Compressed Air Energy Storage-Part I: An Accurate Bi-linear Cavern Model

> REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 1 Abstract — Compressed air energy storage (CAES) is suitable for large-scale energy stora ge and can help to increase the penetration of wind power in power systems. A CAES plant consists of compressors, expanders, caverns, and a motor/generator set . Current ly used ca vern models fo r CA ES are either accurate but highly non-linear or linear but in accurate . Highly non-linear cavern models cannot be di rectly utilized in power syst em optimization problems. In this regard, a n a ccurate bi -linear cavern model for CAES is proposed in this first paper of a two- part series. The charging and discharging processes in a cavern are divided into several virtual sta tes an d then the first law of thermodynamics a nd ideal gas law a re used to derive a cavern model, i.e., model for th e variation of temperature and pressure in these processes. Therea fter, the heat transfer b etween the air in the c avern and the cavern wall is considered and integ rated in to the c avern model. By subsequently eliminating sever al negligible terms, the cavern m odel reduc es to a bi - linear (linear) m odel for CAES with multiple (sin gle) time steps. The accuracy of the proposed cavern model is verified via comparison with an a ccurate non-linear model. Index Terms — Bi -linear cavern model; compressed air energy storage (CAES); heat transfer ; ideal gas law; thermodynamics. N OMENCLATURE a, ht Both adiabatic process and heat transfer are considered (superscript) ht Heat transfer ( superscript)   Parameters ,      Parameters ,     , representing the left -hand side of (3) an d ( 10 ), respectively   Constant volu me specific heat (J/(k g K))   Heat transfer co efficient (W/(m 2 K) )  A constant eq ual to 1.4   Mass of air in the cavern (kg)   Mass of air in virtual container 2 as shown in Fig. 3 (kg)   Mass of air char ged into the cavern (kg)  󰇗  Mass flow rate char ged into a caver n (kg/s)  󰇗  Mass flow rate dischar ged out of a cavern (kg/s)   Pressure of the air in the cavern (bar)   Pressure of the air in the cavern af ter th e charging, discharging , and id le processes fo r   2, 3, an d 4, respectively (b ar)   Pressure of th e air charged into the cavern (bar)   Pressures in virtual states as sho wn in Fig. 2 ,    (bar)   Surface area of the cavern wall (m 2 )  Total intern al energy (J)  Specific air con stant (J/(kg K))   Temperatur e of the air in the cavern (K)   Temperatur e of the air in the cavern after t he charging, disch arging, and idle processes for   2, 3, and 4, respectiv ely (K)   Temperatur e of the cavern wall (K)   Temperatur e of the air charged into the cavern (K)   Temperatur es in virtual states as shown in Fig. 2,    (K)       Volumes of v irtual containers as sho wn in Fig s. 2 and 3 (m 3 )   Volume of a cavern (m 3 )  Work (J)   Average air density in a cav er n (kg/m 3 )  Time interv al (s)  Change in in ternal energy (J) I. I NTRODUCTION NERGY stor age tech nologies are valuable to power systems, especially co nsidering th e penetration of renewable generation is g rowing rapidly, e.g., the win d po wer share of global electricity demand will incr ease from 4% in 2015 to 25 - 28% in 2050 [1] . En ergy storage ca n provide different kind s of services [2] , e.g., electric energy time-shift, electric supply capacity, reg ulation, power reliability, etc. T he current global installed electricity storag e capacity is abo ut 141 GW and an estimated 310 GW of additional capacity would be needed in the United States, Europe, China, and India [ 3] to support the massive increase of renewable generation in the fu ture. There are cur rently two k inds of large -scale energy storage, i.e., pumpe d-hydro storage and compressed air en ergy storage (CAES) , that can be installed at the gr id scale . CAES is a high power an d energy storage techno logy and has relatively lo w capital, operational, an d maintenance costs [4] . The power rating of a large -scale CAES plant can re ach 3 00 or even 1000 MW and the rated energy ca pacity can r each 1 000 or even 2 860 MWh [4] . Currently, there a re two comm ercialized CAES plants. The wo rld’s first CAES p lant was installed in Huntorf, Germany in 197 8. A schematic of the Huntorf CAES plant is shown in Fig . 