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Long-term Scheduling of Battery Storage Systems inEnergy and Regulation Markets Considering Battery’s

lifespanMostafa Kazemi and Hamidreza Zareipour, Senior Member, IEEE

Abstract—This paper presents a new method for scheduling ofbattery storage systems for participation in frequency regulationand energy markets, simultaneously. Unknown automatic genera-tion control (AGC) signal of regulation market is modeled throughrobust optimization. In addition, the complex effect of participationin regulation market on battery’s lifespan is modeled through adynamic procedure. For this purpose, a long-term optimizationprocess is proposed in which, the short-term participation strategydefines battery’s lifespan. In order to prevent fast depreciation ofbattery due to frequent and deep charges/discharges, a new limitingmethod is introduced here, which would be useful for participationin regulation market. The proposed long-term model is linearizedby implementation of Benders’ decomposition. Optimum values forlimiting factors are determined in the master problem, while thedaily operation strategies are decided by sub-problems. Finally,the applicability of the proposed method is investigated using anillustrative case study.

Index Terms—Battery storage systems, Battery’s lifespan, Ben-ders’ decomposition, Regulation market, Robust optimization

I. NOMENCLATUREA. Set and indices

d Index of days during the annual long-term horizont, i Index of hours during the daily operationk, l Index of sub-intervals during an hour for regulation

marketm,n Index of iterationY, T,K Set of days, hours and sub-intervals, respectively

B. Parameters

CB Capital cost of the battery storage system [$]CF100 Maximum number of full cycle to failureE0 Initial energy of the battery storage system [MWh]Em Maximum capacity of the battery storage system

[MWh]En Minimum capacity of the battery storage system

[MWh]h Sub-interval’s duration [h]Pm Maximum power of the battery storage system [MW]α Level of robustness in modeling uncertainty of the

AGC signalλCd,t Regulation capacity price at day d and hour t [$/MWh]λDd,t Energy price at day d and hour t [$/MWh]

This work was partially supported by NSERC Energy Storage TechnologyNetwork (NEST).

M. Kazemi is with the Faculty of Electrical Engineering, University ofShahreza, Shahreza, Iran, e-mail: (m kazemi@shahreza.ac.ir)

H. Zareipour is with the Department of Electrical and Computer Engineering,University of Calgary, Alberta, Canada, e-mail: (h.zareipour@ucalgary.ca).

λMd,t,k Regulation movement price at day d, hour t and sub-interval k [$/MWh]

C. Variables

CE100d,t,k The full cycle equivalent of partial cycle Dd,t,k

CUd,t,k Linear approximation of CE100

d,t,k with respect to lim-iting factors

Dd,t,k The partial cycle of battery at day d, hour t and sub-interval k [MW]

LD Limiting factor of participation in energy market[MW]

LR Limiting factor of participation in regulation market[MW]

PDd,t Offered capacity to energy market at day d and hour

t [MW]PRd,t Offered capacity to regulation market at day d and

hour t [MW]PDepd,t,k Requested power in regulation market at day d, hour

t and sub-interval k [MW]Prd Profit of day d [$]Pran Annual long-term profit [$]αDd Linearized coefficient of energy limiting factor of

day dαRd Linearized coefficient of regulation limiting factor of

day dη

(.)d,t, µd,t Lagrangian coefficient of limiting constraints

II. INTRODUCTION

AS of March 2016, 50 MW of battery storage systems wereunder construction with the purpose of participating in

regulation market in the USA [1]. One of the main reasons forthis growth is that the dropping trend in the cost of batterystorage systems driven by the activities in other sectors (e.g.,electric car industry) [2]. For example, it is expected that thecapital cost of battery storage systems will drop by 25% by2017 [3].

The activity in grid-scale battery energy storage systems isalso attributed to the approval of the federal energy regulatorycommission (FERC) order 755 [4]. Based on this order, reg-ulation resources are paid not only for providing capacity forregulation services, but also for their performance in tracking theautomatic generation control (AGC) signal. The fast response ofbattery storage systems and their quality in tracking the AGCsignal makes them an attractive resource for providing regula-tion services [5]. Following FERC Order 755, a new pay-for-performance structure for the regulation market is implemented

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in PJM and New York markets. For example, the requestedmileage by AGC signal is used to calculate the performancepayment in the PJM regulation market [6]. In a similar way,the movement of regulation resource in tracing AGC signal isused for determining performance payment in the New Yorkregulation market. In addition, in both markets, a new score isevaluated for each regulation resource, based on their historicalperformance [7], [8]. This score is used in the clearing processof the regulation market. This score naturally makes offers offast response resources, such as, battery storage systems, morelikely to be accepted.

Various applications of energy storage systems can be cat-egorized into three main groups, namely, 1) improving systemreliability [9], 2) reducing the fluctuation of renewable resourcesby joint operation [10], and 3) participating in electricity marketas a merchant facility [11]–[16]. The scheduling problem ofenergy storage systems in the first two categories is mainlyaddressed from the system operator’s viewpoint; however, aprivate owner’s perspective is considered for a merchant facilityin the third group. This paper belongs to the last group inwhich, a standalone merchant energy storage system participatesin regulation markets. A pay-for-performance mechanism isconsidered for the market structure of this paper [17], [18].

