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Using inventory models for sizing energy storage systems: An interdisciplinary approach
G Model EST 79 No. of Pages 10
Journal of Energy Storage xxx (2015) xxx–xxx
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Journal of Energy Storage journal homepage: www.elsevier.com/locate/est
Using inventory models for sizing energy storage systems: An interdisciplinary approach Maximilian Schneidera,* , Konstantin Bielb , Stephan Pfallera , Hendrik Schaedea , Stephan Rinderknechta , Christoph H. Glockb a b
Institute for Mechatronic Systems in Mechanical Engineering, Technische Universität Darmstadt, Otto-Berndt-Straße 2, 64287 Darmstadt, Germany Institute of Production and Supply Chain Management, Technische Universität Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany
A R T I C L E I N F O
A B S T R A C T
Article history: Received 30 September 2015 Received in revised form 19 February 2016 Accepted 19 February 2016 Available online xxx
This paper adopts a single-period newsvendor model with supply uncertainties to be used for optimally sizing an electrical energy storage system (EESS) for an apartment house with a photovoltaic system. Hence, typical inventory cost components and supply chain characteristics are translated into the EESS application. The results show that existing research approaches for inventory management can be adapted to determine the optimal size of an EESS. The proposed EESS-sizing procedure takes into account the total cost of the storage system including energetic losses as well as the costs for energy supply from the own energy systems and from the energy supplier. ã 2016 Elsevier Ltd. All rights reserved.
Keywords: Energy storage PV home storage Sizing Newsvendor model Inventory model
1. Introduction The demand for energy storage capacities has been soaring in recent years [1]. This development stems from both economic and technological reasons [2]. The economic perspective arises from the fact that the electricity generation cost which utilities incur vary with customer demand. These demand-dependent cost discrepancies originate in very high operating cost of electricity generation facilities, which utilities employ to cover peak demand loads (e.g. gas-fired power plants), as opposed to low operating cost of electricity generation facilities, which utilities use to provide the base load (e.g. coal-fired power plants) [3]. As a consequence, both utilities and in some cases the customer may use energy storages to store energy in low-price periods and to make use of the stored energy in high-price periods [2]. Despite the wish to relocate energy demand to low-price periods, the increasing use of renewable energies in power supply has also contributed to a rising demand for energy storage capacities [1]. Even for small households with photovoltaic (PV) modules, an electrical energy storage system (EESS) can be profitable if it increases the self-consumption of the building. Furthermore, with a higher penetration of renewable energies in the power grid, EESSs will be necessary to provide valuable grid
* Corresponding author. Fax: +49 6151 16 23264. E-mail address: [email protected] (M. Schneider).
services in addition to their main task [4,5]. [6], for instance, estimated that 27 TWh (two days) of storage size would be required in Europe if the energy demand was completely satisfied from renewable energy sources and 10% backup capacity from power plants existed. Yet, without backup capacity, 30–90 days of storage capacity would be required for a stable national grid without international exchange, as opposed to 7–30 days for a stable European grid. Following the increasing use of EESSs, the question ultimately arises how much capacity an EESS should have in a particular application. To answer this question, researchers and practitioners have recently started to discuss methods that support decision makers in determining the optimal size of an EESS [7–13]. The approach proposed in this paper is based on analogies between inventory management in classical supply chains and the sizing of EESSs. As will be illustrated below, both applications show strong parallels in terms of the basic assumptions and general conditions. A plethora of methodologies has been developed for costefficiently planning inventories within a supply chain. This paper adopts such a methodology to optimally size an EESS for an apartment house with a PV system as the building’s own source of energy with respect to different cost parameters. Before this methodology is explained in detail, Section 2 gives an overview of procedures which are currently used to size EESSs, shortly elaborates on the fundamentals of inventory management and presents areas of interdisciplinary research which have already capitalized on the use of inventory models. Section 3
http://dx.doi.org/10.1016/j.est.2016.02.009 2352-152X/ ã 2016 Elsevier Ltd. All rights reserved.
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Nomenclature
a
C C phys ~ C c cGeneration cStorage co cu CaAC CyAC D DoA DoD Echar;i Edis;i ED;i ED;mean Ed;mod;i ES;i
h Fw fD lEESSLT NoC p Q Qw
RC S S SoC TC TCoE v w
Self-discharge ratio for 24 h in % Optimal usable capacity of the EESS in kWh Optimal physical capacity of the EESS in kWh Current capacity of the EESS during the parameter variation in kWh Purchase price in s/kWh Cost for generating solar energy in s/kWh Cost for storing energy in s/kWh Overage cost in s/kWh Underage cost in s/kWh Cost due to calendric aging of the EESS in s Cost due to cyclical aging of the EESS in s Demand in kWh Degree of autarky in % Depth of discharge in % Discrete amount of energy charged to the EESS in time interval i in kWh Discrete amount of energy discharged from the EESS in time interval i in kWh Discrete amount of energy demand in time interval i in kWh average amount of energy demand in kWh Modified discrete amount of energy demand in time interval i in kWh Discrete amount of energy supply in time interval i in kWh Single-sided conversion efficiency in % Cumulative distribution function of w Density function of D Calendric lifetime of the EESS in years Average number of cycles in cycles/day Sales price in s/kWh Optimal order quantity in kWh Stochastic correction quantity of demand D depending on the cost parameters co and cu in kWh Ratio of cost parameters Order-up-to level in kWh Optimal cyclical order-up-to level in kWh State of charge of the EESS in kWh Total cost of newsvendor problem in s Total cost of energy in s Salvage value in s/kWh Random yield in kWh
2.1. Existing EESS-sizing procedures Since energy storage systems of different types are established in numerous applications, methods have been developed in the past that support finding suitable storage systems for different tasks. Most of these methods use search heuristics or optimization algorithms based on input parameter variations to identify the appropriate dimensions (i.e. rated power and energy) of the EESS. Examples of such approaches can be found in [10–12]. An example for an analytical approach, where the objective function can be solved for the dimensions of the EESS, can be found in [13]. The intermittent character of most demand and production profiles has led to promising stochastic approaches. Examples can be found in [7–9]. Yet, to the best of the authors’ knowledge, none of these procedures make use of inventory models to size and optimize an EESS. 2.2. Inventory models and their interdisciplinary applicability Research on the lot-sizing problem was initiated by the seminal work of Harris [14]. Harris proposed the economic lot size model which answers the question of how many parts to produce in one production run. To this end he considers variable inventory holding cost and fixed cost incurred each time when setting up a machine or when placing an order [14]. By relaxing or altering one or more of the underlying assumptions, numerous inventory models have been developed over the last 100 years. [15] provides a comprehensive review of the research streams that emerged from Harris’ economic lot size model. These approaches are applied to various supply chain management issues. Typically, inventory models are concerned with problems arising in a production environment. Yet, there are also other areas of application where the use of inventory models has been of value. One of these areas is the health care sector. Examples of such approaches can be found in [16–18]. Furthermore, inventory models are employed in disaster relief management [19–22]. Some inventory models also include inventory capacity considerations as limited inventory capacity may harm a smooth production flow [23–25]. In all applications, the question remains which storage capacity is economically most reasonable. 3. Modeling EESSs using inventory models In this chapter, approaches used to address the lot-sizing problem in inventory management are applied to the sizing of energy storages. Section 3.1 explains the analogies between inventory management and EESS-sizing. In addition, Section 3.2 defines requirements which energy storage systems impose on the model to be developed. Sections 3.3 and 3.4 deal with the selection and adaption of a suitable model. 3.1. Analogies between inventory management and EESS-sizing
defines requirements which the operation of an EESS imposes on the model. Based on these requirements, a model is selected and adjusted accordingly. Section 4 presents a numerical example of the developed model, and Section 5 concludes the paper. 2. Literature overview In consequence of the interdisciplinary nature of the proposed EESS-sizing approach, the developed model builds on traditional EESS-sizing procedures as well as on classical inventory models. Hence, Section 2.1 identifies relevant papers on sizing of EESSs. Subsequently, Section 2.2 outlines the evolution of inventory management and refers to various interdisciplinary applications of inventory models.
