Social cognitive optimization with tent map for combined heat and power economic dispatch Jiaze Sun a,b *, Yang Li c

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1 Socal cogntve optmzaton wth tent map for combned heat and power economc dspatch Jaze Sun a,b *, Yang L c a School of Computer Scence and Technology, X an Unversty of Posts and Telecommuncatons, X an , Chna b Shaanx Key Laboratory of Network Data Intellgent Processng, X an Unversty of Posts and Telecommuncatons, X an , Chna c School of Electrcal Engneerng, Northeast Electrc Power Unversty, Jln , Chna ABSTRACT: Combned Heat and Power Economc Dspatch (CHPED) problem s a sophstcated constraned nonlnear optmzaton problem n a heat and power producton system for assgnng heat and power producton to mnmze the producton costs. To address ths challengng problem, a novel Socal Cogntve Optmzaton algorthm wth Tent map (TSCO) s presented for solvng the CHPED problem. To handle the equalty constrants n heat and power balance constrants, Adaptve Constrants Relaxng (ACR) rule s adopted n constrant processng. The novelty of our work les n the ntroducton of a new powerful TSCO algorthm to solve the CHPED ssue. The effectveness and superorty of the presented algorthm s valdated by carryng out two typcal CHPED cases. The numercal results show that the proposed approach has better convergence speed and soluton qualty than all other exstng state-of-the-art algorthms. Keywords: socal cogntve optmzaton algorthm; combned heat and power economc dspatch; tent map; adaptve constrants relaxng 1. Introducton Despte the reducng fossl fuels and the ncreasng need for electrc energy, the effcency of the converson of fossl fuels nto electrcty s stll less than 60%. Plenty of thermal energy as wasted heat s drectly released nto the envronment, whch leads to severe envronmental polluton problems. To make full use of wasted heat, Combned Heat and Power (CHP) unts are ntroduced to smultaneously produce electrcty and heat energy n the power producton by conventonal thermal plants. CHP unts can generate converson effcency above than 90% through usng wasted heat n thermal plant [1], so the cogeneraton unts, power unts, and heat unts are combned to satsfy power and heat demands n CHP plants. To acheve the comprehensve utlzaton of CHP unts, Combned Heat and Power Economc Dspatch (CHPED) optmzaton s mplemented for obtanng an optmal power and heat generaton schedulng, whch mnmzes the overall cost of supplyng power and heat demand whle satsfyng system all nequalty and equalty constrants. The CHPED optmzaton problem s a multmodal, non-convex and constraned nonlnear programmng problems [2]. Consderng couplng of power and heat, the valve pont effects and transmsson losses between multple plants, the sophstcaton of the CHPED problem ncreases further [3]. The lteratures [1, 3, 4] show that varous optmzaton methods have been ntroduced to solve the CHPED problem. Several tradtonal optmzaton approaches, such as gradent descent approach, nonlnear optmzaton method, have been ntroduced to work out the CHPED problems, but the tradtonal determnstc approaches are very dffcult to precsely obtan the optmal soluton because the model of the sophstcated CHPED problem s always non-smooth wth many complex equalty and nequalty constrants. Moreover, ths constraned nonlnear global optmzaton problem has proven to be non-determnstc polynomal-tme hard (NP-hard) [1], up to now there s stll no effectve determnstc method to solve them n practce. When solvng the multmodal, non-convex and non-dfferentable optmzaton problem, Swarm Intellgence (SI) optmzaton algorthms have been dentfed to have effectve performance. Recent research demonstrates that the NP-hard problems can be approxmately settled by metaheurstc or heurstc algorthms [4, 6, 7]. Many effectve meta-heurstc or heurstc algorthms are mplemented to search the optmal soluton for the CHPED optmzaton problem [1]. The metaheurstc algorthms appled n CHPED optmzaton problem from the lteratures nclude

