Compressed Air Energy Storage-Part II: Application to Power System Unit Commitment

> REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 1 Abstract — Unit commitment (UC) is one of the most important power system operation problems. To integrate higher penetrat io n of wind power into power systems, more compressed air energy storage (CAES) plants are being built . Existing cavern models fo r the CAES used in power system optimization problems are not accurate, which m ay lead to infeasib le solution s, e. g. , the air pressure in the cavern is o utside its operating range. In this regard, an accurate CAES model is proposed for the UC problem based on the accurate b i -linear caver n model proposed in the first paper of this two-part series. The minimum switch tim e between the charging and disc ha rging processes of CAES is considered. The whole m odel, i.e., the UC model with an accurate CAES mod el, is a la rge-scale mix ed integer bi-linear p rogramming problem . To reduce th e com plexity of the who le model, th ree strategies are proposed to reduce the number of bi -linear terms without sacrificing accuracy. McCormick relaxation and piecewise linearization a re then used to li nearize the whole model. To decrease the solution t ime, a method to obtain a n initial solution of the linearized model is proposed. A modified RTS-79 system is used to verify the effectiveness of the whole model and the solut ion methodology. Index Terms — Accurate bi-linear cavern model; compressed air energy storage; initial solution; linearization; unit commitment. N OMENCLATURE Sets/Indices  ,   Bus index an d set of all bus indices, respectively  Index for injections (including conventional generation u nits, wind generation units, compressed air energy storag e (CAES))  ,   Scenario index and set of all scen ario indices, respectively  ,   Line index and set of all line indices, resp ectively  ,   Time index and set of all time indices, resp ectively   󰇝      󰇞 where   represent s the num ber of time periods   󰇝     󰇞   Set of indices o f injections connected to bus    Set of con ventional generation units   Set of all load indices   Set of CAES units   Set of wind generation units   Set of indices of all scenarios consider ed at time     Indices of all units available for d ispatch in scen ario   at time t Parameter s   Constant volum e specific heat (J/(kg K))   Load dem and (MW)    The maximum p ower flow of line  ( MW)   Heat transfer co efficient (W/(m 2 K))  A constant eq ual to 1.4    Average m ass of air in the cavern ( kg)   Pressure of the air charged into a cavern (bar)    ,    Maximum and minimum pressures in a cavern for optimal oper ation of CAES (bar)   Surface area of the cavern wall (m 2 )    ,    Startup and shutdown costs, respectively ($)     ,     Cost co efficients of upward and d ownward load- following r amp reserve, respectively ($/MW)    ,    Cost coefficient s of conventional gener ators    ,    Charging and discharging costs, respectively ($/MWh)    Wind sheddin g cost ($/MWh)    Reserve cos t ($ /MWh)   The element in the  th row and the  th co lumn of a nod e-branch incidence matrix     ,     Maximum and minimum ch arging power of CAES, respectively (MW)     ,     Maximum and minimum dischargin g power of CAES, respectively (MW)    Power reserve required (MW)  Gas constant (           )   Temperatur e of the air injected into a cavern (K)   Temperatur e of the ca vern wall (  )    ,    Maximum and minimum temperature of air inside a cavern (K)   Volume of the storage (   )    Maximum wind power that can be generated at scenario  (MW)    Susceptance of a line on right- of -way  (Siem ens)    ,    Upward and downward ramping limits, respectively (MW)   Probability o f scenario  at time   Time interval (second) Variables    Load shed ding (MW)   Total active po wer flow on line  (MW)  󰇗   ,  󰇗   Rate of flow of air mass charged into and discharged from a cavern, respectively (kg/s)    ,       Pressure (b ar), temp erature (  ), an d mass (kg ) of air stored in the cavern, respectively Compressed Air Ener gy Storage -Part II: Application to Power System Unit Commitment Junpeng Zhan, M em be r, IEEE, Y unfeng Wen, Me mb er , IEEE, Osama Aslam An sari , Stud ent Member, IEEE, and C. Y. Chung, F e ll o w , I EEE The work was supp orted in part by t he Natural Sciences and Engineeri ng Research Council (NSERC) of Canada and the Sask atchewan Power Corporation (Sas kPower). The authors we re with the Department of Electrical and Computer Engineering, Universi ty of Saskatche wan, Saska toon, SK S7N 5 A 9, Canada (e -mail: zhanjunpeng@ gmail.com , y.f.wen@us ask.ca , oa.ansari@usas k.ca , c.y.chung@us ask.ca ). > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 2  󰇛  󰇜 󰇛   󰇜  Pressure after a (xx) process wh ere (xx) ca n be ‘ch’, ‘ dch ’, and ‘ idl ’ , which represent ch arging, discharging , and idle, respectively (bar)   Unit on/off status; 1 if unit is on, 0 otherwise   ,   Binary startup and shu tdown states, respectively ; 1 if unit  has startup/shu tdown events, 0 otherwise   Power outp ut from conventional unit  (MW)    ,    Charging and d ischarging power, respectively (MW)  󰇛  󰇜 󰇛   󰇜  Temperatur e after a (xx) process where (xx) can be ‘ch’, ‘ dch ’, and ‘ idl ’ , which represent charging, discharging , and idle, respectively (K)   󰇛   󰇜  Temperatur e at time 󰇛    󰇜 if the  th per iod is either a charging o r discharging process (K)   Scheduled win d power generation (MW)    ,      Binary variable indicat ing the ch arging an d discharging processes, respectively    ,    Upward and downward load- following ramping reserve nee ded from un it  at time  for transition to time    , respectively (MW)      Phase angle of from-side node of right- of -way  (r ad)      Phase angle o f to-side node of right - of -way  (r ad) I. I NTRODUCTION NIT comm itment (UC) is a key power system operation proble m [1]-[4] that determines the un it on/off status ahead of time to supply suf ficient