# A Sizing Method for PV–Battery–Generator Systems for Off-Grid Applications Based on the LCOE

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions, employing relatively fast evolutionary algorithms. Results showed that the best pareto fronts included a diesel generator that contributes to the overall economic and environmental performance. In Maleki et al. [7], a combinatorial optimization method based on the harmony search algorithm for sizing off-grid PV–battery and generator systems was presented. In this study, the proposed method outperformed a simulated annealing method, displaying extremely fast runtimes, although battery technology and various battery specifics such as depth of discharge and cycle life were not considered. Simulated annealing genetic algorithms were employed by Wei et al. [8] to optimize coal-fired boiler operation so as to reduce NOx emissions, thereby exhibiting accurate solutions in a low computing time. Similarly, in [9], Ghafoor et al. presented a deterministic and straightforward off-grid PV–battery configuration model for a residential case study in Pakistan accompanied by a lifecycle cost analysis without considering battery technology.

_{2}emissions, and grid voltage for sizing interconnected PV–battery–diesel combinations.

## 2. Materials and Methods

#### 2.1. Off-Grid PV–Battery–Generator System

#### 2.1.1. Preference of the DC-Coupled Architecture

#### 2.1.2. Selection of the System Components

^{2}, resulting in a module energy efficiency of approximately 19%, under STCs. According to the manufacturer’s datasheet, the peak power temperature coefficient is −0.40%/°C. The manufacturer product and linear power warranties are 15 years and 30 years, respectively. More specifically, the performance warranty for 15 years is 91.2% and that for 30 years is 80.6% of rated power. Cost of the module was inquired online, and offers ranging from 0.45 EUR/Wp to 0.35 EUR/Wp were collected from selected distributors.

#### 2.2. Annual Electricity Generation Estimation

#### 2.3. Reference Load Profile

#### 2.4. System Operation and Sizing Algorithm

_{pv}) and to the initial battery state of energy (SOE) available. The P

_{pv}at this point included converter losses. A matching load profile was synthesized, as described in Section 2.3, leading to an hourly power demand (P

_{load}) for an entire year. Figure 5 illustrates the system operation flow in terms of P

_{pv}usage, P

_{load}coverage, and hourly battery SOE.

- Case A.
- Surplus energy case and a fully charged battery

_{sur}, should either be used to charge the battery or be discarded.

_{inv}is the inverter efficiency and can be acquired from the inverter’s efficiency curve [26]. The SOE

_{t−1}is greater than or equal to 100%, which means that the battery is fully charged, and the excess of PV energy remains unused; thus, potential E

_{sur}at this step is wasted and characterized as E

_{waste}. With the SOE remaining at the higher level (100%), the algorithm proceeds to the next hourly step with the following equations:

- Case B.
- Surplus energy case and a partially charged battery

_{t-1}is not greater than or equal to 100%, the battery is in a partially charged state and keeps charging by E

_{sur}(Equation (1)). Then, SOE

_{t}which is initially scaled at 100% is defined as follows:

_{sur}) can be totally absorbed by the battery, where n

_{bat}is the charging efficiency of the battery.

_{t}, calculated by Equation (5), Ebat

_{t}> B

_{nec}applies, then the amount of energy that cannot be absorbed is discarded.

- Case C.
- Energy shortage case

_{short}.

_{t-1}is greater than the threshold percentage (100%—DOD) which means that the battery is in a partially charged state and keeps discharging as it contributes to the load coverage. Consequently, the SOE

_{t}is defined again by Equation (4), using this time the remaining amount of energy Ebat

_{t}, defined by the following equations:

- Case D.
- Backup case

_{t-1}is less than the threshold value, which means that the battery is in a discharged state, the backup source needs to start charging the battery in the next hourly step and keep the entire load covered until the battery is fully charged again. This recurring case consists of a loop in which the hourly load is primarily covered by the generator, for loads up to the maximum generator output (P

_{backupMAX}). Secondarily, the battery is charged with the sum of the power of the remaining generator output on the DC side (P

_{backup->battery}) and the available PV power output. Hence, no energy is being discarded (Ε

_{waste}= 0). In this case, the accumulated Ebat

_{t}is described in Equation (13).