2 Genetc Algorthm (GA) [8-9], Partcle Swarm Optmzaton (PSO) algorthm [10-11], Ant Colony Optmzaton (ACO)algorthm [12], Bee Colony Optmzaton (BCO) [13], Cuckoo Search (CS)algorthm [14], Grey Wolf Optmzaton (GWO) algorthm [15], Artfcal Immune System (AIS) algorthm [16], Frefly Algorthm (FA) [17], Harmony Search(HS) [18], Dfferental Evoluton (DE) [19], Fsh School Search (FSS) [20], Invasve Weed Optmzaton(IWO) algorthm [21], Group Search Optmzaton(GSO) [22] and Teachng Learnng Based Optmzaton (TLBO) [23]. Although the above swarm ntellgence optmzaton algorthms have had acceptable results to solve the CHPED optmzaton problem, the global optmzaton soluton and ts global convergence are usually dffcult to be guaranteed. Moreover, the other notable meta-heurstc algorthms have not been explored. It s known that human has better socal ntellgence and hgher ftness than the other swarm. Vtalzed by the human cogntve process, Socal Cogntve Optmzaton (SCO) algorthm [24] was devsed as a heurstc swarm algorthm. SCO algorthm ncludes many learnng agents, whch have symbolzng ablty and vcarous ablty. Snce SCO algorthm makes optmum use of the overall socal knowledge, t has better optmal ablty than many other well-known optmzaton algorthms n many felds [25], such as GA,ACO and PSO. To obtan a powerful, stable and global SCO algorthm for the CHPED problem, we combne the socal cogntve optmzaton algorthm wth Tent map and Adaptve Constrants Relaxng (ACR) for addressng the challengng CHPED problem. The man contrbutons of the paper are as follows: (1) A novel socal cogntve optmzaton algorthm based on Tent map and Adaptve Constrants Relaxng s proposed for the CHPED problem n the heat and power producton system. (2) Integratng Tent map nto SCO ntalzaton and neghborhood searchng mproves unform dstrbuton and ncreases the ergodcty of the SCO wth less computaton. Furthermore, ACR rule s adopted n constrant processng to effectvely handle the equaton constrants n heat and power balance constrants. (3) The numercal expermental results that assess the effectveness and convergence of the proposed novel socal cogntve optmzaton wth two benchmarks testng proects are compared wth the other classc heurstc algorthm for the CHPED problem. The remander of ths paper s structured as follows. The CHPED formulaton s detaled n a constrant non-lnear program model n Secton 2. The specfc descrpton of the novel SCO wth tent map for the CHPED problem s provded n secton 3. In secton 4 two classc test cases are exercsed to assess the effectveness and superorty of the novel SCO. In Secton 5 the conclusons are shown. 2. Optmzaton formulaton of CHPED Problem A CHP system ncludes power unts, cogeneraton unts, and heat unts. Mathematcally speakng, the CHPED problem can be formulated as constraned Nonlnear Programmng Problems (NLPs) amng to mnmze total cost of supplyng power and heat demand. 2.1 cost functon In practce, when steam admsson valve starts to open, an ntense fuel loss rasng wll enhance the fuel cost because of the wre drawng outcomes, whch results n a non-smooth and non-convex fuel cost functon. If the valve-pont effects for power unts are neglected, the fuel cost functon s closely a quadratc functon, whch would lead an naccuracy to the CHPED problem. Consequently, the obectve functon of the practcal CHPED problem wth the consderaton of the valve-pont effect s a superposton of quadratc and snusodal functons, whch would ncrease the non-smooth and non-convex characterstc [4]. The cost functon of power unt consderng the valve-pont effect can be represented as follows: 2 mn cost ( p ) ( p ) p P + sn( ( p P a P b p c d e P P )) (1)

