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energies

Article

Proﬁtability of Residential Battery Energy Storage

Combined with Solar Photovoltaics

Christoph Goebel 1,*, Vicky Cheng 2and Hans-Arno Jacobsen 1

1Chair of Business Information Systems, Technical University of Munich, Boltzmannstr. 3, 85748 Garching,

Germany; jacobsen@in.tum.de

2Munich School of Engineering, Technical University of Munich, Lichtenbergstr. 4a, 85748 Garching,

Germany; vicky.cheng@tum.de

*Correspondence: christoph.goebel@tum.de; Tel.: +49-89-289-19452

Academic Editor: Francesco Calise

Received: 9 May 2017; Accepted: 6 July 2017; Published: 11 July 2017

Abstract:

Lithium-ion (Li-Ion) batteries are increasingly being considered as bulk energy storage in

grid applications. One such application is residential energy storage combined with solar photovoltaic

(PV) panels to enable higher self-consumption rates, which has become ﬁnancially more attractive

recently due to decreasing feed-in subsidies. Although residential energy storage solutions are

commercially mature, it remains unclear which system conﬁgurations and circumstances, including

aggregator-based applications such as the provision of ancillary services, lead to proﬁtable consumer

investments. Therefore, we conduct an extensive simulation study that is able to jointly capture these

aspects. Our results show that, at current battery module prices, even optimal system conﬁgurations

still do not lead to proﬁtable investments into Li-Ion batteries if they are merely used as a buffer for

solar energy. The ﬁrst settings in which they will become proﬁtable, as prices are further declining,

will be larger households at locations with higher average levels of solar irradiance. If the batteries

can be remote-controlled by an aggregator to provide overnight negative reserve, their proﬁtability

increases signiﬁcantly.

Keywords: lithium-ion batteries; solar photovoltaics; ancillary services; economics

1. Introduction

The generation of electricity using residential-size solar photovoltaics (PV) installations has

reached grid parity in 19 markets globally, including countries with relatively low levels of solar

irradiance such as Germany [

1

]. Today’s levelized cost of solar energy without subsidies is therefore

smaller than the cost of purchasing energy from the utility. Until recently, PV systems were mainly

installed to feed the generated electricity directly into the grid at a ﬁxed feed-in remuneration that

is guaranteed for usually 20 years. Today, the solar PV feed-in remuneration ranges below the

end-consumer prices in many countries worldwide, which is making self-consumption of solar power

more attractive.

Although solar energy and most of the demand in households occurs in the daytime, the

simultaneity of solar power and demand is limited. Besides controlling deferrable loads, e.g., hot water

heaters and washing machines, battery energy storage is increasingly being considered as an effective

way to increase self-consumption. Several companies, including electric vehicle manufacturer Tesla

(San Carlos, CA, USA) [

2

], offer lithium-ion (Li-Ion) batteries as a buffer for solar energy so that excess

PV generation can be stored in the battery for self-consumption during times when demand exceeds

supply. Li-Ion battery cell prices are declining rapidly, while their lifetime is slowly increasing [

3

].

Therefore, investing in Li-Ion battery storage will become ﬁnancially more attractive in the future.

However, it still remains an open research question how systems should be sized and operated to

Energies 2017,10, 976; doi:10.3390/en10070976 www.mdpi.com/journal/energies

Energies 2017,10, 976 2 of 17

achieve positive business cases. Moreover, considering the high cost of Li-Ion battery cells, being able

to utilize residential batteries for more than just buffering solar energy could be a major driver of

proﬁtability. Their ability to quickly adjust (dis)charging power makes batteries a potential provider of

ancillary services. Controlled by an aggregator or directly by the system operator, distributed batteries

could, for instance, help to provide frequency reserve.

In this paper, we investigate the ﬁnancial impact of coupling state-of-the-art Li-Ion batteries with

solar PV panels in residential settings. We make the following contributions:

1.

We advance beyond conventional methodology by simulating stochastically the electricity demand

of different households and PV generation at different locations in high temporal resolution.

2.

In addition to solar energy buffering, we investigate the use of batteries to provide negative

reserve and its impact on proﬁtability, which is a largely unexplored topic.

3.

We implement a detailed model to control the operation of a residential energy system and derive

the net present value (NPV) of different system conﬁgurations.

4.

Based on the results of an unprecedented large number of experiments with different inputs, we

conduct a sensitivity study to identify optimal system conﬁgurations and the drivers of energy

storage proﬁtability.

5.

Our study provides fresh insights that facilitate investment decisions, in particular which

system conﬁguration should be chosen based on location, household size, and current battery

module price.

This paper is organized as follows: in Section 2, we review related literature. Section 3provides

details on the different models and data we use. In Section 4, we summarize the setup and results of

a comprehensive simulation study. Section 5discusses our work, including its limitations and possible

future work. Finally, in Section 6, we provide summarized conclusions of our study.

2. Related Work

Several recent studies [

4

–

6

] attempted to examine the proﬁtability and optimal sizing of residential

PV-battery systems, but most suffer from major limitations that impede the robustness of the results.

A common shortcoming stems from the use of low time resolution electricity demand and PV

generation models, which fail to capture the short-time peaks that are intrinsic to real generation

and load curves. The time-averaging effects arise from coarse resolution models, lead to inaccurate

representation of the instantaneous matching of electricity supply and demand, and, consequently,

undermine the capacity sizing and economic assessment of PV-battery systems. Based on the

simulation of a UK household over a summer day, the authors of [

7

] reported signiﬁcant errors

in the estimation of on-site solar fraction by coarse time resolution models. An hourly resolution

model overestimated the PV supply and demand matching capability by over 60% compared to

a minute-resolution model. Ried et al. [

8

] modeled the proﬁtability of PV-battery systems for

a sample of households and reported signiﬁcant underestimation of battery cycle life using coarse

time resolution models. An hourly resolution model on average underestimated the battery cycle

by 9% and overestimated the battery life by up to three years compared to a minute-resolution

model. Furthermore, past studies neglected the stochasticity in residential electricity demand and

drew conclusions merely based on simulations using standard load proﬁles of a single household.

