Economic optimization of component sizing for residential battery storage systems

energies Article Economic Optimization of Component Sizing for Residential Battery Storage Systems Holger C. Hesse 1,*, Rodrigo Martins 2, Petr Musilek 2,3, Maik Naumann 1, Cong Nam Truong 1 and Andreas Jossen 1 1 Department of Electrical and Computer Engineering, Technical University of Munich (TUM), 80333 Munich, Germany; [email protected] (M.N.); [email protected] (C.N.T.); [email protected] (A.J.) 2Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada; [email protected] (R.M.); [email protected] (P.M.) 3Electrical Engineering and Computer Science, VSB-Technical University Ostrava, 70800 Ostrava, Czech Republic *Correspondence: holger[email protected]; Tel.: +49-89-289-26964 Academic Editor: Haolin Tang Received: 7 May 2017; Accepted: 12 June 2017; Published: 22 June 2017 Abstract: Battery energy storage systems (BESS) coupled with rooftop-mounted residential photovoltaic (PV) generation, designated as PV-BESS, draw increasing attention and market penetration as more and more such systems become available. The manifold BESS deployed to date rely on a variety of different battery technologies, show a great variation of battery size, and power electronics dimensioning. However, given today’s high investment costs of BESS, a well-matched design and adequate sizing of the storage systems are prerequisites to allow profitability for the end-user. The economic viability of a PV-BESS depends also on the battery operation, storage technology, and aging of the system. In this paper, a general method for comprehensive PV-BESS techno-economic analysis and optimization is presented and applied to the state-of-art PV-BESS to determine its optimal parameters. Using a linear optimization method, a cost-optimal sizing of the battery and power electronics is derived based on solar energy availability and local demand. At the same time, the power flow optimization reveals the best storage operation patterns considering a trade-off between energy purchase, feed-in remuneration, and battery aging. Using up to date technology-specific aging information and the investment cost of battery and inverter systems, three mature battery chemistries are compared; a lead-acid (PbA) system and two lithium-ion systems, one with lithium-iron-phosphate (LFP) and another with lithium-nickel-manganese-cobalt (NMC) cathode. The results show that different storage technology and component sizing provide the best economic performances, depending on the scenario of load demand and PV generation. Keywords: battery energy storage system; battery aging; linear programming; size optimization; Lithium-Ion battery; cost analysis; photovoltaic panel; economic analysis; residential battery 1. Introduction and Related Work Battery energy storage systems (BESS) are considered for a variety of applications in modern power grids [ 1 ]. As these systems decline drastically in cost, commercial and customer interest for this type of storage is growing. As a result, the combination of residential photovoltaic (PV) systems with battery storage (“PV-battery energy storage systems”, PV-BESS) and grid connection (grid-connected PV-BESS) have attained significant growth rates [2–4]. Such systems enable customers to avoid the retail electricity tariff for all energy fostered by surplus PV generation via buffering in the BESS, instead of selling surplus power at the feed-in tariff. This is a potentially profitable scenario in countries where the electricity retail tariff exceeds PV feed-in Energies 2017,10, 835; doi:10.3390/en10070835 www.mdpi.com/journal/energies Energies 2017,10, 835 2 of 19 tariffs, e.g., Australia, Canada, regions in the USA, and multiple countries in Europe. Academia has analyzed the economic value of PV-BESS for various individual systems [ 5 – 7 ], and small but positive business cases seem to be in reach for specific usage scenarios. In addition, several online tools are available free of charge and are capable of analyzing the benefit for specific BESS with respect to load and PV size variation [ 8 – 11 ]. These multiple approaches provide a sensitivity analysis for given BESS systems but are unable to guide residential customers to find the economically best-suited storage and inverter combination for their specific needs. Despite the fact that PV-BESS is still a niche market at present, various automotive companies have started to enter the market and have announced products with drastically lower price tags, e.g., Tesla, Mercedes-Benz, and Nissan [ 12 – 14 ], making PV-BESS potentially economic in multiple regions around the world [4]. Interestingly, currently available and announced PV-BESS rely on different battery technologies and show a strong variation of storage size [ 15 ]. At a first glance, there appears to be a market trend towards lithium-ion based systems with storage capacity above 5 kWh coupled with inverter sizes of nominal power (P N ) often exceeding P N =3kW[ 3 ]. Nevertheless, lead-acid (PbA) systems still hold an appreciable market share of over 10% for new system installations, and there is a strong competition within the category of lithium-ion batteries within which different cell chemistries differ significantly in performance, cost, and aging [3]. Despite the market availability of these various systems, there is still an obvious lack of accurate quantitative assessment tools to determine return on investment (ROI)-optimal storage solutions for individual households with particular PV generation and load demand. The methods and evaluation tools presented in this paper will help to determine the best-suited storage technology and system dimensioning for a variety of BESS application settings. While most existing studies assess the economic value of residential battery storage using sensitivity analysis, there is lack of system size optimization studies considering technology specific parameters and aging information [ 16 ]. Nevertheless, numerous significant contributions in the literature describe the usage of