1 [5] . A CAES plant is comp rised of compressors, turbines, a motor/gen erator set, and large repositories, e.g., und erground salt caver ns. CAES uses off- peak electricity (up to 60 MW for the Huntorf CAES plant) to compress the air to hig h pressure and stor e it in a lar ge repository. CAES generates electrici ty (up to 290 MW for the Huntorf CAES plant) by releasing the stored compressed air , which is combusted with fuel to dr ive the turbines. The second commercialize d CAES plan t is the McIntosh p lant in McIntosh, Alabama, U. S. [6 ] . This p lant, which b ecame op erational in Compressed Air Ener gy Storage-Part I: An Accurate Bi -li near Cavern Model Junpeng Zhan, M e m b e r , IEEE, Osama Aslam An sari , Student Member, IEEE, and C. Y. Chung, F el lo w , IEEE E The work was supp orted in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Saskatchewan Power Corporation (Sas kPower). The authors were with the Departmen t of Electrical and Computer Engineering, Universi ty of Saskatchewan, Saskatoo n, SK S7N 5A9, Canada (e -mail: zhanjunpe ng@gmail.com , oa.ansari @usask.ca , c.y.chung@us ask.ca ). > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 2 1991, can pr oduce an output o f 110 MW electricity for up to 2 6 hours. The plant efficiencies of the Huntorf plant an d McIntosh plant are ~42 % and ~54%, r espectively [7]. The ro und-trip efficiency of th e CAES is ~80% [8 ]. According to the Electric Power Research Institute (EPRI) , about 75% of th e U.S. has geologic s ites s uitable for C AES [7] , [8]. Northern Europe is also replete with suitable salt deposits. For example, nearly 50 0 salt cav erns are curren tly being used for natural gas stor age. Therefore, it is f easible to install CAES in many different locations . A number of CAES plants are being constructed as a result o f increasing renewable energy utilization and several adv antages offered by CAES. For example, a 160 MW CAES plant near Saskatchewa n-Alb erta border in Canada [ 9] is expec ted to be completed in a few years and combined with the interconnection between the Saskatchewan an d Alberta power grids . From 2009 to 2013 , Pacific Gas & Electric received US$50 million in fund ing for a demonstration project to validate the design, performance, and reliability of a 300 MW CAES plant in Kern County, California [10] . Several further examples ar e provided in [10] . Thermodyn amic properties such as variations of temperature and pressure in the caverns of a CAES plant are important factors that affec t the overall plant operation an d performance [11] - [13] . Two kin ds of cavern models for CAES are currently described in the literature. The first co nsists o f accurate bu t h ighly nonlinear mod els [11] - [13] . I n [11 ] , complex a nd simplified real gas models are developed for an ad iabatic caver n for CAES, b oth of which adequately represent the thermodynamic p roperties of the air. Reference [12] developed an ac curate d ynamic simulation model for a CAES cavern that incorpo rates an accurate heat transfer mod el. I n [12] , heat tran sfer is shown to play an important r ole in the thermodynamic behavior o f the cavern and therefore the propo sed mod el can accurately simulate the actual cavern beh avior. In [13] , a simp lified and unified an alytical solution con sidering heat transfer is proposed for tem perature and pressure var iations in CA ES caverns. The mo del proposed in [13] is v alidated using rea l data from the Huntorf plant trial tests and the results calcu lated fr om the m odels in [ 11] and [12] , demonstrating that the proposed solution is ca pable of adequately calculating the th ermodynamic beh avior of CAES caverns. All three models in [11] - [13] are accurate but highly non -linear , and therefore ca nnot be used in large- scale power system optimizat ion problems. The second kind of ca vern model assume s that the air temperature in the cavern is constant. This kind of model ha s been adopted in diff erent power system o peration pro blems , e.g., transmis sion congestion relief [14] , bidding and offering strategy [15] , and unit comm itment [16] . The constant temperature mo del is linea r b ut in accurate, which ca n result in non -optimal or even infeasible solution s. In this regar d, a novel bi -linear cavern m odel based on the ideal g as law a nd the first law of thermodynam ics is proposed in this paper (the first in a two - pa rt series) , where the hea t transfer between the air in the caver n and the cavern wall is considered . The advan tages of the bi -linear mod el over the existing two types of models mention ed above are two-fold: 1) it is accurate, as will be verified in this paper, and 2) it can be integrated in to large-scale p ower system o ptimization pr oblems , as will be demon strated in the seco nd p aper of this two -part series. The main contribution o f this p aper is th e proposed accurate bi -linear cavern model o f the CAES. The rest of th is pap er is o rganized as follows. Section II details the ded uction of the accurate bi -linear cavern model. Section III v erifies the effectiveness of the p ropo sed cavern model. Section IV presents the conclusions drawn from the results. Fig. 1. Schematic of the Huntorf CAE S plant. II. A CCURATE B I - LINEAR C AVERN M ODEL FOR CAES In this paper, constant-volume caverns of CAES ar e considered , as they are used in the ex isting CAES plants. A. Charging Process In the charging proce ss , a certain amount of air is injected by compressors into a cavern, as shown in Fig. 1 . To facilitate the model deduction given in Sections II -A1 to II- A3 , the charging process is divided into four states and the air is ass umed t o be stored in the ( virtual) container s as shown in Fig . 