Energy & Buildings 175 (2018) Contents lists available at ScienceDirect. Energy & Buildings

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1 Energy & Buildings 175 (2018) Contents lists available at ScienceDirect Energy & Buildings journal hoepage: Optial battery energy storage investent in buildings Hrvoje Pandži ć Faculty of Electrical Engineering and Coputing University of Zagreb, Unska 3, Zagreb, Croatia a r t i c l e i n f o a b s t r a c t Article history: Received 8 March 2018 Revised 13 June 2018 Accepted 11 July 2018 Available online 21 July 2018 Keywords: Battery energy storage Robust optiization Stochastic optiization Optial investent proble In ost countries, electricity is charged depending on the tie of use, which usually includes two tariffs: the low tariff, during night tie, and the high tariff, during daytie. On top of their energy payent, large electricity consuers pay a onthly fee per kw of peak deand. Therefore, installation of stationary battery storage can reduce electricity payents for large consuers in two ways: by reducing the peak deand and by shifting consuption fro the high tariff to the low tariff. The ai of this paper is to forulate a odel to deterine optial energy and power capacity of a stationary battery storage in order to iniize electricity payents. Since the future electricity consuption is uncertain, after forulating the deterinistic odel, two additional odels that consider uncertainty are introduced. The first one is the stochastic odel, which considers uncertain scenarios, and the second one is the robust odel, where uncertainty is represented by only an uncertainty set, without an assuption on the distribution of uncertainty within this set. The proposed odels are tested and copared on a real-world load data of a hotel in Croatia. The case study indicates that deterinistic and stochastic forulations result in slightly better investent decisions than the robust forulation Elsevier B.V. All rights reserved. 1. Introduction Acronys For easier referencing, the acronys used throughout the text are listed below: AC Alternating Current. CBA Cost-Benefit Analysis. CDF Cuulative Distribution Functions. CHP Cobined Heat and Power. DC Direct Current. EU European Union. Li-ion Lithiu-ion. NaS Sodiu sulfur. PbA Lead-acid. PUN Prezzo Unico Nazionale, Italian for single national price. USA Unites States of Aerica. Electricity consuers can be divided into three categories: doestic, coercial and industry consuers. All of the purchase electricity in retail arket fro a chosen supplier. The supplier forulates a tariff syste for different consuers, depending on the peak load, quantity and quality of electricity, voltage level of the consuer, and other characteristics. Eleents of the coer- cial/industry tariff syste are: consued active energy (kwh), active peak load (kw), over-consuption of reactive energy (kvar), easureent service fee, transission and distribution syste utilization fees, and lately a fee for supporting renewable energy sources. Consued active energy, active peak load and overconsuption of reactive energy can be directly affected by the consuer s behavior and/or investents, while the reaining eleents of the tariff syste are either regulated or related to the consuption, e.g. transission and distribution fees are usually set per kwh of consued electricity The quantity of active energy consued over a tie period, e.g. a onth, is deterined based on readings fro a power eter. Tariff systes usually differentiate between the active energy consued under the high tariff, i.e. during the day, and under the low tariff, i.e. during the night. Although the two-tariff etering systes have been used for any years, they are still in effect in any countries. The highly encouraged deployent of sart eters, essential for adoption of dynaic tariffs, is slow due to high investent costs and questionable benefits [1]. For exaple, the EU Meber States are required to ensure the ipleentation of sart etering, but this ipleentation is subject to a long-ter cost-benefit analysis (CBA) conducted for each country individually. If the results of CBA are positive, the roll-out target for year 2020 is 80% penetration of sart eters [2]. E-ail address: hrvoje.pandzic@fer.hr / 2018 Elsevier B.V. All rights reserved.

