# Screening Curve Method for Economic Analysis
of Household Solar Energy Self-Consumption

Hikaru Hoshino, , Yosuke Irie, Eiko Furutani This work was supported in part
by JST, ACT-X Grant Number JPMJAX210M, Japan.The authors are with the
Department of Electrical Materials and Engineering, University of Hyogo, 2167
Shosha, Himeji, Hyogo, 671-2280, Japan (email: hoshino@eng.u-hyogo.ac.jp).

###### Abstract

The profitability of solar energy self-consumption in households, the so-
called photovoltaic (PV) self-consumption, is expected to boost the deployment
of PV and battery storage systems. This paper develops a novel method for
economic analysis of PV self-consumption using battery storage based on an
extension of the Screening Curve Method (SCM). The SCM enables quick and
intuitive estimation of the least-cost generation mix for a target load curve
and has been used for generation planning for bulk power systems. In this
paper, we generalize the framework of existing SCM to take into account the
intermittent nature of renewable energy sources and apply it to the problem of
optimal sizing of PV and battery storage systems for a household. Numerical
studies are provided to verify the estimation accuracy of the proposed SCM and
to illustrate its effectiveness in a sensitivity analysis, owing to its
ability to show intuitive plots of cost curves for researchers or policy-
makers to understand the reasons behind the optimization results.

###### Index Terms:

Solar PV, battery storage, screening curve method, feed-in tariffs.

††publicationid: pubid:

##  I Introduction

The installation capacity of solar photovoltaics (PV) in the world has
increased over the past decades owing to the use of support policies such as
feed-in tariffs [1, 2]. Recently, the price of PV has declined dramatically,
and the grid parity has been reached in several countries [3], which means
that PV can produce electricity at a price below the price of electricity
purchased from the grid. In these regions or countries, not only selling
electricity to the grid but also self-consumption of PV-produced electricity
can bring economic benefit to the consumer. Because of this, the deployment of
PV based on household solar energy self-consumption, which is called as PV
self-consumption [4], has attracted great interest. As one of the policies
based on the concept of PV self-consumption, many countries introduced net-
metering or net-billing mechanisms [5], where the surplus electricity fed into
the grid is rewarded with credits that can be applied to offset consumption.
However, there have been intense debates on net‑metering policies regarding
recovery of network costs and equity of cost allocation [6, 7], and several
setbacks and cancellations in the application of these policies have occurred
[3]. Furthermore, various new self-consumption schemes, such as collective or
distributed self‑consumption [8, 9], are proposed or implemented in several
countries, and thus appropriate regulatory frameworks based on the concept of
self-consumption remain to be discussed.

To contribute to such discussions, economic analysis for evaluating the
profitability of PV self-consumption has been extensively studied. Existing
methods can be categorized into optimization and simulation methods, depending
on whether the capacities of PV and battery storage are optimization variables
or simulated as exogenous parameters [10]. Since simulation-based methods
might underestimate the economic value of the investment [11], it is important
to correctly determine the sizes of the PV and battery storage according to
the customers’ load profiles, and thus optimization methods are often used
[12, 13, 14]. While these methods can incorporate constraints to describe
various situations, it takes hours or days to obtain optimization results.
Therefore, only a limited number of scenarios (input data and parameters) are
likely to be analyzed to provide inadequate information to infer a trend for
inputs. Besides, these methods gives the optimization results like a “black
box”, and limited intuition can be gained from the obtained results.  Since
there is a high uncertainty in the estimates of parameters such as prices and
costs, the above two issues of computational time and the interpretability of
the results make it difficult for policy-makers and regulators to learn from
these optimization studies.

The Screening Curve Method (SCM) is an intuitive and quick method for
estimating the least-cost generation mix, and it is a good tool to provide
answers to the above issues. SCM was first proposed in the 1960s [15, 16]. It
compares the summations of annualized upfront investment costs (fixed costs)
and production costs (variable costs) of multiple technologies to determine
the optimal mix when no technical constraints are considered. In recent years,
much effort has been made to develop enhanced versions of SCM to estimate the
optimal mix considering technical details. The start-up cost, thermal cycling,
and other miscellaneous short-term operations of thermal generation are
modeled in [17, 18]. Furthermore, ancillary services are considered in [19],
hydro scheduling is modeled in [20], existing capacities are considered in
[21, 22], and planned outages are considered in [23]. All of these SCMs have
been developed for estimating the optimal capacity mix in a bulk power system.

The purpose of this paper is to develop an SCM for economic analysis of
photovoltaic self-consumption to estimate optimal sizes of PV and battery
storage systems at the demand side. It has been known that the framework of
SCM can be applied to the optimal sizing of various sources to satisfy load
demands [24, 25]. However, there has been no existing SCM that can take into
account the intermittent nature of renewable energy sources. In this paper, to
consider the fact that the amount of electricity generated by PV depends on
the solar irradiation at each time, we newly adopt a time-varying scheme to
decompose the load curve, whereas time-invariant load slices have been used in
existing SCMs to decompose the original optimization problem into sub-
problems. It is shown that the proposed SCM can be used to estimate the
optimal solution of a Linear Programming (LP) problem. While an estimation
algorithm has been proposed in [26] for the upper bound of the battery size,
estimation of the optimal value of the battery size as well as the PV size has
not been achieved.

A preliminary version of the proposed SCM was presented in [27], but
periodicity in the input data was assumed on a time scale of 24 hours. The
proposed SCM that can handle arbitrary shapes and lengths of data has been
partly presented in the conference publication [28]. This paper adds the
following important contributions: the proposed algorithm is refined, and an
analytical expression for estimating the economic benefit of battery is newly
introduced; a systematic verification study is presented by comparing the
results with LP optimal solutions under various settings of parameters; and
the effectiveness of the proposed SCM is examined in terms of computational
time.

This paper is structured as follows. Section II introduces the optimal sizing
problem for PV and battery storage systems studied in this paper. The proposed
SCM is presented in Section III after a brief review of the existing SCMs.
Section IV provides a verification study for the estimation accuracy of the
proposed SCM under various settings of parameters. Section V discusses the
advantages and disadvantages of the proposed SCM, and Section VI concludes
this paper.

##  II Optimal Sizing Problem

The proposed SCM is intended to estimate the optimal solution of the LP
problem introduced in this section. The problem formulation follows from [7]
and is based on the following assumptions:

  * •

The sizes of PV and battery storage systems are continuous decision variables,
while several discrete sizes would be the options in practice.

  * •

The electricity generated by a unit capacity of PV is proportional to the
solar irradiation, and the effect of inverter clipping due to oversizing of a
solar array is neglected.

  * •

The profiles of load demand and solar irradiation are completely known a
priori to optimize the operation of the battery storage system.

  * •

The electricity retail price per unit of energy consumed is fixed, and other
practical tariff structures such as three-part tariff are not considered.

The optimal sizes of PV and battery are determined with the aim of minimizing
the annualized electricity-related costs of a household while satisfying the
load demand at each time. This problem can be viewed as a multi-scale
decision-making problem, where the decision variables include both the long-
term decisions on the installation capacities $v_{\mathrm{pv}}$ of PV (in
$\mathrm{k}\mathrm{W}$) and $v_{\mathrm{bat}}$ of battery (in
$\mathrm{k}\mathrm{W}\mathrm{h}$) and the short-term operational decisions on
the amounts of electricity purchased from and sold to the grid at each time
step $k$, denoted by $u_{\mathrm{buy},k}$ and $u_{\mathrm{sell},k}$ (for
$k=1,\,2,\,\dots,\,N_{\mathrm{t}}$), and the amounts of charging and
discharging of battery, denoted by $u_{\mathrm{bin},k}$ and
$u_{\mathrm{bout},k}$, respectively, where $N_{\mathrm{t}}$ stands for the
number of time steps. Figure 1 shows the overview of the PV and battery sizing
problem considered in this paper. The meaning of parameters and these typical
values are listed in Table I.

Figure 1: Optimal PV and battery sizing problem TABLE I: Parameters of the optimization problem Symbol | Meaning | Value
---|---|---
$C_{\mathrm{pv}}$ | Annual fixed cost of PV |  $12\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$ 111It represents annualized cost and corresponds to, e.g., the case where the PV unit price of $200\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}$ with the lifetime of 25 years, the discount rate of 0%, and the maintenance and operation cost of $4000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$.
$C_{\mathrm{bat}}$ | Annual fixed cost of battery |  $4400\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\mathrm{h}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$ 222It corresponds to, e.g., the case where the unit price of the battery is $66\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{h}$, the lifetime of 15 years, and the discount rate of 0%.
$P_{\mathrm{buy}}$ | Price for buying from the grid | $26.0\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{h}$
$P_{\mathrm{sell}}$ | Price for selling to the grid | $6.0\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{h}$
$E_{\mathrm{chg}}$ | Efficiency of charging | $0.9$
$E_{\mathrm{dis}}$ | Efficiency of discharging | $0.9$
$E_{\mathrm{pv}}$ | Performance ratio of PV system | $0.78$
$G_{\mathrm{stc}}$ | STC reference irradiance intensity | $1000\text{\,}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
$S_{k}$ |  Solar irradiation at time $k$ | Given as input data
$D_{k}$ |  Electricity consumption at time $k$ | Given as input data
$M_{\mathrm{pv}}$ | Maximum capacity of PV | $10\text{\,}\mathrm{k}\mathrm{W}$
$F_{\mathrm{anu}}$ | Scaling factor for annualization | $365/\text{\\# of days analyzed}$

The objective function $L$ to be minimized is given as follows:

$\displaystyle L=$ $\displaystyle
F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}(P_{\mathrm{buy}}u_{\mathrm{buy},k}-P_{\mathrm{sell}}u_{\mathrm{sell},k})$
$\displaystyle+C_{\mathrm{pv}}v_{\mathrm{pv}}+C_{\mathrm{bat}}v_{\mathrm{bat}}$
(1)

where the parameters $P_{\mathrm{buy}}$ and $P_{\mathrm{sell}}$ stand for the
prices of electricity purchased from and sold to the grid, respectively, and
the electricity trading costs are annualized using a scaling factor
$F_{\mathrm{anu}}$. The parameters $C_{\mathrm{pv}}$ and $C_{\mathrm{bat}}$
stand for the fixed costs (mainly installation costs) of PV and battery per
unit installation amount per year, respectively.

