Optimal Sizing of Battery Energy Storage System

***Abstract***

There are two view types of BESS owners. The first one is the utility and the second one is a demand-side-BESS-owner. They have different objective of sizing BESS. Utility wants to maximize social welfare, but demand-side-BESS-owner pursues their own profits. Therefore, according to the type of BESS owner, the method for finding optimal size of BESS is different. Also, although the type of BESS owner is same, method or way to find optimal size of BESS can be different according to what authors consider. Methods of determining the optimal BESS size will be summarized in this paper.

***Index Terms***

BESS, DG, RES, Utility, Demand-side-BESS owner, Peak shaving, Mitigating the output of RES, PV, Wind, Size, Optimization, Social welfare

## Introduction

### Introduction to BESS

Battery Energy Storage System (BESS) changes our life. Conventionally, power system is passive and hard to store electricity, so power between generation and demand must always be balanced. However, due to the advent of BESS, electricity can be stored and used afterward. As and when we like. Usually, there are three main important role of BESS.

#### Peak shaving

The first role of BESS is peak shaving. Peak shaving is saving power in BESS when load-level is low (at off-peak time), and make the output when load-level is high (at peak time). Peak shaving helps an owner of BESS to increase social welfare or make profits. Fig. 1 shows a peak shaving using BESS.12

#### Mitigating/maximizing the output of RESs

   Secondly, using BESS, the output of Renewable Energy Sources (RES) can be mitigated or maximized. Usually, the output of RESs, such as a wind generator and a PV generator, is fluctuated because it depends on wind speed, solar radiation, and etc. Fluctuation of the output has deteriorating effect on reliability of a power system. When the output of RESs is fluctuated, the BESS can mitigate the fluctuation and improve the reliability. Most RESs operate in Maximum Power Point Tracking (MPPT) mode to generate maximum power. But, sometimes RESs operate for deloaded-mode which makes the output be lower than MPPT modes to improve the reliability of a power system. However, if BESS is used, RESs can operate near MPPT point because BESS can improve the reliability. Fig. 2 shows how BESS mitigates the output of RESs.

#### Frequency regulation

The third one is frequency regulation. Primary frequency control and secondary frequency control need fast-response sources. Because of fast response features of BESS, it is effective to use BESS in primary/secondary control. Fig. 3 shows how BESS helps frequency regulation.

### Optimal size of BESS

There are many researches related to operation and control using BESS. However, before adapting operation scheme and controller, planning scheme of BESS should be decided first. Planning, such as location and sizing, is the first one to be considered. Benefits and social welfare can be increased by a location of BESS and a size of BESS, therefore, planning of BESS is important problem and it can be an optimization problem. When optimal size of BESS is found, optimal size of BESS used for peak-shaving and mitigating/maximizing of the output of RESs can be obtained. However, the size of BESS used for frequency regulation is different, because BESS is used for control not for dispatching. Determining sizing of BESS for frequency regulation is usually not to find a "optimal" size, but to find a "proper" size. In this paper, we just focus on the BESS used for peak-shaving and mitigating/maximizing of the output of RESs.

### Two types of owner : Utility and demand-side-BESS-owner

Sizing of BESS should be decided toward maximizing benefits and social welfare. There are two types of owners. The first one is utility and the second one is a demand-side-BESS-owner. Utility supervises whole power system and a demand-side-BESS-owner supervises only their area. Utility and a demand-side-BESS-owner(in Fig. 4, customer represents demand-side-BESS-owner) are shown in Fig. 4. Upper figure shows managed area of utility, and lower figure shows managed area of each demand-side-BESS-owner (customer). They have a different purpose for using BESS. The objective of utility is maximizing social welfare. However, the objective of demand-side-BESS-owner is maximizing their profits or benefits. Specific optimization problems are handled below.

## Optimal BESS size : The view of utility

This section is dealing with BESS sizing in a point of BESS owning utilities’ view. Utilities use BESS to make additional social welfare. BESS is used to shave a peak load and mitigate the output from RES.