2 190 H. Pandži ć / Energy & Buildings 175 (2018) Another eleent of the tariff systes for large consuers in any countries is the peak active power, which is priced based on the axiu of all average 15 in or 30 in loads. Depending on the electricity prices and peak load policy in effect, peak load payents can constitute as low as few percent of the overall consuer s electricity bill, all the way to half of the overall electricity bill, e.g. in San Diego Gas & Electric utility [3]. This indicates that an energy storage device could reduce custoer s electricity bill in two ways, first by oving the deand fro the high to the low tariff, and second by reducing the peak load payents. This paper focuses on coercial/industry facilities that are charged both consued active energy and active peak load. The goal is to find optial battery energy storage investent, in ters of kw and kwh, that iniizes the su of overall operating and investent costs. The ain contribution is the forulation of a rigorous atheatical odel that considers the uncertainty of the annual load curve using stochastic and robust optiization. Monte Carlo ethod is used to obtain cuulative distribution functions (CDF) used to copare the perforance of the optiization ethods. 2. Literature review and contributions Energy storage has spurred a lot of interest in the research counity recently. At large scale, i.e. transission level, there have been any studies dealing with integration of energy storage. For exaple, ipact of large-scale battery storage on operating costs in systes with wind uncertainty is exained in [4]. Another ethod for syste-wide siting and sizing of energy storage is presented in [5], which, instead of characteristic days in [4], considers each day of the year individually. Both [4,5] indicate that distribution of wind power plants significantly affect the location and sizes of energy storage. When deciding on energy storage investent, it is essential to consider and appropriate storage technology. A coparative overview of large-scale energy storage technologies is available in [6,7], while a detailed review of battery technologies used for large-scale energy storage is found in [8]. The work presented in this paper is focused on sall-scale energy storage connected behind the eter at the networks user s preises. In order to ipleent optial control of energy flows between a consuer with energy storage and the power grid, a control device, such as the one developed in [9], is needed. An ipleentation of Berkeley Labs Distributed Energy Resources Custoer Adoption Model for optial selection of flexible technology, e.g. heat and cheical storage, efficiency investents, and cobined heat and power (CHP), in coercial buildings is described in [10]. The perfored case study on an iaginary hotel in San Francisco results in a reciprocating engine and an absorption chiller investents. These investents are estiated to provide 11% cost savings and 8% carbon eission reductions. A ultienergy odel for iniizing total operating cost of a building is proposed in [11]. The presented case study shows that theral storage units and water tanks are ost effective in reducing overall energy costs and that battery storage ight becoe attractive in case of reduced investent cost and longer lifetie. Arbitrage potential of a sall-scale battery energy storage in Korean electricity arket is studied in [12]. Charging and discharging schedules are derived using the Hotelling rule. The results indicate that the cost of Li-ion and sodiu sulfur (NaS) batteries is too high and such investent would result in onetary losses. This conclusion is based on energy prices in the observed spot arket. Energy storage perfors arbitrage between the highest and the lowest price throughout the day. If the difference is insufficient, the storage will not retrieve the investent. However, there are soe fundaental differences between energy storage acting in electricity arket, as an independent arket player, and energy storage being installed at the end user s facility: Perforing optial arbitrage in electricity arket is difficult due to unknown arket prices at the bidding stage. In other words, it is difficult to identify the optial hours for selling/buying electricity in the arket. On the other hand, end users are under tie-of-use tariff and electricity prices are known in advance for each hour of the day. However, end users face their uncertain local deand, the issue which large-scale energy storage perforing only arbitrage does not have. Reducing peak load reduces peak deand charges to the end users, while this source of incoe (savings) does not exist in the electricity arket. Both arket player with energy storage and end user with energy storage could take part in ancillary services arket providing up and down reserve. The difference is that the end user cannot provide such services directly (due to insufficient capacity), but only through a ediator, e.g. aggregator. Battery energy storage investent is reported to be econoically feasible in households when cobined with photovoltaic syste in case of a steady growth of the retail electricity prices [13]. Sensitivity analysis is perfored using different investent tiing (reduction of investent cost in the future is assued), different increase of electricity price and various sets of reuneration tariffs. A technique for sizing of battery storage in Belgian households with photovoltaic panels is proposed in [14]. Optial battery capacity is deterined based on correlations with the easured data. The authors indicate the iportance of battery storage in reducing the peak deand. A ore recent econoic assessent of energy storage in households with photovoltaic panels is presented in [15]. The results include sensitivity analysis of the ajor input paraeters, i.e. cost of energy storage, cost of electricity when purchasing and selling, and electricity generated by the rooftop solar syste. The authors conclude that diinishing feed-in tariffs and expected reduction of energy storage investent cost will result in higher integration of energy storage in households. In [16], energy storage is used in a household to reduce electricity charges using the actual Con Edison s electricity tariffs. The authors use an agentbased odel of appliances to randoly generate deand profiles. The case study indicates that the proposed operational strategy can result in up to 48% savings. Tie-of-use pricing has been used for any years to