Several parallels can be identified between the process of sizing an EESS and the process of determining the optimal order or production lot size in batch production processes (cf. Fig. 1). The classical supply chain consists of a supplier, a retailer, and a consumer. The supplier transforms raw materials into finished goods and ships them to the retailer in order sizes determined by the retailer. The retailer stores the finished goods and ultimately satisfies the customer demand from the stored goods. In an energy supply chain with renewable energy, a PV system or wind power plant produces energy that is transported to the consumer using an electricity grid. When installing an EESS, the system would assume the role of the retailer. While the lot-sizing problem aims at balancing setup or order cost and inventory holding cost given
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3
Fig. 1. Analogies between inventory management and EESS-sizing.
specific production and demand parameters, energy storage-sizing aims at finding the optimal storage capacity and power given a specific load profile. Thus, the question at the core of this paper is whether approaches to determine the optimal order quantity may provide the basis for a new procedure for sizing energy storages. 3.2. Requirements for modeling EESSs To select a suitable inventory model, the requirements imposed on a technically correct model of an EESS need to be translated into the language of inventory management. Requirements can be clustered into I) fundamental requirements the model needs to fulfill, II) additional requirements that improve the quality of the model, if fulfilled, but that are not compulsory, and III) application-specific requirements that are valuable for adapting the model to this application. Table 1 summarizes the resulting adaption of requirements.
3.3. Selection of a suitable inventory model As this paper constitutes a first attempt to scale an EESS using inventory models, the presented approach concentrates on complying with the fundamental requirements before refining the model and subsequently integrating additional and application-specific requirements. The first three fundamental requirements can be fulfilled by most inventory models. However, to the best of the author’s knowledge no suitable inventory models exist that cover both stochastic demand and stochastic supply. Therefore either demand or supply has to be assumed to be deterministic and this fundamental requirement cannot be fulfilled at the moment. Among the stochastic types of inventory models, the group of newsvendor models fits best to the above mentioned requirements. Most of these models, however, are a single period approaches which contradicts the first additional requirement. To which extent these shortcomings influence the quality of the
Table 1 Translation of technical properties of EESS into requirements for inventory models. Technical properties of EESS
Properties of inventory model
Fundamental requirements
Additional requirements
multi-period inventory model energy can be stored for multiple cycles EESS incurs energy losses during conversion and storage inventory model with perishable goods or loss of goods operation of EESS has to consider degradation effects via limited depth of discharge consideration of safety stocks
Applicationspecific requirements
energy is stored there is only one EESS parameters of the EESS do not change over time intermittent energy source intermittent energy demand
unsatisfied demand cannot be satisfied in the future a part of the energy demand can be satisfied without disproportional punishment electricity prices can change over time starting a charging/discharging process is cost-neutral charging/discharging can start without delay
single-product inventory model single-stage inventory model inventory model with static model parameters inventory model with stochastic supply inventory model with stochastic demand
shortages are considered as lost sales shortage costs are proportional to the size of the shortage varying prices for supply and demand no fixed order costs no lead time
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model has to be determined at a later stage when detailed information about the application is introduced. The classical newsvendor model optimally solves the problem of how many newspapers a newsvendor should order every day given a purchase price c, a sales price p, a salvage value v, and a stochastic demand D [26]. If the newsvendor orders more newspapers than eventually demanded during the day, s/he will incur so-called overage cost co ¼ c v for every unsold newspaper. Yet, if the newsvendor orders fewer newspapers than eventually demanded, s/he will incur opportunity cost, termed underage cost, cu ¼ p c, for every newspaper s/he could have sold if it had been on stock. The optimal order quantity Q is determined by minimizing the total cost TC as sum of overage and underage costs: Z Q Z 1 TC ¼ co ðQ DÞf D dD þ cu ðD Q Þf D dD; ð1Þ 0
Q
where f D represents the density function of D [27]. As shown in [28], the optimal order quantity solely depends on the overage and underage costs and can be derived from the critical fractile: F D ðQ Þ ¼
cu : co þ cu
ð2Þ
Thus, the critical fractile corresponds to the probability that the order quantity Q will be sufficient to meet the entire demand D. When applying the newsvendor model to the dimensioning of an EESS, the optimal order quantity Q equals the optimal usable
capacity of a loss-free EESS. The sales price p corresponds to the market price of electricity, while the purchase price c corresponds to the sum of the cost for generating energy and the cost of storing energy in the EESS. In its basic form, the newsvendor model is only of limited help to solve the sizing problem of an EESS as it does not comply with the central requirement that the model has to include an uncertain supply. However, there is a stream of research derived from the classical newsvendor problem whose models satisfy this fundamental requirement. These models feature an unreliable supplier, who is backed up by a reliable supplier in some cases [29–31]. Of these models, the one presented in [30] is best suited for determining the optimal dimensions of an EESS as it expresses the supply uncertainty in form of an additive yield. It represents the difference between the amount ordered by the newsvendor and the amount ultimately supplied to her/him. In case of an additive yield, this difference is stochastic, but independent of the amount ordered by the newsvendor. As the standard deviation of the energy supplied to the EESS is independent of the amount of energy the operator would like to charge the EESS with, an additive yield models the supply uncertainty at stake best. Furthermore, the model proposed in [30] differentiates itself from other models which consider an additive yield as it can also be used for a multiperiod planning horizon. Yet, for simplicity, this paper assumes that the energy storage will be empty at the end of each period. Thus, a single-period model with stochastic yield appears to be well suited to support the sizing of an EESS.
Fig. 2. Step-by-step implementation of EESS-sizing approach.
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3.4. Description and adjustment of the selected inventory model [30] replaced the order size Q of the classical newsvendor problem by the sum of the order-up-to level S and the random yield w, whereas w is generally distributed with mean w and independent of S: Q ¼ S þ w:
ð3Þ
As mentioned above [30] assumed the supply to be stochastic and the demand D to be deterministic. Eq. (1) can now be reformulated as follows: Z 1 Z DS ðS þ w DÞf w dw þ cu ðD S wÞf w dw; ð4Þ TC ¼ co DS
1
where f w represents the density function of w. Eventually, [30] derived a closed-form solution for the optimal order-up-to level, which can be interpreted as the optimal usable capacity of a lossfree EESS: co ð5Þ ¼ D Qw; S ¼ D F 1 w co þ cu where F w represents the cumulative distribution function of the additive yield w. Q w equals the additive yield which the random additive yield w is smaller than or equal to with the probability of co =ðco þ cu Þ. Q w is termed stochastic correction quantity of demand D. As a consequence S , only depends on the demand for energy stored in the EESS and the basic cost parameters of the newsvendor model. Clearly, by applying the approach of [30], the requirement of a stochastic demand is relaxed. Therefore this method is only applicable if the demand is fairly stable and well predictable. For most private houses this assumption holds true. Thus, this relaxation does not limit the applicability of the model presented in this article. 4. Implementation of EESS-sizing approach and numerical example To investigate the applicability of the developed EESS-sizing approach, a numerical example is studied next. It focuses on a building with 17 apartments with a 27 kWp PV system on the roof. In total, the PV system generates an amount of energy which is 1.2 times higher than the building’s all-year consumption. On an average day the PV system generates an energy surplus during the midday hours whereas in the morning, the evening and at night there is an additional energy demand. Hence, it would be theoretically possible to achieve complete autarky for the building.