2, wh ere the five containers are indexed b y the numbers i n h eptagons . The characteristics of the air in each container, includ ing the pressure, volume, temp erature, an d mass, are shown in each container in Fig. 2. T he values of the un derlined notations, i.e., the pressure and temperature in containers 2, 4, and 5, are not known wh ile the values of the other notations ar e know n. Containers 1, 2, an d 4 ar e virtual while containers 3 and 5 represent the cavern before and after the air from the compressors is injecte d into it , respec tively. It is assumed that the air coming out of the compr essors is stored in a virtual container , i.e ., cy linder 8 in Fig. 1 an d container 1 in Fig. 2 . The con ditions of contain er 1 represent the therm odynamic pr operties of air at th e outlet o f the compressors. First, let the air in virtual c ontainer 1 go into virtual container 2 . The volume of container 2 is set such that the ratio of the volume of containers 2 to 3 is eq ual to the ratio of the mass of air in containers 2 to 3, i.e.,        󰇗     (1) where    and   represent the volumes of container s 2 and 3,   represents the mass of air in container 3, and  󰇗  represents the flow rate of mass charged into the cavern , which is assumed to be constant during a p eriod of time ,  . The m ass co ming out of the compressor during that period of time, denoted as   , > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 3 can be expressed as     󰇗   . The mass o f air injected i nto the cavern is assum ed to be equ al to   , i.e. , there is no air leakage .   is also the mass of air in both containers 1 and 2. Then, let the air in both co ntainers 2 and 3 go into container 4. The v olume of (mass of air in) container 4 is set to the s um of the volumes of (mass of air in) containers 2 and 3, i.e., container 4 can be seen as a combination of con tainers 2 and 3. Note that the purp ose o f using virtual co ntainers 2 and 4 is to let the wor k be 0 during the p rocess of merging the air in containers 2 and 3 into container 4. Last, let the air in container 4 g o into container 5, which is a n adiabatic co mpressing process as the mass o f air does not change and the volume decreases. The rest of this sub section details the ded uction of the model for the ch arging process. Fig. 2 . Four states and five contai ners used f o r m odel deducti on in t he char ging process. 1) St ate 1 → State 2 The transfer o f air f rom containers 1 to 2 is an adiab atic proce ss and the mass of air does n ot change . According to the ideal gas law f or the air in container 2 , one can obtain         󰇗       . (2) According to the temp erature-pressure relation f or an adiabatic p rocess,     is con stant from containers 1 to 2 [17] and one can obtain 󰇛   󰇜      󰇛   󰇜       . (3) Let   represent the left-hand side o f (3), i.e. ,          .    can be d etermined f rom (1), i.e.,        󰇗     . This leaves only two unknown variables in (2) and (3), i.e.,    and    . Th er efore ,    and    can be obtained from (2) a nd (3):     󰇛   󰇜          (4)     󰇛        󰇜  . (5) 2) St ate 2 → State 3 Now we con sider the proce ss o f the air in b oth containers 2 and 3 going in to con tainer 4. The volum e of container 4 is eq ual to the sum o f the v olumes of containers 2 and 3 . I n this p rocess , the work is zero a nd the total inter nal energy does not change. The change of the internal energy [18] in container s 2 and 3 is     󰇛       󰇜 and     󰇛      󰇜 , respectively . According to the first law of thermodynamics, i.e.,      , one can obtain     󰇛        󰇜      󰇛       󰇜   . (6) Note that    can be obtained fr om (5).    is then the only unknown variable in (6) and can b e expressed as     󰇛           󰇜  󰇛      󰇜 . (7) According to the ideal gas law for th e air in container 4 in Fig. 2 , one can obtain    󰇛       󰇜  󰇛     󰇜    (8) Then, by substituting (7) into (8) , one can obtain     󰇛    󰇜          󰇛          󰇜       (9) Therefore,    and    in contain er 4 are obtained. 3) St ate 3 → State 4 From contain ers 4 to 5, the mass o f air does not change and the volume reduce s from 󰇛       󰇜 to   . T his is an adiabatic compression/h eating process. Thus,     is constant from container s 4 to 5 [17] , i.e., 󰇛    󰇜       󰇛   󰇜     ( 10 ) According to the ideal gas law for th e air in container 5 in Fig. 2 , one has      󰇛 󰇗      󰇜   ( 11 ) Let   represent the left-hand side of ( 10 ), i.e.,    󰇛    󰇜      . Only two variab les are unknown in ( 10 ) an d ( 11 ), i.e.,   and   . Therefore,   and   can be obtained from ( 10 ) and ( 11 ) :    󰇛  󰇗      󰇜    󰇛   󰇜     ( 12 )    󰇛     󰇜    . ( 13 ) Now   and   for container 5 have been obtain ed. By substituting   (   , (5) , (7), and (9) are needed to calculate   ) into ( 12 ) and ( 13 ) , these two equation s can be reform ed as :      󰇡   󰇗     󰇢     󰇛  󰇗      󰇜    󰇗   ( 14 )      󰇡   󰇗     󰇢     󰇛  󰇗      󰇜    󰇗   ( 15 ) where               and               . Equation s ( 14 ) and ( 15 ) show that   and   are non linear function s of  󰇗  , which are linear iz ed as