# Design, Engineering and Optimization of a Grid-Tie Multicell Inverter for
Energy Storage Applications

A. Ashraf Gandomi 1, S. Saeidabadi 1, M. Sabahi 1, M. Babazadeh 1, and Y.
Ashraf Gandomi 2 Manuscript submitted August 26, 2017. Corresponding author:
A. Ashraf Gandomi (email: a.ashrafgandomi91@ms.tabrizu.ac.ir). 1Department of
Electrical and Computer Engineering,
University of Tabriz, Tabriz  2Electrochemical Energy Storage and Conversion
Laboratory,
Department of Mechanical, Aerospace and Biomedical Engineering,
University of Tennessee, Knoxville, Tennessee, USA

###### Abstract

Multilevel converters have found many applications within renewable energy
systems thanks to their unique capability of generating multiple voltage
levels. However, these converters need multiple DC sources and the voltage
balancing over capacitors for these systems is cumbersome. In this work, a new
grid-tie multicell inverter with high level of safety has been designed,
engineered and optimized for integrating energy storage devices to the
electric grid. The multilevel converter proposed in this work is capable of
maintaining the flying capacitors voltage in the desired value. The solar
cells are the primary energy sources for proposed inverter where the maximum
power density is obtained. Finally, the performance of the inverter and its
control method simulated using PSCAD/EMTDC software package and good agreement
achieved with experimental data.

###### Index Terms:

Multicell inverter; Grid-tie multilevel inverter; Maximum Power Point
Tracking; Flying capacitor; Energy storage devices.

##  I Introduction

The application of renewable energy sources (i.e. solar wind energy) into an
electric grid requires high performance energy storage devices along with
various types of power electronics (i.e. rectifiers, converters and
inverters). Figure 1 includes the schematic of a hybrid energy storage system
in which a renewable energy source (here photovoltaic modules) along with an
energy storage device has been implemented to the electric grid via the
utilization the multilevel inverters [1, 2, 3].

Figure 1: Schematic of a hybrid energy system.

As shown in Fig. 1, battery-based devices and hydrogen-based energy storage
technologies are promising. A good review on the battery- and hydrogen-based
energy storage has been provided recently [4]. For hydrogen based energy
storage, an electrolyzer along with a polymer electrolyte membrane (PEM) fuel
cells are utilized. Although being very promising, the water management for
electrolyzers and PEM fuel cells remains a challenging issue to be addressed
[5, 6]. Also, the hydrogen gas storage technologies and infrastructure for
regenerative hydrogen fuel cell systems is required to be addressed
appropriately [7, 8]. Several battery technologies are developed for energy
storage technologies that redox flow batteries are among the most promising
devices . However engineering the redox flow batteries for optimum operation
requires detailed mathematical models [9, 10, 11] and experimental diagnostics
[12, 13].

The application of multilevel converters for the renewable energy systems has
been increased recently [14, 15, 16, 17]. This is mostly because of high
energy output (in the range of MW), high efficiency and low electromagnetic
interference being offered via their application [18, 19, 20].

Three-level converter first proposed by Nabae [21] and the converters with
increased number of levels developed later to obtain higher levels of voltage
and reduce the harmonic content of the output voltage. However, increasing the
number of voltage levels results in increased system complexity and severe
voltage balancing problems. Different topologies for these types of converters
have been utilized including Neutral Point Clamped (NPC) and variant types of
multicell inverters [22, 23, 24]. The NPC topology suffers from diode clamped
and capacitor voltage balancing issues where the multicell inverters do not
suffer from the same issue. Therefore, multicell inverters are promising
candidates to generate increased voltage levels for high power generation
applications. The multicell inverters have different types including the
Cascaded Multicell (CM), Flying Capacitor Multicell (FCM) and Stacked
Multicell (SM) inverters [25, 26, 27]. The FCM inverters along with the SM
inverters have many unique advantages for medium voltage applications such as
operation without transformer and naturally capacitor voltage balancing
ability. A comparison of the different components being used with these
different inverters for generating levels had been provided in Table. I.

Table. I. comparison of different inverter’s components

TABLE I: Comparison of different inverter’s components Type | # DC Sources | Magnitude | # Switches | # Capacitors
---|---|---|---|---
CM | $n$ | $E/n$ | $4n$ | $0$
FCM | $2$ | $E$ | $4n$ | $2n-1$
SM | $2$ | $E$ | $4n$ | $2(n-1)$
DFCM | $1$ | $E$ | $2n+2$ | $n-1$

The advantages of DFCM compared to conventional structures are eliminating the
common point of DC sources, reducing the number of input DC sources, switches
and capacitors. However, this structure is an island and in the grid-tie form
of this inverter with presented control method, some difficulties arise
including the power transfer from grid to input source and the possibility of
discharging flying capacitors in case of connecting to the Photovoltaic (PV)
cells. In this paper a new topology for grid-tie multicell inverter has been
developed. The proposed inverter has been designed based upon the DFCM
inverter and accordingly shares the common advantages associated with the
DFCMs including the elimination of common point of DC source, reduced number
of DC sources and increased number of voltage levels. Also, the new design
assures higher level of safety and via its utilization, the common problems
related to the DFCMs has been eliminated (while extracting the maximum power
from PV cells).

##  II PROPOSED MULTILEVEL INVERTER

The proposed inverter is shown in Fig. 2. In this structure independent of the
number of inverter cells and generated voltage levels, only the switches
$S_{n}$ and $P_{n}$ in $cell_{n}$ are bidirectional. The oscillations of the
PV cells voltage is mostly caused by climatic oscillations. In case of low PV
cells and because of the cascaded diode of PV cells would not be functional
and therefore, the possibility of discharging flying capacitors or grid
voltage on the PV cells increases. Discharging the grid in the positive half
cycle, the negative half cycle and discharging the flying capacitors are shown
in Fig. 3. According to Fig. 3, in the proposed inverter, by using only two
bidirectional switches, there would be no chance of discharging the grid and
the flying capacitors on the PV cells.

Table II, includes the switching states within the proposed inverter for
generating 5-level output voltage. In this table, when the switch is on, the
corresponding state number is one and for the off position it is zero. Also X
in this table due to modes of charging or discharging of flying capacitor
$C_{1}$, is either one or zero. Based on this table, in the states 2 and 4,
charging or discharging of $C_{1}$ happens naturally, while in the other
states, due to value of X, $C_{1}$ can be charged. The voltage balancing
algorithm of $C_{1}$ in positive half cycle is shown in Fig. 3. Based on this
figure, if the measured value of voltage of capacitor $C_{1}$ is lower than
its set value ($v_{C1}$), $P_{2}$ (or $S_{2}$) will be on in the state 1 (or
3). This can continues until the voltage of $C_{1}$ reaches its specified
value then $P_{2}$ will be off.