electric power to customers in a secure and economic manner. A comprehensive review of UC is provided in [5] and [6] . It is benef icial to integrate energy storage systems in to UC problem s [3], [7]-[9]. To hedge the wind power ou tput uncertainty, p umped-storage units ar e in corporated in the UC problem [7] . Reference [8] proposed deterministic and interval UC formulations for the co-o ptimization of controllable generation and pumped hydro energy storage. In [3] and [9] , fast-response b attery energy storage is utilized in UC problems for congestion relief and frequency suppor t, respectively. Compressed air energy sto rage ( CAES) , as m entioned in the first p aper of th is two -part series, is a pr omising large-scale energy storage technology. CAES h as been used to enhance power system op eration by mitigating wind sheddin g [10] , smoothing wind po wer fluctuation [11], p ro viding ancillary service [12] , particip ating in energy and reserve markets [13] , [14], etc. Some preliminary work considering CAES in UC problems has been done [15]- [18] . Reference [15] in tegrates ideal and generic storage devices into stochastic r eal-time UC problems to deal with the stochasticity and intermitten ce of non- dispatchable renewable resources. Reference [16] develop ed an enhanced security constrain ed UC formulatio n consider ing CAES and wind power. In [17] , CAES and sodium s ul fur batteries are used in a U C prob lem to m aximize the win d en ergy penetration level. In [18] , a constant-pressure CAES i s m odeled for the bi -level planning of a microgrid including C AES , where UC with CAES is desc ribed on the lower level. In the papers mentioned above, the temper ature of the air in the cavern of CAES is ass um ed to be co nstant (called constant- temperature cavern model for the CAES). Th e pressur e of the air is th en a lin ear function of the mass of air in the cavern according to the ideal g as law. Th e pressur e of the air in a CAES cavern must be within an operatin g range to ensure stable CAES operation. How ever, solutions obtained from t he constant-tem perature cavern mo del can allo w the pressure of the air in the cavern to fall outside of the oper ating range. That is, th e con stant-temperature cavern model is inaccu rate and may result in an in feasible solution. As mentioned in the fir st paper of this two- part series, ac curate analytical models [19] that have been propo sed fo r the cavern used in CAES are highly non-lin ear and therefore cannot be integrated in to large-scale power system o ptimization problems . I n this regard, the b i-linear accurate cav ern model proposed in the first paper of this series is integ rated into power system operatio n problem s in this seco nd pap er to ensur e the pressure of the air in the cavern is main tained within the operating range. This is an important and urgent task considering two CAES plants are alread y in operation and several more plants are under construction , as men tioned in the first paper. This second paper focuses on integrating the CAES into UC problems. However, the CAES model proposed herein can be easily ex tended to other po wer system optimization problems, e.g., optimal power flow, economic dispatch , etc . In CAES, a single mo tor/generator set is u sed to drive both the compr essor and expander. Therefore, it needs time to s witch between th e chargin g and dischar ging pro cesses . In th e literature, constraints associated with the minimum switch time between charging an d discharging pr oc esses are u sually n ot considered. Reference [17] prop osed a set of c onstrain ts to ensure switch time. In the current paper, a novel method with a smaller number of con straints an d variables than [17] is proposed to en sure the minimum swit ch time. In the proposed CAES model using the accurate bi -linear cavern m odel, th ere are two k inds of bi -linear term s, i.e. , the product of a binary variable and a con tinuous variable (called a binary- continuous bi -linear term) a n d the product of two continuous variables (called a con tinuous bi -lin ear term). These bi -linear terms complicate the who le model, i.e., the UC model with CAES using the accurate bi- li n ear cavern model . To decrease the complexity of solving the whole model, three strategies are pro posed to reduce the num b er of binary- continuous and continuous bi -linear terms. Given t hat piecewise linea rization of continuous bi -linear te rms will introduc e more constraints and binary variables, these strateg ies can significantly redu ce the number of constraints and binary variables added to the original whole model. Therefore, these strategies can significan tly reduce the solu tion time . This paper uses a McCormick relax ation [9] to r eplace the binary- continuous bi-linear term by a new continuous var iable subject to s ever al linear constrai nts. The ad vantages of the McCormick relaxation fo r binary-continuous bi-linear terms include no error, i.e., it is a tight relaxation, and no introduction of new binary /integer variables to the orig inal model . In th e CAES mo del, the mass, pressure, and tem perature of the air in the cavern ar e in volved in the continuous bi -linear terms and have large r anges. Unfo rtun ately, th e McCorm ick relaxation for con tinuous bi -linear ter ms has a relativ ely large error when the ranges of continuous var iables are large . Therefore, the McCorm ick r elaxatio n is not applicable to linearize the continu ous bi -linear terms herein. Piecewise U > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 3 linearization is a widely used and effective method to approximate a non - linear function [20] . Therefore, the piecewise linearization is u sed to linea rize the c on tinuous bi - linear terms. Specifically , the co ntinuous bi -linear term is transformed in to the di fference of two quadratic term s, which are subsequ ently piecewise linearized . It is always valuable to sup ply a