_{t}variation is kept track of by using Equation (4). Breaking out of the loop and returning to normal operation requires that the SOE

_{t}reaches approximately 80%, a safe state that prevents overcharging by an overshooting charge. More specifically, during battery charging by the backup, all the energy generated by the PV system is directed to the battery due to the particular architecture of the system. If we charge the battery further than SOE

_{t}> 80% with the generator, then there is a serious possibility that the contribution of the PV-generated energy gets wasted. Our main concern is always to avoid situations where the energy produced by RES has to be rejected.

_{waste}+ ΣP

_{backup}Δt) for every N and Bnec simulation instance. Therefore, a distinct minimum of ΣΕ

_{waste}+ ΣP

_{backup}Δt signifies an optimal N and Bnec arrangement for which the renewable harvest is maximized. Furthermore, the annually utilized PV output, i.e., the useful energy E

_{us}, is defined in Equation (14).

#### 2.5. System Cost Analysis and the Levelized Cost of Electricity (LCOE)

_{pi}is the total expenditure cost, C

_{b,r}is the discounted battery replacement cost, C

_{m}is the annual system maintenance cost, C

_{eb}is the annual backup energy cost, C

_{es}is the cost of the discarded renewable energy, and E

_{us}is the annual useful energy. Similar to the total economic costs at the numerator, the E

_{us}at the denominator is also multiplied by a discount factor which is described in more detail in Appendix A. The annual real interest rate, here i, is estimated using the Fisher equation [49].

#### 2.5.1. Cost of Battery Replacement (C_{b,r})

_{max}), the number of charge–discharge cycles of the batteries at 100% DOD was estimated according to the total electrical energy delivered by the battery (Pbat

_{d}) in a year, using Equation (17).

_{max}), the estimated TOR is taken into account as the replacement time of the battery. In cases where the calculated TOR exceeds the maximum battery calendar life (TOR > TOR

_{max}), TOR

_{max}is considered as the battery replacement time.

_{max}. Otherwise, the product n × TOR

_{max}is used.

#### 2.5.2. Estimations of Annual C_{m}, C_{eb}, and C_{es}

_{m}, is the sum of the individual operation and maintenance costs of each system component during a year, including the cost of potential services (e.g., equipment insurance, monitoring services).

_{fuel}), defined in Equation (20), is the required electrical energy to preserve load supply and to charge the batteries via the generator set. This energy is converted from chemical energy contained in diesel fuel and expressed in kWh.

_{gen}is the generator efficiency. The annual backup energy cost C

_{eb}, defined in Equation (21), is derived from the product of E

_{fuel}and the fuel cost (Fuel

_{cost}) expressed in EUR/lt, divided by the lower heating value (LHV) of diesel fuel, i.e., 9.85 kWh/lt.

_{es}described in Equation (22) is derived from the product of the annual sum of E

_{waste}in kWh and the average LCOE of PV–battery systems expressed in EUR/kWh.

## 3. Results

#### 3.1. Sizing Simulations

_{optimal}) and the necessary nominal energy of the energy storage system (Bnec

_{optimal}), so as to satisfy, without interruptions, the energy consumption profile of an average household for a typical meteorological year in a specified location, whilst minimizing the LCOE expressed by Equation (15).

