The application of Imperialist Competitive Algorithm to the combined heat and power economic dispatch problem

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1 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 45 The applcaton of Imperalst Compettve Algorthm to the combned heat and power economc dspatch problem H.NORIANFAR 1 AND H.ABDI 2 1,2 Electrcal Engneerng Department, Engneerng Faculty, Raz Unversty, Kermanshah, Iran * Correspondng author:hamdabd@raz.ac.r Manuscrpt receved 20 Aprl, 2018; September 20 July, 2018, accepted 2 October,2018. Paper no. JEMT As the penetraton of multcarrer energy systems n power grd ncreases, the economcal optmzaton analyss are becomng of ncreasng mportance more and more. One of the most challengng ssues n ths context s optmzng the sophstcated combned carrer systems n power system operaton feld, especally solvng the combned heat and power economc dspatch (CHPED) problems. Ths paper presents a soluton for the sophstcated and non-convex CHPED problems applyng the mperalst compettve algorthm (ICA). The dea of the ntroduced algorthm s derved from the socal and poltcal development of human socetes. Dfferent study cases by takng the effects of valve-pont loadng effect (VPLE) and transmsson ohmc losses have been smulated to confrm the effcency of the suggested algorthm. The obtaned results from the ICA-based CHPED problem are compared wth those obtaned by varous algorthms to prove the performance of the proposed algorthm n searchng the optmal soluton. As the results confrm, the ICA has superorty on CPSO, TVAC-PSO, RCGA, RCO, BCO, EP, DE, EMA, CSO, RCGA-IMM and GSA, whch were prevously proposed n ths doman for solvng the descrbed problem Journal of Energy Management and Technology keywords: ICA, combned heat and power economc dspatch, heurstc algorthm, electrcal losses, value-pont effect INTRODUCTION Ths paper ntroduces the applcaton of ICA to CHPED problem, as one of the most mportant problems n the plannng and control ssues of ntegrated power and heat systems. For ths purpose, the defnton of CHPED problem and some relevant tasks are frstly descrbed. The prevous related works are then revewed and fnally, the man contrbutons of ths work are summarzed. A. CHPED Problem concept In recent years, the research focus on power systems has been changed, drastcally. Intally, the man ssues were related to the voltage mprovement and the power network expanson plannng. Although n many developng countres, these domans are stll of nterest, nonetheless, n ndustralzed countres, the most mportant areas of nterests are tacklng wth ncreasng the securty or relablty of the extended power systems. The power system securty s nfluenced by the greater dependence to the power grd, and access to more generaton s related to the nature of the system and deals wth macroeconomc ssues. Recently, several ssues of world-wde energy nfrastructures were evolved, and t s questoned whether these emergng systems can meet our future ncreasng needs for dfferent types of energy? Along wth composte energy transfer systems, many of the equpment wll be close to ther lfetme. In addton, challenges such as supplyng the contnuous growth of energy demand, dependence on fossl fuels, renewal of power system equpment s, and usng of clean and sustanable energy resources rase the need for new energy systems along wth some basc changes n exstng systems. As these changes are appled n power systems, revsng the tradtonal tools appled to power system operaton and plannng ssues are needed. One of the most challengng topcs n today s power system s the optmal operaton of power networks. Ths problem, whch s nomnated as economc load dspatch (ELD) searches the optmal power of generaton unt outputs, to supply the demanded load, satsfy the dfferent constrants, at the lowest cost. Despte the development of renewable energes, a large amount of energy s stll suppled by fossl fuels. The average effcency of fossl-fueled power plants n the Amerca s 33 %. So ncreasng fuel effcency s essental. One of the best solutons s the combnaton of heat and power.

