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Considering Thermodynamic Characteristics of aCAES Facility in Self-scheduling in Energy and

Reserve MarketsSoroush Shafiee, Student Member, IEEE, Hamidreza Zareipour, Senior Member, IEEE, Andrew M. Knight, Senior

Member, IEEE,

Abstract—The efficiency of a compressed air energy storage(CAES) facility deviates significantly from its nominal valuedepending on its thermodynamics and operational conditions.Thus, the thermodynamic characteristics of the facility should beincorporated in its scheduling to model the variation in efficiency.This paper proposes a self-scheduling approach for a CAESfacility that participates in energy, spinning and non-spinningreserves markets. Considering the thermodynamic characteristicsof the facility, the limitations imposed on the facility are modeledwhen devising operations schedules. Thus, the model leads to amore realistic view of the revenues. Numerical simulations areprovided using the historical hourly energy and reserve pricesof the ERCOT electricity market for years 2011 to 2015. Theresults are compared with those of derived from by using theconventional model with constant efficiency parameters.

Index Terms—Compressed air energy storage, self-scheduling,thermodynamic characteristics, energy arbitrage, spinning andnon-spinning reserve, linearization.

NOMENCLATUREindicess Steps of the curve representing the compres-

sion air flow rate versus cavern state of charge(CAFRC) from 1 to nc.

s′ Steps of the curve representing the turbine airflow rate versus discharging rate (TARFC) from1 to nd.

s′′ Steps of the heat rate curve (HRC) from 1 to nh.t Operation intervals running from 1 to T .k Scenario index from 1 to K.

ParametersπEt,k Day-ahead energy price for hour t in scenario k.πsrt,k Day-ahead spinning reserve price for hour t in

scenario k.πnrt,k Day-ahead non-spinning reserve price for hour t

in scenario k.πNG Natural gas price.γk Probability of scenario k.ARF c

s Charge air flow rate corresponding to step num-ber s of the CARFC

ARF ds′ Discharge air flow rate corresponding to step

number s of the TARFCbc,maxs size of step s of the CARFC.bd,maxs′ size of step s′ of the TARFC.bh,maxs′′ size of step s′′ of the HARFC.CAmax Total mass of cushion air in cavern in kg

S. Shafiee, H. Zareipour, and AM. Knight are with the Department of Elec-trical and Computer Engineering, Schulich School of Engineering, Universityof Calgary, Calgary, AB, Canada T2N 1N4 (e-mail: sshafiee@ucalgary.ca;h.zareipour@ucalgary.ca; aknigh@ucalgary.ca).

depsr/nrt,k Status of spinning/non-spinning reserve deploy-

ment at time t in scenario k (1 is deployed and0 is not deployed).

Emax maximum stored energy capacity of air storagecavern in MWh.

ER Nominal energy ratio of CAES facility.HRh

s′′ Heat rate corresponding to step number s of theHRC

HRnom Heat rate of the CAES facility at 100% discharg-ing rate.

P expmax maximum generation capacity of the expander.P cmax maximum compression capacity of the compres-

sor.P expmin minimum generation capacity of expander.P cmin minimum compression capacity of compressor.QSC Quick start capacity of the CAES facility.qd,mins′ Summation of power blocks from step 1 to step

s′ − 1 of TARFC.qh,mins′′ Summation of power blocks from step 1 to step

s′′ − 1 of HRC.SOCmin minimum state of charge (SOC) of air storage

cavern.SOCmax maximum SOC of air storage cavern.SOCinit Initial SOC of air storage cavern.SOCfinalSOC of air storage cavern at the end of the day.Sc,mins Summation of SOC blocks from step 1 to step

s− 1 of CARFC.V OMexpVariable operation and maintenance cost of ex-

pander.V OM c Variable operation and maintenance cost of com-

pressor.

Variables

airch/dt Total amount of air compressed to/ extracted from

the air cavern at time t.bc/d/ht,s/s′/s′′ The fractional value of the SOC/power/power

block corresponding to step s/s′/s′′ of theCAFRC/TARFC/HRC to obtain SOCt/P

dt /P

dt

at time t.CONG

t Cost of natural gas consumption at time t.OCt Operation cost of the plant at time t.P dt Discharging power at time t.P ct Charging power at time t.P sr,xt Spinning reserve power at time t in either modes

x , i.e, discharging (d), or charging (c).Pnrt Non-spinning reserve power at time t.SOCt,k Cavern state of charge at time t in scenario k in

percent.

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uc/d/ht,s/s′/s′′ Binary variable that is equal to 1 if step s/s′/s′′

of CARFC/DAFRC/HRC is the last step to obtainSOCt/P

dt /P

dt and 0 otherwise.

xdt Unit status indicator in discharging mode at timet (1 is ON and 0 is OFF).

xct Unit status indicator in charging mode at time t.

