteness results from hyperbolic geometry imply the finiteness
of the isotropic nef line bundles up to automorphisms.

##  2 Hyperkähler manifolds: basic results

In this section, we recall the definitions and basic properties of hyperkähler
manifolds and MBM classes.

###  2.1 Hyperkähler manifolds

Definition 2.1: A hyperkähler manifold $M$, that is, a compact Kähler
holomorphically symplectic manifold, is called simple (alternatively,
irreducible holomorphically symplectic (IHSM)), if $\pi_{1}(M)=0$ and
$H^{2,0}(M)={\mathbb{C}}$.

This definition is motivated by the following theorem of Bogomolov.

Theorem 2.2:  ([Bo1]) Any hyperkähler manifold admits a finite covering which
is a product of a torus and several simple hyperkähler manifolds.

The second cohomology $H^{2}(M,{\mathbb{Z}})$ of a simple hyperkähler manifold
$M$ carries a primitive integral quadratic form $q$, called the Bogomolov-
Beauville-Fujiki form. It generalizes the intersection product on a K3
surface: its signature is $(3,b_{2}-3)$ on $H^{2}(M,{\mathbb{R}})$ and
$(1,b_{2}-3)$ on $H^{1,1}_{{\mathbb{R}}}(M)$. It was first defined in [Bo2]
and [Bea], but it is easiest to describe it using the Fujiki theorem, proved
in [F].

Theorem 2.3:  (Fujiki) Let $M$ be a simple hyperkähler manifold, $\eta\in
H^{2}(M)$, and $n=\frac{1}{2}\dim M$. Then
$\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, where $q$ is a primitive integer
quadratic form on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ is a rational number.

Definition 2.4: Let $M$ be a hyperkähler manifold. The monodromy group of $M$
is a subgroup of $GL(H^{2}(M,{\mathbb{Z}}))$ generated by the monodromy
transforms for all Gauss-Manin local systems.

It is often enlightening to consider this group in terms of the mapping class
group action. We briefly recall this description.

The Teichmüller space $\operatorname{\sf Teich}$ is the quotient
$\operatorname{\sf Comp}(M)/\operatorname{\sf Diff}_{0}(M)$, where
$\operatorname{\sf Comp}(M)$ denotes the space of all complex structures of
Kähler type on $M$ and $\operatorname{\sf Diff}_{0}(M)$ is the group of
isotopies. It follows from a result of Huybrechts (see [H2]) that for an IHSM
$M$, $\operatorname{\sf Teich}$ has only finitely many connected components.
Let $\operatorname{\sf Teich}_{M}$ denote the one containing our given complex
structure. Consider the subgroup of the mapping class group $\operatorname{\sf
Diff}(M)/\operatorname{\sf Diff}_{0}(M)$ fixing $\operatorname{\sf
Teich}_{M}$.

Definition 2.5:  The monodromy group $\Gamma$ is the image of this subgroup in
$O(H^{2}(M,{\mathbb{Z}}),q)$. The Hodge monodromy group
$\Gamma^{{\operatorname{\sf Hdg}}}$ is the subgroup of $\Gamma$ preserving the
Hodge decomposition.

Theorem 2.6:  ([V1], Theorem 3.5) The monodromy group is a finite index
subgroup in $O(H^{2}(M,{\mathbb{Z}}),q)$ (and the Hodge monodromy is therefore
an arithmetic subgroup of the orthogonal group of the Picard lattice).

###  2.2 MBM classes

Definition 2.7: A cohomology class $\eta\in H^{2}(M,{\mathbb{R}})$ is called
positive if $q(\eta,\eta)>0$, and negative if $q(\eta,\eta)<0$. The positive
cone $\operatorname{Pos}(M)\in H^{1,1}_{{\mathbb{R}}}(M)$ is that one of the
two connected components of the set of positive classes on $M$ which contains
the Kähler classes.

