Exergoeconomic optimization of a thermal power plant using Particle Swarm optimization

Transcription

1 xergoeconomc optmzaton of a thermal power plant usng Partcle Swarm optmzaton by Axel GRONIWSKY* The basc concept n applyng numercal optmzaton methods for power plants optmzaton problems s to combne a State of the art search algorthm wth a powerful, power plant smulaton program to optmze the energy converson system from both economc and thermodynamc vewponts. Improvng the energy converson system by optmzng the desgn and operaton and studyng nteractons among plant components requres the nvestgaton of a large number of possble desgn and operatonal alternatves. State of the art search algorthms can assst n the development of cost-effectve power plant concepts. The am of ths paper s to present how nature-nspred swarm ntellgence (especally PSO) can be appled n the feld of power plant optmzaton and how to fnd solutons for the problems arsng and also to apply exergoeconomc optmzaton techncs for thermal power plants. Keywords: thermal power plant, partcle swarm optmzaton, exergoeconomc optmzaton, exergy, thermodynamc modellng software 1. Introducton Although the very dea of lnng Thermodynamcs and costng consderatons and analysng a system not ust from an energetc but also an economc pont of vew was already explored, not untl Trbus et al. [1] ntroduced Thermoeconomcs for analysng desalnaton systems were thermodynamc analyss and economc optmzaton combned. Though, n the early years of Thermoeconomcs there were several attempts to use energy costng nstead of exergy costng Gaggol [2] demonstrated on a cogeneratng power plant that the use of energy as the measure for the power flow leads to error. In lne wth Gaggol, Tsatsarons [3] suggested the name of xergoeconomcs to pont out that thermoeconomc analyss s based on the Second law. Followng l-sayed, Gaggol and Kanoglu et al. [4], [5], [6], thermoeconomc methods fall nto two categores: algebrac methods and calculus methods. The algebrac methods use algebrac cost-balance equatons derved from conventonal economc analyss and auxlary cost equatons for each subcomponent of any system presented [7]. alculus methods on the other hand are bult on dfferental equatons. ost flows n a system are developed n a ln between optmzaton procedures that are based on Lagrange multplers. The weaness of the calculus method s that f the component fals to acheve thermoeconomc solaton the Lagrange multplers vary from teraton to teraton mang the applcablty of ths method dffcult. Therefore xergoeconomc analyss (A), whch s

2 a subcategory of algebrac method has been chosen to estmate the cost-optmal structure and the costoptmal values of the thermodynamc neffcences n the case study n a later secton. Due to plant performance smulaton software n the feld of energy engneerng the complexty of search spaces s ncreasng and the number of varables s growng. Therefore, nstead of classcal optmzaton technques whch have lmted scope n practcal applcatons heurstc search methods become more and more frequently used tools. volutonary algorthms and especally Genetc Algorthms (GA) are commonly used for mult-crtera optmzaton problems of power plants. Valdes at al. [8] used GA to acheve thermoeconomc optmum n a combned cycle gas turbne (GT). Also Sahoo [9] and Mofda at al. [10] used volutonary programmng and GA to optmze a cogeneraton system. Although lterature shows that evolutonary algorthms, especally GA provdes suffcent results n the feld of exergoeconomc power plant optmzaton, other alternatves of optmzaton technques shall be taen nto consderaton as well to ncrease the varety of avalable tools. One of the most mportant behavourally nspred search algorthms approprate to test for power plants optmzaton problems s the Partcle swarm optmzaton (PSO) [11]. PSO has roots n genetc algorthms and evoluton strateges therefore t shares many smlartes wth evolutonary computng such as random generaton of populatons at system ntalzaton or updatng generatons at optma search but also dffers from t n not usng evoluton operators such as crossover and mutaton or n that each partcle owns memory. Because of the many smlartes PSO has many of the preferable propertes of GA and used successfully n many felds. Yoshda et al. [12] proposed expanded PSO method for reactve power and voltage control consderng voltage securty assessment (VSA). Zara et al. [13] used PSO to solve a constrant economc load dspatch problem for power systems. Snce the orgnal PSO algorthms have no mechansm to handle constrants authors ntroduced several selecton rules and handlng methods. Heo et al. [14] used hybrd PSO, evolutonary PSO, and constrcton factor to fnd optmal mappng between unt load demand and pressure set pont n a fossl fuel power unt (FFPU). Yousef et al. [15] appled a GA hybrd wth PSO (GAHPSO) to fnd the optmal desgn of a plate-fn heat exchanger. Based on lterature PSO has been found to be robust, flexble, and stable. It s nsenstve to local optma or saddle and sutable to solve complex optmzaton problems wth many parameters. PSO s fast n solvng nonlnear, non-dfferentable mult-modal problems [16] and ust le GA t does not requre gradent computaton. Based on ts propertes, PSO seems an approprate choce for power plant optmzaton, however, the fact that each partcle owns memory rases more questons. For a power plant t s very lely that a parameter set contans ncalculable solutons. It has no affect on GA snce ts unsuccessful ndvduals do not partcpate n the producton of the next generaton. However, the partcles of the PSO reman part of the swarm even f they represent ncalculable solutons. Therefore, the am of ths paper s to prove that wth only mnor changes even a conventonal PSO s sutable for optma search n the feld of exergoeconomc power plant optmzaton. Namely, f the slght modfcaton on the structure of the PSO does not nterfere wth the velocty updatng algorthm n the future t wll provde PSO alternatves wthout any constrans, ncreasng the possblty to create more precse adaptatons for dfferent power plant problems. 2. The partcle swarm optmzaton concept