The operation of the battery storage system is subject to the following state
equation:

$\displaystyle
x_{\mathrm{soc},k+1}=x_{\mathrm{soc},k}+E_{\mathrm{chg}}u_{\mathrm{bin},k}-\dfrac{1}{E_{\mathrm{dis}}}u_{\mathrm{bout},k},\quad\forall
k$ (2)

where $x_{\mathrm{soc},k}$ stands for the state of charge (SOC) of the battery
storage at the time step $k$, and the parameters $E_{\mathrm{chg}}$ and
$E_{\mathrm{dis}}$ for the efficiencies of charging and discharging of
battery, respectively. In addition, the following constraints need to be
addressed:

$\displaystyle x_{\mathrm{soc},k}\leq v_{\mathrm{bat}},\quad\forall k,$ (3)
$\displaystyle x_{\mathrm{soc},N_{\mathrm{t}}+1}=x_{\mathrm{soc},1}$ (4)

where (3) imposes an upper bound of SOC being subject to the long-term
decision $v_{\mathrm{bat}}$, and (4) is the boundary condition. Besides, the
electricity demand should be equal to supply at each time step, and the
following equation holds:

$\displaystyle
u_{\mathrm{buy},k}-u_{\mathrm{sell},k}-u_{\mathrm{bin},k}+u_{\mathrm{bout},k}=D_{k}-\dfrac{S_{k}E_{\mathrm{pv}}v_{\mathrm{pv}}}{\color[rgb]{1,0,0}G_{\mathrm{stc}}},\,\forall
k$ (5)

where $D_{k}$ stands for the electricity demand at the time step $k$ in
$\mathrm{k}\mathrm{W}\mathrm{h}$, and $S_{k}$ for the solar radiation in
$\mathrm{k}\mathrm{W}\mathrm{h}\mathrm{/}\mathrm{m}^{2}$. The parameter
$G_{\mathrm{stc}}$ stands for the reference irradiance intensity of the
Standard Test Conditions (STC) [29], which is equal to
$1000\text{\,}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$, and $E_{\mathrm{pv}}$
stands for the performance ratio of PV system [30], which depends on the
efficiencies of the PV panel and the power conditioning system and so on. An
upper bound is imposed on the size of PV:

$\displaystyle v_{\mathrm{pv}}\leq M_{\mathrm{pv}}.$ (6)

This condition is needed to prevent the LP problem becomes unbounded when the
PV price $C_{\mathrm{pv}}$ is lower than a certain value. Finally, the
following condition forces all the decision variables to be non-negative:

$\displaystyle
v_{\mathrm{pv}},\,v_{\mathrm{bat}},\,u_{\mathrm{buy},k},\,u_{\mathrm{sell},k},\,u_{\mathrm{bin},k},\,u_{\mathrm{bout},k},\,x_{\mathrm{soc},k}\geq
0,\,\forall k.$ (7)

Note that the dimension of the above LP problem increases with the increase in
the length of data analyzed. For an analysis over 1 year with a temporal
resolution of 1 hour, the resultant LP problem becomes $2+8760\times 5=43,802$
dimensional optimization problem, and may take hours or days to obtain the
result by a general-purpose optimization solver.

##  III Screening Curve Method

In this section, we introduce the framework of SCM with a brief review of
existing SCMs in Section III-A. The proposed SCM for economic analysis of PV
self-consumption is presented in Section III-B.

###  III-A Review of Existing SCMs

The classical SCM [15, 16] requires two sets of input data: costs and a load
curve for a target year. Cost data includes the annualized upfront investment
cost $C_{\mathrm{fix}}$ per unit capacity (in $\mathrm{k}\mathrm{W}$ or
$\mathrm{M}\mathrm{W}$) and the variable production costs $C_{\mathrm{var}}$
per unit production (in $\mathrm{k}\mathrm{W}\mathrm{h}$ or
$\mathrm{M}\mathrm{W}\mathrm{h}$). The total annualized cost of a unit
capacity of generation technology is represented as follows:

$\displaystyle c=C_{\mathrm{fix}}+C_{\mathrm{var}}T$ (8)

where $T$ stands for the firing hours of the considered thermal generation
technology. The cost representation in (8) is used to draw the so-called
_screening curve_ for each generation technology as shown in the upper part of
Fig. 2, where the horizontal axis is taken as the capacity factor $f$ defined
as the ratio of the firing hours $T$ to the total hours in a year
($f=T/$8760\text{\,}\mathrm{h}$$). Figure 2 illustrates examples of the
screening curves for three candidate technologies: base-load (such as coal and
nuclear), combined-cycle gas turbine (CCGT) and simple-cycle combustion
turbine (GT). Then, the intersections of these screening curves separate the
regions of $f$ in which the different generation technologies become optimal.
The least-cost generation mix for a given load curve can be obtained by
combining them with the load-duration curve as shown in the lower part of Fig.
2, where the load-duration curve is obtained by sorting a chronological load
curve in descending order of the load level. Since the load-duration curve
maps a load level on the vertical axis to a capacity factor on the horizontal
axis, and the screening curves map the capacity factor to the annual costs of
generation alternatives, the optimal generation capacity mix can be read from
the vertical axis of the lower graph as shown in Fig. 2. This analysis gives
the optimal solution to the least-cost problem with no technical constraints
[24].

Figure 2: The classical SCM to determine the optimal thermal generation mix
when no technical constraints are considered.

In recent years, enhanced versions of SCM have been developed to estimate the
optimal generation mix considering more detailed situations. For this purpose,
it was proposed in [17] to relate the load level directly to the generation
cost. With this scheme, a chronological load curve is discretized into
multiple load slices as shown in Fig. 3. For example, if the width of a load
slice is set to be $0.1\text{\,}\mathrm{M}\mathrm{W}$, then a load curve with
a peak of $80\text{\,}\mathrm{M}\mathrm{W}$ will be divided into $800$ load
slices. Then, the original least-cost problem will be decomposed into sub-
problems corresponding to these slices, and a heuristic optimization is
performed to determine which generation technology is used in each slice. This
is done by calculating the annualized total cost to balance the demand in each
load slice with a production by each generation technology to draw screening
curves (cost curves) as shown in Fig. 3. The estimated optimal generation mix
can be derived by simply choosing the least-cost technology for each slice.
With this approach, in addition to the fixed and variable costs in (8), a
simple representation of the start-up costs can be obtained by counting the
number of segments in each load slice and summing up the cost for each start
up. In the example shown in Fig. 3, the costs for three re-starts will be
added. Furthermore, the start-up can be separated into hot starts and cold
starts depending on the shut-down duration before the restart, which is
evaluated from the distance between adjacent segments [17, 18]. With the above
framework of enhanced SCM, other technical details such as thermal cycling
[18], ancillary services [19], hydro scheduling [20], existing capacities [21,
22], and planned outages [23] have been taken into account.

(a) Definition of load slice

(b) Example of screening curves (cost curves)

Figure 3: The framework of enhanced SCMs to estimate the optimal generation
mix when technical constraints are considered.

###  III-B Proposed SCM for PV and battery

Based on the framework of enhanced SCM introduced above, here we develop the
proposed SCM for economic analysis of PV self-consumption.

####  III-B1 Overview

The existing SCMs have been developed mainly for thermal generation
technologies and are not able to take into account intermittent nature of
renewable energy sources. To consider the fact that the amount of electricity
generated by PV depends on the solar irradiation at each time, we newly adopt
a time-varying scheme for load slicing. Specifically, as shown in Fig. 4, the
load curve is discretized into multiple load slices using the generation
profile of a unit capacity of PV, which corresponds to the width of the load
slice. As shown in the lower part of the figure, the PV generation shaded in
red will be directly consumed and the area shaded in blue becomes surplus
electricity, which will be sold to the grid or charged to the battery to
increase self-consumption. Note that the load demand during nighttime is not
directly considered in this method, since there is no power generation by PV
at night.

With this modified approach of load slicing, the overall procedure of the
analysis is followed from the framework of enhanced SCM and is given as
follows:

  1. (i)

Discretize the chronological load curve into time-varying load slices by using
the generation profile of a unit capacity of PV, denoted by $\Delta P$,
corresponding to the width of a load slice.

  2. (ii)

Draw screening curves by calculating the annualized cost to satisfy the demand
in each slice by each of the following three technology options: a) buying
electricity from the grid, b) installing PV without battery, and c) installing
PV with battery.

  3. (iii)

Estimate the optimal PV size by counting the number of slices where the option
b) or c) becomes the least-cost technology.

  4. (iv)

Estimate the optimal battery size by summing up the required amount of battery
calculated for each slice where the option c) becomes the least-cost
technology.

The detailed calculations of the screening curves and required amount of
battery are presented in the rest of this section.  Parameters and variables
used for calculations are summarized in Table II.

Figure 4: Time-varying scheme of load slicing in the proposed SCM TABLE II: Parameters and variables used for SCM Symbol | Meaning
---|---
$\Delta P$ | Unit capacity of PV corresponding to the width of a slice
$N_{\mathrm{t}}$ | Total number of time steps of analyzed data
$N_{\mathrm{d}}$ | Total number of days of analyzed data
$i$ | Index of load slices
$k$ | Index of time steps in chronological order
$j$ | Index of days in ascending order w.r.t. surplus electricity
$q_{\mathrm{load},i,k}$ |  Electricity demand at the time step $k$ in the slice $i$
$q_{\mathrm{sur},i,k}$ |  Surplus electricity at the time step $k$ in the slice $i$
$q_{\mathrm{chg},i,k}$ |  Charged electricity at the time step $k$ in the slice $i$
$q_{\mathrm{bat},i}$ |  Amount of battery required for the slice $i$
$Q_{\mathrm{bat},i,j}$ |  Amount of battery required for the $j$-th day in the slice $i$
$\Delta Q_{\mathrm{bat},i,j}$ |  Incremental battery amount for the $j$-th day in the slice $i$

####  III-B2 Calculation of Screening Curves

For the technology option a), the annualized cost $c_{\mathrm{grid},i}$ of
purchasing electricity from the grid for the slice level $i$ can be calculated
as follows:

$\displaystyle
c_{\mathrm{grid},i}=F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}P_{\mathrm{buy}}q_{\mathrm{load},i,k}$
(9)

where $q_{\mathrm{load},i,k}$ stands for the amount of load demand of the
slice $i$ at the time step $k$. The total amount of electricity
$\sum_{k}q_{\mathrm{load},i,k}$ corresponds to the area shaded in red in Fig.
4. The parameters $P_{\mathrm{buy}}$ and $F_{\mathrm{anu}}$ are the same with
those defined in Section II and stand for the electricity retail price and the
annualization factor, respectively.