#### 3.2. Performance of Systems

#### 3.2.1. Maximizing the Contribution of Solar Energy

#### 3.2.2. Minimizing the LCOE

_{br}column of Table 2.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

ALV | Average load value of diesel generator |

AOI | Angle of incidence |

APT | Annual operation time of diesel generator |

BMS | Battery management system |

Bnec | The required battery energy capacity |

Cb,r | Discounted battery replacement cost, |

Ceb | Annual backup energy cost |

Ces | Cost of the discarded renewable energy |

Cm | Annual system maintenance cost |

Cpi | Total capital cost |

CTU | Maximum continuous time use of diesel generator |

DER | Distributed energy resource |

DHI | Diffuse horizontal irradiance |

DNI | Direct normal irradiance |

DOD | Depth of discharge |

Ebatt | The remaining amount of energy that is transferred to the battery at time t |

EES | Electrical energy storage |

Eshort | Energy shortage when the PV output cannot cover the power demand |

Esur | Surplus energy |

Eus | Annually utilized PV energy output |

Ewaste | Excess solar power that remains unused |

GHI | Global horizontal irradiance |

I0 | Diode reverse saturation current |

IL | Photocurrent |

LCOE | Levelized cost of electricity |

LPG | Liquefied petroleum gas |

MPPT | Maximum power point tracking |

N | The required PV module number |

nbat | Charging efficiency of the battery |

ninv | Inverter efficiency |

nNsVth | Modified diode ideality factor |

P_{backup->battery} | The remaining generator output on the DC side charging the battery |

P_{backupMAX} | Maximum diesel generator output |

PERC | Passivated emitter and rear contact PV module |

Pload | The hourly power demand for in a year. |

Ppv | Hourly PV power output in a year |

PVBG | Photovoltaic system with batteries and generator set |

PVLIB | Solar simulation library for PV energy systems |

Rs | Series resistance |

Rsh | Shunt resistance |

SA | Operation and sizing algorithm |

SOE | Battery state of energy |

SOE | Battery state of energy (SOE) available |

STC | Standard test conditions |

T | Atmospheric temperature |

TC | PV cell temperature |

TM | PV module temperature |

TMY | Typical meteorological year |

WS | Horizontal windspeed |

## Appendix A

## Appendix B

Parameter | Description | |
---|---|---|

p.1 | Bnec | The minimum multiple battery nominal energy capacity, in Wh. |

p.2 | N | The PV module number in an array. |

p.3 | Σ(Ppv) | The sum of hourly PV power output in a year, in Wh. |

p.4 | Σ(Pload) | The sum of the hourly power demand on the consumption side in a year, in Wh. |

p.5 | Σ(Pbat,d) | The annual sum of energy discharging the battery, in Wh. |

p.6 | Σ(Pbat,ch) | The annual sum of energy charging the battery, in Wh. |

p.7 | Σ(Ewaste) | The annual sum of unused (discarded) energy, in Wh. |

p.8 | Σ(Pbackup) | The annual sum of the generator output, in Wh. |

p.9 | Σ(Pbackup->bat) | The annual sum of energy flowing from the generator to the battery, in Wh. |

p.10 | Σ(Backup_Operation) | The sum of operational hours of the generator in a year. |

p.11 | Σ(Pload)_Σ(Ppv)_ratio | The ratio of the annual PV power output to the annual power demand. |

p.12 | Battery_Cycles | Battery_Cycles = abs(Σ(Pbat,d))/Bnec The total charge–discharge cycles of the battery for a year. |

p.13 | Σ(Εwaste) + Σ(Pbackup) | The sum of annual discarded energy and annual energy supplied by the generator, in Wh. |

p.14 | Eus | = Σ(Ppv) − Σ(Ewaste) Useful_Energy The utilized PV output, in Wh. |

p.15 | Bat_Replac_Years | = NBC/Battery_Cycles Where NBC is the nominal battery cycle life stated by the battery manufacturer in the datasheet. |

p.16 | PV_Cost(N) | = N × PV_Module_Cost The capital cost of PV panels. |

p.17 | Battery_Cost_Lead | = (Bnec/1000) × Bat_Cost_Lead–Acid, where Bat_Cost_Lead–Acid is the battery capital cost per kWh. |

p.18 | Battery_Cost_Lithium | = (Bnec/1000) × Bat_Cost_LiFePO4, where Bat_Cost_LiFePO _{4} is the battery capital cost per kWh. |