A. FunctionsΓ(P c

t , SOCt) The amount of air stored in the cavern interm of kg as a function of charging power,P ct , the SOC of the cavern at time t.

AFRc(SOCt)Stepwise decreasing function that indicatesthe charging air flow rate as a function ofthe SOC of the cavern at time t.

AFRd(P dt ) Stepwise decreasing function that indicates

the required discharging air flow rate as afunction of discharging rate, P d

t , at time t.HR(P d

t ) Stepwise decreasing function that indicatesthe heat rate as a function of the dischargingrate, P d

t , at time t.

I. INTRODUCTION

THE total installed electricity storage capacity worldwideis estimated to grow from around 85 GW in 2011

to 460 GW with 27% renewable energy share in annualpower generation by 2050 [1]. Compressed air energy storage(CAES), as one of bulk energy storage technologies, hasa variety of potential applications due to its capability ofstoring large amount of energy as well as its fast response.These applications includes energy time-shifting, facilitatingthe large-scale integration of renewable energy resources, andenhancing power system reliability [2]–[5].

Figure 1 illustrates the schematic diagram of a conventionalCAES plant with a two-stage high pressure and low pressurecompressors and turbines. Large compressors use electricityto compress and store air into a reservoir, typically an under-ground salt cavern. The high pressure air is later heated ina combustor using natural gas fuel and then used to powergas turbines to generate electricity. In order to quantify theeconomics of CAES technology in an electricity market, anappropriate scheduling model for the CAES facility needsto be developed considering efficiency of the componentsand operational characteristics. Several studies develop self-scheduling models of a CAES facility to estimate the en-ergy arbitrage revenue of the CAES technology in differentelectricity markets [6]–[9]. The self-scheduling of generationcompany with a CAES facility as well as thermal units andrenewable resources are addressed in [10], [11].

When participating in competitive electricity markets, alarge merchant storage facility may benefit from energy ar-bitrage. In addition to providing energy arbitrage, the CAEStechnology can also provide spinning and non-spinning re-serves services to the market. Stacking multiple revenuestreams improves the economics of energy storage, and thus,needs to be properly modeled [12], [13]. Previous studies havemodeled participation of energy storage systems in energy andreserve markets [12]–[15]. The additional revenue of a CAESfacility gained by providing ancillary services in different U.S.electricity markets are explored in [12]. The benefits of aCAES facility providing arbitrage and reserves in a powersystem are studied in [16]–[18].

Fig. 1: Schematic diagram of a CAES facility.

The efficiency of a CAES facility is expressed based on itsheat rate and energy ratio [19]. Heat rate expresses the amountof fuel burned per unit of electricity generated by the turbine.Energy ratio indicates the amount of energy that the compres-sor of the plant consumes per unit of energy that the expandergenerates [12]. The energy ratio is calculated based on theair flow rate the compressor compresses and stores as well asthe required discharging air flow rate. In the developed CAESscheduling model in [6]–[14], [16]–[18], the nominal heat rateand energy ratio, i.e., required heat and air flow rate at fulldischarging capacity, is considered for the facility. However, ithas been shown that the efficiency of a CAES facility dependson its operational status [19], [20]. For instance, the heatrate increases for lower discharging powers. Several studieshave concentrated on thermodynamic analysis of the CAEStechnology [20]–[22]. It is demonstrated that the air flow rateduring charging depends on the cavern SOC. Moreover, theheat and air flow rate during discharging vary significantly fordifferent discharging rates [20]. Thus, the varying efficiencyof the CAES facility based on the system thermodynamicsshould be taken into account in the facility scheduling planto avoid costly and unprofitable operations, and consequentlyprevent overestimation of the facility’s revenues.

This paper proposes an optimization-based self-schedulingmodel of a merchant CAES facility participating in day-aheadenergy and reserve markets incorporating the thermodynamiccharacteristics of the CAES technology. Thus, the proposedformulation properly models the changes in the facility ef-ficiency in different operational conditions and optimize itsscheduling accordingly. In doing so, the self-scheduling of aCAES facility providing energy arbitrage, spinning and non-spinning reserves with nominal heat rate and energy ratio isinitially developed. The thermodynamic characteristics of theCAES facility during charge and discharge processes are thentaken into consideration and the model is modified accord-ingly. Since the formulation is non-linear, binary techniquesare used to convert it to its equivalent linear formulation tobe solved by conventional solvers. The main contribution ofthis paper is to incorporate practical limitations of a CAESfacility in operation scheduling model. The significance of thiscontribution is that the prescribed schedules revenue estima-tions are more realistic compared to when the thermodynamiccharacteristics are ignored. Note that in his paper, it is assumedthat the CAES facility is a merchant privately-owned unit,which operates independently in the electricity market. Wedo not consider co-operation of the facility with a wind/solarfarm.

