Electrcal and Electronc Engneerng 013, 3(6): 139-148 DOI: 10.593/j.eee.0130306.01 Computatonal Soluton to Economc Operaton of ower lants Temtope Adefarat 1,*, Ayodele Sunday Oluwole 1, Mufutau Adewolu Sanus 1 Department of Electrcal/Electronc Engneerng, Federal Unversty, Oye Ekt, 3708, gera Department of Electrcal/Electronc Engneerng, Federal ploytechnc Ede Osun State gera Abstract The producton cost of electrcty s a very mportant ndex n natonal development. Electrcty tarff depends on the fuel cost whch carres the hghest percentage of the total operaton cost n any power plant. In order to keep electrcty tarff as low as possble, fuel cost whch carres the hghest percentage of the total operatng cost has to be mnmzed. Economc operaton of the power plants can be acheved through economc load dspatch and unt commtment. Lagrange relaxaton s one of the best solutons n solvng economc load dspatch problem because t s more effcent and easer than other methods. Ths approach has been mplemented to mnmze the fuel cost of generatng electrcty whle takng nto account some techncal constrants. Keywords Economc Effcency, Economc Load Dspatch, Incremental Cost 1. Introducton The optmum economc operaton of electrc power syste m has occuped an mportant poston n the electrc power ndustry. Wth recent power deregulaton all over the world, t has become necessary for power generatng utltes to run ther power plants wth mnmum cost whle satsfyng ther customers load demand (eak and Base load). In order to acheve ths, all the generatng unts n any power plant must be loaded n such a way that optmum economc effcency can be acheved[]. The purpose of economc operaton of any power plant s to reduce the fuel cost whch carres the hghest percentage of the operatng cost whle runnng the plant[1][]. The mnmum fuel cost can only be acheved by applyng economc load dspatch and unt commtment n any nterconnected power system. Hence, Economc load dspatch s a powerful and useful tool to assess optmum operaton as well as the fnancal and electrcal performance of a power plant. economc operaton s to reduce the fuel cost of the operaton of the power system. The optmum operaton of a power plant can only be acheved by economc load schedulng of dfferent unts n the power plants or dfferent power plants n the power system. Economc load schedulng s the determnaton of the generatng output of dfferent unts n a power plant n such way to mnmze the total fuel cost and at the same tme meet the total power demand[],[7]. The economc load dvson between dfferent generatng unts can only be computed f the operatng cost expressed n terms of power output Output n MWx 1000 x100 % Effcency of generatng unt= Input n KJ per second Output n MWx 1000 x3600 x100 % = Input n KJ per hour If stands for the power output n megawatts (MW) and C be the fuel cost, then Fg.1 shows a typcal nput and output characterstc curve of a power plant.. Economc Operaton of ower Systems Economcal producton of electrcty s the most mportant factor n the power system. In any combned power plants, all the generatng unts should be loaded n such a way that optmum effcency can be acheved. The purpose of * Correspondng author: temtope.adefarat@yahoo.com (Temtope Adefarat) ublshed onlne at http://journal.sapub.org/eee Copyrght 013 Scentfc & Academc ublshng. All Rghts Reserved F gure 1. Typcal nput/output Characterstc curve for a sngle unt n a power plant
