###### Abstract

This work studies synergies arising from combining industrial demand response
and local renewable electricity supply. To this end, we optimize the design of
a local electricity generation and storage system with an integrated demand
response scheduling of a continuous power-intensive production process in a
multi-stage problem. We optimize both total annualized cost and global warming
impact and consider local photovoltaic and wind electricity generation, an
electric battery, and electricity trading on day-ahead and intraday market. We
find that installing a battery can reduce emissions and enable large trading
volumes on the electricity markets, but significantly increases cost. Economic
and ecologic process and battery operation are driven primarily by the
electricity price and grid emission factor, respectively, rather than locally
generated electricity. A parameter study reveals that economic savings from
the local system and flexibilizing the process behave almost additive.

Optimal design of a local renewable electricity supply system for power-
intensive production processes with demand response

Sonja H. M. Germscheida,b, Benedikt Nilgesc, Niklas von der Assenc,d,
Alexander Mitsosd,a,e, Manuel Dahmena,∗

  * a

Forschungszentrum Jülich GmbH, Institute of Energy and Climate Research,
Energy Systems Engineering (IEK-10), Jülich 52425, Germany

  * b

RWTH Aachen University Aachen 52062, Germany

  * c

RWTH Aachen University, Institute of Technical Thermodynamics (LTT), Aachen
52062, Germany

  * d

JARA-ENERGY, Aachen 52056, Germany

  * e

RWTH Aachen University, Process Systems Engineering (AVT.SVT), Aachen 52074,
Germany

Keywords: Integrated design and scheduling, stochastic programming, demand
response, local electricity supply system, renewable energy

##  1 Introduction

Renewable electricity has a varying supply that can be utilized by flexible
production processes in a demand response (DR) scheduling (Daryanian et al.,,
1989; Zhang and Grossmann,, 2016; Burre et al.,, 2020; Mitsos et al.,, 2018).
DR savings can be improved by participating on multiple short-term electricity
markets, see, e.g., Leo et al., (2021); Dalle Ave et al., (2019); Simkoff and
Baldea, (2020); Liu et al., (2016); Pandžić et al., (2013); Kwon et al.,
(2017); Golmohamadi and Keypour, (2018); Nolzen et al., (2022); Germscheid et
al., (2022, 2023); Schäfer et al., (2019); Varelmann et al., (2022).
Furthermore, flexible operation should be accounted for at design stage in
order to determine optimal investment decisions for both the production
processes itself, see, e.g., Mitra et al., (2014); Teichgraeber and Brandt,
(2020); Steimel and Engell, (2015); Seo et al., (2023); Leenders et al.,
(2019), and for its local energy supply system, see, e.g., Yunt et al.,
(2008); Zhang et al., (2019); Voll et al., (2013); Baumgärtner et al., (2019);
Langiu et al., (2022); Bahl et al., (2017); Fleschutz et al., (2023).

In local energy supply systems, integrated design and scheduling has already
been used to optimize on-site renewable electricity generation and storage
systems that satisfy a fixed demand profile but can offer flexibility by
combining different electricity generation technologies, see, e.g., Zhang et
al., (2019); Bahl et al., (2017); Fleschutz et al., (2023); Baumgärtner et
al., (2019). Furthermore, combining on-site renewable electricity generation
and storage systems with flexible production processes can reduce both
production cost and emission, see, e.g., Mucci et al., (2023); Wang et al.,
(2020); Allman and Daoutidis, (2018); Martín, (2016). However, to the best of
our knowledge, a generalized assessment about synergies arising from combining
on-site electricity supply systems and DR-capable processes has not been
conducted yet.

In our prior work (Germscheid et al.,, 2022), we conducted a DR potential
assessment of power-intensive production processes by means of the generalized
process model introduced by Schäfer et al., (2020). The generalized process
model can represent a wide range of continuous production processes by means
of few key process characteristics, i.e., oversizing, minimal part load,
ramping limitations, and production storage capacity. We analyzed the benefit
of participating simultaneously on both the day-ahead (DA) and the intraday
(ID) spot electricity market, but neglected potential electricity provision by
on-site renewable electricity generation and storage (Germscheid et al.,,
2022).

In this article, we extend our prior work (Germscheid et al.,, 2022) by
integrating the scheduling of the generalized production process into the
design optimization of a local electricity generation and storage system. In
the resulting multi-stage approach, we optimize the design of the local
renewable electricity supply system considering photovoltaic (PV) power, wind
power, and an electric battery for a location in Germany and optimize the DR
scheduling of both the energy system and the process, together with
electricity market participation. We consider both economic and ecologic
design objectives, i.e., we optimize with respect to the total annualized cost
(TAC) and the global warming impact (GWI), respectively. We study the
influence of the production process flexibility capabilities on the optimal
design of the energy system and the resulting savings. Similar to our prior
work (Germscheid et al.,, 2022), we consider simultaneous market participation
on both the DA and ID electricity market to analyze the benefit of considering
multiple electricity markets in an integrated design and scheduling problem.

The remainder of the article is structured as follows: Section 2.1 explains
the structure of the integrated design and scheduling problem. We specify the
objectives in Section 2.2 and the operational constraints in Section 2.3. The
scenarios and the model parameters are specified in Section 2.4 and Section
2.5, respectively. We discuss the optimal energy system design for a reference
process in Section 3.1, the dependency between process parameters and
potential savings in Section 3.2, and the benefit of considering simultaneous
DA and ID market participation at design stage in Section 3.3. In Section 4,
we conclude our work.

##  2 Methods

###  2.1 Structure of the integrated design and scheduling problem

Integrated design and scheduling problems are often set up as two-stage
stochastic problems (Birge and Louveaux,, 2011) with the design decisions on
the first stage and scheduling decisions and operational constraints on the
second stage, see, e.g., Yunt et al., (2008); Zhang et al., (2019); Langiu et
al., (2022); Mitra et al., (2014); Teichgraeber and Brandt, (2020); Steimel
and Engell, (2015); Seo et al., (2023); Bahl et al., (2018). In this work, we
determine the optimal design of a local electricity supply system for a
flexible industrial production process. We account for simultaneous DA and ID
market participation in the design and scheduling problem by the three-stage
structure shown in Fig. 1. In the first stage, the design decisions for the
energy system are made, i.e., photovoltaic (PV), wind power, and electric
battery capacities are to be determined. The DA trading decisions are taken
the day before operation when the ID price and renewable electricity
generation are still uncertain. Thus, we consider the DA decisions on the
second stage and ID trading and operational decisions on the third stage. In
particular, the operation of the flexible process is adapted on the third
stage in response to realizations of the ID price, renewable electricity
production, and the momentary emission factor of the grid electricity. Note
that we consider a time-varying average grid emission factor similar to
Baumgärtner et al., (2019) and Nilges et al., (2023) and that the emission
factor is uncertain a day before the actual consumption.

Figure 1: Structure of the integrated design and scheduling problem: The
problem structure allows to optimize the design of the local electricity
generation and storage system while considering simultaneous DA and ID market
participation and process DR. The decision tree models the chronological
sequence of the decisions. Each branch represents a scenario and each node
represents a decision point.

Similar to our prior work (Germscheid et al.,, 2022), we consider hourly DA
purchases and quarter-hourly ID purchases and sales and assume a one-day
scheduling horizon. In addition, we allow to sell electricity from the local
generation and storage system on the DA market. For simplicity, we assume that
throughout any quarter-hour time slice, renewable electricity generation is
constant.

In the following, we omit a distinct notation for second- and third-stage
parameters and variables for better readability. In particular, we consider DA
trading decisions $\bm{q}_{\text{DA,$s_{3}$}}$ and DA price
$\bm{c}_{\text{DA,$s_{3}$}}$ on the third stage instead of the second stage
and guarantee equality on the second stage by means of non-anticipativity
constraints:

$\displaystyle\bm{c}_{\text{DA,$s_{3}$}}=\bm{c}_{\text{DA},\hat{s}_{3}}\quad$
$\displaystyle\text{if}\quad
f(s_{3})=f(\hat{s}_{3})\quad\forall(s,\hat{s})\in\mathbb{S}_{3}\times\mathbb{S}_{3},$
(1)
$\displaystyle\bm{q}_{\text{DA,$s_{3}$}}=\bm{q}_{\text{DA},\hat{s}_{3}}\quad$
$\displaystyle\text{if}\quad
f(s_{3})=f(\hat{s}_{3})\quad\forall(s,\hat{s})\in\mathbb{S}_{3}\times\mathbb{S}_{3}$
(2)

Here, $f:\mathbb{S}_{3}\xrightarrow{}\mathbb{S}_{2}$ maps a third-stage
scenario $s_{3}\in\mathbb{S}_{3}$ to the respective second-stage scenario
$s_{2}\in\mathbb{S}_{2}$. In the following, we refer to $s\in\mathbb{S}$ as
third-stage scenario for conciseness. The non-anticipativity constraints allow
to formulate the three-stage problem as a two-stage problem in the following.

###  2.2 Objectives

We consider the total annualized cost (TAC)and the global warming impact
(GWI)as economic and ecologic objective, respectively.

The TAC is defined as

$\displaystyle\text{\lx@glossaries@gls@link{acronym}{tac}{\leavevmode
TAC}{}}=\text{CAPEX}+\text{OPEX}_{el}+\text{OPEX}_{Grid},$ (3) with
$\displaystyle\text{CAPEX}=\sum_{i\in\\{\text{PV,W,B}\\}}\left(\frac{(\gamma_{\text{1}}+1)^{\gamma_{2,i}}\gamma_{1}}{(\gamma_{1}+1)^{\gamma_{2,i}}-1}\text{CAPEX}^{0}_{i}\;Q_{i}+\gamma_{3,i}\;Q_{i}\right),$
(4)
$\displaystyle\text{OPEX}_{el}=365\sum_{s\in\mathbb{S}}\pi_{s}(4\;\Delta{t}\;{\bm{c}_{\text{DA,s}}}\cdot\bm{q}_{\text{DA,s}}+\Delta{t}\;\bm{c}_{\text{ID},s}\cdot\bm{q}_{\text{ID},s}),$
(5)
$\displaystyle\text{OPEX}_{\text{Grid}}=365\sum_{s\in\mathbb{S}}\pi_{s}\sum_{t=1}^{T}\text{OPEX}_{\text{Grid},s,t},$
(6) $\displaystyle\text{OPEX}_{\text{Grid},s,t}\geq
c_{\text{Fee}}\;\Delta{t}\;(q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}+q_{\text{ID,}s,t}),$
(7) $\displaystyle\text{OPEX}_{\text{Grid},s,t}\geq 0.$ (8)