3 In whch, p cost ( P ) s the cost of power unt, p P s the power generated by the power unt, a, b, and c are the cost parameters of power unt, d and e are the cost parameters of the valve-pont effect. P mn P s the lower power boundares of the power unt n MW. Addtonally, the cost functon of cogeneraton unt s stated as follows: cost ( P, H ) a ( P ) b P c d ( H ) e H f H P (2) c c c 2 c c 2 c c c c c where cost ( P, H ) s the cost functon of cogeneraton unt, and a, b, c, d, e and f are the cost parameters of cogeneraton unt. c P s the power generated by the cogeneraton unt, c H s the heat generated by the cogeneraton unt. Moreover, the double-dp functon can model the cost of heat unt, whch s stated as follows: cost ( H ) a ( H ) b H c (3) h h 2 h k k k k k k k h n whch, cost ( ) k H s the cost functon of heat unt k, and k h H k s the heat generated by heat unt k. a k, b k, and c k are the cost parameters of heat unt k. 2.2 obectve functon Accordng to the above cost functons, the obectve functon of the CHPED problem can be wrtten as cost N p N p + Nc N p + Nc + Nh p c c h k k 1 N p +1 kn p + Nc +1 (4) mnmze f cost ( P ) cost ( P, H ) cost ( H ) where fcost s the total cost ($/h). N p, N c, and N h are the respectve number of power unts, cogeneraton unts, and heat unts. The varables, and k are respectvely the power unt number, cogeneraton unt number, and heat unt number. 2.3 constrant Snce the power transmsson loss exsts factually n the power transmsson system, network transmsson loss s an mportant factor n the CHPED problem. In general, the network losses can be calculated through power generaton of all unts whch s known as B-matrx approach. Consderng transmsson lne losses usng B-matrx method s reflected as follows: N p N p N p N p + Nc N p + Nc N p + Nc N p N p + Nc p p P c c c p c Loss N p +1 N p +1 N p +1 1 N p +1 (5) P P B P P B P P B P B P B P B n whch, PLoss s the power transmsson of the system. Matrx B s the coeffcents of the transmsson power loss wth dmenson ( N + N ) ( N + N ). B 0 s a vector wth dmenson ( N + N ), and B 00 s a real constant number. p c p c The equalty constrants representng the power and heat demands and nequalty constrants representng the capacty boundares are gven as follows: p c

4 Power balance N p N p + Nc p c P P Pd PLoss 1 N p -( ) =0 (6) In whch, P d and P Loss are the ndexes of power demand and the power transmsson losses. Heat balance Np+ Nc Np+ Nc + Nh c h H Hk Hd Np k Np+ Nc - =0 (7) H d s the ndcator of heat demand. Capacty boundary P P P 1,..., Np Pmn p p max (8) P H P P H N +1,..., N + N cmn ( c ) c cmax ( c ) (9) p p c H P H H P N +1,..., N + N cmn ( c ) c cmax ( c ) (10) p p c H H H k N + N +1,..., N + N + N hmn h h max k k k (11) p c p c h The upper and lower power boundares of the power unt are p P max and p P mn (MW). The upper and lower power boundares of cogeneraton unt are c mn c P ( H ) and c max c P ( H ) (MW), whch are functons of heat c H. The upper and lower heat boundares of cogeneraton unt are c mn c H ( P ) and c max c H ( P ) (MWth) whch are functons of power n the cogeneraton unt. The upper and lower heat boundares of the heatng unt k are H and hmax k H hmn k (MWth). For ease of descrpton, the CHPED problem as a typcal constraned NLPs can be converted to the followng tradtonal form: mn f ( X ) s. t g( X ) 0 hx ( ) 0 (12) where X ( x1,, x,, x ) (1 q n, q ) are the decson varables. f (X) s the mnmum obectve q n functon. g(x) [ g1( X),, g ( )] T k X s a vector of k nequalty constrants, and h(x) [ h1 ( X ),, h ( )] T m X s a vector of m equalty constrants. 3 Socal cogntve optmzaton algorthm wth tent map 3.1 Socal cogntve optmzaton (SCO) The socal cogntve theory argues that the manknd studes by observng others wth the surroundng, behavor, and