140 Temtope Adefarat et al.: Computatonal Soluton to Economc Operaton of ower lants The max s lmted by thermal consderaton and a gven power unt cannot produce more power than t s desgned for. The mn s lmted because of the stablty lmt of the machne. If the power output of any generatng unt for optmum operaton of the system s less than a specfed value mn, the unt s not put on the bus bar because t s not possble to generate that low value of power from that unt[7],[8]. Hence the generatng ower cannot be outsde the range stated by the nequalty,.e. mn max.1. Operatonal Cost n a ower lant The man economc factor n the power system operaton s the cost of generatng real power. In any power system ths cost has two components[1]. 1. The fxed cost beng determned by the captal nvestment, nterest charged on the money borrowed, tax pad, labour charge, salary gven to staff and other expenses that contnue rrespectve of the load on the power[1].. The varable cost s a functon of loadng on generatng unts, losses, daly load requrement and purchase or sale of power[1]. The economc operaton of an electrcal power can be acheved by mnmzng the varable factor only whle the personnel n charge of the plant operaton have lttle control over the fxed costs[1]... The Objectves of the Research The objectves of the research are as follows: 1. To formulate a mathematcal model to mnmze the total fuel cost of producng electrcal power n a power plant wthn a stpulated tme nterval. The cost of each generatng unt n a power plant s represented by the quadratc equaton of the second order. The objectve functon of a power plant s the algebrac sum of the quadratc fuel cost of each gener atng unt n a power plant.[6],[1]. The objectve functon of each generatng unt can be expressed as F ( ) =a +b +c, Where a, b and c are the cost coeffcents of generatng unt at bus [6][1].. To develop the best approach that wll help all the power utlty companes to solve the problem of economc load dspatch n an nterconnected power system. 3. To estmate the output power and fuel consumpton of each generatng unt n a power plant whle meetng the load demands at a mnmum fuel cost. 4. To desgn a computer applcaton program to solve the problem of economc dspatch problem n any nterconnecte d power system. 5. To deploy all the avalable resources for power generaton such as natural gas, water, desel, uranum, coal and petrol more effcently and thus handle peak and base loads more effcently and relably wth economc load dspatch..3. Economc Load Dspatch The Economc Load Dspatch s a process of allocatng demand loads to dfferent generatng unts n a power plant at a mnmum fuel cost whle meetng the techncal constrants. It s formulated as an optmzaton process of mnmzng the total fuel cost of all the commtted generatng unts n a power plant whle meetng the load demands and techncal constrants[3],[7]. F =a +b +c (1) The fuel cost functon of a generatng unt s represented by a quadratc equaton of the second order as shown n equaton.1 Where a, b and c are constants of th generatng unt..4. Incremental Cost Incremental cost can be determned by takng the dervatve of the equaton 1.0 Subject to = b + c () λ = b + c (3) λ b = c (4) mn max Sum up the entre of the power system I.e. (5) =1 to (6) D = (7) ε (8) Or D - Where ε = 10 5. If condtons n equaton (5.0) are met, Then Sum up all the (s).e. (9) Error = ABS ( - D ) (10) Error = ε (11) If convergence s not acheved then modfes λ and recompute, the process s contnued untl D - F Ρ Ρ s less than a specfed accuracy or Ρ