2 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 46 CHP s technology whch generates power and captures the heat. Generally the heat may be wasted n order to provdng some knds of thermal energy. The steam or hot water are good examples n ths regard, whch can be appled to space heatng applances, coolng, and some ndustral applcatons. CHP whch s abundantly used as a soluton to ncrease the fuel effcency, has several advantages n compare to conventonal electrcty and thermal energy producton. Ths system retreves wasted heat and applyng t wll be resulted n ncreasng the effcences to upper than 70% [1] for power generaton and useful heat. There are some reports n ths regard whch nsst on achevng the effcences up to 90% [2, 3]. More effcency means burnng less fuel to produce each unt of energy. A CHP unt can reduce about 13-18% the envronment emssons of the generaton [3]. The power dspatch s not a new concept, and CHPED was solved n the thrd decade of the 19th century. However, t stll has some drawbacks, whch means t needs to be mproved and searched to gettng some optmal solutons. The purpose of CHPED problem s mnmzng the fuel cost n such way that all the equalty and non-equalty constrants are met for dfferent unts. The complex and sophstcated structure of the CHPED problem s such that t cannot be easly solved applyng some classcal methods such as lambda-teraton, nteror pont, and quas-newton methods. The performance of any suggested algorthm n ths feld should be compared wth the other well-known algorthms n terms of economc benefts. Ths comparson results n verfyng the algorthms performance n reachng the operatng pont at a lower cost. B. Lterature revew and challenges of CHPED problem The lterature revew shows that there are manly three factors n regard to the CHPED optmzaton problem, whch make t very nonlnear, non-smooth, and non-convex. These factors are descrbed hereafter: 1. VPLE [4] wll cause rpples to the fuel-cost curve. The fuel cost functon wll be smooth and defne as a quadratc curve n the absence of VPLE. It affects the nput-output characterstcs of generatng power unts. It also causes the non-lnear and non-smooth of the generaton unt output. 2. In CHP unts, the power generaton capacty as well as the heat generaton quantty depends on each other [1]. Ths nterdependence of generated power and heat n CHP unts wll restrct the operaton of CHP unts. Ths subject s at the heart of the most mportant research n the CHP operaton feld. 3. The power system losses caused by the realstc AC nature of electrcal networks must be suppled through the generaton unts. The electrcal losses as a man factor should be consdered n power balance equaton. Therefore, the optmzaton methods needed to search for the CHPED problem solutons consderng the ablty to fnd the global mnma. Otherwse they wll be trapped n some local optmum ponts. The lterature proposed dfferent algorthms and methods appled to ths optmzaton problem. VPLEs makes t possble to create more than one local mnma. The method based on gradent generally get trapped n the quas optmal or local extremsm ponts. The two-layer algorthm based Lagrangan relaxaton used n [5] and also, some nonlnear-based approaches such as quadratc and dual programmng [6] cannot to be used for these complex problems. For ths reason, the meta-heurstc algorthms have been addressed to mplement to the CHPED problem. References [2], and [7] employed harmony search algorthm (HSA) to the economc CHP problem wth constrants and regardless of the VPLE. An extended verson of partcle swarm optmzaton (PSO) method known as mproved PSO (IPSO) has been used n ref. [8] for solvng the CHPED consderng multple objectves n a multobjectve optmzaton manner. Also, to solve the mult-objectve problem concludng CHPED and emsson problems, ref. [9] ntroduced a two-stage approach by combnng mult-objectve optmzaton wth ntegrated decson makng. Also, the mentoned mult-objectve problem s solved by consderng the fuel cell (FC), boler, storage system, and heat buffer tank by Nazar- Hers et al. n ref. [10]. Ref. [11] suggested a hybrd algorthm known as IGA_MU to deal wth the mentoned problem. Also, the authors n ref. [12] suggested the CSO algorthm to tackle wth the CHPED n presence of a large amounts of local mnma ponts. Cuckoo search method [13] ntroduced by Mohamed Arezk Mellal for solvng the CHPED problem. In ths regard, an algorthm provdng a better soluton based on PSO algorthm, known as SPSO n reported n [14]. Hossen et al. n [15] proposed a mesh-based algorthm whch drectly search the best ponts n order to fnd the CHPED optmal pont. Rashed et al n [16] ntroduced GSA algorthm and Derafsh and et al successfully appled t to the CHPED problem [17]. In ths feld, the DE n [1], GSA n [17], TVAC-PSO n [18], and BCO n [19] are proposed to solve the mentoned problem concludng the VPLE ncorporatng wth electrcal transmsson ohmc losses. In order to solve the CH- PED problem by consderng prohbted zones of thermal power plants, the transmsson ohmc losses as well as the VPLE, Murugan et al. n [20] and pattanak et al n [21] presented hybrdzng BA and ABC wth Chaotc based Self-Adaptve (CSA) and modfed teachng-learnng-based optmzaton, respectvely. Furthermore, n order to mprove the optmzaton technque, reference [22] have proposed the embeddng CSO and Powell s pattern search method. Also reference [23] appled whale optmzaton algorthm (WOA) to ths problem. The authors reported the smulaton case study regardng to standard cases and proved the ablty and performance of the presented method. Reference [24] appled a meta-heurstc algorthm known as an mproved verson of socal cogntve optmzaton (SCO) algorthm wth tent map known as TSCO to the mentoned problem. The text s enhanced by presentng 2 case study results. Also ths problem s addressed n [25], n whch the frog leapng algorthm (LFA) s used to solvng that problem. The case study smulaton results on standard power systems are detaled by the authors. Furthermore, Ghorban n [26], ntroduced the applcaton of exchange market algorthm (EMA) to CHPED problem. He also detaled the results regardng applyng ths algorthm to fve well-known case studes, and compared the obtaned results wth other evolutonary algorthms. The crsscross optmzaton (CSO) algorthm was mplemented to CHPED by Meng et al. n [27] consderng two nteractng operators, namely horzontal and vertcal crossovers. They also appled the CSO on some standard CHPED problems and demonstrated the proposed algorthm features. The authors n [28] have proposed a real coded genetc algorthm wth mproved Mühlenben mutaton (RCGA-IMM) for solvng CHPED problem. They mplemented a Mühlenben mutaton on basc RCGA for speedng up the convergence and mprovng the optmzaton problem results. The detaled results obtaned by the proposed algorthm on