optimization routines for storage dispatch and size optimization in a distinct but related context. Their overview is presented in Table 1. Table 1. Non-comprehensive overview of literature in the field of battery energy storage system (BESS) analysis and optimization. Application Type/Focus of Research References Vehicle Economic analysis [2] Residential Market analysis [3,15] Techno-economic analysis [5–7,16] Online economic estimation tools [8–11] Size optimization (genetic algorithm) [17] Optimization of power flow (dynamic programming) [18] Inverter size (sensitivity analysis) [19] Co-optimization of electricity and thermal energy flow [20] Commercial Techno-economic analysis [21] Other/grid level BESS for distribution grid support [22,23] BESS microgrid support [24] Various/comparison of applications Technical review [1] Economic value assessment [4] Complex optimization approaches can be applied to storage dispatch optimization in various use cases. Although this helps to reveal possible operation modes of a system, such approaches often require extensive computational resources and may fail to find a globally optimal solution. Geth et al. [ 22 ] show an optimization method for the best positioning and sizing of energy storage in distribution grids. Using a multi-objective optimization method, the authors find an optimized dispatch operation strategy for multiple households with respect to BESS profit generation via energy Energies 2017,10, 835 3 of 19 market trading. They also provide a detailed discussion of concerns of distribution system operators related to security of the energy service, e.g., using voltage control. In a subsequent work, Tant et al. [ 23 ] demonstrate how complex optimization methods can be applied to find the best-suited battery storage system for PV integration in a given distribution grid. The authors analyze in detail the storage dispatch optimization using PbA and lithium-ion batteries. However, this work focuses on multi-objective optimization for peak shaving and voltage regulation, rather than on aspects relevant to a single household cost optimization. Recent work by Merei et al. [ 21 ] concentrates on commercial applications of BESS. The authors use sensitivity analysis to study the maximization of energy self-consumption via storage integration. The techno-economic analysis reveals that, for most commercial applications, BESS is not favored economically when battery degradation is taken into account. Other previous work by Magnor and Sauer [ 17 ] and Merei et al. [ 24 ] analyzed the optimal sizing of storage in the context of island grids and home storage systems. A genetic algorithm-based method allows the modelling of a non-linear set of equations including battery-aging models. However, the solver results may not find a globally optimal solution to the described problem, and the studies do not provide design rules for future storage systems. In contrast, others have used sensitivity studies to reveal the optimal size of storage system components. For example, Weniger et al. [ 19 ] provide a detailed analysis of power conversion efficiency of state-of-art battery home storage inverters. However, this work does not consider the economic impact of component sizing. Muenzel et al. [ 18 ] investigate the economics of residential storage systems with a dynamic programming derived operation strategy and screen payback periods achievable for several storage system sizes in Australia. Using generalized parameters for the inverter, cost, and degradation of an unspecified litium-ion battery type, they anticipate a positive return on investment in the near future. In general, such sensitivity analyses commonly fail if various parameters are to be screened and optimized at the same time. This issue can be effectively resolved using linear optimization approaches, which have been successfully applied to energy storage optimization. For example, Lauingera et al. provide a framework for electrical and thermal storage integration in households [ 20 ]. Based on linear programming, the energy dispatch of a residential building is optimized. However, this work does not consider the sizing optimization of storage and periphery, and battery storage aging is not part of the model. In contrast to the aforementioned publications, this work conducts a comprehensive power flow analysis, implements technology-specific battery degradation, and presents a highly reproducible and easily adaptable linear optimization approach to assess both the cost and the maximum profit attainable for residential BESS. Parameterized with conditions matching the German regulatory framework as well as detailed cost and aging information for three commonly deployed battery technologies (a PbA and two lithium-ion systems), this approach allows the best storage type and power electronics size to be selected for households with rooftop-mounted PV generators. The presented results also provide design rules applicable to residential PV-BESS around the world. The remainder of this paper is organized as follows: Section 2introduces the system layout and parameters necessary as input for the subsequent optimization procedure. The linear programming methodology, including equations and constraints for optimization, is described in Section 3. The subsequent Section 4describes obtained results and discusses the findings. The final Section 5 summarizes major conclusions and outlines possible directions for future research. 