In Eq. 3, we compute the TAC as the sum of investment cost, CAPEX, and
operational cost, OPEX. According to current legislation in Germany (Status
2023), the grid fee has to be paid in addition to the market price for
electricity removed from the electricity grid (Bundesministerium der Justiz
der Bundesrepublik Deutschland,, 2022). Thus, we consider both annual
operational cost from electricity procurement, $\text{OPEX}_{el}$, as well as
annual grid fee cost, $\text{OPEX}_{Grid}$. Eq. 4 specifies the investment
cost of the local electricity generation and storage system considering
photovoltaic (PV), wind power (W), and electric battery (B). Similar to
Baumgärtner et al., (2019), we calculate the annualized CAPEX based on the
total investment cost $\text{CAPEX}^{0}_{i}$, the present value factor with
interest rate $\gamma_{1}$ and life time $\gamma_{2,i}$ (Broverman,, 2010), a
maintenance factor $\gamma_{3,i}$, and the installed capacity of the
respective technology $Q_{i}$. Note that in contrast to Baumgärtner et al.,
(2019), we consider a component-specific life time $\gamma_{2,i}$. Eq. 5
defines the electricity cost $\text{OPEX}_{el}$ by the purchases and sales on
hourly DA and quarter-hourly ID electricity market $\bm{q}_{\text{DA,s}}$ and
$\bm{q}_{\text{ID},s}$, the DA and ID electricity price $\bm{c}_{\text{DA,s}}$
and $\bm{c}_{\text{ID},s}$, the time step size $\Delta t=0.25$h, and the
probability $\pi_{s}$ of scenario $s$. In Eq. 6, the grid cost
$\text{OPEX}_{\text{Grid}}$ is derived from the sum of the grid cost
$\text{OPEX}_{\text{Grid},s,t}$ of each scenario $s$ and time step $t$ with a
total of 96 time steps for the one-day scheduling horizon, i.e, $T=96$. Eq. 7
and Eq. 8 constitute lower bounds for $\text{OPEX}_{\text{Grid},s,t}$, which
ensure that $\text{OPEX}_{\text{Grid},s,t}$ is equal to zero in case of
electricity injection into the grid and greater or equal to the grid fee with
the grid fee cost $c_{\text{Fee}}$ in case of electricity removal from the
grid. $\text{OPEX}_{\text{Grid},s,t}$ is equal to the respective lower limit,
i.e., zero or the grid fee, when minimizing the TAC. Note that in
$\text{OPEX}_{\text{Grid},s,t}$, $q_{\text{ID,}s,t}$ varies quarter-hourly and
$q_{\text{DA,}s,t}$ varies hourly.

The expected annual GWI is computed as:

$\displaystyle\text{\lx@glossaries@gls@link{acronym}{gwi}{\leavevmode
GWI}{}}=$ $\displaystyle
365\sum_{s\in\mathbb{S}}\pi_{s}\sum_{t=1}^{T}\Bigl{(}\text{GWI}^{\text{el}}_{s,t}\;\Delta{t}\;(q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}+q_{\text{ID},s,t})\Bigl{)}+\sum_{i\in\\{\text{PV,W,B}\\}}\frac{\text{GWI}^{\text{i}}Q_{\text{i}}}{\gamma_{2,\text{i}}}$
(9)

Eq. 9 considers the quarter-hourly average GWI of the electricity from the
grid $\text{GWI}^{\text{el}}_{s,t}$, the GWI of the installed PV, wind power,
and battery capacity, i.e., $\text{GWI}^{\text{PV}}$, $\text{GWI}^{\text{W}}$,
and $\text{GWI}^{\text{B}}$, and their respective life time $\gamma_{2,i}$ in
years. Consequently, the total annual GWI depends on hourly and quarter-hourly
purchases and sales from the DA and ID market $q_{\text{DA,s,t}}$ and
$q_{\text{ID,s,t}}$, respectively, and the installed PV, wind, and battery
capacities, i.e., $\text{Q}^{\text{PV}}$, $\text{Q}^{\text{W}}$, and
$\text{Q}^{\text{B}}$, respectively. Note that we allow for a GWI credit,
i.e., negative GWI, in case of electricity sales accounting for an avoided
emission burden (Horne et al.,, 2009).

###  2.3 Operational constraints

In the following, we shortly describe the generalized process model from our
prior work (Schäfer et al.,, 2020; Germscheid et al.,, 2022) and discuss the
operational constraints specific to the local energy system and the
electricity trading.

The generalized process model (Schäfer et al.,, 2020) relies on few key
parameters to describe the DR capabilities. In this work, we consider the key
characteristics oversizing, minimal part load, product storage capacity with
cyclic storage constraints, and ramping limitation. Note that without
efficiency losses, the production rate of the process scales directly with the
process power intake. For a detailed explanation of the generalized process
model including the model equations, we refer to Germscheid et al., (2022).

In the electricity generation and storage system, we consider the DA and ID
purchases and sales $q_{\text{DA,s,t}}$ and $q_{\text{ID,s,t}}$, respectively,
with positive values corresponding to purchases. In addition, we consider PV
power $q_{\text{PV,}s,t}$, wind power $q_{\text{W,}s,t}$, and battery charge
and discharge $q_{\text{in,}s,t}$ and $q_{\text{out,}s,t}$ that together
equate to the power intake $p_{s,t}$ of the production process by the energy
balance:

$\displaystyle p_{s,t}$
$\displaystyle=q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}+q_{\text{ID,}s,t}+q_{\text{PV,}s,t}+q_{\text{W,}s,t}-q_{\text{in,}s,t}+q_{\text{out,}s,t}$
(10)

Additionally, we consider operational constraints similar to Baumgärtner et
al., (2019):

$\displaystyle q_{\text{PV,}s,t}$
$\displaystyle=\bar{q}_{\text{PV,}s,t}\;Q_{\text{PV}},$ (11) $\displaystyle
q_{\text{W,}s,t}$ $\displaystyle=\bar{q}_{\text{W,}s,t}\;Q_{\text{W}},$ (12)
$\displaystyle 0$ $\displaystyle\leq q_{\text{in,}s,t}\leq Q_{\text{B}}/\tau,$
(13) $\displaystyle 0$ $\displaystyle\leq q_{\text{out,}s,t}\leq
Q_{\text{B}}/\tau,$ (14) $\displaystyle 0$ $\displaystyle\leq SOC_{s,t}\leq
Q_{\text{B}},$ (15) $\displaystyle SOC_{s,t+1}$
$\displaystyle=SOC_{s,t}+(\eta^{\text{in}}q_{\text{in,}s,t}-\frac{q_{\text{out,}s,t}}{\eta^{\text{out}}})\Delta{t},$
(16) $\displaystyle SOC_{s,1}$ $\displaystyle=SOC_{s,T+1}$ (17)

Here, Eq. 11 and Eq. 12 define the PV and wind power generation,
$q_{\text{PV,}s,t}$ and $q_{\text{W,}s,t}$, by multiplying the relative power
output, $\bar{q}_{\text{PV,}s,t}$ and $\bar{q}_{\text{W,}s,t}$, with the
installed PV and wind capacity, $Q_{\text{PV}}$ and $Q_{\text{W}}$,
respectively. Eq. 13 and Eq. 14 constrain the battery charge and discharge,
i.e., $q_{\text{in,}s,t}$ and $q_{\text{out,}s,t}$, respectively, by the
installed battery capacity $Q_{\text{B}}$ and the allowed charging and
discharging rate $\tau$. Eq. 15 constrains the state of charge $SOC_{s,t}$ by
the installed battery capacity $Q_{\text{B}}$. Eq. 16 relates charging and
discharging with respective efficiency losses $\eta^{\text{in}}$ and
$\eta^{\text{out}}$ and the state of charge. Additionally, we consider the
cyclic constraint, Eq. 17, requiring that the state of charge is the same at
the beginning and at the end of the scheduling horizon.

We consider trading electricity on both DA and ID market while making use of
the local energy system:

$\displaystyle-q_{\text{PV,}s,t}-q_{\text{W,}s,t}-Q_{\text{B}}/\tau$
$\displaystyle\leq q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1},$ (18)
$\displaystyle-q_{\text{PV,}s,t}-q_{\text{W,}s,t}-SOC_{s,t}/\Delta{t}$
$\displaystyle\leq q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1},$ (19)
$\displaystyle-
q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}-q_{\text{PV,}s,t}-q_{\text{W,}s,t}-Q_{\text{B}}/\tau$
$\displaystyle\leq q_{\text{ID},s,t},$ (20) $\displaystyle-
q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}-q_{\text{PV,}s,t}-q_{\text{W,}s,t}-SOC_{s,t}/\Delta{t}$
$\displaystyle\leq q_{\text{ID},s,t},$ (21) $\displaystyle
q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}$ $\displaystyle\leq
P_{\text{nom}}(1+\theta_{\text{max}})+Q_{\text{B}}/\tau,$ (22) $\displaystyle
q_{\text{DA,s,}\lfloor\frac{t-1}{4}\rfloor+1}$ $\displaystyle\leq
P_{\text{nom}}(1+\theta_{\text{max}})+\frac{Q_{\text{B}}-SOC_{s,t}}{\Delta{t}},$
(23) $\displaystyle q_{\text{ID},s,t}$ $\displaystyle\leq
P_{\text{nom}}(1+\theta_{\text{max}})+Q_{\text{B}}/\tau,$ (24) $\displaystyle
q_{\text{ID},s,t}$ $\displaystyle\leq
P_{\text{nom}}(1+\theta_{\text{max}})+\frac{Q_{\text{B}}-SOC_{s,t}}{\Delta{t}}$
(25)

Here, Eqs. 18, 19, 20 and 21 constrain DA and ID sales by produced PV and wind
power, the current state of charge, and maximum discharging capabilities of
the installed battery. Similar to our prior work (Germscheid et al.,, 2022),
Eqs. 20 and 21 allow to sell previously purchased DA electricity on the ID
market. Eqs. 22, 23, 24 and 25 constrain DA and ID purchases by the maximum
power consumption of the production process and the battery. The maximum
consumption of the process is defined by the nominal consumption
$P_{\text{nom}}$ and the process oversizing $\theta_{\text{max}}$. The maximum
consumption of the battery is given by the state of charge and the charging
capabilities of the battery.

###  2.4 Scenarios

We base our scenarios for the electricity price, wind power generation, PV
power generation, and grid emission factor on historical time series.
Specifically, for the electricity price, we use data from the German DA and ID
spot market (Fraunhofer Institute for Solar Energy Systems ISE,, 2023),
assuming that the consumer can purchase and sell electricity at DA market-
clearing price and ID index price. We refer to Germscheid et al., (2022) for a
detailed explanation about these assumptions. Fig. 2 shows the annual mean and
the annual mean daily standard deviation of the ID electricity market price
and the market deviation, i.e., the difference between the DA and the ID
price. The corresponding figure for the DA price shows similar characteristics
as the one of the ID price and can be found in Section 1 of the supporting
material. Fig. 2(a) reveals a price decrease in 2020 that can be attributed to
the initial phase of the COVID-19 pandemic (Halbrügge et al.,, 2021) and an
increase of mean and standard deviation in 2021 and 2022 due to the conflict
in the Ukraine (Haucap and Meinhof,, 2022). Fig. 2(b) reveals that in 2020 and
2021 the mean ID price was slightly larger than the mean DA price as indicated
by the positive market deviation. Moreover, the standard deviation of the
market deviation has significantly increased in 2021 and 2022. Note that DR
scheduling optimization necessitates electricity price time series rather than
an average annual electricity price. In our analysis, we pragmatically
consider the time series of the years 2020, 2021, and 2022, assuming that
these represent scenarios for low, medium, and high future electricity prices.

(a) ID electricity price

(b) Market deviation

Figure 2: Mean and mean daily standard deviation of ID electricity price (a)
and market deviation (b), i.e., the price difference between the DA and the ID
price.