5 cognton as the man nteractve development factors. Human learnng owns the capablty to symbolze, learn from others, plan alternatve strateges, regulate one's own behavor, and engage n self-reflecton. Socal cogntve experts argue that manknd has greater socal ntellgence and ftness than nsect socety. By adoptng socal cogntve theory n an artfcal system, SCO algorthm was proposed n lterature [24]. In SCO teraton process, knowledge lbrary s composed of many knowledge ponts, whch nclude the locatons and ftness values. Learnng agents take part n observatonal learnng by the local neghbor searchng. Referrng to x 1, the local neghbor searchng for x 2 s to obtan another pont ' x, whch s formulated for d dmenson as follows. ' x U x1 x2 x1 (,2 ) (13) where U( u, v) s a unform dstrbuton, whch s always produced by the lnear congruental approach n the nterval [ uv, ]. Furthermore, t s necessary to ensure that x ' satsfy all the constrants. To select an approprate pont from the specal set, tournament selecton chooses the better pont for neghborhood searchng from the knowledge lbrary and chooses the worse pont for refreshng the knowledge lbrary. More detaled procedure about the SCO algorthm can be found n lterature [24]. 3.2 Tent map Chaos s a type of ubqutous nonlnear phenomena n lots of actual systems. Chaotc movement can reach every state n certan scale accordng to ts own regularty and ergodcty, whch s better than a smple stochastc search algorthm. The chaotc search algorthm s featured wth randomcty, ergodcty, and regularty. So, n many optmzaton algorthms, t s usually ntroduced to ntalze the ntal soluton or to terate local search to enhance the ergodcty of the soluton and accelerate the global optmal convergence. Chaos search algorthm has many mplement models ncludng Tent map, Kent map, and Logstc map. Tent map teraton s faster and more sutable for a computer than Logstc map and Kent map [25]. Meanwhle, the dstrbuton of Tent map s very even, and ts ntal senstvty s weak. After Bernoull shft transformaton, Tent map can be stated n the nterval [0, 1] 2xk, 0 xk 1/ 2 x k 1 (14) 2(1 xk ), 1/ 2 xk 1 In Eq. (14), there are mnor rotaton ponts n tent map, such as four cycles: 1/5, 2/5,3/5 and 4/5, and unstable cycle ponts, such as 0, 1/4, 1/2, 3/4 and 1, whch wll fall nto the fxed pont 0. The chaotc seres n the nterval ( VV, ) based on Tent map s produced as follows: l r Step1: gve an ntal value whch must avod takng a few specal values of 0, 1/4, 1/2, 3/4 and 1. Through m (about x m 300) tmes teraton, numercal values n (0, 1) are gotten. In an teratve procedure, f x {0,0.25, 0.5, 0.75, 1} or x k {1,2,3,4}, then x x1 to avod varable rotaton ponts and mnor rotaton ponts. s a very x k -3 small real, for example 10. Step2: calculatng chaotc varables n each teraton accordng to Eq. (15) where V r s the rght value of the nterval, and 3.3 Coordnatng Tent map and SCO V l f ( x) V x. V V (15) l m r l s the left value of the nterval. To obtan even dstrbuton and mprove the ergodcty of SCO ntalzaton and local searchng, chaotc search algorthm [26] s appled to lbrary ntalzaton and local searchng of SCO nstead of the common random approach n the classcal SCO algorthm. In the socal agents ntalzaton phrase, the ntal poston of agents s generated by the chaotc search algorthm n the feasble space rather than the stochastc method. In the local neghbor searchng

6 process, the local neghbor searchng of the learnng agents adopts the chaotc search algorthm to generate the new agent s poston. Because the Tent map algorthm has more benefts than Kent map and Logstc map, the common random approach wll be shfted by Tent map searchng algorthm. Referrng to, the formulaton of the local searchng for x 2 s x 1 where Tent( u, v) ' x x1 x2 x1 s a stochastc value generated by Tent map wthn [u, v]. Tent(,2 ) (16) 3.4 TSCO for CHPED problem The decson varables n the CHPED problem are power and heat dspatch outputs values. The poston of learnng agent represents a feasble CHPED output value scheme n constrant space and ts ftness functon. The ftness ncludes two parts: CHPED obectve functon and constrants volaton value. Each agent shows the possble soluton of the CHPED problem, and all the agents form knowledge lbrary. The th knowledge pont s represented as: L [ P,..., P, P,..., P, H,..., H, H,..., H ] p p c c c c c c 1 N p ( Np1) ( N p Nc ) ( Np1) ( N p Nc ) ( N p Nc 1) ( N p Nc Nh ) The length of the knowledge pont s N N N p c h. Learnng agents n possesson of knowledge ponts operate observatonal learnng by the local Tent map searchng and model selecton va tournament selecton. Fg. 1 llustrates the flowchart of Socal Cogntve Optmzaton wth Tent map (TSCO). The TSCO for CHPED problem s depcted as follows: Step 1: ntalzaton 1) Set parameters: the number of knowledge ponts teraton tmes 2) Create T, the vcarous wdth W N L knowledge ponts n knowledge lbrary ( K N L, the tournament wdth, the number of learnng agents N, maxmum a B. ) wth Tent map, and calculate ther ftness values ncludng obectve functon value and constrants volaton value, then preserve the global optmal pont G P. 3) Every learnng agent s assgned to a known pont n K Step 2: every learnng agent L : 1) Tournament selecton: Choose the best-known pont lbrary K, not same wth L tself. 2) Observatonal learnng: After the pont T P T P at random, but not reduplcatvely. from random B knowledge ponts n knowledge s confronted wth the pont L, the better pont s selected as a central pont to generate the new pont T accordng to Eq. (16) wth Tent map by referrng to the worse. O If the pont T O s better than G, P T O s assgned to G. P 3) Lbrary refreshment: Choose the worst pont T from random w W ponts n knowledge lbrary K. If the ftness value of a pont T s better than the ftness value of O T, the pont w T wll shft pont O T. w Step 3: terate Step 2 untl the stop condton s satsfed. The overall calculaton tme s T NL N * T. e a