mixed integer linear programming (MI LP) solver with an initial s olu tion to solve an MILP problem, which can reduce the solution time a n d increase the solution accuracy [21] . In this regard, a metho d to o btain an initial solution for th e linea rized whole model is proposed that utilizes the solution o btained from the UC with CAES using a constant-tem perature cavern model; it ca n significantly reduce the solution tim e required to solve the linea rized whole model. In summary, the contrib utions of this secon d paper include a novel b i-linear CAES model for the UC an d its linearization, strategies to simplify the bi-linear CAES model fo r the UC, and a method to obtain an initial solutio n for the linearized whol e model. The rest of the paper is organized as fo llows. Section II details the UC mod el considering CAES. Section III describes the m odel reformulation and solution method used to so lve the whole UC m odel. Simulation results are given in Section IV and conclusion s are drawn in Section V. II. U N IT C O M M IT M E N T C ON S I D E R I N G C O M P R E S SE D A IR E N E R G Y S T OR A GE A. Unit Co mmitment Model In this subsection, the UC model proposed in [22] is adopted and modified to include CAES. 1) Objective Fu nction The objective function is giv en in ( 1) . The first ter m is the startup and shu tdown costs of conventional gener ators; the second term is the c ost of l oad -following ramp reserves of conventional generators; the third term includes the power generation cost, charging and disch arging costs, and the penalt y cost of wind shedding; and the last term is the cost of spinnin g reserves.      󰇛            󰇜          󰇟   󰇛    󰇜          󰇛    󰇜 󰇠                            󰇛             󰇜                               󰇛         󰇜                            (1) 2) DC Power Flow Constraint (2) rep resents the power balance at each bus and (3) represents the DC power flow [22] . Con straint (4) represen ts the capacity lim it of each transmission line.                                             (2)                              (3)                   (4) 3) Spinning R eserve The spin ning reser ve can be expressed as (5) [23] . Note that    is a load but it do es not appear on the right-hand side of (5) because the ch arging load of CAES can be curtailed immediately when reserve is required .                              󰇛   󰇜        ,          (5) where    represents the requir ed spinning r eserve at tim e  , which is set to the power capacity o f the largest u nit in the system. 4) Intertemporal Constraints: Load-followin g ramping limit s and reserves are given in ( 6 )- (9) accordin g to [22] .          (6)          (7)       󰇛   󰇜      󰇛  󰇜               (8)  󰇛   󰇜           󰇛  󰇜               (9) The other constraints used include the startup and shutdown constraints, the minim um up and d own times f or conventional generator s [22] , an d the lo wer and upper bounds for the outpu t of conven tional generators and wind farms. B. CAES Co nstraints The mass flow rate in and out, i.e.,  󰇗   and  󰇗   , can be expressed as linear function s of th e charging power (    ) an d discharging power (    ), respectively, according to [24] :  󰇗                    ( 10 )  󰇗                    , ( 11 ) where the values of the coefficients   and   are adopted from [24] . There is an op timal operating r ang e for the pressure of the air in the cav ern, which can be expressed as                . ( 12 ) The CAES can not be in charging and d ischarging processes at the same time, which can be modeled as                  ( 13 ) where   and   are binary variables used to rep resent the charging and discharging processes, resp ectively. The idle process can be r epresented as 󰇛        󰇜 as the CAES should be in one and only one of the ch arging, discharging, a nd idle processes at a tim e. This rep resentation can reduce the number of va riab les and equality constraints compared to using another binary variable to indicate the status of the idle process . The lower and upper b ou nds of the charging power an d discharging power can be expressed as ( 14 ) and ( 15 ) , respectively. If the CAES is not in the charging (disch arging) process, then     (     ) and there fore the charging (discharging) p ower is 0.                            ( 14 )                            ( 15 ) The following constraint e ns ures that, when it is in idle process, i.e.,      and      , the ind icator for the idle process is eq ual to 1 , i.e.,          . Note that       and       , i.e.,              . 󰇛      󰇜                 +          ( 16 ) C. Minimum Switch Time Between Charging and Dischargin g In a CAES plant, a single mo tor/generator set is used to drive both the compressor and expander. Thus, a minimum time, i.e., 20 m inutes, is required to switch between ch arging and discharging [24] . Th e following two constraints are proposed to > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 4 guarantee this switch time:     󰇛   󰇜                       ( 17 )  󰇛  󰇜                          ( 18 ) where   is an integer and       . Compared to [17] , these switch time constraints do not introdu ce extra binary/integ er variables and the number of constraints is reduced . That is, these constraints are simpler , which can reduce t he complexity of the CAES model . D. Temp erature and Pressure Model s During Charging , Discharging , and Idle Proce ss es When    ,    (    ,    ) rep resents the initial mass (temperatur e, pressure) o f th e air in the cavern and is the sam e for each scenario  . All of the o ther notations with superscript    or 󰇛    󰇜   in ( 19 )-( 28 ) and ( 43 )- ( 47 ) are variables. For the mass (    ), temperature (    ), and pressure (    ) of air, the values are instantaneous. For all of the other variables involved in ( 19 )-( 28 ) and ( 43 )-( 47 ) , the values are assumed to be constant for a given period of time . According to th e first paper of this two -part series, the temperature (pressure) o f the air in the cav ern in the charg ing , discharging , an d idle processes can be expressed as ( 19 ) , ( 21 ) , and ( 23 ) (( 20 ) , ( 22 ), and ( 24 )), resp ectively.       