Figure 2: (a) Proposed grid-tie multicell inverter and (b) Proposed 5-level grid-tie multicell inverter. Figure 3: (a) Discharging the grid in the positive half cycle, (b) Discharging the grid in the negative half cycle and (c) Discharging the flying capacitors. TABLE II: Switching states in the proposed inverter States | $S_{1}$ | $S_{2}$ | $S_{3}$ | $P_{1}$ | $P_{2}$ | $P_{3}$
---|---|---|---|---|---|---
1 | $1$ | $1$ | $1$ | $0$ | $X$ | $0$
2 | $1$ | $0$ | $1$ | $0$ | $1$ | $0$
3 | $1$ | $1$ | $0$ | $0$ | $X$ | $1$
4 | $0$ | $1$ | $0$ | $1$ | $0$ | $1$
5 | $0$ | $X$ | $0$ | $1$ | $1$ | $1$
Figure 4: Algorithm of charge and discharge of capacitor in the positive half
cycle of output voltage.

##  III PROPOSED CONTROL METHOD

In the control method, it is assumed that the inverter output current tracks
the favorite sinusoidal reference current. Also as it is shown in Fig. 5, it
is assumed that the proposed inverter (base on IEEE 1547 standard) does not
exchange any reactive power with the grid and only active power exchanges,
therefore:

Figure 5: Proposed control method.

$v_{s}(t)=V_{m}\sin{(wt)}$ (1)

$i_{r}(t)=I_{m}.\sin{(wt)}$ (2)

Figure 6 shows five output voltage levels that have been considered for the
proposed inverter. The proposed control method should be capable of injecting
power from inverter to grid at different operating conditions and therefore,
the maximum voltage generated through the inverter ($E_{1}$) should be greater
than the grid’s maximum voltage ($V_{m}$).

Figure 6: The output voltage levels of proposed inverter.

Due to generating output voltage with low Total Harmonic Distortion (THD), the
values of output voltages are considered as follows:

$E_{2}=\frac{E_{1}}{2},E_{3}=0,E_{4}=\frac{-E_{1}}{2},E_{5}=-E_{1}$ (3)

The grid voltage and reference current is calculated according to (4) and (5)
assuming the circuit current in the constant $t_{k}$ is $i_{s}(t_{k})$:

$v_{s}(t_{k})=V_{m}\sin{(wt_{k})}$ (4)

$i_{r}(t_{k})=I_{m}\sin{(wt_{k})}$ (5)

The equivalent circuits of the proposed inverter are shown in Fig. 7.

Figure 7: (a) Equivalent circuit of the system, (b) Equivalent circuit of
capacitor discharging state, (c) Equivalent circuit of capacitor discharging
state and (d) Equivalent circuit of capacitor charging.

In Fig. 7, $R_{l}$ and $L_{l}$ are the line resistance and inductance between
inverter and grid and $V_{0}$. According to Fig. 7(a):

$L_{1}\frac{di_{s}(t)}{dt}+R_{l}i_{s}(t)=V_{0}+v_{s}(t)=V_{0}+V_{m}\sin{(wt)}$
(6)

Then:

$i_{s}(t_{k})=I_{k}$ (7)

$i_{s}(t)=K_{1}e^{\frac{-R_{l}t}{L_{l}}}+K_{2}\sin{(wt)}+K_{3}\cos{(wt)}+\frac{V_{0}}{R_{l}}$
(8)

$K_{1}=\frac{I_{k}-\frac{V_{m}}{R_{l}^{3}+L_{l}^{2}w^{2}R_{l}}+\frac{V_{m}L_{l}w}{R_{l}^{2}+L_{l}^{2}w^{2}}}{e^{\frac{-R_{l}t}{L_{l}}}}$
(9)

$K_{2}=\frac{V_{m}}{R_{l}^{3}+L_{l}^{2}w^{2}R_{l}}$ (10)

$K_{3}=\frac{-V_{m}L_{l}w}{R_{l}^{2}+L_{l}^{2}w^{2}}$ (11)

In constant $t_{k}$, $i_{s}(t_{k})$ is measured. Control system using (8),
calculates five current for a five-level output voltage of inverter during the
next switching period. Therefore, for five output voltage levels, five cost
functions can be written as follow:

$\Delta{I}=|i_{r})t_{k}+T_{s})-i_{s}(t_{k}+T_{s})|$ (12)

From (12), proper switching state and consequently proper output voltage can
be applied. Initially, when the inverter connects to the network, the initial
voltage of the capacitor is zero. At this point, no impulse current is drawn
until the capacitor voltage reaches the proper value where the grid current is
bypassed by $S_{1}$, $S_{2}$ and $P_{3}$ and simultaneously input source
charged the capacitor by $S_{2}$ and $P_{2}$. When the capacitor voltage
reaches its proper value, control method will follow their normal operation.
In Fig. 8(a) and (b), upper waveforms are inverter output voltage and the
lower waveforms are the inverter output current. Based on Fig. 8(a), if the
initial control method does not apply, 60% of the original period (0.012
seconds) would be required for inverter to reach its own stable performance.
This situation also results in over-drawn current as high as four times of the
maximum nominal current through the circuit, which potentially damages the
whole circuit. However, this situation has improved in Fig. 8(b) where the
response time has decreased to about 0.0005 seconds. Therefore, the inverter
only requires approximately 2.5% of the original period to reach its own
stable performance. Also in this case, the drawn current is within the safe
range.

Figure 8: Voltage stability (a) without initial control method and (b) with
initial control method.

##  IV FLYING CAPACITOR VOLTAGE BALANCING METHOD

In this section, the method of capacitor voltage balancing is investigated.
Based on Table II, the capacitor is discharged in the duration that output
voltage equals to $E/2$ with positive current of circuit and output voltage
equal to $-E/2$ with negative current of circuit. Based on the equivalent
circuit of capacitor discharging states (Fig. 7(b) and (c)):

$v_{s}(t)=V_{m}\sin{(wt)}$ (13)

$i_{s}(t)=I_{m}\sin{(wt)}$ (14)

$v_{s}(t)=V_{c}(t)+L_{l}\frac{di_{L}(t)}{dt}+R_{l}i_{s}(t)$ (15)

$v_{c}(t)=\sqrt{(V_{m}-R_{l}I_{m})^{2}+(L_{l}I_{m})^{2}}\sin[wt+\\\
arcsin(\frac{-L_{l}I_{m}}{\sqrt{(V_{m}-R_{l}I_{m})^{2}+(L_{l}I_{m})^{2}}})]$
(16)

In the equations formulated earlier ((13) to (16)), is the grid voltage and
$v_{c}(t)$ is the capacitor voltage. Also $i_{s}(t)$ is the circuit current
that should be the sine wave because of tracking the reference sine wave.
According to Table II, unlike the case of the capacitor discharging state that
was possible only in two operating modes, charging of capacitor happens in all
five operating modes. When the inverter output voltage is equal to $E/2$ with
negative current of circuit and equal to $-E/2$ with positive current of
circuit, capacitor is charged naturally. But in the duration that output
voltage is equal to $E$ or zero, charging of capacitor is controlled by switch
$P_{2}$ and within the range that output voltage is equal to $-E$ or zero,
charging of capacitor is controlled by switch $S_{2}$. The capacitor charging
circuit in all five modes is shown in Fig. 7(d). In this figure, $R_{s}$ is
the resistance of switches per current loop, therefore the following equations
for a capacitor charging state can is obtained:

$E=R_{s}i_{c}(t)+v_{c}(t)$ (17)

$i_{c}(t)=c\frac{dv_{c}(t)}{dt}$ (18)

$v_{c}(t_{1})=V_{1}$ (19)

$v_{c}(t)=E+\frac{V_{1}-E}{e^{\frac{-t_{1}}{R_{s}C}}}e^{\frac{-t}{R_{s}C}}$
(20)

In the (17) and (18), $v_{c}(t)$ and $i_{c}(t)$ are the voltage and current of
capacitor respectively. The value of $R_{s}$ is very small, therefore the time
of capacitor charging is short and capacitor is charged rapidly. However,
capacitor discharging state is a swinging relation. Thus, capacitor voltage
can be adjusted to the desired value just by controlling the charging state of
capacitor voltage.