II. CAES SELF-SCHEDULING FORMULATION

In this section, the self-scheduling of a CAES facility partic-ipating in day-ahead energy and reserve markets is developed.The storage facility is assumed to be a price-taker and cannot

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alter the market price by its operation. This assumption is onlyvalid when the CAES capacity is very small compared to thesize of the supply-side in the market. Modeling the behavior ofa price-maker CAES facility considering the thermodynamiccharacteristics is the subject of the authors’ future work.

A. Objective function

The goal of the CAES plant is to maximize its profit throughenergy arbitrage as well as offering spinning and non-spinningreserves as a participant in the day-ahead energy and reservesmarkets. In order to take price uncertainty into account, wegenerate different price scenarios on historical price data. Wealso generate different spinning and non-spinning deploymentscenarios to consider the uncertainty of their deployment.Then, the expected value of the profit for the day is calculated,in line with [14]. The objective function is expressed asfollows

max

K∑k=1

γk

T∑t=1

[(P dt − P c

t )× πEt,k + (P sr,d

t + P sr,ct )× πsr

t,k

+ Pnrt × πnr

t,k + [(P sr,dt + P sr,c

t )× depsrt,k+ Pnr

t × depnrt,k]× πEt,k −OCt,k] (1)

The objective functions (1) consists of five terms. The firstterm represents the energy arbitrage revenue, i.e. the profit ofselling electricity to the market minus the cost of purchasingthe electricity from the market to power the compressor. Thesecond term is the spinning reserve revenue determined bythe spinning reserve price and the spinning reserve capacityoffered during either charging or discharging modes. Notethat a responsive load can offer spinning reserve service inelectricity markets such as ERCOT and NYISO [23]. The thirdterm is the non-spinning reserve revenue. The forth term is therevenue comes from the real-time spinning and non-spinningreserves deployment respectively assuming a probability ofdeployment depsrt,k and depnrt,k in scenario k. In this study,based on the deployment probability for each reserve, depsrt,kand depnrt,k are vectors consist of 0 and 1 elements, in which1 at time t states total deployment of offered spinning or non-spinning reserve at that time in scenario k. It is assumed thestorage facility is paid by the energy price (πE

t,k) in case it isdeployed. The fifth term in the objective functions shows theoperation cost of the CAES facility at time t in scenario k. Itis stated as follows:

OCt,k = [(P dt + P sr,d

t × depsrt,k + Pnrt × depnrt,k)

× (HRnom × πNG + V OMexp)]

+ [(P ct − P

sr,ct × depsrt,k)× V OM c] ∀t ∈ T, ∀k ∈ K (2)

The operation cost is expressed in two terms in (2). Thefirst term shows the operation cost during discharging. This isthe cost of burning natural gas and the variable operation andmaintenance cost of expander to provide energy offered in theenergy market P d

t plus the spinning or non-spinning reservesduring discharging if deployed at time t. The second term in(2) is the total variable operation and maintenance cost duringcompression.

B. Power Capacity ConstraintsEquation (3) states the CAES facility can operate in only

one specific mode at a time. The limits on the charging anddischarging power and the spinning reserve during chargingand discharging are presented in (4)-(7) based on the minimumand maximum capacity of compressor and expander. The non-spinning reserve capacity is limited by the quick start capacityof the CAES facility, as expressed in (8).

xct + xdt ≤ 1 ∀t ∈ T (3)P ct ≤ P c

max.xct ∀t ∈ T (4)

P cmin.x

ct ≤ P c

t − Psr,ct ∀t ∈ T (5)

P dt + P sr,d

t ≤ P expmax.x

dt ∀t ∈ T (6)

P expmin.x

dt ≤ P d

t ∀t ∈ T (7)

0 ≤ Pnrt ≤ QSC × [1− (xct + xdt )] ∀t ∈ T (8)

C. Energy Capacity ConstraintsParticipation in energy and reserve markets creates opera-

tional constraints based on cavern capacity. These constraintsmust be taken into account when scheduling operation inmultiple markets. In [12], [16], this issue is not addressed andonly the effect of charging and discharging power is consideredin energy capacity constraints.

During discharging, the CAES facility needs to store suf-ficient compressed air in the cavern to not only follow itsdischarge schedule in the energy market, but also provide an-cillary service in response to the system operator’s deploymentdispatch, as specified in (9). The effect of discharging on thecavern SOC is a function of the energy ratio. Additionally,the storage cavern must have sufficient available capacity tobe able to follow the charge schedule in case the spinningreserve during charging is not deployed by the system operator.This constraints is defined in (10). Equation (11) calculatesthe state of charge for the next hour (SOCt+1,k) based on thecurrent SOC, the level of charging and discharging power (P c

t

and P dt ), as well as the amount of spinning and non-spinning

reserves deployed in that hour. The deployed spinning reserveduring charging is calculated as (P sr,c

t ×depsrt,k) in scenario k.If depsrt,k = 1 at time t, i.e, the spinning reserve is deployed,the CAES facility must decrease its charging level by P sr,c

t ,as stated in (11). Similarly, if the spinning reserve is deployedduring discharging, the facility must increase its discharginglevel by P sr,d

t . In a similar way, the effect of non-spinningreserve deployment on SOC is calculated. The initial levelfor the air storage cavern is specified by (12). In order tohave sufficient energy at the end of the day to be able to takeadvantages of energy arbitrage opportunities in the next day, aminimum level for SOC is considered for the end of the day,as stated in (13).