󰇛  󰇜             󰇗  󰇛  󰇜            󰇗  󰇛   󰇜   󰇛   󰇜  󰇗  󰇛   󰇜                     ( 19 )       󰇛   󰇜          󰇛    󰇜      󰇗  󰇛   󰇜               󰇗  󰇛   󰇜   󰇛   󰇜  󰇗  󰇛   󰇜      󰇛   󰇜     󰇛   󰇜  󰇗  󰇛   󰇜   󰇛   󰇜            ( 20 )      󰇛  󰇜          󰇛   󰇜     󰇗  󰇛   󰇜   󰇛   󰇜      󰇛   󰇜        ( 21 )      󰇛   󰇜            󰇗  󰇛   󰇜      󰇛   󰇜             󰇛   󰇜     󰇗  󰇛   󰇜   󰇛   󰇜            󰇛   󰇜          󰇗  󰇛   󰇜        ( 22 )                   󰇛   󰇜                 󰇛   󰇜       󰇛     󰇜      ( 23 )                                        󰇛   󰇜   󰇛   󰇜                ( 24 ) where   -   and   -   are parameters defined in the Appendix. The first 4 , 5, 3, 5, 1 , and 2 term s in ( 19 ) , ( 20 ) , ( 21 ) , ( 22 ) , ( 23 ), and ( 24 ), respectively , are bi-linear terms. E. Rela tionship Between Two Consecutive Time Periods for Temperature, Pressu re, and Mass of Air in the Cavern The temper ature and p ressure of the air in the c avern at time    can be expressed using ( 25 ) a n d ( 26 ), respectively, which are eq ual to the v alues in the ch arging, discharging, o r idle processes acco rding to the value s of   󰇛  󰇜  and   󰇛   󰇜  .   󰇛   󰇜    󰇛  󰇜     󰇛   󰇜    󰇛  󰇜    󰇛   󰇜      󰇛  󰇜    󰇛  󰇜     󰇛   󰇜       ( 25 )   󰇛   󰇜    󰇛  󰇜      󰇛  󰇜    󰇛   󰇜    󰇛   󰇜   󰇛   󰇛  󰇜    󰇛  󰇜  󰇜   󰇛   󰇜       ( 26 ) T he relationship between the mass of air in the cavern at tw o consecutive tim e intervals ca n be expressed as   󰇛   󰇜        󰇛  󰇜   󰇗  󰇛   󰇜     󰇛   󰇜   󰇗  󰇛  󰇜       ( 27 ) Note that it is guaranteed by ( 10 ) , ( 11 ) , and ( 13 )-( 15 ) that  󰇗  󰇛   󰇜    in th e charging process and  󰇗  󰇛   󰇜    in the di scharging process. Ther efore,  󰇛  󰇜  and  󰇛   󰇜  in ( 27 ) can be deleted, i.e., the bi-linear constrain t ( 27 ) is equivalent to the linear co nstraint:   󰇛   󰇜        󰇗  󰇛   󰇜     󰇗  󰇛   󰇜        ( 28 ) Therefore, the optimizatio n model of UC co nsidering CAES is complete and can be formed as Minimize: (1 ), s.t. (2)-( 26 ) and ( 28 ) III. M ODEL R EFORMULATIO N AND S OLUTION M ETHOD A. McCormick Linearization o f ( 25 ) and ( 26 ) Both ( 25 ) and ( 26 ) contain four binary -continuous bi-lin ear terms. The McCormick rela xation [9] is used to linearize the binary- continuous bi-linear terms withou t any error. Here, a gen eral term  is used to repr esent the binary- continuous bi-linear terms in ( 25 ). Replace it by a new variable, i.e.,    , where  should satisfy           ( 29 )   󰇛   󰇜       󰇛   󰇜  . ( 30 ) where   and   are the lower and u pper bounds of  , respectively ,  is a binary variable, and  a nd  are con tinuous variables. When    , ( 29 ) becomes      , i.e.,    . When    , ( 30 ) b ecomes      , i.e.,    . Th erefore, ( 29 ) and ( 30 ) en sure that  is equivalent to  . That is, ( 25 ) can be linearized by replacing ea ch bin ary-continuous bi-linear term by a new variab le,  , subject to ( 29 )-( 30 ), which has no error. Similarly, a g eneral term  is used to represent the b inary- continuous bi-linear terms in ( 26 ). Replace it by a new variable, i.e.,    , where  sho uld satisfy       ( 31 )   󰇛   󰇜      ( 32 ) where   is the upp er bound of  ,  is a binar y variable, and  a nd  are continuous v ariables. When    , ( 31 ) becomes       , i.e.,    . When    , ( 32 ) becomes      , i.e.,    . Therefore, ( 31 ) and ( 32 ) en sure that  is equivalent to  . That is, ( 26 ) can be linearized by replacing ea ch bin ary-continuous bi-linear term by a new variab le,  , subject to ( 31 )-( 32 ), which has no error . B. Piecewise L inearization of Continuous Bi -linear Terms In ( 19 )-( 24 ), there are continuo us bi -linear terms, i.e., a product o f two co ntinuous variab les. I n this subsecti on, reformulation and p iecewise linearization are used to linearize these bi -linea r terms and reduce the complexity of solvin g the whole model g iven in Section II. R ef erence [25] compared different formulations of piecewise linear approximations for non -linear functions, including convex combination, multiple choice, increm ental, etc., and conclud ed that the incremental format consumed the least time for all three cases consider ed . Therefore, piecewise linear ization u sing an incremental fo rmat is used in this pap er. T he continuous bi -linea r term is r epresented by a g eneral term,  . First, the bi -linear ter m is reformulated as the difference of two quadratic terms:   󰇛    󰇜    󰇛    󰇜   ( 33 ) > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 5 The right-h and side of ( 33 ) is th en piecewise linearized. Let    󰇛  󰇜 and equally divide the range of   into    segments with each divide point represented by                   . The value s of    and 󰇛    󰇜  can then be obtained. Usin g a piecewise linearization method with an incremental format, 󰇛    󰇜   can be represented by the right-h and side of ( 34 ) subject to ( 35 )-( 37 ) . 󰇛   󰇜   󰇛    󰇜     󰇛󰇛     󰇜   󰇛    󰇜  󰇜        ( 34 )    󰇛  󰇜       󰇛         󰇜        ( 35 )                 󰇝  󰇞       ( 36 )             ( 37 ) where    is a continu ous v ariable while    is a binar y v ariable ,     󰇝       󰇞 , and     󰇝         󰇞 . Constraints ( 36 ) and ( 37 ) en sure             if      . Similarly, let    󰇛   󰇜 and equally d ivide the range of   into    segments with each divi de point represented by                   . The value s of    and 󰇛    󰇜  can then be obtained. Then 󰇛    󰇜   can be represented by the right-h and side of ( 38 ) subject to ( 39 )-( 41 ) . 