One of the important issues related to scheduling of batterystorage systems is the lifespan of the battery. Unlike othertypes of energy storage systems, a battery’s lifespan is highlydependent upon its operation mechanism [19]. Frequency anddepth of charges/discharges are the main factors contributingto a battery’s degradation [14]. Thus, the strategy of a batterysystem for participating in regulation markets, i.e., a short-termoperation, needs to take into accounts the battery’s lifespan anddegradation, i.e., a long-term scheduling strategy. For example,a short-term strategy including sharp and frequent chargesand discharges may be profitable from a short-term operationperspective; however, it could significantly impact battery’s lifein the long run. In this paper, a long-term scheduling approachthat considers the life of the battery is proposed for short-termoperation of a battery system.

In order to create an approximated relation between short-termand long-term scheduling of battery storage systems two mainapproaches are presented in the literature, namely, objectivebased [9], [14], [20] and constrained based methods [21]. Inthe objective based methods, each cycle brings in a portion ofbattery’s capital cost to the objective considering its impact onreducing battery’s lifespan. In [20] and [9], the capital cost ofbattery storage system is assigned as a short-term operation costto each cycle. In this way, based on the number of cyclingper day and their depth of discharges, an approximation ofbattery’s lifespan is evaluated and the related cost is assignedto the operation strategy. In order to avoid the approximationof previous methods, constrained based methods are presentedand used in practical projects [21]. In this group, constraintsare inserted in the short-term scheduling problem to reflect thebattery’s lifespan. For example, in [21], only one full cycleis allowed in daily operation. By setting this constraint, theshort-term scheduling is limited to avoid fast depreciation ofbatteries. In this category, a long-term optimization processshould evaluate the optimum value of limiting factors that make

them complicated. The present paper is in line with the secondgroup in which, a long-term problem is solved to evaluate theoptimum limiting factors.

Considering a battery’s cycling limitations and impacts couldbe more complicated in the case of participation in regulationmarkets. Providing regulation service requires high frequencycycling and usually low levels of depth of discharge. Thus, adiscrete limitation constraint on the number of cycles per day[21] is not applicable. In addition, the approximations made in[9], [14], [20] may not be straightforward given the randomnature of the AGC signal. In this paper, a new approach basedon robust optimization is proposed to model the impact of AGCsignal on a battery’s lifespan. The advantage of the proposedapproach is that instead of limiting the number of cyclingoperation, limitations are imposed on the depth of discharge.

In this paper, a new method for scheduling the operation ofa battery storage system is proposed. The main focus of thispaper is participation in pay-for-performance regulation markets.Lifespan of the battery and the impacts of cycling the batteryon its life are modeled. To do so, the long-term scheduling ofthe battery storage system is tied to its short-term operation bydefining new limiting constraints in daily operation. In addition,in order to model the impact of the random and sharp variationsof the AGC signal on a battery’s lifespan, a new model basedon robust optimization for energy deployment through trackingthe AGC signal is proposed. Finally, Bender’s Decompositionis employed to solve the long-term scheduling problem, whichis a nonlinear and time consuming problem. By applying thismethod, this large-scale and complex problem is converted to alinear and simpler master and sub-problems that could be solvediteratively.

Compared to [9]–[13] and in line with [14]–[16], this paperis about participation of battery storage systems in regulationmarkets. Compared to [15], [16] and in line with [9], [14],[20], [21], the lifespan of the battery is modeled in this paper.Compared to [9], [14], [20], a new method for modeling theimpact of the fluctuations of the AGC signal on cycle life of thebattery is proposed here. Compared to [21], new limiting factorsare presented in this paper that tie the short-term and long-termissues. Compared to [9], [14], a linear formulation is devisedthat reduces computational burden.

The rest of this paper is organized as follows. The short-term scheduling problem without modeling battery’s lifespan isintroduced in Section III. The method of calculating battery’slifespan is presented in Section IV and the long-term schedulingmodel is proposed in Section V. The illustrative example isdescribed and analyzed by Section VI and finally, the conclusionis presented by Section VII.

III. DAILY OPERATION SCHEDULING

Scheduling a battery storage system for short-term operationin a regulation market considering the longer term impacts ofcycling the battery on its lifespan is the main focus of thispaper. A performance-based day-ahead market structure similarto that of the New York electricity market is considered forthe regulation market. In addition to the regulation market,it is assumed that the battery storage system participates inthe day-ahead energy market in order to buy energy and keep

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its state of charge at an optimum level. It may also considerenergy arbitrage if proved profitable. Therefore, the schedul-ing problem of this paper is about determining the optimumshort-term offering strategy in day-ahead energy and regulationmarkets, when long term lifespan issues are taken into account.Thus, such scheduling model needs to have three components.The first component is a daily profit maximization formulationthat models the short-term operation scheduling of the batterysystem; this component is discussed in this section. The secondcomponent is to determine how daily operation of a batterysystem reduces its lifespan; this is described in Section IV.Finally, a cost-benefit model needs to be built to ensure theprofits gained from daily operation outweigh the costs incurredfrom running the battery and losing its useful capacity overtime. This requires a longer term operation modeling, whichis discussed in Section V.