Electrcal and Electronc Engneerng 013, 3(6): 139-148 141 D = If convergence s acheved, then compute the followng, 1. F = a + b Ρ + c Ρ. for each unt.5. Computat onal Algorthms Step1. Total power demand would be gven. Step. Assgn ntal estmated value of λ (0). Step3. Let ε be equal to 10 5. Step4. F For all the unts would be gven. c Step5. Dfferentate Ρ =λ) Step6. Rearrange F wth respect to ( F Ρ (so that = λ b c F Ρ ) = b + Step7. Compute the ndvdual Unts 1, -------n Correspondng to λ (0). Step8. Compute Step9. Check f the relatonshp (0) =D s satsfed or D - = ε Step10. If the sum s less than total power demand, then assgns a new value λ (1) repeat steps 8 and 9. Step11. If the sum s less than the demand, then assgns a new value λ () and repeat steps 8 and 9. Contnue the teraton untl when t wll converge. unt. D = or D - Step1. Calculate fuel cost and ε for each generatng 3. Modellng of olynomal Equaton for Each Generatng Unt olynomal model for the generatng unts can be acheved through the least square method. 3.1. Least S quare Equatons F = a + b p + c p ( ) F p = a p + b p + c p 3 () p F = a p 3 + b p + c p 4 () 3.. MAT LAB Smulaton Wth MAT LAB smulaton coeffcents a, b and c can be acheved, therefore the polynomal equaton for each generatng unt s expressed as F= a + b + c Table 1. The 1st generatng unt ower ( Mw) Fuel Cost ( $/Hr ) 3 4 xf ( Mw) ( Mw) 3 ( Mw) 4 ( $Mw/Hr) XF ( Mw) ($/Hr) 100 710 10000 1000000 100000000 71000 7100000 150 997.5 500 3375000 506,50,000 14965 443750 00 130 40000 8000000 1600000000 64000 5800000 50 1677.5 6500 1565000 390650000 419375 104843750 300 070 90000 7000000 8100000000 61000 186300000 350 497.5 1500 4875000 1500650000 87415 305943750 400 960 160000 64000000 5600000000 1184000 473600000 450 3457.5 0500 9115000 4100650000 1555875 700143750 500 3990 50000 15000000 6500000000 1995000 997500000 700 19680 960000 378000000 1.5835E+11 7134000 850675000 Table 1 shows the power and fuel characterstc of the frst generatng unt
14 Temtope Adefarat et al.: Computatonal Soluton to Economc Operaton of ower lants START Read F,a,b,c,D,ε,mn,max,ame of Generatng staton,type of generatng Staton and o of buses Intate λ as 0 SET n=1 SOLVE THE EQUATIO FOR b = λ c Check f >max Yes Set =max Check f <mn Yes set=mn Set n=n+1 o Check f all buses have been accounted Calculate = Is <= ε ( D) Yes Calculate cost of generaton for each unt, total optmum fuel cost, total fuel cost for each type of generatng staton and compute the value of ncremental cost o Reduce λ and assgn new value λ(1) o Is Σ > ε Yes End Increase λ and assgn new value of λ() Flow chart for economc load dspatch problems Fgure. Flow chart for economc load dspat ch
Electrcal and Electronc Engneerng 013, 3(6): 139-148 143 By applyng the least square equatons 19680 = 9a + 700b + 960000c () 7134000 = 700a + 960000b + 378000000c () 850675000 = 960000a + 378000000b+ 1.5835x10 11 c () Where a = 40, b = 4 and c= 0.007 Therefore, the polynomal equaton for the frst generatng unt can be expressed as F = 40 + 4 + 0.007 Table. The nd generatng unt ower Fuel Cost ( Mw) ( $/Hr ) ( Mw) 3 ( Mw) 3 4 ( Mw) 4 xf XF ( Mw) ( $Mw/Hr) ($/Hr) 50 673.75 500 15000 650000 33687.5 1684375 75 98.4375 565 41875 31,640,65 6963.81 56598.437 100 1195 10000 1000000 100000000 119500 11950000 15 1473.4375 1565 195315 4414065 184179.6875 30460.94 150 1763.75 500 3375000 50650000 6456.5 39684375 175 065.9375 3065 5359375 93789065 361539.065 6369335.94 00 380 40000 8000000 1600000000 476000 9500000 875 10480.315 16875 034375 346171875 1509101.56 40436475.3 Table shows the power and fuel characterstc of the second generatng unt By applyng the least square method 10480.315 = 7a + 875b + 16875c () 1509101.56 = 875a + 16875b +034375 c () 40436475.3 = 16875a + 034375b+ 346171875c () Where a = 00, b=9 and c=0.0095 Therefore, the polynomal equaton for the second generatng unt can be expressed as F = 00 + 9 + 0.0095 Table 3. The 3rd generatng unt ower Fuel Cost ( Mw) 3 ( Mw) 3 4 ( Mw) 4 xf ( $Mw/Hr.) XF ( Mw) ( Mw) ($/Hr) ($/Hr.) 80 733.6 6400 51000 40960000 58688 4695040 100 880 10000 1000000 100,000,000 88000 8800000 150 177.5 500 3375000 50650000 19165 8743750 00 170 40000 8000000 1600000000 344000 68800000 50 07.5 6500 1565000 390650000 551875 137968750 300 740 90000 7000000 8100000000 8000 46600000 350 3317.5 1500 4875000 1500650000 116115 406393750 1430 1876.1 353900 98387000 959710000 317313 9000190 Table 3 shows the power and fuel characterstc of the thrd generatng unt
