90 CHAPTER 7 STOCHASTIC ECOOMIC EMISSIO DISPATCH-MODELED USIG WEIGHTIG METHOD 7.1 ITRODUCTIO early 70% of electrc power produced n the world s by means of thermal plants. Thermal power statons are the major causes of atmospherc polluton because of the hgh concentraton of pollutants they cause. The man draw back of generatng electrcty from fossl fuel releases several contamnants such as Sulphur Oxdes, trogen Oxdes, and Carbon-d-Oxde nto the atmosphere and generate partculates. The contamnants cause atmospherc polluton. The polluton mnmzaton has attracted a lot of attenton due to the publc demand for clean ar. In recent years rgd envronmental regulaton forced the utlty planners to consder emsson control as an mportant objectve. In ths work Economc Emsson Dspatch (EED) s consdered as a stochastc problem. The cost and emsson coeffcents are consdered as random varables and a stochastc model s developed. Due to consderaton of the problem as stochastc there are three objectves to be mnmzed, cost, emsson, varance of power from ts expected value. A mult-objectve problem s formulated consderng all the three objectves.
91 7. PROBLEM FORMULATIO The frst objectve functon to be mnmzed s the total operatng cost for thermal generatng unts n the system. A quadratc operatng cost curve s assumed. 1 1 F a p b p c (7.1) where F 1 s the cost functon to be mnmzed a,b,c are the cost coeffcents of the th generator s the total number of generators P s the power output of th generator. 7..1 Stochastc Model A Stochastc model of functon F 1 s formulated by consderng cost coeffcents and load demand as random varables. By takng expectaton the stochastc model can be converted nto ts determnstc equvalent. The random varables are assumed to be normally dstrbuted and statstcally dependent on each other. As the random varables are statcally dependent both varance and covarance of generated power exsts. The expected value of operatng cost s obtaned by expandng the operatng cost functon usng Taylor s seres, about mean. The expected cost s: E(F 1) E ap bp c (7.) 1 1 1 F [(E(a P ) E(b P ) E(c )] (7.3) 1 1 F [(E(a )E(P ) E(b )E(P ) E(c )] (7.4)
9 1 1 F (a [P var(p )] P cov(a,p )) b (P cov(b,p )) c (7.5) where a,b,c are the expected cost coeffcent of the th generator P s the expected value of power generated by th generator F 1 s expected cost functon to be mnmzed var(p ) s varance of power P cov(a,p ) s the covarance of random varable a and P cov(b,p ) s the covarance of random varable b and P. The expected operatng cost functon s represented as: 1, 1 F a p b p c a var(p ) P cov(a,p ) cov(b p ) (7.6) By substtutng for varance and covarance by ts coeffcent of varaton and correlaton coeffcent the equaton (7.6) can be rewrtten as: 1 P ap a p bp b P 1 F (1 C R C C )a P (1 R C C )b P c (7.7) where C a,c b,c s coeffcent of varaton of random varables a P, b, P R a P s correlaton coeffcent of random varable a and P R b P s correlaton coeffcent of random varable b and P. 7.. Expected O x Emsson The amount of O x emsson s gven as a functon of generator power output P whch s quadratc.
93 1 F d p e p f (7.8) where F s the emsson functon to be mnmzed d, e, f are the emsson coeffcents of the th generator s the total number of generators P s the generator power output of th generator. A stochastc model of functon F s formulated by consderng emsson coeffcents and load demand as random varables.. E(F ) E dp ep f 1 (7.9) 1 E(F ) E(d P ) E(e P ) E(f ) (7.10) 1 F [d P d var(p ) P cov(d,p ) (7.11) e P cov(e,p ) f ] where d,e,f are the expected emsson coeffcent of the th generator P s the expected value of power generated by th generator F s emsson functon to be mnmzed var(p ) s varance of power P cov(d,p ) s the covarance of random varable d and P cov(e,p ) s the covarance of random varable e and P.