Ried et al. [

8

] reported considerable errors in the estimation of battery cycle life and overall cost-savings

when standard load proﬁles as opposed to measured household demand curves are used. In the

following, we provide several detailed reviews of key papers related to our work.

Naumann et al. [

4

] investigate the costs of Li-Ion battery storage for a 4-person single family house

under the current regulatory regime in Germany. They scale and linearly interpolate a low granularity

standard load proﬁle (15 min averages, 4-person household) to obtain one-minute household demand.

Instead of searching the optimal combination of PV and battery sizes, they ﬁx the PV capacity (4.4 kWp)

and conduct a sensitivity study of the return on investment (ROI) of the system in three battery size

Energies 2017,10, 976 3 of 17

scenarios. With the PV size constraint, they reported an optimal battery capacity of 4.4 kWh while the

ROI varies depending on assumptions in the state of health and aging behavior of the batteries.

Weniger et al. [

5

] attempt to ﬁnd optimal PV and Li-Ion battery sizes for a 3-person single family

house in Germany. They use measured data of typical daily load proﬁles from one household in

one minute resolution based on VDI (verein deutscher ingenieure) guideline 4655 [

9

], which they

assemble into a one year long proﬁle. The main caveat of the study arises from the very conservative

assumptions regarding the battery modules, in particular costs ranging between 3000 EUR/kWh and

600 EUR/kWh, state of charge ranging between 20% and 80%, and a cycle life of 5000 equivalent

full cycles. As a result, PV-battery systems are found to be economically viable only in the long term

assuming PV system costs of 1000 EUR/kWp, battery costs of 600 EUR/kWh, and feed-in tariffs as low

as 0.02 EUR/kWh. The authors reported an optimal conﬁguration consists of (per MWh of demand) a

0.8 kWp PV system and 1.1 kWh usable battery capacity.

The authors of [

6

] conduct a very detailed study of the proﬁtability of residential lead-acid

batteries. In particular, they use several learning curve models to forecast system costs (one for each

major system component) and different price development scenarios to investigate how net present

values would evolve if investments were made anytime between 2013 and 2022. In contrast to the

authors of [

4

], they consider optimal PV panel and battery sizing. However, they use hourly solar data

and concatenated standard load proﬁles (15 min resolution) as input for their simulations.

Zhu at al. [

10

] investigate the use of residential batteries for exploiting time of use (TOU)

tariffs. TOU tariffs are end-customer contracts that result in different electricity prices based on when

electricity is consumed. In contrast to us, they consider two different battery types with different cost

characteristics being operated simultaneously, a lead-acid and a lithium-ion battery. This can lead to

higher cost-efﬁciency due to optimal coordination. Moreover, they do not consider optimal battery

sizing and combined use with solar PV, which is the focus of this paper.

The authors of [

11

] developed a method for optimally sizing PV-battery systems based on

a probabilistic (“chance constraint”) approach. Similar to [

12

], which their formal analysis is based on,

they consider autonomous energy systems without grid connection. In contrast to the grid-connected

systems investigated in our paper, such scenarios call for an investigation of how a probabilistic metric

of autonomy, i.e., how likely a certain system conﬁguration would be able to cover all demand, can be

provided. In [

11

], such a method is proposed. However, to be used in practice, one needs to carry out

stochastic simulations based on actual data, such that the complex time-dependent system behavior

can be considered.

In this paper, we try to combine the strengths and overcome the weaknesses of the past studies.

In particular, we use recent data-driven models to simulate stochastically the electricity demand of

households and PV generation in high time resolution in an effort to accurately assess the proﬁtability

of residential PV-battery systems. We also consider a large sample of households in different sizes and

PV generation in different locations to ensure generalizable results. In fact, our results show that these

contextual parameters are decisive to the proﬁtability of PV-battery systems. Similar to [

5

], our work

considers the important ﬁnancial trade-offs resulting from different PV panel and battery sizes by

searching for optimal conﬁgurations instead of ﬁxing them ex ante. However, we use more realistic

input assumptions and stochastic residential load proﬁles that are lacking in [

4

,

5

]. Furthermore, our

paper also investigates a realistic extension of the standard residential battery use case, namely the

provision of negative reserve. This application is recognized to have growing market potential

worldwide but has not been considered in previous publications [13].

3. Models and Data

3.1. Household Load Model

We use the residential electricity demand model described in detail in [

14

] to generate random

load proﬁles. The model uses an activity-based modeling technique which combines data from

Energies 2017,10, 976 4 of 17

a time use survey with smart meter data of household appliances to derive realistic load proﬁles

on the household level at one-minute resolution. Load is computed using a bottom-up approach.

Empirical data about timing and duration of activities (e.g., cooking, watching TV, etc.) is combined

with empirical data on load distributions depending on household size to derive so-called activity

load proﬁles of households. These proﬁles are stochastic, since the use of each activity-based load is

simulated by sampling from the corresponding probability distributions. One of the special merits of

this model is that it also considers the shape of standard load proﬁles. It achieves this by making sure

that the mean of many simulated household load proﬁles converges to the standard load proﬁle. The

authors of [

14

] have validated the generated household load traces by comparing them to actual smart

meter measurements based on critical statistical properties.

3.2. Solar PV Model

We apply models described in [

15

,

16

] to obtain realistic data traces for the DC power generated

by residential solar crystalline silicon PV panels. PV module efﬁciency

ηPV

can be approximated using

Equation (1) [

15

], where

Tc

and

G

are input variables denoting the PV cell temperature and incident

solar irradiance on the PV array, respectively. The remaining parameters are constants as provided

in Table 1:

ηPV =ηre f (1−β(Tc−Tc,re f ) + γlog(G

Gre f

). (1)

The PV module temperature

Tc

can be modeled according to Equation (2) [

16

], where

Ta

and

Vw

are variables representing ambient temperature and wind speed, respectively:

Tc=Ta+G

Gno

·9.5

5.7 +3.8Vw

·(Tno −Ta,no )·(1−ηre f

τα ). (2)

The total DC power output of a PV module can be calculated as

PDC

PV =Ac·G·ηPV

, where

Ac

is

the module area.