7 Start Parameters set,lbrary ntalzaton and learnng agent assgnment Tournament selecton Observatonal learnng wth Tent map Lbrary refresh and record best pont N Satsfy stop condton Y Obtan the optmal soluton End FIG. 1 Flowchart of TSCO 3.5 Constrant handlng Most lterature apples a penalty functon method to handle the constrants for the CHPED problem [1]. But the penalty handlng strategy needs the tough penalty coeffcent settngs and feasble ntal pont. Basc Constrant Handng (BCH) strategy s ntroduced to handle the constrants n the swarm ntellgence optmzaton algorthm. However, BCH strategy s dffcult to cope wth equalty constrants n the swarm, snce the volaton values of the equalty constrants are usually very small and easy to fall nto the rdge functon class landscape [27]. Because there are some power and heat balance equatons constrants n CHPED, we adopt the relaxed quas-feasble space to approxmately meet the equaton n the early teraton perod. The actual feasble space s adaptvely entered n the later teratons perod [27]. The followng formulaton s ntroduced to calculate the constrant volaton values: m F ( X ) max{, max{0, h ( X )} max{0, g ( X )}} CON R =1 =1 where R 0 s the relaxng threshold value. In the teratons, fundamental rato-keepng latent rules and forcng latent rules [28]. 4. Case studes k (17) R constrnges adaptvely to zero based on the In ths secton, the performance of TSCO for resolvng the CHPDE problems n the feld of qualty of soluton and convergence speed s studed. Two typcal test cases are chosen from lterature [1] for comparson, whch are used to evaluate the algorthm n most lterature of studyng the CHPDE problems. The frst case s the classcal smple representaton of the CHPDE problem, and the second case s the classcal sophstcated representaton. The program has been mplemented on the Eclpse Kepler SR1 IDE n Java language and executed on an Intel U CPU@ 2.4 GHz 2.39 GHz PC wth 8 GB RAM (thank Dr. Xe for sharng the source code of tradtonal SCO algorthm n the webste: If dfferent values are not clearly descrbed n the test cases, the parameters of the TSCO for all test cases are set as follows: N L =150, N a =50, T =100, =4, W B =2. Cost s n $, heat output s n MWth, and power output s n MW n all the test cases. The numercal expermental results usng tradtonal SCO and TSCO n ths paper make a