For the technology option b), the annualized cost $c_{\mathrm{pv},i}$ of
installing PV without battey for the slice $i$ is given as follows:

$\displaystyle c_{\mathrm{pv},i}=C_{\mathrm{pv}}\Delta
P-F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}P_{\mathrm{sell}}q_{\mathrm{sur},i,k}$
(10)

where $q_{\mathrm{sur},i,k}$ stands for the amount of surplus electricity for
the slice $i$ at the time step $k$. The first term of the right-hand side of
(10) represents the investment cost for PV, and the second term the revenue by
selling surplus electricity to the grid. The total amount of electricity
$\sum_{k}q_{\mathrm{sur},i,k}$ corresponds to the area shaded in blue in Fig.
4. The parameters $C_{\mathrm{pv}}$ and $P_{\mathrm{sell}}$ stand for the
annualized fixed cost of PV and the price of selling electricity to the grid,
respectively.

For the technology option c), the cost $c_{\mathrm{pv/bat},i}$ of installing
PV and using battery for increasing self-consumption is given by

$\displaystyle c_{\mathrm{pv/bat},i}=$ $\displaystyle\,C_{\mathrm{pv}}\Delta
P+C_{\mathrm{bat}}q_{\mathrm{bat},i}$ $\displaystyle-
F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}P_{\mathrm{sell}}\left(q_{\mathrm{sur},i,k}-q_{\mathrm{chg},i,k}\right)$
$\displaystyle-
F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}P_{\mathrm{buy}}E_{\mathrm{dis}}E_{\mathrm{chg}}q_{\mathrm{chg},i,k}$
(11)

where $q_{\mathrm{bat,}i}$ stands for the required amount of battery for the
slice $i$, and $q_{\mathrm{chg},i,k}$ stands for the amount of charged
electricity for the slice $i$ at the time $k$. The procedure of estimation of
these values will be presented later. The second term of the right-hand side
of (III-B2) represents the investment cost for battery. The third term
represents the revenue by selling surplus electricity. The fourth term
represents the economic benefit due to the increase in self-consumption by
charging surplus electricity and using it in e.g. nighttime. While the profile
of charging is modeled in the proposed method, the discharging will not be
explicitly considered.

####  III-B3 Estimation of Battery Capacity and Charging Profile

To estimate the values of $q_{\mathrm{bat},i}$ and $q_{\mathrm{chg},i,k}$, we
will consider the economic benefit of battery in a sequential manner. To this
end, let us define an index $j$ that indicates the ascending order of the days
with respect to the total amount of surplus PV generation in each day. That
is, as shown in Fig. 5(a), the day with the index $j=1$ has the smallest
amount of surplus electricity, and the day with $j=N_{\mathrm{d}}$ has the
largest amount, where $N_{\mathrm{d}}$ stands for the number of days analyzed.
This order may vary depending on the slice $i$, but we do not explicitly show
it in the notation. Then, the amount of battery required to store all the
surplus electricity of the $j$-th day can be given as

$\displaystyle
Q_{\mathrm{bat},i,j}:=E_{\mathrm{chg}}\sum_{k\in\mathcal{K}_{j}}q_{\mathrm{sur},i,k}$
(12)

where the symbol $\mathcal{K}_{j}$ stands for the set of indices of the time
step $k$ belonging to the $j$-th day. Note that this assumes that there is no
discharging of battery during day time, otherwise the required amount of
battery could be smaller.

(a) Definition of the index $j$

(b) Charged electricity for the case of $j=1$

(c) Charged electricity for the cases of $j=2$ and $3$

Figure 5: Examples of charging profile of electricity for each value of
$j=1,\,2$, and $3$.

Now the economic benefit of the battery can be sequentially estimated for each
slice. First, consider to install a battery storage system of the amount of
$Q_{\mathrm{bat},i,1}$ for the slice $\\#i$. Then, the surplus electricity can
be charged up to this amount every day (see Fig. 5(b)), and the economic
benefit $B_{1}$ associated to this use of battery can be written as

$\displaystyle B_{1}=$ $\displaystyle
F_{\mathrm{anu}}N_{\mathrm{d}}P_{\mathrm{buy}}E_{\mathrm{dis}}Q_{\mathrm{bat},i,1}$
$\displaystyle-
C_{\mathrm{bat}}Q_{\mathrm{bat},i,1}-F_{\mathrm{anu}}N_{\mathrm{d}}P_{\mathrm{sell}}Q_{\mathrm{bat},i,1}/E_{\mathrm{chg}}$
(13)

where the first term represents the amount of payment of electricity that can
be avoided, the second term the installation cost of the battery, and the
third term the lost revenue that could have been earned by selling this
electricity. Next, for $2\leq j\leq N_{\mathrm{d}}$, if the amount of battery
is increased to $Q_{\mathrm{bat},i,j}$, the marginal economic benefit $B_{j}$
due to the incremental amount of battery defined by $\Delta
Q_{\mathrm{bat},i,j}:=Q_{\mathrm{bat},i,j}-Q_{\mathrm{bat},i,j-1}$ can be
given by

$\displaystyle B_{j}=$ $\displaystyle
F_{\mathrm{anu}}(N_{\mathrm{d}}-j+1)P_{\mathrm{buy}}E_{\mathrm{dis}}\Delta
Q_{\mathrm{bat},i,j}$ $\displaystyle-C_{\mathrm{bat}}\Delta
Q_{\mathrm{bat},i,j}$ $\displaystyle-
F_{\mathrm{anu}}(N_{\mathrm{d}}-j+1)P_{\mathrm{sell}}\Delta
Q_{\mathrm{bat},i,j}/E_{\mathrm{chg}}.$ (14)

Figure 5(c) shows examples for the cases of $j=2$ and $j=3$. Note that the
economic benefit of the incremental battery capacity $\Delta
Q_{\mathrm{bat},i,j}$ can occur only for $N_{\mathrm{d}}-j+1$ days because the
surplus electricity over the first to $(j-1)$-th days is already accounted in
$B_{1},\dots,\,B_{j-1}$. By rewriting Eq. (14), we have

$\displaystyle B_{j}=$
$\displaystyle\\{F_{\mathrm{anu}}(N_{\mathrm{d}}-j+1)(P_{\mathrm{buy}}E_{\mathrm{dis}}-P_{\mathrm{sell}}/E_{\mathrm{chg}})$
$\displaystyle-C_{\mathrm{bat}}\\}\Delta Q_{\mathrm{bat},i,j}.$ (15)

Thus, the value of $B_{j}/\Delta Q_{\mathrm{bat}i,j}$ decreases as the
increase in $j$ when
$P_{\mathrm{buy}}>P_{\mathrm{sell}}/(E_{\mathrm{chg}}E_{\mathrm{dis}})$
holds333Note that this condition holds for the setting in Table I and is
necessary for a self-consumption policy to work. Otherwise $B_{j}<0$ for all
$j=1,\dots,N_{\mathrm{d}}$.. Since the economic benefit of the incremental
amount of battery $\Delta Q_{\mathrm{bat},i,j}$ occurs when $B_{j}>0$, the
installation capacity of battery $q_{\mathrm{bat},i}$ in (III-B2) can be given
by

$\displaystyle
q_{\mathrm{bat},i}:=Q_{\mathrm{bat},i,J}=E_{\mathrm{chg}}\sum_{k\in\mathcal{K}_{J}}q_{\mathrm{sur},i,k}$
(16)

with

$\displaystyle J:=\max\\{j\in\mathbb{Z}_{[0,N_{\mathrm{d}}]}~{}|~{}B_{j}\geq
0\\}$ (17)

where $\mathbb{Z}_{[0,N_{\mathrm{d}}]}$ stands for the set of integers ranging
from $0$ to $N_{\mathrm{d}}$, and for $j=0$, we define $B_{0}:=0$ and
$Q_{\mathrm{bat},i,0}:=0$. From the observation that the value of
$B_{j}/\Delta Q_{\mathrm{bat},i,j}$ decreases with $j$, the value of $J$ can
be analytically described as

$\displaystyle J=\max\left(0,\,\left\lfloor
N_{\mathrm{d}}+1-\dfrac{C_{\mathrm{bat}}E_{\mathrm{chg}}}{F_{\mathrm{anu}}(P_{\mathrm{buy}}E_{\mathrm{dis}}E_{\mathrm{chg}}-P_{\mathrm{sell}})}\right\rfloor\right)$
(18)

where $\lfloor\cdot\rfloor$ stands for the floor function.

Given $J$ and $q_{\mathrm{bat},i}$ above, we can arbitrarily determine the
charging profile $\\{q_{\mathrm{chg},i,k}\\}_{k=1}^{N_{\mathrm{t}}}$ used in
(III-B2) as long as the total charged amount of electricity at each day does
not exceed the upper limit given by $q_{\mathrm{bat},i}/E_{\mathrm{chg}}$.
Furthermore, by defining the total amount of electricity $Q_{\mathrm{chg},i}$
charged to the battery as

$\displaystyle
Q_{\mathrm{chg},i}=\sum_{j=1}^{J}\sum_{k\in\mathcal{K}_{j}}q_{\mathrm{sur},i,k}+(N_{\mathrm{d}}-J)\sum_{k\in\mathcal{K}_{J}}q_{\mathrm{sur},i,k},$
(19)

the cost (III-B2) can be rewritten as

$\displaystyle c_{\mathrm{pv/bat},i}=$ $\displaystyle\,C_{\mathrm{pv}}\Delta
P+C_{\mathrm{bat}}E_{\mathrm{chg}}\sum_{k\in\mathcal{K}_{J}}q_{\mathrm{sur},i,k}$
$\displaystyle-
F_{\mathrm{anu}}\sum_{k=1}^{N_{\mathrm{t}}}P_{\mathrm{sell}}q_{\mathrm{sur},i,k}$
$\displaystyle-
F_{\mathrm{anu}}(P_{\mathrm{buy}}E_{\mathrm{dis}}E_{\mathrm{chg}}-P_{\mathrm{sell}})Q_{\mathrm{chg},i}.$
(20)

Thus, the screening curve of $c_{\mathrm{bv/bat},i}$ can be directly
calculated from the profile
$\\{q_{\mathrm{sur},i,k}\\}_{k=1}^{N_{\mathrm{t}}}$ by using (18), (19), and
(20) without calculating the values of $B_{j}$.

##  IV Verification Study

This section examines the estimation accuracy of the proposed SCM by comparing
the results with the LP problem presented in Section II. The python source
code for the implementation of the proposed SCM is available upon request.