SOCmin ≤ SOCt,k −(P d

t + P sr,dt + Pnr

t )× EREmax

∀t ∈ T, ∀k ∈ K (9)

SOCt,k +P ct

Emax≤ SOCmax ∀t ∈ T, ∀k ∈ K (10)

SOCt+1,k = SOCt,k +(P c

t − Psr,ct × depsrt,k)

Emax

−(P d

t + P sr,dt × depsrt,k + Pnr

t × depnrt,k)× EREmax

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Fig. 2: The variations of air flow rate and compressor power duringcharge process [20].

∀t ∈ T, ∀k ∈ K (11)

SOC1,k = SOCinit ∀t ∈ T, ∀k ∈ K (12)

SOCfinal ≤ SOCT+1,k ∀t ∈ T, ∀k ∈ K (13)

III. INCORPORATING THERMODYNAMICCHARACTERISTICS IN SELF-SCHEDULING FORMULATION

In Section II, constant heat rate and energy ratio areassumed for the facility, disregarding the facility operationalconditions. This is the case for previous studies focusingon self-scheduling of the CAES technology in an electricitymarket. However, the efficiency depends on its operationalstatus [19], [20]. In [20], the charge/discharge process analysisof a conventional compressed air energy storage system isconducted. In that study, the air storage cavern operatesbetween the pressure range of 7.2 MPa and 4.2 MPa andthe rated powers of the two-stage compressor and two-stageturbine are 60 MW and 290 MW, respectively. Developingthe thermodynamic equations for different components, thethermodynamic analysis is carried out for charging processand also for a range of discharging rate, i.e., 30% to 100%. Itis shown that the cavern SOC affects the air flow rate duringcharging. Moreover, the heat rate and required air flow rateduring discharging varies significantly for different dischargingrates.

A. Effect of SOC in Charging ProcessReference [20] investigates the variation of air flow rate

and compressor power during charging time with rated com-pression power when the compressor fully charges the storagecavern with initial minimum SOC. It is shown in Fig. 2. Asshown in Fig. 2, the air flow rate of the compressor dropsas the SOC and consequently cavern pressure increase. Thisis because as more air is stored in the cavern, it gets moredifficult to compress air in a higher pressure. As seen in Fig. 2,compressor power increases before decreasing. This is becauseof the increased compressor outlet temperature and decreasedair flow rate [20].

Based on the above discussion, the impact of charging onthe cavern SOC is not constant. Depending on the level ofSOC, the amount of air which could be stored in the cavernwith the power P c

t directly depends on the level of SOC. Inother words, the more amount of air is compressed and stored,the lower level of air flow could be stored in the cavern due tothe higher pressure level of the cavern. Thus, if at time t, thecompressor is operating with the power level of P c

t , the massof air stored in the cavern in kg depends on P c

t as well asthe current cavern SOC. This is presented as a function of P c

t

and SOCt: say Γ(P ct , SOCt). Thus, the SOC equation can be

expressed as follows:

SOCt+1 = SOCt +Γ(P c

t , SOCt)

CAmax−Φ(P d

t ) ∀t ∈ T (14)

Fig. 3: The level of air flow rate per MW of charging versus cavernSOC.

Fig. 4: Variations of air flow rate under different generation levels[20]

Φ(P dt ) represents the effect of discharging on the cavern

SOC. It will be explained later in this section. The functionΓ(P c

t , SOCt) is defined based on the information presentedin Fig. 2. Based on this figure, the air mass flow rate ofcompressors per MW of energy consumed by the compressor,AFRc, is calculated by dividing the air flow rates by thecompressor power. The cavern SOC at time t of charging isalso calculated, which is the cumulative mass of air stored bythe time t, i.e., integral of airflow rate curve up to t. By meansof this information, AFRc versus SOC is illustrated in Fig.3. Note that, based on the air flow rate curve shown in Fig.2, the total amount of compressed air is 6.4 million kg. Totalmass of cushion air in cavern is reported to be 9.48 millionkg [20]. Thus, it can be concluded that, in order to maintainthe minimum required pressure for the cavern, the minimumamount of 3.08 million kg of compressed air must remain inthe cavern. In other words, the minimum cavern SOC mus be33%.