2. Photovoltaic-Battery Energy Storage Systems Layout, Storage Model and Parametrization This section summarizes all parameters relevant for BESS optimization. It describes the system layout, overviews technical parameters of the storage systems under investigation, and specifies the economic framework considered in this study. Energies 2017,10, 835 4 of 19 2.1. System Layout The schematic diagram of Figure 1shows the system configuration as well as electrical connections and power flows for the PV-BESS system under study. All variables necessary for subsequent modeling are explained in more detail later, along with the definition of the optimization problem. The arrows in Figure 1indicate the directions of power flows allowed for all component links. For this work, the optimization approach is confined to an alternate current (AC) coupling of battery storage, which offers the broadest flexibility in system design and is also suitable for the retrofitting of existent PV installations [ 6 ]. It is worth mentioning that a variety of different direct current (DC) system coupling topologies (e.g., generator coupled or converter link topology) have also been proposed for PV-BESS. Although such differing topologies have their individual strengths and weaknesses, an overall consistent trend for choice of best technology and storage system sizing is expected [6]. Energies 2017, 10, 835 4 of 18 2.1. System Layout The schematic diagram of Figure 1 shows the system configuration as well as electrical connections and power flows for the PV-BESS system under study. All variables necessary for subsequent modeling are explained in more detail later, along with the definition of the optimization problem. The arrows in Figure 1 indicate the directions of power flows allowed for all component links. For this work, the optimization approach is confined to an alternate current (AC) coupling of battery storage, which offers the broadest flexibility in system design and is also suitable for the retrofitting of existent PV installations [6]. It is worth mentioning that a variety of different direct current (DC) system coupling topologies (e.g., generator coupled or converter link topology) have also been proposed for PV-BESS. Although such differing topologies have their individual strengths and weaknesses, an overall consistent trend for choice of best technology and storage system sizing is expected [6]. Figure 1. Schematic illustration of the investigated alternate current (AC) topology photovoltaicbattery energy storage systems coupling. Arrows indicate the direction of possible power flows between the individual components. 2.2. Storage System Technical Parameters, Cost Assumptions and Battery Aging Model This study analyzes the economic potential and technical capabilities of three commonly used battery technologies for PV-BESS; a typical vented PbA system and two lithium-ion systems with lithium-iron-phosphate (LFP) and lithium-nickel-manganese-cobalt (NMC) cathodes, respectively. Table 2 provides an overview of the characteristic parameters for the individual technologies under investigation (Appendix A provides a more detailed survey of common performance data and citations to literature references for all battery technologies under consideration). It is worth mentioning here that data on aging and lifetime predictions are highly sensitive to various influencing factors (e.g., cell construction type, sealing quality, electrolyte additives) and test conditions. Furthermore, the values are likely to vary between batteries of individual manufacturers. However, it is not the focus of this work to question the correctness of the available lifetime data. The trends are well in accordance with the literature, and lifetime estimations derived in this work match well with the expert knowledge of BESS manufacturers. The battery efficiency () is given as an averaged number of round trip Watt-hour retention using typical low charge and discharge rates of 0.1 C (capacity-rate) and ambient temperature (approximately 25 °C). These conditions correspond well to the scenarios commonly present for a typical home storage system. Self-discharge () values considered in the optimization model are also listed in Table 1 and taken into account during simulations. However, having relatively small values, self-discharge plays a minor role, especially for the lithium-ion based battery chemistries. Figure 1. Schematic illustration of the investigated alternate current (AC) topology photovoltaic-battery energy storage systems coupling. Arrows indicate the direction of possible power flows between the individual components. 2.2. Storage System Technical Parameters, Cost Assumptions and Battery Aging Model This study analyzes the economic potential and technical capabilities of three commonly used battery technologies for PV-BESS; a typical vented PbA system and two lithium-ion systems with lithium-iron-phosphate (LFP) and lithium-nickel-manganese-cobalt (NMC) cathodes, respectively. Table 2provides an overview of the characteristic parameters for the individual technologies under investigation (Appendix Aprovides a more detailed survey of common performance data and citations to literature references for all battery technologies under consideration). It is worth mentioning here that data on aging and lifetime predictions are highly sensitive to various influencing factors (e.g., cell construction type, sealing quality, electrolyte additives) and test conditions. Furthermore, the values are likely to vary between batteries of individual manufacturers. However, it is not the focus of this work to question the correctness of the available lifetime data. The trends are well in accordance with the literature, and lifetime estimations derived in this work match well with the expert knowledge of BESS manufacturers. The battery efficiency ( ηbatt ) is given as an averaged number of round trip Watt-hour retention using typical low charge and discharge rates of 0.1 C (capacity-rate) and ambient temperature (approximately 25 ◦ C). These conditions correspond well to the scenarios commonly present for a typical home storage system. Self-discharge ( SDbatt ) values considered in the optimization model Energies 2017,10, 835 5 of 19 are also listed in Table 1and taken into account during simulations. However, having relatively small values, self-discharge plays a minor role, especially for the lithium-ion based battery chemistries. Table 2. Performance parameters of BESS using three different battery technologies. The data was derived from a literature survey (see Appendix A). Terms state of charge (SOC) and full equivalent cycles (FEC) are further explained in the text. Parameter Unit Battery Technology PbA LFP NMC ηbatt : Battery round-trip efficiency % 85 98 95 SDbatt: Self-discharge per day % 0.17 0.02 0.02 (SOCmin −SOCmax): Usable SOC % 50–100% 5–95% 5–95% Li f e80% Cal : Calendric life indicator in years (years) 10 15 13 Li f e80% Cyc : Cycle life indicator in FEC (FEC) 1500 10,000 4500 Cvar,bat: Variable battery price €/kWh 271 752 982 Cf ix : Fixed price for storage (price for housing, cooling, and periphery) €1182 1723 580 In contrast, the aging of storage devices cannot be neglected. In fact, the deterioration of storage is a major cost driver during storage operation. It is common to differentiate between cyclic and calendric aging processes for battery degradation, as described in detail for PbA [ 25 ] and lithium-ion batteries [ 26 ]. The battery cyclic and calendric lifetime indicators ( Li f e80% Cyc ; Li f e80% Cal ) specify a battery usage scenario, until a certain capacity fade for a battery cell becomes evident. As obvious from the nomenclature, values provided in the table are linked to the remaining state of health (SOH) of 80% nominal capacity, matching a typical replacement criterion for automotive applications. In this paper, we use a simple estimate for solely time-dependent calendric aging processes as well as a charge throughput-dependent cyclic aging model. The values of calendric lifetime ( Li f e80% Cal ) provide a reference value for storage degradation to 80% SOH at 20 ◦C temperature, when no charge throughput is applied. To describe the cyclic aging ( Li f e80% Cyc ) caused by energy throughput in the battery storage, a correlation with full equivalent cycles (FEC), based on the definition by Fuchs et al. [27], is used: FEC =0.5 ×1 tZSOC(t)dt ≈0.5 ×R|Pbatt|dt Enom batt . (1) The factor of 0.5 results from the conversion of charge throughput to full cycle counting consisting of one charging and one discharging process. SOC denotes the state of charge, Pbatt the power flow via the battery, and Enom batt the nominal energy capacity of the battery. A theoretical maximum charge throughput is defined via Li f e80% Cyc , i.e., the number of FEC until 80% capacity is reached if there were no calendric aging. To formulate battery aging for subsequent modeling, the following equations are derived: agingcal =∆t Li f e80% Cal . (2) agingcyc =0.5 ×R|Pbatt|dt Li f e80% Cyc ×Enom batt . (3) In accordance with Schmalstieg et al. [ 28 ], a superposition principle is used to estimate the overall aging: agingtot =agingcal +agingcyc. (4) As such, a parameter value of agingtot = 0 corresponds to a fresh, unused battery, whereas at agingtot = 1, the remaining capacity of the battery is 80% of its original value as a result of calendric time and battery use. Further use of the storage system with agingtot > 1 might be allowed if the replacement of storage is set to a remaining capacity below 80%, as further described below. Energies 2017,10, 835 6 of 19 A detailed analysis and validation of battery performance and aging models in the context of such techno-economic applications is given in [29]. Table 2also summarizes the economic parameters of the storage system. In this study, the investment costs of different battery types for BESS and the inverter coupling are analyzed independently. The values listed in the table are derived from a recent detailed market survey with n= 445 storage systems using a fit to the systems with lowest purchase prices [ 15 ]. They are discussed in more detail in a separate publication [ 30 ]. We attribute the lower fixed price for NMC storage compared to the price offset determined for LFP and PbA systems mainly to synergy effects attainable for storage systems that have been developed for the electric vehicle automotive market (relying mostly on NMC-based battery chemistry). For the sole battery storage investment without an inverter, the following price structure is considered: CBatt(Enom batt )=Cf ix +Cvar,Bat ×Enom batt . (5) where CBatt represents the price of the battery, Cf ix is the fixed price including all peripheries and housing of storage, and Cvar,Bat is the energy specific price of a storage system. The storage maintenance cost within the battery lifetime is negligible and not considered herein. Furthermore, the separate installation cost of the storage system is not taken into account; such costs are strongly linked to the PV installation cost and may vary strongly for individual households. For the inverter systems, the following assumptions are made: one way conversion efficiency ηinv = 97.5%, calendric life of 20 years, and a variable cost of approximately Cvar,inv = 155 /kW (see also Table 3). Data used was derived from an internet market survey on standalone DC/AC inverters and expert interviews with leading brand inverter manufacturers [ 15 , 31 ]. As most PV-BESS package batteries and inverters are in one casing, no separate fixed costs for inverters are assumed but are given as part of the overall storage fixed cost Cf ix. Table 3. Inverter performance and price information derived from literature survey [15,31]. Inverter Data Unit Value ηinv: Average one way inverter efficiency % 97.5 Tinv: Assumed inverter lifetime in years (years) 20 Cvar,inv: Cost of inverter per nominal power €/kW 155 As such, the overall cost Cstorage for the energy storage system including battery storage with energy content Enom batt , inverter with nominal power Pnom inv , and all peripherals cost Cf ix totals to: CstorageEnom batt ,Pnom inv =CbatteryEnom batt +CinverterPnom inv  =Cf ix +Cvar,batt ×Enom batt +Cvar,inv ×Pnom inv .