### (1) Optimal sizing of BESS which is used for peak shaving

In  and , mixed-pass dynamic programming (MPDP) is used to find an optimal size of BESS. The objective function for an optimal size of BESS is presented as a ratio of the fuel cost saving over the capital cost of BESS as 3.1.1.(1).

m
a
x

F
S

C
P
L

⋯

{\displaystyle max{\frac {FS}{CPL}}\cdots }

3.1.1.(1)

F
S

{\displaystyle FS}

: Fuel cost savings due to BESS,

C
P

{\displaystyle CP}

: Capital cost of BESS

The fuel cost saving is generated from a peak shaving by using BESS. The capital cost of BESS consists of a cost of the power capacity (W) and a cost of the energy capacity (Wh) as shown in 3.1.1.(2)

C
# P

C

E

E

r
a
t
e
d

C

S

S

r
a
t
e
d

⋯

{\displaystyle CP=C\_{E}E^{rated}+C\_{S}S^{rated}\cdots }

3.1.1.(2)

C

E

{\displaystyle C\_{E}}

 : cost of energy capacity related to amount of capacity,

C

S

:

{\displaystyle C\_{S}:}

 cost of rated power related to

BESS,

E

r
a
t
e
d

{\displaystyle E^{rated}}

 : energy capacity,

S

r
a
t
e
d

{\displaystyle S^{rated}}

 : rated power

The power capacity cost is related to a cost of power converter system, and a cost energy capacity is related to the amount of batteries, costs of infrastructures and facilities, and engineering costs. BESS spends an operation and a management (O&M) cost for its life time and also generate additional profits. But these are not considered in this paper. Because these papers are the earliest research in BESS size optimizing area, constraints merely include energy and a capacity of BESS, power balance and power limits.

In , the objective function is formed as a generation cost. Mixed integer programming model treating power/energy size as a decision variable is used to find the optimal size of BESS. Switchable loads are included, and the turning off penalty of switchable loads is given as an average electricity rate. Power loss in a power balance equation, spinning reserve requirements, unit minimum up/down time, and ramp constraints are considered as constraints.

There are papers, , , , additionally considering distribution network system (DNS), contrary to above mentioned papers. In , objective function is to minimize daily deviation of a load power. Security constraints of voltage and frequency are complemented. Economical aspects are not considered in choosing capacity of BESS.

In , the objective function is formed as minimizing the sum of squared voltage deviations, power loss and fuel cost. Voltage support of BESS and network losses are considered in objective function with BESS site and size as variables. But O&M cost of BESS is not included in the papers.

In , savings in DNS and life time cost of BESS are considered. Objective function is established with increased savings subtracted by BESS costs. Total saving consists of carbon emission cost saving and distribution network savings, and fuel cost savings. Distribution network saving include savings obtained from providing reactive power and savings from peak shaving. To reflect life-cycle cost of BESS, operating and maintenance costs respectively related to rated power and annual discharge energy variables are added to previously presented capital cost of BESS. Size of BESS is calculated with heuristic-based approach that includes iterations dealing power/energy capacity as parameters. Because the solution is solved by heuristic method that use BESS capacity as parameters, the result is feasible but not always optimal.

In , savings on transmission facilities and DNS facilities are additionally considered. Not only saving of peak shaving and providing reactive power is included, but also upgrades deferral savings and reduction of annual invoice for utilization of power transmission network are included. As the amount of power flow becomes larger year by year, facilities of substation need to be improved. But if a peak load is shaved with BESS, substation upgrade can be deferred and lead to saving cost which is the function of load consumption growth rate, peak load reduction ratio. In France, customers subscribe to power of demand and annual invoice is charged according to contraction power. If customer overuse power more than contracted, penalty is charged. But this penalty can be avoided by using BESS. Cost is considered same as , life-cycle cost of BESS. To formulate net present value (NPV) as objective function, inflation and discount rate are considered. RESs are not included in this paper, which have intermittent and fluctuating output that can be mitigated/maximized when integrated with BESS.

