# Influence of Efficiency, Aging and Charging Strategy on the Economic Viability and Dimensioning of Photovoltaic Home Storage Systems

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

**Table 1.**Literature review on the simulation and sizing of PV home storage systems with a special focus on the consideration of efficiencies, aging and charging strategy.

Analysis of | ||||||||
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Author | Technology | Varied Input Parameters | Inverter and Battery Efficiency | Battery Aging/PV Degradation | Different Charging Strategies | Evaluation Parameters | Method | Load-Profile |

Braun et al., 2009 [9] | LIB | Storage size, electricity tariff, technology cost, FIT degression rate | cs | c (Batt) | simple | IRR, payback period | Simulation & selection | Sim |

Li et al., 2009 [57] | LAB, Fuel cell | PV system size, technology cost, component efficiency | cs | - | complex (two batteries) | Cost of electricity | Simulation & selection | |

Mulder et al., 2010 | LIB, LAB | PV system and storage size | - | - | simple | Energy sent to the grid, covered peak demand | Simulation & selection | Meas |

Battke et al., 2013 [34] | LAB, LIB, RFB, NAS | Storage cost, storage roundtrip efficiency, life time and cycle life | cs | - | complex (two batteries) | Cost of electricity | Monte Carlo Simulation | Sim (Load profile of the battery) |

Mulder et al., 2013 [10] | LIB, LAB | PV system and storage size, electricity tariff, year of investment, technology cost (PV and Batt), Incentives and electricity sale price | c | - | simple | NPV, battery throughput cost | Simulation & selection | Meas |

Ru et al., 2013 [58] | LAB | Storage size | cs | c (Batt) | simple | Cost of electricity | Exact optimization problem | Constant |

Bruch and Müller (2014) [8] | LIB, LAB, RFB | Battery technology, storage size, electricity tariff, user behavior | cs | cs (Batt) | simple | Profit (after tax), Return | Simulation & selection | Meas |

Hoppmann et al., 2014 [7] | LAB | PV system and storage size, technology cost, excess household to wholesale market, electricity tariff, electricity sale price, nominal discount rate, O&M cost, further economic parameters and learning curves | cs | cs (Batt) | simple | NPV | Simulation & selection | Slp |

Waffenschimdt (2014) [36] | N/A | PV system and storage size, load profile, charging strategy | cs | - | simple, reduction of PV generation peaks (v) | SCR, cut off PV energy | Simulation & selection | Slp |

Weniger et al., 2014 [27] | LIB | PV system and storage size, technology cost (PV and Batt) | cs | cs (Batt + PV) | simple | SSR, SCR, cost of electricity | Simulation & selection | Slp |

Yang et al., 2014 [33] | LIB | Storage size, PV penetration, | cs | c (Batt) | complex (voltage regulation and peak load shaving) | Annual cost | Simulation & selection | Meas |

Meunier et al., 2015 [39] | LIB | Storage size, load profiles | cs | - | simple | ROI, SCR, Cost after 26 years, investment roundtrips, | Simulation & selection | Slp |

Moshövel et al., 2015 [11] | LIB | PV system and storage size, Battery price, electricity tariff, FIT, household | - | cs | simple | NPV, SCR | Simulation & selection | Sim |

Naumann et al., 2015 [12] | LIB | Storage size, Batt cost, electricity tariff, aging | c | cs + v (Batt) | simple | ROI | Simulation & selection | Slp (H0) |

Ried et al., 2015 [35] | LIB | Battery inverter size | cs | - | simple | NPV | Simulation & selection | Meas, Sim |

Beck et al., 2016 [41] | LIB | PV system, storage and battery inverter size | cs | - | simple | Cost of electricity, SSR, SCR | MILP | Meas |

Chiaroni et al., 2016 [32] | LIB, LAB | Technology cost (PV and Batt), SCR, SSR, dept capital | cs | - | - | NPV | Simulation & selection | - |

Cuchiella et al., 2016 [24] | LAB | PV system and storage size, technology cost (PV and Batt), battery lifetime, electricity tariff, electricity sale tariff, level of insolation, tax deduction, shares of SCR, increase of SC through a battery | - | c (PV), cs + v (Batt) | simple | NPV, DCF | Simulation & selection | - |

Du et al., 2016 [52] | LIB | Storage size | c (Batt) | c (Batt) | simple | Time to EOL | Simulation | Meas |

Magnor and Sauer (2016) [26] | LIB | Technology cost (Batt) | c (power electronics) | c (Batt) | simple | LCOE | Genetic algorithm | Slp |

Nyholm et al., 2016 [37] | LIB | PV system and storage size, Load profiles | cs | cs (PV) | simple | SSR, SCR | Simulation & selection | Meas, Sim |

Quoilin et al., 2016 [30] | LIB | PV system and storage size, max chg/dischg power, storage cost | cs | cs (Batt) | simple | LCOE, SSR, SCR | Simulation & selection | Meas |

Truong et al., 2016 [14] | LIB | Battery aging, electricity tariff, household size, coupling of storage system, subsidies | c | c + v (Batt) | simple | NPV, ROI | Simulation & selection | Meas |

Weniger et al., 2016 [42] | LIB | PV system, storage and inverter size | c | - | simple | charge and discharge energy | Simulation & selection | Meas, Sim |

Bertsch et al., 2017 [13] | LIB | PV system and storage size, load profiles, cycle stability, electricity tariff, FIT, technology cost (Batt) | cs | cs + v (Batt) | simple | IRR | Simulation & selection | Meas, Sim |

Goebel et al., 2017 [48] | LIB | PV system and storage size, location, storage cost, household size, battery providing reserve | c | cs (Batt) | simple, grid charging | NPV | Simulation & selection | Sim |

Sani Hassan et al., 2017 [25] | LIB | FIT | cs | - | simple | Annual cost of electricity | MILP | Meas |

Sharma et al., 2017 [23] | LIB | PV system and storage size, electricity tariff | cs | - | simple | Total life cycle cost | One dimensional optimization | Meas and scaled |

Wu et al., 2017 [59] | N/A | Storage size, charging strategy | cs | - | simple | Cost of electricity | Exact (convex optimization) | Meas |