offset soe of the consuption during peak hours to the lowconsuption periods, i.e. night hours. In tie-of-use pricing, each day is divided in blocks, which consists of one or ore hours, with the sae price of electricity. The price of electricity varies between consecutive blocks, but not within any of the blocks [17]. The siplest tie-of-use tariff syste coprises of two blocks, the lowcost block during the night hours and the high-cost block during the daytie. The cost of electricity in each block is known ahead of tie and is the sae every day. Even this siple static tie-ofuse tariff, which is still coonly used by the suppliers and utilities, results in iplicit deand response, e.g. when a consuer decides to turn on a hoe appliance during the low-cost period. A proposal of a four-day tariff syste for industrial loads is carried out in [18]. In this four-day tariff syste, the week is divided in four day types: Monday, Tuesday-Friday, Saturday and Sunday. The authors report that the proposed tariff benefits both the utility copany and the industrial custoer with energy storage. Paper [19] develops a pricing schee for energy storage operated by custoers. The designed energy storage operation strategy iniizes electricity costs to the custoers using responsive tie-ofuse tariffs. Techno-econoic analysis of battery energy storage in a student doritory in South Africa under the tie-of-use tariff is presented in [20]. The results show that none of the three battery

3 H. Pandži ć / Energy & Buildings 175 (2018) technologies considered in the case study (PbA, NaS, and Li-ion) is econoically viable for the current electricity prices and tariff syste. The authors point out that the ain reason for this is the high cost of installation of battery storage syste, while the ipact of peak electricity price is lower. This is confired by the econoic assessent of behind-the-eter energy storage in Italy presented in [21]. This study considers li-ion, advanced PbA, zinc-oxide, NaS and flow battery technologies. Although zinc-oxide, li-ion and flow batteries result in best rate of return, it is still insufficient to ake the econoically attractive. A ethod for benefit assessent of an energy storage in a counity with renewable sources is proposed in [22]. The results indicate that the optial capacity of a battery energy storage is the one ensuring a full discharge during the peak load period. Another study on optial energy storage investent in local counities with high adoption of photovoltaics is presented in [23]. The case study, perfored on actual locations in Cabridge, USA, shows that the optial energy storage capacity at the counity level is 35% lower than the cobined energy storage at the individual households level. Additionally, electricity exchange between the counity and the distribution network is reduced. An interested reader can find a coprehensive review of challenges and perspectives of counity energy storage in [24]. An electricity cost iniization strategy for a ediu-scale public facility with energy storage is presented in [25]. The authors assue hourly real-tie pricing at the retail level and use NaS batteries to ove consuption to low-cost tie periods. The results of the Italian case study indicate that the return of investent is 10 years in case of correlation of the retail prices with the Sicilian zonal prices and 30 years in case of correlation with the PUN ( Prezzo Unico Nazionale ) values. A probabilistic approach for optial sizing of battery energy storage under tie-of-use tariff is proposed in [26]. The proposed Monte Carlo odel considers uncertainties of energy prices, load profile, and interest rate. A nuber of saples generated fro rando uncertain input data is applied to discrete battery energy storage capacities. After obtaining the average total cost over all Monte Carlo saples for each battery storage capacity, the optial battery capacity is identified as the one that yields the iniu overall cost of electricity, including the battery investent cost. Optial sizing proble of a battery storage in industrial facility is addressed in [27]. In the presented approach, different investent alternatives, i.e. battery energy storage capacities, are applied to different future realizations of deand profiles. Aong the obtained results, the best investent is identified as the iniu of the total operating cost, in-ax regret, or a cobination of the two. The authors conclude that the probabilities of the assued deand profiles significantly affect the investent decisions. With respect to the literature review above, the contributions of this paper are: 1. Forulation of a linear atheatical progra that finds optial battery storage investent in a coercial building. The capacity of battery energy storage is used to decrease the electricity bill in two ways: (i) it offsets a part of the deand fro the high-cost tariff to the low-cost tariff; (ii) it decreases the peak load payents. The odel above is extended to accoodate the uncertainty of the local annual load curve. Two odels under uncertainty are forulated: the stochastic odel and the robust odel. 2. Application of the proposed odels on an actual hotel load curve. The investents obtained using all three odels, i.e. deterinistic, stochastic and robust, are applied to a nuber of Monte Carlo scenarios (realizations of the uncertain annual load curve) in order to quantify the perforance of the proposed odels. The presented odels should be useful to owners of coercial and industry facilities interested in reducing their operating cost by investing in new technologies, i.e. batteries. 