8 comparson wth the typcal lterature [1] reports n the feld of convergence speed and soluton qualty. 4.1 Test case 1 Ths typcal test case ncludes four unts whch are one conventonal power unt, one heat unt and two cogeneraton unts, and t s the classcal representaton of smple CHPDE problems and used to evaluate the algorthms n most lterature of the CHPDE problems. In order to compare the proposed algorthm wth the typcal algorthms, we choose the classcal smple test case as the benchmark, and we also do not consderate the valve effect as the same as n the other referred lteratures. The test case power demand P d s 200MW and the test case heat demand 115MWth. In ths test case, the power transmsson loss and valve-pont effects are gnored as the same as n all the other lteratures. The test system has sx decson varables ( P1, P2, P3, H2, H3, H 4). The fuel cost formulaton of the power unt 1, co-generaton unt 2, co-generaton unt 3 and heat unt 4 are gven: cost ( P) 50P cost ( P, H ) P P 4.2H 0.03H 0.031P H H d s cost ( P, H ) P P 0.6H 0.027H 0.011P H cost ( H ) 23.4H The domans of sx decson varables are stated: P1 [0,150], P2 [81,274], P3 [40,125.8], H2 [0,180], H3 [0,135.6], H4 [0,2695.2] Subected to the power and heat balance equaton constrants: h1: P1 + P2 + P3 - P d =0 h2: H2+ H3+ H4- H d =0 Dual dependency constrants of co-generaton unt 2 g1: P H g2: 98.8-P H2 0 H2 [0,104.8] P H H2 (104.8,180] Dual dependency constrants of co-generaton unt 3 g3: P H [0, 32.4] 3 3 P3 H3 H (32.4,135.6] 44-P3 0 H3[0,15.9] g4: -P H H3 (15.9, 75] - P H H3 (75,135.6] To demonstrate the tweakng mpact of the key parameters NL and Na of the TSCO, 20 trals has been executed usng dfferent knowledge lbrary szes and the learnng agent szes n test case, and the obtaned solutons are presented n Table 1. Table 1. Impact of parameters NL and Na on test case 1 NL Na executon tme (s) Obectve functon value mnmum maxmum mean

9 From Table 1, we can see that the learnng agent sze plays a more mportant role than the knowledge lbrary sze n the executon process, and the tme costs manly depend on the learnng agent sze. The reason for ths phenomenon s that the overall calculaton tme of TSCO s the sum of NL and. And furthermore, Table 1 suggests that global solutons can be acheve very well wth approprate tme n the test case 1 when the knowledge lbrary sze and learnng agent sze are respectvely taken as 150 and 50. In addton, there s no great sgnfcant mprovement f the szes are ncreased beyond those values, but t ncreases the executon tme whch s not desrable n real tme problems. Therefore, the parameters NL and Na are chosen as 150 and 50 to acheve optmal performance n ths test case. The optmal soluton of the test case 1 attaned by the TSCO algorthm s $ , whch s showed n Table 2 wth focus on the comparson of TSCO n the feld of mnmum fuel cost and computatonal tme wth the earler lterature algorthms. From Table 2, the TSCO algorthm obtans global optmal fuel cost wth less computatonal tme than the tradtonal SCO and other methods. The reason for ths phenomenon s that human has the ablty of observatonal learnng and tournament learnng and has more ntellgence than other smarms. Meanwhle, the Tent map local searchng mproves the ergodcty of the SCO. Therefore, the concluson can be drawn that TSCO algorthm s an effectve way to solvng the CHPED problem. Na Table 2 Optmal solutons of CHPED for test case 1. * T Output P1 P2 P3 H2 H3 H4 Mn Tme(s) ACSA [12] GA [29] RGA [30] EP [31] FA [17] IWO [21] CPSO [32] RCGA-IMM [8] HS [33] IGA-MU [34] SARGA [35] EMA [36] TVAC_PSO [32] Drect method [37] GWO[3] MCSA[14]

10 CSA[14] CSO[38] SCO TSCO Consderng that GA s the most classc and representatve heurstc algorthm, GWO s an up-to-date bo-nspred optmzaton algorthm, SCO s the latest human learnng-based swarm optmzaton algorthm, these typcal heurstc algorthms GA, GWO and SCO are chosen to evaluate the performances of the TSCO for solvng CHPED ssues. For ease of analyss, the populaton szes are set to 50 n all these algorthms. In the GA, the strng length, crossover probablty and mutaton probablty are respectvely set to 72, 0.90, and The convergence speeds of the four algorthms n CHPED problem are shown n Fg. 2. FIG. 2. Convergence comparson n case 1 by GA, GWO, SCO and TSCO From Fg. 2, t can be observed that the TSCO has the faster convergence speed than the other three classcal algorthms, especally the GA. Meanwhle, the GA doesn t acheve the optmal soluton n the 100 teratons, the soluton qualty of the TSCO s superor to the other three algorthms. Durng the teraton process, because the basc constrant handlng rule s that any feasble pont s preferred over any unfeasble pont and the pont havng less constrant volaton s preferred among the unfeasble ponts, some pont s obectve values n the later teraton are greater than the value n the former teraton. The TSCO shows faster convergence speed and upper steady ablty n the teratons of evoluton than tradtonal SCO, GA and GWO. The reason for ths phenomenon s that TSCO has the hgh ntellgence and more global convergence performance and ntegrates Tent map nto TSCO ntalzaton and neghborhood searchng to mprove unform dstrbuton and ncrease the ergodcty of the SCO. Therefore, the concluson can be drawn that the TSCO has good global convergence ablty and Tent map s an effectve way n ntalzaton and neghborhood searchng. To reasonably compare the soluton performances and stabltes of the four algorthms, 20 ndependent trals have been executed by usng each algorthm. Fg. 3 s the boxplot of the cost dfference values between the optmal value