3 In the frame of multvarable optmzaton problems, the swarm s assumed to be of specfed sze wth each partcle located ntally at random locatons wth zero velocty n the multdmensonal desgn space. Partcles of the swarm represent possble solutons n the search space, havng the two before mentoned parameters as changeable propertes [17]. ach partcle eeps trac of ts postons n the search space and ts behavour wll depend on the best poston (hghest ftness value) that t has dscovered and on the best overall poston that any member of the swarm has acheved so far. Unle GA only the best partcle shares nformaton (poston) wth the others. Snce ts ntroducton many researchers have wored on mprovng the performance of PSO by modfyng the velocty updatng strategy of the orgnal algorthm. anoncal PSO, whch has been used here only dffers from the orgnal algorthm n the use of nerta weght at velocty updatng. onsderng the thermoeconomc optmzaton as an unconstraned D-dmensonal mnmzaton problem as follows: 1 D Mn f (X), X [x,...x,...x ], (1) where X, as a member (partcle) of the swarm s a soluton to be optmzed n a form of a D- dmensonal vector. Assumed that x s the poston and v s the velocty of the th partcle on the th dmenson ther values can be updated by teraton [10][17][18] as follows: v wv c1 rand1 pbest x c2 rand2 gbest x, (2) x x v t, (3) D where X x 1 D... x... x and V v 1... v... v D the th partcle n the D-dmensonal search space whle pbest pbest 1... pbest... pbest D gbest gbest 1... gbest... gbest represents the poston and velocty, respectvely of and represents the best poston of the th partcle and the overall best poston of the swarm dscovered so fare. t refers to the tme steps between two teratons and can be consdered as 1. The acceleraton constants c 1 and c 2 are the cogntve and socal learnng rates, respectvely, denotng the relatve mportance of pbest and gbest postons. rand1 and rand2 are randomly generated numbers n the range [0,1]. The nerta weght w s used to balance the global and local search abltes. A large nerta weght s more approprate for global exploraton of new areas, meanwhle small nerta weght facltates local search. Snce ts ntroducton, several varants of nerta weght have been proposed. One of the lnearly descendng nerta weghts appled here s: w w max mn w w max max, (4) where w max and w mn are the ntal and fnal values of the nerta weght, respectvely, and max s the maxmum number of teratons. Besdes v max maxmum velocty has to be gven to determne constrants [18]: max mn v mn v,max v, v. (5) 3. Implementaton of PSO for power plants optmzaton problems Power plant optmzatons are ether desgned to help decson maers or to acheve a more suffcent and more effectve operaton.