󰇛   󰇜   󰇛    󰇜     󰇛󰇛     󰇜   󰇛    󰇜  󰇜        ( 38 )    󰇛  󰇜       󰇛         󰇜        ( 39 )                 󰇝  󰇞       ( 40 )             ( 41 ) where    is a continu ous v ariable while    is a binar y v ariable ,     󰇝       󰇞 , and     󰇝         󰇞 . Constraints ( 39 ) and ( 40 ) ensure             if      . Now,  can be expr essed by 󰇛   󰇜   󰇛   󰇜  , i.e.,   󰇛    󰇜     󰇛󰇛     󰇜   󰇛    󰇜  󰇜        󰇛    󰇜     󰇛󰇛     󰇜   󰇛    󰇜  󰇜        ( 42 ) which is a lin ear expression. C. Strategies to Red uce the Number of Bi -linear Terms Now, t he op timization model o f UC considering CAES can be formed as Minimize: (1 ), s.t. (2)-( 26 ) and ( 28 ) where the c on tinuous bi -linear term s in ( 19 ) , ( 20 ) , ( 21 ) , ( 22 ) , ( 23 ), an d ( 24 ) are represented by the right -hand side of ( 42 ) and the binary-con tinuous bi-linear terms in ( 25 ) and ( 26 ) are linearized using the McCormick relaxation described in Section III - A. This is referr ed to a s Linearized Model I . Considering that there are 20   continuous bi-linear ter ms , Linearized Mo del I is difficult to so lve f or a larg e-scale power system. Note that piecewise linearization of continuous bi- linear ter ms introduces extra constraints and binary/con tinuous variables, which increases the complexity of the model. In this regard , thr ee strateg ies are p roposed to reduce the number of both binary -continuous and continuo us bi -linear terms: 1) In stead of using ( 19 )-( 24 ) to represent th e temperature an d pressure, ( 19 ) , ( 21 ), ( 23 ) and ( 43 ) are used:   󰇛   󰇜    󰇛  󰇜      󰇛   󰇜         ( 43 ) 2) The bi -linear term,       , in ( 19 ) , ( 21 ), and ( 23 ) is replaced by        acco rding to ( 43 ). Then ( 23 ) becomes a linear con straint:                    󰇛   󰇜                 󰇛   󰇜       󰇛            󰇜     ( 44 ) 3) Constraints ( 19 ) and ( 21 ) ar e merged into a single constraint:      󰇛  󰇜           󰇛   󰇜  󰇗  󰇛   󰇜      󰇛   󰇜     󰇗  󰇛   󰇜   󰇛   󰇜  󰇗  󰇛   󰇜         󰇗  󰇛   󰇜                     ( 45 ) where   󰇛   󰇜  is used to represent the temperature at time 󰇛    󰇜 if the  th period is either a char ging or discharg ing process. If the  th p eriod is a charging pr ocess, then the 6 th term in ( 45 ) is ze ro as  󰇗  󰇛   󰇜    ,   󰇛   󰇜  is equ ivalent to    󰇛   󰇜  , an d ( 45 ) is equivalent to ( 19 ). If the  th period is a discharging process, then the 3 rd -5 th terms in ( 45 ) ar e ze ro as  󰇗  󰇛   󰇜    ,   󰇛   󰇜  is equivalen t to    󰇛   󰇜  , and ( 45 ) is equivalent to ( 21 ). Using ( 45 ) in stead of ( 19 ) and ( 21 ) results in one less continuous bi -linear term. Now, ( 25 ) beco mes   󰇛   󰇜    󰇛  󰇜    󰇛   󰇜    󰇛   󰇜      󰇛  󰇜    󰇛   󰇜    󰇛   󰇜       ( 46 ) which can b e linearized in a similar way to ( 29 )-( 30 ). Now, the im proved model (referr ed to as Linearized Model II ) can be ex pressed as Minimize: (1 ), s.t. (2)-( 18 ), ( 28 ) , ( 43 )-( 46 ) where ( 46 ) is linearized in a similar way to ( 29 )-( 30 ) and th e continuous bi -linear term s in ( 43 ) and ( 45 ) are represented b y the right -hand side of ( 42 ). Linearized Model I linearizes 20   continu ous bi-linear terms while Linearized Mo del II linearizes 5   terms, i.e., the number of continuous bi -linear terms is r educed by a factor of four . Moreov er , Linea rized Mod el I I avoids ( 26 ). Th e advantage of Linearized M odel I I over Linearized Model I is significant and will be shown in Sectio n IV -D. D. Con stant-Temperature Mod el In the literatu re, the temp erature of the air in the cavern is assumed to be consta nt [15]-[18] . That is, the ca ver n model can be modeled as   󰇛   󰇜     󰇛   󰇜             ( 47 ) where    represents the temperature of the air in the cavern. The corresponding optimization m odel of UC con sidering CAES using a constant air temperature model can be formed as Minimize: (1 ), s.t. (2)-( 18 ) , ( 28 ) , ( 47 ) E. Metho d to Obtain an Initial Solution Solving Linearized Mo dels I and I I for 2 4 hours is time consuming . To sh orten the solutio n time, a metho d to obtain an initial solution fo r both models is pr oposed; a flo wchart of the method is giv en in Fig . 1. Step 1 is to solve the constant- temperature cavern model given in Section I II -D and obtain a solution, named solution 1. Step 2 is to solve the CAES mod el (i.e., ( 10 )-( 28 )) where the charging and discharging power of C AES are fix ed to be the same as solution 1. The maximum an d minimum air pr essure s obtained in step 2 are den oted as   and   , respectively. If both   and   are within the optimal operatin g r ange as given in ( 12 ) and described in box 3 of Fig. 1, then go to step 5; other wise go to step 4. In step 4, decrease and increase the maximum a nd minim um p ressures at the two en ds o f ( 12 ) , respectively, according to box 4 o f Fig. 1. In step 5, Linearized Model I o r II is s olv ed by setting the charging/discharg ing power equal to so lution 1 ; the solution obtained in this step is > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 6 c) a) b) b) a) denoted as solution 2. In step 6, so lution 2 is used as an initial solution to solve Linearized Mo del I or II without fixing the charging /discharging po wer. The solution obtained in step 6 is the final solutio n of Linearized Mod el I or II. Note that wh en the c harging and discharg ing power are fixed ,    ,  󰇗  󰇛   󰇜  , and  󰇗  󰇛   󰇜  can be directly calculated from (28), (10), and (11), respec tively, i.e ., th ey b ecome parameters instead of variables. Ther efore, the bi -linear co nstraints ( 19)- (24), (43), and (45) -(46) become linear constraints and can b e solved very fast , i.e., steps 2 and 5 consume very little time. 1 . So l ve the consta n t-temp erat u re m o del and o bt ai n so lut ion 1 2 . So l ve the C AES m o d el b y se tti ng t h e cha rg in g/d isc h argin g pow er eq u a l to sol ution 1 , and obtai n the m a x im u m and m inimu m pres sures o f air o ve r all th e tim e pe rio ds , den o te d as p m a x 1 and p m in1 , resp e c t iv el y 3. ( p m a x 1 ≤ p m a x 0 ) &(p m in 1 ≥ p m in0 ) ? 