Recall e.g. from [M2] that the positive cone is decomposed into the union of
birational Kähler chambers, which are monodromy transforms of the birational
Kähler cone $\operatorname{BK}(M)$. The birational Kähler cone is, by
definition, the union of pullbacks of the Kähler cones
$\operatorname{Kah}(M^{\prime})$ where $M^{\prime}$ denote a hyperkähler
birational model of $M$ (the “Kähler chambers”). The faces111A face of a
convex cone in a vector space $V$ is the intersection of its boundary and a
hyperplane which has non-empty interior in the hyperplane. of these chambers
are supported on the hyperplanes orthogonal to the classes of prime uniruled
divisors of negative square on $M$.

The MBM classes are defined as those classes whose orthogonal hyperplanes
support faces of the Kähler chambers.

Definition 2.8:  A negative integral cohomology class $z$ of type $(1,1)$ is
called monodromy birationally minimal (MBM) if for some isometry $\gamma\in
O(H^{2}(M,{\mathbb{Z}}))$ belonging to the monodromy group,
$\gamma(z)^{\bot}\subset H^{1,1}_{{\mathbb{R}}}(M)$ contains a face of the
Kähler cone of one of birational models $M^{\prime}$ of $M$.

Geometrically, the MBM classes are characterized among negative integral
$(1,1)$-classes, as those which are, up to a scalar multiple, represented by
minimal rational curves on deformations of $M$ under the identification of
$H_{2}(M,{\mathbb{Q}})$ with $H^{2}(M,{\mathbb{Q}})$ given by the BBF form
([AV1], [AV3], [KLM]).

The following theorems summarize the main results about MBM classes from
[AV1].

Theorem 2.9: ([AV1], Corollary 5.13) An MBM class $z\in H^{1,1}(M)$ is also
MBM on any deformation $M^{\prime}$ of $M$ where $z$ remains of type $(1,1)$.

Theorem 2.10: ([AV1], Theorem 6.2) The Kähler cone of $M$ is a connected
component of $\operatorname{Pos}(M)\backslash\cup_{z\in S}z^{\bot}$, where $S$
is the set of MBM classes on $M$.

In what follows, we shall also consider the positive cone in the algebraic
part $NS(M)\otimes{\mathbb{R}}$ of $H^{1,1}_{{\mathbb{R}}}(M)$, denoted by
$\operatorname{Pos}_{{\mathbb{Q}}}(M)$. Here and further on, $NS(M)$ stands
for Néron-Severi group of $M$.

Definition 2.11: The ample chambers are the connected components of
$\operatorname{Pos}_{{\mathbb{Q}}}(M)\backslash\cup_{z\in S}z^{\bot}$ where
$S$ is the set of MBM classes on $M$.

One of the ample chambers is, obviously, the ample cone of $M$, hence the
name.

In the same way, one defines birationally ample or movable chambers as the
connected components of the complement to the union of orthogonals to the
classes of uniruled divisors and their monodromy transforms, cf. [M2], section
6. These are also described as intersections of the biratonal Kähler chambers
with $NS(M)\otimes{\mathbb{R}}$.

Remark 2.12: Because of the deformation-invariance property of MBM classes, it
is natural to introduce this notion on $H^{2}(M,{\mathbb{Z}})$ rather than on
$(1,1)$-classes: we call $z\in H^{2}(M,{\mathbb{Z}})$ an MBM class as soon as
it is MBM in those complex structures where it is of type $(1,1)$.

###  2.3 Morrison-Kawamata cone conjecture

The following theorem has been proved in [AV2].

Theorem 2.13: ([AV2]) Suppose that the Picard number $\rho(M)>3$. Then the
Hodge monodromy group has only finitely many orbits on the set of MBM classes
of type $(1,1)$ on $M$.

Since the Hodge monodromy group acts by isometries, it follows that the
primitive MBM classes have bounded square (using the deformation argument, one
easily extends this last statement from the case of $\rho(M)>3$ to that of
$b_{2}(M)\neq 5$, but we shall not need this here). In [AV1] we have seen that
this implies some apriori stronger statements on the Hodge monodromy action.