follows. According to Newton’s generalized binomial theorem [19] , one has 󰇛    󰇜    󰇡   󰇢            󰇛   󰇜      ( 16 ) where 󰇡   󰇢   󰇛   󰇜 󰇛  󰇜   ,  can be an y real n umber, and  is an integer. That is, 󰇡   󰇗     󰇢  i n ( 14 ) can be expressed as   󰇛  󰇜  󰇗      󰇛   󰇜󰇛  󰇜  󰇡  󰇗     󰇢     . Considering that  󰇗   is much smaller th an   , the second and high er orders of 󰇛  󰇗    󰇜 can be ignored. Then , ( 14 ) can be reformed as      󰇡  󰇛  󰇜  󰇗     󰇢    󰇛   󰇜    󰇗   ( 17 ) Note that th e second term in ( 14 ) is replaced by   󰇛   󰇜   󰇗   , which h as neg ligible error as   is m uch smaller than   (e.g.,    46~66    and    1.04    for the Huntorf CAES p lant) a nd  󰇗   is much smaller than   . Similarly, ( 15 ) can be reformed as      󰇡  󰇛  󰇜  󰇗     󰇢    󰇛   󰇜   󰇗   ( 18 ) When ( 17 ) an d ( 18 ) are u sed in a one-step optimization problem,   is a linear fun ction o f  󰇗  in ( 17 ) and   is a linear functio n of  󰇗  in ( 18 ) as   and   have a known initial status. When used in a multi-step op timization problem, ( 17 ) and ( 18 ) ar e b i-linear eq uations as   and   become decision variables. > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 4 B. Discharging Process Fig. 3 . Three c ontainers used f or model de duction in the discharg ing process . In the dischar ging process, th e air is discharged from a cavern , as s hown in Fig. 1 . To fac ilitate the model deduction given in the rest o f th is subsection , the caver n b efore disch arging is divided into two containers, i. e., containers 1 and 2, while the cavern after dischar ging (cy linder 7 in Fig. 1 ) is represented as container 3, as shown in Fig. 3 . The three c ontainers in Fig. 3 are indexed by the numbers in hep tagons. The ch aracteristics of the air in each container, including the p ressure, volume, temperature , and mass, are shown in each container in Fig. 3. The valu es of the u nderlined n otations, i.e., th e pressure an d temperature in container 3, are not known while the values of the other notatio ns are know n. The dischar ging process can be divided into two virtual steps. First, the air in co ntainer 2 is taken out of the cavern. Then , the air in container 1 exp ands to the whole cavern, i.e. , air goes from con tainers 1 to 3 . The deductio n of the model for the second step is giv en in the rest of this subsection . The volume of container 2 is set such that the ratio of the volume of containers 2 to 1 is equal to the ratio of the mass of air in contain ers 2 to 1, i.e.    󰇛      󰇜     󰇛      󰇜 . ( 19 ) Note that the purpose of using virtual container 2 is to let the temperature and pressure of t he air in container 1 be the same as the air in th e cavern before disch arging a nd to let air go ing from con tainers 1 to 3, i.e., step 2, be an adiab atic expansion process. Let  󰇗  represent the flow rate of mass discharged from the cavern , w hich is assumed to be con stant during a period of time,  . The n, the mass of air discharged from the cav ern, denoted as   , during that period of time can be expressed as     󰇗   . Air expan sion from container s 1 to 3 is an adiabatic process. Then,     is constan t from containers 1 to 3: 󰇛   󰇜      󰇛   󰇜     . ( 20 ) According to the ideal gas law for th e air in container 3 in Fig. 3, o ne has      󰇛     󰇜   . ( 21 ) There are only two variables u nknown in ( 20 ) and ( 21 ), i.e.,   and   . Therefore,   and   can be obtain ed from ( 20 ) and ( 21 ):    󰇛    󰇗    󰇜    ( 22 )    󰇛    󰇗    󰇜     . ( 23 ) According to the Newton’s generalized binomial theorem , when  󰇗   is much smaller than   , ( 22 ) and ( 23 ) can be respectively reformed as    󰇛     󰇗    󰇜   ( 24 )    󰇛   󰇛  󰇜 󰇗     󰇜   ( 25 ) Similar to ( 17 ) and ( 18 ) , ( 24 ) and ( 25 ) are linea r ( bi -linear ) equations when used in a one-step (multi-step) optimization problem . C. Chargin g Process Considering Heat Transfer In Section s II -A and II-B, the heat transfer between the air and the cav ern wall is n ot co nsider ed. Howev er, the heat transfer play s an important role in th e variation of the air temperature/p ressure in the cavern [ 12] . Therefore, th e heat transfer is considered in this and the following two subsections. In this subsection, the temperature as a function of time is fir st deduced . The pressure as a f unction of time is then obtain ed via the ideal gas law . Last, the temperatu re/pressure as a function of time is linea rized to obtain a bi -linear model. According to [1 3] , t he air density (   ) in the cavern and the cavern wall temperature (   ) can be assum ed t o be co nstant and the heat transfer between t he air and the cavern wall can b e modelled as              󰇛    󰇜 ( 26 ) 󰇛 󰇜             󰇛    󰇜  ( 27 ) Equation ( 18 ) can be written as   󰇛  󰇜    󰇡  󰇛    󰇜  󰇗     󰇢    󰇛   󰇜   󰇗   . ( 28 ) By substituting ( 28 ) into ( 27 ), i.e. , replac ing  o n the right- hand side of ( 27 ) by th e right-hand side of ( 28 ), one can ob tain    󰇛  󰇜             󰇡     󰇡  󰇛    󰇜  󰇗     󰇢    󰇛   󰇜   󰇗  󰇢  ( 29 ) where super script ‘ ht ’ represents ‘ heat transfer ’ . By