4 . p m a x = p m a x - m ax(0 , p m a x 1 -p m a x 0 ) P m in = p m in + m a x(0 , p m in0 -p m in1 ) 0 . Le t p m a x 0 = p m a x and p m in 0 = p m in 5 . Le t p m a x = p m a x 0 and p m in = p m in0. So l ve Lin e a rized Model I or II b y s ettin g th e ch arging /discha rging pow e r equal to sol ut io n 1 and obtai n solutio n 2 6 . Us e so lu tio n 2 as a n initial solutio n to s o lve Lin eari z e d Mo del I or II Y N Fig. 1. Flowchar t of the sol ution method using an initial solution. IV. S IMULATIO N A. Test System To verify th e effectiven ess o f the proposed model and solution method, Linearized Models I and II, the constant - temperature model, and the UC mo del without CAES are solved separ ately on a modified RTS- 79 system with 3 3 conventional generator s [15 ]. Three same -capacity win d farms are added t o the system and loca ted at Buses 1, 4, an d 6, respectively. The maximu m load demand is set to 310 0 MW and the maximum wind penetration is set to 35%. The load and wind p rofiles are given in Fig. 2. Th e wind profile comes from the real o utput of a wind f ar m in Saskatchewa n, Canada. T he wind power for scenar ios 1 and 3 is set to 0.8 and 1.2 times that of scen ario 2, respectively. All of the models ar e solved using MATLAB® on a Lenovo® ThinkStation with two Intel Xeon E5 -2650 V4 processors. Both the charging and discharging costs are set to 3 $/MWh, the wi nd sheddin g co st is set to 100 $/MWh, and the r eserve cost is set to 3 $/MWh. All o ther data used can be obtained from the RTS- 79 system [15] and MatPower [22] . The parameter of the CAES p lant co mes f rom the Huntorf CAES p lant as described in the first paper of th is two-part series an d the optimal op erating range o f the air pressure in the cavern is 46 - 66 bar which is used in (12). The linearized m odel is an MI LP problem and is solved using CPLEX. Th e relative mixed-integer programming ( MIP) gap in the CPLEX is set to 0 .1%. The time in terval of CAES mod el is set to 20 minute s which will be further discu ssed in Section IV - C. Th e time resolu tions for the unit on/off schedule an d generatio n dispatch are one hour and 20 minutes, respectively . B. UC wi th/without CAES The resu lts obtained from Linearized Model II are given in Figs. 3 and 4a. T he total load dem and and the total output of all of th e conventional g enerators in the three scenarios are g iven in Fig. 3a. The total power capacity and the total output of the three wind f arms are depicted in Fig. 3b. In scenario 1, all of the wind power can be integrated. Scenario 2 (3) features some (much mor e) wind shedding. The ch arging/discharging power of the CAES is giv en in Fig. 3c. Fig. 3 shows that the CAES discharges in low -wind periods, i.e., periods 3 2-41 and 50 - 63 , and ch arges in the oth er hours. Fig. 4a shows the UC result where each row (co lumn) is associated with a u nit (a period of time), and a unit is on (off) if it is filled (blank ). Fig. 2. Load and w ind profiles. Fig. 3 . Results obtained from Linearized Model I I: a) l oad and total o utput fr om conventional units, b) total wind power capacity and total wind power output, c) charging/dischar ging power of CA ES. Fig. 4. UC result obtained from a) Linea riz ed Model II and b) the UC model without CAES. To see the benefit of CAES, the UC without CAES is also solved and the results are s h own in Figs. 5 and 4b. Fig. 5 sh ows that the wind power generation drops in p eriods 37 -39 and 5 5- 63. To satisfy th e load in th ese low -wind p eriods, more units are tur ned on to generate more power as shown in Figs. 5a and 4b. Comparing Fig. 3a with Fig. 5a sho ws that CAES reduces the po wer output from conventio nal gener ators in low -win d periods. Comparing Fig. 3b with Fig. 5b shows th at CAES help s to reduce wind sh edding, esp ecially in scen arios 2 and 3 t hat have more wind p ower . Comparin g Figs . 4a with 4b shows that CAES reduces th e number of times conventional units are turned on and o ff . To inv estigate the impacts of wind power penetration on the benefits of C AES, the UC problems with an d without CAES are solved separately by setting th e wind power penetration to 32, 0 10 20 30 40 50 60 70 75 1000 2000 3000 3500 Time p erio d (20 m in s .) Power (M W) Load Co nv . output sc e. 1 Co nv . output sc e. 2 Co nv . output sc e. 3 0 10 20 30 40 50 60 70 75 50 200 400 600 800 1,000 1,200 1,400 Time p erio d (20 m in s .) Power (M W) W P c apa. s ce. 1 W P c apa. s ce. 2 W P c apa. s ce. 3 W P output sc e. 1 W P output sc e. 2 W P output sc e. 3 0 10 20 30 40 50 60 70 75 -80 0 100 200 300 Time p erio d (20 m in s .) Char. /Disc har. Power (MW) CAES sc e. 1 CAES sc e. 2 CAES sc e. 3 > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 7 a) b) a) b) d) c) 35, and 38%. The results in terms o f total cost and wind power shedding a re tabulated in Table I, which indicate s that CAES can reduce wind p ower shed d ing by 392.8, 754.4, and 9 92.6 MWh and total co sts by 2.8, 5.3, and 6.0 % for the three d ifferent wind power penetrations, respectively. T hat is, the b enefit attributed to CAES incr eases as the win d p ower p enetration increases. Fig. 5. Results obtained from the UC model without CAES: a) load and t otal output from conventio n al units, b) total wind power capacity and total wind power output. TABLE I T OTAL C OST AND W IND S HEDDING OF S OLUTIONS O BTAINED FROM UC WITH AND WITHOUT CAES U NDER D IFFERENT L EVELS OF W IND P OWER P ENETRATION . Penetr ation Linearized Mod el II UC without CAES Cost ($) Wind shed. (MW h ) Cost ($) Wind shed. (MW h ) 32% 698159 313.0 717711 705.8 35% 732886 823.9 771870 1578.3 3 8% 797294 1599 .0 845019 2591.6 C. Comparison Between the Linearized Model II and the Constant-temp erature Model To sho w the superiority of the pro posed model, the pressure and temper ature results obtained from Linearized Model II given in Section I II - C (constant-temperatur e model [1 6] given in Section I II -D) ar e p lotted in Figs. 6a and 6c (Figs. 6b an d 6d) , respectively. Furthermore , the charging/dischargin g power obtained from the bi -linear model (constant-temp erature model) is used by the accurate model [19] to calculate the pressure and temperature , which are also plotted in Figs. 6a and 6c (Figs. 6b and 6d) . That is, the accurate model is used to v erify the accuracy of t he bi -linear and the constant -temperature models. Figs. 6a an d 6c show th at the pr essure/temperature obtained from Linearized Model II and the analy tical mod el of CAES [19] are quite close to one another . Note that only scen ario 2 is shown in Fig. 6; scenarios 1 and 3 are similar but not sho wn as the space of the paper is limited. The averag e relative er rors between the pressure (tem per ature) o btained b y t he two mod els are 0.27, 0.28, and 0.28% (0.27, 0.27, and 0.28%) for scenarios 1, 2, and 3, respectively . That is, the bi -linear model is accur ate. Therefore, the time interval of the caver n model for CAES can be set to 2 0 minutes an d there is no nee d to decrease th is tim e interval to further in crease accuracy at the expense of a higher computation al burden. Fig s . 