p.19 | Power_Electr_Cost(N) | = Inverter_Cost + Monitoring_Cost + N × 50, in EUR. The capital cost for power electronics depends on the number of PV panels installed. The above function is an approximation and can usually be derived by the power electronics distributor pricelist. |

p.20 | Mounting_Cost(N) | = N × 1_Panel_Roof_Mounting_Cost. The capital cost for PV mounting systems can vary significantly depending on site-specific individualities. Although, in this work, a simple dependency on the amount of PV panels was acceptable, a more precise cost estimation must be considered in demanding installation sites. |

p.21 | Installation_Service_Cost(Ν) | = Electrical_Install_Cost + Mounting_Cost(N), where Electrical_Install_Cost is the service cost for the indoor electrical construction, i.e., the power electronics–battery–wiring setup. Installation_Service_Cost(Ν) is also dependent on the amount of installed PV panels since more modules generally mean more converters, batteries, cabling, and mounting stands to install. This is an empirically determined quantity and scaled to the size of the total PV installation. |

p.22 | Electr&Install_Materials(N) | = N × 12.5 + Electrical_Install_Cost, in Euro. The material purchasing and installation cost for the electrical wiring, and construction, consisting mainly of low voltage protection and control equipment (Miniature circuit breakers (MCBs), wires, enclosure, switchboards, etc.). This is also an empirically determined quantity, scaled to the number of PV panels and dependent on the service cost for the indoor electrical construction. |

p.23 | Σ_COST_(PV&Lead-Acid) | = [PV_Cost(N) + Battery_Cost_Lead + Power_Electr_Cost(N) + Mounting_Cost(N) + Installation_Service_Cost + Electr&Install_Materials] |

p.24 | Σ_COST_(PV&LiFePO4) | = [PV_Cost(N) + Battery_Cost_Lithium + Power_Electr_Cost(N) + Mounting_Cost(N) + Installation_Service_Cost + Electr&Install_Materials] |

p.25 | Σ_Bat_Repl_Cost_Lead | Accumulation of net present value battery replacement cost for lead–acid batteries. |

p.26 | Σ_Bat_Repl_Cost_Lithium | Accumulation of net present value battery replacement cost for LiFePO_{4} batteries. |

p.27 | Efuel | = Σ(Pbackup)/ngen The fuel energy required by the generator to cover the load and battery demand, in kWh per year. |

p.28 | Ces | = Σ(Εwaste) × LCOE_{PVBAT}/1000The unit cost of the discarded energy, based on the estimated LCOE of a typical PV-Battery setup, approximately 0.26 EUR/kWh. |

p.29 | Ceb | = Efuel × Cost_of_Fuel/Fuel_energy_vol Cost_of_Fuel = 1.175 EUR/L and Fuel_energy_vol = 9850 |

p.30 | Cost_of_Energy | = [(Ces + Ceb) × ((1 + Real_Interest) ^ (Project_Years + 1) − 1)/(Real_Interest × (1 + Real_Interest) ^ Project_Years)] |

p.31 | Total_Cost | = T + Generator_Cost + Σ_Bat_Repl_Cost + Cost_of_Energy, where T = [Σ_COST_(PV&Lead–Acid) or Σ_COST_(PV&LiFePO _{4})] |

p.32 | TC_UE_ratio | = Total_Cost/ΣUseful_Energy (EUR/kWh) |

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**Figure 1.**Flow diagram depicting the basic steps of the proposed photovoltaic (PV)–battery sizing method. Steps 7 to 10 are iterated for a specified range of PV modules and battery energy capacities, subsequently highlighting optimal values for N and Bnec at minimum levelized cost of electricity (LCOE).

**Figure 3.**The synthesized electricity consumption of an average four-person household. The estimated baseload of the lowest 3 months was approximately 300 kWh.