solving the integral equation ( 29 ), one can o btain    󰇛  󰇜            󰇡      󰇡  󰇛    󰇜  󰇗      󰇢    󰇛   󰇜   󰇗    󰇢 ( 30 ) Addi ng ( 28 ) and ( 30 ) together gives     󰇛  󰇜    󰇡  󰇛    󰇜  󰇗     󰇢    󰇛   󰇜    󰇗               󰇡      󰇡  󰇛    󰇜  󰇗      󰇢    󰇛   󰇜    󰇗     󰇢 ( 31 ) where superscript ‘ a,ht ’ indicates that both the adiabatic process and heat tr ansfer are considered. Ac cording to [ 19] , i.e.,  󰇛  󰇜   󰇛   󰇜   󰆒 󰇛   󰇜  󰇛    󰇜 , one can l inearize 󰇛   󰇜   and 󰇛   󰇜  at   as 󰇛   󰇜   󰇛    󰇜   󰇛    󰇜󰇛    󰇜   󰇛      󰇜 ( 32 ) 󰇛   󰇜   󰇛    󰇜   󰇛    󰇜󰇛    󰇜   󰇛      󰇜 ( 33 ) whe re    is a fixed v alue, i.e.,         . Then, by u sing ( 32 ) and ( 33 ) , ( 31 ) can b e reformed as       󰇛  󰇜    󰇛    󰇛    󰇜  󰇗   󰇜     󰇗  󰇛 󰇛    󰇜   󰇛    󰇜󰇛    󰇜  󰇛      󰇜󰇜         󰇛        󰇛    󰇛    󰇜  󰇗        󰇜     󰇗    󰇛󰇛    󰇜   󰇛    󰇜󰇛    󰇜   󰇛      󰇜 󰇜  󰇜 ( 34 ) Equation ( 34 ) rep resents the change of the temperature during the charg ing process as a fun ction of time  an d charg in g mass flow rate  󰇗  , wher e both the ad iabatic process and the heat transfer process are considered . According to the ideal gas law, one can obtain     󰇛  󰇜  󰇛     󰇗   󰇜      󰇛󰇜   ( 35 ) which can be expande d to ( 36 ) by sub stituting ( 31 ) therein.     󰇛  󰇜  󰇛    󰇗   󰇜     󰇡  󰇛    󰇜  󰇗     󰇢 > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 5  󰇛    󰇗   󰇜      󰇛   󰇜   󰇗    󰇛    󰇗   󰇜               󰇡          󰇛    󰇜  󰇗      󰇢  󰇛    󰇗   󰇜              󰇡  󰇛   󰇜    󰇗     󰇢 ( 36 ) The four terms in ( 36 ) are each on a separate line. Th e second term of ( 36 ) can be rep laced b y        󰇛   󰇜   󰇗   because      is small and  󰇗   is much smaller than   . The las t term of ( 36 ) can be ignored because both           and      are small. According to [19] , i.e.,  󰇛  󰇜   󰇛   󰇜   󰆒 󰇛   󰇜  󰇛    󰇜 , one can linearize    at    :                󰇛       󰇜 ( 37 ) Now, by using ( 37 ) , ( 36 ) ca n be reformed as       󰇛  󰇜    󰇛    󰇛  󰇜 󰇗   󰇜     󰇗   󰇡            󰇛       󰇜 󰇢         󰇛     󰇗   󰇜            󰇛  󰇜 󰇗          ( 38 ) D. Dischargi ng Pro cess Considering Heat Transfer In this subsection, the temperature as a function of time is first dedu ced . The pressure as a f unction of time is then obtain ed via the idea l gas la w . Last, the temperature/p ressure as a function of tim e is linearized to obtain a bi -linear model. Equation ( 25 ) can be written as 󰇛 󰇜     󰇛  󰇜   󰇗      ( 39 ) By substituting ( 39 ) into ( 27 ), i.e. , replac ing  o n the right- hand side of ( 27 ) by the right-hand side of ( 39 ) , one can obtain    󰇛 󰇜             󰇡      󰇛  󰇜   󰇗      󰇢  ( 40 ) By solving the integral equation ( 40 ), one ca n obtain    󰇛 󰇜            󰇧 󰇛      󰇜   󰇛    󰇜    󰇗       󰇨 ( 41 ) Adding ( 39 ) and ( 41 ) together gives     󰇛󰇜     󰇛  󰇜   󰇗                 󰇧 󰇛      󰇜   󰇛    󰇜    󰇗       󰇨 ( 42 ) Equation ( 42 ) represents the chan ge in temperature during the d ischarging process as a function o f time  and discharging mass flow rate  󰇗  , where both th e adiab atic process and t he heat transfer pr ocess are consid ered. Consider ing that           is very small (around     ) ,             can be rep laced by       and therefore ( 42 ) can be reformed as       󰇛󰇜       󰇛    󰇜    󰇗          󰇛      󰇜            󰇛    󰇜    󰇗    ( 43 ) Note that there are four v ariables in ( 43 ), i.e.,   ,  󰇗  ,     󰇛󰇜 , and   . According to the ideal gas law, one can obtain     󰇛  󰇜  󰇛     󰇗   󰇜      󰇛󰇜   ( 44 ) which can be expanded as follows by substituting ( 42 ) therein:     󰇛  󰇜  󰇛    󰇗    󰇜      󰇛    󰇗   󰇜    󰇛  󰇜   󰇗                 󰇡 󰇛    󰇗   󰇜    󰇛      󰇜   󰇛    󰇗   󰇜    󰇛    󰇜    󰇗       󰇢 ( 45 ) Note that 󰇛     󰇗   󰇜  󰇗     in the second term of ( 45 ) can be replaced by    󰇗     because  󰇗   is much smaller th an   . Equation ( 45 ) can be r eformed as       󰇛  󰇜  󰇛      󰇗   󰇜               󰇛󰇛   󰇗   󰇜 󰇛      󰇜    󰇛    󰇜    󰇗    󰇜 ( 46 ) Comparing ( 46 ) with ( 24 ), we kno w th at the first term in ( 46 ) represents the adiabatic process inside the caver n and the second term is associated with the heat transfer between the air in the cav ern and the cavern wall. No te that there are f ive variables in ( 46 ), i.e.,   ,  󰇗  ,     󰇛  󰇜 ,   , and   . E. I dl e Process Co nsidering Heat Tran sfer When in the idle p rocess, i.e., neither charg ing n or discharging occurs, heat transfer occurs between the air and the cavern wall if there is a tem perature differen ce between them. By solv ing th e integral equation ( 27 ), the change of te mperature in the cav ern in the idle process can be expressed as    󰇛󰇜  󰇛      󰇜                 ( 47 ) where   is the initial temperature of the air in the ca vern in the idle process. According to the ideal gas law, on e can obtain    󰇛  󰇜        󰇛 󰇜  ( 48 ) which can be expanded into ( 49 ) by substituting ( 47 ) therein:    󰇛  󰇜     󰇛      󰇜                        ( 49 ) which can b e reformed as    󰇛  󰇜                       󰇛               󰇜  ( 50 ) The              in ( 50 ) can be expressed as            , which can be lin earized as follows                         󰇛      󰇜 ( 51 ) where              . By substituting ( 51 ) into ( 47 ) a nd ( 50 ), one can obtain    󰇛󰇜  󰇛      󰇜󰇛          󰇛      󰇜   󰇜    ( 52 )    󰇛  󰇜    󰇛          󰇛       󰇜    󰇜        󰇡              󰇛      󰇜󰇢 ( 53 ) There are three v ariables in ( 52 ), i.e.,    󰇛󰇜 ,   , and   . There are also three variables in ( 53 ), i.e.,    󰇛  󰇜 ,   , and   . In sum m ar y, the bi-linear cavern models include ( 34 ) and ( 38 ) for the charg ing process , ( 43 ) and ( 46 ) for the discharging process, and ( 52 ) and ( 53 ) for the idle process. Equation s ( 34 ) , ( 38 ) , ( 43 ) , ( 46 ) , ( 52 ) and ( 53 ) ar e linea r ( bi -linear) equations wh en used in a one-step (multi-step) o ptimization problem . III. S IMULATI ON Parameters for the Huntorf CAES plant ar e used for t he calculations in this paper . The Hun torf CAES plant features tw o caverns with volumes of 1 41,000 a nd 169,000 m 3 , respectively. > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 6 Note that the m aximum mass flo w rate in the charg ing process (  󰇗  ) is 108 k g/s for th e whole plant and 49.1226 kg/s for the first cavern , wh ich is calculated fr om 108     . Similarly, the maximum mass flow rate in the discharging process (  󰇗  ) is 4 17 kg/s for the whole plant and 189.6677 kg/s for the first cavern, which is ca lculated fr om 417     . In this paper, th e first cavern is used f or calculations. The other parameters for the Huntorf CAES plant are given in Table I [12] , [13] . TABLE I P ARAMETERS FOR THE H UNTORF CAES P LANT .          25 , 000 m 2 718.3 J/(kg K) 30 W/(m 2 K) 1.4 66 bar        286.7 J/(kg K) 141,000 m 3 40 °C 50 °C Three processes are defined as follows and used to verify the accuracy of the proposed bi -linear mo del in the rest o f this section: • Charging process: Set the initial pressure (temperature) of the air in th e cavern to 46 b ar ( 20 °C ). Charge the first cavern (141,000 m 3 ) continuously for 16 ho urs at the maximum m ass flow rate, i.e.,  󰇗   49.1226 kg/s . • Discharging process: Set the initial pressure (temperature) of the air in the ca vern to 66 bar (40 °C ). Discharge th e cavern co ntinuously for 4 hours at the max imum mass flow rate, i.e.,  󰇗   189.6677 kg /s . • Idle process: Set the initial pressure (temperature) o f the air in the cavern to 60 bar (45 °C ). Let th e cavern be in the idle process f or 16 hours. The status of the air in the ca vern includes the temperature (   ), pressure (   ), and mass (   ). When the status at the start of a time interval is known, w hich is used in th e right -hand s ide of ( 34 ) , ( 38 ) , ( 43 ) , ( 46 ) , ( 52 ), and ( 53 ), then th e statu s at the en d of that time interval can be calculated, i.e., the left-hand side of ( 34 ) , ( 38 ) , ( 43 ) , ( 46 ) , ( 52 ), and ( 53 ), respectively. The status of the air in the cavern in each interval of each process defin ed above is calculated in a recursive way . That is, given the status of the air at the start o f a time inter val, the status of the air at the end of that time interval is ca lculated via ( 34 ) , ( 38 ) , ( 43 ) , ( 46 ) , ( 52 ), and ( 53 ) as described in th e pr evious paragraph , and is then used as the status of the air at the start of the next time interval. Reference [13] compares several existing C AES mo dels with the measured data from Huntorf. The analytical model in [13] is accurate and simpler th an other existing an alytical models. Thus, in the rest of this section, th e analytical mo del in [13] is used as a b enchmark model to verify the ac curacy of the proposed bi -linear cavern model. A. Model Verificatio n In this section, the tim e interval is set to 1 second, i.e.,  is equal to 1 secon d in ( 34 ) , ( 38 ) , ( 43 ) , ( 46 ) , ( 52 ), and ( 53 ) . The pressure and temperature for each time interval o f the charging (discharging, idle) process obtain ed from bo th th e proposed bi-linear model and the an alytical model given i n [13] are plotted in Fig. 4 (Fig. 5, Fig. 6 ). Figs. 4-6 show that the pressure/temper ature results obtain ed f rom both the proposed bi -linear model and the analytical mo del are q uite clo se to one another . The mean absolute percentage err or between the results, in terms of pressure or temperature , obtained from the bi -linear model and the an alytical model during the charging , discharging , and idle p rocesses is tabulated in Table II. The last column o f Table II sh ows that the idle part of the bi -linear model, i.e., ( 52 ) and ( 53 ), is almost as accurate as th e analytical model. The 2 nd and 3 rd columns of Tab le II show that t he accuracy of the charging/dischargin g parts of th e bi -linear model, i.e., ( 34 ) , ( 38 ) , ( 43 ), and ( 46 ), is around 0.12 %. TABLE II T HE M EAN A BSOLUTE P ERCENTAGE E RROR B ETWEEN THE R ESULTS O BTAINED BY THE B I - LINEAR M ODEL AND THE A NALYTICAL M ODEL G IVEN IN [13] IN E ACH OF THE T HREE P ROCESSES . C ha rging process Discharging proces s Idle process Pressure 0.0013 0.0012 1.61    Temperature 0.0012 0.0012 1.61    Fig. 4 . Results obtained by the proposed bi -linear