As seen in Fig. 3, AFRc is a function of cavern SOC.Thus, the function Γ(P c

t , SOCt) and (14) can be expressed asfollows:

Γ(P ct , SOCt) = P c

t ×AFRc(SOCt)× 3600 ∀t ∈ T(15)

SOCt+1 = SOCt +P ct ×AFRc(SOCt)× 3600

CAmax− Φ(P d

t )

(16)

B. Effect of Generation level on SOC

During discharging, the required air flow rate of a CAESfacility is a function of generation power - higher generationpower requires higher air flow rates through the expander tomeet the energy requirement. Based on the data in [20], thisrelationship can be plotted, as shown in Fig. 4. It can be seenthat generation level and air flow rate are linearly related.

An interesting point can be found from the data presented inFig. 4. Dividing the air flow rates by the generation power fordifferent generation power, the required air flow rate per unitof generated electricity at different generation levels is derived,which is depicted in Fig. 5. According to this figure, the airflow rates per MW of generated electricity increases with forlower generation levels. For instance, when generating at 30%generation level, the required air flow rate per MW is 2.30

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Fig. 5: The variations of air flow rate per unit of generated electricityunder different generation levels

Fig. 6: The variations of turbine efficiency under different generationlevels [20].

kg/s.MW , which is 42% higher than that of generating withfull capacity, 1.38 kg/s.MW . This shows that the impacts ofgenerating electricity during discharging mode on cavern SOCis not constant. Conversely, it significantly depends on thegeneration level. The lower the generation level is, the higherair flow rates per unit of electricity is required. Based on datain [20], the reason behind this issue is implied in Fig. 6. Thisfigure shows the turbine efficiency under different generationlevel conditions. As seen in this figure, the efficiency of thehigh pressure (HP) turbine decreases significantly with thedecrease in generation level. Hence, in order to compensate thedecrease in turbine efficiency for lower generation level, higherair flow rate is required to generate one unit of electricity.

Based on the above discussion, the required mass of airper MW of generated electricity, released from the cavern, togenerate a certain level of power is not constant and directlydepends on the discharging level as shown in 5. These issuesmust be incorporated in the self-scheduling model of a CAESsystem. The mass of air released from the cavern at time t togenerate P d

t is P dt ×AFRd(P d

t ). AFRd is the discharging airflow rate per MW, which is a function of generation power,as shown in Fig. 5. Thus, the cavern SOC equation (16) isupdated as follows:

SOCt+1 =SOCt +P ct ×AFRc(SOCt)× 3600

CAmax

− P dt ×AFRd(P d

t )× 3600

CAmax∀t ∈ T (17)

C. Effect of Discharging Rate on Heat Rate (HR)During discharging, the required heat rate is a function of

generation level. Based on the data presented in [20], Fig. 7plots the variations of HR for different generation levels. Asseen in this figure, the heat rate increases noticeably for lowergeneration levels. For instance, the heat rate increases by 26%from the rate value when operating at 30% generation level.The reason for this increment in HR is implied in Fig. 6. Dueto decrease in turbine efficiency for lower generation level,higher air flow rate and consequently higher fuel flow rate isrequired to generated one unit of electricity.

According to Fig. 7, HR is a function of generation leveland thus, the cost of natural gas when discharging at P d

t canbe expressed as follows:

CONGt = [P d

t ×HR(P dt )× πNG

t ] ∀t ∈ T (18)

Fig. 7: The variations of HR under different generation levels [20].

Fig. 8: Linearization process for (a) Compression air flow rate versuscavern SOC, (b) Dischrging air flow rate versus discharging rate, (c)heat rate versus discharging rate.

IV. EQUIVALENT LINEAR FORMULATION

The developed equation for the cavern SOC in (17) and thecost of natural gas in (18) are non-linear due to the productsbetween the variables. In this section, their equivalent linearformulations are presented.

A. Linearizing the effect of SOC in Charging ProcessIn (17), the term P c

t × AFRc(SOCt) shows the amountof air compressed and stored at time t, which causes non-linearity. In the following, the equivalent linear constraints ofthe term is developed.

As an example, the curve in Fig. 3 is represented by a fourstep decreasing curve as shown in Fig. 8-(a). A set of binaryvariables uct,s is defined for each hour. Then, based on thevariables bct,s, u

ct,s and parameters Sc,min

s , bc,maxs , shown in

Fig. 8-(a), the set of following equations are developed to findthe corresponding step the level of cavern SOC is at.

SOCt =

nc∑s=1

(bct,s + uct,sSc,mins ) ∀t ∈ T (19)

0 ≤ bct,s ≤ uct,sbc,maxs ∀t ∈ T, ∀s ∈ nc (20)

nc∑s=1

uct,s = 1 ∀t ∈ T (21)

In (19), SOCt is linearly expressed as a function ofvariables bct,s and uct,s, shown in Fig. 8-(a). Equation (20)expresses the limit on the block of the curve shown in Fig.8-(a) in every hour, which is between zero and the size of thatstep. Equation (21) states that in every hour, only one instanceof the variable uct,s is nonzero, which shows the correspondingstep of the curve the cavern SOC is at that hour.