3. Matheatical forulation 3.1. Noenclature Indices t Index of hours in the year, belonging to set T. Index of onths in the year, belonging to set M. w Index of stochastic scenarios, belonging to set W Paraeters C t, e Cost of energy at hour t in onth, Eur/kWh. C p Cost of peak onthly load, Eur/kW. D t, Average load during hour t in onth, kw. K e Cost of energy storage installed energy capacity, Eur/kWh. K p Cost of energy storage installed power capacity, Eur/kW. L Energy storage lifetie. P distr Maxiu power drawn fro the distribution network, kw. P ax Maxiu allowed battery power capacity, kw. R Interest rate. S ax Maxiu storage state of charge. S in Miniu storage state of charge. η ch Battery charging efficiency. η dis Battery discharging efficiency. T Tie step, i.e. 1 h Variables d t, Average load during hour t in onth, kw (used only in robust forulation). e bat Battery energy capacity, kwh. p bat Battery power capacity, kw. p ch t, Average battery charging power during hour t of onth, kw. p dis t, Average battery discharging power during hour t of onth, kw. p peak Peak power withdrawn fro the network during onth, kw. r t, Auxiliary variable used only in the robust odel. s t, Battery state of charge at hour t of onth, kwh. u t, Auxiliary variable used only in the robust odel. v t, Auxiliary variable used only in the robust odel Deterinistic odel Objective function of the deterinistic odel iniizes the annualized battery storage investent cost ( P 1 ), energy payents throughout the year ( P 2 ), and onthly peak load payents throughout the year ( P 3 ): where t Miniize P 1 + P 2 + P 3, (1) ( ) P 1 = p bat K p + e bat K e ( 1 + R ) L, (2) L P 2 = C t, e T (D ) t, + p ch t, /ηch p dis t, ηdis, (3) P 3 = C p p peak, (4) The cost of battery energy storage installation P 1 consists of two parts: installation of power capacity, p bat K p, which reflects the cost of power converter that interfaces the AC power grid and DC battery syste (in kw), and installation of energy capacity, e bat K e,

4 192 H. Pandži ć / Energy & Buildings 175 (2018) which represents the cost of battery stack (in kwh). Since the energy storage investent is ade at the beginning of the observed period, net present value is used to levelize the investent cost to annual basis. Energy payents, P 2, are the result of energy drawn fro the grid at each hour. This energy is used to supply the load and to charge the battery. Variable p ch denotes only power used to increase the battery state of charge. However, the power withdrawn fro the network needs to include inefficiency of the power converter and the battery charging process. These inefficiencies are included in paraeter η ch, which is lower than 1. Accordingly, battery discharge inefficiencies are captured in η dis. Peak load payents for the entire year, P 3, are calculated based on the peak onthly load. The deterinistic odel is subject to the following constraints: p peak D t, + p ch t, /ηch p dis t, ηdis, t T, M (5) 3.3. Stochastic odel Stochastic forulation iniizes objective function (1) over a set of stochastic scenarios w. Constraint (2) does not change, since the investent is the sae across all stochastic scenarios. However, all the other constraints need to be updated to accoodate the stochastic scenarios. Therefore, the stochastic forulation of constraints (3) (12) is: P 2 = π w w t C e t, T (D t,,w + p ch t,,w /ηch p dis t,,w ηdis ), w (13) P 3 = π w C p p peak, 2, (14) p peak, 2 D t,, 2 + p ch t,, 2 /ηch p dis t,, 2 ηdis, t T, M, w W (15) s t, = s t 1, + p ch t, T p dis t, T, t T, M (6) SOC in e bat s t, SOC ax e bat, t T, M (7) p ch t, p bat, t T, M (8) p dis t, p bat, t T, M (9) D t, p dis t, ηdis 0, t T, M (10) D t, + p ch t, /ηch P distr, t T, M (11) e bat, p bat, p ch t,, p dis t,, p peak, s t, 0, t T, M (12) Peak onthly load, p peak, used to calculate onthly peak load payents is calculated in constraint (5). Eq. (6) calculates battery storage state of charge, which is liited fro the lower and the upper side in (7). Battery charging and discharging powers are liited in (8) and (9) by the installed battery power capacity. Battery odel (6) (9) does not include degradation costs as they are already included in Eq. (2). Paraeter L denotes the assued battery lifetie considering the expected nuber of cycles per day and overall nuber of cycles the battery can perfor. For exaple, if the battery can perfor 3650 cycles before its capacity is reduced to an unacceptable level and one charging/discharging cycle per day is expected, paraeter L will be set to 10 years. After this period, the battery will be considered worn out and should be replaced. Constraint (10) disables reverse power flow, i.e. flow of energy into the distribution network. This is needed as otherwise the consuer would need to register as a producer as well. Constraint (11) liits the power drawn fro the distribution network. Finally, non-negativity of variables is defined in (12). The odel above yields optial solution in case all the paraeters are known ahead of tie. However, annual load curve cannot be perfectly forecasted. Therefore, its uncertainty should be incorporated into the odel. The ost coon technique for accoodating uncertainty in optiization odels is stochastic optiization, where the uncertain paraeter is represented with a set of stochastic scenarios, each with an assigned probability of occurrence. Another technique is the robust optiization, which does not consider stochastic scenarios, but only the range of uncertainty. This eans there is no characterization of uncertainty within the uncertainty set. The following subsections forulate the stochastic and the robust odels. s t,,w = s t 1,,w + p ch t,,w T p dis t,,w T, t T, M, w W (16) SOC in e bat s t,,w SOC ax e bat, t T, M, w W (17) 3.4. Robust odel p ch t,,w p bat, t T, M, w W (18) p dis t,,w p bat, t T, M, w W (19) D t,,w p dis t,,w ηdis 0, t T, M, w W (20) D t,,w + p ch t,,w /ηch P distr, t T, M, w W (21) e bat, p bat, p ch t,,w, p dis t,,w, p peak,w, s t,,w 0 (22) Robust optiization takes a different approach towards uncertainty as copared to the stochastic optiization. Instead of stochastic scenarios, robust odels are based on uncertainty sets [28]. This eans there is no assuption on the uncertainty distribution within the uncertain range. The goal of the robust optiization is to optiize against the worst realization of uncertainty within this uncertain range. To achieve this, the initial deterinistic odel (1) (12) can be expressed in the following general ter: subject to J J Miniize c j x j (23) j=1 a i, j x j b i, i I (24) j=1 x j 0, j J (25) In proble (23) (25) soe of the coefficients a i,j are uncertain [ ] and take values within range a i, j, a i, j + a ˆ i, j, where a ˆ i, j indicates a deviation of uncertain paraeter a i,j fro its desirable value. Budget of uncertainty, coonly denoted as Ɣ i, is used to control which [ values ] of uncertain paraeters can be taken fro the range a i, j, a i, j + a ˆ i, j, thus controlling the conservatis of solution. Ɣ { } i takes values fro interval [0, J ], where J = j a ˆ i, j 0. The ost optiistic solution is achieved for Ɣ = 0, since no deviations fro the ost desirable value of uncertain paraeter are considered. On the other hand, the ost pessiistic solution is achieved for Ɣ = J, when the worst deviation is considered.