Astaneh et al., 2018 [49] | LIB | PV system and storage size, battery price | c | c (Batt) | strategy to minimize aging | LCOE, NPV, met load %, days of autonomy | Simulation & selection | Meas |

de la Torre et al., 2018 [50] | LIB | Storage size | cs | c (Batt) | simple | Storage cost | Simulation/optimization-based | Meas |

Dietrich and Weber 2018 [16] | LIB | PV system, storage and battery inverter size, investment cost, electricity demand, interest rate, electricity tariff, technology cost (PV and Batt), FIT, VAT | cs + cs (standby) | c (Batt + PV) | simple | SSR, SCR, NPV | MILP | Sim |

Schopfer et al., 2018 [29] | LIB | Load profiles, technology cost (Batt and PV) | cs | cs (Batt) | simple | NPV | Machine learning, simulation | Meas |

Tervo et al., 2018 [31] | LIB | PV system and storage size, location, technology cost (PV and Batt), interest rate, ITC, efficiency, | cs + v | cs + v (Batt + PV) | simple | LCOE (System, Batt, PV) | Simulation & selection | Sim |

Boeckl and Kienberger (2019) [38] | LIB | PV system and storage size, household load | c | - | simple | SSR, PV system and storage size | Simulation & selection | Sim |

Comello et al., 2019 [43] | LIB | Storage and battery inverter size, location; year of installation | cs | - | simple | LCOE | Simulation & selection | Sim |

Heine et al., 2019 [51] | LIB | Storage size, location, storage cost | cs | c (Batt) | simple | NPV | Simulation & selection | Sim |

Koskela et al., 2019 [19] | LIB | PV system and storage size, investment cost, electricity tariff | cs | - | simple | Annual profit, annual cost savings, IRR | Simulation & selection | Meas |

Li et al., 2019 [28] | LAB | PV system and storage size, location | - | - | simple | Maximum demand, Electricity cost reduction | Genetic algorithm | Meas |

Sandelic et al., 2019 [53] | LIB | - | cs | c (Batt + inverter) | simple | lifetime (Batt + inverter) | Simulation | Meas |

Sharma et al., 2019 [22] | LIB | PV system and storage size, electricity tariff, FIT | cs | - | simple | Annual energy cost | One dimensional optimization | Meas and scaled |

Tang et al., 2019 [18] | LIB | Storage size, load data, electricity tariff | cs | - | simple | NPV | Simulation & selection | Meas |

Alavi et al., 2020 [21] | LIB | PV system and storage size, electricity tariff, FIT, storage cost | cs | c (Batt cyclic aging) | simple | Annual profit loss | Simulation & selection | Meas |

Beltran et al., 2020 [54] | LIB | PV system and storage size, household, irradiance, location, cell chemistry | cs | c (Batt) | predictive control (energy arbitrage, peak shaving) | Lifetime of the battery, calendar and cyclic aging | MILP, simulation | Meas and Sim |

Gagliano et al., 2020 [60] | LIB | PV system and storage size, household consumption | cs | - | simple | NPV, IRR, payback period, SSR, SCR | Simulation & selection | Sim |

Pena-Bello et al., 2020 [15] | LIB | Electricity consumption, location including electricity tariff | cs | cs (Batt) | simple, demand load-shifting, avoidance of PV curtailment, demand peak shaving, (individually and jointly) | NPV, SCR, peak shaved | Simulation | Meas |

Withana et al., 2020 [20] | LIB | Storage and inverter size, electricity tariff | - | - | simple | NPV | Simulation & selection | Meas (average monthly load demand) |

Zhang et al., 2020 [17] | LIB | PV system and storage size, electricity tariff | cs | cs (Batt) | simple | SSR, SCR, LCOE, payback time | Simulation & selection | Artificial synthesized load |

Ayuso et al., 2021 [40] | LIB | PV system and storage size, LIB cell type | cs | c (Batt) | complex (high and low grid tariffs) | Annual savings, net payback, NPV | MILP | Maes and Sim |

Current Paper | LIB | c + v | c + v (Batt + PV) | strategy to minimize aging, simple (v) | Cost of electricity, Annual energy cost, SSR | Simulation & selection | Maes and Sim |

## 3. Input Data and Methods

#### 3.1. Input Data

#### 3.1.1. PV and Load Data

#### 3.1.2. Battery and Inverter Efficiency and Standby Consumption

_{BAT2AC}for battery discharging, η

_{AC2BAT}for battery charging and η

_{PV2AC}for the conversion of the DC PV power into AC power.

_{BAT,RT}) for Li-ion home storage systems are usually between 90% and 100%. Many systems are in the range between 95% and 97% [44,63]. For this reason, a value of 96% is selected for the reference case. The efficiency during (η

_{BAT}) charging or discharging can be determined with Equation (1). By simple measurement, only the roundtrip efficiency of storage systems can be determined. How much of this is accounted for by charging and how much by discharging is difficult to determine. For this reason, the same efficiency was assumed for both directions in the present work.

_{min}and SOC

_{max}). The 8 W peripheral consumption (P

_{AC,periph}) represents a high value of the shown storage systems at minimum SOC and a low value at maximum SOC. Since peripheral consumption always occurs, 8 W of P

_{AC,periph}results in 70.3 kWh of P

_{AC,periph}per year. According to Munzke et al. [44], the P

_{AC,periph}for home storage systems per year is between 13 kWh and 144 kWh. The 8 W assumed here therefore represent a good average value. The DC standby consumption (P

_{DC,BAT,stby,dischg}) of the 4 storage systems ranges from 0 W to 19 W, regardless of whether the battery is full or empty. For the references case of the simulation, 8 W are selected. The AC standby consumption (P

_{AC,BAT,stby}) is significantly higher in some systems with a fully charged battery than with a completely discharged battery. In other systems, it is about the same in both cases. Overall, the standby consumption lies between 0 W and 40 W, depending on the system and state. For the simulation, 12 W is selected for the reference case. This leads to standby losses of 60 kWh per year on average in the simulation, which is comparable to the values in Munzke et al. [44].