144 Temtope Adefarat et al.: Computatonal Soluton to Economc Operaton of ower lants By applyng the least square equatons 1876.1 = 7a + 1430b + 353900c () 317313 = 1430a + 353900b +98387000c () 9000190 = 353900a + 98387000b+ 959710000c () Where a = 0,b = 5.7 and c= 0.009 Therefore, the polynomal equaton for the 3rd generatng unt can be expressed as F = 0 + 5.7 + 0.009 Table 4. The 4th generatng unt ower Fuel Cost ( Mw) 3 ( Mw) 3 4 ( Mw) 4 XF ( Mw) xf ( $Mw/Hr.) ( Mw) ( $/Hr. ) ($/Hr.) 50 77.5 500 15000 650000 3865 193150 75 1075.65 565 41875 31,640,65 80673.75 6050531.5 100 1390 10000 1000000 100000000 139000 13900000 15 1715.65 1565 195315 4414065 14453.15 6806640.63 150 05.5 500 3375000 50650000 307875 4618150 500 7006.75 5650 6875000 8888150 78066.875 94869671.88 Table 4 shows the power and fuel characterstc of the fourth generatng unt By applyng the least square equatons 7006.75 = 5a + 500b + 5650c () 78066.875 = 500a + 5650b +6875000c () 94869671.88 = 5650a + 6875000b+ 8888150c () Where a = 00, b = 11 and c= 0.009 Therefore, the polynomal equaton for the fourth generatng unt can be expressed as F = 00 + 11 + 0.009 Table 5. The 5th generatng unt Fuel Cost ower 3 XF ( Mw) ( Mw) ( Mw) 3 4 ( Mw) 4 xf ( Mw) ($Mw/Hr.) ($/Hr. ) ($/Hr.) 50 730 500 15000 650000 36500 185000 75 1000 565 41875 31,640,65 75000 565000 100 180 10000 1000000 100000000 18000 1800000 15 1570 1565 195315 4414065 19650 453150 150 1870 500 3375000 50650000 80500 4075000 175 180 3065 5359375 93789065 381500 6676500 00 500 40000 8000000 1600000000 500000 1100000 875 11130 16875 034375 346171875 1597750 164818750 Table 5 shows the power and fuel characterstc of the ffth generatng unt
Electrcal and Electronc Engneerng 013, 3(6): 139-148 145 By applyng the least square equatons 11130 = 7a + 875b + 16875c () 1597750 = 875a + 16875b +034375c () 164818750 =16875a + 034375b+ 346171875c () Where a = 0,b = 9.8 and c= 0.008 Therefore, the polynomal equaton for the ffth generatng unt can be expressed as F = 0 + 9.8 + 0.008 ower ( Mw) Fuel Cost ( $/Hr. ) Table 6. The 6th generatng unt 3 4 xf ( Mw) ( Mw) 3 ( Mw) 4 ($Mw/Hr.) XF ( Mw) ($/Hr.) 50 858.75 500 15000 650000 4937.5 146875 60 997 3600 16000 1,960,000 5980 358900 70 1136.75 4900 343000 4010000 7957.5 5570075 80 178 6400 51000 40960000 1040 817900 90 140.75 8100 79000 65610000 17867.5 11508075 100 1565 10000 1000000 100000000 156500 15650000 110 1710.75 1100 1331000 146410000 18818.5 0700075 10 1858 14400 178000 07360000 960 67550 680 1085 6000 5984000 603560000 980080 94098700 Table 6 shows the power and fuel characterstc of the sxth generatng unt By applyng the least square equatons 1085 = 5a + 875b + 16875c () 980080 = 875a + 16875b +034375c () 94098700 =16875a + 034375b+ 346171875c () Where a = 190, b = 13 and c= 0.0075 Therefore, the polynomal equaton for the sxth generatng unt can be expressed as 4. Test System Ths system has 6 unts whle the Unts Cost data and system load demand are gven respectvely n Table 7. F1= 40 + 4 + 0.007 ($/Hr) F= 00 + 9 + 0.0095($/ Hr) F3= 0 + 5.7 + 0.009($/Hr) F4= 00 + 11 + 0.009($/ Hr) F5= 0 + 9.8 + 0.008($/Hr) F6= 190 + 13 + 0.0075 ($/Hr) F = 190 + 13 + 0.0075 Ta ble 7. Test Syst em Data Unt mn max A B C MW MW $/Hr $/MWHr $/MW Hr 1 100 600 40 4 0.007 50 600 00 9 0.0095 3 80 800 0 5.7 0.009 4 50 500 00 11 0.009 5 50 650 0 9.8 0.008 6 5 300 190 13 0.0075 Table 7 shows the quadratc fuel cost for the sx generatng unts