### (2) Optimal sizing of BESS which is used for mitigating/maximizing output of RES

In , the authors focus on the spilled wind energy. As mentioned before, RESs in deloaded mode generate non- maximum output because of reliability of power system. Reduced wind energy for reliability is the spilled wind energy. The objective function is the cost of spilled wind energy. The spilled energy is related to the size of BESS, and the rated power and energy capacity of BESS are decision variables. The purpose is to maximize the objective function. In other words, maximizing the spilled wind energy is related to deciding optimal size of BESS. However, in , authors determine the optimal size of BESS just considering the spilled wind energy. They don’t consider cost of BESS and just focus on how the energy can be saved most.

In , author considers the cost of BESS when the optimal size of BESS is decided. The system consists of conventional generator, BESS, wind farm, and loads. BESS is used for mitigating output of wind farm. It means wind farm can generate the desired output and wind farm can be dispatched and can participate in electricity market with using BESS. Scheduled value of wind farm is decisions variables, and the size of BESS can be expressed by dispatch value. The objective function is presented as 3.1.2.(1).

m
a
x

F

l
t

C

c
a

C

o
p

⋯

{\displaystyle max{\frac {F\_{lt}}{C\_{ca}+C\_{op}}}\cdots }

3.1.2.(1)

F

l
t

{\displaystyle F\_{lt}}

: Output of BESS life time function,

C

c
a

{\displaystyle C\_{ca}}

: BESS capital cost,

C

o
p

{\displaystyle C\_{op}}

: BESS operating cost

Output of life time function is equivalent value of life time of BESS which is estimated by the numerical approach. Life time function and operating cost are function of the size of BESS. The optimal size of BESS is decided through optimization function. But, the method for estimating life time function is not quite elaborate.

## Optimal BESS size : The view of demand-side-BESS-owner

In this section, we focus on the research related to the view of demand-side-BESS-owner. The objective of researches is to maximize owner’s profit. In the most researches, it is assumed that demand-side-BESS-owners have loads, ESS and DGs(sometimes, they have conventional generators). Demand-side-BESS-owner can be just customer or community which is a big customer, such as LSEs, or distribution network operators. The energy storage sizing problem related to Demand-side-BESS-owner has been addressed in the literature ,,,,.

### (1) Optimal sizing of BESS which is used for peak shaving

Especially, in ,,, Optimal size of BESS is found in order to maximize benefits of customers and community which have BESS and loads.

In , authors assume that customers are Time Of Use(TOU) rates customer. TOU rates consist of three time period : Peak load hour, medium load hour, and light load hour. The objective is to maximize saved electricity cost considering the capital cost. Saved cost can be expressed by total electricity charge without BESS minus total electricity charge with BESS. Capital cost is comprised of battery cost, converter cost, and balance of plant cost and maintenance cost. Therefore, the objective function is saved cost/capital cost and goal is to maximize the objective function. Capacity and rated power of BESS are closely related and they are the decisions variable. Through maximizing objective function, optimal capacity and rated power of BESS can be determined. Adavnced MPDP is used for optimizing method which is an improvement of multi pass dynamic programming (MPDP). However, authors don’t consider essential constraints and BESS capacity loss.

In , authors approach similar way to , but they compose the objective function different way. The objective function is expressed as benefits minus capital cost, and authors assume benefits are proportional to required maximum power to shave. Likewise, through maximizing objective function, optimal capacity and rated power of BESS can be determined. Dynamic Programming(DP) method is used for optimization. Like [10], many constraints don’t be considered.

In , demand-side-BESS-owner is not just load customer but little community having BESS. Community having loads and BESS is connected with the main grid. The community owner can import power from the external main grid. There is contract between the external main grid and community. If power deviation of import power from schedule value occurs, community must pay the penalty for the deviation. Hence, The objective function consists of the total cost which is the sum of the cost of imported power, the cost of the BESS and the penalty due to imported power deviation from the scheduled power. The main purpose is to minimize the objective function, and power and energy rating of the BESS are decisions variables. 3.2.1.(1) shows the objective function of this paper. Power balance, power rating limits of BESS, energy rating of BESS, and imported power limits should be considered and these are presented as the constraints. Mixed integer programming (MIP) is used for optimization. Although it is novel idea to consider the penalty for deviation from schedule value, the paper doesn’t consider the BESS capacity loss which contains life and degradation of performance.