#### 3.1.3. Chosen Parameters for the Battery, the PV System and the Power Electronic Components

#### 3.1.4. Cost

#### 3.2. Methods

#### 3.2.1. Simulation of Power Flows Including Different Charging Strategies

_{DC,PV}) is used for the simulation (see Figure 3). These are converted to AC power (P

_{AC,PV}) using Equation (4). By subtracting the PV power from the load including peripheral consumption (P

_{Load}), potential charging (P

_{AC,BAT,chg,pot}) and discharging (P

_{AC,BAT,dischg,pot}) powers can be determined (see Equations (5) and (6)). By Equations (7) and (8), these can also be converted to potential battery power (P

_{DC,BAT,chg,pot}and P

_{DC,BAT,dischg,pot}). It should be noted that the DC power is limited by the selected inverter power (rated power). This applies to both charging and discharging.

_{BAT}).

_{stby,DC,BAT}) may occur when the battery is fully charged or discharged. This is the case when a part of the standby consumption of the inverter is covered by the battery and the battery is slightly discharged. To be able to simulate this, the variable E

_{BAT,real}is introduced to avoid immediate recharging of the battery. It refers to the actual state of charge of the battery. In case of a fully charged battery, the following applies. If E

_{BAT,real}has fallen below a certain limit (97.5% SOC) or the battery is discharged in the next time step, E

_{BAT}is set equal to E

_{BAT,real}. The limit is determined by a percentage (Pct

_{rechg}) of the actual total battery capacity left (E

_{CapaBAT}(y)). In the first case the battery is charged again, in the second case the energy discharged from the battery due to standby consumption is taken into account in further operation. To simplify the calculation of the actual charged and discharged power, the additional variable BAT

_{chg,dischg}is introduced. Whenever E

_{BAT}is determined with the help of E

_{BAT,real}, it is set equal to 1. The battery is only discharged to 5% SOC in the simulation during regular operation. This means that the battery is completely discharged at a SOC of 5%. However, it can be further discharged by DC standby losses. As soon as 0% SOC is reached due to standby consumption, the battery is charged from the grid with 500 W (P

_{DC,BAT,rechg}) until a SOC of 5% is reached.

**Figure 5.**Calculation of the energy stored in the battery and the DC standby consumption—T stands for true and F for false.

_{chg},

_{d}is introduced. This factor is determined for each day (t

_{2}) and depends on the ratio of the battery capacity and its charging efficiency to the PV energy that is not directly consumed by the load (see Equation (9)). When calculating the power charged in the battery, the potential energy available is multiplied by fact

_{chg,d}and thus reduced. Thus, the battery is charged more slowly over the course of the day. This can lead to the battery not being completely full at the end of the day although sufficient energy is available during the day. To prevent this, all days are identified on which fact

_{chg,d}is smaller than 1 and the SOC of the battery does not reach 100%. fact

_{chg,d}is then increased by 0.01 for all days affected by this and the calculation from Figure 5 is repeated. This is done until the battery is fully charged on all days where sufficient energy is available. If the simple charging strategy is used, fact

_{chg,d}in Figure 5 is set to 1.

_{DC,BAT,chg}) and discharge (P

_{DC,BAT,dischg}) power can be determined using BAT

_{chg,dischg}, E

_{BAT}and E

_{BAT,real}. If BAT

_{chg,dischg}is 1, the actual charging power is determined from the difference between E

_{BAT}and E

_{BAT,real}of the previous time step (see Equations (10) and (11)). It must be taken into account that there are situations in which power is required from the battery, but it is already empty and must be recharged from the grid. This must be considered when calculating the actual discharge power. With Equations (12)–(14), the AC charge (P

_{AC,BAT,chg}) and discharge (P

_{AC,BAT,dischg}) power as well as the AC recharge power (P

_{AC,BAT,rechg}) of the battery can be determined.

**Figure 6.**Calculation of the grid consumption and the grid feed-in—T stands for true and F for false.

#### 3.2.2. Aging Model

_{Cell,Cal}) in years per SOC. This is shown as a dashed line in Figure 7, right panel, as a function of the SOC. The following functional relationship between life time in years and SOC can be derived by fitting from the measurement (see Equation (17)).

_{Cal}(y)) (see Equation (18)). A

_{Cal}(y) represents the percentage that the battery has degraded due to calendar aging in a given year. 100% degradation corresponds to EOL of the battery with a SOH of 80%.

_{Cycl}represents a factor that can be 1 or 0.5, depending on whether the corresponding cycle is a half or full cycle and Cycl the number of cycles per year. A

_{Cycl}(y) represents the percentage that the battery has degraded due to cyclic aging in a given year. 100% degradation corresponds to EOL of the battery with a SOH of 80%. The total aging results from the sum of the calendar aging and the cyclic aging (see Equation (21)). It is important that the calendar aging was previously subtracted from the cyclic aging. As mentioned, a value of 80% SOH is considered as the end of life criterion. The SOH at the end of a year can be calculated with Equation (22).

#### 3.2.3. Economic Evaluation

_{N,C,X}) for the PV system, the power electronics and the battery can be determined using Equations (23)–(27). In this context, X as a variable stands for one of the mentioned components. For the PV system, a lifetime of 20 years is assumed, which corresponds to the usual depreciation period for PV systems. Since the simulation period T also corresponds to 20 years, no calculation of replacement investments and residual value is necessary. For the inverters it is assumed that they have to be replaced after 10 years. Thus, the cash values (A

_{i}

_{,PE}) of the replacement investments can be determined with Equation (24), where i represents the number of the replacement and T

_{N}the number of years of the depreciation period. The replacement of the battery depends on the aging of the battery. The degree of aging depends on various factors, such as the battery size, the load profile and the PV system. How to determine the aging of the battery and when to replace it is described in Section 3.2.2. Equation (25) can be used to calculate the cash value of the replacement investment of the battery, and Equation (26) to calculate the residual value. y

_{n}

_{,BAT,repl}represents the year in which the respective investment takes place. The residual value of the battery depends on the remaining capacity (E

_{Capa,BAT}(y)) at the end of the simulation. r in the equations stands for the respective price change factor. Interest rates can be taken into account in the calculation via the interest factor q. Although the load and PV curves in the simulation remain the same each year, the amount of electricity fed into (E

_{to,Grid}) or taken from the grid (E

_{from,Grid}) changes over the years. This is due to battery aging and degradation of the PV system. The annuity of the annual remuneration for electricity fed into the grid (A

_{N,R}) and the electricity costs of the electricity purchased from the grid (A

_{N,EC}) can be determined using Equations (28) and (29). Where y stands for the respective year in which the costs or the remuneration occur. In the given case, the annuity of the total annual payments (A

_{N}) is the difference between A

_{N,R}and the sum of the capital-related annuities and A

_{N,EC}(see Equation (30). To calculate the cost per kWh (C

_{E}

_{,total}), the annuity is divided by the energy consumed per year (E

_{Load,y}) (see Equation (31)). The costs (C

_{E}

_{,total,ref}) can be compared with reference costs that would arise if no plant is built and the entire electricity demand would need to be covered by the grid (see Equation (32)).