(6) 2.3. Economic and Legal Framework for Battery Energy Storage Systems For the economic framework refer to Table 4. A retail energy price of 28.69 ct € /kWh and feed-in tariff of 12.31 ct € /kWh are assumed, in accordance with a retail price analysis and EEG (“Erneuerbare Energien Gesetz”—German renewable energies act), which granted feed-in tariffs for PV installations in Germany for 2016 [ 32 , 33 ]. Furthermore, in accordance with German regulations, a feed-in limit of fnoEES = 70% has to be taken into account for all residential PV installations. This means that power exceeding the feed-in limit Pf eed,max =fnoEES ×Ppeak,PV may not be injected from the household back to the grid. Instead, this additional power can be either stored in a battery or an unfavorable curtailment becomes effective (i.e., regulatory forced dissipative energy loss at the PV generator/inverter). It is worth mentioning that, for storage installations taking advantage of a government funded subsidy program on home storage systems, the PV grid feed-in regulation is enforced with a more strict curtailment rate of 50% [ 34 ]. For such partially subsidized systems, a discount of storage system Energies 2017,10, 835 7 of 19 investment may be obtained. As such, for subsidized systems with discount rate rsubsidy , the storage investment cost is given as: Csubsidy storage =Cstorage ×1−rsubsidy. (7) Table 4. Remuneration and retail energy prices for households in Germany (2016), and legal framework for PV-grid feed-in. Economic and Legal Framework Variable Value Retail energy price cbuy 28.69 ct€/kWh Feed-in energy reimbursement tariff csell 12.31 ct€/kWh Maximum feed-in ratio (without BESS subsidy) fnoEES 0.7 Maximum feed-in ratio (with BESS subsidy) fEES 0.5 Government subsidy rate for storage systems rsubsidy 0.22 3. Linear Optimization of Photovoltaic-Battery Energy Storage Systems The structure of the optimization problem addressed in this study can be represented by a mathematical model. The objective function and the constraints have linear relationships, meaning that the effect of changing a decision variable is proportional to its magnitude. This makes linear programming (LP) well suited to solve the optimization problem considered here, due to the linearity of the decision variable on electricity price, feed-in tariff, and other parameters. While some aspects of battery system operation are not linear, they can be linearized to fit the requirements of LP. e.g., models of BESS aging processes can be reductively applied to obtain a linearized degradation function. In addition, linear optimization provides unambiguous, repeatable results with modes and controllable computational effort, compared to other optimization methods typically based on heuristics or meta-heuristics. The economically optimal battery storage component sizing for a household equipped with PV and an energy storage system is obtained using LP. The load demand and PV-generation profiles considered in this study cover one full year, to capture all seasons with their characteristic, distinct patterns of PV-generation, storage, and grid energy transfers. As the intent is to minimize electricity cost and maximize the revenue generation on the profit side, two types of profits are considered: the profit attainable by feeding energy into the grid, and the avoided cost stemming from the reduced need to purchase energy when a storage system is installed. On the annual cost side, a fraction of the total cost of the energy storage system Ctot is considered. This fraction is determined based on a battery storage technology-specific aging analysis as further described below. The presented cost flow analysis takes into account the discounted storage cost caused by degradation. Data used for simulations was averaged with a resolution of ∆tres = 15 min , a value that provides a reasonable compromise between the accuracy of the obtained results and computational speed [ 35 ]. As such, the one-year simulation time frame covers a total of 35040 time intervals, indexed with variable i. All variables and parameters considered in this study are described in Table 5. The locally generated PV power ( Ppvi ) is first used to satisfy the local demand. When the local power production is greater than the demand, the surplus power is preferably transferred to the battery ( Ppv−batti) and stored for later use. If there is still additional energy available, the surplus power is injected into the grid ( Ppv−gridi ) or curtailed via feed-in limitation ( Pcurtaili ). The following equation considers all power flows from the PV generator: Ppvi=Ppv−loadi+Ppv−batti+Ppv−gridi+Pcurtaili. (8) Energies 2017,10, 835 8 of 19 Table 5. Variables and parameters used for the battery modeling and optimization routines. Battery Modelling Parameter Variable Unit Constraints/Comments Load demand (historical data) PloadikW ≥0; input data PV power generated (historical data) PpvikW ≥0; input data Nominal power of the battery inverter Pnom inv kW subject to optimization Nominal battery capacity Enom batt kWh subject to optimization Bidirectional power flow from/to the battery PbattikW result of optimization PV power fed to the load Ppv−loadikW ≥0; see Equations (8) and (10) PV power stored in the battery Ppv−battikW ≥0; see Equation (8) PV power exported to the grid Ppv−gridikW ≥0; see Equation (8) Power transferred from the battery to the load Pbatt−loadikW see Equation (10) Power exported from the battery to the grid Pbatt−gridikW ≥0; see Equation (13) Power imported from the grid to the load Pgrid−loadikW ≥0; see Equation (10) Surplus power-curtailed according to regulations PcurtailikW ≥0; see Equation (9) State of health SOHip.u. see Equations (18) and (19) Battery energy content at time i EbattikWh see Equations (14) and (15) State of charge SOCip.u. [SoCmin;SoCmax] To avoid back-feeding of power injected into the grid from PV system owners, a feed-in limitation is enforced. Any power above the limitation threshold value must be discarded as a curtailment loss, i.e., Ppv−gridi+Pcurtaili≤Pf eed,max. (9) To meet the electrical demand ( Ploadi ) the system first attempts to use power from local generation ( Ppv−loadi ). If this is not sufficient, power is drained from the battery ( Pbatt−loadi ). As the last resource, the system draws power from the grid ( Pgrid−loadi ). Consequently, demand is comprised of the following three components: Ploadi=Ppv−loadi+Pbatt−loadi+Pgrid−loadi. (10) The bidirectional power flow from the storage inverter to