Table 1. Constant model parameters.

Param. Value Param. Value

ηre f 0.21 Tc,ref 25 ◦C

Gre f 1.0 kW/m2β0.0048

γ0.12 Gno 0.8 kW/m2

Tno 25 ◦CTa,no 20 ◦C

τα 0.9 rc1.0

rd1.0 ηB0.95

CAPEO L

B0.74 Lcal

B20 years

Lcyc

B8.000

We use Meteonorm [

17

] to create data traces for a typical year (1991–2010) in one-minute

resolution. Meteonorm provides simulated ground measurements of solar irradiance for typical

years, i.e., it considers the short-term effect of moving clouds. Meteonorm was conﬁgured to simulate

G

over a year, assuming a south facing PV panel tilted by 30

◦

, which represents the optimal ﬁxed tilt

position in Central Europe. We also use Meteonorm to obtain data traces of the ambient temperature

Ta

and wind speed

Vw

. Data was generated for two locations in Germany, Bremen and Munich. We have

chosen these locations since Bremen has one of the lowest, whereas Munich one of the highest solar

irradiance levels in Germany. The results obtained for these two extreme locations enable us to estimate

the impact of solar availability on the proﬁtability of PV-battery systems. We scale the resulting DC

power generation traces based on the maximum value observed during one year and the desired peak

power capacity, C APPV.

Energies 2017,10, 976 5 of 17

3.3. Battery Model

Our battery model can be deﬁned using the following parameters: The battery efﬁciency

ηB

(charging and discharging), the nominal battery storage capacity

CAPB

, and the C-rates for charging

and discharging, rcand rd. In addition, we deﬁne the usable range of the state of charge (SOC) of the

battery via SOCmin and SOCmax .

The overall lifetime of Li-Ion batteries can be approximated based on two parameters, the calendar

lifetime

Lcal

B

and the maximum number of full battery cycles until decommissioning

Lcyc

B

. Lifetime

parameters differ depending on the battery chemistry. In this work, we consider the lithium iron

phosphate (

LiFePO4

) battery chemistry, which is used by several solar battery providers [

18

,

19

] due to

their long cycle life and operational robustness (depth of discharge, safety, temperature, etc.). The exact

speciﬁcations were based on the Sony Fortelion cell (Tokyo, Japan) [

20

]. The capacity of

LiFePO4

batteries decreases approximately linearly with time until a certain reﬂection point, beyond which

capacity rapidly declines. We denote the end of life battery capacity by

CAPEOL

B

. The SOC range of

these cells are not restricted in practice, because, in contrast to other battery chemistries, depth of

discharge does not play a major role in the aging process of LiFePO4cells [21].

3.4. Power Electronics Model

To transform the DC power from the solar PV panel and the battery into AC power used by

household appliances and the grid, an inverter is required. Likewise, charging the battery with grid

power requires rectiﬁcation of AC grid power to DC battery charging power. Power electronics

efﬁciency

ηPE (Pl

,

Pr)

usually peaks at its rated output power and decreases in other load conditions.

To accurately consider this behavior, we use the following model provided in [4]:

ηPE (Pl,Pr) = p

p+0.0072 +0.0345 ·p2;p=Pl

Pr. (3)

The rated output power

Pr

is treated as a constant. In the following, we will model two power

electronic devices: the inverter that converts DC power from the panel and the battery into AC power

and the rectiﬁer required to convert AC grid power to DC charging power for the battery. The nominal

power electronics capacity is chosen to be equal to expected peak load. Therefore,

Pr=CAPPV

in the

former and rc·CAPBin the latter case.

3.5. Battery Control

We combine the different models described above to perform a comprehensive evaluation of

state-of-the-art residential battery energy storage. In addition to using the battery as a way to store

excess solar PV energy and use the stored energy to (partially) replace power drawn from the grid,

we also investigate the case of charging the battery using grid power during times of negative

reserve deployment.

Figure 1shows the algorithm of method

control

, which takes the following inputs: the current

time

t

, solar PV generation

PDC

PV

, household demand

PAC

HH

, and the battery’s SOC. It controls the

battery power during the control time interval

∆S

by determining a feasible value for the DC battery

power

PDC

B

. Thus, the control frequency depends on the length of

∆S

, which is set to 1 min in our

evaluations. We assume that all control inputs are measured or are accurately estimated based on

available measurements. Inputs have to be provided to the algorithm at the same frequency as controls

are computed. Apart from computing

PDC

B

, the battery control algorithm also returns the remaining

power

PAC

D

that needs to be transferred to or from the grid. If

PAC

D

is positive, additional power needs

to be drawn from the grid to cover the demand. If negative, excess energy produced by the solar PV

panels that cannot be absorbed by the battery is fed back into the grid.

Lines 3–9 of algorithm

control

determine the battery dispatch, i.e., the value of

PDC

B

. The method

getReservePower

provided in Figure 2determines a feasible reserve contribution of the battery,

Energies 2017,10, 976 6 of 17

which requires the battery state, and can otherwise be based on time, reserve requirements, etc.