#### 3.3. Parameters Studied

## 4. Results

#### 4.1. System Design Single Family Household

#### 4.1.1. Effects of the Size of the PV Field, the Battery and the Battery Inverter

**Figure 9.**Electricity costs as a function of self-sufficiency, the size of the PV system is shown in color. The simulation is based on an annual electricity price increase of 1%.

**Figure 10.**Electricity costs as a function of self-sufficiency, the capacity of the battery is shown in color. The simulation is based on an annual electricity price increase of 1%.

**Figure 11.**Electricity costs as a function of self-sufficiency, the size of the battery inverter is shown in color. The simulation is based on an annual electricity price increase of 1%.

#### 4.1.2. Effects of Electricity Price Increase and Feed-In Tariff Decrease

**Figure 12.**Electricity costs as a function of self-sufficiency. The size of the PV system is shown in color in the

**left**panel and the capacity of the battery is shown in color in the

**right**panel. The simulation is based on an annual electricity price increase of 0%.

**Figure 13.**Electricity costs as a function of self-sufficiency. The size of the PV system is shown in color

**left**panel and the capacity of the battery is shown in color in the

**right**panel. The simulation is based on an annual electricity price increase of 1% and the expected feed-in tariff in June 2021 (7.47 cent/kWh for PV systems up to 10 kWp and 7.25 cent/kWh for PV systems up to 40 kWp).

#### 4.2. Sensitivity Analysis of Technical Parameters on Electricity Costs of the System Operator

#### 4.2.1. Effects of the Efficiency of Power Electronics

**Figure 16.**Change in electricity costs per kWh of the reference case to the 3 other simulations with the battery inverter efficiency curve of system C, A and E as a function of the battery system size (

**left**panel). Electricity costs per kWh as a function of the PV system size (

**right**panel). Simulations were performed with the efficiency curves of systems A and D with an average pathway efficiency of 95.51 and 96.29% for the path PV2AC.

#### 4.2.2. Effects of the Efficiency of the Battery

#### 4.2.3. Effects of Standby Consumption

**Figure 21.**Electricity costs per kWh as a function of battery system size (

**left**panel) and PV system size (

**right**panel). The results for different levels of DC standby consumption are shown in color. The reference case with an DC standby consumption of 8 W is shown in green.

**Figure 22.**Electricity costs per kWh as a function of battery system size (

**left**panel) and PV system size (

**right**panel). The results for different levels of peripheral consumption are shown in color. The reference case with a peripheral consumption of 8 W is shown in green.

#### 4.2.4. Effects of Battery Aging and PV Degradation

**Figure 23.**Electricity costs per kWh as a function of battery system size (

**left**panel) and the change in cost as a function of battery size (

**right**panel). The results for different degrees of calendrical aging are shown in color. The reference case is shown in green.

**Figure 24.**Electricity costs per kWh as a function of battery system size (

**left**panel) and the change in cost as a function of battery size (

**right**panel). The results for different degrees of cyclic aging are shown in color. The reference case is shown in green.

**Figure 25.**Electricity costs as a function of the PV System size (

**left**panel) and the change in cost as a function of PV system size (

**right**panel). Shown in color are different levels of degradation of the PV system.

#### 4.2.5. Effects of an Intelligent Charging Strategy

**Figure 26.**Electricity costs per kWh as a function of battery system size (

**left**panel) and the change in cost as a function of battery size (

**right**panel). Shown are the results for a and an intelligent charging strategy to prevent calendar aging. The reference case is shown in green.

## 5. Discussion

**Figure 29.**Impact on electricity costs due to the change in technical parameters. The

**left**panel shows the PV system battery combination of 15 kWp and 10 kWh, the

**right**one of 15 kWp and 5 kWh. While the upper figures show all data points, the lower figures show only parts of the data in order to better illustrate even small changes in relation to the reference case.

**Figure 30.**Impact on electricity costs due to the change in technical parameters. The

**left**panel shows the PV system battery combination of 10 kWp and 10 kWh, the

**right**one of 10 kWp and 5 kWh. While the upper figures show all data points, the lower figures show only parts of the data in order to better illustrate even small changes in relation to the reference case.

**Figure 31.**PV system battery combination of 5 kWp and 5 kWh. While the

**upper**figures show all data points, the

**lower**figures show only parts of the data in order to better illustrate even small changes in relation to the reference case.

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | annuity factor |

A_{0,X} | investment amount of the component X/€ |

A_{1,X}…. A_{m,X} | cash value of the first, second, …, mth procured replacement/€ |

A_{cal}(y) | calendar aging per year |

A_{Cycl}(y) | cyclic aging per year |

A_{i,X} | cash value of a procured replacement/€ |

A_{N} | annuity of the total annual payments/€ |

A_{N,C,X} | annuity of the capital related costs/€ |

A_{N,EC} | annuity of the costs of the electricity purchased from the grid/€ |

A_{N,R} | annuity of the annual remuneration for electricity fed into the grid/€ |

aw | Wöhler parameter |

BAT_{chg,dischg} | variable that can be 1 or 0 depending on whether surplus energy is available to charge the battery or is needed to power the load |

bw | Wöhler parameter |

c | considered |

C_{BAT} | battery cost (Invest)/€ |

C_{E}_{,total} | electricity cost per kWh/€/kWh |

C_{E}_{,total,ref} | reference electricity costs per kWh/€/kWh |

C_{PV} | PV system cost (Invest)/€ |

cs | considered in a simplified way |

Cycl | total number of encountered cycles during one year |

DCACBAT | battery inverter |

DCACPV | PV inverter |

DoD | depth of discharge |

E_{BAT} | energy charged within the battery only taking battery charging and discharging into account, variable is needed to determine P_{DC,BAT,chg} and P_{DC,BAT,dischg}/kWh |