model and the analytical model in [13] duri ng the chargi ng process: a) press ure, b) temperat ure. Fig. 5 . Results obtained by the p ropose d bi-linear model and the analytical model in [13] duri ng the dischargi ng process: a) pr essure, b) temperature. Fig. 6. Results obtained by the proposed bi -linear model and the analytical model in [13] duri ng the idle process: a) pressure , b) temperature . b) b) b) a) a) a) > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 7 a) c) e) d) b) f) B. Impact o f Heat Transfer an d Temperature To observ e the imp act of the heat tran sfer, the r esults obtained from the bi -linear model with and witho ut considering heat transfer in the charging (discharging) process are p lotted in Fig. 7 ( Fig. 8). The heat tran sfer clearly has a significant impact on the temperature and pressure. Th erefore, it is important to consider the heat transfer in the cavern model. The results obtain ed from the constant tem perature model in the ch arging (dischar ging) process are also plotted in Fig. 7 (Fig. 8 ). Obviously, th e pressure and temperature o btained from t he constant temp erature model are quite differ ent fr om the bi - linear model, which indicates that the constant temperature cavern model is inaccurate. Therefo re, it is necessary to use an accurate ca vern model instead o f the constant tem perature cavern model. Fig. 7. Re sults obtained from the bi-linear cavern model with and without considering heat transfer and the constant temperatur e cavern model in the charging process: a) pressure of the air in the cavern, and b) temper ature of the air in the caver n. Fig. 8. Res ults obtained from the bi-linear cavern model with and without considering heat transfer and the constant temperature cavern model in the discharging process: a) p ressure of the air in the cavern, and b) temperatur e of the air in the ca vern. C. Different Time Intervals In power system o peration problems, th e tim e in terval can be longer th an one seco nd , e. g., the time in tervals of econ omic dispatch and u nit commitment are usually 15 min utes and 1 hour, respectively. Therefore, it is nece ssary to kno w whether the proposed bi-linear model is accurate for different time intervals. In this regard , the status of the air in the cavern is calculated for different tim e interval s (1 second, 1 minu te, 5 minutes, 10 minutes, 20 minutes, and 6 0 minutes) using the same initial status and the sam e recursive approach as outlined in the second paragraph b efore Section III- A. The fi nal temperature and pressure of the char ging, discharging, and idle processes obtained by the bi -linear mod el and th e analytical model are plotted in Fig . 9. The er ror and relative error (values given in rou nd brac kets) betw een the results, in term s of final temperature and pressure, o btained from th e two models are shown in Table I II , where the 2 nd -5 th , th e 6 th -9 th , and the 10 th - 13 th rows show the error /relative error in the charging, discharging , an d idle processes, respectively. In Table I II , ‘E - 4’ and ‘E - 6’ represent ‘    ’ a nd ‘    ’ , respectively . Fig s . 9a-9 d show that the accuracy of the bi -linear model in the charging and discharging p rocesses d ecreases as th e ti me interval increases . Figs. 9e-9 f show that the accuracy of th e bi - linear mo del in the idle process does not chan ge with the ti me interval. Table III sho ws that the er ror and relative er ror o f th e temperature a nd pressure in both the charg ing and disch arging processes are small when the time interval is smaller than or equal to 5 minutes. When the time interva l is equal to 10 or 20 minutes, the errors (relative errors) of the pressure in the charging process are 0.0416 and 0.0993 bar (0.06 and 0.1 4%), respectively, which are relativ ely small. When the time interval is eq ual to 10 or 20 m inutes, th e err ors ( relative e rrors) of the pressure in the discharging process are 0.0298 and 0.0740 bar (0.06 and 0.16%), respectiv ely, which are also relatively small. When the time interval is equal to 60 min utes, th e error (relative error) in both the ch arging and discharging processes is relatively large. Therefore, Fig. 9 and Table III show that the accuracy o f the bi -linear model is high , moderate, and relatively low when the time interval is between 1 secon d and 5 m inutes, between 10 and 20 minu tes, and equal to 6 0 minutes, respectively. Fig. 9. Final tempera ture/press ure obtained by both the analytical model and the bi -linear model in different processes using different time i ntervals: a) final temperature in the charging process, b) f in al pressure in t he chargin g process , c) fi nal tem perature in the dischar ging process, d) final pressure in the discharging process, e) final temperat ure in the idle process, f) fi nal pressur e in the idle process. b) b) a) a) > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 8 TABLE I II E RROR (R ELATIVE E RROR IN R OUND B RACKETS ) B ETWEEN THE S OLUTION O BTAINED BY THE B I - LINEAR M ODEL AND THE A NALYTICAL M ODEL IN D IFFERENT P ROCESSES U SING D IFFERENT T IME I NTERVALS (2 ND -5 TH R OWS FOR C HARGING P ROCESS , 6 TH -9 TH R OWS FOR D ISCHARGING P ROCESS , 10 TH - 13 TH R OWS FOR I DLE P ROCESS ) Interval 1 s 1 m in 5 m in 10 m in 20 m in 60 m in Tempera ture ( °C ) -0.0173 -0.0323 -0.0936 -0.1706 -0.3255 -0.9574 (-0.04%) (-0.07%) (- 0. 