Using the defined parameters and variables, the total amountof air compressed when compressor is operating at the powerP ct can be expressed as follows:

aircht = P ct ×

[ nc∑s=1

uct,sAFRcs

](22)

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Based on the fact that at time t only one instance ofthe variable uct,s is one, the summation in (22) shows thecompressor air flow rate. The equation (22) is still non-lineardue to the products between the variables P c

t and uct,s. Thefollowing constraints are the equivalent linear constraints of(22). The method used here to resolve the non-linearity of(22) is the extension of big M method presented in [24]. Thedetails of this method can be found in [24].

aircht +M ≥ AFRcs.P

ct + uct,s ×M,∀t ∈ T, ∀s ∈ nc (23)

aircht −M ≤ AFRcs.P

ct − uct,s ×M,∀t ∈ T, ∀s ∈ nc (24)

where M is a positive big enough number.

B. Linearizing the effect of Discharging Process on SOCIn (17), the term P d

t ×AFRd(P dt ) is the required amount of

air released from the cavern to generate P dt . In the following,

the equivalent linear constraints of the term is presented. Theprocess is in line with the approach used in [25], [26].

Figure 8-(b) illustrates the linearization process for a sam-ple four step discharging air low rate curve. Based on thisapproach, the linearization process may be written as follows:

P dt =

nd∑s′=1

(bdt,s′ + udt,s′qd,mins′ ) (25)

0 ≤ bdt,s′ ≤ udt,s′bd,maxs′ (26)

nd∑s′=1

udt,s′ = xdt (27)

airdt =

nd∑s′=1

AFRds′ × (bdt,s′ + udt,s′q

d,mins′ ) (28)

Figure 8-(a) shows the variables, i.e., bdt,s′ , udt,s′ , and param-

eters, i.e., AFRds′ , q

d,mins′ , bd,max

s′ , used to linearize the amountof air as a function of hourly discharging power. The shadedarea in this figure represents the total required amount of air,which is the discharging power multiplied to the air flow rateper MW. In (25), the discharging power is linearly expressedas a function of variables bdt,s′ , u

dt,s′ , shown in 8-(b). Equation

(26) expresses the limit on the block of the curve, which isbetween zero and the size of that step. Equation (27) specifiesthat in every hour of discharging, only one instance of thevariable udt,s′ is nonzero, which shows the corresponding stepof the discharging air flow rate the storage is operating at thathour. Based on (27), all instance of the variable udt,s′ are zero attime t if storage is not in discharging mode at that hour. Basedon (26) and (27), during a discharging hour, only one instanceof the variable bdt,s′ could vary between zero and the size ofselected step of the curve. All the others are forced to be zero.Based on above discussion, for each hour, the total amount ofair required to discharge at P d

t is linearly expressed as (28).Therefore, the term P d

t × AFRd(P dt ) in (17) is replaced by

the variable airdt and the constraints (25)-(28) are added tothe optimization problem.

C. Linearizing the cost of natural gas in discharging Dis-charging Process

The cost of natural gas in (18), which is used in theoperation cost constraint, is non-linear. A similar approach, de-scribed in Section IV-B, is used to develop the equivalent linear

constraints of (18). Figure 8-(c) illustrates the linearizationprocess for a sample four step heat rate curve. The linearizationprocess is expressed as follows:

P dt =

nh∑s′′=1

(bht,s′′ + uht,s′′qh,mins′′ ) (29)

0 ≤ bht,s′′ ≤ uht,s′′bh,maxs′′ (30)

nh∑s′′=1

uht,s′′ = xdt (31)

CONGt =

nh∑s′′=1

HRhs′′ × (bht,s′′ + uht,s′′q

h,mins′′ ) (32)

The process of incorporating thermodynamic characteris-tics of the conventional CAES technology in self-schedulingformulation and the the linearization processes, developed insections III and IV, respectively, are used to update the CAESself-scheduling optimization problem presented in section II.

V. NUMERICAL RESULTS

Daily and yearly numerical simulations are performed fora CAES facility with 100 MW of discharging power, 60 MWof charging power, and 8 hours of full discharging capabilityas the storage capacity. Minimum discharging, charging andSOC levels are respectively, 30 MW, 10 MW, and 33%. Thecharging and discharging air flow rate and the heat rate curvespresented in Section III are also used for this case study. Theminimum level of SOC at the end of the day is assumed tobe at least 60% to have enough compressed air in the cavernto take advantage of the opportunities in the next day. Theenergy and reserve prices of the ERCOT market for a five yearperiod from 2011 to 2015 are used for the yearly analysis.Based on the available installed generation capacity of theERCOT market, which is 77,000 MW [27], considering a 100MW storage facility as price-taker is a reasonable assumption.The proposed mixed integer linear model is implementedin generalized algebraic modeling systems (GAMS) softwarepackage and solved using CPLEX solver. The solution timefor daily scheduling on a PC with an Intel Core 7 CPU (2.8GHz) and 8.0 GB RAM is in the order of few seconds. GAMSand MATLAB are used to solve the model for the five yearperiod on a daily basis.