A. Deterministic model

The optimal operation scheduling of a battery system formaximizing its profit in day-ahead energy and regulation marketsis as follows [14], [15]:

maxΩ

Prd =∑t∈T

λDd,tP

Dd,t + λCd,tP

Rd,t+∑

k∈K λMd,t,k|PDepd,t,k|

(1a)

Subject to:

− Pm ≤ PDd,t ≤ Pm ∀t ∈ T (1b)

0 ≤ PRd,t ≤ Pm ∀t ∈ T (1c)

− Pm ≤ PDepd,t,k + PD

d,t ≤ Pm ∀t ∈ T, ∀k ∈ K (1d)

En ≤ E0 +

t∑i=1

(−PDd,i +

l∑k=1

−PDepd,i,kh) ≤ Em

∀t ∈ T, ∀l ∈ K (1e)

− PRd,t ≤ P

Depd,t,k ≤ P

Rd,t ∀t ∈ T, ∀k ∈ K (1f)

The scheduling model for dth day of the year for a batterystorage system is presented in (1). In the objective of (1a),the profit is maximized by determining the set of short-termvariables, i.e., Ω = PD

d,t, PRd,t. Participation in day-ahead

energy market is modeled through the first term of the objective,i.e., λDd,tP

Dd,t. As it is stated by (1b), battery storage system

in energy market can be either a seller or a buyer. Hence, itis possible for the battery storage system to gain profit fromarbitrage in day-ahead energy market as well.

The remaining terms of the objective model the profit ofparticipation in the regulation market. In this formulation, itshould be noted that, the regulation market is dispatched basedon sub-hourly intervals. Thus, the parameters and variablesrelated to this market have three time indexes, d as the indexof day, t as hourly intervals and k as sub-hourly intervals. Forexample, the regulation market of New York is cleared eachfive minutes, which means that each hour includes 12 sub-intervals in the regulation market. It can be seen in (1a) that,the regulation market is modeled through two terms. The firstterm, i.e., λCd,tP

Rd,t, models the capacity payment for preparing

capacity of PRd,t in the regulation system. The second term, i.e.,

λMd,t,k|PDepd,t,k|, is the movement payment related to tracing of the

AGC signal. It should be noted that the response of a battery

storage system to the AGC request, i.e., PDepd,t,k, can be either

up or down in the range defined by the offered capacity (1f);however, the related payment is independent of the directionof response. Therefore, the absolute value of PDep

d,t,k is used incalculation of movement payment.

The capacity limits of the battery storage system are presentedby (1b)-(1d). It is stated in these equations that, not only theoffered capacity to the energy and regulation market should bein the feasible capacity range of battery storage system, butalso, their aggregated response to their commitment in day-aheadenergy and regulation market should be in this feasible region(1d). Finally, the energy constraint of battery storage system isstated through (1e).

The optimization problem of (1) would obtain the optimumcapacity for participation in regulation and energy markets,provided that, the deployment capacity PDep

d,t,k be known as aparameter. However, this parameter is completely proportionalto the AGC signal and prediction of AGC signal is not straight-forward. This problem is handled in the next subsection by usingrobust optimization.

B. Robust-based model

The main challenge of (1) is about modeling the deployedenergy from regulation market, i.e., PDep

d,t,k, due to tracking theAGC signal. This signal is provided as the input control forregulation service. The AGC signal is highly correlated to therandom errors of demand forecasting. Therefore, PDep

d,t,k is notpredictable, neither is the AGC signal. In this work, we assumethe average of the regulation up and down deployments over adaily scheduling window is zero. While this assumption maynot always be true in reality, we will demonstrate later thatsince robust optimization operates based on worst case scenario,this assumption does not jeopardize the modeling approach. Theother assumption here is that historical AGC data is availableand could be used to drive the more likely range of regulationdeployment over the scheduling period. For example, one maydecide that the most likely range of deployment is three standarddeviation around the average. The deployment signal’s expectedstandard deviation can be easily determined using historical data.Thus, based on those assumptions, interval based optimizationmethods such as robust optimization, are good candidates formodeling the uncertainty of PDep

d,t,k.In robust optimization, the decision maker should only define

the variation ranges of uncertain parameter, known as robustnessgap [22]. This method tries to evaluate the worst case of uncer-tainty’s occurrence inside of the robustness gap; and the decisionis made based on this worst case analysis [23]. Therefore,robust optimization has no assumption on the probability densityfunction of the uncertainty. In the case of modeling uncertaintyof energy deployment from the regulation market in (1), we needto estimate the fluctuation range of PDep

d,t,k around zero; this canbe easily achieved using historical AGC data. Thus, the dailyoptimization scheduling problem of (1) can be rewritten as (2),by including a robust-based model of the regulation market. Theuncertainties associated with the market prices, i.e., λDd,t, λ

Cd,t,

are not considered in this work; because, modeling the battery’slifespan is the main focus of this paper. However, their effectcan be easily applied using the same approach that we used for

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modeling uncertain regulation deployment.

maxΩ

minPDep

d,t,k

Prd =∑t∈T

λDd,tP

Dd,t + λCd,tP

Rd,t+∑

k∈K λMd,t,k|PDepd,t,k|

(2a)

Subject to:

− PRd,tα ≤ P

Depd,t,k ≤ P

Rd,tα ∀t ∈ T, ∀k ∈ K (2b)

(1b)-(1e)

As can be seen in (2), the robust-based model is a max-minoptimization problem in which, battery storage system’s profitis maximized with respect to the decision variable set Ω andminimized with respect to PDep

d,t,k. The minimization part is withthe purpose of evaluating the worst case of PDep

d,t,k within theconfidence interval of [−PR

d,tα, PRd,tα] (2b). In other words, in

(2) it is assumed that, fluctuations of AGC signal would be inthe range of [−α, α] in which, α is the robustness parameterdetermined by analyzing historical data of AGC signal.