###  IV-A Base-Case Analysis

The base-case analysis is performed with the setting of parameters shown in
Table I. A synthetic hourly load profile was generated from real measurement
data of a household in Japan [31] by concatenating data for February (Winter),
June (Rainy season), and August (Summer) to construct a three-month data
including different seasons in a year. A generation profile for a unit
capacity of PV was generated by using solar irradiation data of METPV20 [32]
for corresponding days and the location of the load profile. The daily
profiles of these input data are shown in Fig. 6. It can be seen that the
solar irradiation is generally low in winter, and there are many days with low
solar irradiation in the rainy season. On the other hand, demand is higher
during winter than other seasons. Although we use demand and solar irradiation
data from Japan, it contains wide spectrum of situation of weather and
seasons, and we aim to verify that the proposed method works also for other
regions or countries.

(a) PV data

(b) Demand data

Figure 6: Daily profiles of the input PV and demand data for February (1–28th
days), June (29–58th days), and August (59–89th days).

(a) Cost curves

(b) Cumulative amount of battery

Figure 7: Base-case result of the proposed SCM

The screening curves (cost curves) obtained by the proposed SCM is shown in
Fig. 7(a). The _green_ and _blue_ lines in the figure show the cost curves for
using PV with and without battery storage calculated by (20) and (10),
respectively, and the _purple_ line the cost curve for buying electricity from
the grid calculated by (9). The cost curve of “PV+battery” (green line) takes
least value from $l=0$ to $5.19\text{\,}\mathrm{k}\mathrm{W}$ and has an
intersection with the cost curve of “Grid” (purple line) at the point shown by
the cross symbol ($\times$) in the figure. Thus, the installation capacity of
PV can be estimated as $5.19\text{\,}\mathrm{k}\mathrm{W}$.  The amount of
battery is calculated for each slice by using (16). Figure 7(b) shows the
cumulative amount of battery $\sum_{i}q_{\mathrm{bat},i}$ until corresponding
slice level. The total amount of battery can be derived by considering the
level of $l=$5.19\text{\,}\mathrm{k}\mathrm{W}$$, and is given as
$4.01\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$. The optimal PV and battery
capacities obtained by solving the LP problem are
$5.15\text{\,}\mathrm{k}\mathrm{W}$ and
$4.06\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$, respectively, and the proposed
SCM approximates these optimal values appropriately.

In Fig. 7(a), the screening curves of PV and PV+battery start at the same
point (same annualized cost at $0\text{\,}\mathrm{k}\mathrm{W}$ slice level).
This is because at $0\text{\,}\mathrm{k}\mathrm{W}$ slice level, the surplus
electricity $q_{\mathrm{sur},0,k}$ is zero for all $k=1,\dots N_{\mathrm{t}}$
(see Fig. 4), and thus only the first terms of Eqs. (10) and (20) become
nonzero. When the screening curves are plotted in Fig. 7(a), the annualized
costs are scaled by dividing by $\Delta P$, and the above two screening curves
start with
$C_{\mathrm{pv}}=$12\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$$.
Further, the option of PV+battery is always cheaper than the option of just
using PV, since the amount of battery is determined such that the economic
benefit $B_{j}$ becomes non-negative. The difference between these two
screening curves represents the amount of economic benefit owing to the
battery.

(a) Electricity sold to the grid

(b) Electricity charged to the battery storage

Figure 8: Amounts of electricity soled and charged for each day.

Figure 8 shows the daily profiles of (a) electricity sold to the grid and (b)
electricity charged to the battery. The profiles obtained by the SCM (shown by
_blue_) capture the optimal profiles by the LP problem (shown by _orange_). It
can be seen that, in many days, the amount of charged electricity takes its
maximum value
($$4.06\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$/E_{\mathrm{chg}}=$4.51\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$$)
meaning that the battery is fully charged. Although the SCM does not consider
the profile of discharging, the ratio of the amount of sold electricity and
the amount of charged electricity is properly reproduced throughout the
analysis period, including the days when the amount of charge is low as
typically seen in winter or rainy season.

###  IV-B Estimation Accuracy under Various Settings

Next, the estimation accuracy of the proposed SCM is examined under various
settings of parameters. To reduce the computational time for obtaining the
optimal solutions of the LP problem, the load demand and PV generation
profiles were shortened by taking the first ten days of each month in the
three-month data used above to construct a one-month long data. Among the
parameters shown in Table I, the four parameters $C_{\mathrm{pv}}$,
$C_{\mathrm{bat}}$, $P_{\mathrm{buy}}$, and $P_{\mathrm{sell}}$, which are
directly relevant to the regulatory discussion of the PV self-consumption, are
changed from the setting of the base-case analysis.

Figure 9 shows the comparison between the estimated and optimal solutions
under varying prices of PV and battery storage systems. Figure 9(a) shows the
results of changing the setting of the parameter $C_{\mathrm{pv}}$. The _blue
circles_ show the PV capacities estimated by the proposed SCM, and they are
close to the results of the LP problem shown by the dotted line. Similarly,
the estimated battery capacities shown by the _orange triangles_ are close to
the optimal capacities over the entire range of $C_{\mathrm{pv}}$.

(a) Changes in $C_{\mathrm{pv}}$

(b) Changes in $C_{\mathrm{bat}}$

Figure 9: Results with various setting of PV and battery prices.

Next, Fig. 9(b) shows the result of changing the setting of the battery fixed
cost $C_{\mathrm{bat}}$. While the estimated capacity of PV is close to the
optimal value over the entire range of $C_{\mathrm{bat}}$, the installation
capacity of battery is overestimated for the region where $C_{\mathrm{bat}}$
is lower than
$4000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{h}\mathrm{/}\mathrm{y}\mathrm{r}$.
A plausible reason for this is that, while the proposed SCM does not take into
account the profile of discharging, the battery is not sufficiently discharged
each day, and there is no room for charging as much as calculated by the SCM.
Thus, the solution derived by the SCM becomes suboptimal for the household.
Nevertheless, the proposed SCM capture the overall trend of the changes in the
optimal battery size.

(a) Change in $P_{\mathrm{buy}}$

(b) Change in $P_{\mathrm{sell}}$

Figure 10: Results with various setting of electricity prices.

Figure 10 shows the comparison of the estimated and optimal capacities under
various setting of electricity prices. Figure 10(a) shows the results where
the retail price $P_{\mathrm{buy}}$ is changed. It can be seen that the
proposed SCM captures the overall trend of the changes in the optimal values
of PV and battery, while the estimation accuracy of the battery capacity
deteriorates in the region where the price $P_{\mathrm{buy}}$ is higher than
$27\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{h}$.
As in the case of Fig. 9(b), it starts to overestimate the economic benefit of
the battery when the battery capacity becomes greater than
$15\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$. Finally, Fig. 10(b) shows the
results where the price $P_{\mathrm{sell}}$ is changed. The estimated and
optimal values are closed to each other for both PV and battery capacities.

To summarize, the proposed SCM can estimate the optimal installation
capacities of PV and battery for a wide range of parameter settings of
$C_{\mathrm{pv}}$, $C_{\mathrm{bat}}$, $P_{\mathrm{buy}}$, and
$P_{\mathrm{sell}}$, except for a region where the battery size becomes larger
than a certain level. While it was assumed that there is no discharging of
battery during day time, it did not largely affect the estimation accuracy.
However, it should be noted that this assumption may become a limitation when
the prices of buying and selling electricity do not remain the same during the
day and at night.

##  V Discussion

In this section, we sort out advantages and disadvantages of the proposed SCM
and discuss its effectiveness.

###  V-A Economic Analysis Using Screening Curves

Here we illustrate how the cost curves obtained by the proposed SCM can be
used for understanding and predicting the changes in the optimal capacities of
PV and battery storage systems. Although similar examples have been presented
in [27], we provide recalculated results using the SCM proposed in Section III
to show its effectiveness in a self-consistent way.

Figure 11 shows the screening curves calculated under the three settings of
the price of PV:
$C_{\mathrm{pv}}=$18\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$$,
$15\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$,
and
$12\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$.
Since the cost curve of $c_{\mathrm{grid},i}$ is independent of
$C_{\mathrm{pv}}$, it is common for all the above settings and shown by the
_black_ line. The cost curves of $c_{\mathrm{pv},i}$ and
$c_{\mathrm{pv/bat},i}$ are shown by _broken_ and _solid_ lines, respectively,
and they move down as $C_{\mathrm{pv}}$ increases without changing their
shapes. This is because the parameter $C_{\mathrm{pv}}$ only affects the first
term of (10) or (20), and they do not depend on the slice level $i$. It can be
seen that the intersection of the cost curves of $c_{\mathrm{pv},i}$ and
$c_{\mathrm{grid},i}$ moves to the right as $C_{\mathrm{pv}}$ decreases,
implying the increase in the installation capacities of PV and battery.
Especially, when $C_{\mathrm{pv}}$ takes a value near
$12\,000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{y}\mathrm{r}$,
the slope of the cost curve $c_{\mathrm{grid},i}$ at the intersection becomes
small, and thus the optimal capacities of PV and battery drastically changes
as also can be seen in Fig. 9(a).

Next, Fig. 12 shows the cost curves calculated under the settings of
$C_{\mathrm{bat}}=$5500\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\mathrm{h}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$$,
$5000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\mathrm{h}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$,
and
$4500\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\mathrm{h}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$.
The cost curves of $c_{\mathrm{grid},i}$ and $c_{\mathrm{pv},i}$ are invariant
and shown by _solid_ and _broken black_ lines, respectively. The cost curve of
$c_{\mathrm{pv/bat},i}$ goes down as $C_{\mathrm{bat}}$ decreases. The
difference between $c_{\mathrm{pv},i}$ and $c_{\mathrm{pv/bat},i}$ describes
the economic benefit of the self-consumption owing to battery storage, and it
can be explained from this figure that the capacity of the battery storage
begins to increase from zero at a point around
$5000\text{\,}\mathrm{y}\mathrm{e}\mathrm{n}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{W}\mathrm{h}\cdot\mathrm{y}\mathrm{r}\mathrm{)}$
as can be seen in Fig. 9(b).

Figure 11: Cost curves with different values of $C_{\mathrm{pv}}$ Figure 12:
Cost curves with different values of $C_{\mathrm{bat}}$

As shown in these examples, the effect of changes in a parameter on each cost
curve can be easily predicted from their definitions in (9), (10), and (20),
and similar implications can also be obtained when other parameters change. By
considering the behavior of each cost curve separately, the changes in the
resultant optimal capacities in PV and battery storage systems can be
understood in an intuitive and consistent way.