##  V DETERMINATION OF CAPACITANCE OF FLYING CAPACITOR

The capacitor charging and discharging states is shown in Fig. 7(b)-(d). The
value of $R_{s}$ is very small, therefore the time of capacitor charging is
short and capacitor’s capacitance determination is applied only for
discharging state of capacitor.

$i_{c}=c\frac{\Delta{V_{c}}}{\Delta{t}}$ (21)

Here, $\Delta{t}$ is the time of single switching duty cycle. $\Delta{V_{c}}$
is the capacitor rated voltage drop in one switching duty cycle which is
assumed five percent of capacitor rated voltage. Consequently:

$T_{s}=\Delta{t}$ (22)

$\Delta{v_{c}}=\%5\frac{E}{2}$ (23)

Circuit current in Fig. 7, ($i_{s}$), is the sine wave that can be formulated
as follow:

$i_{s}=i_{m}\sin{(wt)}$ (24)

$i_{c}$ in (21) is equal to maximum value of current in (24), $I_{m}$, and
therefore (25) is utilized to calculate the capacitor’s capacitance.

$c=\frac{i_{c}\Delta{t}}{\Delta{v_{c}}}$ (25)

##  VI MPPT IN THE PROPOSED INVERTER

Maximum power of PV cells in any constant is shown by $P_{PV}$. The efficiency
of the system is $\eta$, the output is $P_{ac}$. Therefore:

$P_{ac}=\eta{P_{pv}}$ (26)

And:

$P_{ac}=V_{s}I_{s}$ (27)

Here, $V_{s}$ and $I_{s}$ are the RMS value of grid voltage and current
respectively. Therefore, if one assigns any values for $P_{PV}$ and $V_{s}$,
the appropriate value for $I_{ref}$ capable of tracking $I_{s}$ determined and
this guarantees the maximum output power from solar cells.

##  VII VOLTAGE LOSS CALCULATION

In this section, the loss calculation has been done for the proposed topology.
The loss calculation contains switches losses and capacitor losses.

###  VII-A Switches losses

Mainly conduction and switching losses are calculated for switches. Every
power switch contains a transistor (MOSFET) and a diode, and therefore, the
instantaneous conduction loss of these elements is calculated using (28) and
(29), respectively [21]:

$P_{c,T}(t)=[R_{T}i(t)]i(t)$ (28)

$P_{c,D}(t)=[V_{D}+R_{D}i(t)]i(t)$ (29)

Where $V_{D}$ is considered as forward voltage drop of the diode. $R_{T}$ and
$R_{D}$ are equivalent resistance of the transistor and diode. At any time
$N_{T}(t)$ transistor and $N_{D}(t)$ diodes are in current path, so the
average conduction power losses can be written as:

$P_{C}=\frac{1}{2\pi}\int_{0}^{2\pi}[N_{T}(t)P_{c,T}(t)+N_{D}(t)P_{c,D}(t)]dt$
(30)

The switching losses occur during turn on and turn off period of switches. For
simplicity, the linear approximation of voltage and current of switches during
the switching period is considered. Based on this assumption, the switching
loss is calculated as follows:

$E_{off,J}=\int_{0}^{t_{off}}(v(t)i(t))dt=\\\
\int_{0}^{t_{off}}[(\frac{V_{sw,J}t}{t_{off}})(-\frac{I^{\prime\prime}(t-t_{off}}{t_{off}})]dt=\frac{1}{6}V_{sw,J}I^{\prime\prime}t_{off}$
(31)

$E_{on,J}=\int_{0}^{t_{on}}(v(t)i(t))dt=\\\
\int_{0}^{t_{on}}[(\frac{V_{sw,J}t}{t_{on}})(-\frac{I^{\prime}(t-t_{on}}{t_{on}})]dt=\frac{1}{6}V_{sw,J}I^{\prime}t_{on}$
(32)

$E_{off,J}$ and $E_{on,J}$ are turn off and turn on loss of the switch $J$,
$I’’$ is the current through the switch before turning off, $I’$ is the
current through the switches after turning on and $V_{sw,J}$ is the off-state
voltage on the switch. The switching power loss is equal to the sum of all
turns off and turns on energy losses in the fundamental frequency of the
output voltage; as a result, average switching power loss can be calculated as
follow:

$P_{sw}=f\sum_{J=1}^{N_{switch}}(\sum_{k=1}^{N_{on,J}}{E_{on,JK}}+\sum_{k=1}^{N_{off,J}}{E_{off,JK}})$
(33)

$f$ is the fundamental frequency, $N_{on,J}$ and $N_{off,J}$ are the number of
turning on and off the $J^{th}$ switch during fundamental frequency.
$E_{on,JK}$ is the energy loss of the $J^{th}$ switch during the $k^{th}$
turning on and $E_{off,JK}$ is the energy loss of the $J^{th}$ switch during
the $k^{th}$ turning off.

###  VII-B Capacitor losses

The capacitor losses consist of capacitor voltage ripple losses $P_{rip}$ and
conduction losses $P_{cond}$. The voltage ripple of capacitor is calculated as
follow:

$\Delta{V_{rip}}=\frac{1}{C}\int_{t_{-}}^{t_{+}}i_{C}dt$ (34)

Where $i_{C}$, the transient current of the capacitor, equals to the line
current. $t_{-}$ to $t_{+}$ is the interval that the capacitor is connected in
series with DC voltage sources. So, voltage ripple loss is written as follow:

$P_{rip}=C\Delta{V_{rip}^{2}f}$ (35)

$f$ is the frequency of the output voltage. The conduction losses can be
written as follow:

$P_{cond}=2f\int_{t_{-}}^{t_{+}}r_{C}i_{C}^{2}dt$ (36)

Where $r_{c}$ is the internal resistance of the capacitor. Total loss of the
proposed topology is equal:

$P_{loss}=P_{C}+P_{sw}+P_{rip}+P_{cond}$ (37)

Finally, the efficiency of converter is written as follow:

$\eta=\frac{P_{out}}{P_{in}}=\frac{P_{in}-P_{loss}}{P_{in}}$ (38)

Where $P_{out}$ and $P_{in}$ are output power and input power of converter.

##  VIII SIMULATION RESULTS AND EXPERIMENTAL VERIFICATION

The simulation has been done by PSCAD / EMTDC. The circuit used in the
simulation and the experimental set up and the value of elements used in both
of them are shown in Fig. 9 and Table III. Due to the limitation of laboratory
components, DC voltage source is used instead of PV cells.