3 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 47 four standard case are reported and compared wth the other algorthms n ths area. Contrbuton of ths work Recently, the authors of reference [29] suggested a socal and poltcal-based algorthm whch models the man behavor known as ICA. Ths new algorthm has been employed to the several optmzaton problems such as Transmsson Expanson Plannng (TEP) matchng [30], DG plannng [31], optmal desgn of heat exchangers [32] and some electromagnetc problems [33]. All of the referred works clamed on good performance of the ICA. So, ths proposed approach can be a good opton for solvng the complex problem of CHPED. Ths paper deals wth applyng a new algorthm, ICA, to solve CHPED problem. The abovementoned algorthm has been adopted to varous problems n dfferent area of power system studes, such as plannng and operaton. As the authors know ths algorthm has not been reported before to be appled to CHPED problem. So, the novelty of ths paper manly comes from ts applcaton to the CHPED for the frst tme. Furthermore, for checkng the precson and performance of the suggested algorthm, dfferent test systems have been selected. Current manuscrpt s structured as follows: Secton 2 explans the CHPED problem formulaton. The ICA structure revews n secton 3. The detals regardng the proposed ICAbased CHPED problem are addressed n later secton. Secton 5, compares the obtaned smulaton results on dfferent standard cases by dfferent algorthms to verfy the effectveness of the presented method. Eventually, the man concluson of ths work are presented n secton MATHEMATICAL FORMULATION APPLIED TO CH- PED PROBLEM Bascally, the CHPED problem manly contans an objectve cost functon as well as some practcal constrants. Ths problem s completely non-convex and non-lnear n nature. So, the tradtonal and mathematcal methods may not be resulted to fndng optmal solutons. The general vew of ths problem can be easly found n the followng: A. Objectve cost functon The man goal of a CHPED problem s mnmzng the total fuel cost functons of power-only unts (POUs), heat-only unts (HOUs) and CHP unts. The objectve cost functon s concludng three terms regardng the POUs, HOUs, and CHP unts as follows: Mn N P C (P p ) + N c C j (Pj c, Hc j ) + N h C k (Hk h) (1) =1 j=1 k=1 where C (P p ) = α (P p )2 + β P p + γ C j (Pj c, Hc j ) = a j(pj c)2 + b j Pj c + c j + d j (Hj c)2 + e j Hj c + f jpj chc j C k (Hk h) = α k(hk h)2 + β k Hk h + γ k where C (P p ) wth = 1, 2,..., N p express the fuel cost of the th POU n $ for tme horzon of 1 h, and α,β and γ are the relevant cost parameters descrbng the cost functon. Also, the fuel cost of a CHP unt j s presented by C j (Pj c, Hc j ) wth j = 1, 2,..., N c n $/h n whch a j, b j, c j, d j, e j, and f j descrbng the relevant cost factors; C k (Hk h ) dentfes the relevant cost functon of kth HOU wth k = 1, 2,..., N h n $/h and α k, β k &γ k descrbe the relevant cost parameters; the number of POUs, CHP unts, and HOUs s demonstrated by N p, N c, and N h ; Hj c, Hh k, and Pp, Pc j are the heat generaton (n MWth) and unt power generaton (n MW), respectvely. Indexes h, k, c, j, and p,, descrbng the relevant ndexes for HOUs, CHP, and POUs. Consderng the mpact of VPLE, add a snusodal form secton to the objectve cost functon C (P p ). Consequently, the objectve cost functon addng the VPLE wll be as follows: C (P p ) = α (P p )2 + β P p + γ + λ sn(ρ (P p,mn P p )) (2) In whch the relevant cost coeffcents of the VPLE mpact are mentoned by λ and ρ. B. Equalty constrants Equaton (3) descrbes the power balance equaton by consderng the electrcal ohmc losses n transmsson system. Generally these losses are calculated applyng B matrx method. Ths approach known as Kron s loss formula [34]. In current work we used the mentoned method n order to electrcal loss calculatons of power system. Furthermore, the total heat generated by the CHP unts, and HOUs must be equal to the heat demand, whch s stated n equaton (4). As mportant note, the heat losses are not modeled n ths study. It should be mentoned that the B Matrx coeffcents for the standard cases are known and here we used the standard data n ths regard. N p P p N c N + P c p N p j = P d + =1 j=1 =1 l=1 P p B lp p l + N p N c =1 j=1 P p Pc j + N c N c Pj cb jmpm c j=1 m=1 (3) N c H c N j + h Hk h = H d (4) j=1 k=1 In whch H d s the total demanded heat and P d s the total demanded power of the combned power and heat network, respectvely. C. Inequalty constrants Because of some practcal, operatonal, and economcal ssues, the generator capactes are restrcted to some predefned power and heat lmts. These constrants come from the dfferent operaton lmtatons of generaton unts. These lmtatons for POUs, CHP unts, and HOUs are represented by the nequalty constrants (5)-(8). P p,mn P p P p,max = 1, 2,..., N p (5) P c,mn j (Hj c) Pc j P c,max j (Hj c) j = 1, 2,..., N c (6) H c,mn j (Pj c) Hc j H c,max j (Pj c) j = 1, 2,..., N c (7) H h,mn k In whch P p,mn H h k Hh,max k k = 1, 2,..., N h (8), H h,mn k and P p,max, H h,max k represent the relevant maxmum and mnmum power and heat generatons, respectvely. We should keep n mnd that n the CHP unts, the power and the heat generaton lmts depends on the heat and power producton, respectvely. Ths means that the correspondent lmts are affected by each other.