In the next subsections, the CAES scheduling modelwith constant efficiency parameters and the proposedthermodynamic-based model are respectively referred to asCM and TBM (conventional model and thermodynamic-basedmodel).

A. CAES Self-scheduling: a Demonstrative CaseThe energy and reserves price profiles for a typical 24 hours

period is depicted in Fig. 9. The resulted schedule for energyand reserve markets from CM is shown in Fig. 10-(a). As seenin this figure, the facility charges at low price hours and alsoparticipates in the spinning reserve market during chargingperiods. Moreover, the facility decides to discharge partiallywith the minimum capacity in all discharging periods andoffers the remaining capacity in the spinning reserve market tomaximize its profit. The estimated profit in this case is $9,800.

The schedule obtained from the CM, shown in Fig. 10-(a), isimported to TBM to see whether the storage facility is capable

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Fig. 9: energy, spinning reserve, and non-spinning reserve priceprofiles.

Fig. 10: (a) CAES scheduling obtained from the simple model, (b)Actual CAES scheduling when following simple model scheduleconsidering thermodynamic characteristics, (c) CAES schedulingresulted from the developed thermodynamic-based model.

of following the schedule and what the actual profit is based onthis schedule considering the thermodynamic characteristics.The actual schedule is depicted in Fig. 10-(b). According tothis figure, the CAES facility should also charge with 20 MWat hour 4 to compressed more air because of two reasons.First, due to the effect of SOC on the compression air flowrate, which is not considered in CM, not enough air is stored.Secondly, due to partial discharge during discharging periods,the required air flow rate increases due to drop in turbineefficiency, which means more amount of air than estimatedis required to follow the discharge schedule. As shown in Fig.10-(a), the storage facility fails to follow the schedule in hours14, 22, and 24 due to faster depletion of the air storage cavernthan what estimated and accordingly lack of compressed airin the air storage cavern. Moreover, due to partial discharging,the heat rate also increases, which imposes higher cost ofburning natural gas. Therefore, the actual revenue gained fromthis schedule is $5,600, which is $4,200 lower than what isestimated.

The optimal scheduling of the CAES facility resulted fromthe TBM is shown in Fig. 10-(c). Comparison of Fig. 10-(a)and Fig. 10-(c) demonstrates that with thermodynamic-basedscheduling, the partial discharging operations in some hoursare curtailed. Furthermore, the facility discharges with higherlevel in hours 18, 20, and 21; since due to decrease in turbineefficiency and higher required air and fuel flow rates in partialoperation, it is not profitable to operate partially. This scheduleleads to $6,940 operation profit, which is $1,340 higher thanthe actual profit gained from the CM. Therefore, the resultsshow that

taking into account the CAES thermodynamic character-istics would lead to a more efficient scheduling, with loweroperation hours and higher operating profit.

Fig. 11: The annual profit of the CAES facility providing energyarbitrage when using CM and TBM.

Fig. 12: Dispatch characteristic of a CAES facility during 2011providing energy arbitrage when scheduling with CM and TBM.

B. Participating in Energy Market: Five year Analysis

In this section, it is assumed that the CAES facility onlyparticipates in the energy market providing energy arbitrage.The CM and TBM are applied sequentially on a daily basisto the energy price of the ERCOT market during years 2011-2015.

Figure 11 illustrates the annual estimated profit gained fromthe CM, the actual gained profit from the CM when theresulted schedules in CM are imported to the TBM, and theTBM profit during year 2011 and 2015. As shown in thisfigure, the actual annual profit decreases slightly from theestimated profit when using CM. Moreover, the improvementin profit obtained from the TBM compared to the actualprofit is small. This implies that in case of providing onlyenergy arbitrage in the market, the CM, widely used in theliterature, has acceptable accuracy. The dispatch characteristicsof the storage facility during the year shows why this happens.As an example, Fig. 12 shows the percentage of time thestorage facility is charging, discharging or idle during 2011.As shown in this figure, in case of using CM (left bar),the CAES facility mostly operates at full discharge whenproviding energy arbitrage, in which the facility operates withthe nominal efficiency. Thus, as shown in the middle bar ofFig. 12, the CAES facility is able to follow all dischargesscheduling. It should only charge for 2% more hours to beable to follow the schedule due to a few percentage of thehours it discharges partially and also not considering the effectof SOC in charging process. This causes a small decrease inthe actual profit compared to the estimated one. Moreover, asshown in the right bar of Fig. 12, the dispatch characteristicof the storage when providing energy arbitrage using TBM issimilar to that of CM. Therefore, the gained profit out of CMand the TBM are close to each other.