To compute the corresponding historical wind and PV power time series, we use
weather data for Aachen, Germany, from the German weather service (Deutscher
Wetterdienst,, 2023; Gutzmann and Motl,, 2023). Specifically, we pre-process
the measured wind speed, global radiation, and diffuse radiation similar to
Bahl et al., (2017) and obtain the relative PV and wind power generation as
discussed in detail in Section 2 of the supporting material. Section 1 of the
supporting material shows the mean and the standard deviation of the
historical weather data, from which it can be seen that mean and standard
deviation of wind speed and solar irradiance stay within rather narrow ranges,
with 2020 as a windier year and 2022 as a sunnier year.

Following Baumgärtner et al., (2019) and Nilges et al., (2023), we determine
the average emission factor of electricity from the German grid for every time
step, i.e., $\text{GWI}^{\text{el}}_{s,t}$ used in Eq. 9, by considering the
momentary mix of power sources and their respective emission factors based on
data of Bundesnetzagentur|smard.de, (2023) and the ecoinvent database (Wernet
et al.,, 2016), respectively.

Figure 3: Approach to determine scenarios based on historical time series
data: The colored dots refer to the multi-dimensional data points that
represent (concatenated) time series data.

We determine the scenarios and the mapping between second and third stage of
the optimization problem based on a clustering as depicted in Fig. 3. First,
we preprocess the data by standardizing the wind and PV power time series and
the GWI time series using the z-score (Mohamad and Usman,, 2013) and
concatenating the daily profiles of wind power, PV power, and GWI time series
to daily energy profiles. Next, we apply k-means clustering using the scikit-
learn module in Python (Pedregosa et al.,, 2011) treating each daily energy
profile as a multi-dimensional data point. This leads to a clustering of
similar energy profiles. We transfer the obtained clustering to the DA price
and market deviation time series and create an average DA profile as DA price
scenario for each cluster. We add the market deviations from all constituents
of a respective cluster to the average DA price scenario to obtain ID price
scenarios. Note that using the market deviation allows to account for the
inter-market correlation similar to our prior work (Germscheid et al.,, 2022).
Finally, we use the mapping resulting from the clustering to connect the
second and third stage of the optimization problem. Note that on the second
stage, the probability of the DA scenarios depends on the cluster size. In
contrast, the scenarios on the third stage are equi-probable, i.e.,
$\pi_{s}=1/|\mathbb{S}|$, as the clustering is used on the third stage for
establishing the mapping but not for data reduction. We show in Section 3 of
the supporting material that the within-cluster sum-of-squares does not allow
to derive an obvious decision on a suitable number of clusters for the given
data. For our application, we look for a compromise between the number of
clusters and number of scenarios per cluster. Pragmatically, we consider 20
clusters, which leads to roughly 18 scenarios per cluster on average.

###  2.5 Model specifications and evaluation

We consider the parameters given in Tab. 1 for the CAPEX. Similar to Sass et
al., (2020), we consider an interest rate $\gamma_{1}=8\%$. In Section 4 of
the supporting material, we list the resulting annual PV and wind electricity
generation cost showing that wind power has lower production cost than PV due
to a higher average utilization rate. For the GWI of the electricity
generation and storage system, we use data of the ecoinvent database 3.9.1
(Wernet et al.,, 2016) that we specify in Section 4 of the supporting material
for reproducibility. For the battery, we consider a charging and discharging
rate of 4h (Tesla,, 2023), i.e., $\tau=4$h, and a round-trip efficiency of
$90\%$ (Hecking et al.,, 2018), i.e.,
$\sqrt{\eta_{\text{in}}}=\sqrt{\eta_{\text{out}}}=\sqrt{0.9}$. Moreover, we
consider the average grid fee cost of 2022 for industrial consumers in
Germany, i.e., $c_{\text{Fee}}=29.6$ EUR/MWh (Bundesnetzagentur und
Bundeskartellamt,, 2022). Tab. 2 lists reference parameters of the generalized
process that we also used in our prior work (Germscheid et al.,, 2022).
Furthermore, we choose the nominal capacity such that the power-intensive
production process is classified as an industrial consumer, i.e., 24 GWh per
year (Bundesnetzagentur und Bundeskartellamt,, 2022), which allows the process
operator to benefit from lower grid fees compared to non-industrial consumers
(Bundesnetzagentur und Bundeskartellamt,, 2022).

Table 1: Component-specific life time, investment and maintenance cost for Germany based on Hecking et al., (2018). | Lifetime $\gamma_{2,i}$ | $\text{CAPEX}^{0}_{i}$ | Annual maintenance cost $\gamma_{\text{2,i}}$
---|---|---|---
Roof-top PV | 25 a | 927 EUR/kWp | 17 EUR/kWp
Onshore wind | 25 a | 1113 EUR/kWp | 13 EUR/kWp
Battery | 15 a | 550 EUR/kWh | 20 EUR/kWh
Table 2: Parameters of the generalized process model: Process oversizing, minimal part load, and ramping limit are given related to the nominal power intake. The product storage capacity refers to the time necessary to fill an empty product storage considering production at nominal power intake.  Parameter | Reference values
---|---
Nominal power intake | 2.74 MW
Process oversizing | 20%
Minimal part load | 50%
Product storage capacity | 3h
Ramping limit | 25%/h

We expect the maximum allowed capacities of the energy system to have an
impact on the optimization results. Pragmatically, we first restrict the
admissible capacities for wind power and PV by the nominal power intake, i.e.,
$Q_{\text{W}}^{\text{max}}=Q_{\text{PV}}^{\text{max}}=P_{\text{nom}}$ and
choose the admissible battery size such that the maximum discharge rate
corresponds to the nominal power intake of the production process, i.e.,
$P_{\text{nom}}=Q_{\text{B}}^{\text{max}}/\tau$. In Section 3.2, we then
analyze the impact of the maximum allowed energy system capacities on the TAC
in detail.

We implement the model in Pyomo (Hart et al.,, 2011) and use the solver Gurobi
9.5.0 (Gurobi Optimization, LLC,, 2020) with default settings on an Intel Core
i7-9700 processor and 32GB RAM. We formulate the multi-stage problem by means
of its deterministic equivalent.

##  3 Results

In the following, we analyze the synergies between the local energy system and
the flexible production process and the benefit of considering simultaneous
market participation at design stage. To this end, we first consider market
participation only on the ID market and discuss the optimal design and savings
of the local energy system (Section 3.1) as well as the impact of the process
flexibility on the potential savings (Section 3.2). Note that we select the ID
market instead of the DA market, as the ID market allows to adapt the
electricity procurement in response to realization of the uncertainty in the
renewable electricity supply. We then show the difference between single and
simultaneous market participation and discuss the energy system design in the
context of DR scheduling with simultaneous DA and ID market participation
(Section 3.3).

###  3.1 Design and operation for single market participation

In the following, we evaluate the energy system design considering only the ID
market for the reference process defined in Tab. 2.

Fig. 4 shows Pareto-optimal energy system designs based on the time series for
2020, 2021, and 2022. For all three cases, the ecologic and economic
objectives lead to competing solutions. Interestingly, the TAC-optimal
solutions do not contain a battery, as potential savings from operating the
battery do not outweigh the battery investment cost. High electricity prices
in 2021 and 2022 incentivize both on-site wind and PV electricity generation
in the TAC-optimal solutions, whereas only wind electricity generation is used
in 2020. The preference for wind can be explained by the higher average
utilization rate (see Section 4 of the supporting material). In contrast,
battery, PV, and wind generation capacities are built at maximum capacity for
all GWI-optimal solutions, irrespective of the year studied. The integration
of a battery, however, leads to a large increase in TAC with only a small
improvement in GWI, as can be seen from the shape of the Pareto front.

Figure 4: Energy system design for ID market-only participation: Pareto-
optimal solutions are given for 2020 (left), 2021 (center), and 2022 (right).
For each year, the TAC-optimal, the GWI-optimal, and three intermediate
Pareto-optimal solutions are shown, which are equi-distant with respect to
GWI. GWI and TAC (lower part) are given with vertical gray dashed lines
pointing to the respective optimal capacities of the local electricity supply
system (upper three parts). Additionally, the TAC\- and GWI-optimal DR as well
as the steady-state operation without a local energy system are given for
comparison.

(a) Electricity price and emission factor.

(b) Renewable electricity generation.

(c) Operation of the flexible reference process.

(d) Battery operation.

Figure 5: Process and energy system operation for an exemplary day in 2022
(ID-only market participation): An energy system design with maximum
admissible PV, wind power, and battery capacities is considered. The operation
is shown for the ID electricity price and emission factor (a) and the
renewable electricity generation (b). The process operation determines the
electricity consumption (c). The battery operation determines the state of
charge (d).

Fig. 4 reveals that for 2022, the two designs with the lowest GWI have
identical PV, wind power, and battery capacities but have notable differences
in TAC and GWI, indicating different operating strategies. Fig. 5 shows the
operation for an exemplary day given a fixed energy system design and confirms
these findings. In particular, the electricity price and grid emission factor
(Fig. 5(a)) set different incentives for TAC-optimal and GWI-optimal operation
of the process (Fig. 5(c)) and the battery (Fig. 5(d)). Fig. 5(c) additionally
shows the process operation without a local energy system, revealing a similar
DR schedule as for a process with a local system. Similarly, Fig. 5(d) shows
the battery operation with and without local renewable electricity generation
revealing a similar operating pattern with only minor differences. We
attribute this behavior to the time-varying incentives for DR at the
operational level, i.e., the electricity price and the grid emission factor.
The incentives predominantly influence the operation of the process and the
local energy system, while on-site generated electricity has a minor
influence.

Fig. 4 suggests that the difference between steady-state operation and DR
without local energy system remains rather similar. In contrast, the
difference between DR without energy system and DR with a local energy system
has been increasing each year in particular with respect to the TAC. Tab. 3
compares ecologic and economic savings due to DR and the local energy system
and confirms these findings. The absolute economic savings from DR in
comparison to steady-state operation have been increasing due to the
increasing standard deviation of the electricity price (Fig. 2(a)). The
relative and absolute economic savings resulting from the energy system have
been increasing due to the increasing grid electricity cost. In particular,
the savings increased significantly from 1.4% in 2020 to 22.8% in 2022.
Looking at the GWI-optimal solution, the ecologic savings resulting from the
energy system are much larger than the savings from DR in comparison to
steady-state operation. Furthermore, the ecologic savings are somewhat similar
in all years, i.e., between 30.3% and 35.6%, and the variance can be
attributed to the natural variability of PV and wind power production and the
varying grid emission factor.

Table 3: Economic and ecologic savings due to DR and local electricity generation and storage. The savings are compared for the TAC-optimal solutions (upper part) and the GWI-optimal solutions (lower part). Single market participation on the ID market is considered.  | 2020 | 2021 | 2022 |
---|---|---|---|---
TAC-optimal solution |  |  |
DR vs steady-state operation (no energy system) |
Relative economic savings | 4.3% | 4.2% | 4.6% |
Absolute economic savings [kEUR] | 56 | 123 | 288 |
DR with energy system vs DR without energy system |  |
Relative economic savings | 1.4% | 8.8% | 22.8% |
Absolute economic savings [kEUR] | 17 | 245 | 1349 |
GWI-optimal solution |  |  |
DR vs steady-state operation (no energy system) |
Relative ecologic savings | 2.5% | 2.2% | 2.6% |
Absolute ecologic savings [kt${}_{{CO}_{2}}$/a] | 0.3 | 0.3 | 0.3 |
DR with energy system vs DR without energy system |  |
Relative ecologic savings | 35.4% | 30.3% | 35.6% |
Absolute ecologic savings [kt${}_{{CO}_{2}}$/a] | 3.6 | 3.5 | 4.3 |

###  3.2 Parameter study of process flexibility and energy system capacity

In the following, we analyze which impact the process flexibility and
admissible energy system capacities have on TAC and GWI.