∑
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a

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(

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−

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)

2

)

b

1

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2

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b

⋯

{\displaystyle \sum (aP\_{im}+F\_{p}(P\_{im}-P\_{sch})^{2})+b\_{1}C\_{b}+b\_{2}E\_{b}\cdots }

3.2.1.(1)

a
:

{\displaystyle a:}

 unit cost of imported power

P

i
m

{\displaystyle P\_{im}}

: imported power

F
p

{\displaystyle F\_p}

 : penalty for the deviation from the scheduled power

P

s
c
h

{\displaystyle P\_{sch}}

: scheduled power

b

1

,

b

2

:

{\displaystyle b\_{1},b\_{2}:}

 unit cost of rating of BESS

C

b

,

E

b

{\displaystyle C\_{b},E\_{b}}

: power/energy rate of BESS

In , demand-side-BESS-owner is a customer who has BESS, PV, and loads. In a different way to ,,, it contains RES. Time of use electricity pricing is used and customer sells electricity when the price is high and imports electricity when the price is low. The objective is to minimize the cost associated with purchasing from(or selling back) the electricity grid and the BESS capacity loss while at the same time satisfying the load and reducing the peak electricity purchase from the grid. For this reason, the objective function depends on the chosen battery size. However, in , just lower and upper bound are determined.

In , the author who is the same author as  proposes optimal sizing of BESS. In similar way to , author finds the bound of BESS. But, Many scenarios are assumed and find the exact optimal size of BESS in each case.

### (2) Optimal sizing of BESS which is used for mitigating/maximizing output of RES

In , demand-side-BESS-onwer has also PV, wind and wave power. BESS is used for smoothing the output of demand-side-BESS-owner area. Because of RES, the power which imports/exports to main grid is fluctuated. For matching scheduled power, the size of BESS is determined. When authors determine the size of BESS, authors consider the dispatch level and the default time rate which actual power supplied to the grid does not match bid within a given tolerance. The default time rate(per year) should be below 5% in this paper. The optimal size of BESS is found Below given default time for maximizing the benefits of demand-side-BESS-owner. However, deciding default time rate can be a controversial problem,

In , demand-side-BESS-owner is a Distribution Network Operator(DNO). Authors approach different way from . The authors want to find optimal size of BESS for maximizing outputs of DGs. Because of fluctuation of RESs, there is reliability problem. For increasing reliability, DGs do not generate their full rated power although DGs can generate more power. For this reason, purpose is to maximize DGs output using BESS. Curtailment of DGs means full rated power minus actual generating power. Purpose is to minimize curtailment. The authors use OPF. At first, set the desired curtailment level. Then, multi period AC OPF is performed. The objective function is to minimize capital cost of BESS. Constraints are about 1) storage facilities, 2) controllable DG Plants constraint, 3) On-Load Tap Changers, and 4) general AC OPF constraints. Optimal solution could be optimal energy size/rated power of ESS. Using the optimal size of BESS, find the optimal actual curtailment of DGs. Then, compare actual curtailment with desired curtailment which is set at first. If the difference is bigger than criteria, iteration repeats. Otherwise, optimal solution can be gotten. Fig. 5. is the algorithm of . 3.2.2.(1),3.2.2.(2) are the objective function of first stage and second stage. It covers a lot of things which other papers don’t consider and the size of multiple BESSs can be determined.

m
i
n
∑

C

E

E

r
a
t
e
d

C

S

S

r
a
t
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d

⋯

{\displaystyle min\sum C\_{E}E^{rated}+C\_{S}S^{rated}\cdots }

 3.2.2.(1)

m
a
x
∑

p

n

⋯

{\displaystyle max\sum p\_{n}\cdots }

 3.2.2.(2)

C

E

{\displaystyle C\_{E}}

 : relative cost of energy capacity,

C

S

:

{\displaystyle C\_{S}:}

 relative cost of rated power,

E

r
a
t
e
d

{\displaystyle E^{rated}}

 : energy capacity,

S

r
a
t
e
d

{\displaystyle S^{rated}}

 : rated power,

p
n

{\displaystyle p\_n}

 : output power of DGs

## Conclusion

Methods of determining the optimal BESS size are summarized in this paper. The main roles of BESS is peak shaving, mitigating/maximizing the output of RESs, and frequency regulation. In this paper, BESS for peak shaving and mitigating/maximizing the output of RESs is only considered. There are two view types of BESS owners. The first one is the utility and the second one is a demand-side-BESS-owner. They have different purpose of sizing BESS. Utility wants to maximize social welfare, but demand-side-BESS-owner pursues their own profits. Therefore, according to the type of BESS owner, the method for finding optimal size of BESS is different. Also, although the type of BESS owner is same, method or way to find optimal size of BESS can be different according to what authors consider. Table. 1 shows papers which determine the optimal BESS size. The objective and contribution are also summarized.

## References

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2. ↑ C. Lo and M. D. Anderson, "Economic dispatch and optimal sizing of battery energy storage systems in utility load-leveling operations," IEEE Trans. Energy Convers., vol. 14, no. 3, pp. 824-829, 1999.
3. ↑ 3.0 3.1 3.2 3.3 3.4 M. Alhaider and L. Fan. "Mixed integer programming based battery sizing." Energy Systems Springer, 2014, pp. 787-805.
4. ↑ 4.0 4.1 W. Bingying, Z. Buhan, M. Biao, and Z. Jiajun, “Optimal capacity of flow battery and economic dispatch used in peak load shifting,” IEEE International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, pp.1395-1400, 2011.
5. ↑ 5.0 5.1 M. Nick, M. Hohmman, R. Cherkaoui, and M. Paolone, "Optimal location and sizing of distributed storage systems in active distribution networks," in Proc. IEEE PowerTech, 2013, pp. 1–6.
6. ↑ 6.0 6.1 6.2 S. Chakraborty, T. Senjyu, H. Toyama, A. Y. Saber, and T. Funabashi, "Determination methodology for optimising the energy storage size for power system." IET Generation, Transmission & Distribution, vol. 3, no. 11, pp. 987–999, 2009.
7. ↑ F. A. Chacra, P. Bastard, G. Fleury, and R. Clavreul, "Impact of energy storage costs on economical performance in a distribution substation." IEEE Trans. Power Syst., vol. 20, no. 2, pp. 684–691, May 2005
8. ↑ 8.0 8.1 Y. M. Atwa, E. F. El-Saadany, "Optimal allocation of ESS in distribution systems with a high penetration of wind energy,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1815–1823, Nov. 2010.
9. ↑ Q. Li, S. S. Choi, Y. Yuan, and D. L. Yao, “On the determination of battery energy storage capacity and short-term power dispatch of a wind farm,” IEEE Trans. Sustain. Energy, vol. 2, no. 2, pp. 148–158, Apr. 2011.
10. ↑ 10.0 10.1 10.2 10.3 10.4 T. Y. Lee and N. Chen "Determination of optimal contract capacities and optimal sizes of battery energy storage systems for time-of-use rates industrial customers," IEEE Trans. Energy Convers., vol. 10, no. 3, pp. 562–568, Sep. 1995.
11. ↑ 11.0 11.1 11.2 11.3 A. Oudalov, R. Cherkaoui and A. Beguin, "Sizing and Optimal Operation of Battery Energy Storage System for Peak Shaving Application,”IEEE Powertech Conf., pp. 621–625, July 2007.
12. ↑ 12.0 12.1 12.2 12.3 12.4 Y. Ru, J. Kleissl, S. Martinez, "Storage size determination for grid-connected photovoltaic systems," IEEE Trans. Sustain. Energy, 2013
13. ↑ 13.0 13.1 Y. Ru, "Exact sizing of battery capacity for photovoltaic systems", European Journal of Control, 2014
14. ↑ 14.0 14.1 L. Bridier, "A heuristic approach for optimal sizing of ESS coupled with intermittent renewable sources systems", Renewable Energy, 2016
15. ↑ 15.0 15.1 S. W. Alnaser, L. F. Ochoa, "Optimal sizing and control of energy storage in wind power-rich distribution networks", IEEE Trans. Power Syst., Vol. 31, No. 3, May 2016