6b and 6 d clearly show th at the p ressure and temperature obtained from the constant -temperature mod el are inaccurate. The average relative er rors between the pr essure (temperatur e) obtained by the two mo dels are 1.55 , 1.49, and 1.39% (1.55, 1 .49, and 1.39%) for scenarios 1, 2, and 3 , respectively, which are about 5 -5.5 times the errors of Linearized Mo del II . Even worse, the pressure ob tained f rom the accurate model goes below the lower bound (46 bar ) o f its operating ran ge, i.e., the solu tion obtained from th e constant - temperature model actually allows the cavern of C AES plant to operate outside th e optimal pressure region (i.e., 46 -66 bar). However, the solution obtained from the bi -linear mo del en sures the cavern of CAES plan t o perates within the optimal pressure region. Therefore, it is necessary to use the proposed Linearized Model II to obtain an accu rate and feasible solution . Fig. 6 . Temperature res ult obtained from a) Linearized M odel II and the analytical model [1 9] , and b) the c onstant-temperature model a nd the analytical model [19] ; Pressure res ult obtained from c) Lineariz ed Mo del I I and t he analytical model [ 19 ] , and d) the constant-temper ature mode l and the ana lytical model [ 19 ] . D. Comp arison Between Linea rized Model II and Linearized Model I To show the effectiveness of the proposed Linearized Model II and the method to obtain an initial solution, the time consumed to solve the linearized models direc tly or using the method to obtain an initial solu tion as d escribed in Section III- E is tabulated in Tab le II (w h ere ‘ ---- ’ indicates ‘it does not converge after running for 7 days’). In Table II ,   and   represent th e time consum ed to solv e Linearized Mod el II directly and to obtain initial solution , respec tively;   a nd      re present the time consumed in step 6 to solve Linearized Mo del I and Linearized Model II, re spectively. Note that the final so lution of Linearized Model II given in previous subsection s is ob tained in step 6, as described in Section III-E. The initial s olu tion does not affect the optimality of th e final solution o f the linearized whole model as the optimality is determined by the termin ation cond ition of the MILP solver, i.e ., the relative MIP gap go es below 0.1%. The 2 nd colu mn of Table II s hows that solv ing Linearized Model II directly is fast when the nu mber of hour s is small but intractable as the number of hours increa ses. Note that it is m ore difficult to solve the model as the numb er of hours increases . The 3 rd and 4 th columns of Table II show that t he initial solution can be o btained in a relatively short time and th at it is a near - optimal solution with an optimality gap of aro und 3%. The last two columns of Table II show that solving Linearized Model II is much easier than Linearized Model I, especially when t h e number of ho urs is large , which indicate s the effectiveness of the proposed three strategies . Comparing the 6 th and the 2 nd columns indicate s that the initial solution significantly reduces the solution time. Therefor e, the three strategies, linear ization, and the initial solution are quite effective and necessary, which 0 10 20 30 40 50 60 70 75 1000 1500 2000 2500 3000 3500 Time p erio d (20 mins.) Power (M W) Load Con v . ou t put s ce. 1 Con v . ou t put s ce. 2 Con v . ou t put s ce. 3 0 10 20 30 40 50 60 70 75 50 200 400 600 800 1,000 1,200 1,400 Time p erio d (20 mins.) Power (M W) W P c apa. sc e. 1 W P c apa. sc e. 2 W P c apa. sc e. 3 W P output s c e. 1 W P output s c e. 2 W P output s c e. 3 > REPLACE THIS LINE WI TH YOUR PAPER IDENT IFICATION NUM BER (DOUBLE - CLICK HERE TO EDIT) < 8 helps to solve the whole model effectiv ely by converting it into Linearized Mo del II. TABLE II T IME C ONSUMED TO S OLVE THE L INEARIZED M ODELS D IRECTLY OR U SING AN I NITIAL S OLUTION . No. of hours     Gap of initial s o lu.          3 22 s 2.9 s 3.00% 58 s 11 s 5 37673 s 5.7 s 3.10% 27248 s 46 s 24 ---- 319 s 2.87% ---- 1565 s V. C ONCLUSION A UC model considering CAES has b een proposed in this paper using th e accurate b i-linear model propo sed in th e first paper o f this two -part series. Simulation results show that the bi -linear cavern model is more accu rate and avoids violating the optimal operating range of the air pressure in the ca vern compared to the constant -temperature cavern model . Therefore, it is necessary and benef icial to use the acc urate bi -linear cavern model. However, the bi -linear terms complicate th e who le model. To address this issue, three strategies have been p roposed to reduce the number of bi -linear terms in the whole model. Thereafter, the McCormick relax ation and piecewise linearization are used to linearize the binary -continuous and continuous bi -linear terms, respectively . Moreover, a m ethod to generate an initial solution, b ased on the solution of the UC with CAES using a constant-tem perature cavern model, for the whole model has been propo sed. Simulation results show that t he th ree strategies redu ce