E_{BAT,real} | actual energy charged within the battery/kWh |

E_{Capa,BAT} | installed battery capacity/kWh |

E_{Capa,BAT}(y) | battery capacity left at the beginning of the year/kWh |

E_{Capa,BAT,y} | remaining battery capacity of a respective year/kWh |

EEG | German Renewable Energy Sources Act |

E_{from,Grid} | amount of electricity taken from the grid/kWh |

E_{Load,y} | energy consumed per year/kWh |

EOL | end of life of the battery |

E_{to,Grid} | amount of electricity fed into the grid/kWh |

F | false |

fact_{chg},_{d} | factor to reduce the charging power, so that the battery is not fully charged before the end of the day |

i | number of the replacement |

LAB | lead-acid battery |

LIB | Li-ion battery |

LT_{Cell,Cal} | battery lifetime in years |

Meas | measured data |

n | time resolution of the simulation |

N(ΔSOC) | number of equivalent full cycles at a certain ΔSOC |

NAS | sodium–sulfur battery |

n_{Cycl} | a certain cycle during one year |

ni_{Cycl} | factor that can be 1 or 0.5, depending on whether the corresponding cycle is a half or full cycle |

P_{AC,BAT} | AC battery power (AC-coupled system)/kW |

P_{AC,BAT,chg} | actual AC charge power of the battery/kW |

P_{AC,BAT,chg,pot} | power on the AC side of the battery inverter which is available to charge the battery (potential) |

P_{AC,BAT,dischg} | actual AC discharge power of the battery/kW |

P_{AC,BAT,dischg,pot} | power on the AC side of the battery inverter with witch the battery should be discharged to cover the load (potential) |

P_{AC,BAT,rechg} | AC battery power with witch the battery is recharged when the SOC has reached 0% |

P_{AC,BAT,stby} | AC standby power of the battery inverter/kW |

P_{AC,periph} | peripheral consumption/kW |

P_{AC,PV} | AC PV power (AC-coupled system)/kW |

P_{DC,BAT} | DC battery power/kW |

P_{DC,BAT,chg} | actual DC charge power of the battery/kW |

P_{DC,BAT,chg,pot} | power on the DC side of the battery inverter which is available to charge the battery (potential) |

P_{DC,BAT,dischg} | actual DC discharge power of the battery/kW |

P_{DC,BAT,dischg,pot} | power on the DC side of the battery inverter with witch the battery should be discharged to cover the load (potential) |

P_{DC,BAT,rechg} | DC battery power with witch the battery is recharged when the SOC has reached 0% |

P_{DC,BAT,stby} | standby consumption on the DC side of the battery inverter/kW |

P_{DC,BAT,stby,dischg} | standby consumption on the DC side of the battery inverter (battery is discharged)/kW |

P_{DC,PV} | DC PV power/kW |

P_{from,GRID} | power fed into the grid/kW |

P_{GRID} | power measured at the grid connection point/kW |

P_{LOAD} | load of the household/kW |

P_{PV,kWp} | installed PV power/kW |

P_{SYS} | power of the whole PV home storage system/kW |

P_{to,GRID} | power supplied by the grid/kW |

q | interest factor (1 + interest rate) |

RFB | redox-flow battery |

R_{V,X} | residual value/€ |

r_{X} | price change factor |

SCR | Self-consumption/% |

Sim | simulated data |

Slp | Standard load profile |

SOC | state of charge |

SOC_{max} | maximum SOC of the battery |

SOC_{min} | minimum SOC of the battery |

SOH | state of health |

SSR | Self-sufficiency rate/% |

t | timestep of the calculation |

T | true |

T | simulation period |

t_{2} | timestep of the calculation of the intelligent charging strategy, t_{2} is equal to one day |

T_{N} | service life (in years) of the installation component (PV, battery, inverter)/y |

v | varied |

X | placeholder for the different components (PV, battery, inverter) |

y | year |

y_{n}_{,BAT,repl} | year in which a replacement investment of the battery takes place (1 to 20) |

ΔSOC | depth of discharge |

η_{AC2BAT} | efficiency of the conversion path AC2BAT/% |

η_{BAT} | battery efficiency during charging or discharging |

η_{BAT,RT} | battery roundtrip efficiency/% |

η_{BAT2AC} | efficiency of the conversion path BAT2AC/% |

η_{PV2AC} | efficiency of the conversion path PV2AC/% |

## Appendix A

**Table A1.**Calculation of the PV system cost per kWp: Costs from Märtel [64] are used as the initial basis for calculating the PV system costs. These can be found in the table in row 1. The costs in the simulation needed only include the costs for the PV system including construction without the inverter. In the simulation the costs for the inverters are 200 €/kW. Thus, the cost of the inverter was deducted from the cost of Märtel. To obtain a simple functional relationship between the PV system size and the PV system cost, the values of 3 kWp and 9 kWp were omitted and the value of 17.5 kWp was corrected to a lower value of 1300 €/kWp. The omitted values are marked in red in the table. The coefficient of determination is 0.9513.