In order to solve the max-min problem of (2), the methodof [24] is used here. It is explained in [24], [25] that, uncer-tain variable affects the optimization problem not only in theobjective, but also in the constraints. For example in our case,PDepd,t,k appears in the objective (2a) and constraints (1d)-(1e).

On the other hand, the worst case of PDepd,t,k is not the same

for these equations. For example, the worst case of PDepd,t,k from

the perspective of the upper side of (1d) is its upper bound,whereas the worst case of PDep

d,t,k from the perspective of thelower side of (1d) is the lower bound of (2b). We use themethodology introduced in [24], [25] to solve this problem. Themethodology simply evaluates a worst case for each equation,instead of determining one worst case for all of the equations.Thus, the problem is solved for the worst case among the worsecase scenarios of PDep

d,t,k. The worst case of PDepd,t,k with respect

to the objective (2a) is zero, i.e., PDepd,t,k = 0; with respect to

the upper side of (1d) and (1e) is the upper bound of (2b), i.e.,PDepd,t,k = αPR

d,t; and finally, with respect to the lower side of(1d) and (1e) is the lower bound of (2b), i.e., PDep

d,t,k = −αPRd,t.

Based on these explanations, formulation (2) can be rewrittenas follows.

maxΩ

Prd =∑t∈T

(λDd,tP

Dd,t + λCd,tP

Rd,t

)(3a)

Subject to:

E0 +

t∑i=1

(−PDd,i +

l∑k=1

αPRd,ih) ≤ Em ∀t ∈ T, ∀l ∈ K (3b)

E0 +t∑

i=1

(−PDd,i −

l∑k=1

αPRd,ih) ≥ En ∀t ∈ T, ∀l ∈ K (3c)

− PDd,t + αPR

d,t ≤ Pm ∀t ∈ T (3d)

− PDd,t − αPR

d,t ≥ −Pm ∀t ∈ T (3e)

(1b)-(1c)

By solving (3), which is a linear optimization problem, theshort-term schedule of battery storage system for participationin regulation and energy market can be evaluated. It should benoted that the risk level associated with the uncertain AGC signal

Fig. 1. The effect of macro and combined cycling on the remaining capacityof Saft’s Lithium-ion batteries implemented in photovoltaic systems [26].

can be controlled by holding a fraction of the committed capac-ity, i.e., αPR

d,t. In this way, more capacity would be available tobe used in the optimization process. On the other hand, if thebattery storage system could not provide the requested servicein the real-time, it will be responsible to provide the shortagequantity from the real-time regulation market. Therefore, bysetting a proper value for α, the real-time shortage cost andthe day-ahead profit would be balanced. In order to model thereal-time shortage cost, a two-stage optimization approach isrequired in which, the here-and-now stage is the model of day-ahead market and the wait-and-see stage is the model of real-time market. This paper only deals with the day-ahead marketand modeling the real-time consequences of decisions made inthe day-ahead market is beyond the scope of this paper. For thesake of simplicity, the uncertainty associated with the marketprices are not considered.

IV. IMPACT OF BATTERY OPERATION ON ITS USEFULCAPACITY

Battery capacity degradation has two components [19]. Firstis natural calendar fading, i.e., losing capacity over time evenif the battery is not used at all. This component is namedcalendar life or float life. The second component depends onhow the battery is operated and cycled over time. Typically,manufacturers provide a “useful capacity” curve that relatescycling numbers and remaining capacity of a battery (e.g., seeFig. 1, curtesy of Saft [26]). This component, which is namedcycle life, is the focus of this paper because it could be managedby an optimal operation plan. Life of batteries is determined asthe minimum of their calendar life and cycle life. Thus, thereis a trade-off between maximizing the short-term optional profitof battery storage systems, versus losing its useful life. In otherwords, while frequent cycling of a battery brings in revenue,it entails battery replacement costs in the longer term. In orderto maximize the long-term profit of running a battery storagesystem, short-term operation limits need to be set in a way thatminimizes the longer term replacement costs of the system.

There are two important factors affecting battery’s lifespan,number of cycling per day and depth of discharge (DOD) ineach cycle. An approximation of Rainflow Counting Algorithm[27] is used here for calculating battery’s life cycle, which is inline with [9], [14]. This method is presented mathematically by

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Fig. 2. Definition of partial half-cycles in a manipulated curve of state ofcharge.

(4).