5 H. Pandži ć / Energy & Buildings 175 (2018) The robust version of the initial deterinistic odel (1) (12) is forulated as follows: subject to t Miniize P 1 + P 2 + P 3 (26) ( ) P 1 = p bat K p + e bat K e ( 1 + R ) L, (27) L P 2 = C t, e T (d ) t, + p ch t, /ηch p dis t, ηdis, (28) P 3 = C p p peak, (29) p peak d t, + p ch t, /ηch p dis t, ηdis, t T, M (30) Fig. 1. Hourly load levels for the deterinistic case during January (1), May (5), August (8) and Deceber (12). s t, = s t 1, + p ch t, T p dis t, T, t T, M (31) SOC in e bat s t, SOC ax e bat, t T, M (32) p ch t, p bat, t T, M (33) p dis t, p bat, t T, M (34) d t, p dis t, ηdis 0, t T, M (35) d t, + p ch t, /ηch P distr, t T, M (36) e bat, p bat, p ch t,, p dis t,, p peak, s t, 0, t T, M (37) d t, = D in t, + u t, Ɣ t, + v t,, t T, M (38) u t, + v t, ( D ax t, D in t, ) r t,, t T, M (39) u t,, v t, 0, r t, 1, t T, M (40) In the above odel, forulation (26) (37) correspond to the deterinistic odel (1) (12), with an exception that new variable d t, now represents the deand instead of paraeter D t,. Value of d t, is deterined in (38) using auxiliary variables u t, and v t,, as well as the robustness paraeter Ɣ t,, which takes values between 0 and 1. Robustness paraeter Ɣ t, is used to control the level of conservatis of the solution by controlling the expected load level ranging between D t, ax and D in t,, where D ax t, is the axiu load and D t, in the iniu load observed over all scenarios at tie period t of onth. In the ost optiistic case Ɣ t, is equal to 0, i.e. the load is equal to D t, in at all tie periods of each onth. On the other hand, in the ost pessiistic case Ɣ t, is equal to 1, i.e. the load is equal to D t, ax at all tie periods of each onth. Any value in between 0 and 1 can be chosen for Ɣ t, to fine tune the level of conservatis of the robust optiization odel. Note that Ɣ t, can take different values at different tie periods. Auxiliary variables u t, and v t, are non-negative, while r t, is greater or equal to 1. Detailed inforation on how constraints (38) (40) are derived, along with atheatical proofs, can be found in [29]. 4. Case study The presented deterinistic, stochastic and robust forulations for finding optial battery energy storage capacity are applied to an actual hotel in the coastal part of Croatia. Ten stochastic scenarios of the recorded one-year hourly loads are used for stochastic forulation 1 Deterinistic forulation is perfored on a single scenario obtained by the forward selection scenario reduction technique [31]. Deterinistic hourly load levels for soe distinct onths are visualized in Fig. 1. The hotel is closed for the ost of Deceber (12) and the load levels are thus very low. The load starts picking up in the second week of January (1), when the preparations for a new season begin. The deand in May (5) is already quite high and reaches its axiu in August (8), after which it gradually decreases until it reaches its iniu in Deceber. All the data on electricity cost, network charges and peak load charges used in this case study are collected fro the website of the Croatian Power Utility - HEP [32]. Cost of electricity is Eur/kWh in the low tariff and Eur/kWh in the high tariff. During the daylight saving tie, the low tariff starts at 22:00 and ends at 8:00, while during the winter tie it starts at 21:00 and ends at 7:00. Peak load is priced at 7.5 Eur/kW. Battery energy storage installation cost is 300 Eur/kW and 100 Eur/kWh, which is the price expected by the year 2025 [33]. Interest rate is 6% and expected battery lifetie 10 years [34]. Both charging and discharging efficiencies are 0.95, resulting in 0.9 roundtrip efficiency [35]. Two different peak load payent policies are exained in this case study. In the first one, all hours of the day are subject to peak load payents, while the second one considers only the high tariff hours when calculating the peak load payents. This eans that, theoretically, in the second case all the load could be oved to the low tariff hours and peak load payents would be zero. This is visualized in Fig. 2, where the optial load profiles are derived solely on the peak load payents (other deand profiles ay be optial in case energy payents are considered as well). There are any real-life exaples of both of these policies. For exaple, the peak load payents applicable to all hours policy is used in Energex s 2 NTC tariff [36], while the peak load payents applicable only to high-tariff hours policy is eployed in Energex s 1 As shown in [30], the choice of ten stochastic scenarios represents a suitable trade-off between coputational coplexity and cost perforance of a stochastic odel. 2 Energex is a power distribution copany serving over 1.4 illion consuers, a subsidiary of Energy Queensland Liited, which is owned by the Australian governent.