PV System Size/kWp | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12.5 | 17.5 |
---|---|---|---|---|---|---|---|---|---|---|

Cost per kWp without VAT/€ [64] | 1.730 | 1.570 | 1.530 | 1.490 | 1.470 | 1.370 | 1.430 | 1.340 | 1.240 | 1.360 |

Cost per kWp with VAT/€ | 2.059 | 1.868 | 1.821 | 1.773 | 1.749 | 1.630 | 1.702 | 1.595 | 1.476 | 1.618 |

Cost of the inverter/€ | 600 | 800 | 1.000 | 1.200 | 1.400 | 1.600 | 1.800 | 2.000 | 2.500 | 3.500 |

Total cost of the PV system without inverter/€ | 5.576 | 6.673 | 8.104 | 9.439 | 10.845 | 11.442 | 13.515 | 13.946 | 15.945 | 24.822 |

Cost of the PV system per kWp without inverter/€ | 1.859 | 1.668 | 1.621 | 1.573 | 1.549 | 1.430 | 1.502 | 1.395 | 1.276 | 1.418 |

**Table A2.**Calculation of the batter cost per kWh: The battery cost function was derived using data from Figgener et al. [66]. These already include the VAT and the inverter costs. The approach is described in the table. In the first step, the data from Figgener et al. was fitted (see Equation (A1)). For all battery sizes between 2 kWh and 10 kWh, the inverter costs are then subtracted from the battery system costs. According to Figgener et al. [66], the average inverter size for storage systems between 5 kWh and 10 kWh is 3.5 kW. The inverter costs are assumed to be 200 €/kW. Thus, for the calculation, for all batteries larger than 5 kWh 700 € was deducted from the storage system costs. For all systems smaller than 5 kWh, only 400 € were deducted for an inverter with a size of 2 kW. From this a functional relationship can be derived, which is represented by Equation (3). The coefficient of determination is 0.9956.

Approach | Data | ||||||||
---|---|---|---|---|---|---|---|---|---|

Data from Figgener et al. [66] | Mean battery size/kWh | 3 | 7.5 | 12.5 | 17.5 | ||||

Mean battery cost including VAT and the battery inverter/€/kWh | 1625 | 1100 | 900 | 825 | |||||

Derived equation (R^{2} = 1) | Equation (A1) | ||||||||

Battery size/kWh | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Cost of the battery storage system/€ | 3616 | 4875 | 5875 | 6684 | 7364 | 7965 | 8529 | 9087 | 9663 |

Cost of the battery without an inverter of 2kW/€ | 1608 | 1492 | 1369 | ||||||

Cost of the battery without an inverter of 3.5 kW/€ | 1197 | 1111 | 1038 | 979 | 932 | 896 | |||

Derived Equation (3) to calculate battery cost (R^{2} = 0.9956) |

**Table A3.**Electricity costs in €/kWh depending on the change of different technical parameters. The selected PV system and battery combination is the combination that is the most favorable for the respective boundary conditions.

AC Stby | DC Stby | Pperiph | η Battery | η Battery Inverter | η PV Inverter | Charging Strategy | Calendar Aging | Cyclic Aging | PV Degradation | |
---|---|---|---|---|---|---|---|---|---|---|

V1 | 31.14 | 31.29 | 31.39 | 31.61 | 32.65 | 31.78 | 30.37 | 36.67 | 34.67 | 32.60 |

V2 | 31.32 | 31.40 | 31.59 | 31.67 | 31.95 | 31.50 | 31.50 | 33.25 | 32.79 | 32.03 |

Ref. | 31.50 | 31.50 | 31.50 | 31.50 | 31.50 | 31.50 | 31.50 | 31.50 | ||

V3 | 31.67 | 31.88 | 31.70 | 31.62 | 31.57 | 30.85 | 31.41 | |||

V4 | 31.85 | 32.00 | 31.90 | 31.75 | 30.46 | 31.17 |

**Table A4.**Electricity costs per year in € depending on the change of different technical parameters. The selected PV system and battery combination is the combination that is the most favorable for the respective boundary conditions.

AC Stby | DC Stby | Pperiph | η Battery | η Battery Inverter | η PV Inverter | Charging Strategy | Calendar Aging | Cyclic Aging | PV Degradation | |
---|---|---|---|---|---|---|---|---|---|---|

V1 | 1312 | 1318 | 1323 | 1332 | 1376 | 1339 | 1280 | 1545 | 1461 | 1374 |

V2 | 1319 | 1323 | 1331 | 1334 | 1346 | 1327 | 1327 | 1401 | 1382 | 1350 |

Ref. | 1327 | 1327 | 1327 | 1327 | 1327 | 1327 | 1327 | 1327 | ||

V3 | 1334 | 1343 | 1335 | 1332 | 1330 | 1300 | 1323 | |||

V4 | 1342 | 1348 | 1344 | 1338 | 1283 | 1313 |

**Table A5.**Total electricity costs over a period of 20 year in € depending on the change of different technical parameters. The selected PV system and battery combination is the combination that is the most favorable for the respective boundary conditions.

AC Stby | DC Stby | Pperiph | η Battery | η Battery Inverter | η PV Inverter | Charging Strategy | Calendar Aging | Cyclic Aging | PV Degradation | |
---|---|---|---|---|---|---|---|---|---|---|

V1 | 26,240 | 26,367 | 26,453 | 26,638 | 27,514 | 26,782 | 25,592 | 30,902 | 29,210 | 27,471 |

V2 | 26,390 | 26,457 | 26,623 | 26,689 | 26,927 | 26,540 | 26,540 | 28,015 | 27,634 | 26,992 |

Ref. | 26,540 | 26,540 | 26,540 | 26,540 | 26,540 | 26,540 | 26,540 | 26,540 | ||

V3 | 26,690 | 26,861 | 26,709 | 26,646 | 26,601 | 25,998 | 26,470 | |||

V4 | 26,840 | 26,967 | 26,879 | 26,753 | 25,663 | 26,262 |

**Table A6.**Overview of the maximum changes regarding electricity cost through V1 to V4 of the individual technical parameters compared to the reference case. To calculate the total cost for the calculation period the maximum cost increase and decrease needs to be multiplied by 20.