Life =CF100∑

d∈Y∑

t∈T∑

k∈K CE100d,t,k

(4a)

CE100d,t,k =

1

2(Dd,t,k

Em)kp ∀d ∈ Y, ∀t ∈ T, ∀k ∈ K (4b)

In (4a), CF100 is the maximum number of full charge-discharge, i.e., at DOD of 100%, resulting in failure of thebattery. However, battery storage system operates at partialcharge-discharge, specifically in regulation market. Thus, CE100

d,t,k

is used here, to convert the partial cycling to its equivalentcycling at DOD of 100%. By counting CE100

d,t,k during one yearof operation of battery storage system, the life of battery, inyears, could be calculated by (4a). The nonlinear equation of(4b) is used to calculate the equivalent of partial half-cycle, i.e.,Dd,t,k, in term of cycling at DOD of 100%. In (4b), kp is aconstant in the range of 0.8 to 2.1 [9]. For example, in Fig. 2,the state of charge of a 30 MW battery storage system is plottedin a typical daily operation. Assuming this pattern for one yearoperation of the battery storage system, the life of battery basedon (4) would be as follows.

Life =CF100

3652 (

D1

30 )kp + (D2

30 )kp + (D3

30 )kp + (D4

30 )kp + (D5

30 )kp

V. LONGER TERM SCHEDULING CONSIDERINGLIFE-OPERATION DEPENDENCIES

Short-term scheduling of battery storage systems was pre-sented by (3). It was shown that the useful life of a batteryis affected by how it is operated. In this section, a longer termscheduling model of a battery storage system is proposed withthe aim of maximizing its overall long-term operation profits.This is achieved by finding an optimal operation plan wherethe short term profits and long-term useful life degradation arebalanced. The proposed model is formulated as follows:

maxΩ,LD,LR

Pran = − CB

Life(E)+∑d∈Y

Prd (5a)

− LD ≤ PDd,t ≤ LD : ηUd,t, η

Dd,t ∀d ∈ Y,∀t ∈ T (5b)

0 ≤ PRd,t ≤ LR : µd,t ∀d ∈ Y,∀t ∈ T (5c)

In the above formulation, the annual profit is maximizedassuming the battery is replaced at the end of its life. Thus,the annual investment cost of battery’s replacement can beobtained as the first term of the objective (5a), i.e., − CB

Life(E) .

The life of battery is calculated using (4). The next term ofthe objective is the summation of daily profits over a year,obtained from (3). In order to tie the long-term and the short-term scheduling problems, constraints (5b) and (5c) are enforced.We have introduced two limiting factors, i.e., LD and LR, indaily scheduling to avoid sharp charges-discharges that woulddegrade the useful life of the battery. The offered capacity to theenergy market is limited with LD through (5b), and the offeredcapacity to the regulation market is limited with LR through(5c). Therefore, the most important outputs of the longer termscheduling model are the limiting factors LD, LR, which shouldbe used in short-term scheduling of the battery storage system,with the purpose of increasing the battery’s useful lifespan. Theregulation market is designed to respond to the unpredictableportion of demand fluctuations, whereas the energy market isto satisfy the predictable demand. Thus, the amount of energytransacted in the energy market is much more than the onein regulation market. Because of this fact, the limiting factorsof these markets are considered, separately. The Lagrangiancoefficients of (5b) and (5c), i.e., ηUd,t, η

Dd,t, µd,t, are presented

following by a colon.There are two challenges in solving the optimization model

(5). The first challenge comes from the fact that (5) is a nonlinearmodel given (4). Also, the problem is solved over a year. Thismakes a large-scale non-linear optimization. The second problemis calculating Life(E) of the battery. It is described by (4) that,the battery’s life is a function of its energy fluctuation. Hence,the exact energy level of battery is required for calculatingits lifespan. However, energy level is impacted by regulationmarket deployments, which is uncertain by nature, and only themaximum and minimum energy levels of the battery is evaluatedby (3b) and (3c).

Both of these challenges can be addressed using a decomposi-tion algorithm. Benders’ decomposition is used here, to solve thelong-term scheduling problem [28]. By implementing Benders’decomposition, we can evaluate the complicating variables LD

and LR in the master problem, and then, the daily scheduleoptimization problem can be solved separately for each day. Theformulation of the master problem is provided by (6).

maxLD(m),LR(m)

Zup(m) (6a)

Zup(m) ≤ − CB

Life(n)+ Zlow(n)

+ (LD(m) − LD(n))

(∑d∈Y

∑t∈T

(ηU(n)d,t + η

D(n)d,t )

)+

(LR(m) − LR(n))

(∑d∈Y

∑t∈T

µ(n)d,t

)∀n ≤ m− 1 (6b)

Life(n) =CF100∑

d∈Y∑

t∈T∑

k∈K CU(n)d,t,k

(6c)

CU(n)d,t,k = (LD(m) − LD(n))α

D(n)d + (LR(m) − LR(n))α

R(n)d

+ CE100(n)d,t,k ∀d ∈ Y,∀t ∈ T, ∀k ∈ K (6d)

LDmin ≤ LD(m) (6e)

LRmin ≤ LR(m) (6f)

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Fig. 3. Flowchart of implemented Benders’ decomposition.

In the above formulation, the master problem is solved inthe mth iteration with the aim of evaluating limiting factorsLD(m) and LR(m). Benders’ cut is stated by (6b). It should benoted that, all of the parameters with superscript n are definedin previous iterations. Zlow(n) is the short-term revenue of nth

iteration obtained by solving sub-problem (3) constrained to thelimiting constraints (5b) and (5c).