For a parameter study on the process flexibility, we fix all process
parameters to their respective reference values as defined in Tab. 2 and vary
one process parameter at a time between the value corresponding to an
inflexible process and the value corresponding to twice the flexibility of the
process parameter. Fig. 6 shows the impact of varying degrees of process
flexibility on the TAC-optimal DR without a local energy system, the economic
savings enabled by a local energy system, and the optimal capacities for local
electricity generation and storage. Fig. 6(a) (top) shows exemplary for 2020
that without a local energy system, the flexible process particularly benefits
from oversizing. Behaviors for 2021 and 2022 are similar and thus the
corresponding figures are omitted. The results confirm the findings of our
prior work (Germscheid et al.,, 2022), where we considered price data of 2019.
Furthermore, Fig. 6(a) (bottom) shows the optimal capacities of the energy
system components and reveals that the optimal wind capacity for 2020 slightly
decreases with a high degree of process oversizing, showing that the process
flexibility can impact the optimal energy system capacity. Corresponding
figures for 2021 and 2022 are omitted, as no impact of the process flexibility
on the resulting optimal designs can be found.

(a) TAC-optimal solutions for 2020.

(b) Relative economic savings due to a local energy system for 2021 and 2022

Figure 6: Economic impact of different degrees of process flexibility: The
TAC-optimal cost of DR without energy system (a, top), the savings due to a
local energy system (a, center), and optimal energy system design (a, bottom)
for 2020 are shown. Furthermore, the savings due to a local energy system are
shown for 2021 (b, top) and 2022 (b, bottom).

Fig. 6 reveals that the range of the economic savings due to a local energy
system for any given year is rather narrow, i.e., varying the process
flexibility does not impact the relative savings in a strong manner. Varying
the process oversizing has the largest leverage on the savings. Interestingly,
the impact of the process parameters may differ depending on the investigated
year, as higher process flexibility actually leads to lower relative savings
in 2020 (Fig. 6(a), center) whereas higher relative savings are recorded for
2021 and 2022 (Fig. 6(b)). However, an analysis of the cost contributions
shows exemplary for varying oversizing that the absolute TAC monotonously
decreases, irrespective of the case with or without a local energy system (see
Section 5 of the supporting material).

Fig. 7 shows the optimal TAC for varying admissible energy system size and
process oversizing, the latter being the largest flexibility leverage for the
economic savings. Specifically, we vary the maximum allowed energy system
capacities, i.e., $Q_{\text{W}}^{\text{max}}$, $Q_{\text{PV}}^{\text{max}}$,
and $Q_{\text{B}}^{\text{max}}$, by a joint scaling factor. Corresponding
optimal capacities of the energy system can be found in Section 5 of the
supporting material. For 2020, process oversizing has a larger leverage on the
TAC than local electricity generation and storage. Furthermore, the TAC
remains constant for scaling factors larger than one, as a cost-optimal
maximum of the wind power capacity is attained (see Section 5 of the
supporting material). For 2021 and 2022, it can be seen that local electricity
generation and storage is more economically attractive than process
flexibility.

Interestingly, the absolute savings of the TAC-optimal solution enabled by a
higher process flexibility and by a larger energy system behave roughly
additive, which is shown exemplary for 2022 in Fig. 7. Section 5 of the
supporting material shows the relative difference between the optimal TAC and
the estimated TAC, the latter being defined as the sum of the absolute savings
from process flexiblization and installation of a local energy system. The
difference being rather small, i.e., less than 0.5%, means that a quick first
approximate economic assessment can be performed by considering the savings
from DR and the cost savings from a local energy system independently.

Figure 7: Optimal TAC [EUR] with varying process oversizing and maximum
admissible energy system capacity for 2020 (left), 2021 (center), and 2022
(right): For 2022, the approximately additive behavior of the absolute savings
is shown exemplary in red.

(a) GWI-optimal DR without a local energy system

(b) Relative ecologic savings due to a local energy system

Figure 8: Ecologic impact of varying process flexibility for 2020: GWI-optimal
DR without a local energy system (a) and relative savings due to installation
of a local system (b) are shown.

Fig. 8 shows the impact of varying degrees of process flexibility on the GWI-
optimal solution exemplary for 2020. Figures for the other years show similar
behavior and are therefore omitted. Fig. 8(a) shows the case of DR without a
local energy system and reveals that the process oversizing and the product
storage capacity have the largest impact on the GWI. This finding is
consistent with the results of our prior work (Schäfer et al.,, 2020), where
we considered the residual load as ecologic objective instead of the GWI. Fig.
8(b) shows that process oversizing and product storage capacity have the
largest leverage on the ecologic savings. Note that, in general, the degree of
process flexibility has a rather low impact on the range of the ecologic
savings.

###  3.3 Simultaneous market participation

Finally, we evaluate the benefit of considering simultaneous DA and ID market
participation in the integrated design and scheduling problem. Tab. 4 lists
the TAC-optimal designs for the reference process. Designs for 2020 and 2021
do not reveal differences to the case of ID-only market participation, i.e.,
the price differences between the markets in these years are not large enough
to incentivize the installation of a battery. In contrast, a battery is built
for the simultaneous market participation in 2022 as the battery offers
trading capacity for exploitation of large price differences between the DA
and ID market. The Pareto-optimal energy system designs of 2022 vary with
respect to the battery capacity for the simultaneous DA and ID market
participation compared to the ID-only case and are shown in Section 6 of the
supporting material.

Table 4: System capacities and electricity trading amounts for ID-only participation (upper part) and simultaneous DA and ID market participation (lower part): In all cases, TAC-optimal DR of the reference process (Tab. 2) with a local energy system is considered. The stated values of the ID-only participation correspond to the TAC-optimal solution from Fig. 4.  | 2020 | 2021 | 2022
---|---|---|---
ID-only participation |
PV capacity | - | 2.74 MW | 2.74 MW
Wind capacity | 2.74 MW | 2.74 MW | 2.74 MW
Battery capacity | - | - | -
Total purchases | 17,316 MWh | 16,008 MWh | 14,776 MWh
Total sales | 62 MWh | 203 MWh | 357 MWh
Simultanous market participation |
PV capacity | - | 2.74 MW | 2.74 MW
Wind capacity | 2.74 MW | 2.74 MW | 2.74 MW
Battery capacity | - | - | 5.31 MWh
DA purchases | 18,582 MWh | 16,389 MWh | 18,874 MWh
ID purchases | 6,796 MWh | 7,442 MWh | 12,944 MWh
Total purchases | 25,378 MWh | 23,831 MWh | 31,818 MWh
DA sales | 452 MWh | 954 MWh | 5,947 MWh
ID sales | 7,678 MWh | 7,080 MWh | 11,083 MWh
Total sales | 8,130 MWh | 8,034 MWh | 17,030 MWh

Tab. 4 compares single and simultaneous market participation with respect to
the electricity purchases and sales. It shows that for ID-only participation
more electricity is purchased in 2020, compared to 2021 and 2022, which are
years with larger optimal PV and wind power capacities. For the simultaneous
participation in 2020, the majority of the electricity is purchased on the DA
market due the positive price difference (Fig. 2(b)). The combination of PV
and wind enables higher DA sales in 2021. Moreover, total purchases and sales
significantly increase for the simultaneous participation in 2022 due to an
increased trading capacity enabled by the battery.

Tab. 5 shows that the relative savings of simultaneous market participation
compared to single market participation stay within a similar range,
irrespective of the considered year. In contrast, the absolute savings have
been increasing each year due to the increased variance of the market
deviation (Fig. 2(b)). Tab. 5 reveals that both the absolute and relative
savings of simultaneous market participation increase with both process
flexibilization and the integration of local electricity generation and
storage.

Table 5:  The TAC savings from simultaneous market participation compared to single market participation for an inflexible process without an energy system (top), a flexible process without an energy system (center), and a flexible process with an energy system (bottom). The flexible process is the reference process (Tab. 2).  | 2020 | 2021 | 2022
---|---|---|---
Inflexible process without energy system
Relative savings | 3.0% | 2.0% | 2.0%
Absolute savings | 39.3 kEUR | 57.4 kEUR | 122.7 kEUR
Flexible process without energy system
Relative savings | 3.8% | 2.5% | 2.5%
Absolute savings | 47.1 kEUR | 68.9 kEUR | 147.2 kEUR
Flexible process with energy system
Relative savings | 3.9% | 2.9% | 4.4%
Absolute savings | 47.9 kEUR | 72.8 kEUR | 198.7 kEUR

Tab. 6 shows the contributions of cost savings for the year 2022 considering
TAC-optimal simultaneous market participation. Here, an inflexible process
without an energy system is modified by adding a battery, renewable
electricity generation, and process flexiblization, separately. Tab. 6 reveals
that the main cost savings result from the on-site electricity generation
followed by process flexiblization. The integration of a battery accounts only
for a small fraction of the savings. Note that similar to ID-only
participation (Section 3.2), summing up the absolute savings from battery
installation, renewable electricity generation, and flexiblization,
separately, allows for a good overall savings estimate.

Table 6: Savings contributions of simultaneous market participation in the TAC-optimal case for 2022: The savings are related to the inflexible process without an energy system (first row). The battery capacity of the inflexible process with the battery (third row) is identical to the flexible process with an energy system (second row).  | TAC | Savings
---|---|---
Inflexible process without energy system | 6073.5 kEUR | -
Flexible process with energy system | 4360.9 kEUR | 1712.6 kEUR
Inflexible process with battery | 6044.7 kEUR | 28.8 kEUR
Inflexible process with wind and PV power | 4702.5 kEUR | 1371.0 kEUR
Flexible process without energy system | 5761.2 kEUR | 312.3 kEUR

Finally, similar to Section 3.2, we vary the degree of process flexibility.
Fig. 9 shows the impact of process flexibility on the optimal battery capacity
for 2022. An analysis of the optimal capacities of PV and wind power is
omitted as these quantities do not vary in response to a varying degree of
process flexibility. It can be noted that larger batteries are built for less
flexible processes, in particular for processes with part load and ramping
restrictions. DR without energy system and economic savings of DR with an
energy system behave similar to the case of single market participation
(Section 3.2), irrespective of the investigated year. Thus, respective figures
are omitted.

Figure 9: TAC-optimal battery capacity for simultaneous market participation
for 2022 with varying degree of process flexibility.

##  4 Conclusion

We assessed the optimal design of a local electricity generation and storage
system for a generalized continuous, power-intensive production process that
is capable of performing demand response and can act on both the day-ahead and
intraday electricity market. In a three-stage stochastic problem, we optimized
the capacities of photovoltaic power, wind power, and electric battery with an
integrated demand response scheduling of the production process. Building on
our prior work (Schäfer et al.,, 2020; Germscheid et al.,, 2022), we used a
generalized process model with few flexibility-defining parameters, i.e.,
process oversizing, minimal part load, product storage, and ramping
limitation. In a bi-objective optimization, we considered both economic and
ecologic objectives. We considered scenarios of low, intermediate, and high
electricity prices for a plant location in Germany as well as a time-varying
grid emission factor.