4 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue THE ICA STRUCTURE ICA derved from socal and poltcal behavor of mperalsm and colones. Generally, the ICA starts n an assumed populaton whch s known as the ntal countres. Ths algorthm belongs to meta-heurstc category and has been proposed to solve the complcated and non-lnear optmzaton problem. Such as the other meta-heurstc algorthms, ICA starts wth a feasble soluton and by evaluatng n each teraton the soluton wll be close to optmal soluton. The man steps of the proposed ICA are llustrated n Fg.??. Some descrpton regardng varous steps of ICA are detaled n followng sub-sectons. A. Step 1: Generatng Intal Empres At frst, countres are randomly assgned at a dstance between ther lower and upper bounds: x (0),j = x,mn + rand.(x,max x,mn ) (9) where x (0),j specfes the startng value of the th varable regardng to the jth country; x,mn and are the relevant mn-max permssve values for the th varable. Also, the rand s a random number n the dstance [0, 1] whch s produced applyng a random operator generator n MATLAB software [34]. The relevant power for each country s calculated accordng to the objectve cost functon (n the context of optmzaton s proportonal to the nverson of ts cost value, countres wth lower costs are more powerful). N mp whch descrbes the numbers of the strongest countres are chosen as mperalsts. The value of the N mp can be selected from 1 to N country. However, t s usually chosen to be 0.1 ncountry. The relevant colones regardng these mperalsts are formed by the rest of countres. N country denotes the total number of countres. Selectng a large amount for N country makes the results more accurate but wll be resulted n ncreasng the smulaton tme. There are dfferent ways to allocate countres to an mperalst. In one way, after the determnaton of mperalsts, countres can be dvded equally between them. The jth mperalst together wth ther colones form the jth Empre. In ths paper, all the countres are dvded among the mperalsts accdentally and a populaton of 100 countres ncludng 10 empres has been used. B. Step 2: movement of colones All the colones move toward ther mperalst. Ths dsplacement by addng a unformly random dstrbuted value n the dstance 0 and β d, to the value of ts varables s modeled as follows [29]: {x} new = {x} old + U (0, β d) (10) {x} new = {x} old + β d {rand} (11) In whch the β s a parameter whch s greater than one, and d s the mathematcal dstance between mperalst and the colony. For ncreasng the searchng space around the mperalst, a predefned random amount of devaton wll be consdered to be added to the movement drecton. θ s a random unform dstrbuton number as follow: θ = U ( γ, +γ) Where γ ndcates the range of the devaton from the drecton. In most of the applcatons, some predefned values about 2 for β [19] and approxmately 0.1 (Rad) for Γ, are proposed n order to avodng the trappng n local optmum and enhance the performance ablty of the proposed algorthm. In ths paper, an ntal country (X0) s taken to mprove and accelerate ICA convergence. Any arbtrary value for varables of ths country can be selected n order to fndng an optmal pont wth good convergence. It should be mentoned that the country s selected as follow: At frst, the mnmum value of varables s consdered, then they are changed n such a way that the constrants of equalty and nequalty are met. C. Step 3: The Imperalst and a Colony Exchangng Postons When colones moves toward the mperalst, the colony and the mperalst wll exchange ther poston, f a colony n ts updated poston s better than ts relevant mperalst. Ths subject s evaluated applyng the defned objectve cost functon. Accordngly, the rest colones move to the updated poston. D. Step 4: Total Power Calculaton for an Empre and Imperalst In mperalstc competton, some of the colones of the weakest empre are taken and all empres compete for possess these colones. In each competton, the empres wll have a lkelhood for takng possesson of the relevant colones based on ther total power. In ths paper, ths colony wll be gven to the most powerful empre. The correspondent power to an empre s largely the result of mperalst power. It should be noted that the power of colones also has lttle mpact n ths context. Thus, the total power of an empre, ncludng the power of mperalst and ts colones, s as follows: TC j = f (mp,j) cos t + ξ.mean( f col,j cos t ) (12) Where TC j s specfed as the total