5
For this scenario a lithium-ion battery is sized considering all relevant cost parameters. The aim of this numerical example is to illustrate step by step how the EESS-sizing approach developed in this paper can be implemented to calculate an optimal capacity of an EESS, given a load profile of the electricity consumers and a generation profile of the renewable energy sources as well as cost parameters for generating, storing, and externally sourcing electricity. Fig. 2 gives an overview of this procedure. 4.1. Load and generation profiles The load profile of the apartment building and the generation profile of the PV system both cover a time span of 365 days with one average value per hour. By subtracting the generation profile from the load profile, the residual load profile shown in Fig. 3 is obtained. Positive values represent times with higher energy consumption than energy generation. The excess demand can be satisfied from the grid or taken from an EESS. Negative values imply that the generation exceeds the consumption. The excess energy can be either fed into the grid or stored in an EESS. 4.2. Analysis of the residual load profile In order to further analyze the residual load profile and to prepare the data for a later use in the developed EESS-sizing model, aggregated positive and negative energies per day are calculated. This is done by integrating the power curve of the residual load profile within intervals defined by the zero-crossings of the power curve. This leads to two vectors, one featuring positive energies Eþ and one featuring negative energies E (cf. Fig. 4). Without an EESS, positive amounts of energy are satisfied from the grid. Therefore, positive energies can be interpreted as demands ðDÞ towards the EESS. Negative amounts of energy are excess production of the PV system and available for charging the EESS. They can be interpreted as supplies for the EESS ðS þ wÞ. Fig. 5 shows the resulting frequency distributions for the residual load profile introduced above. Here, the relative frequency can be understood as the probability that a specific point or class in the corresponding distribution function occurs. The relative cumulative frequency, however, represents the cumulative distribution function of the considered discrete and continuous variable, in this case the computed amounts of energy. As can be seen in Fig. 5, the resulting demands ED;i (ca. 20–80 kWh) vary less than the supplies ES;i
Fig. 3. Residual load profile of the building with PV system.
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Fig. 4. Definition of positive and negative energies within the residual load profile.
(ca. 0–180 kWh). This fact supports the assumption of a deterministic demand in the proposed model without considering stochastic influences. The average demand ED;mean amounts to 45.2 kWh. 4.3. Formulation of the model The calculation of the optimal cyclical order-up-to level S according to [30] can basically be divided into two parts, the deterministic demand D and the amount to compensate the occurring random yield Q w (cf. Eq. (5)). The deterministic oneperiodical demand D of the building can be equated with the mean value ED;mean of all single demands ED;i . Since ED;i does not consider any conversion or self-discharge losses of the EESS, it has to be adjusted by the respective efficiency
factors. Without this adjustment, the capacity would be systematically underestimated. The occurring losses can be translated into an additional demand of the building. In order to calculate the real losses, temporal aspects along with the specific charging and discharging powers of the battery would need to be considered. Conversion losses are implemented via the single-sided conversion efficiency h ¼ 0:9. In contrast to conversion losses, self-discharge losses depend on the battery’s state of charge. In order to compute the self-discharge losses, the assumption is made that the average energy content of the EESS equals half the demand in the corresponding period of 24 h. This value is multiplied by the selfdischarge ratio for 24 h, a ¼ 0:168%. Due to the fact that only energy sums are considered in this approach, the demand D has to be augmented to account for storage losses. This leads to the modified demands ED;mod;i using
Fig. 5. Frequency distributions for energy demands ED,i with highlighted average demand ED,mean (top) and supplies ES,i. (bottom).
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the simplified relationship: D¼
I 1X
I
i¼1
1
h2 a2 h2
ED;i ¼
I 1X E ; I i¼1 D;mod;i
ð6Þ
where I is the total number of intervals defined by the zero crossings of the residual load profile as shown in Fig. 4. The cumulative distribution function of the additive yield F w is gained by subtracting the modified demands ED;mod;i from the related supplies ES;i for every consecutive interval i. The resulting quantities are termed EYield;i and can be both positive and negative. This way, the model does not only consider undersupply, but also oversupply, and therefore becomes more realistic regarding the analyzed scenario. Since the cumulative distribution function F w represents an empirically generated, discrete distribution function, Q w equals the smallest valid boundary value of a class of yields in the cumulative distribution function F w that fulfills the following condition [30]: F w ðQ w Þ
co : co þ c u
C ¼ hS :
ð8Þ
As the condition h < 1 always holds, the optimal capacity of the EESS is always smaller than the optimal order-up-to level S . In addition to the capacity of the EESS, its electrical power needs to be determined. In order to be able to charge and discharge the complete energy amounts calculated before, the required electrical power of the EESS is specified as the highest hourly average power value in the residual load profile. 4.4. Cost parameters The inventory model proposed in [30] basically requires two cost parameters, the overage and underage cost, according to Eq. (5). Due to still high investment cost of a lithium ion battery and still comparatively low energy cost in many countries, PV home storage systems are on the edge of profitability. In order to be able to evaluate the model, cost and price parameters are chosen based on parameter values that are expected to be realistic in three to seven years. The purchase price c for energy taken from the EESS equals the cost for generating and storing energy. The cost for generating 1 kWh of solar energy, cGeneration , can be calculated based on the investment cost of a PV system. They are estimated at 1000 s/kWp [32]. Furthermore, the energy yield is assumed to have an average value of 1000 kWh per kWp and year [33]. Finally, the PV system is assumed to be operated for 25 years. Hence, cGeneration can be computed: cGeneration ¼
The cost for storing energy, cStorage , is regarded as cost per kWh of energy, which is charged to the EESS. The investment cost cEESS are roughly estimated at 800 s per kWh of storage capacity for a lithium ion battery. Furthermore, the maximum cycle life of the lithium ion battery is assumed to cover 6000 full cycles. The Depth of Discharge DoD is set to 70% of the total storage capacity [34]. Hence, cStorage can be calculated as: cStorage ¼
800 s=kWh s : ¼ 0:19 0:76000 kWh
1000 s=kWp 1 kWp s : ¼ 0:04 kWh 25 years1000 kWh=year
ð9Þ
ð10Þ
The salvage value v of energy first stored and then discharged to the grid is assumed to be 0 s/kWh. Assuming a further increase of electricity prices, this parameter is estimated at p ¼ 0:35 s=kWh. With these input data, the overage and underage costs for sizing a building’s EESS with PV system are calculated as follows: s ; co ¼ c v ¼ cGeneration þ cStorage v ¼ 0:23 kWh
ð7Þ
With Q w , all parameters are available to calculate the cost-optimal periodical order quantity S according to Eq. (5). To ultimately derive the optimal capacity C of the EESS, the optimal order quantity S that can be interpreted as the amount of energy that is used for charging has to be adjusted by the conversion losses incurred when charging the EESS:
7
cu ¼ p c ¼ 0:12
s : kWh
ð11Þ
ð12Þ
The cost for operating the PV system and the EESS are neglected here because they are relatively small as compared to their investment costs. 4.5. Sizing results With the numerical values for the different parameters described in Section 4.4, the results shown in Table 2 can be calculated. Fig. 6 shows the frequency distributions of the calculated random yields used to determine Q w at the ratio of cost parameters RC. The high deterministic demand is adjusted by stochastic correction quantity Q w according to Eq. (5) and by conversion losses according to Eq. (8), leading to an optimal usable storage capacity C of about 26 kWh. This capacity can be interpreted as useable storage capacity and has to be adapted by the assumed DoD of 70% in order to obtain the optimal physical capacity of the EESS C Phys of about 37 kWh. Lithium-ion batteries can easily be charged with C-rates up to 1 leading to a power of the designed EESS of about 37 kW. The highest average hourly value in the residual load profile is 25 kW (cf. Fig. 3). Thus the available power of the EESS is sufficient. 4.6. Time domain simulation A time domain simulation is performed in order to validate the gained results using the inventory model. The same residual load profile and cost parameters are used to simulate different storage sizes. Different performance indicators (PIs) are used to evaluate the sensitivity of the problem. While varying the usable storage capacity, the maximum power was adjusted accordingly to assure maximum C-rates of 1. To measure the effectiveness of the EESS in exploiting the energy produced by the PV system, the building’s Degree of Autarky DoA and the Total Cost of Energy TCoE of the building are analyzed. DoA is defined as the quotient of self-produced energy
Table 2 Numerical results for the model’s output parameters. Ratio of cost parameters co/(co+cu)
Deterministic demand D
Stochastic correction quantity Qw
Cyclical order-up-to level S*
Optimal usable capacity C*
0.66
55.87 kWh
27.41 kWh
28.46 kWh
25.61 kWh
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Fig. 6. Frequency diagrams for yields EYield;i .