Corollary 2.14:  The Hodge monodromy group has only finitely many orbits on
the set of faces of the Kähler chambers, as well as on the set of the Kähler
chambers themselves.

For reader’s convenience, let us briefly sketch the proof (for details, see
sections 3 and 6 of [AV1]). It consists in remarking that a face of a chamber
is given by a flag $P_{s}\supset P_{s-1}\supset\dots\supset P_{1}$ where
$P_{s}$ is the supporting hyperplane (of dimension $s=h^{1,1}-1$), $P_{s-1}$
supports a face of our face, etc., and for each $P_{i}$ an orientation
(“pointing inwards the chamber”) is fixed. One deduces from the boundedness of
the square of primitive MBM classes that possible $P_{s-1}$ are as well given
inside $P_{s}$ by orthogonals to integral vectors of bounded square, and it
follows that the stabilizer of $P_{s}$ in $\Gamma^{{\operatorname{\sf Hdg}}}$
acts with finitely many orbits on those vectors; continuing in this way one
eventually gets the statement.

By Markman’s version of the Torelli theorem [M2], an element of
$\Gamma^{{\operatorname{\sf Hdg}}}$ preserving the Kähler cone actually comes
from an automorphism of $M$. Thus an immediate consequence is the following
Kähler version of the Morrison-Kawamata cone conjecture.

Corollary 2.15: ([AV2]) $\operatorname{Aut}(M)$ has only finitely many orbits
on the set of faces of the Kähler cone.

Remark 2.16:  As the faces of the ample cone are likewise given by the
orthogonals to MBM classes, but in $\operatorname{Pos}_{{\mathbb{Q}}}(M)$
rather than in $\operatorname{Pos}(M)$, one concludes that the same must be
true for the ample cone.

##  3 Hyperbolic geometry and the Kähler cone

###  3.1 Kleinian groups and hyperbolic manifolds

Definition 3.1: A Kleinian group is a discrete subgroup of isometries of the
hyperbolic space ${\mathbb{H}}^{n}$.

One way to view ${\mathbb{H}}^{n}$ is as a projectivization of the positive
cone ${\mathbb{P}}V^{+}$ of a quadratic form $q$ of signature $(1,n)$ on a
real vector space $V$. The Kleinian groups are thus discrete subgroups of
$SO(1,n)$. One calls such a subgroup a lattice if its covolume is finite.

Definition 3.2: An arithmetic subgroup of an algebraic group $G$ defined over
the integers is a subgroup commensurable with $G_{\mathbb{Z}}$.

Remark 3.3:  From Borel and Harish-Chandra theorem (see [BHCh]) it follows
that when $q$ is integral, any arithmetic subgroup of $SO(1,n)$ is a lattice
for $n\geqslant 2$.

Definition 3.4: A complete hyperbolic orbifold is a quotient of the hyperbolic
space by a Kleinian group. A complete hyperbolic manifold is a quotient of the
hyperbolic space by a Kleinian group acting freely.

Remark 3.5: One defines a hyperbolic manifold as a manifold of constant
negative bisectional curvature. When complete, such a manifold is uniformized
by the hyperbolic space ([Th]).

The following proposition is well-known.

Proposition 3.6:  Any complete hyperbolic orbifold has a finite covering which
is a complete hyperbolic manifold (in other words, any Kleinian group has a
finite index subgroup acting freely).

Proof: Let $\Gamma$ be a Kleinian group. Notice first that all stabilizers for
the action of $\Gamma$ on ${\mathbb{P}}V^{+}$ are finite, since these are
identified to discrete subgroups of a compact group $SO(n)$. Now by Selberg
lemma $\Gamma$ has a finite index subgroup without torsion which must
therefore act freely.

Remark 3.7: If $M$ is an IHSM, the group of Hodge monodromy
$\Gamma^{{\operatorname{\sf Hdg}}}$ is an arithmetic lattice in
$SO(H^{1,1}(M,{\mathbb{Q}}))$ when
$\operatorname{rk}H^{1,1}(M,{\mathbb{Q}})\geqslant 3$. The hyperbolic manifold
${\mathbb{P}}(H^{1,1}(M,{\mathbb{Q}})\otimes_{\mathbb{Q}}{\mathbb{R}})^{+}/\Gamma^{\operatorname{\sf
Hdg}}$ has finite volume by Borel and Harish-Chandra theorem.