2%) (- 0. 37 %) (- 0. 7%) (-2 .1 %) Pressure (bar) -0.0162 -0.0105 0.0126 0.0416 0.0993 0.3292 (-0.02%) (-0.02%) (0.02%) (0.06%) (0.14%) (0.48%) Tempera ture ( °C ) -0.0848 -0.0475 0.1050 0.2973 0.6879 2.3371 (- 0. 38 %) (- 0. 21 %) ( 0. 47 %) (1.3%) (3.1%) (10%) Pressure (bar) -0.0132 -0.0090 0.0081 0.0298 0.0740 0.2572 (-0.03%) (-0.02%) (0.02%) (0.06%) ( 0. 16 %) ( 0. 56 %) Tempera ture ( °C ) 0 0 0 0 0 0 (0) (0) (0) (0) (0) (0) Pressure (bar) 5E -4 5E -4 5E -4 5E -4 5E -4 5E -4 (8.5E-6) (8.5E-6) (8.5E-6) (8.5E-6) (8.5E-6) (8.5E-6) IV. C ONCLUSION This pap er has proposed an accurate bi-linear cavern model for CAES based o n the ideal gas law an d the first law of thermodynam ics. An accurate analytical m odel in the literature is used as benchmark model to verify the accuracy of the proposed bi -linear cavern mode l. Simulation results sho w that the error betwe en t he bi -linear model and the accurate analytical model is ar ound 0.12% when the time in terval is set to 1 second . The accuracy of th e proposed bi- linear model decreases as the time interval increases . For time inter vals between 1 second an d 5 minutes , between 10 and 20 minutes, and 6 0 minutes o r long er, t he bi - linear cavern model has high, m oderate, and relative ly l ow accuracy, respectively. Simulation r esults also sho w that h eat transfer has an obvious eff ect on the variatio n of temperature and pressure of the air in the cavern. Ther efore, it is necessary to co nsider heat transfer in the cavern model. The constant- temperature cavern model is al so shown to be inaccurate, which emphasizes the necessity of the proposed b i-linear cavern model for power system optimization problems. By pro perly setting the time interval, the p roposed b i -linear cavern m odel is accu rate and suitable for use in power system optimization p roblems, as will be demonstrated i n the second paper of this two -part series. R EFERENCES [1] Global wind energy council, Global Wind Energy Outlook 2016, Available: http: //www.gwec. net. [2] U. S. D epartment of Energy, “ EPRI -DOE H andbook of Energy St orage for Transmission & Di stribution A pplications: Final Report, ” 2003. [3] International Energ y Agency, “Technology Roadmap Energy Storage,” [Online]. Avai lable: https://www. iea.org, accesse d on 19 Mar ch 2014. [4] X. Luo, J. Wang, M. Doon er, and J. 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Delhi, In dia: Pear son Educa tion, 20 09. [19] Gilbert Stra ng, Calculus , W elle sley, MA, US A: Wellesle y-Cambridge Pre ss, 2010. Junpeng Zhan (M’16) received B.Eng. and Ph.D. degrees in electrical engineering from Zhejiang Universit y, Hangzhou, China in 2009 and 2014, respectively. He wa s a Postd octoral Fellow in the D epartme nt of Electrical and Computer Engineering, Un iversity of Saska tchewan, Saskat oon, SK, Canada. H is current research interes ts include the integration of the energy storage systems, dynamic thermal rating and renewable electric en ergy so urces in power systems. Osama Aslam Ansari (S ’16) recei ved the B.Eng. degree in electrical engineering from National University of Sciences and Technology (NUS T), Islamabad, Pakista n, in 2015. He is c urrentl y work ing towar d th e M. Sc. degree in electrical engi neering at the Departme nt of Electrical and Co mput er Engineering, University of Saskatchewan, Saskatoon, S K, Canada. His current research in terests i nclude the energy storage sys tems and power system relia bility. C. Y. Chun g ( M’01 - SM’07 - F’ 16) received B.Eng. (w ith First Class Hono rs) and Ph.D. degrees in electr ical engineering from The Hong Kong Polytechnic University, H ong Kong, Ch ina, in 1995 an d 1999, resp ectively. He has w o rked for Powertech Labs, Inc., Surrey, BC, Canada ; the University of Alberta, Edmonton, AB, Canada; and The Hong Kong Polytechnic University, China. He is c urrently a P rofess or, t he NSE RC/SaskPow er (Senior) Industrial Research Chair in S mart Grid Technologies, a nd the SaskPower Chair in Power Sys tems E ngineering in the Dep artment of Electri cal and Computer Engineering a t the Uni versity of Saskat chewan, Saskato on, S K, Canada. His res earch inter ests inclu de smart gri d technologies , renewabl e energy, power system stability/contr ol, planning and operation, co mputational intelligence app lications, an d power marke ts. Dr. C hung is an Editor of IEEE Transactions on Power Systems , IEEE Transactions on Sustainable Energy , a nd IE EE Power E ngineeri ng Lett ers and an Associate Editor of IET Generati on, Transmiss ion, an d Distribu tion . He is also an IEEE PES D istinguished Lecturer and a Member- at -Lar ge (Global Outreach) of t he IEEE PES Go verning Board.