that was self-consumed and the total consumed energy of the building. TCoE equals the sum of the yearly cost of energy generated from the PV system, the yearly storage cost, and the yearly cost of electricity drawn from the grid using the above defined cost parameters and is normalized to the maximum value occurring during the parameter variation. The yearly storage cost equals the maximum of the cost due to cyclical aging of the EESS CyAC and the cost due to calendric aging of the EESS CaAC. CyAC can be calculated as follows: ( ) I I X X Edis;i ; Echar;i cStorage : ð12Þ CyAC ¼ min i¼1
i¼1
On the contrary, CaAC is calculated as follows: CaAC ¼
cEESS C Phys : lEESSLT
ð13Þ
As in Section 4.4,cEESS is assumed to equal 800 s per kWh, while the calendric lifetime of the EESS lEESSLT is assumed to equal 16 years. As further reference values, the average State of Charge SoC and the average Number of Cycles NoC per day are used. NoC can be calculated using the following relationship: nXI o XI min E ; E i¼1 dis;i i¼1 char;i NoC ¼ ; ð14Þ ~ 365 d C
where Edis;i is the energy discharged from the storage system in the time interval i and Echar;i is the energy charged to the storage ~ is the current storage capacity system in the time interval i. C during the parameter variation. Fig. 7 shows how the introduced ~ in the studied example. The vertical dashed PIs react to varying C line marks the optimal storage capacity C calculated using the inventory model introduced above. Fig. 7 reveals that the calculated storage capacity covers the area with a steep increase in DoA. To the right of the calculated capacity, every incremental capacity increase becomes more and more expensive. When looking at TCoE, the optimal usable capacity C determined with the help of the proposed model (vertical dashed line) is close to the minimum of the curve at a storage capacity of around 32 kWh. The discrepancy can be explained by the different calculation approaches. The single period inventory model optimizes the capacity based on cumulative distribution functions of the load profiles whereas the time domain simulation is a multi-period approach where energy can be stored for several consecutive days. The inventory model-based approach, however, accounts for the uncertainties of the energy supply from the PV system. With NoC of around 1, the cyclical aging of the battery is moderate and a long cycle life is possible. An average SoC of around 0.3 indicates that SoC is very low at times. However, due to the unsteady supply of solar power, particularly during autumn and winter, this seems acceptable.
Fig. 7. Behavior of performance indicators with increasing storage capacity (step range 1 kWh) and optimal usable capacity C determined by the proposed inventory model.
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5. Discussion and conclusion This paper contributes to the interdisciplinary research on sizing of EESSs. A comparison of properties of energy storage models with inventory models showed that the latter provided a promising starting point for a new domain of EESS-sizing approaches. In this paper, a single-period newsvendor model with supply uncertainties introduced by [30] was adapted for sizing EESSs. In a numerical analysis, the developed model was used to optimally size an EESS for an apartment building with a PV system. The sensitivity of the solution was then investigated with respect to different performance indicators using time domain simulation. The results show that inventory management and energy storagesizing can be brought well in line with each other. The optimal EESS capacity based on the developed sizing procedure is very close to the capacity that the time domain simulation would deem optimal a posteriori with regard to the total cost of energy. Hence, the proposed sizing approach is able to handle the uncertainty attached to the energy supply from PV systems. The applicability of the developed methodology for sizing energy storage systems in practice is further highlighted by the fact that it supports the objective of increasing the building’s degree of autarky as long as this is profitable. Another advantage of the presented approach is its robustness against variations of the load profiles. As long as the probability distributions of production and supply remain close to their original values, the storage size is optimal. In contrast to that, results derived from time-domain simulations are only valid for the used time series. Due to the use of probability distributions instead of time-domain data, computing time can be significantly decreased. The distribution of energy supply just has to be evaluated at the probability defined by the ratio of cost parameters. The model itself, however, is only valid under the assumption that the random yield follows a general distribution. For most practical cases only empirical distributions based on measurements or simulated data are available. The general applicability and optimality of the selected method for different distributions still needs to be proven. Another downside of the introduced approach is that due to the focus on sums of energy no procedure is provided to determine the optimal power of the EESS. Yet, as long as the analyzed application is characterized by C-rates lower than 1, this is not a problem. However, for high power applications, the approach proposed in this paper needs to be adjusted. In future studies, the developed method should be extended to a multi-period time frame that can also be modeled using the approach described in [30]. This way, the operating strategy of an EESS can be integrated into the EESS-sizing decision and a truly optimal EESS capacity can be calculated. The notion that the EESSsizing decision and the determination of the EESS operating strategy should ultimately not be decoupled was also supported by [35]. [35] studied under which conditions it is cost-efficient to employ an EESS to store electricity generated from manufacturing waste heat in times of low electricity prices and dispatch it to support the energy supply of a manufacturing system in times of high electricity prices. In a numerical analysis, they found that the EESS can only be capitalized on if the capacity and the performance of the EESS are tailored to the actual pattern of use of the EESS [35]. Thus, a fundamental integration of EESS-operating strategies into the developed sizing model should be addressed in future research. In addition, the basic as well as possible extended models should be applied to further scenarios and applications for detailed evaluation. The numerical example studied in the paper at hand solely focused on the task of finding the building’s cost optimal degree of autarky. When maximizing the profitability of storage systems, approaches considering multiple goals, e.g. additional grid relief, have to be taken into account. As the presented
9
approach is rather generic, it can serve as a starting point for extended models seeking to cope with complex operating strategies or multi use case applications.
Acknowledgements The second and the sixth author wish to acknowledge the support of the Carlo and Karin Giersch-Stiftung in funding their research.