11 and the cost soluton obtaned by the four algorthms GA, GWO, SCO and TSCO. The vertcal coordnate represents the dfference values and the horzontal coordnate represents the correspondng algorthm of the boxes. FIG. 3. The cost boxplot of four algorthms for testcase 1 It can be seen from Fg. 3 that the mnmum, maxmum, medan, lower quartle and upper quartle of the cost dfference values of the four classcal algorthms are sgnfcantly dfferent wth each other. From the dstrbuton of cost dfference values, we can see that the box of TSCO s lower and shorter than those of the GA, GWO and SCO. Hence the TSCO algorthm has hgher stablty and better performance than the other three algorthms. Fg.4 llustrates the constrant volaton value convergence comparson of the BCH and ACR strategy n TSCO algorthm for the CHPED problem. The vertcal coordnate represents constrant volaton value logarthm to the base of ten, and the horzontal coordnate represents the teraton number. Because the zero logarthm does not exst, the curve of the ACR vanshes from the forty-eghth teraton. From the convergence curve, the constrant volaton values of the BCH are better than those of the ACR n the frst 16 teratons, because the ACR strategy ntroduces the relaxed quas-feasble space. But the constrant volaton values of ACR are better than those of the BCH and quckly move to zero at about forty eghth teraton. The constrant volaton value of the BCH never equals zero, so the algorthm wth the BCH strategy does not obtan the feasble soluton wthn 100 teratons. The ACR strategy only needs 48 teratons from the unfeasble solutons to feasble soluton area. So, the concluson can be drawn that adaptve constrants relaxng (ACR) strategy s a very effectve method to handle the equaton constrants n the CHPED problem. FIG. 4. Constrant volaton value comparson n case 1 by ACR and BCH

12 4.2 Test case 2 In the test case 2, the valve-pont effects of the power unt and power net transmsson losses n the CHPED problem are consdered as the same as n all the other referred lteratures [3]. The test case 2 s a classcal representaton of sophstcated CHPDE problems and used to evaluate the algorthms n most lterature. Test case 2 conssts of four conventonal power unts, two co-generaton unts, and one heat unt. The test case total heat demand s 150MWth and the power demand s 600MW. The nne decson varables of the test case can be represented as ( P1, P2, P3, P4, P5, P6, H5, H6, H 7) The fuel cost formulaton of four conventonal power unts, two co-generaton unts, and heat unt are gven: cost ( P) P +2 P sn(0.042 (10- P)) cost ( P ) P +1.8 P sn(0.04 (20- P )) cost ( P ) P +2.1 P sn(0.038 (30- P )) cost ( P ) P +2 P sn(0.037 (40- P )) cost ( P, H ) P P 4.2H 0.03H 0.031P H cost ( P, H ) P P 0.6H H 0.011P H cost ( H ) H H P (49P 14P 15P 15P 20P 25 P ) P 10 Loss (14P 45P 16P 20P 18P 19 P ) P (15P 16P 39P 10P 12P 15 P ) P (15P 20P 10P 40P 14P 11 P ) P (35P 34P P 39 P ) ( P P P P P P ) The domans of nne decson varables are lsted: P1 [10,75], P2 [20,125], P3 [30,175], P 4 [40,250], P 5 [81,247], H5 [0,180], P6 [40,125.8], H6 [0,135.6], H7 [0,60] Subected to the power and heat balance equaton constrants: h1: h2: P -P - P =0 d d Loss H - H =0 Dual dependency constrants of co-generaton unt 5 g1: P H g2: 98.8-P H 0 H [0,104.8] P5 H5 H (104.8,180] Dual dependency constrants of co-generaton unt 6