PV System/Battery | Max Cost Increase/Decrease due to the Variations V1–V4 in Comparison to Ref. | AC Stby | DC Stby | P Periph | Eff Battery | Eff Battery Inverter | Eff PV Inverter | Charging Strategy | Calendar Aging | Cyclic Aging | PV Degradation |
---|---|---|---|---|---|---|---|---|---|---|---|

15 kWp/10 kWh | max cost increase cent/kWh | 0.16 | 0.10 | 0.28 | 0.31 | 1.19 | 0.29 | 8.91 | 4.90 | 1.14 | |

max cost increase €/y | 6.95 | 4.02 | 11.96 | 13.10 | 49.98 | 12.37 | 375.48 | 206.29 | 48.10 | ||

max cost decrease cent/kWh | −0.16 | −0.09 | −0.28 | −0.22 | −0.32 | −0.13 | −2.94 | −0.12 | 0.55 | ||

max cost decrease €/y | −6.94 | −3.97 | −11.85 | −9.35 | −13.32 | −5.28 | −123.89 | −4.86 | 23.32 | ||

15 kWp/5 kWh | max cost increase cent/kWh | 0.31 | 1.40 | 0.87 | 1.10 | 1.26 | 0.29 | 5.93 | 3.57 | 1.11 | |

max cost increase €/y | 12.98 | 59.10 | 36.83 | 46.55 | 53.08 | 12.09 | 249.73 | 150.31 | 46.82 | ||

max cost decrease cent/kWh | −0.31 | −0.09 | 0.19 | 0.14 | 0.51 | −1.27 | −1.20 | −0.16 | 0.54 | ||

max cost decrease €/y | −12.97 | −3.78 | 7.90 | 5.90 | 21.57 | −53.54 | −50.56 | −6.73 | 22.69 | ||

10 kWp/10 kWh | max cost increase cent/kWh | 0.19 | 0.12 | 0.30 | 0.30 | 1.17 | 0.20 | 8.86 | 2.29 | 0.69 | |

max cost increase €/y | 8.11 | 5.04 | 12.73 | 12.50 | 49.49 | 8.59 | 373.14 | 96.33 | 29.05 | ||

max cost decrease cent/kWh | −0.19 | −0.11 | −0.30 | −0.21 | −0.30 | −0.30 | −3.30 | −1.62 | 0.33 | ||

max cost decrease €/y | −8.10 | −4.82 | −12.58 | −8.93 | −12.63 | −12.64 | −139.07 | −68.22 | 13.89 | ||

10 kWp/5 kWh | max cost increase cent/kWh | 0.32 | 0.18 | 0.39 | 0.28 | 1.30 | 0.20 | 4.11 | 2.63 | 0.65 | |

max cost increase €/y | 13.57 | 7.69 | 16.55 | 11.74 | 54.76 | 8.47 | 172.99 | 110.63 | 27.32 | ||

max cost decrease cent/kWh | −0.32 | −0.19 | −0.39 | −0.20 | −0.26 | −1.43 | −2.00 | −1.07 | 0.31 | ||

max cost decrease €/y | −13.56 | −7.82 | −16.31 | −8.45 | −10.96 | −60.07 | −84.25 | −44.97 | 13.04 | ||

5 kWp/5 kWh | max cost increase cent/kWh | 0.38 | 0.23 | 0.21 | 0.12 | 0.53 | 0.12 | 2.79 | 1.19 | 0.36 | |

max cost increase €/y | 15.99 | 9.84 | 8.94 | 5.05 | 22.16 | 5.21 | 117.73 | 50.19 | 15.31 | ||

max cost decrease cent/kWh | −0.38 | −0.22 | −0.44 | −0.44 | −0.25 | −1.92 | −2.10 | −1.78 | 0.18 | ||

max cost decrease €/y | −15.98 | −9.37 | −18.52 | −18.40 | −10.43 | −80.71 | −88.36 | −74.98 | 7.40 |

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**Figure 1.**Schematic of an AC-coupled PV home storage system including possible power flows and the corresponding symbols from Munzke et al. [44].

**Figure 2.**Efficiency of the power conversion pathways AC2BAT (battery charging—

**left**) and BAT2AC (battery discharging—

**right**) as a function of the output power, for 5 PV storage systems.

**Figure 3.**Efficiency of the power conversion pathways PV2AC as a function of the output power, for 4 PV inverters.

**Figure 4.**Standby consumption and peripheral consumption (W) of 4 PV storage systems the systems measured when the battery was completely charged (

**left**) or discharged (

**right**) peripheral consumption.

**Figure 7.**Result of calendar aging tests at 25 °C at different SOCs (10%, 30%, 50%, 70%, 90% and 100%) of a 53 Ah NMC cell (

**left**). Functional relationship between the lifetime in years until EOL at different SOCs (

**right**).

**Figure 8.**Scaled Wöhler curves based on the Wöhler curve presented in Angenendt et al. of the LG ICR18650MF1 lithium-ion battery cell [3]. aw for all presented curves is −0.968423. The

**right**panel shows a section of the

**left**panel.

**Figure 14.**Electricity costs per kWh as a function of the battery system size (

**left**panel) and the PV system size (

**right**panel). Simulations were performed with the efficiency curves of systems C, A, E and D with an average pathway efficiency of 83.70, 89.35, 94.32 and 92.50% and for the AC2BAT path and of 83.70, 89.35, 94.32 and 92.50% for the path BAT2AC.

**Figure 15.**Electricity costs per year as a function of the battery system size (

**left**panel) and the PV system size (

**right**panel). Simulations were performed with the efficiency curves of systems C, A, E and D with an average pathway efficiency of 83.70, 89.35, 94.32 and 92.50% and for the AC2BAT path and of 83.70, 89.35, 94.32 and 92.50% for the path BAT2AC.

**Figure 17.**Electricity costs per kWh as a function of battery system size (

**left**panel) and PV system size (

**right**panel). Results for the different battery efficiencies are shown in color. The reference case with a battery efficiency of 96% is shown in green.

**Figure 18.**Change in electricity costs per kWh of the reference case compared to the 4 other simulations, with a battery efficiency of 92, 94, 98, 99% as a function of the battery system size.

**Figure 19.**Electricity costs per kWh as a function of battery system size (

**left**panel) and PV system size (

**right**panel). The results for different levels of AC standby consumption are shown in color. The reference case with an AC standby consumption of 12 W is shown in green.