In Germany, there exist separate markets for positive and negative secondary reserve, from 8 a.m. to

8 p.m., and from 8 p.m. to 8 a.m. the next day. The underlying market mechanism is a pay-as-bid

auction for capacity and actual energy delivery. The corresponding Internet-based market platform [

22

]

is run jointly by the German Transmission System Operators (TSOs). Any business party able to fulﬁll

the technical qualiﬁcation criteria (including minimum power and response time) can participate in

these markets. More details about the German ancillary services markets can be found in [

23

]. In this

paper, we evaluate negative reserve provision in

∆S=

15 min time intervals between

tR=

8 p.m.

and

tR=

8 a.m., which corresponds to the current rules for participating in the secondary reserve

market in Germany. Furthermore, we assume that, during each of these 12 h intervals, batteries can

charge for at most

∆R=

1 h, which can result in one full recharge at a charging C-rate of one. If the

battery is scheduled to deliver reserve energy, i.e., the determined reserve power level

PDC

R

is not equal

to zero, the battery is exclusively used for this purpose. Otherwise, it is used to absorb excess solar

energy or provide energy if the household demand is higher than PV generation.Figure 3speciﬁes the

algorithm we use to determine feasible battery power based on the SOC, the required power

Preq

, and

the time interval ∆used for charging or discharging.

1: Input: t,PDC

PV ,PAC

HH ,SOC

2: PDC

B←0

3: if CAPB>0then

4: PDC

R←getReservePower(t,SOC)

5: if PDC

R=0then

6: Preq ←PAC

HH

ηPE (PAC

HH ,C APPV )−PDC

PV

7: PDC

B←getBatPower(SOC,Preq,∆S)

8: else

9: PDC

B←PDC

R

10: if PDC

B>0then

11: SOC ←SOC −PDC

B·∆S

ηB

12: else

13: SOC ←SOC +ηB·PDC

B·∆S

14: if PDC

R=0then

15: PDC

G←PDC

PV +PDC

B

16: else

17: PDC

G←PDC

PV

18: PAC

D←PAC

HH −ηP E(PDC

G,CAPPV )·PDC

G

19: return PAC

D

Figure 1. Algorithm of control(t,PDC

PV ,PAC

HH ,SOC).

1: Input: t,SOC

2: if t=tRthen

3: ∆R←∆R

4: PDC

R←0

5: if Pneg

GR (t)≥Pne g

GR ∧∆R>0∧t∈tR,tR−∆Sthen

6: PDC

R←getBatPower(SOC,−∞,∆S)

7: ∆R←max [∆R−∆S, 0]

8: return PDC

R

Figure 2. Algorithm of getReservePower(t,SOC).

Energies 2017,10, 976 7 of 17

1: Input: SOC,Preq,∆

2: PDC

B←0

3: if Preq <0then

4: Pmax ←rc·C APB

5: Ptheo ←(SOCmax −SOC)·C APB/(ηB·∆)

6: PDC

B← −min Pmax ,Ptheo ,−Preq

7: else if Preq >0then

8: Pmax ←rd·C APB

9: Ptheo ←(SOC −SOCmin )·(ηB·C APB)/∆

10: PDC

B←min Pmax ,Ptheo,Preq

11: return PDC

B

Figure 3. Algorithm of getBatPower(SOC,Preq,∆).

Figure 4shows selected input traces and simulation results to demonstrate the effect of the

applied battery control strategies. Ten single-person household traces are plotted at once to reveal the

stochasticity of the demand and how it leads to different battery utilization patterns. One can also see

that negative reserve is required very often, i.e., the chances that batteries can be controlled to provide

it during the allowed time interval are very high.

.

Figure 4.

Selected input traces and modelled results of 10 single-person households for the simulation

case

CAPPV =

1 kWp and

CAPB=

1 kWh. Each household is represented by a different colour and

the plot illustrates the stochasticity of household electricity consumptions.

Energies 2017,10, 976 8 of 17

4. Simulation Study

To perform a comprehensive proﬁtability evaluation of the residential battery energy storage,

we run a large number of simulations with different parameter conﬁgurations. Each simulation run

covers one entire year in one minute detail, i.e., the computational steps summarized in algorithm

control (cf. Figure 1) have to be executed 365

×

24

×

60

=

525, 600 times in each run. Since we intend

to consider a typical investment horizon of 20 years, we scale the PV module and battery capacity

to average values, i.e.,

CAPPV =CAPPV ·(

1

−(rPV ·LPV )/

2

)

and

CAPB=CAPB·(

1

+CAPEOL

B)/

2,

where

rPV =

0.5% is the annual (linear) performance loss of solar PV cells according to [

24

]. Thus,

for the NPV calculations the full detail of minute calculations is considered because the energy values

EB

,

ED

,

EHH

, and

EPV

are calculated using the method described above. However, we simulate one

year per conﬁguration, and then use the resulting energy values to estimate net present value (NPV)

for longer time periods.

To compute the ﬁgures of merit presented in the following, we collect several aggregated metrics

in each simulation run, including the total energy consumed by the household EHH , the energy from

the grid

ED

and the excess solar energy being fed into the grid,

EPV

. In addition, we compute the total

energy EBbeing provided by the battery.

4.1. Metrics

We evaluate the proﬁtability of different combinations of solar PV panel and battery sizes by

computing the NPVs of the corresponding investments. NPV is one amongst several ways to determine

the return of an investment (ROI). In contrast to the most basic ROI formula, i.e., the difference between

the gain from an investment and the cost of investment, and the cost of investment, the use of NPV

allows us to consider the time value of cash ﬂows in subsequent time periods, which is particularly

important for long investment horizons. In our case, signiﬁcant investments are due right at the

beginning of the considered investment time period, whereas revenues are constantly being generated

during a relatively long time (20 years). Thus, even relatively low interest rates may play a crucial role

in the decision. Furthermore, the NPV method provides a monetary value instead of merely measuring

the efﬁciency of an investment in terms of a percentage rate and is thus more informative.

The corresponding ﬁnancial parameters are summarized in Table 2. All price values exclude the

value added tax (VAT).

Table 2. Financial parameters.