objectve cost of the jth Empre and ξ s assumed to be a postve number less than 1. It should be mentoned that the power of each country s proportonal to the nverson of ts cost value. Whatever the value of ξ s larger, the effect of power the colones wll be hgher. Whle choosng a small value for t causes the total power of an empre to be determned by just the mperalst. In most mplementatons, the value of 0.1 for ξ s a proper choce [29]. E. Step 5: Elmnaton of the Powerless Empres Durng the mperalstc competton, powerless empres lose ther colones. When an empre loses all of ts colones, t s collapses and elmnates. Under these condtons, other empres compete for possess ths empre as a colony. In ths paper the empre wll be gven to the most powerful empre. F. Step 6: Revoluton In ths step, the probablty of revoluton s compared wth a unformly random value that dstrbuted between 0 and 1. In ths paper, the probablty of revoluton s 0.5. If there has been possblty of revoluton, a random amount s added to one of the colony varables randomly. Otherwse, the colony wll not change. G. Step 7: Stoppng crteron The stoppng crteron can be ether reachng a sngle empre or reachng the maxmum teraton. All of the above mentoned steps contnue untl the teraton number reaches to a predefned

5 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 49 value of Iteraton max. As n mportant not, n ths paper, the maxmum number of teratons s supposed to be equal to APPLYING THE ICA TO THE PROBLEM The man steps of soluton the mentoned problem applyng the ICA algorthm as a meta-heurstc algorthm are as follows: The man steps of soluton the mentoned problem applyng the ICA algorthm as a meta-heurstc algorthm are as follows: Step 1. Introducng the relevant parameters of ICA such as N country, N mp, X0, β, ξ, γ, and Iteraton max. Step 2. Determnng the ntal poston of countres. In CH- PED problem, the heat and power generaton of dfferent unts concludng POUs, HOUs, and CHPs are the varables of each country whch determne ther poston. They can be stochastcally selected usng the equatons (13) (16) [18] to satsfy all relevant nequalty constrants: P p j = Pp,mn j + rand (P p,max j P p,mn j ) (13) j = 1, 2,... N p Pj c = Pc,mn j (.) + rand (P c,max j (.) P c,mn j (.)) (14) j = 1, 2,..., N c Hj c = Hc,mn j (Pj c ) + rand (Hc,max j (Pj c ) Hc,mn j (Pj c )) (15) j = 1, 2,..., N c Hj h = Hh,mn j + rand (H h,max j H h,mn j ) (16) j = 1, 2,.., N h Step 3. Calculatng the power of each country accordng to the objectve cost functon of the problem. In order to satsfy the equalty constrants, the method of penalty coeffcents has been used as follows: N p cos t = C (P p ) + N c C j (Pj c, Hc j ) + N h C k (Hk h)+ =1 ( j=1 k=1 ) Np w 1 abs P p + N c P c P d P loss + w 2 ( =1 j=1 ) Nc abs Hj c + N h Hk h H d j=1 k=1 (17) Where w1 and w2 are the penalty coeffcents. In ths paper, w1=50, w2 = 30 s a proper settng. Step 4. Defnton of mperalsts and empre formaton. Step 5. Movng the colones of each empre towards ther mperalst and exchange the poston of a colony and the mperalst f t s stronger. Step 6. If possble, the revoluton wll be carred out. Step 7. Updatng the f col,j (mp,j) cos t, and f cos t for each set of country. Then, calculate the total cost usng (12) for all empre. Fnally, the empres compete wth each other and f an empre wll be collapsed n case t loses all ts relevant colones. Step 8. Repeatng the steps 5-7 as long as the number of teratons reaches Iteratonmax. Otherwse, go to Step 9. Step 9. Introducng the varables of the strongest mperalst and the relevant cost as the best soluton. Table 1. Case study data used n ths work CASE STUDY DESCRIPTION # 1 Rosenbrock Functon n MATLAB software # 2 # 3 # 4 # 5 5 unts: one POU, 3 CHP unts, and one HOU, gnorng the VPLE and transmsson ohmc losses 7 unts: 4 POUs, 2 CHP unts, and one HOU, consderng transmsson ohmc losses based on B-Matrx and VPLE. 24 unts: 13 POUs, 6 CHP unts, and 5 HOUs, consderng the VPLE 48 unts: 26 POUs, 12 CHP unts, and 10 HOUs. 5. COMPARISON OF NUMERICAL RESULTS OF DIFFER- ENT ALGORITHMS Generally for testng a suggested metaheurstc algorthm performance, t should be mplemented on some standard case systems. In ths work we selected fve case studes as the test study systems for verfyng the performance and convergence of the suggested approach. It should be noted that these systems are generally has been used n the lterature revew. Furthermore the