We find that batteries are mainly built to lower the global warming impact,
however, leading to a significant increase in total annualized cost. Economic
and ecologic process and battery operation primarily respond to the time-
varying electricity price and grid emission factor, but only to a little
extent to the on-site generation of renewable electricity. Varying the degree
of process flexibility, we find a rather small impact on the achievable
relative economic and ecologic savings that come with local electricity
generation and storage. Moreover, we show that the absolute cost savings from
flexiblizing the process and installing a local energy system are
approximately additive. Comparing intraday-only and simultaneous day-ahead and
intraday market participation, we find that the energy system designs are
similar for the investigated scenarios, except when high price differences
between the markets incentivize the installation of a battery. The cost-
optimal battery capacity significantly depends on the available process
flexibility, enables large volumes for trading on the markets, but comes with
only minor economic savings.

In order to use our approach to evaluate the benefit of optimally sizing a
local electricity generation and storage system for a specific process and
location, the process characteristics must be known, respective local weather
data is required, and electricity price scenarios have to be created.
Considering the long lifetime of a process and the energy system equipment,
long-term electricity price scenarios with measures of uncertainty and the
associated financial risk should be accounted for, e.g., similar to Xuan et
al., (2021); Vieira et al., (2021).

## Authorship contribution

Sonja H. M. Germscheid: Conceptualization, Methodology, Software,
Investigation, Data curation, Writing - original draft, Visualization.
Benedikt Nilges: Data curation, Writing – review & editing. Niklas von der
Assen: Funding acquisition, Writing – review & editing. Alexander Mitsos:
Writing - review & editing, Supervision, Funding acquisition. Manuel Dahmen:
Conceptualization, Methodology, Writing - review & editing, Supervision,
Funding acquisition.

## Declaration of Competing Interest

We have no conflict of interest.

## Acknowledgements

SG gratefully acknowledges the financial support of the Helmholtz Association
of German Research Centers through program-oriented funding (POF) and the
grant Uncertainty Quantification – From Data to Reliable Knowledge (UQ) (grant
number: ZT-I-0029). AM and MD acknowledge funding from the Helmholtz
Association of German Research Centers through program-oriented funding (POF).
BN gratefully acknowledges the financial support of the Kopernikus project
SynErgie (grant number 03SFK3L1-2) by the Federal Ministry of Education and
Research (BMBF). This work was performed as part of the Helmholtz School for
Data Science in Life, Earth and Energy (HDS-LEE). We kindly thank Yifan Wang
(RWTH Aachen University, Institute of Technical Thermodynamics) for providing
the wind turbine performance curve developed by Bahl et al., (2017) which was
used for pre-processing of the wind data.

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# The change of vertex energy when joining trees.

Octavio Arizmendi  Department of Probability and Statistics, Centro de
Investigación en Matemáticas, Guanajuato, México  Saylé Sigarreta  Department
of Probability and Statistics,Benemérita Universidad Autónoma de Puebla,
Puebla, México

###### Abstract

In this manuscript we study how the vertex energy of a tree is affected when
joined with a bipartite graph. We find an alternating pattern with respect to
the coalescence vertex: the energy decreases for vertices located at odd
distances and increases for those located at even distances.

Keywords: graph energy, graph join, trees.

MSC classes: 05C05, 05C09

##  1 Introduction

Gutman introduced in 1978, the energy of a graph, [7], motivated by notion of
$\pi$-electron energy in conjugated hydrocarbon molecules. Forty years later,
inspired by Non-Commutative Probability, Arizmendi and Juárez-Romero [4]
introduced the concept of energy of a vertex (or vertex energy). Since we can
calculate the energy of a graph by adding the individual energies of its
vertices, it follows that the energy of a vertex must be understood as the
contribution of this vertex to the energy of the graph. In this sense,
studying vertex energies constitutes a tool to achieve a better understanding
of the energy of a graph. The basics of the theory were developed in [3],
where the authors derived fundamental inequalities and gave examples and
counterexamples of natural conjectures for the vertex energy.

Taking into account the developed theory and in search of an interpretation
that would obey intuition and allow us to understand the interaction between
local and global energy in graphs, in [2] the authors defined the graph energy
game. An important conclusion is that, by considering the vector of vertex
energies as the payoff vector of the cooperative game, an interpretation of
how the total energy is distributed among the vertices of the graph can be
offered.

Motivated by the above, in the present work we continue the research in this
direction. Specifically, the main objective of this manuscript is to study the
effect produced in the vertex energies of a tree after being merged with a
bipartite graph. Our main contribution is to show that the change in vertex
energy has an alternating behavior, when considering the distance with the
merging vertex. Let us state our main result, the definitions involved in the
statement can be seen in Section 2.

######  Main Theorem 1.

Let $T$ be a tree and $B$ be a bipartite graph. Consider the coalescence of
$T\circ B$ where we have identified the vertices $v$ in $T$ and $u$ in $B$
where $deg_{B}(u)\geq 1$. Let $w$ be a vertex in $T$ and $\hat{w}$ its
corresponding copy in $T\circ B$ then

  * •

If $d(v,w)$ is odd, $\mathcal{E}_{T}(w)>\mathcal{E}_{T\circ B}(\hat{w}).$

  * •

If $d(v,w)$ if is even, $\mathcal{E}_{T}(w)<\mathcal{E}_{T\circ B}(\hat{w})$.

As we explain below, in Section 3, this is consistent with the intuition that
vertex energy acts as a payoff in a cooperative game.

##  2 Preliminaries

In this section we introduce basic results on Graph Theory, Graph Energy and
Vertex Energy for further reference.

###  2.1 Basic definitions in Graph Theory

In this work we will only consider simple finite undirected graphs. A simple
undirected graph is a pair $G=(V,E)$, where $E\subseteq V\times V$ is a set
such that $(v,v)\notin E$ for all $v\in V$, and $(v,w)\in E$ implies that
$(w,v)\in E$. $V$ is called the vertex set and $E$ the edge set. The degree
$deg_{G}(v)$ of a vertex $v$ is the number of edges at $v$. The distance
$d_{G}(u,v)$ in $G$ of two vertices $u,v$ is the length of a shortest $u-v$
path in $G$. A connected graph that does not contain any cycle is called a
tree and a graph that does not contain cycles of odd length is called
bipartite. Given $G$ and $H$ two graphs with disjoint vertex sets, $u\in G$
and $v\in H$, the graph $G\circ H$ is known as the coalescence of $G$ and $H$
with respect to $u$ and $v$. It is constructed from copies of $G$ and $H$ by
identifying the vertices $u$ and $v$.

Let $G=(V,E)$ be a graph of order $n$, where
$V=\\{v_{1},v_{2},\dots,v_{n}\\}$. The adjacency matrix
$A(G)=(a_{ij})_{n\times n}$ of $G$ is defined by $a_{ij}=1$ if
$(v_{i},v_{j})\in E$ and 0 otherwise. The characteristic polynomial of $G$,
denoted by $\phi_{G}(x)$, is defined as the characteristic polynomial of
$A(G)$, that is, $\phi_{G}(x)=det(xI-A)=:\sum_{k=0}^{n}a_{k}x^{n-k}$, where
$I$ is the identity matrix of size $n\times n$. In particular, it is verified
that, $\phi_{G\circ H}(x)=\phi_{G}(x)\phi_{H\setminus v}(x)+\phi_{G\setminus
u}(x)\phi_{H}(x)-x\phi_{G\setminus u}(x)\phi_{H\setminus v}(x).$ Also, if $G$
is a bipartite graph, then its characteristic polynomial is of the form
$\phi_{G}(x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^{k}b_{2k}(G)x^{n-2k}$, where
$b_{2k}(G)=|a_{2k}(G)|$.

###  2.2 Graph energy

The energy of $G$, denoted by $\mathcal{E}(G)$, is defined as
$\mathcal{E}(G)=Tr(|A(G)|),$ where $|A(G)|=(A(G)A(G)^{\ast})^{1/2}$. In
particular, when $G$ is bipartite, the Coulson's integral formula [6]
motivates the following definition of a quasi-order for bipartite graphs. For
two bipartite graphs $G_{1}$ and $G_{2}$, we define the quasi-order $\preceq$
and write $G_{1}\preceq G_{2}$ if $b_{2k}\left(G_{1}\right)\leq
b_{2k}\left(G_{2}\right)$ for all $k$. If, in addition, at least one of the
inequalities $b_{2k}(G_{1})\leq b_{2k}(G_{2})$ is strict, then we write
$G_{1}\prec G_{2}$. The importance of this order lies in the fact that it
allows to compare the energy of two graphs.

While developing the main results of this paper, we will need the following
results.

######  Lemma 2.1.

(Theorem 1.3, [8]) Let $uv$ be an edge of $G$. Then

$\phi_{G}(x)=\phi_{G\setminus
uv}(x)-\phi_{G\setminus\\{u,v\\}}(x)-2\displaystyle\sum_{C\in\mathscr{C}(uv)}\phi_{G\setminus
C}(x),$

where $\mathscr{C}(uv)$ is the set of cycles containing $uv$.

######  Lemma 2.2.

(Theorem 4.18, [8]) Let $G$ and $H$ be graphs. Then $\mathcal{E}(G\circ
H)\leq\mathcal{E}(G)+\mathcal{E}(H).$

######  Lemma 2.3.

(Theorem 4.20, [8]) If $F$ is an edge cut of a simple graph G,
$\mathcal{E}(G\setminus F)\leq\mathcal{E}(G).$

###  2.3 Energy of a vertex

The energy of the vertex $v_{i}\in V(G)$, which is denoted by
$\mathcal{E}_{G}(v_{i})$, was defined in [4], as follows:
$\mathcal{E}_{G}(v_{i})=|A(G)|_{ii}$, for $i=1,2,\dots,n$. Equivalently,

$\mathcal{E}_{G}\left(v_{i}\right)=\sum_{j=1}^{n}p_{ij}\left|\lambda_{j}\right|,\quad
i=1,\ldots,n,$ (1)

where $\lambda_{j}$ denotes the $j$-eigenvalue of the adjacency matrix of $G$
and the weights $p_{ij}$ satisfy

$\sum_{i=1}^{n}p_{ij}=1\text{ and }\sum_{j=1}^{n}p_{ij}=1.$

More precisely, $p_{ij}=u_{ij}^{2}$ where $U=(u_{ij})^{n}_{i,j=1}$ is the
orthogonal matrix whose columns are given by the eigenvectors of $A(G)$. On
the other hands,

$M_{k}(G,i)=\sum_{j=1}^{n}p_{ij}\lambda_{j}^{k},\quad i=1,\ldots,n,$ (2)

where $M_{k}(G,i)$ is equal to the number of $v_{i}-v_{i}$ walks in $G$ of
length $k$.

By definition and guided by the results obtained for $\mathcal{E}(G)$, in [5],
a Coulson-type integral formula to calculate the energy of a vertex was found
and as a consequence, it allowed to elucidate the structural interpretation of
the energy of a vertex and its relationship with the energy of a bipartite
graph, since, it provided a method to compare the energy of two vertices in
such graphs.