94 P dp d P ep e P 1 F (1 C R C C )d P (1 R C C )e P f (7.1) where C d,c e,c s coeffcent of varaton of random varables d P, e, P R d P s correlaton coeffcent of random varable d and P R e P s correlaton coeffcent of random varable e and P. Covarance of bvarate random varables s consdered postve or negatve. Covarance s represented by correlaton coeffcent; t s vared from 1.0 to 1.0. One par of random varable s consdered at a tme whle the rest of random varables are consdered ndependent of each other (uncorrelated). 7..3 Expected power devaton Generator outputs P are treated as random varables. There s varaton n random varable from ther expected value whch results n surplus or defct power. Expected devatons are proportonal to the expectaton of the square of the unsatsfed load demand. F3 var P E P (PD P L) 1 1 (7.13) where P D s the expected power demand P L s expected transmsson loss. Substtutng power balance equaton (7.4) n (7.13) we get 3 (7.14) 1 1 F E[ P P ]
95 F3 E (P P ) (7.15) 1 F3 E (P P ) (P P )(Pj P j) 1 1 j1 j (7.16) As random varables are assumed as normally dstrbuted and statstcally dependent expected devatons are gven by equaton (7.18) (7.17) F var(p ) cov(p,p ) 3 j 1 1 j1 j 3 P P j Pj P Pj 1 1 j1 j (7.18) F C P R C C P P where R P P j s correlaton coeffcent of random varable P and P j 7..4 Expected Transmsson Loss The transmsson power loss expressed through the smplfed well known loss formula expresson as a quadratc functon of the power generaton are gven by equaton (7.19) as: P PB P (7.19) L j j 1 j1 where P L s the transmsson loss B j s the loss coeffcent.
96 Takng expected value of transmsson loss E[P L] E PB jpj 1 j1 (7.0) PL E P B E PB jp j 1 1 j1 j (7.1) The expected transmsson loss s represented as equaton (7.) L j j j j 1 1 1 j1 1 j1 j j P P B B var P P B P B cov(p,p ) (7.) where PL s the expected transmsson loss Bj s the expected loss coeffcent. Substtutng for varance and covarance equaton (7.) can be wrtten as equaton (7.3). L P p P P j Pj 1 1 j1 j j j (7.3) P (1 C )B P (1 R C C )P B P 7..5 Equalty and Inequalty Constrants by equaton (7.4). Real power balance s the equalty constrant to be satsfed s gven P PD PL (7.4) 1
97 Expected power generaton lmts are the nequalty constrants to be satsfed. mn max P P P (7.5) where mn P and power output. max P are expected lower and upper lmts of th generator The mult-objectve problem s formulated as follows: Mnmze 1 P ap a P bp b P 1 F (1 C R C C )a P (1 R C C )b P c P dp d P ep e P 1 F (1 C R C C )d P (1 R C C )e P f 3 P j P Pj P Pj 1 1 j j F C P R C C P P Subject to 1 D L mn max 3 k1 P P P P P P ( 1...) w 1 w 0 k k (7.6) To generate the non-nferor soluton of mult objectve optmzaton problem, PSO s used. w K s the levels of the weghtng coeffcents. The values of weghtng coeffcents vary from 0 to 1 for each objectve. The weght w 1, w, w 3 are vared n the range 0 to 1 n such a way that ther sum s 1.0. Ths approach yelds meanngful result to the decson maker when solved many tmes for dfferent values of w k, (k = 1,,3). Ths provdes K number of non-nferor soluton whch are pareto optmal non domnatng. To dentfy one optmal soluton out of K soluton fuzzy membershp satsfacton ndex k D s used.