**Figure 4.**Hourly sequence of the annual load profile, based on the daily electrical appliance usage from a four-person household.

**Figure 5.**System operation flow according to the to the hourly battery state of energy and PV energy usage.

**Figure 8.**Marker chart depicting 14 simulation instances with constant Bnec and varying PV module number. The descending change of the backup energy and the ascending change of discarded surplus energy indicate the optimal PV panel quantity for which the solar generation is maximized.

**Figure 9.**Each marker chart unfolds the optimal N–Bnec arrangement in terms of maximal solar energy usage (

**a**,

**c**) and LCOE minimization (

**b**,

**d**). Charts 9a and 9b show all the simulation instances of the lead–acid system, while charts 9c and 9d show the results of the lithium-ion system.

**Figure 10.**In this marker chart, six simulation instances per EES technology are presented and their calculated LCOEs are directly compared.

**Figure 11.**Comparison of the annually utilized PV output E

_{us}, as defined in Equation (14), of every lithium-ion (

**a**) and lead–acid (

**b**) simulation instance and the corresponding theoretically optimal photovoltaic output P

_{pv}(black marks).

**Table 1.**Input data of two off-grid PV–battery–generator sizing simulations employing different electrical energy storage (EES) technologies. DOD, depth of discharge; SOE, state of energy; BMS, battery management system; LCOE, levelized cost of electricity; LHV, lower heating value.

Input Parameters | Sim Data A | Sim Data B | Units | Input Type | |
---|---|---|---|---|---|

1 | Number of PV modules, N | 6 to 34 | 6 to 34 | Physical model assumptions | |

3 | Nominal PV module power, P_{mpp} | 310 | 310 | Wp | |

2 | Nominal energy capacity, Bnec | 5 to 30 | 5 to 30 | kWh | |

4 | Nominal battery cycle life | 500 | 2500 | Cycles (100% DOD) | |

5 | Calendar battery life | 5 | 10 | Years | |

6 | Depth of discharge | 50 | 80 | % | |

7 | SOE_{t=0} | 100 | 100 | % | |

8 | Inverter efficiency, n_{inv} | 95 | 95 | % | |

9 | Battery charging efficiency, n_{bat} | 85 | 98 | % | |

10 | P_{backup_max} | 4000 | 4000 | W | |

11 | Number of inverters | 1 | 1 | ||

12 | P_{load_max} | 4000 | 4000 | W | |

13 | Battery type | Lead–acid | Lithium-ion | ||

14 | Generator efficiency, n_{gen} | 80 | 80 | % | |

15 | Battery cost | 154 | 574 | EUR/kWh | Economic cost assumptions |

16 | PV module cost | 110 | 110 | EUR | |

17 | Inverter cost | 1600 | 1600 | EUR | |

18 | Monitoring and BMS cost | 250 | 500 | EUR | |

19 | PV mounting system cost | 50 | 50 | EUR/mod | |

20 | Electrical installation cost | 1000 | 1000 | EUR | |

21 | Generator cost | 1300 | 1300 | EUR | |

22 | LCOE_{pvbat} | 0.26 | 0.26 | EUR/kWh | |

23 | LHV | 9.85 | 9.85 | kWh/lt | |

24 | Cost of fuel | 1.175 | 1.175 | EUR/lt | |

25 | Real interest rate | 0.06919 | 0.06919 | % | |

26 | Maintenance cost, C_{m} | 160 | 160 | EUR/year | |

27 | Project years | 25 | 25 | Years |

**Table 2.**The values of the parameters used to calculate the LCOE at the point where the minimum appeared.

Battery Type | N (mods) | Bnec (kWh) | Component Cost | Installation Cost (EUR) | Maintenance Cost | Operation Cost | Eus (kWh/year) | LCOE (EUR/kWh) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PV (EUR) | Battery (EUR) | PE&M * (EUR) | Generator (EUR) | C_{br}(Discounted) (EUR) | C_{m}(EUR/year) | C_{es}(EUR/year) | C_{eb}(EUR/year) | ||||||