######  Lemma 2.4.

(Theorem 4, [5]) Let $G$ be a graph, then for a vertex $v_{i}$ in $G$,

$\mathcal{E}_{G}\left(v_{i}\right)=\frac{1}{\pi}\int_{\mathbb{R}}1-\frac{\mathbf{i}x\phi\left(G-v_{i};\mathbf{i}x\right)}{\phi(G;\mathbf{i}x)}dx.$

######  Lemma 2.5.

(Lemma 6, [5]) Let $G$ a bipartite graph and $v,w\in G$. If $G\setminus
w\succeq G\setminus v$, then $\mathcal{E}_{G}(w)\leq\mathcal{E}_{G}(v)$.
Moreover if $G\setminus w\neq G\setminus v$ then
$\mathcal{E}_{G}(w)<\mathcal{E}_{G}(v)$.

The following result which is a consequence of Lemma 2.5 will be often used in
our proofs below.

######  Lemma 2.6.

(Theorem 7, [5]) Let $G_{1}$ and $G_{2}$ two disjoint bipartite graphs with
$w\in G_{2}$ and $v\in G_{1}$. If $G_{1}\cup\left(G_{2}\setminus
w\right)\succeq G_{2}\cup\left(G_{1}\setminus v\right)$, then
$\mathcal{E}_{G_{2}}(w)\leq\mathcal{E}_{G_{1}}(v)$. Moreover if
$G_{2}\setminus w\neq G_{1}\setminus v$ then
$\mathcal{E}_{G}(w)<\mathcal{E}_{G}(v)$.

The remaining two lemmas will also be used in what follows.

######  Lemma 2.7.

(Proposition 3.9, [3]) Let $G$ be a bipartite graph with parts $V_{1}$ and
$V_{2}$. Then

$\displaystyle\sum_{v\in V_{1}}\mathcal{E}_{G}(w)=\displaystyle\sum_{v\in
V_{2}}\mathcal{E}_{G}(w).$

######  Lemma 2.8.

(Theorem 3, [1]) Let $v_{i}$ and $v_{j}$ be connected vertices of a simple
(undirected) graph G. Then
$\mathcal{E}\left(v_{i}\right)\mathcal{E}\left(v_{j}\right)\geq 1$.

##  3 Main Results

In this section we prove the main theorem. To prepare for the main part of the
proof, we need some notation.

######  Notation 3.1.

Let $T$ be a tree and consider a given path $P=\\{v_{1}\sim
v_{2}\sim\cdots\sim v_{n}\\}$ in $T$. If $n\geq 2$, then for $1\leq i\leq
n-1$, let $e_{i}=v_{i}v_{i+1}$.

  1. 1.

For $n=1$, $\tilde{T}$ is the empty graph. For $n\geq 2$, $\tilde{T}$ denotes
the connected component of $T\setminus v_{1}$ which contains $v_{2}$.

  2. 2.

$A_{i}$ denotes the connected component of $T\setminus e_{i}$ which contains
$v_{i}$.

  3. 3.

For $i=1$, $\tilde{A}_{i}$ is the empty graph. For $i\geq 2$ (which implies
that $n\geq 3$), $\tilde{A}_{i}$ denotes the connected component of
$A_{i}\setminus v_{1}$ that contains $v_{2}$.

The following simple lemma is crucial for the proof of our main theorem.

######  Lemma 3.2.

Let $T$ be a tree and $P=\\{v_{1}\sim v_{2}\sim\cdots\sim v_{n}\\}$ be a path
in $T$ with $n\geq 2$. Then for $1\leq i\leq n-1$. The following order is
satisfied in the quasi-order $\preceq$:

  * •

If $i$ is even, $T\cup\tilde{A}_{i}\prec\tilde{T}\cup A_{i}.$

  * •

If $i$ is odd, $T\cup\tilde{A}_{i}\succ\tilde{T}\cup A_{i}$.

###### Proof.

We will get the proof via mathematical induction on $i$. Our base case will
be, $i=1,2$. First, for $i=1$, by Lemma 2.1 applied to $T$ with $e_{1}$, we
can observe that

$b_{2k}(T\cup\tilde{A}_{1})=b_{2k}(T)=b_{2k}(\tilde{T}\cup
A_{1})+b_{2(k-1)}(\widetilde{\textit{{T}}}\cup(A_{2}\setminus\\{v_{1},v_{2}\\})),$

where $\textit{{T}}=\tilde{T}$ and $\textit{{P}}=\\{u_{1}=v_{2}\sim
u_{2}=v_{3}\sim\dots\sim u_{n-1}=v_{n}\\}$; which completes this part. For
$i=2,$ by Lemma 2.1 applied to $A_{2}$ and $T$ with $e_{1}$, respectively, we
obtain

$b_{2k}(\tilde{T}\cup A_{2})=b_{2k}(\tilde{T}\cup\tilde{A}_{2}\cup
A_{1})+b_{2(k-1)}(\textit{{T}}\cup(A_{2}\setminus\\{v_{1},v_{2}\\}))$,

and

$b_{2k}(T\cup\tilde{A}_{2})=b_{2k}(\tilde{T}\cup\tilde{A}_{2}\cup
A_{1})+b_{2(k-1)}(\widetilde{\textit{{T}}}\cup{\textit{{A}}}_{1}\cup(A_{2}\setminus\\{v_{1},v_{2}\\}))$.

Then, if now with T we apply the same argument used when $i=1$, we obtain the
conclusion. In the following, we will show the statement for $i$. By using
Lemma 2.1 for $A_{i}$ and $T$ with $e_{1}$, the following equalities hold

$b_{2k}(\tilde{T}\cup A_{i})=b_{2k}(\tilde{T}\cup\tilde{A}_{i}\cup
A_{1})+b_{2(k-1)}(\textit{{T}}\cup\widetilde{\textit{{A}}}_{i-1}\cup
A_{2}\setminus\\{v_{1},v_{2}\\}$),

and

$b_{2k}(T\cup\tilde{A}_{i})=b_{2k}(\tilde{T}\cup\tilde{A}_{i}\cup
A_{1})+b_{2(k-1)}(\widetilde{\textit{{T}}}\cup{\textit{{A}}}_{i-1}\cup
A_{2}\setminus\\{v_{1},v_{2}\\})$.

Finally, applying the induction hypothesis for
$\textit{{T}}\cup\widetilde{\textit{{A}}}_{i-1}$ and
$\widetilde{\textit{{T}}}\cup{\textit{{A}}}_{i-1}$ the proof is complete.

∎

######  Remark 3.3.

The result seen above is also valid for bipartite graphs when the path $P$ is
formed solely by edge cuts.

Now we can prove the main theorem of the paper that we state again for the
convenience of the reader.

######  Theorem 3.4.

Let $T$ be a tree and $B$ be a bipartite graph. Consider the coalescence of
$T\circ B$ where we have identified the vertices $v$ in $T$ and $u$ in $B$
where $deg_{B}(u)\geq 1$. Let $w$ be a vertex in $T$ and $\hat{w}$ its
corresponding copy in $T\circ B$ then

  * •

If $d_{T}(v,w)$ is odd, $\mathcal{E}_{T}(w)>\mathcal{E}_{T\circ B}(\hat{w}).$

  * •

If $d_{T}(v,w)$ if is even, $\mathcal{E}_{T}(w)<\mathcal{E}_{T\circ
B}(\hat{w})$.

###### Proof.

By Lemma 2.6 it is enough to prove that

$(T\circ B)\cup(T\setminus w)\prec T\cup((T\circ B)\setminus\hat{w})$,

when $d_{T}(v,w)$ is odd and,

$T\cup((T\circ B)\setminus\hat{w})\prec(T\circ B)\cup(T\setminus w)$,

when $d_{T}(v,w)$ is even. Firstly, for $d_{T}(v,w)\geq 1$, let
$P=\\{v_{1}=v\sim v_{2}\sim\dots\sim v_{d(v,w)+1}=w\\}$ be the path in $T$
connecting the vertices $v$ and $w$ and note that

$T\cup((T\circ B)\setminus\hat{w})=T\cup(A_{d(v,w)}\circ B)\cup Q,$

and

$(T\circ B)\cup(T\setminus w)=(T\circ B)\cup A_{d(v,w)}\cup Q,$

where $Q$ is the disjoint union of the connected components of $T\setminus
v_{d(v,w)+1}$ that do not contain $v_{d(v,w)}$ and $A_{d(v,w)}\circ B$ is the
coalescence with respect to $v$ and $u$. Now, by computing the characteristic
polynomial of $A_{d(v,w)}\circ B$ and $T\circ B$, respectively, we have that

$b_{2k}(T\cup(A_{d(v,w)}\circ B))=$

$b_{2k}(T\cup A_{d(v,w)}\cup B\setminus u)+b_{2k}(T\cup A_{d(v,w)}\setminus
v\cup B))-b_{2k}(T\cup A_{d(v,w)}\setminus v\cup B\setminus u)),$

and

$b_{2k}((T\circ B)\cup A_{d(v,w)}))=$

$b_{2k}(T\cup A_{d(v,w)}\cup B\setminus u)+b_{2k}(T\setminus v\cup
A_{d(v,w)}\cup B))-b_{2k}(T\setminus v\cup A_{d(v,w)}\cup B\setminus u)).$

Since, the first term is the same in both expressions and $B\setminus u\prec
B$, all boil down to compare $b_{2k}(T\cup A_{d(v,w)}\setminus v)$ and
$b_{2k}(T\setminus v\cup A_{d(v,w)})$. Rewriting the last two expressions in
the following way

$b_{2k}(T\cup A_{d(v,w)}\setminus v)=b_{2k}(T\cup\tilde{A}_{d(v,w)}\cup
A_{1}\setminus v),$

and

$b_{2k}(T\setminus v\cup A_{d(v,w)})=b_{2k}(\tilde{T}\cup A_{d(v,w)}\cup
A_{1}\setminus v).$

By Lemma 3.2, this part is completed. Finally, if $w=v$ then

$T\cup((T\circ B)\setminus\hat{v})=T\cup B\setminus\hat{v}\cup T\setminus v,$

and

$(T\circ B)\cup(T\setminus v)=(T\circ B)\cup T\setminus v.$

Hence, the statement is verified, if in a similar way we calculate the
characteristic polynomial of $T\circ B$. ∎

The central problem of cooperative games is to examine how the benefits
obtained with the coalition will be distributed among all players. Now, since
given a graph, we have that
$\mathcal{E}_{G}(G)=\sum_{i=1}^{n}\mathcal{E}_{G}(v_{i})$. Then, it is natural
to think this as a partitioning problem. With this approach, in [2], the
authors define the energy game as follows: Let $G=(V,E)$ be a simple
undirected graph, for each subset $S\subseteq V$, they define the
characteristic function as $w(S):=$ $\mathcal{E}(I(S))$ where $I(S)$ is the
graph induced by $S$. Furthermore, they show that the vertex energy is a
payoff vector that belongs to the core. Thus, one potential theoretical
explanation for this behavior may be derived through the application of a Game
Theory perspective: Consider the path $P=\\{v_{1}\sim v_{2}\cdots\sim
v_{n}\\}$ in $T$, where we merge a bipartite graph $B$ in $v_{1}$. First, the
player who is used for the merger, increases its options to establish
alliances. It is expected then to increase its valuation as well, since, it
will have a larger negotiating margin. On the other hand $v_{2}$ now has more
competition, that is, the new neighbours of $v_{1}$ hence lower negotiating
margin leading to a decrease in its payoff. By applying inductively the same
reasoning on the remaining players in the path, we should obtain the
alternating behavior observed in Theorem 3.4 (Figure 1).