obtaned smulaton results are compared wth the other proposed algorthms n ths context whch appled to ths problem. In the frst Test system, a smple and sample functon n MAT- LAB software s chosen to show and llustrate the ablty of the algorthm. In other test cases, whch completely were reported before n the relevant references, n order to prove the effcency and ablty of the presented algorthm, the fndngs are detaled and the relevant results obtaned by dfferent algorthms were detaled n terms of objectve cost functon wth other algorthms. It should be mentoned that all smulatons are run 50 tmes. As an mportant note we should keep n the mnd that none of the non-equalty constrants are volated n these smulatons. Also, the generaton and load (plus the transmsson ohmc losses n cases that the electrcal losses are modeled) balance constrant can be easly checked by consderng the total generated power and heat reported n these tables n order to ensure the constrant satsfacton. The relevant data of the smulated case studes are depcted n Table 1. It should be mentoned that the detaled data can be easly fnd n ref. [18]. A. Test system 1 At the frst secton and n order to prove the proposed algorthm ablty, we consdered the rosenbrock, whch conssts of two varables. General functon of ths optmzaton problem s as: F rosenbrock (x) = 100(x 2 1 x 2) 2 + (x 1 1) 2 Where 10 x 1, x 2 10 The system conssts of 38 countres ncludng 5 mperalsts and 33 colones. Fg. 1 llustrates how the strongest mperalst forms as the fnal answer. In ths fgure, the mperalsts are

6 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 50 depcted n the form of stars where the colones are shown n the form of some ponts. As can be seen from the fgure, the powerless empres collapse wth ncreasng number of teratons. Fg. 1. The formaton of the most powerful empre as the ultmate answer for Test system 1. B. Test system 2 The smulaton results found by the ICA are compared wth the other addressed algorthms n ths feld. These algorthms (R1_C9) are concludng: HS [2], GA [2], EDHS [7], TVAC-PSO [18], EMA [6], RCGA-IMM [28], and CPSO [18]. It should be mentoned that dfferent load profles are consdered n ths regard, whch are demonstrated n table 2. The reported results show that the ICA algorthm s superor to other algorthms. Assumng the constant load level n one year, annual savngs resultng from applyng the proposed algorthm wll be equal to 36,792 and 213,130.8$ n comparson wth GSA [17] for load profles 2, and 3, respectvely. In the case of load profle 1, GSA [17] results n less cost, but equalty constrants are not met. Furthermore the convergence curve of the proposed algorthm under studed condtons s dsplayed n Fg. 2. Fg. 2. Convergence curve of the suggested algorthm regardng studed case 2. C. Test system 3 The respectve demanded heat and power n ths case are supposed to be equal to 150 MWth, and 600 MW, respectvely. Table 3 shows the smulaton results obtaned from applyng ICA algorthm. Also, the reported results regardng DE [1], PSO [1], EP [1], GSA [17], CPSO [18], TVAC-PSO [18], BCO [19], EMA [26], CSO [27], and RCGA [19] are detaled. It expresses that the proposed approach has the best results among the obtaned results from all algorthms. Also, assumng the constant demanded load for all the year, annual savngs resultng from applyng the ICA algorthm wll be equal to 148,657.2$ n compare to EMA [26]. Furthermore, the electrcal transmsson ohmc losses are the lowest value, as well. Fg. 3 demonstrates the convergence curve of the suggested algorthm. Fg. 3. Convergence curve of the suggested algorthm regardng case 3. D. Test system 4 Total demanded heat s supposed to be equal to 1250 MWth. Ths tem s assumed to be equal to 2350 MW for demanded power n ths case. A comparson between the results of ICA and the other proposed algorthms n ths feld such as GSA [17], TVAC- PSO [18], EMA [26], CSO [?], RCGA-IMM [28], and CPSO [18] are presented n Table 4. It shows that the smulaton results from the suggested algorthm are better than generally form the other ntroduced algorthms n ths work The annual cost savngs arsng n the proposed method compared to CPSO [18], TVAC- PSO [18], RCGA-IMM [28], CSO [27], and GSA [17] are , , , , and $, respectvely. Fg. 4, shows the convergence curve of ths case. At the last four rows of ths table, the mnmum, maxmum, mean, and standard devaton costs of dfferent algorthms applyng the CHPED problem are