98 7.3 PSO ALGORITHM FOR ECOOMIC EMISSIO DISPATCH The followng steps are nvolved n solvng the determnstc and stochastc model of economc and emsson dspatch problem. Step 1: Read total number of thermal unts, cost coeffcents, emsson coeffcents, B coeffcents, coeffcent of varaton of each plant, correlaton coeffcents of random varables, maxmum number of teratons, populaton sze, acceleraton constants c 1 and c, nerta weght w mn and w max Step : Confne the search space. Specfy the lower and upper lmts of each decson varable. Step 3: Intalze the ndvdual of populaton. The velocty and poston of each partcle should be ntalzed wth n the feasble decson varable space X =[X 1, X, X 3 X ]. Step 4: Feed or generate the weght (w k =1, n) where n s the number of objectves. Here n=3. Sum of w 1, w, w 3 must be equal to one. Step 5: For each ndvdual X of the populaton the transmsson loss P L s calculated usng B-coeffcents. domnance. Step 6: Evaluate the ftness of each ndvdual X n terms of pareto them n archve. Step 7: Record the non domnated solutons found sofar and save Step 8: Intalze the memory of each ndvdual where the personnel best poston (t) p best d s stored.
99 poston as g (t) best d. Step 9: Fnd the best partcle out of the populaton and store ts Step 10: Set teraton count t=1 Step 11: Update the velocty of each partcle X usng the equaton (4.1). Step 1: Update the poston of each partcle X usng the equaton (4.). Step 13: Check whether the new partcles are wth n the feasble regon. If any element volates nequalty constrants then the poston of the ndvdual s fxed to ts mnmum/maxmum lmts accordng to equaton (4.5). Step 14: Check the power balance constrants. Any volatons are penalzed by addng penalty. Step 15: Calculate the ftness functon of the new ndvdual. soluton. Step 16: Update the archve whch stores the non domnated Step 17: Update memory of each partcle usng the equaton (4.6). Compare new partcles ftness (t 1) X d wth partcles p (t) best d. If the current value s better then the prevous value then set (t 1) p best d to the new value and ts locaton equal to current locaton n d dmensonal space. Compare new partcle s ftness wth the populaton s overall bestg partcle ftness. If the ftness value of the new partcle s better, reset ndex and value. (t 1) g best d (t) best d to current partcle array
100 Step 18: Iteraton = Iteraton +1 Step 19: The algorthm repeats Step 11 to Step 18 untl a suffcent good ftness or a maxmum number of teratons/epochs are reached. Once termnated, the algorthm outputs the ponts of soluton. (t) best d g and f( g (t) best d ) as ts 7.4 TEST SYSTEM AD RESULTS The valdty of the proposed method s llustrated on a sx generator sample system Dhllon et. al (1993) and the result obtaned from the proposed algorthm s compared wth that of ewton-raphson teratve method. For determnstc case coeffcent of varaton and correlaton coeffcent s zero. The coeffcent of varaton and correlaton coeffcent assumed are: C a = C b = C d = C e = C p = 0.1 ( = 1...6). Two cases are consdered: Case 1: Wth dependent varables: In ths case all the random varables are consdered dependent on each other R R R R R 1.0 (=1,.6,j=1,,.6) ap bp cp dp P Pj Case : Wth ndependent varables: In ths case all the random varables are consdered ndependent of each other R R R R R 0.0 ( =1, 6, j = 1, 6). ap bp cp dp P Pj The best soluton obtaned for expected demand of 500 MW, 700 MW, 900 MW for both the cases s shown n Table 7.4.