The first term of equation (6b), i.e., − CB

Life(n) , states theannual replacement cost of battery with respect to its lifespan. Itshould be noted that, the battery’s useful lifespan is a functionof limiting factors LD(m) and LR(m) decided in the masterproblem. For example, when the limiting factors are increasedin the master problem, the rate of charge/discharge of batterystorage system defined by the daily operation sub-problem of(3) would also be increased; this decreases the battery’s lifespan.Therefore, we need to rewrite (4a) as a function of the limitingfactors when working with iteration-based approaches such asBenders’ decomposition. The updated format of (4a) is stated in(6c) and (6d) to define the relation of Life(n) to the limitingfactors LD(m) and LR(m). From the previous iterations, weknow the values of CE100(n)

d,t,k . These values are used as the basepoints of linear approximation of (6d). Observe from (6d) thatthe updated equivalent full cycling, i.e., CU

d,t,k, is defined in away to be linearly related to the two limiting factors LD(n) andLR(n). Hence, a slight change in LD(n) and LR(n) would alsochange CU

d,t,k linearly. This fact is explained mathematically,by (6d) using the linearized factor of αD(n) and αR(n). Thelinearized factors are obtained from a sensitivity analysis on thelimiting factors (7). It should be noted that, in the first iterationsthe linear approximation would not be a good assumption;however, in the final iterations where LD(m) and LR(m) changesare small, the limiting factors and CU

d,t,k are related linearly. Theiterative procedure of solving the Benders’ model is describedby the flowchart of Fig. 3.

αDd =

∑t∈T

∑k∈K ∂CE100

d,t,k

∂LD∀d ∈ Y (7a)

αRd =

∑t∈T

∑k∈K ∂CE100

d,t,k

∂LR∀d ∈ Y (7b)

From this figure, the optimum value of the two limitingfactors are determined by master problem (6); then LD and LR

are sent to the sub-problems. Sub-problems are the short-termscheduling problems (3) constrained to (5b)-(5c). By solving thesub-problems, the optimum daily schedule and the Lagrangiancoefficients of (5b)-(5c) could be evaluated. The Lagrangiancoefficients are then sent back to the master problem to makenew Benders’ cut for the next iteration. Then, the optimum dailyschedule is used to determine the life of the battery (4).

In order to evaluate the impact of daily operation scheduleson battery life, the DOD associated with each cycle shouldbe calculated first. The DOD of each cycle can be calculatedby the energy transacted in the energy and regulation markets.This procedure is handled by the third box of the flowchart ofFig. 3. As can be seen in this flowchart, the third box receivesthe daily operation schedule, i.e., PD

d,t and PRd,t, from the sub-

problem. The energy transacted in the energy market can beevaluate by PD

d,t, which is a known value. However, the energydeployment in the regulation market can be upside or downsidedepending to the AGC signal. In this step, the worst case ofenergy deployment in regulation market resulting in maximumdeviation of battery’s energy level is calculated. This meansthat the minimum possible lifespan of battery associated withthe given daily schedule by the sub-problems is calculated. Forthis purpose, the regulation energy deployment is defined bythe energy transacted in the energy market in that hour. If thebattery storage system sells energy in hour t, i.e., PD

d,t ≥ 0,the regulation deployment is assumed to be upside in fullcapacity, i.e., PDep

d,t,k = PRd,t to result in a higher level of

DOD. If the battery storage system buys energy in hour t, i.e.,PDd,t ≤ 0, the regulation deployment is assumed to be downside

in full capacity, i.e., PDepd,t,k = −PR

d,t for the same reason. Ifthe battery storage system has no offer in energy market inhour t, the direction of energy deployment in the regulationmarket is evaluated by its previous energy transaction. If itsprevious energy trend was increasing, the regulation deploymentis assumed to be downside in full capacity, i.e., PDep

d,t,k = −PRd,t ,

to keep the trend and increase level of DOD in that cycle. If itsprevious energy trend was decreasing, the regulation deploymentis assumed to be upside in full capacity , i.e., PDep

d,t,k = PRd,t ,

for the same reason. In this way, the corresponding DOD ofeach cycle, i.e., Dd,t,k can be evaluated by the third box of theflowchart of Fig. 3. These DODs are used to calculate CE100

d,t,k

based on (4b), and this generated feedback is sent back as CE100d,t,k

to the master problem for the next iteration. This repetitiveBenders’ process would stop when the change in the objectiveof master problem becomes negligible.

Using Benders’ decomposition, the long-term scheduling ofbattery storage systems, which is a large-scale nonlinear prob-lem, is decomposed into many small sub-problems. In otherwords, by the use of the Benders’ decomposition the dailyoperation schedule problems can be solved separately, which sig-nificantly reduces the size of optimization problem. Furthermore,by implementing the linearization method of (6d), the non-linearlife model of the battery is replaced with a linear model. In thisway, instead of using non-linear solvers we can use linear solversand determine the optimal solutions. Also, the effect of uncertain

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Fig. 4. The short-term operation schedule for a typical weekday in July 2015for three scenarios, a) LD = 30, LR = 30 MW b) LD = 1, LR = 1 MW c)LD = 6.9, LR = 26.5 MW

energy deployment from the regulation market on the lifespanof battery is determined through the repetitive procedure ofBenders’ decomposition. After each iteration, the worst case ofregulation energy deployment from the perspective of battery’suseful life is calculated and used in the next iteration. Thismodel can be easily implemented by linking MATLAB R© andGAMS [29], using CPLEX [30] solver. The combination ofBenders’ method and the linearization reduce the computationalcomplexity of the model significantly.