146 Temtope Adefarat et al.: Computatonal Soluton to Economc Operaton of ower lants T me (Hr) Table 8. The results of the smulaton Load (MW) Fuel Cost ($/Hr) Incremental Cos t($/mwhr) 0100HRS 1600 16845.93 13.08 000HRS 1800 19517.6 13.63114 0300HRS 000 98.39 14.18009 0400HRS 100 3750.1 14.45456 0500HRS 00 5189.3 14.7903 0600HRS 50 5949.18 14.8666 0700HRS 300 6675.93 15.0035 0800HRS 350 7449.53 15.14073 0900HRS 400 8190 15.7797 1000HRS 450 8977.33 15.415 1100HRS 500 9731.5 15.5544 100HRS 550 3053.57 15.68967 1300HRS 600 31300.49 15.8691 1400HRS 650 3115.7 15.96414 1500HRS 750 3375.4 16.3861 1600HRS 800 3450.77 16.37585 1700HRS 900 3619.07 16.6503 1800HRS 950 3708.0 16.78755 1900HRS 3000 37850.83 16.9479 000HRS 3100 39557.03 17.1996 100HRS 300 4190.68 17.47373 00HRS 3100 39557.03 17.1996 300HRS 900 3619.07 16.6503 400HRS 750 3375.4 16.3861 The smulaton results are analysed n Table 8, the table shows the load emand, fuel cost and ncremental cost for the sx generatng unt Fuel Cost($/Hr) 46000 41000 36000 31000 6000 1000 16000 1600 100 600 3100 Fgure 3. shows the relatonshp between the load demand and fuel cost n a power system. There s a lnear relatonshp between fuel cost and load demand F gure 3. ower generated vs. Fuel cost Fuel Cost($/MWHr) 46000 41000 36000 31000 6000 1000 16000 ower (MW) 0 10 0 30 Hour(Hr) Based on the smulated results, fgure 4 shows the hourly fuel cost n the power system F gure 4. Fuel Cost per hour Incremental Cost($/MWHr) 16.08 15.08 14.08 13.08 1600 100 600 3100 3600 ower (MW) Fgure 5 shows that the syst em ncrement al cost λ s drect ly proportonal to the demand load F gure 5. ower generated vs. Incremental Cost Table 9. The effect of unt commtment n a power plant Unt s Type of Generatng unt Output ower MW Fuel Cost $/Hr Economc Effcency $/MWHr 1 Base load generatng unt 648.788 5780.859 8.91106 Base load generatng unt 4.858 57.1 1.0101 3 Base load generatng unt 410.14 407 9.9874 4 eak load generatng unt 115.68 159.89 13.7698 5 Base load generatng unt 05.1377 567.001 1.51355 6 eak load generatng unt 5.4803 61.4681 47.711 Total 16000 16846.4301 104.84447
Electrcal and Electronc Engneerng 013, 3(6): 139-148 147 Incremental Cost($/MWHr) 17.08 16.58 16.08 15.58 15.08 14.58 14.08 13.58 13.08 16845.93 1845.93 6845.93 31845.93 36845.93 Fuel Cost($/MW) 5. Effect of Unt Commtmen Fgure 6 shows that fuel cost s drectly proportonal to the system ncremental cost λ The frst aspect of the Economc Load Dspatch s the unt commtment problem where t s requred to select optmally out of the avalable generatng sources to operate, to meet the expected load and provde a specfed margn of operatng reserve over a specfed perod of tme[10]. From the analys s as shown n Table 9, the base load generatng unts are 1,, 3 and 5 due to ther low fuel consumpton and optmum economc operaton whle peak load generatng unts are 4 and 6. Table 10. ower plant fuel Consumpton by consderng unt commtment ower Generated (MW) FCWO($/Hr) FCW($/Hr) F gure 6. Fuel cost vs. Increment al Cost Fuel Cost($/Hr) 1600 16845.93 16606.9 39.03 1800 19517.6 19406 111.3 000 98.39 165. 