Figure 9: (a) Circuit of simulation of 5-level proposed inverter and (b) Picture of experimental set up. TABLE III: Quantitative values used for simulation Parameter | Value | Attribute
---|---|---
$E$ | $60$ | Inverter Input Voltage
$V_{m}$ | $48$ | Grid maximum voltage
$R_{l}$ | $0.1$ | Line Resistance
$L_{l}$ | $0.005$ | Line Inductance
$C_{1}$ | $500$ | Flying Capacitor
$f_{s}$ | $10$ | Switching Frequency
$f_{g}$ | $50$ | Grid Frequency
$R_{D-S,on}$ | $0.4$ | Transistor Drain-Source On-State Resistance

For further analysis of inverter’s performance, input DC voltage source is
changing from 60 V to 65 V in 0.04 s and from 65V to 55 V in 0.06 s, the
results are shown in Fig. 10. According to Fig. 10, the proposed inverter with
proposed control method produces 5-level voltage and tracks the reference
current properly in the variation of DC source. Also, Flying capacitor’s
voltage is balanced in its specified value (half of the input DC source).

Figure 10: Input DC source (E), grid voltage (Vs), inverter output voltage
(Eo) and inverter output current (Io).

In the experimental circuit, inverter output voltage and current for 5-level
output voltage is shown in Fig. 11. This figure indicates that the proposed
inverter produces an output voltage waveform with 5 levels with maximum output
level of 60 V. Also in this inverter, output current of approximately 2 A
tracks the reference current properly that is in phase with grid voltage,
therefore the proposed inverter can exchange active power with the grid. In
the laboratory condition the measured output and input power are about 48 (W)
and 52.8 (W), respectively. Therefore, the efficiency is equal 90.91%. Based
on the efficiency calculation, it is about 91.13%. Accordingly, the computed
efficiency has good accordance with the measured efficiency. Voltage and
voltage ripples of flying capacitor are shown in Fig. 12. Base on Fig. 12, the
voltage of flying capacitor is balanced on the desired value with 0.2 ripples.

Figure 11: Grid voltage (50V/div), inverter output voltage (50V/div) and
current (4A/div), 10ms/div. Figure 12: Voltage ripples (200mV/div) and voltage
of flying capacitor (20V/div), 5ms/div.

##  IX Conclusion

In the proposed inverter compared to the structure of DFCM, independent of the
inverter cell number and the number of generated voltage levels, by replacing
only two unidirectional switches with two bidirectional switches, network
connectivity with high safety is provided. Also, this inverter with proposed
control method has unique abilities such as capacitor voltage balancing in any
desired value and obtaining maximum power from PV cells. Theoretically there
is no limitation for the number of flying capacitors for balancing the
voltage. The proposed control method tracks maximum power point of PV cells
through assuring the equal magnitude for the input and output power. Through
this unique controlling methodology, the MPPT is performed just by the
inverter itself. Finally, the mathematical analysis and simulation results are
used to verify the operation of proposed inverter and control method. The
design, engineering and optimization procedure explained in this work provides
further insight for designing multilevel inverters to be used in hybrid
systems equipped with various energy storage devices (i.e. redox flow
batteries and regenerative hydrogen fuel cell systems).

TABLE IV: Abbreviations DC | Direct Current
---|---
MPPT | Maximum Power Point Tracking
NPC | Neutral Point Clamped
CM | Cascaded Multicell
FCM | Flying Capacitor Multicell
SM | Stacked Multicell
DFCM | Double Flying Capacitor Multicell
PV | Photovoltaic
THD | Total Harmonic Distortion
MOSFET | Metal–Oxide–Semiconductor Field-Effect Transistor
TABLE V: Symbols $n$ | Number of cells
---|---
$E$ | Magnitude of DC voltage source
$v_{s}$ | Grid voltage
$i_{r}$ | Reference current
$V_{m}$ | Grid maximum voltage
$i_{s}$ | Grid current
$\omega$ | Angular frequency of grid voltage
$R_{s}$ | Resistance of switches per current path loop
$N_{T}(t)$ | Number of Transistor in current path at any time
$N_{D}(t)$ | Number of Diode in current path at any time
$r_{C}$ | Internal resistance of capacitor
$f_{g}$ | Grid frequency
$f_{s}$ | Switching frequency

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# Barrow holographic dark energy in the Brans-Dicke cosmology

S. Ghaffari<EMAIL_ADDRESS>Research Institute for Astronomy and
Astrophysics of Maragha (RIAAM), University of Maragheh, P.O. Box 55136-553,
Maragheh, Iran  G. G. Luciano<EMAIL_ADDRESS>Applied Physics Section of
Environmental Science Department, Universitat de Lleida, Av. Jaume II, 69,
25001 Lleida, Catalonia, Spain  INFN, Sezione di Napoli, Gruppo collegato di
Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy  S.
Capozziello<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”,
Università di Napoli “Federico II”, Via Cinthia 9, I-80126 Napoli, Italy
Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138 Napoli, Italy
Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, Via Cinthia 9,
I-80126 Napoli, Italy

###### Abstract

We construct a holographic model for dark energy in the Brans-Dicke cosmology
by using the holographic principle considering the Barrow entropy instead of
the standard Bekenstein-Hawking one. The former arises from the effort to
account for quantum-gravitational effects in black-hole physics and, according
to the gravity-thermodynamic conjecture, in the cosmological framework. In
order to explore the cosmological consequences of our model, we consider the
Hubble horizon as the IR cutoff. We investigate both the non-interacting and
interacting cases with the sign-changeable and linear interactions, showing
that they can explain the present accelerated phase of the Universe expansion,
in contrast to the standard Holographic Dark Energy model. We then perform the
classical stability analysis using the squared sound speed. We find that,
whilst the non-interacting model is unstable against the small perturbations,
the sign-changeable interacting one can be stable only for suitable values of
the model parameters. On the other hand, the linear interacting model always
predicts a stable Universe. The consistency of our model with cosmological
observations is discussed.

##  I Introduction

The recent technological progress has ushered a new era for observational
cosmology, the so-called Precision Cosmology Primack:2006it  allowing for
significant advances in our understanding of the Universe. Yet, most of the
Universe matter-energy composition remains elusive. It is a fact that about 25
percent of matter content consists of an enigmatic substance - the Dark Matter
(DM) Zwicky:1933gu  \- the existence of which is established by astrophysical
observations at large scales. Possibly even more mysterious is the Dark Energy
(DE) component (roughly 70 percent), which is invoked to account for the
current accelerating expansion of the Universe. In spite of strong evidences
from Supernova SNIa observations, baryon acoustic oscillations and
gravitational waves, a complete explanation for the origin of DE is an open
issue Nojiri:2006zh ; Bamba:2012cp . Undoubtedly, this reveals the necessity
of novel physics to be solved.