the battery is stored in an auxiliary variable (Pbatti)and correlated with the inverter efficiency ηinv: Pbatti=ηinv ×Ppv−batti−1 ηinv ×(Pbatt−loadi+Pbatt−gridi). (11) where ηinv is the average one-way efficiency of the inverter. The reciprocal efficiencies are the battery charge power Ppv−batti and the discharge power Pbatt−loadi+Pbatt−gridi , both limited by the nominal power flow from the inverter to the battery: 0≤Ppv−batti≤Pnom inv . (12) 0≤Pbatt−loadi+Pbatt−grid ≤Pnom inv . (13) where Pnom inv corresponds to the inverter nominal size. The battery energy content at time step i( Ebatti ) satisfies the recurrence relation: Ebatti=Ebatti−1×SDbatt d+ηbatt ×Pbatti×1h ∆tres . (14) where SDbatt represents the self-discharge factor of the battery and d= 96 is a conversion factor of the number of time steps per day. The energy content of the storage system is further confined by an upper boundary that decreases upon usage and aging according to the SOH: Ebatti≤Euseable batt ×SOHi. (15) Energies 2017,10, 835 9 of 19 The SOH is defined as the irreversible capacity fade over time, related to the nominal battery capacity, and Euseable batt is a fraction of the total energy content of the battery installed: Euseable batt =Enom batt ×(SOCmax −SOCmin). (16) For battery usage in stationary and automotive applications, it is useful to define an end of life (EOL) criterion, which is often linked to the SOH with a certain percentage value αReplace [ 36 ]. This percentage value also defines the time of battery replacement: EOL →SOHt=EOL ≤αReplace. (17) In many cases, αreplace = 80% or 70% is used for automotive applications. However, lower values are often stated for less demanding residential storage applications [ 6 , 37 ]. In this study, αreplace = 60% is used as the replacement parameter, matching e.g., the warranty conditions of the Tesla ® Powerwall product. Using the definition of agingtot =1 at SOH =80%, the SOH condition reads: SOH =1−agingtot ×0.2. (18) The time evolution of SOH also satisfies the recurrence relation: SOHi=SOHi−1−agingcali+agingcyci×0.2. (19) Using Equations (1) and (2), the calendric and cyclic aging can be estimated as: agingcali=i×∆tres Li f e80% Cal . (20) agingcyci=agingcyci−1+0.5 ×Pbatti×∆tres Ebatti ×1 Li f e80% Cyc . (21) As such, the additional cyclic aging degradation of time step i is estimated by the energy throughput in that time step ( Pbattj×∆tres ) divided by the energy content of the system Ebatti . Additionally, it is normalized with the factor 0.5 and the technology specific cycle life indicator Li f e80% Cyc . Similarly, the SOC can be expressed as: SOCi=Ebatti Euseable batt ×SOHi . (22) The optimal solution must satisfy all constraints described above. It aims to reduce the overall cost by minimizing the expenses for energy purchase and the implicit cost caused by battery degradation. This cost model is divided into three components, i.e., minimize Ctot =Cbuy_energy −Rsell_energy +Csubsidy storage_deg . (23) subject to constraints in equations and inequalities (8)–(22). The first component Cbuy_energy comprises the cost of energy purchased from the grid, while the second component Rsell_energy is the revenue from PV energy generation exported to the grid. These two components are evaluated as follows: Cbuy_energy =∑ i Cbuy ×Pgrid−loadi. (24) Rsell_energy =∑ i Csell ×(Ppv−gridi+Pbatt−grid). (25) Energies 2017,10, 835 16 of 19 Acknowledgments: The authors acknowledge financial support from the Technical University Munich and Nanyang Technological University funded International Center for Energy Research (ICER) Project and the Bavarian funded EEBatt project, the Science without Border PhD grant (BEX 13301/13-6), and the Natural Sciences and Engineering Research Council (NSERC) of Canada. This work was also 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: Holger C. Hesse established the mathematical framework for the techno-economic analysis of energy storage systems. Rodrigo Martins developed the optimization model and executed the simulation experiments. Maik Naumann and Cong Nam Truong contributed to the result analysis and sensitivity study. Petr Musilek and Andreas Jossen provided overall guidance for the study and contributed with many fruitful discussions on the methodology. Holger C. Hesse wrote the paper with contributions of all co-authors. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations BESS Battery energy storage system EOL End of life LFP Lithium-ion battery with graphite anode and iron (Fe)-phosphate cathode LP Linear programming (mixed integer LP) NMC Lithium-ion battery with graphite anode and nickel-manganese-cobalt cathode PbA Lead (Pb)-(sulfuric)-acid battery PV Photovoltaic generator ROI Return on invest SOH Battery state of health Appendix A Table A1. Literature review for battery performance parameters used in this study. For all table fields the value used for simulations is given first. In some cases, other values are given in brackets—these are for information to the reader only, but not further used in the paper. Parameter Variable Unit PbA LFP NMC Battery round trip efficiency ηbatt %85 [41,42] * (80 [43]) 98 [41,42] * (95 [43]) 95 [43] ** Battery self-discharge SD %/day 0.17 [41] (0.2 [1,44] 0.1 [1]) 0.02 [41,42] * (0.33 [44] 0.1 [1,44]) 0.02 [45] Calendric lifetime Li f e80% Cal (years) 10 [41] (5 [1] 8 [46]) 15 [1] (12–20+ [42]) 13 [44,45] Cyclic lifetime Li f e80% Cyc FEC 1500 [46] *** (200–1300 [1,41]) 10,000 [45] **** (6000 [42,47] 1000–10,000+ [1,48]) 4500 [29] (700–1000 [49]) * Experiments conducted at the following parameters: 1/10 C, 25 ◦ C, 50% SOC; ** Experiments conducted at the following parameters: 1 C, 25 ◦C, 50% SOC; *** Derived at 50% DoD; **** Tested at 60–100% DoD. Energies 2017,10, 835 17 of 19 Appendix B Energies 2017, 10, 835 16 of 18 Table A1. Literature review for battery performance parameters used in this study. For all table fields the value used for simulations is given first. In some cases, other values are given in brackets—these are for information to the reader only, but not further