###  V-B Computational Time

TABLE III: Computational times Data length | Comp. Time (SCM) | Comp. Time (LP)
---|---|---
1 months | $1.29\text{\,}\mathrm{s}$ | $57.73\text{\,}\mathrm{s}$
2 months | $2.46\text{\,}\mathrm{s}$ | $276.15\text{\,}\mathrm{s}$
3 months | $2.93\text{\,}\mathrm{s}$ | $784.18\text{\,}\mathrm{s}$
4 months | $3.49\text{\,}\mathrm{s}$ | $1439.31\text{\,}\mathrm{s}$
5 months | $3.96\text{\,}\mathrm{s}$ | $3289.80\text{\,}\mathrm{s}$
6 months | $4.56\text{\,}\mathrm{s}$ | $6089.13\text{\,}\mathrm{s}$
Figure 13: Computational times with SCM and LP

Since the computations of cost curves in the proposed SCM only requires
numerical operations on arrays, it is faster than solving a least-cost problem
using a general-purpose optimization solver. Table III and Fig. 13 show the
comparison of the computational times between the proposed SCM and the LP
problem. These results have been obtained in simulations with a Core i9-10850K
based desktop computer. The proposed SCM is much faster than solving the LP
problem. The LP problem has been solved by using an interior-point method
provided by linprog in the python package SciPy, and the computational time
could be reduced by using much more powerful commercial solvers. However, when
the length of data is increased, the rapid increase in the computational time
cannot be avoided due to the increase in the dimension of the problem. The
proposed SCM enables quick estimation of the optimization results without any
special commercial solver and its computational time increases only linearly
to the length of the data. The short computational time is useful to perform
sensitivity analysis or other studies that require multiple simulations with
different settings of parameters or input data.

###  V-C Advantages and Disadvantages

The proposed SCM estimates the economic benefit of the photovoltaic self-
consumption using battery storage. It does not consider other types of battery
usage, such as energy arbitrage in response to variations in electricity
prices [33, 34] or provision of ancillary services for voltage regulation or
congestion relief of distribution grid [35]. There are also limitations to the
proposed SCM. For example, it cannot describe discrete installation
capacities, existing capacities, or a common three-part tariff, while some of
these issues including ancillary services and existing capacities may be
resolved by further modifications as performed with the SCM for thermal
generation [18, 21].

One advantage of the proposed SCM is that it can provide intuitive plots of
cost curves to understand the reasons behind the results of optimal
installation capacities. In addition, the computational time of the SCM is
short (few seconds) compared to optimization models (usually more than hours).
These facts are useful to explore cost sensitivity or other studies that
require multiple simulations. Thus, the proposed SCM is more useful for
researchers or policy makers studying incentive policy for supporting PV or
tariff regulations, who want to draw policy implications from an intuitive and
quick trend analysis, rather than residential or commercial consumers who want
to derive an optimal energy systems design with detailed specifications taken
into account. Nevertheless, the SCM can be used to derive an initial estimate
prior to the detailed design or to screen out cases where there is no economic
benefit of installing PV or battery storage systems.

##  VI Conclusion

This paper developed a method of economic analysis of PV self-consumption with
and without battery storage. The development is based on a generalization of
the framework of Screening Curve Method (SCM), which has been utilized for
generation expansion problem of a bulk power system. The proposed SCM enables
quick and intuitive estimation of the installation capacities of PV and
battery storage systems. The estimation accuracy has been validated through
numerical simulations, and high accuracy has been achieved when the battery
installation capacity is not large. The method is useful to explore cost
sensitivity or other studies that require multiple simulations.

Future directions of this study include further improvements of the method to
consider, e.g., the usage of battery storage for ancillary services or the
effect of existing capacities. Another one is to perform practical economic
analyses by using the proposed SCM.

## Acknowledgments

The authors wish to thank K. Okada and K. Furusawa (Central Research Institute
of Electric Power Industry; CRIEPI) for feedback during the course of this
work.

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# Eigenvalues of the normalized complex Laplacian on finite electrical
networks

Anna Muranova  and  Robert Schippa  Anna Muranova: 5050 Institut für discrete
Mathematik, Steyrergasse 30/III, 8010 Graz, Austria<EMAIL_ADDRESS><EMAIL_ADDRESS>Robert Schippa: Fakultät für Mathematik, Karlsruher
Institut für Technologie, Englerstrasse 2, 76131 Karlsruhe, Germany
<EMAIL_ADDRESS>
###### Abstract.

The spectrum of the normalized complex Laplacian for electrical networks is
analyzed. We show that eigenvalues lie in a larger region compared to the case
of the real Laplacian. We show the existence of eigenvalues with negative real
part and absolute value greater than $2$. An estimate from below for the first
non-vanishing eigenvalue in modulus is provided. We supplement the estimates
with examples, showing sharpness.

*Corresponding author

Keywords: finite weighted graphs, complex Laplacian, electrical networks,
spectral estimates

Mathematics Subject Classification 2010: 05C22, 05C50, 39A12, 34L15

##  1\. Introduction

The purpose of this note is to analyze the complex Laplacian for a class of
finite weighted graphs. In the following we consider finite _electrical
networks_ (cf. [13, 14, 15]). These consist of finite sets of vertices and
edges $(V,E)$, where loops are excluded. For any edge $xy$, $x,y\in V$, there
are non-negative real numbers $L_{xy}$, $D_{xy}$, $R_{xy}$ satisfying

$L_{xy}+D_{xy}+R_{xy}>0.$

For $s\in\mathbb{C}$ with $\operatorname{Re}s>0$, we consider the finite
weighted graph $(V,E)$ with weights given by the _admittances_

(1) $\rho^{(s)}(x,y)=\frac{s}{L_{xy}s^{2}+R_{xy}s+D_{xy}}.$

Since $s$ will be clear from context, we write $\rho_{xy}$ for
$\rho^{(s)}(x,y)$ for the sake of brevity. We set $\rho_{xy}\equiv 0$ if $xy$
is not an edge. Then, any electrical network is uniquely determined by the
pair $(V,\rho)$. Let

$\rho(x)=\sum_{y\in V}\rho_{xy}.$

The admittance in form (1) corresponds to the case of electrical networks with
passive elements (coils, capacitors, and resistors). In this case $s$
corresponds to a complex frequency (cf. [4, 8, 9]). We consider the normalized
complex-weighted Laplacian:

(2) $\widetilde{\Delta}_{\rho}f(x)=f(x)-\dfrac{1}{\rho(x)}\sum_{y\in
V}f(y)\rho_{xy}.$

Set ${\rho_{xy}=\tau_{xy}+i\sigma_{xy}}$, where
$\tau_{xy},\sigma_{xy}\in\mathbb{R}$ and ${\rho(x)=\tau(x)+i\sigma(x)}$.
Clearly, $\tau(x)=\sum_{y}\tau_{xy}$ and $\sigma(x)=\sum_{y}\sigma_{xy}$.

Since admittance is positive real function (cf. [4, 13, 15]), i.e.,
$\operatorname{Re}s>0$, whenever $\rho^{(s)}_{xy}>0$ (in particular,
$\rho^{(s)}_{xy}$ has no poles), the normalized complex-weighted Laplacian is
well-defined. For the sake of completeness, we show this in Proposition 5.

The real Laplacian ($L_{xy}=D_{xy}=0$ for any $x,y$) arises in both electrical
networks with resistors and random walks (cf. [7, 10, 12, 16]). The complex
Laplacian arises in AC electrical networks and corresponds to Kirchoff’s law
(cf. [8, 11, 14, 15]). Note that if $s\in\mathbb{R}^{+}$, then the resulting
Laplacian is still real and corresponds to some random walk.

For $s\in\mathbb{R}^{+}$ the resulting Laplacian is a self-adjoint, non-
negative operator acting on a Hilbert space with scalar product depending on
$\rho$. The real Laplacian is studied extensively (cf. [3, 6, 10]). In this
case, the eigenvalues lie in the interval $[0,2]$. In Section 2 we recall
among basic facts how $2$ is attained as an eigenvalue for bipartite graphs.

In Section 3 we show the following generalization:

######  Theorem 1.

Let $\lambda$ denote an eigenvalue of (2). Then, we find the following
estimate to hold:

(3) $|1-\lambda|\leq\left(\frac{|s|}{\operatorname{Re}s}\right).$

Secondly, we show that eigenvalues are additionally confined to the region
described below:

######  Theorem 2.

Let $\Delta_{\rho}$ be a normalized complex-weighted Laplacian. Then the
following holds:

  * •

All its eigenvalues with positive real part lie in the circle with center at
$(1,|\operatorname{Im}s|/\operatorname{Re}s)$ and radius
$\sqrt{1+(\operatorname{Im}s/\operatorname{Re}s)^{2}}$.

  * •

All its eigenvalues with negative real part lie in the circle with the center
at $(1,-|\operatorname{Im}s|/\operatorname{Re}s)$ and radius
$\sqrt{1+(\operatorname{Im}s/\operatorname{Re}s)^{2}}$.

  * •

All its real eigenvalues lie in $[0,2]$.

We shall see that for the linear graph $P_{4}$ a particular choice of weights
exhausts the radii of the circles from the above theorem.

In Section 4 we prove the following bound on the smallest eigenvalue in
modulus:

######  Theorem 3.

Let $\lambda_{1}$ denote the smallest eigenvalue in modulus for the complex
Laplacian $\widetilde{\Delta}_{\rho}$, and let $D$ denote the diameter of the
underlying graph. Let

$\displaystyle
C_{1}=\min_{x}\sum_{y}\big{(}\frac{1}{L_{xy}+R_{xy}+D_{xy}}\big{)},\quad
C_{2}=\sum_{x,y}\frac{1}{L_{xy}+R_{xy}+D_{xy}}$

and suppose that

(4)
$C_{1}\operatorname{Re}s\min(|s|^{2},|s|^{-4})-C_{2}\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\frac{\max(1,|s|^{2})}{\operatorname{Re}s}>0.$

Then, we find the following estimate to hold:

$|\lambda_{1}|\geq\frac{C_{1}(\operatorname{Re}s)^{2}}{D\cdot
C_{2}\cdot\min(1,|s|^{2})}.$

_Outline of the paper._ In Section 2 we recall basic identities, which are
needful for the proofs in later sections. Furthermore, we record symmetries of
the problem. In Section 3 we prove Theorems 1 and 2. The eigenvalues for the
linear graph $P_{4}$ are computed for a particular choice of weight. In
Section 4 we prove Theorem 3.

##  2\. Preliminaries

We start with the following repetition concerning the real Laplacian:

######  Proposition 4.

Let $(V,E)$ be a finite connected graph with real weights and
$\widetilde{\Delta}_{\rho}$ be the real Laplacian. Then, we find the following
to hold:

  * 1.

The eigenvalues lie in the interval $[0,2]$ of $\widetilde{\Delta}_{\rho}$,
where $0$ is a simple eigenvalue.

  * 2.