4 Improvng the energy converson system by optmzng the desgn and studyng nteractons among plant components requres the nvestgaton of a large number of possble desgn alternatves. State of the art search algorthms can assst process desgners n the development of a cost-effectve power plant concept. To optmze process structure, a superstructure can be developed whch ncludes a lmted number of the most lely desgn alternatves wth estmated values of the process varables. onsderng the superstructure as a search space, wth the applcaton of a sutable search algorthm an optmal set of process structure can be determned. In general, the cost of electrcty s more senstve to changes n the confguraton of the process structure than the modfcaton of the values of process varables [19]. As the mpact of the process varables s not as sgnfcant as the mpact of the modfcaton of process structure superstructures do not requre accurate process varables. Improvng the effectveness of operaton however requres a more specfed and detaled modellng of the equpment of the operatng power plant. The accuracy of the provded soluton of the search algorthm heavly depends on the thermodynamc model (search space) created n the power plant smulaton software. To establsh an accurate thermodynamc model not ust a desgn case but also an off-desgn case shall be created to be able to predct the performance of a fxed plant desgn as condtons vary. Fne-tunng of the process varables s one of the smplest ways to reduce expenses wthout nvestng on desgn restructurng. After obectve functon (reducng fuel costs at constant load) and constrants (envronmental consderatons) of the problem are determned and search space (off-desgn case of the thermodynamc model) s created, a PSO varant shall be chosen. The algorthm chosen shall fulfl the crtera of robustness and t shall eep balance between dversty and convergence speed. To decde the number of dmensons of the power plants optmzaton problem the number of degrees of freedom shall be determned. The number of degrees of freedom refers to the ndependent process varables of the offdesgn model, havng mpact on operatng condtons. The operatng range of the equpment of the actual power plant determnes the lower and upper lmtng values of the process varables Drawbacs of PSO n the feld of power plant optmzaton In 2011, n hs revew paper Pezzn [20] summarzed the State of the art optmzaton technques appled n the feld of energy engneerng. A great number of ssues are shown where PSO was appled however, power plant optmzaton problems were solved manly wth GA. PSOs are manly neglected because of the senstvty to ncalculable solutons of the parameter sets durng optma search. The basc concept n applyng numercal optmzaton methods for power plants optmzaton problems s to combne a heurstc search algorthm wth a thermodynamc smulaton software. In a thermodynamc smulaton software energy converson cycles are created by the components of a system. ach pece of equpment represents an energy- and a mass balance. These equatons wth other relatons for thermodynamc propertes form an ndependent set of equatons. Ths system of nonlnear equatons, where the number of equatons and the number of unnown parameters depend on the qualty and quantty of the components and process varables, s solved n an teratve way. Therefore stablty and convergence for both search algorthms and numercal models depend heavly on parameter set and constrans. Although reasonable parameter set provdes stablty for a search algorthm t wll stll not guarantee the same for the solver of the thermodynamc smulator. Namely, the search space representng all theoretcally possble parameter set s greater than the set of