###  3.2 The cone conjecture and hyperbolic geometry

Recall that the rational positive cone $\operatorname{Pos}_{\mathbb{Q}}(M)$ of
a projective hyperkähler manifold $M$ is one of two connected components of
the set of positive vectors in $NS(M)\otimes{\mathbb{R}}$.

Replacing $\Gamma^{{\operatorname{\sf Hdg}}}$ by a finite index subgroup if
necessary, we may assume that the quotient
${\mathbb{P}}\operatorname{Pos}_{\mathbb{Q}}(M)/\Gamma^{{\operatorname{\sf
Hdg}}}$ is a complete hyperbolic manifold which we shall denote by $H$.

By Borel and Harish-Chandra theorem (see 3.1), $H$ is of finite volume as soon
as the Picard number of $M$ is at least three.

Let $S=\\{s_{i}\\}$ be the set of MBM classes of type $(1,1)$ on $M$. The
following is a translation of the Morrison-Kawamata cone conjecture into the
setting of hyperbolic geometry.

Theorem 3.8: The images of the hyperplanes $s_{i}^{\bot}$, $s_{i}\in S$, cut
$H={\mathbb{P}}\operatorname{Pos}_{\mathbb{Q}}(M)/\Gamma^{Hdg}$ into finitely
many pieces. One of those pieces is the image of the ample cone (up to a
finite covering, this is the quotient of the ample cone by
$\operatorname{Aut}(M)$) and the others are the images of ample cones of
birational models of $M$. The closure of each one is a hyperbolic manifold
with boundary consisting of finitely many geodesic pieces.

Proof: According to 2.3, up to the action of $\Gamma^{{\operatorname{\sf
Hdg}}}$ there are finitely many faces of ample chambers. Each face is a
connected component of the complement to $\cup_{j\neq i}s_{j}^{\bot}$ in
$s_{i}^{\bot}$ for some $i$. It is clear that the images of the faces do not
intersect hence, being finitely many, cut $H$ into finitely many pieces which
are images of the ample chambers. We have already mentioned that an element of
$\Gamma^{{\operatorname{\sf Hdg}}}$ preserving the Kähler cone is induced by
an automorphism. Finally, the whole $H$ is covered by the birational ample
cone (since the other birational ample chambers are its monodromy transforms)
and thus each part of $H$ obtained in this way comes from an ample chamber.

Let us also mention that the same arguments also prove the following result
(cf. [MY]).

Corollary 3.9: There are only finitely many non-isomorphic birational models
of $M$.

Proof: Indeed, the Kähler (or ample) chambers in the same
$\Gamma^{{\operatorname{\sf Hdg}}}$-orbit correspond to isomorphic birational
models, since one can view the action of $\Gamma^{{\operatorname{\sf Hdg}}}$
as the change of the marking (recall that a marking is a choice of an isometry
of $H^{2}(M,{\mathbb{Z}})$ with a fixed lattice $\Lambda$ and that there
exists a coarse moduli space of marked IHSM which in many works (e.g. [H1])
plays the same role as the Teichmüller space in others).

##  4 Cusps and nef parabolic classes

Definition 4.1: A horosphere on a hyperbolic space is a sphere which is
everywhere orthogonal to a pencil of geodesics passing through one point at
infinity, and a horoball is a ball bounded by a horosphere. A cusp point for
an $n$-dimensional hyperbolic manifold ${\mathbb{H}}/\Gamma$ is a point on the
boundary $\partial{\mathbb{H}}$ such that its stabilizer in $\Gamma$ contains
a free abelian group of rank $n-1$. Such subgroups are called maximal
parabolic. For any point $p\in\partial{\mathbb{H}}$ stabilized by
$\Gamma_{0}\subset\Gamma$, and any horosphere $S$ tangent to the boundary in
$p$, $\Gamma_{0}$ acts on $S$ by isometries. In such a situation, $p$ is a
cusp point if and only if $(S\backslash p)/\Gamma_{0}$ is compact.