reported. It s clear that the ICA can be mentoned as a superor evolutonary algorthm n solvng the complex, non-lner and non-convex problems, such as CHPED. One of the most mportant features regardng the evolutonary algorthms, such as ICA, s due to selectng the algorthm parameters, or parameter settng. As t was demonstrated n the prevous sectons, dfferent references have proposed dfferent strateges for settng the algorthm parameters. Here, and n order to clarfy ths task, a bref senstvty analyss regardng the ICA parameters s reported. As a soluton, the senstvty of the obtaned results to dfferent values of the ICA parameters are studed. A summary of results are gven n Table 5. As t s shown n ths Table, the best parameter settng to obtan the optmal results n a reasonable tme s demonstrated n the thrd row of ths Table. It should be mentoned that we appled these preferred values as the parameter settng n ths paper. E. Test system 5 The total demanded power s assumed to be 4700 MW and total demanded heat s supposed to be equal to 2500 MWth [17]. Table 6 descrbes the results obtaned from the proposed method, CPSO [18], TVAC-PSO [18], CSO [27], and GSA [17]. The annual cost that the proposed method can save compared to CPSO, TVAC-PSO, CSO [27], and GSA algorthms s , , , and $, respectvely. Fg. 6 ndcates the convergence characterstc curve of the suggested

7 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 51 Table 2. Obtaned numercal results for case 2 Method P1 P2 P3 P4 H2 H3 H4 H5 Total Heat Total Power Total cost GA\cte{2} HS\cte{2} EDHS\cte{7} * CPSO\cte{18} TVAC-PSO\cte{18} RCGA-IMM\cte{28} * EMA\cte{26} GSA\cte{17} ICA Power = 300 MW Load profle number 1 Heat = 150 MWth GA\cte{2} HS\cte{2} EDHS\cte{7} * CPSO\cte{18} TVAC-PSO\cte{18} RCGA-IMM\cte{28} * EMA\cte{26} GSA\cte{17} ICA Power = 250 MW Load profle number 2 Heat = 175 MWth GA\cte{2} HS\cte{2} EDHS\cte{7} * CPSO\cte{18} TVAC-PSO\cte{18} RCGA-IMM\cte{28} EMA\cte{26} GSA\cte{16} ICA Power = 160 MW Load profle number 1 Heat = 220 MWth nvald

8 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 52 Table 3. Obtaned numercal results for case 3. Generaton EP [1] PSO [1] DE [1] GSA [17] CPSO [18] TVAC-PSO [18] CSO [27] EMA [26] BCO [19] RCGA [19] Proposed ICA p p p p p p h h h ploss Total cost Total Power Total Heat Invald Ths paper focused on solvng the CHPED problem by proposng a new approach known as ICA. As the lterature confrms, ths s the frst tme that the ICA has been adopted to the mentoned problem. The presented approach takes nto account the VPLE as well as electrcal transmsson ohmc losses. Its performance and effectveness were verfed by applyng to fve test case and comparng ts numercal results wth other algorthms showed the ablty of the mentoned algorthm to fnd the optmal soluton. As t was detaled n the smulaton case studes, ICA has the ablty n fndng the optmal operaton ponts n comparng the other well-known algorthms whch were suggested before n ths feld. In the future plan, completng the CHPED problem concludng the other real-world condtons such as the prohbted zones and ramp rate lmts should mentoned n the problem formulaton. Furthermore, applyng the proposed algorthm to solve some mult-objectve sophstcated CHPED problems should be studed n detal. REFERENCES 1. M. Basu, CHPED by usng DE, Electr. Power Comp. Syst. 38 (2010) A. Vaseb, M. Fesanghary, S. Bathaee, CHPED by HAS, Int. J. Electr. Power Energy Syst. 29 (2007) S. Kark, M. Kulkarn, M.D. Mann, H. Salehfar, Effcency mprovements through CHP for on-ste dstrbuted generaton technologes, Cogener. Dstrb. Gener. J. 22 (2007) Fg. 4. Convergence curve of the suggested algorthm regardng case system 4 algorthm. 6. CONCLUSION 4. G.L. Decker, A.D. Brooks, Valve pont loadng of turbnes, Trans. Am. Inst. Electr. Eng. Power Appar. Syst. Part III 77 (1958) Guo T, Henwood MI, van Oojen M. An algorthm for CH- PED. IEEE Trans Power Syst 1996; 11(4): Roojers FJ, van Amerongen RAM. Statc economc dspatch for co-generaton systems. IEEE Trans Power Syst 1994; 9(3): E. Khorram, M. Jaberpour, HAS for solvng CHPED problems, Energy Convers. Manage. 52(2011) Wang L, Sngh C. Stochastc CHPED based on multobjectve PSO. Electr Power Energy Syst 2008; 30: L, Yang, Jnlong Wang, Dongbo Zhao, Guoqng L, and Chen Chen. "A Two-Stage Approach for Combned Heat and Power Economc Emsson Dspatch: Combnng Mult- Objectve Optmzaton wth Integrated Decson Makng." Energy (2018).