13 P H [0, 32.4] g3: 6 6 P H H6 (32.4,135.6] 44-P6 0 H6 [0,15.9] g4: -P H H6 (15.9, 75] - P H H6 (75,135.6] The smulaton result obtaned by the TSCO algorthm s $ , whch s showed n Table 3 wth focus on the comparson of TSCO n the feld of mnmum fuel cost and computatonal tme wth exstng state-of-the-art algorthms. Table 3 Optmal solutons of CHPED for test case 2. Method P1 P2 P3 P4 P5 P6 H5 H6 H7 Mn ($) Tme(s) RCGA[8] PSO[32] EP[31] AIS[15] CPSO [32] DE [19] BCO [13] ECSA [39] KHA [40] EMA[36] TLBO[23] CSO[38] SCO TSCO From Table 3, t can be observed that the CPU tme of TSCO algorthm s much less than all other compared algorthms. In addton, the TSCO algorthm obtans less fuel cost than the algorthm reported n the exstng lterature [6, 11, 13, 17, 21, 29, 30, 34, 36, 37, 38], and almost equal fuel cost wth the CSO [36] algorthm. As shown n Table 2, the power producton and the heat producton are MW and 150 WMth, respectvely. The power net transmsson loss s MW. Transparently, the output result completely fulflls the heat and power demands. The reason for ths phenomenon s that human has the ablty of observatonal learnng and tournament learnng and has more ntellgence than other smarms. Meanwhle, the Tent map local searchng mproves the ergodcty of the SCO. Therefore, the result of the CPU computaton tme of TSCO s much less than that of the classcal SCO. Therefore, the concluson can be drawn that TSCO algorthm s an effectve way to solvng CHPED problem. The cost convergence qualty acqured by TSCO and SCO s presented n Fg.5.

14 FIG. 5. Convergence comparson n case 2 by SCO and TSCO As shown n Fg. 5, the TSCO shows faster convergence speed and upper steady ablty than tradtonal SCO n the teratons of evoluton. The reason for ths phenomenon s that TSCO algorthm ntroduces Tent map to mprove unform dstrbuton and avod the prematurty convergence of the algorthm. Therefore, the concluson can be drawn that Tent map s very effectve n ntalzaton and neghborhood local searchng. The constrant volaton value convergence comparson of the BCH and ACR strategy of the TSCO algorthm n the sophstcated CHPED problem s llustrated n Fg. 6. From the convergence curve n Fg. 6, the constrant volaton values of the BCH are better than those of the ACR n the frst 8 teratons. The reason for ths phenomenon s that the ACR strategy ntroduces the relaxed quas-feasble space and obtans the unfeasble soluton n the early teratons perod. But the constrant volaton values of ACR are better than those of the BCH n the later teratons perod and quckly move to zero after about forty seventh teraton. The constrant volaton value of the BCH never equals zero, so the algorthm wth the BCH strategy does not obtan the feasble soluton wthn 100 teratons. The ACR strategy only needs 47 teratons from the unfeasble solutons to feasble soluton area. The reason for ths phenomenon s that the ACR strategy enhances the movement to the feasble space n the later teratons perod. So, the concluson can be drawn that adaptve constrants relaxng (ACR) strategy s very effectve to handle the equaton constrants n the sophstcated CHPED problem. FIG. 6. Constrant volaton value comparson n case 2 by ACR and BCH 5. Conclusons To acqure the best utlzaton of CHP unts, a socal cogntve optmzaton algorthm wth tent map for the CHPED problem s presented. The presented algorthm s verfed by two classcal test cases. The conclusons can be drawn as follows: (1) TSCO algorthm can effectvely settle the CHPED problem, and t has better soluton characterstc, less CPU computaton tme and upper convergence speed than many ntellgence optmzaton algorthms.

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