$2$ is an eigenvalue if and only if $V$ is _bipartite_ , i.e., there is a
partition of vertices $V=V_{+}\cup V_{-}$, $V_{+}\cap V_{-}=\emptyset$ such
that for any $x\in V_{+}$ $x\sim y$ implies that $y\in V_{-}$.

  * 3.

$V$ is bipartite if and only if for any eigenvalue $\lambda$ we find
$2-\lambda$ to be another eigenvalue.

Recall the following Green’s formula ([14, 15]):

(5) $\sum_{x\in
V}\widetilde{\Delta}_{\rho}f(x)g(x)\rho(x)=\dfrac{1}{2}\sum_{x,y\in
V}(\nabla_{xy}f)(\nabla_{xy}g)\rho_{xy},$

where we denote

$\nabla_{xy}f=f(y)-f(x).$

Applying Green’s formula to $g=\overline{f}$ (complex conjugate), we get

(6) $\sum_{x\in
V}\widetilde{\Delta}_{\rho}f(x)\overline{f(x)}\rho(x)=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}\rho_{xy}.$

Note, that the right hand side of the last equation corresponds to the
_complex power_ (cf. [1, 2, 5, 15]). For estimates on the eigenvalues we will
need estimates for $\rho_{xy}$. These are collected in the following
proposition:

######  Proposition 5.

Let $\rho_{xy}$ be as in (1) with the notations from above. Then, the
following holds:

  * 1.

If $x$ and $y$ are related, then $\operatorname{Re}\rho_{xy}>0$.

  * 2.

(7) $|\rho_{xy}|\leq\frac{|s|}{\operatorname{Re}s}\operatorname{Re}\rho_{xy}.$

  * 3.

(8)
$|\rho(x)|\leq\sum_{y}\frac{1}{L_{xy}+D_{xy}+R_{xy}}\frac{\max(1,|s|^{2})}{\operatorname{Re}s}.$

###### Proof.

  * 1.

Note that for any complex number $z\in\mathbb{C}$, $\operatorname{Re}z>0$ if
and only if $\operatorname{Re}\frac{1}{z}>0$. We have

$\operatorname{Re}\frac{1}{\rho_{xy}}=L_{xy}\operatorname{Re}s+R_{xy}+D_{xy}\operatorname{Re}\frac{1}{s}>0,\mbox{
whenever }\operatorname{Re}s>0.$

Therefore, $\operatorname{Re}\rho_{xy}>0$.

  * 2.

$\displaystyle\dfrac{\operatorname{Re}\rho_{xy}}{|\rho_{xy}|}=$
$\displaystyle|\rho_{xy}|\operatorname{Re}\frac{1}{\rho_{xy}}=\dfrac{1}{\left|L_{xy}s+R_{xy}+\frac{D_{xy}}{s}\right|}\left(L_{xy}\operatorname{Re}s+R_{xy}+D_{xy}\operatorname{Re}\frac{1}{s}\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{\left|L_{xy}s+R_{xy}+\frac{D_{xy}}{s}\right|}\left(L_{xy}\operatorname{Re}s+R_{xy}+D_{xy}\frac{\operatorname{Re}s}{|s|^{2}}\right)$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{L_{xy}|s|+R_{xy}+\frac{D_{xy}}{|s|}}\left(L_{xy}\operatorname{Re}s+R_{xy}+D_{xy}\frac{\operatorname{Re}s}{|s|^{2}}\right)$
$\displaystyle\geq$
$\displaystyle\dfrac{1}{L_{xy}|s|+R_{xy}+\frac{D_{xy}}{|s|}}\left(L_{xy}\operatorname{Re}s+R_{xy}\frac{\operatorname{Re}s}{|s|}+D_{xy}\frac{\operatorname{Re}s}{|s|^{2}}\right)=\dfrac{\operatorname{Re}s}{|s|},$

from which (7) follows.

  * 3.

$\displaystyle|\rho(x)|=$
$\displaystyle\left|\sum_{y}\rho_{xy}\right|\leq\sum_{y}|\rho_{xy}|\leq\sum_{y}\dfrac{1}{\left|L_{xy}s+R_{xy}+\frac{D_{xy}}{s}\right|}\leq\sum_{y}\dfrac{1}{\operatorname{Re}\left(L_{xy}s+R_{xy}+\frac{D_{xy}}{s}\right)}$
$\displaystyle=$
$\displaystyle\sum_{y}\dfrac{1}{\left(L_{xy}\operatorname{Re}s+R_{xy}+D_{xy}\frac{\operatorname{Re}s}{|s|^{2}}\right)}=\sum_{y}\dfrac{1}{\operatorname{Re}s\left(L_{xy}+R_{xy}\frac{1}{\operatorname{Re}s}+D_{xy}\frac{1}{|s|^{2}}\right)}$
$\displaystyle\leq$
$\displaystyle\sum_{y}\dfrac{1}{\operatorname{Re}s\left(L_{xy}+R_{xy}+D_{xy}\right)\min\left(\frac{1}{\operatorname{Re}s},\frac{1}{|s|^{2}},1\right)}=\sum_{y}\dfrac{\max\left({\operatorname{Re}s},{|s|^{2}},1\right)}{\operatorname{Re}s\left(L_{xy}+R_{xy}+D_{xy}\right)}$
$\displaystyle=$
$\displaystyle\sum_{y}\dfrac{\max\left({|s|^{2}},1\right)}{\operatorname{Re}s\left(L_{xy}+R_{xy}+D_{xy}\right)},$

where in the last line we have used the fact that either
$\operatorname{Re}s\leq 1$ or
$1<\operatorname{Re}s<(\operatorname{Re}s)^{2}\leq|s|^{2}$.

∎

######  Proposition 6.

$0$ is a simple eigenvalue of the complex Laplacian.

###### Proof.

It is clear that $f\equiv 1$ is an eigenfunction with eigenvalue $0$. On the
other hand, for an eigenfunction $f$ with eigenvalue $0$, (6) yields

$0=\frac{1}{2}\sum_{x,y\in V}|\nabla_{xy}f|^{2}\rho_{xy}.$

Since $\operatorname{Re}\rho_{xy}>0$ provided that $x$ and $y$ are related,
taking the real part of the above display we find that $|\nabla_{xy}f|=0$
whenever $x$ and $y$ are related. Since the graph is connected, $f\equiv c$.
The proof is complete. ∎

######  Proposition 7.

The sum of the all eigenvalue of Laplacian, counted with algebraic
multiplicities, is equal to the number of vertices in the graph, i.e

(9) $\lambda_{0}+\lambda_{1}+\dots+\lambda_{n-1}=n,$

where $n=|V|$.

Moreover,

(10) $\max_{i}\operatorname{Re}\lambda_{i}\geq\dfrac{n}{n-1}>1,$

and

$\min_{i}\operatorname{Re}\lambda_{i}\leq\dfrac{n}{n-1}.$

###### Proof.

The first fact follows from the consideration of the trace of the Laplacian
matrix $A$. Indeed, since $A_{ii}=1$, $i=1,\ldots,n$, we find

$\lambda_{0}+\lambda_{1}+\dots+\lambda_{n-1}=\operatorname{trace}A=n,$

where $n=|V|$.

Further, since $\lambda_{0}=0$, we have

$n=\operatorname{Re}\lambda_{1}+\dots+\operatorname{Re}\lambda_{n-1}\leq(n-1)\max_{i}\operatorname{Re}\lambda_{i}.$

and

$n=\operatorname{Re}\lambda_{1}+\dots+\operatorname{Re}\lambda_{n-1}\geq(n-1)\min_{i}\operatorname{Re}\lambda_{i}.$

Therefore,

$\max_{i}\operatorname{Re}\lambda_{i}\geq\dfrac{n}{n-1}>1$

and

$\min_{i}\operatorname{Re}\lambda_{i}\leq\dfrac{n}{n-1}.$

∎

######  Remark 8.

Note that if we assume that

$0=|\lambda_{0}|<|\lambda_{1}|\leq|\lambda_{2}|\leq\dots\leq|\lambda_{n-1}|,$

then from (10) follows that

$|\lambda_{n-1}|\geq\dfrac{n}{n-1}.$

Moreover, from (9) follows that

$\operatorname{Im}\lambda_{1}+\operatorname{Im}\lambda_{2}+\dots+\operatorname{Im}\lambda_{n-1}=0,$

and, therefore,

$\min_{i}\operatorname{Im}\lambda_{i}\leq
0\leq\max_{i}\operatorname{Im}\lambda_{i}.$

Next, we show that switching to the dual network conjugates eigenvalues and
eigenvectors:

######  Proposition 9.

If we consider the dual network, i.e., the network with the weights
$\overline{\rho}_{xy}$, eigenvalues (eigenvectors) of its complex Laplacian
will be conjugated to the corresponding eigenvalues (eigenvectors) of the
Laplacian of the original network.

###### Proof.

Let $\lambda f(x)=\widetilde{\Delta}_{\rho}f(x)$. Then

$\displaystyle\overline{\lambda}\overline{f(x)}=$
$\displaystyle\overline{\widetilde{\Delta}_{\rho}f(x)}=\overline{f(x)}-\dfrac{1}{\overline{\rho(x)}}\sum_{y}\overline{f(y)}\overline{\rho}_{xy}=\widetilde{\Delta}_{\overline{\rho}}\overline{f(x)}.$

∎

The claims about eigenvalues for bipartite graphs in the real case generalize
as follows:

######  Proposition 10.

With the above notations, let $\widetilde{\Delta}_{\rho}$ be the complex
Laplacian of an electrical network. Suppose that the underlying graph is
bipartite. Then for any eigenvalue $\lambda$ we find $2-\lambda$ to be another
eigenvalue.