Param. Values Param. Values

pPV 750 EUR/kWp [6]pPE 170 EUR/kWp [6]

pBOS 640 EUR/kWp [6]fEPC 0.08 [6]

fOP 0.015 [6]pF I 0.10 EUR/kWh [4]

pEL 0.30 EUR/kWh [4,25]iI NV 0.02 [4]

iVAT 0.19 TIN V 20 years

LPV 25 years [6]LPE 10 years

Based on these values, we calculate initial investments into the different system components based

on their respective sizes, in particular

CAPPV

and

CAPB

. The nominal capacity of the PV inverter and

grid-to-battery rectiﬁer used during reserve provision are derived from these values. Thus, the initial

investment costs for the solar modules (Equation (4)), the battery (Equation (5)), and the additional

power electronic equipment (Equation (6)) can be calculated as follows:

C0,PV = (1+iVAT)·(pPV +pPE +pBOS )·(1+ (( fEPC)−1−1)−1)·CAPPV , (4)

Energies 2017,10, 976 9 of 17

C0,B= (1+iVAT)·pB·CAPB, (5)

C0,PE+=IRE ·(1+iVAT)·pPE ·C APB. (6)

Since both the power electronic equipment and the battery modules usually have a shorter

lifetime than the assumed investment horizon of 20 years, we consider their replacement costs in

the NPV calculation. Furthermore, since we examine a limited investment horizon, we also consider

the residual value of assets at the end of the investment period. Otherwise, the replacement of

a system component shortly before the investment horizon ends would distort the NPV. Replacement

costs and residual values include VAT, but exclude additional (de)installation costs. We denote

the annual cash ﬂow resulting from solar PV system replacement costs and residual value in year

t

by

Ct,PV

. The cash ﬂow resulting from the investment in the battery modules is denoted by

Ct,B

, the cash ﬂow corresponding with the investment in additional power electronics required for

reserve provision by

Ct,PE+

. Replacement times are determined from the parameters

LPV

,

LPE

, and

LB=min nLcal

B,(Lcyc

B·CAPB)/EBo

. Residual values are determined proportionally, i.e., the residual

value of device

D

that costs

CD

EUR, has lifetime

LD

, and was installed or last replaced at time

trep

,

would be equal to

CD·(LD−tend +trep )/LD

. In addition to the equally distributed replacement costs,

we consider annual PV system maintenance costs according to Equation (7):

Ct,OP =fOP ·C0,PV. (7)

Positive cash ﬂows correspond to revenues resulting from the grid electricity cost savings and the

revenue from PV feed-in. In Germany, households are paid a ﬁxed feed-in tariff for the PV energy they

deliver to the grid. The tariff is determined at installation time and remains valid for 20 years. More

details can be found in the corresponding Act on the Development of Renewable Energy Sources [

26

].

Since households have to pay VAT on self-consumed electricity, the complete annual revenue can be

obtained using Equation (8):

Rt= (1+iVAT)·EPV ·pFI +pEL ·(EH H −ED). (8)

The net present value of an investment into a residential energy system that is composed

of PV module, battery, and power electronic equipment is given in Equation (9). If this value

is positive, the investment would be beneﬁcial compared to the non-investment alternative,

and disadvantageous otherwise:

NPV =C0,PV +C0,B+C0,PE++

TIN V

∑

t=1

Rt−Ct,OP −Ct,B−Ct,PE+

(1+iI NV )t. (9)

In addition to net present values, we compute the number of annual storage cycles according to

Equation (10) and the self-sufﬁciency rate according to Equation (11):

ASC =EB/C APB, (10)

SSR =100 ·(1−ED/EH H ). (11)

In the case of reserve provision, i.e.,

IRE =

1, the total energy drawn from the distribution grid

includes the reserve energy. Thus, we still deﬁne self-sufﬁciency as the percentage of used energy

that is produced on-site, even if charging during reserve provision is assumed to be free. We do not

assume that the aggregator pays the home owner extra for being able to control battery charging,

i.e., the NPVs computed for

IRE =

1 represent lower bounds. Market participation operations of the

aggregator, in particular the control of many batteries in concert with other energy resources forming a

virtual power plant, are beyond the scope of this paper [

27

]. Provided that the aggregator fulﬁlls the

Energies 2017,10, 976 10 of 17

requirements for participating in the reserve market, it should be able to obtain (positive) revenues

from the participation. However, based on historical price data, these revenues should be rather

limited today [

28

]. The deployment bids would have to be sufﬁciently low, such that the aggregator

would be instructed to deliver reserve energy relatively often, which, in turn, is required to recharge

the batteries overnight.

4.2. Setup

Table 3lists the sensitivity parameters considered in this study. All parameters except the battery

module price pBrequire independent simulations. To account for the stochasticity of household load

and its impact on self-consumption and battery operation, we perform ten independent simulations

for each parameter conﬁguration and report the mean metrics.

Table 3. Sensitivity parameters.

Param. Input Range Param. Input Range

CAPPV 1, 2, ..., 6 kWp CAPB0, 1, ..., 10 kWh

nHH 1, ..., 4 LOC Munich, Bremen

IRE 0, 1 pB800–100 EUR

4.3. Implementation

Our simulation and data processing procedures can be divided into three phases.

In the ﬁrst phase, we use appropriate models to generate the input data traces. We use the

original implementation of the household load model described in Section 3.1 to generate a large

number of year-long household consumption traces in one-minute resolution for different household

sizes. Using the weather data traces generated by Meteonorm, we compute corresponding PV power

generation using our own Python implementation of the models described in Section 3.2. The reserve

traces were obtained from [29].

In the second phase, we simulate the PV-battery systems over the course of a year in one minute

detail. Even at 1 kWp increments of PV capacity and 1 kWh increments of battery capacity, we still need

to perform 1056 simulation runs. Due to the high time granularity, the inclusion of many independent

household demand traces per run (cf. Figure 4), and the required computation in each simulation

step (cf. Section 3.5), each simulation run takes several hours on commodity hardware. Using two

powerful computer servers with 32 cores each, we were able to parallelize simulations such that the

full experimental setup was feasible in reasonable time.