$\hat{v}$$\hat{w}$$B\circ
T$$\dots$$\uparrow$$\uparrow$$\downarrow$$\downarrow$$\uparrow$$\downarrow$$\uparrow$$\downarrow$
Figure 1: Graphical representation of the behavior observed in the Theorem
3.4.

######  Remark 3.5.

According to Remark 3.3 and the proof seen above, the result shown in Theorem
3.4 is also valid for $B_{1}\circ B_{2}$ respect to $u_{i}\in V(B_{i})$ with
$B_{i}$ bipartite graphs if we take either $w=u_{i}$ or any $w\in
V(B_{i})\setminus u_{i}$ connected to $u_{i}$ by means of a path formed by
edge cuts (which tells us that the path is unique). Moreover, the energy of
vertices in $B_{i}$ that are not connected with $u_{i}$ is the same after
coalescence.

Next, by directly applying the Theorem 3.4 and Remark 3.5 we can obtain the
following corollary; which gives us information about the behavior of vertex
energy when we perform the opposite operation, that is, when we remove edges
or vertices, and as expected it shows a contrary behavior.

######  Corollary 3.6.

Let $T$ be a tree and $B$ be a bipartite graph. Consider the coalescence
$T\circ B$ where we have identified the vertices $v$ in $T$ and $u$ in $B$,
respectively. Given $v_{1}v_{2}\in E(T)$, we will denote by $T_{v_{i}}$ the
connected component of $T\setminus v_{1}v_{2}$ containing $v_{i}$. Then taken
a vertex $w$ in $T_{v_{i}}$ we have that:

  * •

If $d(w,v_{i})$ is odd, $\mathcal{E}_{(T\circ B)\setminus
v_{2}v_{1}}(\hat{w})>\mathcal{E}_{T\circ B}(w).$

  * •

If $d(w,v_{i})$ is even, $\mathcal{E}_{(T\circ B)\setminus
v_{2}v_{1}}(\hat{w})<\mathcal{E}_{T\circ B}(w)$.

It is important to highlight that in the previous corollary we covered all the
vertices in $T$, since, $T=T_{v_{1}}\cup T_{v_{2}}$. In particular, when $B$
is the graph formed by an isolated vertex, it shows us what is the behavior in
the trees after removing an edge; and as consequence, after eliminating
vertices.

The preceding theorems compare the energies before and after coalescence.
Another pertinent issue regarding the merging vertex is to compare the energy
of the vertex after coalescence with their individual energies before
coalescence. This is exactly what the following theorem addresses.

######  Theorem 3.7.

Given $G$ and $H$ two bipartite graphs with disjoint vertex sets. Let $u\in
G$, $v\in H$ and $w$ its copy in $G\circ H$, then

$\mathcal{E}_{G\circ H}(w)\leq\mathcal{E}_{G}(u)+\mathcal{E}_{H}(v).$

Moreover, equality is attained if and only if either $u$ is an isolated vertex
of $G$ or $v$ is an isolated vertex of $H$ or both.

###### Proof.

By Lemma 2.4, we have

$\displaystyle\mathcal{E}_{G\circ H}(v)$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{ix\phi(G\circ
H-v,ix)}{\phi(G\circ H,ix)}\right)dx$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{ix\phi(G-v,ix)\phi(H-v;ix)}{\phi(G\circ
H;ix)}\right)dx$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}}\left(1+\frac{\phi(G\circ
H,ix)}{\phi(G\circ H,ix)}-\frac{\phi(G;ix)\phi(H-v;ix)}{\phi(G\circ
H;ix)}-\frac{\phi(H;ix)\phi(G-v)}{\phi(G\circ H;ix)}\right)dx$
$\displaystyle=\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(G;ix)\phi(H-v;ix)}{\phi(G\circ
H;ix)}\right)dx+\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(H;ix)\phi(G-v;ix)}{\phi(G\circ
H;ix)}\right)dx.$

For the first integral notice that,

$\displaystyle\frac{\phi(G;ix)\phi(H-v;ix)}{\phi(G\circ H;ix)}$
$\displaystyle=\frac{(ix)^{n+m-1}\sum_{k\geq
0}(-1)^{k}b_{2k}(G)(ix)^{-2k}\sum_{k\geq
0}(-1)^{k}b_{2k}(H-v)(ix)^{-2k}}{(ix)^{n+m-1}\sum_{k\geq
0}(-1)^{k}b_{2k}(G\circ H)(ix)^{-2k}}$ $\displaystyle=\frac{\sum_{k\geq
0}b_{2k}(G)x^{-2k}\sum_{k\geq 0}b_{2k}(H-v)x^{-2k}}{\sum_{k\geq
0}b_{2k}(G\circ H)x^{-2k}},$

where $|V(G)|=n$ and $|V(H)|=m$. Besides,

$\frac{\phi(H-v;ix)ix}{\phi(H;ix)}=\frac{\sum_{k\geq
0}b_{2k}(H-v)x^{-2k}}{\sum_{k\geq 0}b_{2k}(H)x^{-2x}},$

and

$\left(\sum_{k\geq 0}b_{2k}(H)x^{-2k}\right)\left(\sum_{k\geq
0}b_{2k}(G)x^{-2k}\right)=\sum_{k\geq 0}b_{2k}(G\cup H)x^{-2k}.$

Since $b_{2k}(G\cup H)\geq b_{2k}(G\circ H)\geq 0$, it is verified that

$\frac{\sum_{k>0}b_{2k}(G)x^{-2k}\sum_{k>0}b_{2k}(H-v)x^{-2k}}{\sum_{k>0}b_{2k}(G\circ
H)x^{-2k}}\geq\frac{\sum_{k>0}b_{2k}(H-v)x^{-2k}}{\sum_{k>0}b_{2k}(H)x^{-2k}}.$

Thus,

$\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(G)\phi(H-v)}{\phi(G\circ
H)}\right)dx\leq\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(H-v)}{\phi(H)}\right)dx=\mathcal{E}_{H}(v)$
(3)

By conducting a similar process with the other summand we obtain

$\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(H)\phi(G-v)}{\phi(G\circ
H)}\right)dx\leq\frac{1}{\pi}\int_{\mathbb{R}}\left(1-\frac{\phi(H-v)}{\phi(H)}\right)dx=\mathcal{E}_{G}(v)$
(4)

which completes the proof. ∎

######  Remark 3.8.

Note that the above result is a local version of Lemma 2.2 and when is
contrasted with Theorem 3.4 and Remark 3.5 tells us how much the energy of the
vertex (where coalescence is made) can increase.

###  3.1  Successive coalescence

Now, we consider the effect of connecting successively a sequence of bipartite
graphs. Later we will study the limiting behavior.

######  Corollary 3.9.

Let $T$ be a tree, $\\{B_{n}\\}_{n\geq 0}$ a sequence of bipartite graphs,
$v\in V(T)$ and $\\{b_{n}\\}_{n\geq 0}$ a sequence of vertices such that
$b_{n}\in V(B_{n})$. Now, we will construct a sequence of graphs
$\\{G_{n}\\}_{n\geq 0}$ using the following recursive method.

  * •

For $n=1$, by identifying the vertex $v$ in $T$ and the vertex $b_{1}$ in
$B_{1}$, we define $G_{1}=T\circ B_{1}$ and let $v_{1}$ be the copy of $v$ in
$G_{1}$.

  * •

For $n\geq 2$, let $v_{n-1}$ be the copy of $v$ in $G_{n-1}$, we define
$G_{n}=G_{n-1}\circ B_{n}$ where we have identified the vertices $v_{n-1}$ and
$b_{n}$, respectively.

Let $w$ be a vertex in $T$ and $\hat{w}$ its corresponding copy in $G_{n}$
then the following results hold:

  * •

If $d(v,w)$ is even, then $\\{\mathcal{E}_{G_{n}}(\hat{w})\\}_{n\geq 0}$ is an
increasing sequence and $\mathcal{E}_{T\setminus v}(w)$ is a upper bound
except when $d(v,w)=0$.

  * •

If $d(v,w)$ is odd, then $\\{\mathcal{E}_{G_{n}}(\hat{w})\\}_{n\geq 0}$ is an
decreasing sequence and $\mathcal{E}_{T\setminus v}(w)$ is a lower bound.

###### Proof.

Since by definition $G_{n}=G_{n-1}\circ B_{n}$, if we apply Theorem 3.4 it is
easy to show that $\\{\mathcal{E}_{G_{n}}(\hat{w})\\}_{n}$ is an increasing
and a decreasing sequence when $d(v,w)$ is even and odd, respectively. On the
other hand, given $w\in V(T\setminus v)$, denoting by $P=\\{u_{1}=v\sim
u_{2}\sim\dots\sim u_{d(v,w)+1}=w\\}$ the unique path in $T$ which connects
$v$ and $w$. If we remove from $G_{n}$ the copy of $u_{1}u_{2}$, by Corollary
3.6 the result follows.

∎

Corollary 3.9 has an important implication which is not obvious: Since for
each $w\in V(T\setminus v)$, $\\{\mathcal{E}_{G_{n}}(\hat{w})\\}_{n}$ is a
monotone sequence which is bounded, it is a convergent sequence. A natural
question is of course what is this limit. For the particular case where
$B_{n}=K_{2}$, and then $G_{n}=S_{n+1}\circ T$, we can calculate such limit.
This is the content of next theorem.

######  Proposition 3.10.

Let $v_{c}$ and $v_{h}$ be the center and the leaves of the start $S_{n+1}$ in
the graph $S_{n+1}\circ T$ with $n\in\mathbb{N}$. Then, let $v\in
V(S_{n+1}\circ T)\setminus\\{v_{c},v_{h}\\}$, $\mathcal{E}_{S_{n+1}\circ
T}(v)\rightarrow\mathcal{E}_{T\setminus v_{c}}(v)$ when $n\rightarrow\infty.$
Moreover, $\mathcal{E}_{S_{n+1}\circ T}(v_{h})\rightarrow 0$ and
$\mathcal{E}_{S_{n+1}\circ T}(v_{c})\rightarrow\infty$ when
$n\rightarrow\infty.$

###### Proof.

First, note that applying Theorem 3.4 we have that $\mathcal{E}_{S_{n+1}\circ
T}(v_{h})\leq\frac{1}{\sqrt{n}}=\mathcal{E}_{S_{n+1}}(v_{h})$ and
$\mathcal{E}_{S_{n+1}}(v_{c})=\sqrt{n}\leq\mathcal{E}_{S_{n+1}\circ
T}(v_{c})$. Then, $\mathcal{E}_{S_{n+1}\circ T}(v_{h})\rightarrow 0$ and
$\mathcal{E}_{S_{n+1}\circ T}(v_{c})\rightarrow\infty$ when
$n\rightarrow\infty.$ On the other hand, by Lemmas 2.2 and 2.3, we have

$2\sqrt{n}=\mathcal{E}(S_{n+1})\leq\mathcal{E}(S_{n+1}\circ
T)-\mathcal{E}(T\setminus
v_{c})\leq\mathcal{E}(S_{n+deg_{T}(v)+1})=2\sqrt{n+deg_{T}(v)}$.