used in the paper. Parameter Variable Unit PbA LFP NMC Battery round trip efficiency  % 85 [41,42] * (80 [43]) 98 [41,42] * (95 [43]) 95 [43] ** Battery self-discharge  %/day 0.17 [41] (0.2 [1,44] 0.1 [1]) 0.02 [41,42] * (0.33 [44] 0.1 [1,44]) 0.02 [45] Calendric lifetime  % (years) 10 [41] (5 [1] 8 [46]) 15 [1] (12–20+ [42]) 13 [44,45] Cyclic lifetime  % FEC 1500 [46] *** (200–1300 [1,41]) 10,000 [45] **** (6000 [42,47] 1000–10,000+ [1,48]) 4500 [29] (700–1000 [49]) * Experiments conducted at the following parameters: 1/10 C, 25 °C, 50% SOC; ** Experiments conducted at the following parameters: 1 C, 25 °C, 50% SOC; *** Derived at 50% DoD; **** Tested at 60–100% DoD. Appendix B Figure A1. PV generation and load profile used for this simulation study. Both profiles are scaled according to the scenarios described in the paper. The load profile was taken from [38]. References 1. Luo, X.; Wang, J.; Dooner, M.; Clarke, J. Overview of current development in electrical energy storage technologies and the application potential in power system operation. Appl. Energy 2015, 137, 511–536. 2. Nykvist, B.; Nilsson, M. Rapidly falling costs of battery packs for electric vehicles. Nat. Clim. Chang. 2015, 5, 329–332. 3. Kairies, K.; Haberschusz, D.; van Ouwerkerk, J.; Strebel, J.; Wessels, O.; Magnor, D.; Badeda, J.; Sauer, D. Wissenschaftliches Mess—Und Evaluierungsprogramm Solarstromspeicher: Jahresbericht 2016; Speicher Monitoring: Aachen, Germany, 2016. 4. Fitzgerald, G.; Mandel, J.; Morris, J. The Economics of Battery Energy Storage, 2015. Available online: http://www.rmi.org/ELECTRICITY_BATTERY_VALUE (accessed on 26 October 2016). 5. Naumann, M.; Karl, R.C.; Truong, C.N.; Jossen, A.; Hesse, H.C. Lithium-ion Battery Cost Analysis in PVhousehold Application. Energy Procedia 2015, 73, 37–47. 6. Truong, N.C.; Naumann, M.; Karl, C.R.; Müller, M.; Jossen, A.; Hesse, C.H. Economics of Residential Photovoltaic Battery Systems in Germany: The Case of Tesla’s Powerwall. Batteries 2016, 2, 14. 7. Garimella, N.; Nair, N.K.C. Assessment of battery energy storage systems for small-scale renewable energy integration. In Proceedings of the TENCON 2009–2009 IEEE Region 10 Conference, Singapore, 23–26 January 2009. Figure A1. PV generation and load profile used for this simulation study. Both profiles are scaled according to the scenarios described in the paper. The load profile was taken from [38]. References 1. Luo, X.; Wang, J.; Dooner, M.; Clarke, J. Overview of current development in electrical energy storage technologies and the application potential in power system operation. Appl. Energy 2015 ,137, 511–536. [CrossRef] 2. Nykvist, B.; Nilsson, M. Rapidly falling costs of battery packs for electric vehicles. Nat. Clim. Chang. 2015 ,5, 329–332. [CrossRef] 3. Kairies, K.; Haberschusz, D.; van Ouwerkerk, J.; Strebel, J.; Wessels, O.; Magnor, D.; Badeda, J.; Sauer, D. Wissenschaftliches Mess—Und Evaluierungsprogramm Solarstromspeicher: Jahresbericht 2016; Speicher Monitoring: Aachen, Germany, 2016. 4. Fitzgerald, G.; Mandel, J.; Morris, J. The Economics of Battery Energy Storage. 2015. Available online: http://www.rmi.org/ELECTRICITY_BATTERY_VALUE (accessed on 26 October 2016). 5. Naumann, M.; Karl, R.C.; Truong, C.N.; Jossen, A.; Hesse, H.C. Lithium-ion Battery Cost Analysis in PV-household Application. Energy Procedia 2015,73, 37–47. [CrossRef] 6. Truong, N.C.; Naumann, M.; Karl, C.R.; Müller, M.; Jossen, A.; Hesse, C.H. Economics of Residential Photovoltaic Battery Systems in Germany: The Case of Tesla’s Powerwall. Batteries 2016,2, 14. [CrossRef] 7. Garimella, N.; Nair, N.K.C. Assessment of battery energy storage systems for small-scale renewable energy integration. In Proceedings of the TENCON 2009–2009 IEEE Region 10 Conference, Singapore, 23–26 January 2009. 8. Quaschning, V. Unabhängigkeitsrechner. 2015. Available online: pvspeicher.htw-berlin.de/unabhaengigkeitsrechner/ (accessed on 9 February 2017). 9. Benz, M. Online-Rechner für SENEC.IES Stormspeicher. Available online: www.speicher-rechnen.de (accessed on 4 May 2017). 10. Piepenbrinck, A. E3/DC System Calculator. Available online: http://s10.e3dc.com/E3dcWeb/SystemCalculator/ syscalc.php (accessed on 4 May 2017). 11. Varta-storage GmbH. Varta-Storage Berechnungstool. 2017. Available online: https://www.varta-storage. com/de/produkte/heimspeichersysteme/berechnungstool.html (accessed on 4 May 2017). 12. Tesla Motors. The Tesla Home Battery. Available online: https://www.tesla.com/powerwall (accessed on 4 May 2017). 13. Mercedes-Benz. Mercedes-Benz Energy Storage. Available online: https://www.mercedes-benz.com/en/ mercedes-benz-energy/products/ (accessed on 4 May 2017). 14. Nissan. xStorage by Nissan-Clean Power Energy. 2017. Available online: https://www.nissan.co.uk/ experience-nissan/electric-vehicle-leadership/xstorage-by-nissan.html (accessed on 4 May 2017). 15. Fuhs, M. Marktübersicht Home-Speicher. PV-Magazine 2016,2016, 35–39. Energies 2017,10, 835 18 of 19 16. Hoppmann, J.; Volland, J.; Schmidt, T.S.; Hoffmann, V.H. The economic viability of battery storage for residential solar photovoltaic systems–A review and a simulation model. Renew. Sustain. Energy Rev. 2014 , 39, 1101–1118. [CrossRef] 17. Magnor, D.; Sauer, D.U. Optimization of PV Battery Systems Using Genetic Algorithms. Energy Procedia 2016,99, 332–340. [CrossRef] 18. Muenzel, V.; Mareels, I.; Hoog, J.D.; Vishwanath, A.; Kalyanaraman, S.; Gort, A. PV generation and demand mismatch: Evaluating the potential of residential storage. In Proceedings of the 2015 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT), Washington, DC, USA, 18–20 February 2015. 19. Weniger, J.; Tjaden, T.; Bergner, J.; Quaschning, V. Sizing of Battery Converters for Residential PV Storage Systems. Energy Procedia 2016,99, 3–10. [CrossRef] 20. Lauinger, D.; Caliandro, P.; van Herle, J.; Kuhn, D. A linear programming approach to the optimization of residential energy systems. J. Energy Storage 2016,7, 24–37. [CrossRef] 21. Merei, G.; Moshövel, J.; Magnor, D.; Sauer, D.U. Optimization of self-consumption and techno-economic analysis of PV-battery systems in commercial applications. Appl. Energy 2016,168, 171–178. [CrossRef] 22. Geth, F.; Tant, J.; Haesen, E.; Driesen, J.; Belmans, R. Integration of energy storage in distribution grids. In Proceedings of the 2010 IEEE Power and Energy Society General Meeting, Providence, RI, USA, 25–29 July 2010; pp. 1–6. 