In the ﬁnal phase, we calculate the ﬁgures of merit, in particular, net present value, annual

storage cycles, and self-sufﬁciency rates using Python scripts. This enables us to perform sensitivity

studies using different ﬁnancial parameters (cf. Table 2) without having to repeat the computationally

expensive simulations of the second phase.

4.4. Results

Figure 5contains plots of net present values and number of annual storage cycles for two-person

households in Munich. These plots only represent a fraction of the total results we obtained. The ﬁgures

reveal several important facts about the optimal sizing of system components and the inﬂuence of

battery prices and reserve provision, which are outlined in the following. At current battery prices

of 800 EUR/kWh and without reserve provision, it would be optimal to invest only into a system

consisting of a small PV module without a battery. Investing into energy storage would only reduce

proﬁtability in this case. Even at very low battery prices of 200 EUR/kWh, NPV peaks at rather small

storage capacities in the range of 1 kWh–4 kWh. The possibility of a free overnight recharge resulting

from negative reserve provision signiﬁcantly increases proﬁtability, despite approximately doubling

Energies 2017,10, 976 11 of 17

the number of annual storage cycles. In this case, the optimal energy storage capacity also increases,

but still peaks well below 10 kWh.

Across all cases, it becomes clear that relatively small PV module capacities (1 kWp–2 kWp) are

more beneﬁcial from a ﬁnancial perspective. Although additional energy storage capacity can, as

expected, signiﬁcantly increase the value of PV capacity, system conﬁgurations with more than 2 kWp

of PV capacity will remain less proﬁtable than smaller systems, irrespective of energy storage. Similar

trends are observed in other household scenarios and for Bremen.

In Figure 6, colors refer to different household sizes (1 person–4 persons), whereas solid and

dashed lines designate different metrics, including NPV, battery capacity and self-sufﬁciency rate.

Figure 6a–d show the best NPVs (solid lines) within the search space deﬁned in Table 3, as battery prices

decrease from 800 EUR/kWh to 100 EUR/kWh. The ﬁgures reveal at which price levels investments in

energy storage become proﬁtable in the studied cases. Taking Figures 6a as an example, for a 3-person

household, it is more proﬁtable to have a solar PV system without a battery when the battery price is

above 600 EUR/kWh. This can be observed as the optimal battery capacity (red dash line) stays at zero

when the value of the x-axis is beyond 600. When the battery price drops to between 300 EUR/kWh

and 600 EUR/kWh, a small battery of 1 kWh capacity is optimal (red dash line reaches 1). The optimal

battery capacity increases in steps up to 6 kWh as the battery price drops to 100 EUR/kWh. The

contextual factors also play a crucial role as NPVs increase substantially with larger household size

and higher solar availability.

Interestingly, irrespective of how low the prices per kWh of energy storage capacity become,

capacities greater than 6 kWh are never an optimal choice in the regular self-consumption use

case. If the batteries deliver negative reserve, however, optimal battery sizes approximately double.

Furthermore, although battery owners are not ﬁnancially compensated directly, reserve provision

signiﬁcantly increases proﬁtability.

0 1 2 3 4 5 6

Battery capacity [kWh]

−6000

−5000

−4000

−3000

−2000

−1000

0

1000

Net present value [EUR]

nHH = 2,pB= 800 EUR, IRE = 0, Munich

(a)

0 1 2 3 4 5 6

Battery capacity [kWh]

−3000

−2500

−2000

−1500

−1000

−500

0

500

1000

Net present value [EUR]

nHH = 2,pB= 200 EUR, IRE = 0, Munich

(b)

0 2 4 6 8 10

Battery capacity [kWh]

−10000

−8000

−6000

−4000

−2000

0

2000

Net present value [EUR]

nHH = 2,pB= 800 EUR, IRE = 1, Munich

(c)

0 2 4 6 8 10

Battery capacity [kWh]

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

Net present value [EUR]

nHH = 2,pB= 200 EUR, IRE = 1, Munich

(d)

1 2 3 4 5 6

Battery capacity [kWh]

50

100

150

200

250

300

350

400

450

Annual number of storage cycles

nHH = 2,IRE = 0, Munich

(e)

1 2 3 4 5 6 7 8 9 10

Battery capacity [kWh]

200

300

400

500

600

700

800

Annual number of storage cycles

nHH = 2,IRE = 1, Munich

CAPP V = 1 kWp

CAPP V = 2 kWp

CAPP V = 3 kWp

CAPP V = 4 kWp

CAPP V = 5 kWp

CAPP V = 6 kWp

(f)

Figure 5.

NPVs (

a

–

d

) and annual number of battery cycles (

e

,

f

) for different solar PV and battery capacities.

Figure 6e,f show self-sufﬁciency rates for the cases without and with reserve provision (solid and

dashed lines, respectively). Whereas the use of additional energy storage capacity at decreasing

Energies 2017,10, 976 12 of 17

battery prices leads to higher self-sufﬁciency, the opposite is true if batteries deliver negative reserve.

The reason is that the amount of energy being charged into the battery in response to reserve provision

starts dwarﬁng the amount of solar energy being buffered. A self-sufﬁciency rate of 0% indicates that

it is ﬁnancially optimal not to invest in either solar PV modules or battery storage.

100200300400500600700800

Battery capacity price [EUR/kWh]

0

500

1000

1500

2000

2500

Best net present value [EUR]

IRE = 0, Munich

nHH = 1

nHH = 2

nHH = 3

nHH = 4

0

1

2

3

4

5

6

Optimal battery capacity [kWh]

(a)

100200300400500600700800

Battery capacity price [EUR/kWh]

0

100

200

300

400

500

600

700

Best net present value [EUR]

IRE = 0, Bremen

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Optimal battery capacity [kWh]

(b)

100200300400500600700800

Battery capacity price [EUR/kWh]

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Best net present value [EUR]

IRE = 1, Munich

1

2

3

4

5

6

7

8

9

10

Optimal battery capacity [kWh]

(c)

100200300400500600700800

Battery capacity price [EUR/kWh]

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Best net present value [EUR]

IRE = 1, Bremen

0

2

4

6

8

10

Optimal battery capacity [kWh]

(d)

100200300400500600700800

Battery capacity price [EUR/kWh]

0

10

20

30

40

50

60

70

Self-sufﬁciency rate [perc.]