###### Proof.

If $V_{+}$ and $V_{-}$ are suitable partitions of the graph, and $(\lambda,f)$
is an eigenpair, then the function $g$, given by

$g(x)=\begin{cases}f(x),x\in V_{+},\\\ -f(x),x\in V_{-},\end{cases}$

is the eigenfuction, corresponding to the eigenvalue $2-\lambda$. Indeed, for
$x\in V_{+}$,

$\displaystyle{\widetilde{\Delta}_{\rho}g(x)}=$
$\displaystyle{g(x)}-\dfrac{1}{{\rho(x)}}\sum_{y}{g(x)}\rho_{xy}={f(x)}+\dfrac{1}{{\rho(x)}}\sum_{y}{f(x)}\rho_{xy}$
$\displaystyle=$ $\displaystyle
2{f(x)}-\left(f(x)-\dfrac{1}{{\rho(x)}}\sum_{y}{f(x)}\rho_{xy}\right)$
$\displaystyle=$ $\displaystyle
2f(x)-{\widetilde{\Delta}_{\rho}f(x)}=(2-\lambda)f(x)=(2-\lambda)g(x).$

The case $x\in V_{-}$ can be treated analogously. ∎

##  3\. Eigenvalue regions

###  3.1. Proof of Theorems 1 and 2

Firstly, we show Theorem 1:

###### Proof of Theorem 1.

We choose an eigenfunction $f$ of $\widetilde{\Delta}_{\rho}$ with $|f(y)|\leq
1$ for any $y\in V$ and $\max_{x}|f(x)|=1$. By (7), we find

$\displaystyle|1-\lambda||f(x)|$
$\displaystyle\leq\left|\sum_{y}\frac{\rho_{xy}}{\rho(x)}f(y)\right|$
$\displaystyle\leq\sum_{y}\frac{|\rho_{xy}|}{|\rho(x)|}|f(y)|$
$\displaystyle\leq\frac{|s|}{\operatorname{Re}s}\sum_{y}\frac{\operatorname{Re}\rho_{xy}}{|\rho(x)|}=\frac{|s|}{\operatorname{Re}s}\frac{\operatorname{Re}\rho(x)}{|\rho(x)|}\leq\frac{|s|}{\operatorname{Re}s}.$

∎

Note how (3) generalizes the claim for real $s$. The eigenvalues of the
Laplacian in the real case lie in the interval $[0,2]$; hence, the estimate is
clearly sharp by comparison with the real case.

Next, we prove Theorem 2:

###### Proof of Theorem 2.

Let ${\lambda=u+iw}$ be an eigenvalue of the normalized complex-weighted
Laplacian $\widetilde{\Delta}_{\rho}$ with the eigenfunction $f$, i.e,

$\lambda f(x)=\widetilde{\Delta}_{\rho}f(x).$

Then by Green’s formula we have

(11) $\sum_{x\in V}\lambda|f(x)|^{2}\rho(x)=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}\rho_{xy}.$

Let us write separately real and imaginary parts of the last equality,
assuming ${\rho_{xy}=\tau_{xy}+i\sigma_{xy}}$ and
${\rho(x)=\tau(x)+i\sigma(x)}$. Note that in this case
$\tau(x)=\sum_{y}\tau_{xy}$ and $\sigma(x)=\sum_{y}\sigma_{xy}$.

Then,

$\lambda\sum_{x\in V}|f(x)|^{2}\rho(x)=(u+iw)\sum_{x\in
V}|f(x)|^{2}(\tau(x)+i\sigma(x)),$

and

$\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}\rho_{xy}=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}(\tau_{xy}+i\sigma_{xy}).$

Therefore, for the real part of (11), we have

(12) $u\sum_{x\in V}|f(x)|^{2}\tau(x)-w\sum_{x\in
V}|f(x)|^{2}\sigma(x)=\dfrac{1}{2}\sum_{x,y\in V}|\nabla_{xy}f|^{2}\tau_{xy},$

and for the imaginary part,

(13) $u\sum_{x\in V}|f(x)|^{2}\sigma(x)+w\sum_{x\in
V}|f(x)|^{2}\tau(x)=\dfrac{1}{2}\sum_{x,y\in V}|\nabla_{xy}f|^{2}\sigma_{xy}.$

Multiplying (12) by $u$ and (13) by $w$ and summing the obtained equalities
up, we get

(14) $(u^{2}+w^{2})\sum_{x\in V}|f(x)|^{2}\tau(x)=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}(\tau_{xy}u+\sigma_{xy}w).$

Let us estimate the left-hand side, using the obvious inequality

$|\nabla_{xy}f|^{2}=|f(y)-f(x)|^{2}\leq 2(|f(y)|^{2}+|f(x)|^{2}).$

Moreover, the right-hand side of (14) is positive, this means also the left-
hand side is positive and at least one $(\tau_{xy}u+\sigma_{xy}w)$ is
positive.

Let $w>0$. Then, we have

$\displaystyle(u^{2}+w^{2})\sum_{x\in V}|f(x)|^{2}\tau(x)$
$\displaystyle=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}\tau_{xy}(u+\dfrac{\sigma_{xy}}{\tau_{xy}}w)$
$\displaystyle\leq\sum_{x,y\in
V}(|f(y)|^{2}+|f(x)|^{2})\tau_{xy}(u+\dfrac{\sigma_{xy}}{\tau_{xy}}w)$
$\displaystyle\leq\sum_{x,y\in
V}(|f(y)|^{2}+|f(x)|^{2})\tau_{xy}(u+w\max_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}})$ $\displaystyle\leq 2\sum_{x\in
V}|f(x)|^{2}\tau(x)(u+w\max_{x\sim y}\dfrac{\sigma_{xy}}{\tau_{xy}}).$

Therefore,

$\displaystyle u^{2}+w^{2}$ $\displaystyle\leq 2(u+w\max_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}}),$

i.e.,

$\displaystyle(u-1)^{2}+\left(w-\max_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}}\right)^{2}$ $\displaystyle\leq
1+\left(\max_{x\sim y}\dfrac{\sigma_{xy}}{\tau_{xy}}\right)^{2}.$

Let $w<0$. Then, we have

$\displaystyle(u^{2}+w^{2})\sum_{x\in V}|f(x)|^{2}\tau(x)$
$\displaystyle=\dfrac{1}{2}\sum_{x,y\in
V}|\nabla_{xy}f|^{2}\tau_{xy}(u+\dfrac{\sigma_{xy}}{\tau_{xy}}w)$
$\displaystyle\leq\sum_{x,y\in
V}(|f(y)|^{2}+|f(x)|^{2})\tau_{xy}(u+\dfrac{\sigma_{xy}}{\tau_{xy}}w)$
$\displaystyle\leq\sum_{x,y\in
V}(|f(y)|^{2}+|f(x)|^{2})\tau_{xy}(u+w\min_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}})$ $\displaystyle\leq 2\sum_{x\in
V}|f(x)|^{2}\tau(x)(u+w\min_{x\sim y}\dfrac{\sigma_{xy}}{\tau_{xy}}).$

Therefore,

$\displaystyle u^{2}+w^{2}$ $\displaystyle\leq 2(u+w\min_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}}),$

i.e.,

$\displaystyle(u-1)^{2}+\left(w-\min_{x\sim
y}\dfrac{\sigma_{xy}}{\tau_{xy}}\right)^{2}$ $\displaystyle\leq
1+\left(\min_{x\sim y}\dfrac{\sigma_{xy}}{\tau_{xy}}\right)^{2}.$

Let $w=0$. Then, we have

$\displaystyle u^{2}\sum_{x\in V}|f(x)|^{2}\tau(x)$
$\displaystyle=\dfrac{1}{2}\sum_{x,y\in V}|\nabla_{xy}f|^{2}\tau_{xy}u$
$\displaystyle\leq 2\sum_{x\in V}|f(x)|^{2}\tau(x)u.$

Therefore,

$\displaystyle u^{2}$ $\displaystyle\leq 2u,$

which means that the real eigenvalues of the normalized complex Laplacian lie
in $[0,2]$.

From (7) follows by squaring

$\dfrac{\operatorname{Re}^{2}\rho_{xy}}{\operatorname{Re}^{2}\rho_{xy}+\operatorname{Im}^{2}\rho_{xy}}\geq\dfrac{\operatorname{Re}^{2}s}{\operatorname{Re}^{2}s+\operatorname{Im}^{2}s}.$

This yields

$\dfrac{\operatorname{Im}^{2}\rho_{xy}}{\operatorname{Re}^{2}\rho_{xy}}\leq\dfrac{\operatorname{Im}^{2}s}{\operatorname{Re}^{2}s},$

and hence,

$-\left|\dfrac{\operatorname{Im}s}{\operatorname{Re}s}\right|\leq\dfrac{\operatorname{Im}\rho_{xy}}{\operatorname{Re}\rho_{xy}}\leq\left|\dfrac{\operatorname{Im}s}{\operatorname{Re}s}\right|.$

The proof is complete. ∎

###  3.2. Examples and sharpness of the estimates

With the following example we illustrate the estimates from above and point
out how we cannot scale down the radii of the circles in Theorem 2 . We
consider the linear graph $P_{4}$ with four vertices and edges weighted $s$,
$1/s$, $s$. The complex normalized Laplacian is given by the matrix

(15) $A=\begin{pmatrix}1&-1&0&0\\\
\frac{-s^{2}}{1+s^{2}}&1&\frac{-1}{s^{2}+1}&0\\\
0&\frac{-1}{s^{2}+1}&1&\frac{-s^{2}}{s^{2}+1}\\\ 0&0&-1&1\end{pmatrix}.$

We find for the eigenvalues

(16) $\frac{1}{1+s^{2}},\quad 0,\quad 2,\quad\frac{1+2s^{2}}{1+s^{2}}.$

The corresponding eigenvectors are computed as

$(-1,\frac{-s^{2}}{1+s^{2}},\frac{s^{2}}{1+s^{2}},1),\;(1,1,1,1),\;(-1,1,-1,1),\;(1,\frac{-s^{2}}{1+s^{2}},\frac{-s^{2}}{1+s^{2}},1).$

On the Figure 1 we green circles correspond to the estimate in Theorem 2, blue
circle corresponds to Theorem 1 ( everything is drawn for the case $s=1+2i$)
and red dots are the eigenvalues $0$, $2$, $-1/10-2i/10$, $21/10+2i/10$. Note,
that in this case there is an eigenvalue with real part, larger than $2$, and
an eigenvalue with negative real part.

Let $s=s_{1}+is_{2},\;s_{1},s_{2}>0$. We shall show sharpness of Theorem 2
considering the eigenvalue

$z=1+\frac{s^{2}}{1+s^{2}}.$

For eigenvalues of the complex Laplacian with positive imaginary part $w$ and
real part $u$ we have shown the estimate

$(u-1)^{2}+(w-m)^{2}\leq 1+m^{2},\quad
m=\max\frac{\operatorname{Im}\rho_{xy}}{\operatorname{Re}\rho_{xy}}.$

Note that in our example the maximum ratio is $m=s_{2}/s_{1}$. We show that
for any $\varepsilon>0$ the estimate

(17) $(u-1)^{2}+(w-m)^{2}\leq 1+m^{2}-\varepsilon,$

fails, choosing a sufficiently large value of $s_{1}$.