So far, several theoretical approaches to DE problem have been developed in
the literature. The possibility that DE could be modeled by the cosmological
constant acting as source of the vacuum energy was first considered as a
natural way out of the DE puzzle. Nevertheless, this scenario seems to clash
with our field theoretical understanding of the quantum properties of vacuum.
Alternative explanations were later offered by modified gravity theories based
on the introduction of either geometric terms or extra degrees of freedom in
the Einstein-Hilbert action (see Copeland:2006wr ; Capozziello:2003tk ;
Capozziello:2002rd ; Nojiri:2010wj ; Capozziello:2011et ; Clifton:2011jh ;
Joyce:2014kja  for reviews). Among these theories, the Brans-Dicke (BD)
gravity is an extension of General Relativity built upon the replacement of
the gravitational coupling constant $G$ by the inverse of a varying scalar
field $\phi$ Weinberg . The first motivation for this choice was to achieve a
theory more in agreement with the Mach Principle than GR Brans ; Faraoni .
While being able to describe the accelerating expansion of the current
Universe, this model predicts a value for the BD ($\omega$) parameter that is
lower than the observational limit Banerjee:2000mj ; Acquaviva:2007mm , thus
requiring to use different DE sources Capoz96 ; Gong:1999ge ; Banerjee:2000gt
; Gong:2004 ; Lamb ; Kim:2005gk ; Setare:2006yj ; Xu:2008sn ; Khodam-
Mohammadi:2014wla  to be reconciled with phenomenology.

In this respect, a useful paradigm is given by the Holographic Dark Energy
(HDE) Cohen:1998zx ; Horava:2000tb ; Thomas:2002pq ; Li:2004rb ; pnp ;
Bamba:2012cp ; Ghaffari:2014pxa ; Wang:2016och . In the primary formulation,
such an approach provides a precise quantitative analysis of DE, originating
from the holographic principle with Bekenstein-Hawking (BH) entropy and the
Hubble horizon as its IR cutoff. However, the shortcomings in describing the
history of a flat Friedmann-Robertson-Walker (FRW) Universe have motivated
some tentative changes of this model. For instance, HDE has been employed to
solve the DE problem in the BD framework Gong:1999ge ; Lamb ; Kim:2005gk ;
Setare:2006yj ; Banerjee:2000gt ; Gong:2004 ; Xu:2008sn ; Khodam-
Mohammadi:2014wla  by considering different IR cutoffs Wang:2016och ;
Ghaffari:2014pxa ; Xu:2008sn  or suitable interactions with DM Jamil:2010ed .

On the other hand, modified HDE models based on deformations of the entropy
area law have been proposed in Tavayef:2018xwx ; Saridakis:2018unr ;
Saridakis:2020zol ; Nojiri:2019skr ; Moradpour:2020dfm ; Drepanou:2021jiv ;
Nojiri:2021jxf ; Fara ; Nojiri:2021iko ; yangluc , while still keeping the
Hubble radius as IR cutoff. Due to the complex (and mostly unknown)
thermostatistical properties of gravity systems, various deformed entropies
have been adopted Tsallis:1987eu ; Kaniadakis:2002 ; Barrow:2020tzx , all of
which recovering BH entropy in a certain limit. In this context, a promising
framework is provided by the Barrow Holographic Dark Energy (BHDE)
Saridakis:2020zol , which relies on a quantum-gravitational deformation of BH
area law in the form Barrow:2020tzx

$S_{\Delta}\,=\,\left(\frac{A}{A_{0}}\right)^{1+\Delta/2}\,,$ (1)

where $A$ is the standard horizon area and $A_{0}$ the Planck area. Quantum
gravity effects are parameterized by the Barrow exponent $0\leq\Delta\leq 1$,
with $\Delta=0$ giving the BH limit, while $\Delta=1$ corresponding to the
maximal entropy deformation. It is worth mentioning that cosmological
constraints on $\Delta$ have been inferred in Anagnostopoulos:2020ctz ;
Leon:2021wyx ; Barrow:2020kug ; Jusufi:2021fek ; Dabrowski:2020atl ;
Saridakis:2020cqq ; Luciano:2022pzg ; Vagnozzi:2022moj . The possibility for a
running $\Delta$ has also been discussed in DiGennaro:2022ykp .

BHDE is inspired by similar generalizations of HDE based on Tsallis
Tavayef:2018xwx ; Saridakis:2018unr ; Nojiri:2019skr  and Kaniadakis
Drepanou:2021jiv  entropies. At a first glance, it appears as a proper model
to trace the evolution of the Universe in the standard cosmological framework.
Just as Tsallis Holographic Dark Energy, however, BHDE suffers from
instability problems. Thus, according to the dynamic behavior of HDE and its
modified-entropy based extensions, it could be more appropriate to study the
cosmological features of BHDE in extended frameworks and, in particular, in
the BD dynamic framework Setare:2006yj ; Jamil:2010ed . In Xu:2008sn , the
authors have shown that the non-interacting HDE with the Hubble radius as IR
cutoff cannot explain the current accelerated expansion of Universe in the BD
theory. While in a similar analysis, it has been shown that the Tsallis and
Kaniadakis Holographic Dark Energy, due to the generalized entropy, with the
Hubble cutoff in the BD gravity, can lead the accelerated Universe phase, even
in the absence of interaction between two dark sectors Ghaffari:2018 ; KBD .

Starting from the above premises, in what follows we examine the implications
of using BHDE to model dark energy in BD cosmology. To this goal, we resort to
the holographic principle with Eq. (1) as horizon entropy and the Hubble
radius as cutoff. In the next section, we extract the BHDE density in BD
gravity and investigate the evolution of its cosmological parameters for a
spatially flat Universe in the non-interacting case. The same study is
developed in Sec. III for the sign-changeable and linear interacting models.
Stability analysis is investigated in Sec. IV. Conclusions and outlook are
finally summarized in Sec. V.

##  II Non-interacting Barrow holographic dark energy in the Brans-Dicke
cosmology

Throughout this work we consider a homogeneous and isotropic Friedman-
Robertson-Walker (FRW) geometry with metric

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\left(\frac{{\rm
d}r^{2}}{1-kr^{2}}+r^{2}{\rm d}\Omega^{2}\right),$ (2)

where $k=0,1,-1$ represent a flat, closed and open maximally symmetric space,
respectively.