C. Participating in Energy and Reserve Markets: Five YearAnalysis

In this section, it is assumed that the CAES facility par-ticipates in both energy and reserve market The developedthermodynamic-based model and the conventional model areapplied to investigate the effect of CAES thermodynamiccharacteristics on its annual operating profit obtained fromenergy and reserve markets.

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TABLE I: Annual error of the estimated profit of CM and the profitimprovement obtained by TBM, when participating in energy andreserve markets

Year 2011 2012 2013 2014 2015Error of estimated profit

compared to actual in CM 10.6% 19.5% 32.5% 23.1% 18.6%

Improvement of TBMcompared to CM 2.50% 2.90% 5.47% 3.18% 3.73%

The annual estimated profit resulted from the CM, the actualgained profit from the CM, and the TBM profit comes fromproviding energy arbitrage, spinning and non-spinning reservesduring year 2011 and 2015 are depicted in Fig. 13. Thenumbers above the bars shows the total profit in each case.As seen in this figure, the actual annual profit for energyarbitrage and spinning reserve decreases noticeably from theprofit estimated by the CM. Table I presents the error of theestimated profit resulted from the CM compared to the actualprofit for the year 2011 to 2015. The results demonstratein case of scheduling for both energy and reserves markets,the CM leads to significant overestimation, which obviouslyaffects the economics of the facility. This shows the impor-tance of considering the thermodynamics of the facility whenscheduling for the energy and reserve markets. Furthermore,the comparison of the actual profit and the TBM profit, shownin Fig. 13, states that by considering the thermodynamics ofthe facility in the scheduling, proposed in the TBM, the profitof the CAES facility is improved, as shown in table I.

Investigating the dispatch characteristics of the storagefacility during the year for each model shows the reasonof such significant error in the profit estimated by the CM.As an instance, Fig. 14 shows the dispatch characteristicsof the facility during the year 2011. As illustrated in thisfigure, in the case of using CM (left bar), the CAES facilitymostly discharge partially to offer the remaining capacity asthe spinning reserve while ignoring the decreasing efficiencyfor lower discharging rate; This is similar to that of reported in[12]. Since, when discharging partially, the required air flowrate increase significantly, the facility needs to charge more tostore more amount of air in the cavern, as shown in the middlebar of Fig. 14. However, in spite of compressing more air, theCAES facility fails to follow all partial discharge schedulesdue to fast depletion of the cavern. Moreover, higher heat rate,required for those hours of partial discharging, imposes higheroperation cost to the facility. Therefore, the actual revenuecomes from the energy and reserve market drops significantly,as depicted in Fig. 13.

As depicted in the right bar of Fig. 14, the dispatchcharacteristic of the storage using TBM is different from thatof the CM. It discharges with full capacity for more number ofhours. Moreover, the number of hours it discharges partially ismuch lower than those of the CM, since the lower efficiencyof the facility for lower discharging rates are considered inthe TBM. Thus, it only discharge partially in those hoursthat the revenue gained by participating in the reserve marketoffsets the high operation cost of partial discharge. Therefore,although in TBM, the facility discharge for less number ofhours compared to those of the CM, due to the efficientscheduling considering the thermodynamics of the system andpreventing unprofitable actions, the obtained profit is higherthan the actual profit gained by the CM, as shown in Fig. 13.

Fig. 13: The annual profit of the CAES facility providing energyarbitrage as well as reserves when using CM and TBM.

Fig. 14: Dispatch characteristic of a CAES facility during 2011providing energy arbitrage and reserves when scheduling with CMand TBM.

VI. CONCLUSION

In this paper, a self-scheduling approach for a merchantCAES facility participating in energy and reserve markets isdeveloped incorporating the thermodynamic characteristics ofthe facility. The developed model is applied to the energyand reserve prices of the ERCOT market to analyze theeffect of the system thermodynamics on the economics ofenergy storage and compare it with the case of conventionalscheduling with nominal constant efficiency parameters. Theresults demonstrates that in case of providing only energyarbitrage, the conventional model with nominal efficiencyhas acceptable accuracy. However, in case of participating inenergy and reserves markets, the results of the conventionalmodel has significant error, which illustrates the importanceof considering CAES thermodynamic characteristics in itsscheduling.

In this paper, it is assumed that the CAES facility is a price-taker and does not change the market price by its operation.Modeling the self-scheduling of a price-maker CAES facilityin energy and reserve markets considering the thermodynamiccharacteristics is the subject of the authors’ future work.Moreover, considering the fact that a CAES plant could affectthe power flow in the grid, the CAES operation in a power gridcould have impact on transmission congestion. Incorporatingthe transmission network and the impacts of CAES operationlimitations on transmission congestion is left to future work.Furthermore, the proposed thermodynamic-based CAES self-scheduling model can be expanded to the scheduling of an en-ergy storage facility in the electricity market co-located with awind or solar farm to investigate what role the thermodynamiclimitations play when co-operation with a wind or solar farmis of interest.

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