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[29] A.J. Schmitt, L.V. Snyder, Z.-J.M. Shen, Inventory systems with stochastic demand and supply: properties and approximations, Eur. J. Oper. Res. 206 (2010) 313–328. [30] A.J. Schmitt, L.V. Snyder, Infinite-horizon models for inventory control under yield uncertainty and disruptions, Comput. Oper. Res. 39 (2012) 850–862. [31] K. Inderfurth, How to protect against demand and yield risks in MRP systems, Int. J. Prod. Econ. 121 (2009) 474–481. [32] H. Wirth, Aktuelle Fakten zur Photovoltaik in Deutschland, Fraunhofer-Institut für Solare Energiesysteme ISE: Freiburg, (2015).
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Journal of Energy Storage xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Journal of Energy Storage journal homepage: www.elsevier.com/locate/est
Using inventory models for sizing energy storage systems: An interdisciplinary approach Maximilian Schneidera,* , Konstantin Bielb , Stephan Pfallera , Hendrik Schaedea , Stephan Rinderknechta , Christoph H. Glockb a b
Institute for Mechatronic Systems in Mechanical Engineering, Technische Universität Darmstadt, Otto-Berndt-Straße 2, 64287 Darmstadt, Germany Institute of Production and Supply Chain Management, Technische Universität Darmstadt, Hochschulstr. 1, 64289 Darmstadt, Germany
A R T I C L E I N F O
A B S T R A C T
Article history: Received 30 September 2015 Received in revised form 19 February 2016 Accepted 19 February 2016 Available online xxx
This paper adopts a single-period newsvendor model with supply uncertainties to be used for optimally sizing an electrical energy storage system (EESS) for an apartment house with a photovoltaic system. Hence, typical inventory cost components and supply chain characteristics are translated into the EESS application. The results show that existing research approaches for inventory management can be adapted to determine the optimal size of an EESS. The proposed EESS-sizing procedure takes into account the total cost of the storage system including energetic losses as well as the costs for energy supply from the own energy systems and from the energy supplier. ã 2016 Elsevier Ltd. All rights reserved.
Keywords: Energy storage PV home storage Sizing Newsvendor model Inventory model
1. Introduction The demand for energy storage capacities has been soaring in recent years [1]. This development stems from both economic and technological reasons [2]. The economic perspective arises from the fact that the electricity generation cost which utilities incur vary with customer demand. These demand-dependent cost discrepancies originate in very high operating cost of electricity generation facilities, which utilities employ to cover peak demand loads (e.g. gas-fired power plants), as opposed to low operating cost of electricity generation facilities, which utilities use to provide the base load (e.g. coal-fired power plants) [3]. As a consequence, both utilities and in some cases the customer may use energy storages to store energy in low-price periods and to make use of the stored energy in high-price periods [2]. Despite the wish to relocate energy demand to low-price periods, the increasing use of renewable energies in power supply has also contributed to a rising demand for energy storage capacities [1]. Even for small households with photovoltaic (PV) modules, an electrical energy storage system (EESS) can be profitable if it increases the self-consumption of the building. Furthermore, with a higher penetration of renewable energies in the power grid, EESSs will be necessary to provide valuable grid
* Corresponding author. Fax: +49 6151 16 23264. E-mail address: [email protected] (M. Schneider).
services in addition to their main task [4,5]. [6], for instance, estimated that 27 TWh (two days) of storage size would be required in Europe if the energy demand was completely satisfied from renewable energy sources and 10% backup capacity from power plants existed. Yet, without backup capacity, 30–90 days of storage capacity would be required for a stable national grid without international exchange, as opposed to 7–30 days for a stable European grid. Following the increasing use of EESSs, the question ultimately arises how much capacity an EESS should have in a particular application. To answer this question, researchers and practitioners have recently started to discuss methods that support decision makers in determining the optimal size of an EESS [7–13]. The approach proposed in this paper is based on analogies between inventory management in classical supply chains and the sizing of EESSs. As will be illustrated below, both applications show strong parallels in terms of the basic assumptions and general conditions. A plethora of methodologies has been developed for costefficiently planning inventories within a supply chain. This paper adopts such a methodology to optimally size an EESS for an apartment house with a PV system as the building’s own source of energy with respect to different cost parameters. Before this methodology is explained in detail, Section 2 gives an overview of procedures which are currently used to size EESSs, shortly elaborates on the fundamentals of inventory management and presents areas of interdisciplinary research which have already capitalized on the use of inventory models. Section 3
http://dx.doi.org/10.1016/j.est.2016.02.009 2352-152X/ ã 2016 Elsevier Ltd. All rights reserved.
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Nomenclature
a
C C phys ~ C c cGeneration cStorage co cu CaAC CyAC D DoA DoD Echar;i Edis;i ED;i ED;mean Ed;mod;i ES;i
h Fw fD lEESSLT NoC p Q Qw
RC S S SoC TC TCoE v w
Self-discharge ratio for 24 h in % Optimal usable capacity of the EESS in kWh Optimal physical capacity of the EESS in kWh Current capacity of the EESS during the parameter variation in kWh Purchase price in s/kWh Cost for generating solar energy in s/kWh Cost for storing energy in s/kWh Overage cost in s/kWh Underage cost in s/kWh Cost due to calendric aging of the EESS in s Cost due to cyclical aging of the EESS in s Demand in kWh Degree of autarky in % Depth of discharge in % Discrete amount of energy charged to the EESS in time interval i in kWh Discrete amount of energy discharged from the EESS in time interval i in kWh Discrete amount of energy demand in time interval i in kWh average amount of energy demand in kWh Modified discrete amount of energy demand in time interval i in kWh Discrete amount of energy supply in time interval i in kWh Single-sided conversion efficiency in % Cumulative distribution function of w Density function of D Calendric lifetime of the EESS in years Average number of cycles in cycles/day Sales price in s/kWh Optimal order quantity in kWh Stochastic correction quantity of demand D depending on the cost parameters co and cu in kWh Ratio of cost parameters Order-up-to level in kWh Optimal cyclical order-up-to level in kWh State of charge of the EESS in kWh Total cost of newsvendor problem in s Total cost of energy in s Salvage value in s/kWh Random yield in kWh
2.1. Existing EESS-sizing procedures Since energy storage systems of different types are established in numerous applications, methods have been developed in the past that support finding suitable storage systems for different tasks. Most of these methods use search heuristics or optimization algorithms based on input parameter variations to identify the appropriate dimensions (i.e. rated power and energy) of the EESS. Examples of such approaches can be found in [10–12]. An example for an analytical approach, where the objective function can be solved for the dimensions of the EESS, can be found in [13]. The intermittent character of most demand and production profiles has led to promising stochastic approaches. Examples can be found in [7–9]. Yet, to the best of the authors’ knowledge, none of these procedures make use of inventory models to size and optimize an EESS. 2.2. Inventory models and their interdisciplinary applicability Research on the lot-sizing problem was initiated by the seminal work of Harris [14]. Harris proposed the economic lot size model which answers the question of how many parts to produce in one production run. To this end he considers variable inventory holding cost and fixed cost incurred each time when setting up a machine or when placing an order [14]. By relaxing or altering one or more of the underlying assumptions, numerous inventory models have been developed over the last 100 years. [15] provides a comprehensive review of the research streams that emerged from Harris’ economic lot size model. These approaches are applied to various supply chain management issues. Typically, inventory models are concerned with problems arising in a production environment. Yet, there are also other areas of application where the use of inventory models has been of value. One of these areas is the health care sector. Examples of such approaches can be found in [16–18]. Furthermore, inventory models are employed in disaster relief management [19–22]. Some inventory models also include inventory capacity considerations as limited inventory capacity may harm a smooth production flow [23–25]. In all applications, the question remains which storage capacity is economically most reasonable. 3. Modeling EESSs using inventory models In this chapter, approaches used to address the lot-sizing problem in inventory management are applied to the sizing of energy storages. Section 3.1 explains the analogies between inventory management and EESS-sizing. In addition, Section 3.2 defines requirements which energy storage systems impose on the model to be developed. Sections 3.3 and 3.4 deal with the selection and adaption of a suitable model. 3.1. Analogies between inventory management and EESS-sizing
defines requirements which the operation of an EESS imposes on the model. Based on these requirements, a model is selected and adjusted accordingly. Section 4 presents a numerical example of the developed model, and Section 5 concludes the paper. 2. Literature overview In consequence of the interdisciplinary nature of the proposed EESS-sizing approach, the developed model builds on traditional EESS-sizing procedures as well as on classical inventory models. Hence, Section 2.1 identifies relevant papers on sizing of EESSs. Subsequently, Section 2.2 outlines the evolution of inventory management and refers to various interdisciplinary applications of inventory models.