Firstly, we compute

$\displaystyle(u-1)^{2}$
$\displaystyle=\big{(}\frac{(s^{2}+\bar{s}^{2})/2+|s|^{4}}{|1+s^{2}|^{2}}\big{)}^{2}=\big{(}\frac{s_{1}^{2}-s_{2}^{2}+(s_{1}^{2}+s_{2}^{2})^{2}}{|1+s^{2}|^{2}}\big{)}^{2},$
$\displaystyle(w-m)^{2}$
$\displaystyle=\big{(}\frac{2s_{1}s_{2}}{|1+s^{2}|^{2}}-\frac{s_{2}}{s_{1}}\big{)}^{2}.$

Multiplying (17) with $|1+s^{2}|^{4}$ we find

$(s_{1}^{2}-s_{2}^{2}+(s_{1}^{2}+s_{2}^{2})^{2})^{2}+(2s_{1}^{2}s_{2}-\frac{s_{2}}{s_{1}}|1+s^{2}|^{2})^{2}\leq|1+s^{2}|^{4}(1+\frac{s_{2}^{2}}{s_{1}^{2}}-\varepsilon).$

Multiplying the above with $s_{1}^{2}$ gives

$s_{1}^{2}(s_{1}^{2}-s_{2}^{2}+(s_{1}^{2}+s_{2}^{2})^{2})^{2}+s_{2}^{2}(2s_{1}^{2}-|1+s^{2}|^{2})^{2}\leq|1+s^{2}|^{4}(s_{2}^{2}+(1-\varepsilon)s_{1}^{2}).$

Subtracting $s_{2}^{2}|1+s^{2}|^{4}$ on both sides and dividing by $s_{1}^{2}$
yields

$(s_{1}^{2}-s_{2}^{2}+(s_{1}^{2}+s_{2}^{2})^{2})^{2}+4s_{1}^{2}s_{2}^{2}-4s_{2}^{2}|1+s^{2}|^{2}\leq(1-\varepsilon)|1+s^{2}|^{4}.$

We consider $s_{2}\ll 1\ll s_{1}$ such that the highest power of $s_{1}$
dominates. But on the left-hand side, we find $s_{1}^{8}$ and on the right-
hand side $(1-\varepsilon)s_{1}^{8}$. Therefore, the inequality (17) fails
choosing $s_{1}$ large enough (for fixed $s_{2}$).

$s_{1}$$s_{2}$$0$$1$ Figure 1. Location of eigenvalues for $s=1+2i$

##  4\. Estimate for smallest eigenvalue in modulus

For the real normalized Laplacian the smallest positive eigenvalue (also
called _spectral gap_) plays a crucial role finding the mixing properties of
the associated random walk. In the real case, a lower bound is provided by the
_Cheeger constant_ (cf. [6]). Roughly speaking, this measures the
connectedness when removing vertices. Evidently, the geometry of the
underlying graph plays the crucial role. Here, we do not know how to
generalize the Cheeger constant to the case of complex weights because
eigenspaces of eigenvalues are not necessarily orthogonal. We can still prove
the lower bound depending on the diameter. We start with a lower bound on
$\operatorname{Re}\rho_{xy}$.

######  Lemma 11.

Let

$\displaystyle
C_{1}=\min_{x}\sum_{y}\big{(}\frac{1}{L_{xy}+R_{xy}+D_{xy}}\big{)},\quad
C_{2}=\sum_{x,y}\frac{1}{L_{xy}+R_{xy}+D_{xy}}.$

Then, we find the following estimate to hold:

(18)
$\begin{split}&\quad\min_{x}\operatorname{Re}(\rho(x))-\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\sum_{x}\operatorname{Re}(\rho(x))\\\
&\geq
C_{1}\operatorname{Re}s\min(|s|^{2},|s|^{-4})-C_{2}\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\frac{\max(1,|s|^{2})}{\operatorname{Re}s}.\end{split}$

###### Proof.

Let $r=\rho_{xy}$ for brevity. Note that

$\operatorname{Re}(r)=|r|^{2}\operatorname{Re}\left(\frac{1}{r}\right)=|r|^{2}\left(\operatorname{Re}(s)L_{xy}+R_{xy}+D_{xy}\frac{\operatorname{Re}s}{|s|^{2}}\right).$

Moreover,

$\displaystyle\big{|}\frac{1}{r}\big{|}$
$\displaystyle=|L_{xy}s+R_{xy}+\frac{D_{xy}}{s}|\leq
L_{xy}|s|+R_{xy}+\frac{D_{xy}}{|s|}$
$\displaystyle\leq(L_{xy}+R_{xy}+D_{xy})\max(|s|,1/|s|).$

Consequently,

$\operatorname{Re}(r)\geq\frac{1}{(L_{xy}+R_{xy}+D_{xy})^{2}\max(|s|^{2},1/|s|^{2})}(L_{xy}+R_{xy}+D_{xy})\min(\operatorname{Re}s,\operatorname{Re}\frac{s}{|s|^{2}})$

because
$\min(1,\operatorname{Re}s,\operatorname{Re}s/|s|^{2})=\min(\operatorname{Re}s,\operatorname{Re}s/|s|^{2})$,
which can be seen by considering $|s|\leq 1$ and $|s|\geq 1$. Hence,

$\displaystyle\operatorname{Re}(\rho_{xy})$
$\displaystyle\geq\frac{\operatorname{Re}s}{L_{xy}+D_{xy}+R_{xy}}\frac{\min(1,|s|^{-2})}{\max(|s|^{2},|s|^{-2})}$
$\displaystyle=\frac{\operatorname{Re}(s)}{L_{xy}+R_{xy}+D_{xy}}\min(|s|^{2},|s|^{-4}).$

Hence, we find by (8)

$\displaystyle\;\min_{x}\operatorname{Re}(\rho(x))-\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\sum_{x}\operatorname{Re}(\rho(x))$
$\displaystyle\geq\operatorname{Re}(s)\big{[}\min_{x}\sum_{y}\big{(}\frac{1}{L_{xy}+R_{xy}+D_{xy}}\big{)}\big{]}\min(|s|^{2},|s|^{-4})-\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\sum_{x}|\rho(x)|$
$\displaystyle=C_{1}\operatorname{Re}s\min(|s|^{2},|s|^{-4})-C_{2}\frac{|\operatorname{Im}s|}{\operatorname{Re}s}\frac{\max(1,|s|^{2})}{\operatorname{Re}s}.$

∎

Thus, for fixed $\operatorname{Re}s$, we can find the above quantity to be
positive choosing $\operatorname{Im}s$ sufficiently small only depending on
$\operatorname{Re}s$ and the electrical network. Moreover, for $s=1$ the lower
bound is attained. We are ready to prove Theorem 3.

###### Proof.

Let $\lambda_{1}$ be the smallest eigenvalue in modulus and let $f$ be a
corresponding eigenfunction. We normalize $f$ such that $f(x_{0})=1$ and
$\max|f(x)|=1$. Green’s formula yields that

$\sum_{x}\widetilde{\Delta}_{\rho}f_{1}(x)\rho(x)=\lambda_{1}\sum_{x}f_{1}(x)\rho(x)=0.$

Indeed, in (5) set $f_{2}\equiv 1$ and integrate by parts. Hence,

$\sum_{x}f_{1}(x)\rho(x)=0.$

Therefore,

$\displaystyle\operatorname{Re}(\rho(x_{0}))+\operatorname{Re}\big{(}\sum_{x\neq
x_{0}}f_{1}(x)\rho(x)\big{)}$ $\displaystyle=0$
$\displaystyle\Leftrightarrow\operatorname{Re}(\rho(x_{0}))+\sum_{x\neq
x_{0}}\operatorname{Re}f_{1}(x)\operatorname{Re}\rho(x)-\sum_{x\neq
x_{0}}\operatorname{Im}f_{1}(x)\operatorname{Im}\rho(x)$ $\displaystyle=0,$

but

$\big{|}\sum_{x\neq
x_{0}}\operatorname{Im}(f_{1}(x))\operatorname{Im}(\rho(x))\big{|}\leq\sum_{x\neq
x_{0}}|\operatorname{Im}\rho(x)|.$

Hence, by Lemma 11 and assumption (4),

$\min\operatorname{Re}\rho(x)-\sum_{x\neq x_{0}}|\operatorname{Im}\rho|>0,$

we find that

$\sum_{x\neq x_{0}}\operatorname{Re}(f_{1}(x))\operatorname{Re}\rho(x)<0.$

Since $\operatorname{Re}(\rho(x))>0$, there is $x_{n+1}$ such that
$\operatorname{Re}(f_{1}(x_{n+1}))<0$.

Another application of Green’s formula gives

$\lambda_{1}\sum|f(x)|^{2}\rho(x)=\frac{1}{2}\sum_{x,y}|f(x)-f(y)|^{2}\rho_{xy},$

which implies

$|\lambda_{1}|=\frac{\frac{1}{2}\big{|}\sum_{x,y}|f(x)-f(y)|^{2}\rho_{xy}}{\big{|}\sum_{x}|f(x)|^{2}\rho(x)\big{|}}$

We estimate the numerator from below by finding a path $x_{0},\ldots,x_{n+1}$:

$\displaystyle\quad\frac{1}{2}\big{|}\sum_{x,y}|f(x)-f(y)|^{2}\rho_{xy}\big{|}$
$\displaystyle\geq\frac{1}{2}\sum_{x,y}|f(x)-f(y)|^{2}\operatorname{Re}\rho_{xy}$
$\displaystyle\geq\frac{1}{2}\sum_{x,y}|f(x)-f(y)|^{2}\min\operatorname{Re}\rho_{xy}$
$\displaystyle\geq\sum_{k=0}^{n}|f(x_{k})-f(x_{k+1})|^{2}\min\operatorname{Re}\rho_{xy}$
$\displaystyle\geq\sum_{k=0}^{n}|\operatorname{Re}f(x_{k})-\operatorname{Re}f(x_{k+1})|^{2}\min\operatorname{Re}\rho_{xy}$
$\displaystyle\geq\frac{\min\operatorname{Re}\rho_{xy}}{n}\big{|}\sum_{k=0}^{n}\operatorname{Re}f(x_{k})-\operatorname{Re}f(x_{k+1})\big{|}^{2}$
$\displaystyle\geq\frac{\min\operatorname{Re}\rho_{xy}}{n}\geq\frac{\operatorname{Re}s\;C_{1}\min(|s|^{2},|s|^{-4})}{D}.$

For the denominator we find by (8)

$\left|\sum_{x}|f(x)|^{2}\rho(x)\right|\leq\sum_{x}|\rho(x)|\leq
C_{2}\frac{\min(1,|s|^{2})}{\operatorname{Re}s}.$

This finishes the proof of Theorem 3. ∎

## Acknowledgements

The second author acknowledges financial support by the Deutsche
Forschungsgemeinschaft (DFG, Deutsche Forschungsgemeinschaft) Project-ID
258734477 -
SFB 1173.

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