5 physcally possble solutons. As Fg. 1 llustrates a search space usually has many ncalculable clouds where, wth the provded parameter set physcal equpment cannot operate. The result of a PSO n each tme step depends on the current poston of the partcles and the velocty updatng algorthm. Asde from hybrd PSOs where velocty update algorthm s complemented wth evolutonary operators velocty ether depends on the partcles own Pbest and swarms Gbest (e.g. canoncal PSO) or depends on the Pbest and Gbest of multple elte examples (e.g. LPSO [18], LPSO [21]). If velocty s calculated wthout the Pbest of elte examples the chance of a soluton becomes proportonal wth the rato of calculable and ncalculable search spaces. If a partcle at ntal step becomes part of an ncalculable cloud nether ftness value nor Pbest can be calculated as the power plant cannot operate among these condtons. Velocty update algorthm wll fal to move partcles outsde the clouds and get stuc. [22] Fgure 1: 2 Dmensonal search space wth ncalculable areas ompared to evolutonary algorthms where new generatons are created from stable and convergent ndvduals PSO eeps all ts ntal partcles for the entre search. Therefore the more partcles stuc n ncalculable clouds at the begnnng, the fewer partcles can search for optmum. It s only possble to use PSOs wth own Pbest dependent velocty update algorthms, n search spaces wth numerous ncalculable clouds f partcles do not get nto these clouds at ntal state. If the number of convergent partcles reaches the number of elte examples, PSOs wth elte example dependent velocty update algorthms wll not have problems wth ncalculable solutons ether. 4. ase study A case study s performed to llustrate that PSOs even wth own Pbest dependent velocty update algorthms are capable of optmzng power plant desgn f a swarm has only convergent members at ntal state. anoncal PSO s chosen for the demonstraton wth a preselected and ntally convergent group of partcles. The populaton sze s set to 30 wth the generaton number of 70. ogntve and socal learnng rates are chosen to be 1 and the maxmum and mnmum Innerta weghts are set to 0.9 and 0.4 respectvely System descrpton To demonstrate the applcablty of PSO, a 10 MW thermal power plant s consdered. When choosng a desgn t s an mportant crteron to create a model typcal for ths small power range. Ths

6 desgn where a turbne has three extracton ponts, two at low pressure to provde deaeraton and extracton steam for a deaerator and for a low pressure feedwater heater respectvely, and one at hgh pressure to drve a hgh pressure feedwater heater s relatvely common for small scale power plants. Fg. 2 shows the structure of a thermal power plant consdered n ths study Steam Water Ar Gas Fluegas F F Fgure 2: Schematc dagram of the thermal power plant PSO can modfy both physcal propertes of the plant and values of the process varables whenever a partcle n the swarm occupes new poston n the search space. In ths model, the pressure and the temperature of the steam produced, the pressure of the extracton steam drvng the hgh pressure feedwater heater, the operatng pressure of the deaerator, and the pressure of the extracton steam drvng the low pressure feedwater heater are consdered as process varables. Process varables are real numbers restrcted by the accuracy of the thermodynamc solver and ther physcal range only. The qualty and quantty of the before mentoned desgn varables are sutable to provde a search space suffcent to test the applcablty of PSO for exergoeconomc power plant optmzaton therefore no other parameter s changed durng optma search. The admssble range of the decson varables consdered for the thermal power plant are as follows: 30 bar<p 1 <120 bar, 400 <T 1 <540, 0.1 bar<p 3 <120 bar, 0.1 bar<p 4 <120 bar, 0.1 bar<p 5 <120 bar. The surface area of the hgh pressure feedwater heater, and that of the low pressure feedwater heater are consdered as structural varables tang dscrete values between 5-35m 2 and 5-50m 2 respectvely. Snce worng range of process varables are chosen to be as wde as physcally possble and the power of the thermal power plant s fxed, at 85 % sentropc expanson effcency n 10 MW the search space contans several parameter sets formng ncalculable clouds. It s assumed that: the system operates at steady state, deal-gas mxture prncples apply for the ar, gas and fluegas, and the reference envronment s consdered to be K and bars xergoeconomc analyss Thermoeconomcs s the branch of engneerng that combnes exergy analyss dentfyng the locaton, the magntude and the sources of thermodynamc neffcences n a thermal system and economc prncples whch help to calculate all the costs assocated wth a power plant nvestment or operaton to provde the system desgner or operator wth nformaton not avalable through conventonal energy analyss and economc evaluatons but crucal to the desgn and operaton of a cost-effectve system.