Several parallels can be identified between the process of sizing an EESS and the process of determining the optimal order or production lot size in batch production processes (cf. Fig. 1). The classical supply chain consists of a supplier, a retailer, and a consumer. The supplier transforms raw materials into finished goods and ships them to the retailer in order sizes determined by the retailer. The retailer stores the finished goods and ultimately satisfies the customer demand from the stored goods. In an energy supply chain with renewable energy, a PV system or wind power plant produces energy that is transported to the consumer using an electricity grid. When installing an EESS, the system would assume the role of the retailer. While the lot-sizing problem aims at balancing setup or order cost and inventory holding cost given
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Fig. 1. Analogies between inventory management and EESS-sizing.
specific production and demand parameters, energy storage-sizing aims at finding the optimal storage capacity and power given a specific load profile. Thus, the question at the core of this paper is whether approaches to determine the optimal order quantity may provide the basis for a new procedure for sizing energy storages. 3.2. Requirements for modeling EESSs To select a suitable inventory model, the requirements imposed on a technically correct model of an EESS need to be translated into the language of inventory management. Requirements can be clustered into I) fundamental requirements the model needs to fulfill, II) additional requirements that improve the quality of the model, if fulfilled, but that are not compulsory, and III) application-specific requirements that are valuable for adapting the model to this application. Table 1 summarizes the resulting adaption of requirements.
3.3. Selection of a suitable inventory model As this paper constitutes a first attempt to scale an EESS using inventory models, the presented approach concentrates on complying with the fundamental requirements before refining the model and subsequently integrating additional and application-specific requirements. The first three fundamental requirements can be fulfilled by most inventory models. However, to the best of the author’s knowledge no suitable inventory models exist that cover both stochastic demand and stochastic supply. Therefore either demand or supply has to be assumed to be deterministic and this fundamental requirement cannot be fulfilled at the moment. Among the stochastic types of inventory models, the group of newsvendor models fits best to the above mentioned requirements. Most of these models, however, are a single period approaches which contradicts the first additional requirement. To which extent these shortcomings influence the quality of the
Table 1 Translation of technical properties of EESS into requirements for inventory models. Technical properties of EESS
Properties of inventory model
Fundamental requirements
Additional requirements
multi-period inventory model energy can be stored for multiple cycles EESS incurs energy losses during conversion and storage inventory model with perishable goods or loss of goods operation of EESS has to consider degradation effects via limited depth of discharge consideration of safety stocks
Applicationspecific requirements
energy is stored there is only one EESS parameters of the EESS do not change over time intermittent energy source intermittent energy demand
unsatisfied demand cannot be satisfied in the future a part of the energy demand can be satisfied without disproportional punishment electricity prices can change over time starting a charging/discharging process is cost-neutral charging/discharging can start without delay
single-product inventory model single-stage inventory model inventory model with static model parameters inventory model with stochastic supply inventory model with stochastic demand
shortages are considered as lost sales shortage costs are proportional to the size of the shortage varying prices for supply and demand no fixed order costs no lead time
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model has to be determined at a later stage when detailed information about the application is introduced. The classical newsvendor model optimally solves the problem of how many newspapers a newsvendor should order every day given a purchase price c, a sales price p, a salvage value v, and a stochastic demand D [26]. If the newsvendor orders more newspapers than eventually demanded during the day, s/he will incur so-called overage cost co ¼ c v for every unsold newspaper. Yet, if the newsvendor orders fewer newspapers than eventually demanded, s/he will incur opportunity cost, termed underage cost, cu ¼ p c, for every newspaper s/he could have sold if it had been on stock. The optimal order quantity Q is determined by minimizing the total cost TC as sum of overage and underage costs: Z Q Z 1 TC ¼ co ðQ DÞf D dD þ cu ðD Q Þf D dD; ð1Þ 0
Q
where f D represents the density function of D [27]. As shown in [28], the optimal order quantity solely depends on the overage and underage costs and can be derived from the critical fractile: F D ðQ Þ ¼
cu : co þ cu
ð2Þ
Thus, the critical fractile corresponds to the probability that the order quantity Q will be sufficient to meet the entire demand D. When applying the newsvendor model to the dimensioning of an EESS, the optimal order quantity Q equals the optimal usable
capacity of a loss-free EESS. The sales price p corresponds to the market price of electricity, while the purchase price c corresponds to the sum of the cost for generating energy and the cost of storing energy in the EESS. In its basic form, the newsvendor model is only of limited help to solve the sizing problem of an EESS as it does not comply with the central requirement that the model has to include an uncertain supply. However, there is a stream of research derived from the classical newsvendor problem whose models satisfy this fundamental requirement. These models feature an unreliable supplier, who is backed up by a reliable supplier in some cases [29–31]. Of these models, the one presented in [30] is best suited for determining the optimal dimensions of an EESS as it expresses the supply uncertainty in form of an additive yield. It represents the difference between the amount ordered by the newsvendor and the amount ultimately supplied to her/him. In case of an additive yield, this difference is stochastic, but independent of the amount ordered by the newsvendor. As the standard deviation of the energy supplied to the EESS is independent of the amount of energy the operator would like to charge the EESS with, an additive yield models the supply uncertainty at stake best. Furthermore, the model proposed in [30] differentiates itself from other models which consider an additive yield as it can also be used for a multiperiod planning horizon. Yet, for simplicity, this paper assumes that the energy storage will be empty at the end of each period. Thus, a single-period model with stochastic yield appears to be well suited to support the sizing of an EESS.
Fig. 2. Step-by-step implementation of EESS-sizing approach.
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3.4. Description and adjustment of the selected inventory model [30] replaced the order size Q of the classical newsvendor problem by the sum of the order-up-to level S and the random yield w, whereas w is generally distributed with mean w and independent of S: Q ¼ S þ w:
ð3Þ
As mentioned above [30] assumed the supply to be stochastic and the demand D to be deterministic. Eq. (1) can now be reformulated as follows: Z 1 Z DS ðS þ w DÞf w dw þ cu ðD S wÞf w dw; ð4Þ TC ¼ co DS
1
where f w represents the density function of w. Eventually, [30] derived a closed-form solution for the optimal order-up-to level, which can be interpreted as the optimal usable capacity of a lossfree EESS: co ð5Þ ¼ D Qw; S ¼ D F 1 w co þ cu where F w represents the cumulative distribution function of the additive yield w. Q w equals the additive yield which the random additive yield w is smaller than or equal to with the probability of co =ðco þ cu Þ. Q w is termed stochastic correction quantity of demand D. As a consequence S , only depends on the demand for energy stored in the EESS and the basic cost parameters of the newsvendor model. Clearly, by applying the approach of [30], the requirement of a stochastic demand is relaxed. Therefore this method is only applicable if the demand is fairly stable and well predictable. For most private houses this assumption holds true. Thus, this relaxation does not limit the applicability of the model presented in this article. 4. Implementation of EESS-sizing approach and numerical example To investigate the applicability of the developed EESS-sizing approach, a numerical example is studied next. It focuses on a building with 17 apartments with a 27 kWp PV system on the roof. In total, the PV system generates an amount of energy which is 1.2 times higher than the building’s all-year consumption. On an average day the PV system generates an energy surplus during the midday hours whereas in the morning, the evening and at night there is an additional energy demand. Hence, it would be theoretically possible to achieve complete autarky for the building.