133.19 00 5189.3 511.3 68.03 300 6675.93 6649.7 6.4 9900 11056.8 109949.1 577.7 FCWO=Fuel Cost wthout consderng unt Commtment FCW=Fuel Cost by consderng unt commtment 6. Results and Dscusson The results for the system ncremental cost and operatng cost were plotted for the varous load levels. From fgure 5 and fgure 6, t shows that operatng cost and ncremental cost rse lnearly wth load values. Therefore, t can be concluded that fuel cost and the system ncremental cost λ are drectly proportonal to the demand load. They follow an approxmately lnear trend n relaton to the load demand. Table 8.0 shows that fuel cost and ncremental costs are drectly proportonal to the demand load n any ntegrated power system. Table 10 also shows the effect of unt comm tment on unts 1,, 3, 4, 5 and 6. From the table 9, the base load generatng unts are 1,, 3 and 5 due to ther effcency and optmum economc operaton whle peak load generatng unts are 4 and 6. In any power plant, the generatng unt wth the cheapest Fuel Cost, effcency and the best optmum economc operaton wll be selected to dspatch frst. As shown n Table 3.0, Generatng Unt o. 1 s the cheapest whle Generatng Unt o. 6 s the most expensve n terms of fuel cost and economc effcency /MWHr. Hence, Generatng Unt o. 1 would be dspatched frst andgenerat ng Unt o.6 last. Generatng unt 1 s the cheapest and t has the best generatng capablty of the system. 7. Conclusons It can be seen that any ncrease n load demand brngs about the same rse n the system fuel cost; a cost that would be passed on to the customers snce fuel cost carres the hghest percentage of the operatng cost of power plants. Hence, t shows that the relatonshp between fuel prces and Load demands s approxmately lnear. Wth the current po wer deregulaton n the world, t s essental to optmse the runnng cost of power plants by reducng the fuelconsumpt on for meetng a partcular load demand. Ths can only be acheved through the economc load dspatch. REFERECES [1] Abhjt Chakrabart, Sunta Halder "ower System Analyss Operaton and Control", HI Learnng rvate Lmted, Thrd Edton, pp548-590, 010 [] Gupta J.B, "A Course on Electrcal ower", S.K.Katara and
148 Temtope Adefarat et al.: Computatonal Soluton to Economc Operaton of ower lants Sons, Fourteenth Edton, 08-1, 01. [3] Manjeet Sngh, Mukesh Garg, Vneet Grdher "Comparatve study of Economc Load Dspatch usng modfed Hopfeld neural network", Internatonal Journal of Computng & Busness Research, ISS (Onlne): 9-6166, pp.1,01. [4]. hanthuna V. hupha. Rugthacharoencheep, and S. Lerdwanttp "Economc Load Dspatch wth Daly Load atterns and Generator Constrants By artcle Swarm Optmzaton "World Academy of Scence, Engneerng and Technology 71 01 pp.133-135 [5] eetu Agrawal, K.K.Swarnkar, Dr. S.Wadhwan, Dr. A. K. WadhwanEconomc Load Dspatch roblem of Thermal Generators wth Ramp Rate Lmt Usng Bogeography - Based Optmzaton Internatonal Journal of Engneerng and Innovatve Technology (IJEIT) Volume 1, Issue 3, March 01 ISS: 77-3754 pp98-10 [6] Tarek Bouktr, Lnda Slman, M. Belkacem A Genetc Algorthm for Solvng the Optmal ower Flow roblem Leonardo Journal of Scences, ISS 1583-033, Issue 4, pp.46, January-June,004. [7] Wadhwa C.L, "Electrcal ower Systems", ew Agenternat onal ublshers, sxth Edton, pp67-661, 010s. [8] Wood, A, Wollenberg, B "ower Generaton Operaton and Control", Wley and Sons, Thrd Edton, pp154-05, 1984 [9] Y. Tng-Fang;. Chun-Hua, Applcaton of an mproved artcle Swarm Optmzaton to economc load dspatch n power plant, n roc IEEE Int. Conf. Advanced Computer Theory and Engneerng, 010. [10] Wadhwa C. L,"Electrcal ower Systems", ew Age nternat onal ublshers, sxth Edton, pp67-661, 010s. [11] Wood, A, Wollenberg, B "ower Generaton Operaton and Control", Wley and Sons, Thrd Edton,pp154-05,1984 [1] Vjayakumar Krshnasamy Genetc Algorthm for Solvng Optmal ower Flowroblem wth UFC, Internatonal Journal of Software Engneerng and Its Applcatons Vol. 5 o. 1,pp.41 January, 011.