The Brans-Dicke field equations for a spatially flat FRW Universe which is
filled by pressureless DM and DE are Banerjee:2000gt ; Gong:2004 ; Kim:2005gk
; Setare:2006yj ; Xu:2008sn

$\frac{3}{4\omega}\phi^{2}H^{2}-\frac{\dot{\phi}^{2}}{2}+\frac{3}{2\omega}H\dot{\phi}\,\phi=\rho_{M}+\rho_{\Lambda},$
(3)
$\frac{-\phi^{2}}{4\omega}\left(\frac{2\ddot{a}}{a}+H^{2}\right)-\frac{1}{\omega}H\dot{\phi}\,\phi-\frac{1}{2\omega}\ddot{\phi}\,\phi-\frac{\dot{\phi}^{2}}{2}\left(1+\frac{1}{\omega}\right)=p_{\Lambda},$
(4)

The scalar field evolution equation is

$\ddot{\phi}+3H\dot{\phi}-\frac{3}{2\omega}\left(\frac{\ddot{a}}{a}+H^{2}\right)\phi=0.$
(5)

where $\omega$ is the BD parameter and $\phi^{2}=\omega/(2\pi G_{eff})$ is the
BD scalar field, with $G_{eff}$ being the effective gravitational constant.
Also, $H=\dot{a}/a$ is the Hubble parameter and $\rho_{m}$, $\rho_{\Lambda}$
and $p_{\Lambda}$ are the pressureless DM density, DE density and pressure of
DE, respectively. Following Banerjee:2000gt ; Gong:2004 ; Kim:2005gk ;
Setare:2006yj ; Xu:2008sn , we consider the BD field as a power law of the
scale factor, $\phi\propto a^{n}$. Then we have

$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle nH\phi,$ (6)
$\displaystyle\ddot{\phi}$ $\displaystyle=$ $\displaystyle
n^{2}H^{2}\phi+n\dot{H}\phi.$ (7)

where dot denotes derivative with respect to the cosmic time. The energy
conservation equations for non-interacting DE and matter are

$\dot{\rho}_{D}+3H\left(1+\omega_{D}\right)\rho_{D}=0,$ (8)

and

$\dot{\rho}_{m}+3H\rho_{m}=0,$ (9)

where $\omega_{D}=\frac{p_{D}}{\rho_{D}}$ denotes the equation of state (EoS)
parameter of DE. With the definition of the critical energy density,
$\rho_{cr}=\frac{3\phi^{2}H^{2}}{4\omega}$, the dimensionless density
parameters can be written as

$\displaystyle\Omega_{m}=\frac{\rho_{m}}{\rho_{cr}}$ $\displaystyle=$
$\displaystyle\frac{4\omega\rho_{m}}{3\phi^{2}H^{2}},$ (10)
$\displaystyle\Omega_{D}=\frac{\rho_{D}}{\rho_{cr}}$ $\displaystyle=$
$\displaystyle\frac{4\omega\rho_{D}}{3\phi^{2}H^{2}}.$ (11)

Using the above definitions (10) and (11), one can rewrite the Friedman Eq.
(3) as follows

$\displaystyle\Omega_{m}+\Omega_{D}=1+2n-\frac{2\omega n^{2}}{3}.$ (12)

Here, by using the holographic principle with the generalized Barrow entropy
(1), we can construct the Barrow holographic dark energy (BHDE) in BD gravity,
in which the apparent horizon in the flat Universe is considered as the IR
cut-off $(L=H^{-1})$, as follows

$\rho_{D}=B\phi^{2+\Delta}H^{2-\Delta}.$ (13)

Here, $B$ is an unknown constant. In the limit of Einstein gravity, where
$G_{eff}\rightarrow G$, the BD scalar field becomes trivial
($\phi^{2}=\frac{\omega}{2\pi G}=4\omega M_{p}^{2}$) and the BHDE in standard
cosmology can be recovered Srivastava . Also for $\Delta\rightarrow 0$ and
$G_{eff}\rightarrow G$, Eq. (13) reduces to the energy density of the original
HDE in standard cosmology Xu:2008sn .

Taking the time derivative of Eq. (13), we have

$\dot{\rho}_{D}=\left(2+\Delta\right)nH\rho_{D}+\left(2-\Delta\right)\frac{\dot{H}}{H}\rho_{D}.$
(14)

By differentiating Eq. (3) and combining the result with Eqs. (9), (10), (11),
(6) and (14), we obtain

$\displaystyle\frac{\dot{H}}{H^{2}}$ $\displaystyle=$
$\displaystyle\Big{[}-\frac{9\Omega_{m}^{0}H_{0}^{2}\phi_{0}^{2}}{4\omega\phi^{2}H^{2}}+(2\omega
n^{2}-6n-3)\frac{n}{2\omega}$ (15) $\displaystyle+$
$\displaystyle\,nB(2+\Delta)\left(\frac{\phi}{H}\right)^{\Delta}\Big{]}$
$\displaystyle\times$
$\displaystyle\Bigg{[}B(\Delta-2)\left(\frac{\phi}{H}\right)^{\Delta}+\frac{3}{2\omega}-n^{2}+\frac{3n}{\omega}\Bigg{]}^{-1},$

where the zero-subscript represents present time. Using Eqs. (8) and (14), we
find the EoS parameter for the BHDE model in the BD gravity as

$\displaystyle\omega_{D}=-1-\frac{\left(2+\Delta\right)n}{3}+\frac{\Delta-2}{3}\frac{\dot{H}}{H^{2}}.$
(16)

For $\Delta=0$ and $n=0$, this equation coincides with the original HDE in the
BD gravity Xu:2008sn . Also in the limiting case $n=0$
$(\omega\rightarrow\infty)$, the EoS parameter for BHDE in standard cosmology
is restored Srivastava .

In the following, using the accelerating parameter relation
$q=-1-\frac{\dot{H}}{H^{2}}$ and Eq. (15), the deceleration parameter can be
extracted as

$\displaystyle q$ $\displaystyle=$
$\displaystyle-1-\Big{[}-\frac{9\Omega_{m}^{0}H_{0}^{2}\phi_{0}^{2}}{4\omega\phi^{2}H^{2}}+\left(2\omega
n^{2}-6n-3\right)\frac{n}{2\omega}$ (17) $\displaystyle+$
$\displaystyle\,nB\left(2+\Delta\right)\left(\frac{\phi}{H}\right)^{\Delta}\Big{]}$
$\displaystyle\times$
$\displaystyle\left[B\left(\Delta-2\right)\left(\frac{\phi}{H}\right)^{\Delta}+\frac{3}{2\omega}-n^{2}+\frac{3n}{\omega}\right]^{-1}$

###  II.1 Cosmological Evolution

In Fig. 1, we plot the evolution of cosmological parameters of the BHDE model
by using a numerical solution of the Hubble differential Eq. (15) for
$H_{0}=70.9,\Omega_{m}=0.3,\omega=1000$ and $n=0.05$ as the initial condition.
In the first panel of this figure, we have $\Omega_{D}\rightarrow 0$ at the
early time (DM dominated phase), while at the late time the DE dominates
$\Omega_{D}\rightarrow 1$, in agreement with cosmological observations Daly ;
Komatsu ; Salvatelli . In the second panel of Fig. 1, we show the behavior of
the deceleration parameter $q(z)$. In contrast to standard HDE, we can see
that non-interacting BHDE with the Hubble radius as IR cutoff can explain the
current acceleration phase. We can see that the Universe undergoes a phase
transition from deceleration to acceleration around $z\approx 0.6$.

Furthermore, as it can be seen from the third panel, the EoS parameter is
always in the quintessence regime ($\omega_{D}>-1$) and, at the late time
$(z\rightarrow-1)$, it tends to the cosmological constant
($\omega_{D}\rightarrow-1$).

Finally, we plot the behavior of the total EoS parameter $\omega_{eff}(z)$ in
the last panel of Fig. 1. We can see that the total EoS parameter can suitably
describe the Universe history, as the pressureless DM is dominant
$(\omega_{eff}\rightarrow 0)$ at early times ($z\rightarrow\infty$), then it
enters the quintessence regime at the present epoch and finally, by decreasing
values of the model parameters, tends to a value very close to $-1$ at late
times ($z\rightarrow-1$).