VI. NUMERICAL RESULTS

A. Data

The proposed model is used in this section for operationscheduling of a Li-ion battery storage system with the capacityof 30MWh when the long-term lifespan impacts are considered.It should be noted that this is a hypothetical case in which, wedid our best to use rational data from the literature and batterymanufacturers (e.g., Saft). We have limited the data to the onesthat are actually needed in our modeling and calculations, andhave left unnecessary data out. The values of Pm, En and E0

are assumed to be 30MW, 1MWh and 15MWh, respectively.The capital cost of the battery is set to 30 m$, i.e., 1000$/kWh[31], [32]. In addition, the calendar life of the battery is assumedto be 20 years, while its cycle life is determined by setting themaximum number of full cycle to 10, 000, i.e., CF100 = 10, 000[14], [26]. The assumed input data provided here may not beapplicable for all types of batteries; however, their values wouldnot affect the justifications provided in this section and the modelcan be easily updated by input data for any type of battery. Therequired price data i.e., day-ahead energy, regulation capacityand regulation movement prices, are employed based on actualvalues observed in the New York electricity market for year2015. For the sake of simplicity and in line with [33], [34],each month is modeled with two days, weekday and weekend.In this way, one year time horizon of this model is reducedto 24 days. In addition, the robustness parameter of regulationenergy deployment is set to 30%, i.e., α = 0.3. The number ofvariables and equations of the proposed model without Benders’decomposition algorithm are 22, 572 and 3, 262, respectively. Byapplying Benders’ decomposition, the model is decomposed into24 separate sub-problems with the size of 985 equations and 145variables. The presented model takes about 26 minutes to solveon a standard desktop computer.

B. Verification of the proposed method

In order to demonstrate how the proposed limiting factorschange the short-term operation of a battery storage system,three different scenarios are compared. In the first scenario, thebattery is scheduled without any limiting factors, i.e., LD = 30MW and LR = 30 MW, which means that no lifespan limitationare considered in short-term scheduling. In the second scenario,the battery is over limited for offering in energy and regulationmarkets by setting LD = 1 and LR = 1 MW, i.e., the batterystorage system operator should not offer more than 1 MW inthose markets, in each hour. For the third scenario, the battery’sshort-term operation is limited based on optimal values of theproposed limiting factors obtained from the proposed model.These optimum limiting factors are determined to be LD = 6.9MW and LR = 26.5 MW. It means that, in the third scenario theoffered capacities in energy and regulation markets are limitedby 6.9 MW and 26.5 MW, respectively.

The optimum scheduling plans, i.e., the outputs of sub-problem (3), are shown in Fig. 4 for a typical weekday in July2015, for the presented scenarios. In the first scenario, i.e., Fig.4.a, the battery storage system participates in energy and regu-lation markets by its whole capacity, resulting in deep chargesand discharges. In other words, the optimum offering strategyfrom short-term perspective would be as Fig. 4.a; however,this strategy may reduce battery’s lifespan, significantly. In thesecond scenario, the offered capacities are over limited to savebattery’s lifespan. As can be seen in Fig. 4.b, by setting limitingfactors to their minimum levels, the participation strategy iscompletely changed. In this scenario, the battery storage systemis frequently charged/discharged with small capacity, instead ofdeep charges/discharges in the first scenario. This scenario maynot be optimum from the perspective of short-term revenue max-imization; however, it would increase battery’s lifespan. Finally,in the last scenario, the battery storage system operates betweenthe two mentioned extreme scenarios. The charging/dischargingstrategy in this scenario is not as deep as in the first scenario;neither is over limited as in the second scenario.

In order to investigate the effect of short-term operation ofbattery storage systems on its lifespan and consequently, its long-term profit, the abovementioned scenarios are compared fromthree different perspectives; those are: 1) the annual revenuegained from participating in energy and regulation markets, i.e.,the annual short-term profit; 2) Annual replacement cost ofbatteries which reflect the battery’s lifespan; 3) The long-termprofit calculated as the difference of the first two factors. Thesethree factors are plotted for each scenario in Fig. 5. Comparingthese scenarios from the viewpoint of the first factor, it canbe seen that, the short-term revenue of the first scenario is

TABLE IEFFECT OF LIMITING FACTORS ON BATTERY’S CYCLE LIFE

Scenario Limiting Factor of Limiting Factor of Cycle lifenumber energy market regulation market of battery

1 LD =30 MW LR =30 MW 10.4 Years2 LD =1 MW LR =1 MW 97.3 Years3 LD =6.9 MW LR =26.5 MW 20 Years

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Fig. 5. Comparing long-term profit and its components for different scenariosof limiting factors.