the complexity of the whole mod el and, hence, significantly red uce the solu tion time required to solve the linearized whole model; the initial solution also su bstantially redu ces the solution time; the whole mod el can be effectively solv ed by using t he three strategies, linear ization, and the initial solution . Simulation results also show that integr ating CAES in th e UC p roblem reduces wind shedding, to tal cost, and the number o f times conventional generators are turned o n and off. The ben efit of CAES increases as th e penetration of wind p ower increases . A PPENDIX               ,               ,    󰇛    󰇜    󰇛    󰇜   󰇛     󰇜 ,              ,       󰇛    󰇜󰇛    󰇜   󰇛    󰇜󰇛    󰇜         ,       󰇛    󰇜   󰇛   󰇜     󰇛    󰇜        ,          ,      󰇛  󰇜     ,              󰇛    󰇜       ,             ,    󰇛     󰇛   󰇜   󰇜󰇛    󰇜 ,           ,             ,    󰇛     󰇜      . R EFERENCES [ 1] J. Wang, M. Shah idehpour, and Z. Li, “Secur ity -constrained unit commitment with volatile win d power generation” , IEEE Trans. P ower Syst. , vol. 23, no. 3, pp. 1319-1327, 2008. [ 2] F. Ami nifar, M. Fo tuhi- Firuzabad, and M . Shahidehpour, “Unit commitment w ith probabilistic s pinning reserve and interrupt ible load considerations” , IEEE Trans. Power Syst. , vol. 24, no. 1, pp. 388-397, 2009. [ 3] Y. Wen, C. Guo, H. Pandžić, and D.S. Kirschen, “Enhanced secur ity - constrained unit commitment with emerging utility -scale energy storage”, IEEE Trans. Pow er Syst. , vol. 31, no. 1, pp. 6 52 -662, 2016 . [ 4] H. Pandžić, Y. Dvorkin, T. Qiu, Y. Wang, and D.S. Kirschen, “Toward cost- efficient and reliable unit commitment under uncertainty”, IEEE Trans. Power Sys t. , vol. 31, no. 2, pp. 970-98 2 , 2016. [ 5] N.P. Padh y, “U nit commitment - a bibliographic al surv ey”, IEEE Trans. Power Syst. , vol. 1 9, no. 2, pp. 1196-1205, 20 04. [6] Q.P. Zheng, J. W ang, and A.L. Liu , “Sto chastic o ptimization for unit commitment - a review”, IEEE Trans. Power Syst. , vol. 30, no. 4, pp. 1913 - 1924, 2015. [ 7] R. Jiang, J . Wang, and Y. Guan, “Ro bust unit commitment wi th win d power and pumped storage hydro”, IEEE Trans. Power Syst. , vol. 27, no. 2, pp. 800-810, 20 12. [ 8] K. Bruninx, Y. Dv orkin, E. Delarue, H. Pandžić, W .D. Haes eleer, and D.S. Kirschen, “Coupling p umped hydro e nergy s to r age with unit commitme nt”, IEEE Trans. Sus tainable Energy , vol. 7, no. 2, pp. 786-796, 2016. [ 9] Y. Wen, W. Li, G. Hua ng, and X. Liu, “Freq uency dynamics constrai n ed unit commitment with battery ener gy storage”, IEEE Tra ns. Power S yst. , vol. 31, no. 6, pp. 5115 -5 125, 2016. [10] B. Cleary, A. Duffy, A . O ’ Connor, M. Conlon, and V. Fthenakis, “Assessing the econ o mic benefits of compres sed air en ergy storage for mitigating wind curtailment”, IEEE Trans. Sust ainable Energy , vol. 6, no. 3, pp. 1021-1028, 2015. [11] B. M. Enis, P. Lieberman, and I. Rubin, “Operation of hybrid wind-tur bine compressed-air system for c onnection to electric grid n etwor ks and cogeneration”, Wind Engineering , vol. 27, no. 6 , pp. 449-459, 2003. [12] H . Kha ni , M.R.D. Zadeh, and A.H. Hajimiragha, “Transmission congestion relief using privat ely owne d large-scale energy s torage syst ems in a competitive ele ctricity market”, IEEE Trans. Power Syst . , vol. 31, no. 2, pp. 1449-1458, 2016. [13] S. Shafiee, H. Zareipour, A.M. Knight, N. Amjady, and B. Mohammadi- Ivatloo, “Risk -constrained bidding and offering strategy for a merc hant compressed air energy storage plant”, IEEE Trans. Power Syst. , vol. 32, no. 2, pp. 946-957, 20 17. [14] S. Shafie e, H. Zareip ou r, an d A. Knight, “Cons idering thermodynamic characteristics of a CAES facility in self-schedulin g i n energy and reserve markets”, IEEE Tr an s. Smart Grid , Early access . [15] D . Pozo, J. Contrer as, and E.E. Sauma, “Unit commitment with ideal and generic energy s torage units”, IEEE Trans. Po wer Syst. , vol. 29, no. 6, pp. 2974-2984, 2014. [16] H . Daneshi, and A.K. Srivastava, “Security -constrained unit commitment with wind generation and compressed air energy storage”, IET Gener. Transm. Distrib. , vol. 6, no. 2, pp. 167-175, 2012. [17] H . Bitaraf, H. Zhong, and S. Rahman, “ Managing large scale energy storage units to mitigate high wind penetration challenges , ” in IEEE PES General Meeti ng , Denver, CO, USA, pp. 1-5, Jul. 20 15. [18] J. Zhang, K.J. Li, M. Wang, W.J. Lee, and H. Gao, “ A bi-level program for the planning of an islanded microgrid including CAES, ” i n I EEE I nd. Ap pl. Society Annual Meeting , Addison, TX, USA , pp. 1-8, Oct. 2015. [19] C. Xi a, Y. Zhou, S. Zhou, P. Zhang, and F. Wang, “A s implified and unified analytical solution for temperature and pressure variations in compressed air energy st orage c averns”, Renewable Energy , vol. 74 , pp. 718-726, 2015. [20] G ei ßler, B., Martin, A., Morsi, A., Schewe, L. Usin g piecewise linear functions for solvin g MINLPs. In: Mixe d Integer Nonlinear Pr ogramming. Springer, pp. 287 – 31 4, 2012. [21] E. Klotz, and A.M. Newman, “Practical guidelines for solving d ifficult mixed integer linear programs”, Surveys in Operations Res. and Management Scie nce , vol. 18, no. 1, pp. 1 8 -32, 2013. [2 2]C.E. Murillo-Sánchez, R.D. Zimmerman, C.L. A nderson, a nd R.J. Thomas, “Secure planning and operations of systems wit h stoch astic s ources, energy storage, and active demand”, IEEE Trans. Smart Grid , vol. 4, no. 4 , pp. 2220-2229, 2013. [2 3] C.Y. Chung, H. Yu , and K .P. Wong, “An advanced quantum -inspired evolutionary algorithm for unit commitment”, IEEE Trans. Power S yst. , vol. 26, no. 2, pp. 847-854, 2011. [2 4] T. Das, V. Krishnan, Y. Gu, and J. D. McCalley, “Compressed air energy storage: state space mode ling and performance a nalysis,” in IEEE PES General Meeti ng, Detroit , USA, pp.1-8 , Jul. 2011. [25] C.M. Correa-Posadaa, and P. Sanchez- M artin, “G as netw ork o ptimization: a comparis o n of piecew ise linear m odels,” Chemical E ngineering Scien ce . pp.1- 24 , Jun. 2014.