If we define $V_{1}=\\{v\in V(T\setminus v_{c})\leavevmode\nobreak\
|\leavevmode\nobreak\ d(v,v_{c})\leavevmode\nobreak\ is\leavevmode\nobreak\
odd\\}$ and $V_{2}=\\{v\in V(T\setminus v_{c})\leavevmode\nobreak\
|\leavevmode\nobreak\ d(v,v_{c})\leavevmode\nobreak\ is\leavevmode\nobreak\
even\\}$, applying Lemma 2.7, Theorem 3.4 and the above inequalities, we
obtain

$0\leq\displaystyle\sum_{v\in V_{1}}\mathcal{E}_{S_{n+1}\circ
T}(v)-\displaystyle\sum_{v\in V_{1}}\mathcal{E}_{T\setminus
v_{c}}(v)\leq\sqrt{n+deg_{v}(T)}-n\mathcal{E}_{S_{n+1}\circ T}(v_{h})$,

and

$\sqrt{n}-\mathcal{E}_{S_{n+1}\circ T}(v_{c})\leq\displaystyle\sum_{v\in
V_{2}}\mathcal{E}_{S_{n+1}\circ T}(v)-\displaystyle\sum_{v\in
V_{2}}\mathcal{E}_{T\setminus v_{c}}(v)\leq 0$.

Since by Theorem 3.4,

$\mathcal{E}_{S_{n+deg_{T}(v)+1}}(v_{h})=\frac{1}{\sqrt{n+deg_{T}(v)}}\leq\mathcal{E}_{S_{n+1}\circ
T}(v_{h}),$

and

$\mathcal{E}_{S_{n+1}\circ
T}(v_{c})\leq\sqrt{n+deg_{T}(v)}=\mathcal{E}_{S_{n+deg_{T}(v)+1}}(v_{c}),$

then, the following limits hold

$\displaystyle\lim_{n\to\infty}(\displaystyle\sum_{v\in
V_{1}}(\mathcal{E}_{S_{n+1}\circ T}(v)-\mathcal{E}_{T\setminus
v_{c}}(v)))=\displaystyle\sum_{v\in V_{1}}(inf_{n}\\{\mathcal{E}_{S_{n+1}\circ
T}(v)\\}-\mathcal{E}_{T\setminus v_{c}}(v))$=0,

and

$\displaystyle\lim_{n\to\infty}(\displaystyle\sum_{v\in
V_{2}}(\mathcal{E}_{S_{n+1}\circ T}(v)-\mathcal{E}_{T\setminus
v_{c}}(v)))=\displaystyle\sum_{v\in V_{2}}(sup_{n}\\{\mathcal{E}_{S_{n+1}\circ
T}(v)\\}-\mathcal{E}_{T\setminus v_{c}}(v))$=0.

Because of Corollary 3.9, we know that $\mathcal{E}_{T\setminus v_{c}}(v)$ is
a lower bound if $v\in V_{1}$ and $\mathcal{E}_{T\setminus v_{c}}(v)$ is a
upper bound if $v\in V_{2}$, hence we conclude the proof. ∎

$\vdots$$\vdots$$S_{n+1}\circ T$$1$$2$$3$$n$$v$ Figure 2:  Graphical
representation of the behavior observed in the Theorem 3.10.

######  Remark 3.11.

By Lemma 2.8 we know that it was sufficient to show that
$\mathcal{E}_{S_{n+1}\circ T}(v_{h})\rightarrow 0$ when $n\rightarrow\infty.$
But, we proved both simultaneously using Theorem 3.4. Another result in the
same direction of Theorem 3.4 is Theorem 7.3 in [3], however, it is not
applicable in the case above.

Here is a possible explanation for the behavior observed in Corollary 3.9 and
Theorem 3.10 by using a game theoretical approach. By recruiting a new player
at each step, the player $v$ increases its value and as a consequence the
payoff of the other players is continually affected. Therefore, it seems
natural to expect that the fair payoff offered in the long run to the
remaining players is at least the payoff they would receive if $v$ was not in
the game, i.e., what they would receive if they all decide to stop cooperating
with $v$ and maintain the other alliances. In search of an adequate payoff,
through the above perspective, we assure the players at odd distance that the
most their payoff could decrease to would be the amount they would receive if
$v$ was not part of the game, thus motivating them to stay in the grand
coalition and thus continue to cooperate with $v$. On the other hand, it seems
logical to think that players who are at an even distance will remain in the
grand coalition as long as we assure them that the minimum their payoff could
increase to would be the amount they would receive if $v$ was not part of the
game. In conclusion, whether their payoff at each step decreases or increases,
gave them the same payoff as they would receive if $v$ was removed from the
game and the other alliances were maintained is strategically the best option.
Since, they end up being given the minimum payoff they "demanded" for
remaining in the grand coalition and the payoff they could receive if $v$ was
not part of the game and the other alliances were maintained. In this way, are
not tempted to stop cooperating with $v$, which implies to leave the grand
coalition. It should be noted that the above tells us that in the asymptotic
behavior the edges incident on $v$ tend to disappear (see Figure 2).

##  4 Examples

Given the above, it is valid to wonder about the possibility of finding
analogous results to Theorem 3.10 for more general graphs $B_{n}$. In this
line of analysis, in order to illustrate potential variations in the behavior
demonstrated in Theorem 3.10, we examine the graph $G_{n}$ constructed as in
Corollary 3.9, taking $T=K_{2}$, $B_{n}=S_{d_{n}+2}$ and $b_{n}$ any leave of
$B_{n}$. Here, $\\{d_{n}\\}_{n\geq 1}$ represents a succession of natural
numbers. Inspired by the behavior of the energy of vertex observed in Theorem
3.10, in this case of study, we will analyze whether it is also verified that
$\mathcal{E}_{G_{n}}(v)\rightarrow\infty$ y $\mathcal{E}_{G_{n}}(u)\rightarrow
0$ when $n\rightarrow\infty$?

To address this objective, let us examine the following. Given
$n\in\mathbb{N}$ if we define $d(n)=\max\\{d_{1},d_{2},\ldots,d_{n}\\}$,
according to the Theorem 3.4, we can state that
$\mathcal{E}_{G_{n}}(v)>\mathcal{E}_{H_{n,d(n)}}(v),$ and
$\mathcal{E}_{G_{n}}(u)<\mathcal{E}_{H_{n,d(n)}}(u),$ where $H_{n,d(n)}$ is
constructed as $G_{n}$ but with the stars $S_{d(n)+2}$. Then, using iterately
that $H_{n,d(n)}=H_{n-1,d(n)}\circ S_{d(n)+2}$ for $n\geq 1$ we can observe
that

$\phi_{H_{n,d(n)}}(x)=x^{n\cdot(d(n)-1)}(x^{2}-d(n))^{n-1}(x^{4}-(n+d(n)+1)x^{2}+d(n)).$

Therefore, the spectra of this graph is
$\\{0,\pm\sqrt{d(n)},\pm\sqrt{\frac{n+d(n)+1\pm\sqrt{(n+d(n)+1)^{2}-4d(n)}}{2}}\\}.$

Since there are only 7 distinct eigenvalues, using (2) the weights $p_{1i}$
with $i=1,2,\dots,7$ for $u$ can be calculated with a 7 by 7 system of linear
equations. Obtaining that $p_{11}=p_{12}=p_{13}=0$,

$p_{14}=p_{15}=\frac{\sqrt{(n+d(n)+1)^{2}-4d(n)}-n-d(n)+1}{4\sqrt{(n+d(n)+1)^{2}-4d(n)}},$

and

$p_{16}=p_{17}=\frac{\sqrt{(n+d(n)+1)^{2}-4d(n)}+n+d(n)-1}{4\sqrt{(n+d(n)+1)^{2}-4d(n)}}.$

Thus, by equation (1)

$\mathcal{E}_{H_{n,d(n)}}(u)=\frac{\sqrt{(n+d(n)+1)^{2}-4d(n)}(\sqrt{d(n)}+1)+n+1-d(n)+\sqrt{d(n)}(n+d(n)+1)}{\sqrt{2}\sqrt{(n+d(n)+1)^{2}-4d(n)}\sqrt{\sqrt{(n+d(n)+1)^{2}-4d(n)}+n+d(n)+1}}.$

Analyzing the previous expression we can see that, if
$\frac{d(n)}{n}\rightarrow 0$ when $n\rightarrow\infty$, then
$\mathcal{E}_{H_{n,d(n)}}(u)\rightarrow 0$ when $n\rightarrow\infty$.
Regarding the graph $G_{n}$, the above tells us that the asymptotic behavior
of $\mathcal{E}_{G_{n}}(v)$ and $\mathcal{E}_{G_{n}}(u)$ depends in part on
how it behaves in the limit $d(n)/n$. Revealing, in particular, that if the
maximum degree of the graphs coalescing with $K_{2}$ can be "controlled" such
that it grows slower than $deg_{G_{n}}(v)=n+1$, then according to Lemma 2.8
the behavior holds, i.e., $\mathcal{E}_{G_{n}}(v)\rightarrow\infty$ y
$\mathcal{E}_{G_{n}}(u)\rightarrow 0$ when $n\rightarrow\infty$. As an
example, when the sequence $\\{d_{n}\\}_{n\geq 0}$ is upper bounded. Rewriting
the above condition in terms of the monotonicity of the sequence we observe
that if it is decreasing, the result is always preserved. On the other hand,
if the sequence is increasing, it does so when it is convergent, that is, when
it tends to stabilize.

Likewise, the result obtained for the graph $H_{n,d(n)}$ give us examples that
allow us to analyze the way in which the energy of vertex behaves when
successive coalescences are made at different vertices of the graph.
Specifically, it shows that if we do not coalesce in an "organized" way by
controlling the degrees and where the coalescence will be made, we can obtain
different results.

At this point, we already have examples where the behavior is maintained;
thus, curiosity is aroused to explore other possibilities. Towards this aim,
if we apply the Theorem 3.7 to the graph $G_{n}$ we get:

$\mathcal{E}_{G_{n}}(v)\leq 1+\sum\limits_{i=1}^{n}\frac{1}{\sqrt{d_{i}+1}}.$

Therefore, if the series converges, we have that
$\mathcal{E}_{G_{n}}(v)\nrightarrow\infty$ when $n\rightarrow\infty$.
Furthermore, the following is corroborated:
$\mathcal{E}_{G_{n}}(u)\nrightarrow 0$, when $n\rightarrow\infty$, according
to Lemma 2.8. Thus, with the help of the series we have a sufficient condition
to find behaviors totally opposite to that of Theorem 3.10. It is worthwhile
to note that the conditions found so far for obtaining the same or totally
opposite behavior are mutually exclusive, as expected. This is because, the
series is lower bounded by $n/(d(n)+1)$.

##  5 Conclusion

In this study, we examined the effect on the energy per vertex of a tree after
it was coalesced with a bipartite graph. In particular, it was shown that the
energies decrease and increase if the vertices are at odd and even distance,
respectively. Thus, an alternating behavior similar to that frequently
observed in physical and chemical properties of organic compounds was
evidenced. On the other hand, when analyzing the long-term impact produced
when successive coalescences are made with respect to a fixed vertex, it was
concluded that, in some cases, the edges incident on the vertex tend to
disappear. In fact, by interpreting the energy of vertices as a payoff vector
in a cooperative game, a possible explanation of the aforementioned behaviors
can be obtained. Finally, it would be pertinent to examine the behavior of
energy per vertex in bipartite graphs, without limiting it only to the merging
vertex or to vertices that lie on paths formed by edge cuts. We hope to
develop this in the future and, at the same time, demonstrate the usefulness
of studying vertex energy using the game theory approach.

### Acknowledgment

The authors were supported by CONACYT Grant CB-2017-2018-A1-S-9764.

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