9 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 53 Table 4. Obtaned numercal results regardng case 4. Generaton GSA [17] TVAC-PSO [18] CPSO [18] EMA 26 RCGA-IMM [28] CSO [27] Proposed Algorthm (ICA) P P P P P P P P P P P P P P P P P P P H H H H H H H H H H H Mn cost Mean cost NA Max cost NA standard devaton NA NA NA NA NA Nazar-Hers, Morteza, Saeed Abapour, and Behnam Mohammad-Ivatloo. "Optmal economc dspatch of FC- CHP based heat and power mcro-grds." Appled Thermal Engneerng 114 (2017): C.T. Su, C.L. Chang, An ncorporated algorthm for CH- PED, Electr. Power Syst. Res. 69 (2004) A. Meng, P. Me, H. Yn, X. Peng, and Z. Guo. Crsscross op-

10 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 54 Table 5. Analyss the senstvty of the results to parameters of algorthms for case 4. State Iteratonmax N country N mp β ξ Mnmum Average cost cost Tme(s) tmzaton algorthm for solvng CHPED problem, Energy Converson and Management (2015) Mellal MA, Wllams EJ. Cuckoo optmzaton algorthm wth penalty functon for CHPED problem. Energy 2015; 93: V. Ramesh, T. Jayabaratch, N. Shrvastava, A. Baska, A novel selectve PSO approach for CHPED, Electr. Power Comp. Syst. 37 (2009) S.S. Sadat-Hossen, A. Jafarnejad, A.H. Behrooz, A.H. Gandom, CHPED by mesh adaptve drect search algorthm, Expert Syst. Appl. 38 (2011) E. Rashed, H. Nezamabad-pour, S. Saryazd, GSA: a gravtatonal search algorthm, Inform. Sc. (N.Y.) 179 (2011) S. D. Begvand, H. Abd, M. La Scala, CHPED problem usng gravtatonal search algorthm, Elect. Power Syst. Res., vol. 133, pp , Apr B. Mohammad-Ivatloo, M. Morad-Dalvand, A. Rabee, CHPED problem soluton usng PSO wth tme varyng acceleraton coeffcents, Electr. Power Syst. Res. 95 (2013) M. Basu, Bee colony optmzaton for combned heat and power economc dspatch, Expert Syst. Appl. 38 (2011) Murugan, R., M. R. Mohan, C. Chrstober Asr Rajan, P. Deva Sundar, and S. Arunachalam. "Hybrdzng Bat Algorthm wth Artfcal Bee Colony for Combned Heat and Power Economc Dspatch." Appled Soft Computng (2018). 21. Pattanak, Jagat Kshore, Mousum Basu, and Deba Prasad Dash. "Modfed Teachng-Learnng-Based Optmzaton for Combned Heat and Power Economc Dspatch." Internatonal Journal of Emergng Electrc Power Systems 18, no. 5 (2017). 22. Narang, Ntn, Era Sharma, and J. S. Dhllon. "Combned heat and power economc dspatch usng ntegrated cvlzed swarm optmzaton and Powell s pattern search method." Appled Soft Computng 52 (2017): Nazar-Hers, M., Mehdnejad, M., Mohammad-Ivatloo, B., & Babamalek-Gharehpetan, G. (2017). CHPED problem soluton by mplementaton of whale optmzaton method. Neural Computng and Applcatons, 1-16.

11 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue 3 55 Table 6. Obtaned numercal results for case 5. GENERATION CPSO18 TVAC-PSO18 GSA17 CSO27 PROPOSED ICA GENERATION CPSO18 TVAC-PSO18 GSA17 CSO27 PROPOSED ICA P P P P P P P P P P P P P P P P P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H P H Algorthm CPSO [18] TVAC-PSO [18] GSA [17] CSO [27] Proposed ICA Total Cost Sun, J., & L, Y. (2018). Socal cogntve optmzaton wth tent map for CHPED. Internatonal Transactons on Electrcal Energy Systems, e Rabee, Abbas, and Mohammad Morad. (2018)"CHP System Operaton Cost Mnmzaton usng Frog Leapng Based Intellgent Search Algorthm." Journal of Energy Management and Technology 26. Ghorban, Naser. "Combned heat and power economc dspatch usng exchange market algorthm." Internatonal Journal of Electrcal Power & Energy Systems 82 (2016): Meng, Anbo, Peng Me, Hao Yn, Xangang Peng, and Zhuangzh Guo. "Crsscross optmzaton algorthm for solvng combned heat and power economc dspatch problem." Energy Converson and Management 105 (2015): Haghrah, A., M. Nazar-Hers, and B. Mohammad-Ivatloo. "Solvng combned heat and power economc dspatch problem usng real coded genetc algorthm wth mproved Mühlenben mutaton." Appled Thermal Engneerng 99 (2016): E. Atashpaz_gargar, c. Lucas, Imperalst Compettve Algorthm: an algorthm for optmzaton nspred by mperalstc competton, (ICA) IEEE CEC Morad, M., Abd, H., Lumbreras, S., Ramos, A., Karm, S. (2016). TEP n the presence of wnd farms wth a mxed AC and DC power flow model usng an Imperalst Compettve Algorthm. Electrc Power Systems Research, 140, Nknam T, Fard ET, Pourjafaran N, Rousta A. An effcent hybrd algorthm based on modfed mperalst compettve algorthm and k-means for data clusterng. Engneerng Applcatons of Artfcal Intellgence 2011; 24(2): 306e Yousef M, Darus A, Mohammad H. An mperalst compettve algorthm for optmal desgn of plate-fn heat exchangers. Internatonal Journal of Heat and Mass Transfer 2012; 55(11):3178e85.

12 Research Artcle Journal of Energy Management and Technology (JEMT) Vol. 2, Issue Coelho L, Afonso L, Alotto P. A modfed mperalst compettve algorthm for optmzaton n electromagnetcs. IEEE Transactons on Magnetcs 2012; 48(2): 579e T. Vctore, A. Jeyakumar, Reserve constraned dynamc dspatch of unts wth VPLE, IEEE Trans. Power Syst. 20 (2005)