6 194 H. Pandži ć / Energy & Buildings 175 (2018) Table 1 Objective function value, costs, installed battery capacity and expected cost based on 500 out-of-saple siulations for all three proble forulations. Deterinistic Stochastic Robust Ɣ = 0. 0 Ɣ = Ɣ = 0. 5 Ɣ = Ɣ = 1. 0 Objective Function, Eur 321, , , , , , ,224 Energy Cost, Eur 275, , , , , , ,460 Peak Load Cost, Eur 27,785 27,825 25,215 26,255 27,421 28,524 29,730 Storage Cost, Eur 18,301 17,847 16,869 16,807 16,812 16,890 17,033 Storage Power, kw Storage Energy, kwh Expected Cost, Eur 322, , , , , , ,434 Fig. 2. Optial load profile under different peak load policies when considering only peak load payents. The original load profile is denoted with black line. If peak load payents apply to all the hours of the day (policy 1), the optial load profile is flat, since it incurs iniu peak load payents. However, if peak load payents are applicable only to the high-tariff hours (policy 2), the optial load profile is zero during the high-tariff period, since it avoids any peak load payents. NTC7400 tariff [36]. The latter policy is also effective in Croatia [37], where the case study is conducted. The following subsections exaine each of the two policies individually Peak load payents applicable to all hours This subsection exaines results when the peak load in constraint (5) considers all hours, both the ones in the low tariff and the ones in the high tariff. The case study results for deterinistic, stochastic and robust forulations are shown in Table 1. The objective function value is divided into energy cost, peak load cost and battery energy storage installation cost. Deterinistic and stochastic forulations result in very siilar objective function values. However, the deterinistic odel invests ore in energy capacity of storage ( kwh copared to kwh), which incurs higher storage cost, but is copensated through lower energy cost. In order to assess the quality of the objective function values fro the first row of Table 1, they are evaluated on a set of 500 out-of-saple scenarios. These scenarios are obtained fro the original 10 scenarios using the rando forest algorith described in [38]. The deterinistic odel (1) (12) is then run for each of the 500 scenarios, which represent possible realizations of uncertainty, for storage power and energy capacities fro Table 1 taken as fixed. In other words, p bat and e bat are now paraeters instead of variables. The resulting expected cost over the 500 out-of-saple scenarios are shown in the last row of Table 1. Again, the deterinistic and the stochastic odels yield alost identical expected cost. This coparison indicates there is no gain in using stochastic forulation over the deterinistic one in ters of the quality of the solution. Furtherore, the stochastic solution is achieved in 5 in 47 s, while the deterinistic one is calculated Fig. 3. Hourly load levels on the 21st day of each onth. in only 9 s. The ain reason for stochastic forulation failing to outperfor the deterinistic one in ters of the quality of the solution is the siilarity of the load curve for each day. Fig. 3 shows hourly load levels on the 21st day of each onth. Regardless on the onth (excluding Deceber, when the load level is flat and very low), the load curves look very siilar, with two characteristic peaks. The only distinct feature is the position of the curve, i.e. higher overall deand in suer onths and lower overall deand in winter onths. Since all the curves have the sae shape and the area under each curve depends solely on the seasonality, there is no gain in solution quality by considering uncertainty of the load curve shape. In case no battery storage is installed, the objective function value in the deterinistic case is 325,565 Eur, which is 1.35% higher as copared to the deterinistic case when the storage is installed. Objective function values of the robust forulation in Table 1 increase as the budget of uncertainty Ɣ increases. When Ɣ = 0, the lowest load at each tie period over the ten stochastic scenarios is assued. Therefore, the expected overall cost is lower than for the deterinistic (or stochastic) forulation. This can be seen in Fig. 4, which shows ten stochastic scenarios, as well as the load curves for three values of Ɣ on May 2nd. The yellow load curve for Ɣ = 0 has lower or equal values than any of the ten stochastic scenarios, including the scenario used in the deterinistic odel. Since overall energy consued is significantly lower than in any stochastic scenario, the objective function value is uch lower as well. Therefore, objective function values of the robust odel should not be directly copared to the values of the deterinistic and stochastic odels. Instead, the results of their out-of-saple analysis should be copared, which is discussed in the following paragraphs. Objective function value of the robust odel increases with the level of conservatis and for Ɣ = 1 it is 9% higher than for the deterinistic (or stochastic) forulation and 20% higher than for