Fig. 6. Comparing components of long-term profit for different approaches.

greater than the other scenarios, due to no limiting in short-term operation. In the first scenario, the battery storage systemparticipates in energy and regulation market with whole capacityresulting in higher level of short-term profit. However, thereplacement cost of the first scenario is much greater than others.It means that, deep charges/discharges in the scheduling strategyrelated to the first scenario reduce battery’s lifespan significantly.The cycle life of the battery corresponding to each scenario isalso presented in Table I. Observe that the battery is deterioratedin the first scenario due to short-term operation strategy, and itslifespan decreased from 20 years to 10.4 years. In the secondscenario, the battery is over limited and its operation plan isconservative enough to increase its cycle life to 97.3 years,despite its service life of 20 years. In the third scenario, thelimiting factors are set in a way that the cycle life of batterybecomes more close to its service life. Thus, the replacementcost of the second and their scenarios are equal in Fig. 5. Finally,observe that the third scenario provides more profit in the long-term. In other words, in the optimal limiting factors balance theshort-term profit and the long-term replacement costs.

TABLE IIEFFECT OF DIFFERENT LIMITING APPROACHES ON BATTERY’S CYCLE LIFE

Scenario Limiting cycle lifetype Factor of battery

Limited cycle 2 Cycles/day 87 YearsLimited cycle 5 Cycles/day 10.1 YearsLimited cycle 6 Cycles/day 8.9 Years

Proposed method LD = 6.8, LR = 23.3 MW 20.8 Years

Fig. 7. Analyzing the effect of battery cost on the cycle life of the battery

C. Comparison with cycle limit method

In this section, we compare the proposed method with theone that puts a strict daily cycle limit on battery operation.This method is used in [21]. In this comparison, we assumed arobustness gap of 10%, i.e., α = 0.1 for the AGC signal; thisencourage more cycles so that we can better compare the out-comes. The resulting optimal limiting factors were determined tobe LD = 6.8, LR = 23.3 MW. We also consider three scenariosfor strictly limiting daily cycles, i.e., 2, 5 and 6 Cycles/day.Similar to the results presented in Fig. 5, the short-term profit,battery’s replacement cost and long-term profit associated witheach scenario are plotted in Fig. 6. In addition, the cycle lifevalues are presented in Table II. For 2 Cycles/day, the batterystorage system is over limited, which results in lower level ofshort-term profit comparing to the other scenarios. This alsoleads to a high cycle life for the battery at 87 years, whichis much greater than the 20 years service life of the battery.For the daily limits of 5 Cycles/day and 6 Cycles/day, observefrom Fig. 6 that the short-term profits are increased; however,the replacement costs of the battery are also higher in thesecases. Also, the replacement duration of the battery are droppedto to 10.1 and 8.9 years, respectively. However, consideringshort-term profits and long-term replacement costs, the proposedmethod results in higher overall long-term profits.

D. Discussion on the cost of battery storage systems

In subsection VI-B, it was shown that the proposed model setslimiting factors in a way that the cycle life of the battery becomesmore close to its calendar life. To elaborate, it should be notedthat the objective (6b) has two terms, named, the replacementcost and the short-term operation profit. In this case study, thebattery replacement cost is the dominant term of (6b). Therefore,the daily operation strategy in subsection VI-B was limited withthe purpose of minimizing the battery replacement cost. In otherwords, the revenue gained in short-term operation is sacrificedto reduce long-term battery replacement cost. This proceduremaybe reversed if the capital cost of batteries is reduced.

In order to analyze the effect of battery cost on the cycle lifeof the battery, Fig. 7 is presented. In this figure the cycle life ofbattery is plotted against the battery cost. This figure is made upof three regions. The first region is defined as the battery costgreater than 100$/kWh. In this region, the optimum values forlimiting factors are LD = 6.9MW and LR = 26.5MW thatresult in the battery cycle life of 20 years. In this region, thebattery cost is still the dominant term in the objective (6b) and

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the limiting factors are set in a way to minimize the replacementcost. The third region is defined as the battery cost lower than27$/kWh. In this region the short-term profit is the dominantterm in the objective. Thus, the battery is operated without limitsin this region, i.e., LD = 30MW and LR = 30MW . Thebattery cost is low enough that we should not be concernedabout its deterioration any more. In the second region defined asthe battery cost between 27$/kWh and 100$/kWh, none of theobjective’s terms are dominant. In this region the optimum cyclelife is decreased by the decrease of the battery cost. Today’sbattery technologies are placed in the first region. Hence, theirshort-term operation should be defined in a way that their cyclelife becomes equal to their calendar life.

VII. CONCLUSION

In this paper, an operation scheduling model for a batterysystem that participates in frequency regulation market is pro-posed. The model optimizes the battery’s schedule to maximizeits profit over a long period. This is done by including theimpact of cycling on battery’s life in the model. We use robustoptimization to handle the uncertainty of the regulation market.The resulting model is a large-scale non-linear model, whichis computationally challenging. We use Benders’ decompositionand a linearized version of the model to ease computationalcomplexity. The model provides two limiting factors, whichessentially define the optimal capacity of the battery to beoffered in the energy and regulation markets. Those factors area fraction of the available capacity. By doing so, the batteryoperator participates in each market such that it can deliver itsmarket commitments, optimize its operation profits and yet usethe battery for a long time.

The numerical results demonstrated that the proposed methodwould result in overall operation profits. We also comparedthe proposed model with a case where a strict daily cyclinglimit was imposed on the battery. The results demonstrated thatthe proposed model outperforms such strict limitation whenparticipation in the regulation market is of concern.

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