7 The Thermodynamc evaluaton of xergoeconomc s based on Second-Law Analyss whch s a useful tool to calculate rreversbltes. The values of the rates of exergy destructon (the rato between the exergy destructon rate of a gven component and the exergy destructon rate wthn the system) and exergetc effcency (the rato between the exergy product rate of a gven component and the exergy fuel rate wthn the system) provde suffcent thermodynamc measures of the system neffcences. A comprehensve ntroducton to the exergoeconomc concept and ts applcatons s provded by Bean et al. [7], and Tsatsarons and Lazzaretto [23]. xergy costng nvolves cost balances formulated for each system component separately. A cost balance appled to the th component q. 6 shows that the sum of cost rates assocated wth all extng exergy streams equals the sum of cost rates of all enterng exergy streams supplemented wth a component dependent cost rate ( Z ) assocated wth nvestment ( Z I ) and wth operatng and Z mantenance expenses ( Z OM ) as t s shown n q. 7: e. e e ) c w, W c q, q, (c ) Z. (6) (c ) c W c (c ) Z. Z (s O & M) P (P ). (7) where s, O&M, P and, are the annual carryng charges, the annual operatng and mantenance costs, the purchased equpment costs and the operatng hours of the power plant per year respectvely. In general, carryng charges decrease whle fuel and O&M costs ncrease wth ncreasng years of operaton. Therefore, levelzed annual values for all cost components should be used when consderng desgn modfcatons. The constant-escalaton levelzaton factor s shown n q. 8: LF 1 RF 1 where RF s the abbrevaton of captal recovery factor, gven by q. 9: and s a constant, gven by q. 10: tot eff n eff 1eff RF n r n 1 eff. The determnaton of relable purchased equpment costs are very dffcult snce vendors are nterested prmarly n sellng ther products for the hghest prce possble causng large dsperson n prce even for the same equpment. To avod nconvenences arsng from the effort of proft maxmzaton thermodynamc property dependent cost functons and constants are used [24], [25]. The most common smple relatonshp between the purchased cost and an attrbute of the equpment related to unts of capacty appled here s gven by q. 11: A A where A 1 and A 2 refers to the requred and base attrbute respectvely, 1 and 2 refers to the purchased costs of the equpment wth the requred attrbute and base attrbute, and n refers to the cost exponent [26]. Wth use of q. 12 the effect of tme on purchased equpment cost were taen nto consderaton: n n (8) (9) (10) (11)

8 I I1 where 1 and 2 refers to the purchased costs of the equpment at base tme when cost s nown and tme when cost s desred, and I refers to the cost ndex to base tme and desred tme respectvely. Snce several cost ndces used by the ndustry to adust for the effects of nflaton based on the suggeston of Turton et al [25] hemcal ngneerng Plant ost Index was appled. The soluton of the system of costng equatons provdes the cost of the unnown streams of the thermal power plant. The state propertes and exerges necessary for solvng costng equatons correspondng to Fg. 2 are gven n Table 1. Table 1: State propertes and exergy of the system for the base case SP m [gs -1 ] T [ ] p [bar] h [Jg] s [Jg -1 K -1 ] [W] A detaled thermoeconomc evaluaton of a thermal system s based on a set of varables calculated for each component of the system. The average unt cost of fuel ( c F, ), average unt cost of product ( c P, ), cost rate of exergy destructon ( D, ), and cost rate of exergy loss ( L, compare nvestment and operaton costs for each component. cf, F, c P, F, P, P, D, c c F, D, (12) ) provde useful tools to (13) (14) c (15) c (16) L, c c F, L,

9 Based on the aforementoned exergoeconomc varables exergoeconomc factor (f) can be calculated whch shows the relatonshp between the monetary mpact of each component s exergy destructon and nvestment cost. Z f Z (17) D, The exergoeconomc varables calculated for each component of the thermal power plant for the base case operatng condtons are summarzed n Table 2. The nvestment cost, operaton and mantenance expenses, and fuel costs are estmated n a detaled economc analyss conducted for each plant separately usng Bean et al. [6], Turton et al. [25] and Petraopouloua et al. [27] for data. Table 3 shows the man estmatons for the economc analyss. Table 2: xergoeconomc parameters of the system for the base case F, P, L, y D, D, D L Z Z D, L, f [MW] [MW] [MW] [%] [%] [$/h] [$/h] [$/h] [-] sys Table 3: Input data for the economc analyss Plant economc lfe (year) 20 Levelzaton perod (year) 10 Average general nflaton rate (%) 3 Average nomnal escalaton rate for natural gas (%) 4 Average real cost of money (%) 10 Data of commercal operaton 2012 Average capacty factor (%) 85 Unt cost of natural gas ($ GJ -1 LHV -1 ) Obectve functon Non-exergy-related costs dependng on nvestment costs and operatng and mantenance expenses and exergy-related costs dependng on component effcency (exergy destructon) show opposte effect on power plant behavour. At hgher total captal nvestment lower operatng and mantenance cost can be expected meanwhle lower total captal nvestment usually causes hgher operatng and mantenance cost. Therefore, the obectve functon expresses the optmzaton crteron as a functon of the dependent and ndependent varables s defned as to mnmze the total cost functon (q. 20).