5
For this scenario a lithium-ion battery is sized considering all relevant cost parameters. The aim of this numerical example is to illustrate step by step how the EESS-sizing approach developed in this paper can be implemented to calculate an optimal capacity of an EESS, given a load profile of the electricity consumers and a generation profile of the renewable energy sources as well as cost parameters for generating, storing, and externally sourcing electricity. Fig. 2 gives an overview of this procedure. 4.1. Load and generation profiles The load profile of the apartment building and the generation profile of the PV system both cover a time span of 365 days with one average value per hour. By subtracting the generation profile from the load profile, the residual load profile shown in Fig. 3 is obtained. Positive values represent times with higher energy consumption than energy generation. The excess demand can be satisfied from the grid or taken from an EESS. Negative values imply that the generation exceeds the consumption. The excess energy can be either fed into the grid or stored in an EESS. 4.2. Analysis of the residual load profile In order to further analyze the residual load profile and to prepare the data for a later use in the developed EESS-sizing model, aggregated positive and negative energies per day are calculated. This is done by integrating the power curve of the residual load profile within intervals defined by the zero-crossings of the power curve. This leads to two vectors, one featuring positive energies Eþ and one featuring negative energies E (cf. Fig. 4). Without an EESS, positive amounts of energy are satisfied from the grid. Therefore, positive energies can be interpreted as demands ðDÞ towards the EESS. Negative amounts of energy are excess production of the PV system and available for charging the EESS. They can be interpreted as supplies for the EESS ðS þ wÞ. Fig. 5 shows the resulting frequency distributions for the residual load profile introduced above. Here, the relative frequency can be understood as the probability that a specific point or class in the corresponding distribution function occurs. The relative cumulative frequency, however, represents the cumulative distribution function of the considered discrete and continuous variable, in this case the computed amounts of energy. As can be seen in Fig. 5, the resulting demands ED;i (ca. 20–80 kWh) vary less than the supplies ES;i
Fig. 3. Residual load profile of the building with PV system.
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Fig. 4. Definition of positive and negative energies within the residual load profile.
(ca. 0–180 kWh). This fact supports the assumption of a deterministic demand in the proposed model without considering stochastic influences. The average demand ED;mean amounts to 45.2 kWh. 4.3. Formulation of the model The calculation of the optimal cyclical order-up-to level S according to [30] can basically be divided into two parts, the deterministic demand D and the amount to compensate the occurring random yield Q w (cf. Eq. (5)). The deterministic oneperiodical demand D of the building can be equated with the mean value ED;mean of all single demands ED;i . Since ED;i does not consider any conversion or self-discharge losses of the EESS, it has to be adjusted by the respective efficiency
factors. Without this adjustment, the capacity would be systematically underestimated. The occurring losses can be translated into an additional demand of the building. In order to calculate the real losses, temporal aspects along with the specific charging and discharging powers of the battery would need to be considered. Conversion losses are implemented via the single-sided conversion efficiency h ¼ 0:9. In contrast to conversion losses, self-discharge losses depend on the battery’s state of charge. In order to compute the self-discharge losses, the assumption is made that the average energy content of the EESS equals half the demand in the corresponding period of 24 h. This value is multiplied by the selfdischarge ratio for 24 h, a ¼ 0:168%. Due to the fact that only energy sums are considered in this approach, the demand D has to be augmented to account for storage losses. This leads to the modified demands ED;mod;i using
Fig. 5. Frequency distributions for energy demands ED,i with highlighted average demand ED,mean (top) and supplies ES,i. (bottom).
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the simplified relationship: D¼
I 1X
I
i¼1
1
h2 a2 h2
ED;i ¼
I 1X E ; I i¼1 D;mod;i
ð6Þ
where I is the total number of intervals defined by the zero crossings of the residual load profile as shown in Fig. 4. The cumulative distribution function of the additive yield F w is gained by subtracting the modified demands ED;mod;i from the related supplies ES;i for every consecutive interval i. The resulting quantities are termed EYield;i and can be both positive and negative. This way, the model does not only consider undersupply, but also oversupply, and therefore becomes more realistic regarding the analyzed scenario. Since the cumulative distribution function F w represents an empirically generated, discrete distribution function, Q w equals the smallest valid boundary value of a class of yields in the cumulative distribution function F w that fulfills the following condition [30]: F w ðQ w Þ
co : co þ c u
C ¼ hS :
ð8Þ
As the condition h < 1 always holds, the optimal capacity of the EESS is always smaller than the optimal order-up-to level S . In addition to the capacity of the EESS, its electrical power needs to be determined. In order to be able to charge and discharge the complete energy amounts calculated before, the required electrical power of the EESS is specified as the highest hourly average power value in the residual load profile. 4.4. Cost parameters The inventory model proposed in [30] basically requires two cost parameters, the overage and underage cost, according to Eq. (5). Due to still high investment cost of a lithium ion battery and still comparatively low energy cost in many countries, PV home storage systems are on the edge of profitability. In order to be able to evaluate the model, cost and price parameters are chosen based on parameter values that are expected to be realistic in three to seven years. The purchase price c for energy taken from the EESS equals the cost for generating and storing energy. The cost for generating 1 kWh of solar energy, cGeneration , can be calculated based on the investment cost of a PV system. They are estimated at 1000 s/kWp [32]. Furthermore, the energy yield is assumed to have an average value of 1000 kWh per kWp and year [33]. Finally, the PV system is assumed to be operated for 25 years. Hence, cGeneration can be computed: cGeneration ¼
The cost for storing energy, cStorage , is regarded as cost per kWh of energy, which is charged to the EESS. The investment cost cEESS are roughly estimated at 800 s per kWh of storage capacity for a lithium ion battery. Furthermore, the maximum cycle life of the lithium ion battery is assumed to cover 6000 full cycles. The Depth of Discharge DoD is set to 70% of the total storage capacity [34]. Hence, cStorage can be calculated as: cStorage ¼
800 s=kWh s : ¼ 0:19 0:76000 kWh
1000 s=kWp 1 kWp s : ¼ 0:04 kWh 25 years1000 kWh=year
ð9Þ
ð10Þ
The salvage value v of energy first stored and then discharged to the grid is assumed to be 0 s/kWh. Assuming a further increase of electricity prices, this parameter is estimated at p ¼ 0:35 s=kWh. With these input data, the overage and underage costs for sizing a building’s EESS with PV system are calculated as follows: s ; co ¼ c v ¼ cGeneration þ cStorage v ¼ 0:23 kWh
ð7Þ
With Q w , all parameters are available to calculate the cost-optimal periodical order quantity S according to Eq. (5). To ultimately derive the optimal capacity C of the EESS, the optimal order quantity S that can be interpreted as the amount of energy that is used for charging has to be adjusted by the conversion losses incurred when charging the EESS:
7
cu ¼ p c ¼ 0:12
s : kWh
ð11Þ
ð12Þ
The cost for operating the PV system and the EESS are neglected here because they are relatively small as compared to their investment costs. 4.5. Sizing results With the numerical values for the different parameters described in Section 4.4, the results shown in Table 2 can be calculated. Fig. 6 shows the frequency distributions of the calculated random yields used to determine Q w at the ratio of cost parameters RC. The high deterministic demand is adjusted by stochastic correction quantity Q w according to Eq. (5) and by conversion losses according to Eq. (8), leading to an optimal usable storage capacity C of about 26 kWh. This capacity can be interpreted as useable storage capacity and has to be adapted by the assumed DoD of 70% in order to obtain the optimal physical capacity of the EESS C Phys of about 37 kWh. Lithium-ion batteries can easily be charged with C-rates up to 1 leading to a power of the designed EESS of about 37 kW. The highest average hourly value in the residual load profile is 25 kW (cf. Fig. 3). Thus the available power of the EESS is sufficient. 4.6. Time domain simulation A time domain simulation is performed in order to validate the gained results using the inventory model. The same residual load profile and cost parameters are used to simulate different storage sizes. Different performance indicators (PIs) are used to evaluate the sensitivity of the problem. While varying the usable storage capacity, the maximum power was adjusted accordingly to assure maximum C-rates of 1. To measure the effectiveness of the EESS in exploiting the energy produced by the PV system, the building’s Degree of Autarky DoA and the Total Cost of Energy TCoE of the building are analyzed. DoA is defined as the quotient of self-produced energy
Table 2 Numerical results for the model’s output parameters. Ratio of cost parameters co/(co+cu)
Deterministic demand D
Stochastic correction quantity Qw
Cyclical order-up-to level S*
Optimal usable capacity C*
0.66
55.87 kWh
27.41 kWh
28.46 kWh
25.61 kWh
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Fig. 6. Frequency diagrams for yields EYield;i .