Figure 1: The evolution of the $\Omega_{D}(z)$, $\omega_{D}(z)$,
$\omega_{eff}(z)$ and $q(z)$ parameters for non-interacting BHDE in BD
gravity.

##  III Interacting BHDE model

Let us now assume that dark components interact each other and one can grow at
the expense of the other. The mutual interaction between two dark components
is initially proposed to solve the coincidence problem Amendola ; Mangano .
Although some observations confirm the mutual interaction between dark matter
and dark energy Bertolami , since the nature of DM and DE are still unknown,
the proposed different interaction models are purely phenomenological. So far,
different models of interaction between the two dark components have been
proposed in literatures to study the dynamics of Universe (for details, one
can look into Quartin ; Pavon ; Boehmer ; Caldera ; Mamon ; Mamon2 ; WeiQq1 ;
WeiQq2 ; Abdollahi Zadeh ; Rocco  and references therein).

In the presence of interaction between two dark components, the conservation
equations are

$\dot{\rho}_{D}+3H\left(1+\omega_{D}\right)\rho_{D}=-Q,$ (18)
$\dot{\rho}_{m}+3H\rho_{m}=Q,$ (19)

where $Q$ represents the interaction term. Positive value of Q indicates that
energy transfers from the BHDE to the DM, while for $Q<0$, the reverse
scenario will occur. In the following, we consider the sign-changeable and
linear models of interaction to investigate the cosmological evolution of BHDE
in the BD gravity.

###  III.1 Sign-changeable Interaction

Following WeiQq2 ; Abdollahi Zadeh , we consider a sign-changeable interaction
term as

$Q=3b^{2}Hq\left(\rho_{m}+\rho_{D}\right),$ (20)

where $b$ is the coupling constant of interaction term and $q$ is the
deceleration parameter. Taking the time derivative of Eq. (3) and using Eqs.
(6), (14) and (19), we get

$\displaystyle\frac{\dot{H}}{H^{2}}=\Big{[}-\frac{9\Omega_{m}^{0}H_{0}^{2}\phi_{0}^{2}\left(1+z\right)^{3}}{4\omega\phi^{2}H^{2}}+\frac{3}{2\omega}\left(2n-\frac{2\omega
n^{2}}{3}-1\right)$
$\displaystyle\times\left(n+\frac{3b^{2}}{2}\right)+nB\left(2+\Delta\right)\left(\frac{\phi}{H}\right)^{\Delta}\Big{]}$
$\displaystyle\times\Bigg{[}B\left(\Delta-2\right)\left(\frac{\phi}{H}\right)^{\Delta}+\left(\frac{3}{2\omega}-n^{2}+\frac{3n}{\omega}\right)\left(1+\frac{3b^{2}}{2}\right)\Bigg{]}^{-1}.$

Combining Eqs. (18) and (14), one can obtain the following expression for EoS
parameter

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-1-\frac{(2+\Delta)n}{3}+\frac{(\Delta-2)\dot{H}}{3H^{2}}$ (22)
$\displaystyle+$
$\displaystyle\,b^{2}\left(1+\frac{\dot{H}}{H^{2}}\right)\left(\frac{1+2n-\frac{2\omega
n^{2}}{3}}{\Omega_{D}}\right)\,.$

In the absence of interaction term $(b^{2}=0)$, these equations reduce to the
corresponding relations of the previous section.

To investigate the effects of interaction term between dark sectors, we have
plotted the behavior of $\Omega_{D}$, the deceleration parameter $q$, the EoS
parameter $\omega_{D}$ and the total EoS parameter $\omega_{eff}$ against the
redshift parameter $z$ for the sign-changeable interacting BHDE in the Fig. 2.
Again, we see that at the early Universe $\Omega_{D}\rightarrow 0$, and at the
late time, the DE dominates, as expected. As it is clear from the second panel
in Fiq. 2, the deceleration parameter $q$ transits from the decelerated phase
$(q>0)$ at the earlier time, to current accelerated phase $(q<0)$. Also by
increasing the values of $\Delta,B$ and $\phi_{0}$ and decreasing the value of
$b^{2}$, the transition occurs earlier (higher redshift). Interestingly, we
found that, as in the non-interacting case, the EoS parameter of sign-
changeable interacting BHDE cannot cross the phantom divide $(\omega_{D}=-1)$
at late time.

Figure 2: The evolution of the $\Omega_{D}$, $\omega_{D}$, $q$ and
$\omega_{eff}$ versus redshift parameter $z$ for sign-changeable interacting
BHDE in BD gravity. We have taken the $H_{0}=70.9$, $\omega_{m}=0.3$,
$n=0.005$ and $\omega=1000$ as the initial condition.

###  III.2 Linear Interaction

In the following, we choose the linear interaction term given as Mamon ;
Mamon2

$Q=H(\alpha\rho_{m}+\beta\rho_{D}),$ (23)

where $\alpha$ and $\beta$ are the coupling constant of interaction term.
Taking the time derivative of Eq. (3) and using Eqs. (6), (14), (19) and (23),
we reach

$\displaystyle\frac{\dot{H}}{H^{2}}=\Big{[}B\left(\frac{\phi}{H}\right)^{\Delta}(2n+\Delta
n+\beta-\alpha)$
$\displaystyle-\,\frac{9\Omega_{m}^{0}H_{0}^{2}\phi_{0}^{2}\left(1+z\right)^{3}}{4\omega\phi^{2}H^{2}}+\frac{3}{2\omega}\left(1+2n-\frac{2\omega
n^{2}}{3}\right)\left(\frac{\alpha}{2}-n\right)\Big{]}$
$\displaystyle\times\Bigg{[}B(\Delta-2)\left(\frac{\phi}{H}\right)^{\Delta}+\frac{3}{2\omega}-n^{2}+\frac{3n}{\omega}\Bigg{]}^{-1}.$
(24)

Substituting Eqs. (14) and (23) into Eq. (18), we find out

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-1-\frac{(2+\Delta)n}{3}+\frac{(\Delta-2)\dot{H}}{3H^{2}}$ (25)
$\displaystyle-$
$\displaystyle\,\frac{\alpha}{3\Omega}\left(1+2n-\frac{2\omega
n^{2}}{3}\right)+\frac{\alpha-\beta}{3}.$

The evolutions of $\Omega_{D}$, $\omega_{D}$, $q$ and $\omega_{eff}$ are
plotted vs. the redshift parameter $z$ for the linear interacting BHDE in Fig.
3. It appears that the linear interacting BHDE, in the BD gravity, can
describe the cosmological evolution of the Universe in a consistent way.
Indeed, from the evolution of dimensionless BHDE density parameter
$\Omega_{D}(z)$ in Fig. 3, we observe that, at early times,
$\Omega_{D}\rightarrow 0$ and at late times $\Omega_{D}\rightarrow 1$. The
behavior of the deceleration parameter $q(z)$ in Fig. 3 indicates that the
Universe has a phase transition from deceleration to acceleration around
$0.3<z<0.8$, consistently with cosmological observations Daly ; Komatsu ;
Salvatelli . The evolution of the EoS parameter $\omega_{D}(z)$ is shown in
the third panel of Fig. 3. As we can observe, the EoS parameter successfully
describes the cosmological evolution of the Universe and shows that the
Universe is in the quintessence dominated phase ($-1/3<\omega_{D}<-1$) at the
current epoch and will enter the phantom regime ($\omega_{D}<-1$) in the far
future.