Lead–Acid | 10 | 5 | 1100 | 770 | 2600 | 1300 | 3125 | 4445 | 100 | 369 | 503 | 3487.4 | 0.61 |

12 | 10 | 1320 | 1540 | 2700 | 1300 | 3350 | 4405 | 100 | 389 | 341 | 4409.8 | 0.48 | |

18 | 15 | 1980 | 2310 | 3000 | 1300 | 4025 | 4289 | 100 | 834 | 215 | 5557.0 | 0.47 | |

14 | 20 | 1540 | 3080 | 2800 | 1300 | 3575 | 5719 | 100 | 247 | 143 | 6021.8 | 0.34 | |

14 | 25 | 1540 | 3850 | 2800 | 1300 | 3575 | 7149 | 100 | 215 | 126 | 6157.9 | 0.36 | |

14 | 30 | 1540 | 4620 | 2800 | 1300 | 3575 | 8579 | 100 | 199 | 117 | 6226.7 | 0.38 | |

Lithium-Ion | 10 | 5 | 1100 | 2870 | 2600 | 1300 | 3125 | 4972 | 100 | 282 | 449 | 3856.2 | 0.57 |

16 | 10 | 1760 | 5740 | 2900 | 1300 | 3800 | 4446 | 100 | 655 | 205 | 5306.1 | 0.51 | |

14 | 15 | 1540 | 8610 | 2800 | 1300 | 3575 | 6669 | 100 | 301 | 129 | 5792.4 | 0.46 | |

14 | 20 | 1540 | 11,480 | 2800 | 1300 | 3575 | 8892 | 100 | 265 | 107 | 5944.1 | 0.51 | |

14 | 25 | 1540 | 14,350 | 2800 | 1300 | 3575 | 11,115 | 100 | 241 | 92 | 6047.0 | 0.57 | |

14 | 30 | 1540 | 17,220 | 2800 | 1300 | 3575 | 13,338 | 100 | 227 | 82 | 6106.0 | 0.63 |

**Table 3.**The maximum continuous time use (CTU), the estimated annual operation time (APT), and the average load value (ALV) of the standby diesel genrator in diverse simulation arrangemnets.

Battery Type | N (mods) | Bnec (kWh) | CTU (h) | APT (h/year) | ALV (kW) |
---|---|---|---|---|---|

Lead–Acid | 10 | 5 | 1 | 873 | 4000 |

12 | 10 | 2 | 571 | 4000 | |

18 | 15 | 2 | 360 | 4000 | |

14 | 20 | 3 | 239 | 4000 | |

14 | 25 | 3 | 211 | 4000 | |

14 | 30 | 4 | 196 | 4000 | |

Lithium-Ion | 10 | 5 | 1 | 784 | 4000 |

16 | 10 | 3 | 343 | 4000 | |

14 | 15 | 4 | 216 | 4000 | |

14 | 20 | 5 | 180 | 4000 | |

14 | 25 | 5 | 158 | 4000 | |

14 | 30 | 6 | 137 | 4000 |

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## Share and Cite

**MDPI and ACS Style**

Kosmadakis, I.E.; Elmasides, C.
A Sizing Method for PV–Battery–Generator Systems for Off-Grid Applications Based on the LCOE. *Energies* **2021**, *14*, 1988.
https://doi.org/10.3390/en14071988

**AMA Style**

Kosmadakis IE, Elmasides C.
A Sizing Method for PV–Battery–Generator Systems for Off-Grid Applications Based on the LCOE. *Energies*. 2021; 14(7):1988.
https://doi.org/10.3390/en14071988

**Chicago/Turabian Style**

Kosmadakis, Ioannis E., and Costas Elmasides.
2021. "A Sizing Method for PV–Battery–Generator Systems for Off-Grid Applications Based on the LCOE" *Energies* 14, no. 7: 1988.
https://doi.org/10.3390/en14071988