that was self-consumed and the total consumed energy of the building. TCoE equals the sum of the yearly cost of energy generated from the PV system, the yearly storage cost, and the yearly cost of electricity drawn from the grid using the above defined cost parameters and is normalized to the maximum value occurring during the parameter variation. The yearly storage cost equals the maximum of the cost due to cyclical aging of the EESS CyAC and the cost due to calendric aging of the EESS CaAC. CyAC can be calculated as follows: ( ) I I X X Edis;i ; Echar;i cStorage : ð12Þ CyAC ¼ min i¼1
i¼1
On the contrary, CaAC is calculated as follows: CaAC ¼
cEESS C Phys : lEESSLT
ð13Þ
As in Section 4.4,cEESS is assumed to equal 800 s per kWh, while the calendric lifetime of the EESS lEESSLT is assumed to equal 16 years. As further reference values, the average State of Charge SoC and the average Number of Cycles NoC per day are used. NoC can be calculated using the following relationship: nXI o XI min E ; E i¼1 dis;i i¼1 char;i NoC ¼ ; ð14Þ ~ 365 d C
where Edis;i is the energy discharged from the storage system in the time interval i and Echar;i is the energy charged to the storage ~ is the current storage capacity system in the time interval i. C during the parameter variation. Fig. 7 shows how the introduced ~ in the studied example. The vertical dashed PIs react to varying C line marks the optimal storage capacity C calculated using the inventory model introduced above. Fig. 7 reveals that the calculated storage capacity covers the area with a steep increase in DoA. To the right of the calculated capacity, every incremental capacity increase becomes more and more expensive. When looking at TCoE, the optimal usable capacity C determined with the help of the proposed model (vertical dashed line) is close to the minimum of the curve at a storage capacity of around 32 kWh. The discrepancy can be explained by the different calculation approaches. The single period inventory model optimizes the capacity based on cumulative distribution functions of the load profiles whereas the time domain simulation is a multi-period approach where energy can be stored for several consecutive days. The inventory model-based approach, however, accounts for the uncertainties of the energy supply from the PV system. With NoC of around 1, the cyclical aging of the battery is moderate and a long cycle life is possible. An average SoC of around 0.3 indicates that SoC is very low at times. However, due to the unsteady supply of solar power, particularly during autumn and winter, this seems acceptable.
Fig. 7. Behavior of performance indicators with increasing storage capacity (step range 1 kWh) and optimal usable capacity C determined by the proposed inventory model.
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5. Discussion and conclusion This paper contributes to the interdisciplinary research on sizing of EESSs. A comparison of properties of energy storage models with inventory models showed that the latter provided a promising starting point for a new domain of EESS-sizing approaches. In this paper, a single-period newsvendor model with supply uncertainties introduced by [30] was adapted for sizing EESSs. In a numerical analysis, the developed model was used to optimally size an EESS for an apartment building with a PV system. The sensitivity of the solution was then investigated with respect to different performance indicators using time domain simulation. The results show that inventory management and energy storagesizing can be brought well in line with each other. The optimal EESS capacity based on the developed sizing procedure is very close to the capacity that the time domain simulation would deem optimal a posteriori with regard to the total cost of energy. Hence, the proposed sizing approach is able to handle the uncertainty attached to the energy supply from PV systems. The applicability of the developed methodology for sizing energy storage systems in practice is further highlighted by the fact that it supports the objective of increasing the building’s degree of autarky as long as this is profitable. Another advantage of the presented approach is its robustness against variations of the load profiles. As long as the probability distributions of production and supply remain close to their original values, the storage size is optimal. In contrast to that, results derived from time-domain simulations are only valid for the used time series. Due to the use of probability distributions instead of time-domain data, computing time can be significantly decreased. The distribution of energy supply just has to be evaluated at the probability defined by the ratio of cost parameters. The model itself, however, is only valid under the assumption that the random yield follows a general distribution. For most practical cases only empirical distributions based on measurements or simulated data are available. The general applicability and optimality of the selected method for different distributions still needs to be proven. Another downside of the introduced approach is that due to the focus on sums of energy no procedure is provided to determine the optimal power of the EESS. Yet, as long as the analyzed application is characterized by C-rates lower than 1, this is not a problem. However, for high power applications, the approach proposed in this paper needs to be adjusted. In future studies, the developed method should be extended to a multi-period time frame that can also be modeled using the approach described in [30]. This way, the operating strategy of an EESS can be integrated into the EESS-sizing decision and a truly optimal EESS capacity can be calculated. The notion that the EESSsizing decision and the determination of the EESS operating strategy should ultimately not be decoupled was also supported by [35]. [35] studied under which conditions it is cost-efficient to employ an EESS to store electricity generated from manufacturing waste heat in times of low electricity prices and dispatch it to support the energy supply of a manufacturing system in times of high electricity prices. In a numerical analysis, they found that the EESS can only be capitalized on if the capacity and the performance of the EESS are tailored to the actual pattern of use of the EESS [35]. Thus, a fundamental integration of EESS-operating strategies into the developed sizing model should be addressed in future research. In addition, the basic as well as possible extended models should be applied to further scenarios and applications for detailed evaluation. The numerical example studied in the paper at hand solely focused on the task of finding the building’s cost optimal degree of autarky. When maximizing the profitability of storage systems, approaches considering multiple goals, e.g. additional grid relief, have to be taken into account. As the presented
9
approach is rather generic, it can serve as a starting point for extended models seeking to cope with complex operating strategies or multi use case applications.
Acknowledgements The second and the sixth author wish to acknowledge the support of the Carlo and Karin Giersch-Stiftung in funding their research.
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