A cusp point $p$ yields a cusp in the quotient ${\mathbb{H}}/\Gamma$, that is,
a geometric end of ${\mathbb{H}}/\Gamma$ of the form $B/{\mathbb{Z}}^{n-1}$,
where $B\subset{\mathbb{H}}$ is a horoball tangent to the boundary at $p$.

The following theorem describes the geometry of finite volume complete
hyperbolic manifolds more precisely.

Theorem 4.2: (Thick-thin decomposition)
Any $n$-dimensional complete hyperbolic manifold of finite volume can be
represented as a union of a “thick part”, which is a compact manifold with a
boundary, and a “thin part”, which is a finite union of quotients of form
$B/{\mathbb{Z}}^{n-1}$, where $B$ is a horoball tangent to the boundary at a
cusp point, and ${\mathbb{Z}}^{n-1}=\operatorname{St}_{\Gamma}(B)$.

Proof: See [Th, Section 5.10] or [Ka, page 491]).

Theorem 4.3:  Let ${\mathbb{H}}/\Gamma$ be a hyperbolic manifold, where
$\Gamma$ is an arithmetic subgroup of $SO(1,n)$. Then the cusps of
${\mathbb{H}}/\Gamma$ are in (1,1)-correspondence with $Z/\Gamma$, where $Z$
is the set of rational lines $l$ such that $l^{2}=0$.

Proof: By definition of cusp points, the cusps of ${\mathbb{H}}/\Gamma$ are in
1 to 1 correspondence with $\Gamma$-conjugacy classes of maximal parabolic
subgroups of $\Gamma$ (see [Ka]). Each such subgroup is uniquely determined by
the unique point it fixes on the boundary of ${\mathbb{H}}$.

The main result of this paper is the following theorem.

Theorem 4.4: Let $M$ be a hyperkähler manifold with Picard number at least 5.
Then $M$ has a birational model admitting an integral nef $(1,1)$-class $\eta$
with $\eta^{2}=0$. Moreover each birational model contains only finitely many
such classes up to automorphism.

Proof: By Meyer’s theorem (see for example [Se]), there exists $\eta\in NS(M)$
with $\eta^{2}=0$. By 4, the hyperbolic manifold
$H:={\mathbb{P}}\operatorname{Pos}_{\mathbb{Q}}(M)/\Gamma^{{\operatorname{\sf
Hdg}}}$ then has cusps, and, being of finite volume, only finitely many of
them. Recall that $H$ is decomposed into finitely many pieces, and each of
those pieces is the image of the ample cone of a birational model of $M$ in
$\operatorname{Pos}_{\mathbb{Q}}(M)$. Therefore a lifting of each cusp to the
boundary of ${\mathbb{P}}\operatorname{Pos}_{\mathbb{Q}}(M)$ gives a BBF-
isotropic nef line bundle on a birational model $M^{\prime}$ of $M$ (or, more
precisely, the whole line such a bundle generates in
$NS(M)\otimes{\mathbb{R}}$). Finally, the number of
$\operatorname{Aut}(M^{\prime})$-orbits of such classes is finite, being
exactly the number of cusps in the piece of $H$ corresponding to $M^{\prime}$:
indeed this piece is just the quotient of the ample cone of $M^{\prime}$ by
its stabilizer which is identified with $\operatorname{Aut}(M^{\prime})$.

Acknowledgements: We are grateful to S. Cantat, M. Kapovich and V. Gritsenko
for interesting discussions and advice.

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Ekaterina Amerik
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Paris-11,
Laboratoire de Mathématiques,
Campus d’Orsay, Bâtiment 425, 91405 Orsay, France

Misha Verbitsky
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Libre de Bruxelles, CP 218,
Bd du Triomphe, 1050 Brussels, Belgium