101 Table 7.1 gves the fuel cost coeffcent and the capacty lmts of the thermal generatng unts. Table 7. gves the emsson coeffcents for the thermal generatng unts. Table 7.3 gves the loss coeffcents of the generator. Table 7.1 Expected fuel cost coeffcents and capacty lmts Generator o. a b c max P mn P 1 0.1574 38.53973 756.79886 10 15 0.10587 46.15916 451.3513 10 150 3 0.0803 40.39655 1049.99770 35 5 4 0.03546 38.30553 143.53110 35 10 5 0.0111 36.378 1658.56960 130 35 6 0.01799 38.7041 1356.6590 15 315 Table 7. Expected O x emsson coeffcents Generator o. d e f 1 0.00419 0.3767 13.8593 0.00419 0.3767 13.8593 3 0.00683 0.54551 40.669 4 0.00683 0.54551 40.669 5 0.00461 0.51116 4.89553 6 0.00461 0.51116 4.89553
10 Table 7.3 Expected loss coeffcents 0.000-0.00086-0.000534-0.000565-0.000454 0.000103-0.00086 0.00343 0.000016-0.000307-0.0004-0.000147-0.000533 0.000016 0.00085 0.000831 0.00003-0.00070-0.000565-0.000307 0.000831 0.00119 0.000113-0.00095-0.000454-0.0004 0.00003 0.000113 0.000460-0.000153 0.000103-0.000147-0.00070-0.00095-0.000153 0.000898 Table 7.4 Comparson of best optmal soluton Load MW Cost (Rs/hr) PSO O x (kg/hr) Rsk Cost (Rs/hr) ewton Raphson O x (kg/hr) Rsk Case 1 500 855.49 304.10 688.05 8550.15 31.51 674.56 700 38951.57 506.6 540.01 39070.74 58.44 5401.18 900 50634.68 798.38 95.77 50807.4 864.06 9110.65 Case 500 8430.9 86.68 566.01 8476.63 87.48 558.75 700 38910.66 49.80 1117.33 39010.74 493.97 1114.8 900 5060.7 798.11 186.71 50854.86 800.6 1861.07
103 Table 7.5 Expected optmal generaton schedule usng PSO Load Case 1 P 1 P P 3 P 4 P 5 P 6 P L 500.0 57.34 37.99 41.03 74.67 179. 18.31 18.5 700.0 88.7 58.51 66.55 110.00 40.8 17.39 36.48 900.0 14.67 9.89 8.07 153.85 86.94 0.09 60.8 Case 500.0 63.8 44.63 35.00 86.43 158.4 19.40 17.6 700.0 88.7 58.51 66.55 110.0 40.8 17.39 36.48 900.0 13.51 89.49 88.49 151.09 88.76 0.65 63.08 Table 7.6 Expected optmal generaton schedules usng R method Load P 1 P P 3 P 4 P 5 P 6 P L Case 1 500.0 59.87 39.65 35.00 7.39 185.4 15.00 17.16 700.0 85.9 60.96 53.90 107.1 50.50 176.50 34.9 900.0 1.00 86.5 59.94 140.95 35.00 0.06 54.49 Case 500.0 59.67 41.41 51.87 83.6 157.83 16.75 0.79 700.0 89.09 65.47 69.71 116.07 3.59 175.70 39.66 900.0 14.61 9.96 87.09 15.74 86.84 0.77 65.03 Table 7.4 gves the comparson of best optmal results obtaned usng PSO and R method. Table 7.5 gves the expected optmal generaton schedule usng PSO. Table 7.6 gves the expected optmal generaton schedule obtaned usng R method.
104 Table 7.7 gves the determnstc results obtaned usng PSO. The correspondng generaton schedule s gven n Table 7.8. Table 7.7 Determnstc results usng PSO Load Cost (Rs/hr) O x (kg/hr) Transmsson loss 500 889.99 86.14 17.67 700 38744. 491.94 36.1 900 50510.3 790. 64.0 Table 7.8 Determnstc generaton schedule usng PSO Load P 1 P P 3 P 4 P 5 P 6 P L 500.0 61.68 38.79 39.11 77.41 17.65 17.67 17.67 700.0 87. 61.3 68.98 108.0 40. 171.4 36.1 900.0 11.1 89.19 89.68 151.87 89.09.41 64.0 Table 7.9 shows the Mnmum, maxmum values of each objectve for Case1. Table 7.10 shows the weght vector for the best optmal soluton n mn 1 PSO and the correspondng k D. F s obtaned by gvng full weghtage to F 1 and neglectng other objectves. Ths s mnmum cost dspatch. mn F s obtaned by gvng full weghtage to F. Ths s mnmum emsson dspatch. mn F 3 s obtaned by gvng full wegtage to F 3. Ths s mnmum rsk dspatch.