**Figure 20.**Change in electricity costs per kWh of the reference case to the simulations with a lower or higher AC standby consumption as a function of the battery system size (

**left**panel). Change in electricity costs per kWh of the reference case to the simulations with a lower or higher DC standby consumption as a function of the battery system size (

**right**panel).

**Figure 27.**Change of energy costs depending on the change of different technical parameters. The selected PV system and battery combination is the combination that is the most favorable for the respective boundary conditions. The

**left**window shows the costs in €/kWh and the

**right**one shows the annual electricity costs.

**Figure 28.**Change of energy costs depending on the change of different technical parameters. The selected PV system and battery combination is the combination that is the most favorable for the respective boundary conditions. The figure shows the total electricity costs over 20 years (calculation period) in €.

Average Pathway Efficiency BAT2AC | Average Pathway Efficiency AC2BAT | |
---|---|---|

System A | 89.35 | 88.68 |

System C | 83.70 | 85.62 |

System D (Ref.) | 92.50 | 91.95 |

System E | 94.32 | 94.66 |

Parameter | Value | Unit |
---|---|---|

Electricity price [67] | 31.89 | cent/kWh |

Calculation period | 20 | y |

PV system cost (C_{PV}) [64] (see Table A1) | Equation (2) | €/kWp |

Battery cost (C_{BAT}) [66] (see Table A2) | Equation (3) | €/kWh |

Inverter cost | 200 | €/kW |

Inverter lifetime [22,68,69] | 10 | y |

Annual battery cost degradation | 3 | % |

Price change factor of the battery | 0.97 | |

Price change factor of the inverters | 1 | |

Annual electricity price increase [4,67] | 0.0/1.0 | % |

Price change factor of the electricity price [4,67] | 1.0/1.01 | |

Feed in tariff (March 2021) for PV systems up to 10 kWp | 7.92 | cent/kWh |

Feed in tariff (March 2021) for PV systems between 10 kWp and 40 kWp | 7.7 | cent/kWh |

Parameter | Variation | Reference Case | Variation | ||
---|---|---|---|---|---|

V1 | V2 | Ref. | V3 | V4 | |

Battery inverter efficiency | System C | System A | System D | System E | - |

PV inverter efficiency | System A | - | System D | - | - |

Battery efficiency | 92% | 94% | 96% | 98% | 99% |

Standby consumption AC | 0 W | 6 W | 12 W | 18 W | 24 W |

Standby consumption DC | 0 W | 4 W | 8 W | 12 W | 16 W |

Consumption of peripheral components | 0 W | 4 W | 8 W | 12 W | 16 W |

Charging strategy | - | - | Simple strategy only to enhance self-consumption | - | Strategy to enhance self-consumption and to reduce battery aging |

Battery calendar aging | 60% | 80% | 100% | 120% | 140% |

Battery cyclic aging | 60% | 80% | 100% | 120% | 140% |

PV degradation per year | - | - | 0.15% | 0.32% | 0.5% |

Battery Capacity/kWh | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|

PV system size/kWp | 15 | 15 | 15 | 15 | 15 | 15 | 15 |

Electricity cost €/kWh | 0.3150 | 0.3178 | 0.3211 | 0.3246 | 0.3293 | 0.3367 | 0.3477 |

Annual cost/€ | 1327 | 1339 | 1353 | 1368 | 1387 | 1418 | 1465 |

Total cost 20 years/€ | 26,540 | 26,776 | 27,060 | 27,356 | 27,744 | 28,368 | 29,297 |

PV system size/kWp | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

Electricity cost €/kWh | 0.3332 | 0.3366 | 0.3405 | 0.3447 | 0.3502 | 0.3582 | 0.3696 |

Annual cost/€ | 1404 | 1418 | 1435 | 1452 | 1475 | 1509 | 1557 |

Total cost 20 years/€ | 28,076 | 28,365 | 28,691 | 29,050 | 29,509 | 30,185 | 31,143 |

**Table 6.**PV system size of the most economic combination of the different simulation runs with the different technical parameter.

AC Stby | DC Stby | Pperiph | η Battery | η Battery Inverter | η PV Inverter | Charging Strategy | Calendar Aging | Cyclic Aging | PV Degradation | |
---|---|---|---|---|---|---|---|---|---|---|

V1 | 15 | 15 | 14 | 14 | 15 | 15 | 15 | 15 | 15 | 15 |

V2 | 15 | 15 | 14 | 14 | 15 | 14 | 15 | 15 | ||

Ref. | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |

V3 | 15 | 14 | 15 | 15 | 14 | 15 | 15 | |||

V4 | 15 | 14 | 15 | 15 | 15 | 13 |

**Table 7.**Parameters used in the literature to evaluate the influence of cycle stability on economic efficiency.

PV System Size | Battery Size | Economical Parameter | Calendar Lifetime | Cycle Stability Variation | Battery Price | |
---|---|---|---|---|---|---|

Bertsch et al. [13] | 6.4 kWp | 12 kWh | IRR | 20 y | 500 until 6000 cycles | 500 €/kWh |

Tervo et al. [31] | 5 kWp | 7 kWh | LCOE | 15 y | 5000 cycles ± 10% | 392 $/kWh + 1700 $ |

Troung et al. [14] | 5 kWp/8 kWp | 6.4 kWh | ROI | 15 y | 5000 & 3000 cycles (−40%) | 781.25 €/kWh |

Naumann et al. [12] | 4.4 kWp | 4 kWh | ROI | 12.5–15 y | 6000 & 3000 cycles (−50%) | <500 €/kWh |

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**MDPI and ACS Style**

Munzke, N.; Büchle, F.; Smith, A.; Hiller, M. Influence of Efficiency, Aging and Charging Strategy on the Economic Viability and Dimensioning of Photovoltaic Home Storage Systems. *Energies* **2021**, *14*, 7673.
https://doi.org/10.3390/en14227673

**AMA Style**

Munzke N, Büchle F, Smith A, Hiller M. Influence of Efficiency, Aging and Charging Strategy on the Economic Viability and Dimensioning of Photovoltaic Home Storage Systems. *Energies*. 2021; 14(22):7673.
https://doi.org/10.3390/en14227673

**Chicago/Turabian Style**

Munzke, Nina, Felix Büchle, Anna Smith, and Marc Hiller. 2021. "Influence of Efficiency, Aging and Charging Strategy on the Economic Viability and Dimensioning of Photovoltaic Home Storage Systems" *Energies* 14, no. 22: 7673.
https://doi.org/10.3390/en14227673