10 Z Z c F, (18) sys Z Z c F, D, c F, L, 4.3. omputer tools The optmzaton process conssts of two parts: thermodynamc analyss and economc calculatons. Both parts are crucal for the exergoeconomc evaluaton and they are performed n every teraton step. The thermal model s developed n Gateycle (G) plant performance montorng software and all mass flow rates, temperatures, pressures, and chemcal compostons for every stream calculated wth ths code by usng JANAF data for the propertes of deal gases and IAPWS-IF97 for the propertes of water and steam. Snce G does not calculate exergy a MATLAB code s developed to provde chemcal and physcal exergy. The cost estmaton and the detaled economc evaluaton are performed n Mcrosoft XL envronment. The PSO algorthm s developed and all optmzaton runs are controlled n MATLAB however the dynamc data exchange s performed va Mcrosoft XL as G s a closed-source software and drect control s not possble. Fgure 3: Structure of the PSO based optmzaton process The followng steps are performed at each teraton: 1. PSO provdes new desgn varables for Gateycle; 2. after smulaton wth new varables, Gateycle provdes thermodynamc propertes for xergy alculaton and PSO search algorthm, and also updated attrbutes to determne Purchased qupment osts; 3. after new P are determned, the algorthm updates the economc evaluaton and calculates new cost rates; 4. based on new thermodynamc and economc data, PSO evaluates the obectve functon and based on the results creates new desgn varables. The structure of the optmzaton process s llustrated n Fg Results and dscusson If the optmzaton s unconstraned and only thermodynamc aspects are consdered maxmum pressure (120 bar) and temperature (540 K) are expected before the steam turbne to acheve the hghest average temperature possble at heat nlet and ncrease cycle effcency. As hgh steam

11 parameters and hgh cycle effcency decreases fuel costs, n the same tme t ncreases nvestment costs. Therefore, the unconstraned optmum case represent the optmum between fuel costs and captal costs. The exergoeconomc parameters for each system components for optmum operatng condtons are summarzed n Table 4. The exergoeconomc factor of the steam turbne and the low and the hgh pressure feedwater heaters are decreased suggestng that cost savngs n the entre system mght be acheved by a decrease n the nvestment costs at the expense of ther exergetc effcency. The exergoeconomc factor of the boler and the deaerator are ncreased suggestng that total cost can be saved by ncreasng exergetc effcency of these equpment va hgher nvestment costs. The exergoeconomc factor of the overall system s decreased from 0.57 to 0.53 whch s consstent wth related lterature. Table 4: xergoeconomc parameters of the system for the unconstraned optmum case F, P, D, L, y D, D L Z Z D, L, f [MW] [MW] [MW] [%] [%] [$/h] [$/h] [$/h] [-] sys Snce the results of the frst optmum search (Optmum case 1) presents a soluton wth a relatvely low qualty at the ext of the condensng secton (88.9 %) causng eroson n the turbne a second optmum search (Optmum case 2) s run wth a 90% qualty restrcton. Due to the fact that qualty of the steam leavng the turbne s only 1.1% below the lmt state propertes and exerges show no great dfference between unconstraned and constraned optmum cases therefore state propertes only presented for the constraned optmum case (Table 5). Table 5: State propertes and exergy of the system for the constraned optmum case SP m [gs -1 ] T [ ] p [bar] h [Jg] s [Jg -1 K -1 ] [W]