Munich

(e)

100200300400500600700800

Battery capacity price [EUR/kWh]

0

10

20

30

40

50

60

70

Self-sufﬁciency rate [perc.]

Bremen

nHH = 1,IRE = 0

nHH = 1,IRE = 1

nHH = 2,IRE = 0

nHH = 2,IRE = 1

nHH = 3,IRE = 0

nHH = 3,IRE = 1

nHH = 4,IRE = 0

nHH = 4,IRE = 1

(f)

Figure 6.

Best NPVs (

a

–

d

) and corresponding self-sufﬁciency rates (

e

,

f

) for different household sizes

and locations.

5. Discussion

Using a high resolution stochastic electricity demand model, we are able to quantify the effects of

household size and solar availability on the proﬁtability of PV-battery systems. Although these

contextual factors have proven to be important in our ﬁndings, they were largely neglected in

previous studies.

Furthermore, we explore the impact of providing negative reserve using the batteries (cf. Section 1),

which is a timely topic in Europe. The way the aggregator controls the batteries to provide reserve in

our use case is simple but realistic. The optimization problem of scheduling large numbers of energy

storage devices in accordance with German market rules has been studied in another publication [

27

],

and would exceed the scope of this work. In this paper, we thus make the simplifying assumption that

batteries can be charged at nominal charging power as long as there is actual demand for negative

reserve (usually many MWs). We used a real one year long trace of negative reserve to make sure that

our results could actually materialize in practice. Given the case we consider, i.e., reserve provision

during a 12 h long time interval with a maximum charging time of 1 h, batteries would almost certainly

be able to recharge every night. The required switching and metering infrastructure on the customer

side would be basic and inexpensive.

Since the power output of lithium-ion batteries can change almost instantaneously based on

the attached load or power supply, considering the ramp rate of the modeled energy storage is not

required here. However, it could be an issue if other types of energy storage are modeled or if the time

granularity of our analysis were much higher. Moreover, considering different reserve provision use

cases, including ones involving the discharge of batteries, could be valuable future work.

Energies 2017,10, 976 13 of 17

The data and assumptions adopted in this study have been carefully selected to cover

a representative range of values in reality in order to ensure generalizable results. Furthermore,

by leveraging an advanced method for generating representative household load traces [

14

], we are

able to take stochastic effects into account. Whereas these features distinguish our study from previous

work, they also lead to high computational complexity, which we manage by deploying our code on

large servers and taking advantage of full parallelization. The results clearly show that the stochasticity

of household consumption patterns are important to consider as they have signiﬁcant impacts on the

NPVs, and in turn, the optimal system conﬁgurations. In this paper, we have not explicitly shown the

impact of data granularity on the matching of solar PV energy production and household demand

because we feel that this is beyond the scope of this paper. The general importance of data granularity

in the context of our study has already been shown in the literature [7].

We do not consider learning curves in this work, which form an integral part of related works [

6

].

Learning curves represent the impact of learning over time, e.g., on costs or prices. In the context of

our study, the prices of residential energy generation and storage systems may decrease over time, e.g.,

due to economies of scale and increasing competition. Instead, we implicitly assume that PV panel and

power electronics prices will not dramatically decrease in the future. Energy storage price, however, is

treated as a sensitivity parameter in our study without making explicit predictions about when which

price level will be reached, since we believe that such predictions are highly speculative at this time.

Our sensitivity study can also be used to assess the impact of subsidies for residential energy

storage. A recently prolonged subsidy program supporting residential battery energy storage

in Germany promises a payment

sB

depending on the size of the solar PV module

CAPPV

and

battery module initial investment cost

C0,B

, which can be calculated according to

sB=CAPPV ·

min n0.25·C0,B

CA PPV , 500o

EUR [

30

]. Thus, assuming the current battery cost of 800 EUR/kWh, the subsidy

for a 1 kWp PV module with a 1 kWh battery would be approximately 200 EUR. Considering the

results shown in Figure 6a,b, this subsidy would be sufﬁcient to make battery investments for larger

households (three and four persons) at locations with high irradiance levels, like Munich, proﬁtable.

However, our results show that it is insufﬁcient to foster storage adoption in other cases.

Time-of-use (TOU) tariffs, i.e., contracts that result in different electricity prices based on when

electricity is used, are common in some countries, in particular in the US [

31

]. Typically, TOU tariffs lead

to higher charges at times of peak demand, i.e., at mid-day, and lower charges otherwise. They require

“smart" meters capable of time-based demand measurement and vary in terms of speciﬁed pricing

periods and levels. In Germany, smart meters have not yet been rolled out at the household level,

thus German utilities do not offer TOU tariffs so far. Since residential energy storage reduces electricity

usage during times of potential peak demand (cf. Figure 4), we expect TOU pricing to have a positive

effect on its proﬁtability. However, since TOU can be very different based on the demand shifting

goal of the utility and a signiﬁcant study is therefore beyond the scope of this paper, we recommend

a self-contained follow-up study that speciﬁcally investigates this issue.

We can conﬁrm that ﬁnancially optimal system conﬁgurations for self-consumption scenarios

imply relatively small solar PV and battery capacities. Even if battery prices halved from today’s

800 EUR/kWh price level to 400 EUR/kWh, optimal battery sizes would still range at approximately

1 kWh (cf. Figure 6a), although much larger residential batteries are offered already today

(e.g., Tesla’s Powerwall has a storage capacity of 6.4 kWh [

2

]). This result generally corresponds

with the ﬁndings of previous studies, in particular [

5

]. Furthermore, we ﬁnd that battery module prices

would have to further decrease from today’s levels to make ﬁrst investments in residential energy

storage proﬁtable. For instance, at locations near Munich, initial investments into energy storage would

become proﬁtable at prices lower than 550 EUR/kWh for all household sizes (cf. Figure 6a), whereas

prices would have to decrease below 200 EUR/kWh to lead to proﬁtable investments near Bremen

(cf. Figure 6b). However, as our detailed results have shown, the exact “break even” battery prices

and the optimal system conﬁgurations vary substantially depending on factors such as household

Energies 2017,10, 976 14 of 17

size and solar availability. We thus believe that it would be dubious to quote such numbers without

considering these contextual features.