23. Tant, J.; Geth, F.; Six, D.; Tant, P.; Driesen, J. Multiobjective battery storage to improve PV integration in residential distribution grids. IEEE Trans. Sustain. Energy 2013,4, 182–191. [CrossRef] 24. Merei, G.; Berger, C.; Sauer, D.U. Optimization of an off-grid hybrid PV-Wind-Diesel system with different battery technologies using genetic algorithm. Sol. Energy 2013,97, 460–473. [CrossRef] 25. Schiffer, J.; Sauer, D.; Bindner, H.; Cronin, T.; Lundsager, P.; Kaiser, R. Model prediction for ranking lead-acid batteries according to expected lifetime in renewable energy systems and autonomous power-supply systems. J. Power Sources 2007,168, 66–78. [CrossRef] 26. Vetter, J.; Novák, P.; Wagner, M.R.; Veit, C.; Möller, K.-C.; Besenhard, J.O.; Winter, M.; Wohlfahrt-Mehrens, M.; Vogler, C.; Hammouche, A. Ageing mechanisms in lithium-ion batteries. J. Power Sources 2005 ,147, 269–281. [CrossRef] 27. Fuchs, G.; Lunz, B.; Leuthold, M.; Sauer, D.U. Technology Overview on Electricity Storage, Overview on the Potential and on the Deployment Perspectives of Electricity Storage Technologies. 2012. Available online: http://www.sefep.eu/activities/projects-studies/120628_Technology_Overview_ Electricity_Storage_SEFEP_ISEA.pdf (accessed on 16 June 2016). 28. Schmalstieg, J.; Käbitz, S.; Ecker, M.; Sauer, D.U. A holistic aging model for Li(NiMnCo)O 2 based 18650 lithium-ion batteries. J. Power Sources 2014,257, 325–334. [CrossRef] 29. Goebel, C.; Hesse, H.; Schimpe, M.; Jossen, A.; Jacobsen, H.-A. Model-based Dispatch Strategies for Lithium-Ion Battery Energy Storage applied to Pay-as-Bid Markets for Secondary Reserve. IEEE Trans. Power Syst. 2016. [CrossRef] 30. Hesse, H.; Müller, M. Price and Market Trends for Battery Stationary Storage Systems. Unpublished work. 2017. 31. Photovoltaik4all. Available online: http://www.photovoltaik4all.de/en/wechselrichter (accessed on 7 September 2016). 32. Bundesverband der Energie-und Wasserwirtschaft e.V. BDEW Strompreisanalyse Mai. 2016. Available online: https://www.bdew.de/internet.nsf/res/886756C1635C3399C1257FC500326489/$file/160524_ BDEW_Strompreisanalyse_Mai2016.pdf (accessed on 4 May 2017). 33. Bundesnetzagentur. Dateneldungen und EEG-Vergütungssätze für Photovoltaikanlagen. 2017. Available online: https://www.bundesnetzagentur.de/DE/Sachgebiete/ElektrizitaetundGas/Unternehmen_Institutionen/ ErneuerbareEnergien/Photovoltaik/DatenMeldgn_EEG-VergSaetze/DatenMeldgn_EEG-VergSaetze_node. html (accessed on 4 May 2017). 34. KfW Bank. Förderprogram Erneuerbare Energien-Speicher. 2016. Available online: https://www.kfw.de/ Download-Center/F%C3%B6rderprogramme-(Inlandsf%C3%B6rderung)/PDF-Dokumente/6000002700_M_ 275_Speicher.pdf (accessed on 14 July 2016). 35. Beck, T.; Kondziella, H.; Huard, G.; Bruckner, T. Assessing the influence of the temporal resolution of electrical load and PV generation profiles on self-consumption and sizing of PV-battery systems. Appl. Energy 2016 , 173, 331–342. [CrossRef] Energies 2017,10, 835 19 of 19 36. Wenzl, H.; Baring-Gould, I.; Kaiser, R.; Liaw, B.Y.; Lundsager, P.; Manwell, J.; Ruddell, A.; Svoboda, V. Life prediction of batteries for selecting the technically most suitable and cost effective battery. In Selected Papers from the Ninth European Lead Battery Conference; Elsevier: Amsterdam, The Netherlands, 2005; Volume 144, pp. 373–384. 37. Battke, B.; Schmidt, T.S.; Grosspietsch, D.; Hoffmann, V.H. A review and probabilistic model of lifecycle costs of stationary batteries in multiple applications. Renew. Sustain. Energy Rev. 2013,25, 240–250. [CrossRef] 38. Quaschning, V. Representative Electrical Load Profiles of Residential Buildings in Germany with a Temporal Resolution of One Second; HTW Berlin—University of Applied Sciences: Berlin, Germany, 2015. 39. Nocedal, J.; Wright, S. Numerical Optimization; Springer Science & Business Media: New York, NY, USA, 2006. 40. Rodway, J.; Musilek, P.; Lozowski, E.; Prauzek, M.; Heckenbergerova, J. Pressure-based prediction of harvestable energy for powering environmental monitoring systems. In Proceedings of the 2015 IEEE 15th International Conference on Environment and Electrical Engineering (EEEIC), Rome, Italy, 10–13 June 2015. 41. Sterner, M.; Stadler, I. Energiespeicher. Bedarf, Technologien, Integration, 1st ed.; Springer: Berlin/Heidelberg, Germany, 2014. 42. European Society for Clinical Nutrition and Metabolism (ESPEN). Potentiale Elektrochemischer Speicher in Elektrischen Netzen in Konkurrenz zu Anderen Technologien und Systemlösungen (ESPEN); EFZN Project Report; ESPEN: Goslar, Germany, 2016. 43. Jossen, A.; Weydanz, W. Moderne Akkumulatoren Richtig Einsetzen, 1st ed.; Reichardt: Untermeitingen, Germany, 2006. 44. Reddy, T. Linden’s Handbook of Batteries, 4th ed.; McGraw-Hill: New York, NY, USA, 2011. 45. Naumann, M.; Keil, P. TUM institute for electrical storage technology internal calendric aging studies. Unpublished work. 46. GNB Industrial Power. Handbuch für Verschlossene Gel-Blei-Batterien; Exide Technologies GmbH.: Büdingen, Germany, 2013. 47. Sony Energy Devices Corporation. Sony Energy Devices Quality, Operations and Management. Available online: http://www.sonyenergy-devices.co.jp/en/csr/quality.php (accessed on 4 May 2017). 48. Omar, N.; Monem, M.A.; Firouz, Y.; Salminen, J.; Smekens, J.; Hegazy, O.; Van Mierlo, J. Lithium iron phosphate based battery—Assessment of the aging parameters and development of cycle life model. Appl. Energy 2014,113, 1575–1585. [CrossRef] 49. Schuster, S.; Bach, T.; Fleder, E.; Müller, J.; Brand, M.; Sextl, G.; Jossen, A. Nonlinear aging characteristics of lithium-ion cells under different operational conditions. J. Energy Storage 2015,1, 44–53. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).