Figure 3: The evolution of the $\Omega_{D}(z)$, $\omega_{D}(z)$, $q(z)$ and
$\omega_{eff}(z)$ for different values of $\Delta,B,\phi_{0},\alpha$ and
$\beta$ as mention on the caption. for linear interacting BHDE in BD gravity.
The initial condition $H_{0}=70.9$, $\omega_{m}=0.3$, $n=0.005$ and
$\omega=1000$ are adopted.

##  IV Stability Analysis

We analyze now the stability of the obtained models against perturbations. So
far we have investigated the dynamical behavior of BHDE model. Now we would
like to see if there are small perturbations in the background, whether the
disturbance grows or collapses. In the perturbation theory, the sign of the
squared sound speed ($v_{s}^{2}$) determines classical stability or
instability. For $v_{s}^{2}>0$, the model is stable because the perturbation
propagates on the background, while, for $v_{s}^{2}<0$, the model is unstable
since every small perturbation grows within the background.

The squared sound speed $v_{s}^{2}$ is given by

$v_{s}^{2}=\frac{dp}{d\rho_{D}}=\frac{\dot{p}}{\dot{\rho}_{D}}.$ (26)

By differentiating $p_{D}$ with respect to the cosmic time, inserting the
result in Eq. (26), and using Eq. (14), we have

$v_{s}^{2}=\omega_{D}+\frac{\omega_{D}^{\prime}}{2\delta
n+2(2-\delta)\frac{\dot{H}}{H^{2}}},$ (27)

for the squared sound speed.

###  IV.1 Non-interacting case

Taking the time derivative of Eq. (16) and using Eqs. (14), (15) and (27), one
can get $v_{s}^{2}$ for the non-interacting BHDE with the Hubble cut-off.
Since the analytic expression of $v_{s}^{2}$ is rather cumbersome to exhibit,
we shall limit to plot it in Fig. 4. As it is clear, from this figure, the
non-interacting BHDE model in the BD gravity is unstable against small
perturbations during the cosmic evolution, which is similar to the result
obtained for BHDE in the standard model Srivastava .

Figure 4: The evolution of the $v_{s}^{2}(z)$ for non-interacting BHDE in BD
gravity. The initial condition $H_{0}=70.9$, $\omega_{m}=0.3$, $n=0.005$ and
$\omega=1000$ are adopted.

###  IV.2 Interacting case

We now calculate the squared sound speed for the BHDE in presence of sign-
changeable and linear interaction terms, respectively. For the sign-changeable
interacting case, by taking the time derivative of Eq. (22) and combining the
result with Eqs. (13), (14), (III.1) and (27), we obtain $v_{s}^{2}$. For the
linear interacting case, we use time derivative of Eq. (25) and Eqs. (13),
(14), (III.2) and (27) to get $v_{s}^{2}$.

We plot $v_{s}^{2}$ for the sign-changeable and linear interacting BHDE for BD
gravity in Figs. 5 and 6, respectively. From Fig. 5, one can clearly see that
the sign-changeable interacting BHDE in the BD cosmology is stable for
suitable values of the model parameters. On the other hand, for the linear
interacting BHDE the squared sound speed is always positive, which means that,
in this case, the Universe is stable.

Figure 5: The evolution of the $v_{s}^{2}(z)$ for sign-changeable interacting
BHDE in BD gravity. The initial condition $H_{0}=70.9$, $\omega_{m}=0.3$,
$n=0.005$ and $\omega=1000$ are adopted. Figure 6: The evolution of the
$v_{s}^{2}(z)$ for linear interacting BHDE in BD gravity. The initial
condition $H_{0}=70.9$, $\omega_{m}=0.3$, $n=0.005$ and $\omega=1000$ are
adopted.

##  V Concluding remarks

We considered the Barrow entropy to develop a holographic DE model within the
framework of Brans-Dicke cosmology. To this goal, we assumed the Hubble
horizon as the IR cutoff and studied the behavior of Barrow Holographic Dark
Energy for both interacting and non-interacting cases. We derived the equation
governing the evolution of Hubble parameter and solved it numerically in order
to study the evolution of corresponding cosmological parameters.

From the behavior of the deceleration parameter $q$, we inferred that, in
contrast to the standard Holographic Dark Energy, our model can explain the
present accelerated phase of the Universe expansion, even in absence of
interactions between the dark sectors of cosmos. Furthermore, for the non-
interacting case, the EoS parameter shows that BHDE model is always in the
quintessence regime ($\omega_{D}>-1$) and, at the late time
$(z\rightarrow-1)$, it tends to cosmological constant
($\omega_{D}\rightarrow-1$) (see Fig. 1). A similar behavior is exhibited in
the sign-changeable interacting model, where the EoS parameter cannot cross
the phantom divide at late times (see Fig. 2). On the other hand, in presence
of a linear interaction, the cosmological evolution of $\omega_{D}$ shows that
the Universe is in the quintessence dominated phase at the current epoch and
will enter the phantom regime ($\omega_{D}<-1$) in the far future (see Fig.
3).

Finally, we performed a classical stability analysis using the squared sound
speed. We found that the non-interacting BHDE model, in the BD gravity, is
unstable against small perturbations during the cosmic evolution (see Fig. 4),
similarly to the result obtained for BHDE in the standard model Srivastava .
On the other hand, the sign-changeable interacting model can be stable only
for certain values of the parameters (see Fig. 5), while the linear
interacting model always predicts a stable Universe (see Fig. 6). This means
that considering an interaction between the dark sectors provides a more
reasonable scenario to trace the cosmological evolution in BHDE with BD
gravity.

Further aspects remain to be addressed. For instance, it is interesting to
examine the correspondence between our model and HDE based on other extensions
of Boltzmann-Gibbs entropy, such as Tsallis Ghaffari:2018 ; LucGUP ;
Lucianoqgen  and Kaniadakis KBD  Holographic Dark Energy in BD cosmology.
Also, since our model relies on a phenomenological attempt to include quantum
gravitational effects through the Barrow entropy, it is important to study
whether our predictions reconcile with more fundamental candidate theories of
quantum gravity, such as String Theory or Loop Quantum Gravity. Work along
these directions is presently under investigation and will be presented
elsewhere.

###### Acknowledgements.

G. G. L. acknowledges the Spanish “Ministerio de Universidades” for the
awarded Maria Zambrano fellowship and funding received from the European Union
- NextGenerationEU. G.G.L. and S.C. acknowledge the participation to the COST
Action CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach”.
S.C. acknowledges the support of INFN sez. di Napoli (Iniziativa Specifica
QGSKY).

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