The weather and time use data used as input to the household load model assume siting in Central

Europe, which determines the results of the second phase of the simulation procedure (cf. Section 4.3).

Furthermore, several ﬁnancial parameters (in particular

pFI

and

pEL

) are Germany speciﬁc. Therefore,

quantitative results assuming other climate zones and jurisdictions would likely differ from the results

presented in Section 4.4. The method itself, including the tools for generating the required input data,

are replicable to any location worldwide.

6. Conclusions

In this paper, we have described a method to investigate the proﬁtability of residential battery

energy storage at the necessary detail, i.e., modeling stochastic demand proﬁles at high time resolution

and considering the monetary trade-offs resulting from different system conﬁgurations. In addition to

the standard solar energy buffering use case of residential batteries, we investigate the intermittent

provision of negative reserve, which turns out to be attractive for home owners, even without direct

ﬁnancial compensation.

In summary, our results conﬁrm the potential of residential battery storage highlighted in previous

studies, but at the same time indicate the need for further battery cost reductions to achieve proﬁtability

under all considered circumstances, e.g., household sizes and locations.

Acknowledgments:

This work was supported by funds from the Alexander von Humboldt Foundation and the

Energy Valley Bavaria Program of the Bavarian State Ministry for Education, Science and the Arts. Its publication

was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the

framework of the Open Access Publishing Program.

Author Contributions:

Christoph Goebel and Vicky Cheng conceived the study and performed the simulation

work. Christoph Goebel wrote the ﬁrst draft of the paper. Vicky Cheng and Hans-Arno Jacobsen reviewed the

paper. Christoph Goebel and Vicky Cheng revised the paper for ﬁnal submission.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

AcSolar PV module area.

AC Alternating current.

ASC Number of annual storage cycles.

βSolar PV module temperature coefﬁcient of power.

C0,BInitial investment cost of battery.

C0,PE+Initial investment cost of additional power electronic equipment.

C0,PV Initial investment cost of solar PV module.

Ct,BAnnual cashﬂow resulting from battery.

Ct,OP Annual solar PV system maintenance cost.

Ct,PE+Annual cashﬂow resulting from additional power electronic equipment.

Ct,PV Annual cashﬂow resulting from solar PV panel.

CAPBNominal energy capacity of battery.

CAPBAverage battery energy capacity during lifetime.

CAPEO L

BEnd of life battery capacity.

CAPPV Peak power capacity of solar PV module.

CAPPV Average peak power capacity of solar PV module during lifetime.

DC Direct current.

∆SDuration of control time interval.

∆RDuration of remaining reserve provision for current reserve interval.

∆RMaximum duration of reserve provisioning per reserve interval.

Energies 2017,10, 976 15 of 17

EBEnergy provided by battery.

EDGrid energy consumed.

EHH Energy consumed by the household.

EPV Excess solar energy fed into the grid.

ηBBattery efﬁciency (charging and discharging).

ηPV Solar PV module efﬁciency.

ηre f Solar PV module efﬁciency.

fEPC Fraction of engineering, procurement, and construction (EPC) cost.

fOP Fraction of annual PV system operations cost.

GSolar irradiance on the PV array.

Gno Incident solar irradiance when normal operating temperature is measured.

Gre f Incident solar irradiance when ηre f is measured.

γSolar PV module solar irradiance coefﬁcient of power.

iIN V Annual interest rate.

iVAT Value added tax (VAT) rate.

IRE Variable indicating whether battery provides reserve.

Lcyc

BCycle life of battery.

Lcal

BCalendar life of battery.

LPE Lifetime of power electronic equipment.

LPV Lifetime of solar panel.

LOC Household location.

nHH Number of household members.

NPV Net present value.

pBBattery price (excl. VAT).

pBOS PV balance of system (BOS) cost.

pEL End-consumer electricity price.

pFI Solar PV feed-in tariff.

pPE Power electronics cost.

pPV Cost of PV module.

PDC

BDC power of battery.

PAC

DAC power to be transferred to or from the grid.

PDC

GDC power generated by household (solar PV plus battery).

Pneg

GR (t)Negative grid reserve required in time interval [t,t+∆S].

Pneg

GR Reserve provision threshold.

PAC

HH AC power demand of household.

PlActual power load of power electronic component.

Pmax Maximum power the battery is able to provide.

PDC

PV Solar PV DC power generation.

PrRated power capacity of power electronic component.

PDC

RReserve power delivered by battery.

Preq Maximum power required from battery.

Ptheo Theoretical power the battery is able to provide.

TcSolar PV module temperature.

Tc,ref PV cell temperature when ηre f is measured.

Tno Normal operating PV cell temperature.

TaAmbient temperature.

Ta,no Ambient air temperature when normal operating temperature is measured.

TIN V Investment horizon.

τα Transmittance and absorbance product.

rcBattery charging C-rate.

rdBattery discharging C-rate.

rPV Solar PV peak power capacity deterioration rate.

RtTotal annual household revenue.

Energies 2017,10, 976 16 of 17

SOCmin Minimum battery state of charge.

SOCmax Maximum battery state of charge.

SOC Actual battery state of charge.

SSR Self-sufﬁciency rate.

tTime index.

tRStart of reserve provisioning time interval.

tREnd of reserve provisioning time interval.

tend Last period in investment horizon.

trep Last replacement time of device.

VwWind speed.

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