105 Table 7.9 Mnmum and maxmum values of objectve functon (Case 1) Load mn F 1 (Rs/hr) max F 1 (Rs/hr) mn F (kg/hr) max F (kg/hr) mn F 3 max F 3 500 817.5 948.39 67.5 354. 576. 881.46 700 38775.48 3976.64 469.3 556.7 5317.6 563.78 900 50470.65 51434.49 766.3 854.3 91.7 9430.77 Table 7.10 Values of weght and k µ D (Case 1) Load w 1,w,w k 3 µ D 500 0.5,0.3,0. 0.0161 700 0.4,0.3,0.3 0.01546 900 0.4,0.4,0. 0.01557 By takng the weght w 1,w,w 3 as 0.4,0.3,0.3 the percentage relatve devaton n F 1 from ther determnstc value wth respect to R a P,R bp,r P P ( j) s calculated and shown n Fgure 7.1. The percentage j devaton n cost ncreases as R PPj s vared from postve to negatve values. There s a decrease n percentage devaton of cost when R ap, R bp s vared from postve to negatve value. Wth the same weght percentage relatve devaton n F from ther determnstc value wth respect to R d P,R e P,R P P ( j) j s calculated and shown n Fgure 7.. The percentage devaton n emsson ncreases as R PPj s vared from postve to negatve values. There s a decrease n percentage devaton of emsson when R dp, R ep s vared from postve to negatve value.
106 Percentage varaton n expected cost 1.3 1.1 0.9 0.7 0.5 0.3 0.1-0.1-0.3-0.5 varatons n Rppj varaton n Rbp varaton n Rap -1 0 0 0.5 1 Correlaton coeffcent Fgure 7.1 Percentage varatons n expected cost wth respect to correlaton coeffcents Percentage varaton emsson 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 varaton n Rppj varaton n Rdp varaton n Rep -1 0 0 0.5 1 Correlaton coeffcents Fgure 7. Percentage varatons n expected emsson wth respect to correlaton coeffcents
107 Fgure 7.3 shows the varaton of the cost wth teraton. Fgure 7.4 shows the varaton of emsson wth teraton. The trade of curve between cost and emsson s shown n Fgure 7.5. Wth varous combnaton of learnng factors the convergence characterstc of the swarm s studed. It s found that the learnng factor of c1 =, c = gves good convergence. Fgure 7.6 shows the convergence for dfferent sets of learnng factor. Fgure 7.7 shows the convergence characterstc for dfferent populaton sze. The populaton sze of 50 gves good convergence. 10 4 Cost Iteraton Fgure 7.3 Fuel cost characterstc of economc emsson dspatch usng PSO
108 Emsson Iteraton Fgure 7.4 Emsson characterstc of economc emsson dspatch usng PSO Emsson Cost Fgure 7.5 Trade off curve between cost and emsson usng PSO
109 10 4 Ftness Iteraton Fgure 7.6 Convergence characterstc for dfferent learnng factor usng PSO 10 4 Ftness Iteraton Fgure 7.7 Convergence characterstc of varous populaton sze for economc emsson dspatch
110 7.5 COCLUSIO PSO s capable of reducng the cost and emsson, but there s ncrease n rsk. Compared to the savng n cost and reducton n the emsson level the ncrease n rsk s tolerable. So the proposed algorthm s effcent n fndng the best optmal soluton. For the loads consdered n ths problem stochastc cost on an average s 0.3% hgher and emsson s 0.865% hgher compared to determnstc results. Table 7.11 gves the average % varaton of stochastc results obtaned usng PSO wth R method. The executon tme s 1 seconds n PIV 3GHz system. Table 7.11 Average results for all the three loads usng PSO Detals Case 1 Case Cost reducton (Rs/h) 0.30% 0.5% O x reducton (kg/h) 4.71% 0.177% Rsk ncrease (MW ) 0.59% 0.606%