7 H. Pandži ć / Energy & Buildings 175 (2018) reduce the peak load and to shift deand fro the high to the low tariff. With couple of exceptions, such as hours 11 and 32, battery storage discharges during the high tariff. Fig. 7 shows a breakdown of electricity payents between the energy cost and the peak load cost by onths. The highest payents appear in suer onths, when the hotel load is the highest (copare to Fig. 3 ). Energy payents are uch higher than the peak load payents and they range fro 82% of the overall bill in 12 to 98% in Peak load payents applicable only to high-tariff hours Fig. 4. Ten stochastic scenarios and load curves for Ɣ = 0. 0, Ɣ = 0. 5 and Ɣ = 1. 0 on May 2nd. Ɣ = 0. For Ɣ = 0. 5, robust forulation results in siilar objective function value as the deterinistic one. Siilar observation is applied to both the energy and power payents, which increase with the overall deand and peak load. However, the storage investent cost is siilar regardless of the uncertainty budget Ɣ and lower than both in the deterinistic and stochastic forulations. This indicates that robust forulation tends to install less storage capacity than deterinistic and stochastic forulations. This is a direct result of the lack of the characterization of uncertainty in the robust optiization. Since the robust forulation only relies on boundary values of the uncertain paraeter, i.e. the load, it tends to ake the load curve soother, which provides less opportunity for storage to reduce peak loads. This phenoenon is visualized in Fig. 4. Clearly, all three load curves used in the robust forulation are less volatile than any of the scenario curves, thus reducing the benefit of battery storage. The flatter profile of the robust load scenarios akes peak shaving ore difficult since ore energy needs to be discharged fro the battery during consecutive hours to achieve significant peak load reduction. As previously said, the only relevant coparison of robust, deterinistic and stochastic solutions is in ters of the expected cost. Although the lowest expected cost of the robust odel is achieved for Ɣ = 0. 0, all the expected costs are close to each other. However, none of the solutions achieved using the robust optiization is as cost-effective as the ones achieved by using deterinistic or stochastic optiization. The resulting cuulative distribution functions are given in Fig. 5. Deterinistic and stochastic forulations result in alost identical curves, which is expected due to very siilar battery storage investents. On the other hand, robust forulation generally results in higher expected overall costs. The best perforance of the robust odel is achieved for Ɣ = 0. 0, while the second best perforance is achieved for Ɣ = 1. 0, which yields the highest battery storage investent aong all robust solutions (see Table 1 ). For Ɣ equal to 0.25, 0.5 and 0.75, battery storage investents are very siilar and the resulting cuulative distribution functions for these three values of the budget of uncertainty are very close to each other. Fig. 6 shows actual load, battery storage charging and discharging actions and net load (actual load plus battery charging power inus battery discharging power) during the first three days of April for the deterinistic case. The optiization set the April peak load at around 249 kw, which eans that storage always discharges when the load is above this value, even during the low tariff, e.g. in hour 9. The battery storage installed power capacity is around 100 kw and the figure shows this capacity is being exploited extensively. Generally, battery storage operates in a way to This subsection analyzes the results of siulations when only the high-tariff hours affect peak load payents. The optiization results are shown in Table 2. The overall costs are 1 2% lower as copared to the case when low-tariff hours affect the peak load payents as well. Again, deterinistic and stochastic forulations yield very siilar results. Energy and peak load costs are drastically reduced by 12% and 43%, respectively. This is a direct result of a heavy investent in battery energy storage, which is ore than tripled (copare to Table 1 ). Much higher capacity of energy storage enables oving larger volues of electricity fro the high-cost periods to the low-cost periods. Energy-to-power ratio of the installed batteries in all forulations is 10, because this corresponds to the nuber of low-tariff hours. Further increase of energy-to-power ratio would not provide additional charging opportunities. The lowest expected cost is achieved for the stochastic optiization. Robust forulation for Ɣ = is the second best, while the deterinistic forulation results in third best expected cost. The worst expected cost is achieved for Ɣ = Again, all the expected costs are very close together, without significant outliers. The battery operation and net load curves in Fig. 8 are uch different than ones in Fig. 6. Since only net load during the hightariff periods (denoted with green background) is considered for peak load payents, ajority of the load is shifted to the low-cost tie periods. The net load during the high-tariff periods is reduced fro 249 kw to only 100 kw. As a consequence, the peak load during the low-tariff periods spikes above 500 kw. This is a direct result of the peak load pricing policy. Battery storage charging curve in Fig. 8 indicates that the storage is charged at the highest possible rate during all low-tariff hours, draatically increasing their net load. The stored energy is used during the day to discharge and keep the peak load at lowest possible level. Utilization of the battery energy storage is very high, as in each hour in Fig. 8 the battery is either charging or discharging. Fig. 9 shows a breakdown of electricity payents between the energy cost and the peak load cost by onths when peak load payents are applicable to only high-tariff hours. When copared to Fig. 7, the energy payents are lower as there is ore energy storage installed, which can ove ore consuption fro the high-tariff to the low-tariff periods. The peak load payents are also reduced and they constitute at ost 7% of the overall bill in 8. Interestingly, in 12 there are no peak load payents as the entire consuption, which is quite low and flat, is oved to the low-tariff periods. 5. Conclusions, liitations and future work Decreasing prices of battery energy storage should ake it an attractive investent for reducing electricity cost in buildings in the near future. Battery energy storage creates a positive effect by reducing consuption in the high-tariff periods, which consequently increases consuption in the low-tariff periods, and by reducing he peak load. Based on the results of the case study, the following conclusions are derived:

8 196 H. Pandži ć / Energy & Buildings 175 (2018) Fig. 5. Cuulative distribution functions based on 500 out-of-saple scenarios for deterinistic, stochastic and robust forulations. Table 2 Objective function value, costs, installed battery capacity and expected cost based on 500 out-of-saple siulations for all three proble forulations when peak load payents consider only high-tariff hours. Deterinistic Stochastic Robust Ɣ = 0. 0 Ɣ = Ɣ = 0. 5 Ɣ = Ɣ = 1. 0 Objective Function, Eur 316, , , , , , ,131 Energy Cost, Eur 240, , , , , , ,333 Peak Load Cost, Eur 15,767 15,236 14,365 14,701 15,297 16,080 16,320 Storage Cost, Eur 59,972 62,170 55,269 57,964 59,959 61,386 65,478 Storage Power, kw Storage Energy, kwh Expected Cost, Eur 316, , , , , , ,934 Fig. 6. Load, battery storage charging and discharging quantities and net load (actual load plus battery charging power inus battery discharging power) during the first three days of April for the deterinistic case when peak load payents are applicable to all hours. Colored background denotes the high-tariff periods. 1. Deterinistic and stochastic forulation result in very siilar objective function values, but soewhat different ener gy storage investents. This can be attributed to the flatness of the objective function. In other words, at the optial point, slightly higher (lower) investent in energy storage results in a siilar reduction (increase) in operating costs. This indicates that the objective function value is not very sensitive to a slight change in the storage investent. 2. Deterinistic and stochastic forulations result in very siilar objective function values because the building used in the case study is a hotel with a consistent ratio of consuption in the low-tariff and the high-tariff periods. Slight changes in Fig. 7. Energy and peak load costs per onth when peak load payents are applicable to all hours. consuption generally just shift the load curve up (in case of higher consuption) or down (in case of lower consuption), but the shape reains alost the sae. Hence, for this type of buildings, there is no additional value in using stochastic forulation as opposed to the deterinistic one. 3. Although robust forulation generally perfors worse than the deterinistic or stochastic forulation, all the expected costs are within 0.1%, which is a negligible difference. This sall difference is because the robust forulation does not capture the load curve shape, but only the range of stochastic load curves. As a result, the worst-case load curve used in objective function of the robust forulation is flatter than any of the scenarios. This drives down the installed battery storage capacity and yields slightly sub-optial investent decisions. In order to achieve the best perforance using the robust forulation,

9 H. Pandži ć / Energy & Buildings 175 (2018) which is not included in the battery investent cost in Eq. (2), the reserve provision odel should include battery degradation costs. A linear and a depth-of-discharge dependent battery degradation odels available in [39] are well-suited for this purpose. Acknowledgeent This work was carried out within the H2020 ERA-Net Sart Grids+ project icrogrid Positioning - ugrip (co-funded by the Croatian Environental Protection and Energy Efficiency Fund) and project Sart Building - Sart Grid - Sart City (co-funded by European Regional Developent Fund through Interreg Danube Transnational Prograe, grant no. DTP Sart). Fig. 8. Load, battery storage charging and discharging quantities and net load (actual load plus battery charging power inus battery discharging power) during the first three days of April for the deterinistic case when peak load payents are applicable to only high-tariff hours. Colored background denotes the high-tariff periods. Fig. 9. Energy and peak load costs per onth when peak load payents are applicable to only high-tariff hours. the budget of uncertainty needs to be carefully selected and verified. 4. When coparing the two peak pricing policies, the one that bases the peak load payents only on the high-tariff hours is ore attractive to the building. This policy results in larger energy storage with higher energy-to-power ratio. The highcapacity battery storage significantly reduces peak load payents and shifts ore consuption to the low-tariff periods. 5. Battery investent cost used in this case study is lower than the current cost, which indicates that, at least for the case study in question, this investent will becoe attractive in the near future, assuing the battery costs will reduce. With current battery cost over 200 Eur [33], the investent is not econoically sound. The presented odel includes only the uncertainty on the consuer s side and does not address the uncertainty of energy prices (set by the supplier) and network tariff uncertainties (set by the utility and approved by the energy regulator). Risk of change in energy prices can be soewhat alleviated by signing long-ter supply contracts. Regardless, an additional sensitivity analysis on the electricity prices should be perfored in order to deterine the value of energy storage at higher or lower electricity prices and differences between the high-tariff cost and low-tariff cost. On the other hand, policy changes that affect network tariffs (payents for utilization of the network and peak load payents) are difficult to anticipate. The future of peak load payents ranges fro their repeal to a draatic change in their structure, see e.g. [36], which includes peak, off-peak and shoulder tariff. 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