12 The decson varables for the base case and both optmum cases are gven n Table 6. Table 6: omparson of the decson varables for the base and optmum cases Propertes Base case Optmum case 1 Optmum case 2 p 1 [bar] T 1 [ ] p 3 [bar] p 4 [bar] p 5 [bar] A 5 [m 2 ] A 8 [m 2 ] Table 7 shows that qualty restrcton causes an ncrease of 2.25 % n the cost of exergy destructon and only a slght decrease n the component dependent cost rates. Table 7: xergoeconomc parameters of the system for the constraned optmum case F, P, L, y D, D, D L Z Z D, L, f [MW] [MW] [MW] [%] [%] [$/h] [$/h] [$/h] [-] sys The costs of the streams n the base case and both optmum cases are gven n Table 8. Snce the energy demands of the pumps are provded by the steam turbne self-consumpton causes a very hgh cost for streams 14, 7 and 11. Otherwse the results meet the expectatons provdng n general the hghest stream costs for the base case and the lowest ones for optmum case 1. Unt cost of electrcty produced s reduced from cents/wh to cents/wh n the frst optmum case and cents/wh n the second one. Although these values seem a bt hgh, consderng the technology, the

13 temperature range of the cycle, and the fact that the unt cost of the electrcty s an average value calculated for the next ten years they are acceptable. Table 8: ost of the streams n the system SP Base case Optmum case 1 Optmum case 2 c [cents/wh] [$/h] c [cents/wh] [$/h] c [cents/wh] [$/h] Fgure 4: Varaton of obectve functon and exergetc effcency of the system durng optmum search

14 The mnmum value of the obectve functon and the correspondng exergetc effcency of the best partcle (Gbest) of each teraton s shown n Fg. 4. Although the exergetc effcency has a maxmum at the second teraton wth a value of 0.29 the result of the total cost functon s relatvely hgh ( $/h) therefore the PSO decreases the nvestment cost of the overall system at the expense of ts exergetc effcency. After the tenth teraton the changes n the decson varables are very small, hence the mprovement after the frst one-thrd teratons becomes small but steady. 6. onclusons The am of an exergoeconomc optmzaton for power plants s to estmate the cost-optmal structure and the cost-optmal values of the thermodynamc neffcences n a system. A detaled thermodynamc analyss maes thermodynamc smulaton software essental. As the search space of an exergoeconomc power plant analyss representng all theoretcally possble parameter set s generally greater than the set of physcally possble solutons t mght contans ncalculable solutons whch cannot be excluded by constrans. The wor shows that although PSO s more senstve to ncalculable clouds at ntal state than other evolutonary algorthms, snce velocty updatng strateges eep all ther ntal partcles for the entre search, PSO stll can be used n conuncton wth exergoeconomc prncples to optmze energy systems. Va a canoncal PSO wth own Pbest dependent velocty update algorthm, whch s more senstve to ncalculable clouds than elte example dependent velocty update algorthms, t s demonstrated that f the ntal partcles are preselected and all partcles have Pbest and the swarm has a Gbest then the PSO wll fnd a soluton even f the rato of calculable and ncalculable parts of the search space s small. The sgnfcance of the result s ncreased by the fact that pre-selecton does not nterfere wth the velocty updatng algorthm thus the soluton provdes PSO alternatves wthout any constrans, ncreasng the possblty to create more precse adaptatons for dfferent power plant problems. Nomenclature c constant, cost per exergy unt ($W -1 h -1 ) cost rate ($/h) s annual carryng charges ($/y) D number of dmenson exergy rate (W) f exergoeconomc factor gbest h m global best specfc enthalpy (J/g) number of teratons, nterest dmenson generaton counter from 1 to max gen mass flow rate (g/sec) max gen maxmum generatons O&M annual operatng and mantenance costs ($/y) p pressure (bar) pbest personal best P purchased equpment costs ($/y) ps populaton sze r escalaton rate rand1, rand2 random numbers rangng over 0 1 s specfc entropy (Jg -1 K -1 ) T temperature ( ) v velocty w nerta weght W wor flow rate (W) X D-dmensonal vector x varable y exergy destructon rato

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