CHAPTER 4 HYDROTHERMAL COORDINATION OF UNITS CONSIDERING PROHIBITED OPERATING ZONES A HYBRID PSO(C)-SA-EP-TPSO APPROACH

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1 77 CHAPTER 4 HYDROTHERMAL COORDINATION OF UNITS CONSIDERING PROHIBITED OPERATING ZONES A HYBRID PSO(C)-SA-EP-TPSO APPROACH 4.1 INTRODUCTION HTC consttutes the complete formulaton of the hyrothermal electrc power generaton n power system operaton. Earler n chapters 2 an 3 t was assume that part of power eman was alreay met usng the avalable hyro generaton an only the remanng loa eman has to be taken over an supple by thermal generators. Thus HTC paves a way to proper strbuton of the requre power eman between the avalable hyro an thermal generators n a gven system. It s a common practce that the avalable hyro power shoul be completely use to meet the maxmum power eman (e. Base loa) an the thermal generators are use to supply the remanng power eman f neee. Thus mnmzng the cost of thermal generaton by maxmally utlzng the hyro generaton s the core feature of the HTC. The Hyrothermal Coornaton problem s a sub-problem of UCP an DEDP solves both thermal unt commtment an hyro scheules. It s therefore, a more complex problem formulaton whch consers the prohbte operatng zones as constrants n aton to other hyro an equvalent thermal unts. Snce the source for hyropower s the natural water resources, the obectve of hyrothermal scheulng s to mnmze the operaton cost of thermal unts n

2 78 gven pero of tme whle preservng all constrants. The hyrothermal scheulng problem s bascally formulate as a nonlnear an nonconvex optmzaton problem nvolvng non-lnear obectve functon an a mxture of lnear an non-lnear constrants. Snce, the conventonal methos are not sutable to solve ths HTC formulaton conserng prohbte operatng zones, t s foun that the propose hybr technque s most aaptve to solve ths HTC. The man obectve of ths chapter s To evelop a heurstc base soluton technque for solvng the HTC. To evelop an algorthm to fn the optmal scheulng of hyro unts conserng hyro constrants. To formulate HTC to nclue all the constrants that were consere n the UCP an DEDP formulate to valate the feasblty of the soluton proceure. The formulatons of the hyrothermal coornaton problem an applcablty of the propose soluton proceures are valate through several numercal smulatons an are llustrate n etal. 4.2 HYDRO THERMAL COORDINATION PROBLEM FORMULATION The hyro generatng unts o not ncur the fuel cost when compare to thermal unts. The hyrothermal scheulng problem s ame at mnmzng the total thermal fuel cost whle makng use of the maxmum avalablty of hyro resource as much as possble. The hyro an thermal constrants nclue generaton loa power balance, real power generaton

3 79 lmt, generatng unt ramp rate lmts an prohbte operatng zones. In aton to that the operatng capacty lmts of the hyro unts, water scharge rate, upper an lower bouns of the reservor volume, water spllage an hyraulc contnuty restrctons are fully taken nto account. The obectve functon an the assocate constrants of the problem are formulate as follows: Obectve functon: The total fuel cost for runnng the thermal system to meet the loa eman n a scheule horzon s gven by Mn T z F = f PT 1 (4.1) where f PT s the fuel cost of the equvalent generator an thermal output at tme pero. PT s the Equalty constrants: The equalty constrants are the power balance, total water scharge an the reservor volume constrants. a) The power balance constrants are escrbe as, PD PT PH PL for = 1, 2, 3 z (4.2) The electrc loss between the hyro plant an loa PL s gven by 2 PL k (4.3) PH In the present work constant hea operaton s assume an the water scharge rate, q, s assume to be a functon of the hyro plant generaton, PH, s gven as

4 PH q g (4.4) 80 b) The total water scharge constrant s gven by q T z 1 n q (4.5) th where q s the water scharge rate n the nterval. c) In the case of a storage reservor wth an gven ntal an fnal volume, the reservor volume constrants s gven as V V 1 n r g PH s for = 1,2,3,.z (4.6) where V s the volume of water n the reservor. n s the span of the nterval, r an s are the water nflow rate an the spllage rate respectvely n the nterval. Inequalty constrants: ) The operatonal range of the thermal plants s boune by ther capablty lmts: PT PT PT (4.7) mn max Where PT PT mn an max are the mnmum an maxmum operatng lmts of the equvalent thermal generator. e) The operatonal range of the hyro plants s boune by ther capablty lmts: PH PH (4.8) mn PH max

5 81 Where PH PH mn an max are the mnmum an maxmum operatng lmts of the hyro generaton n nterval n the scheule horzon. f) Reservor storage lmt: V mn V V (4.9) max Where V mn an V max are the volume lmts n nterval. g) Hyraulc contnuty equaton X X q s r 1 (4.10) h) Water scharge rate lmts: q mn q q max (4.11) ) Reservor ntal an fnal volume: V V 0, n V t V, fn (4.12) ) Generaton lmts conserng Prohbte operatng zones: The lterature has shown the nput-output characterstcs of thermal unt conserng POZ as nequalty constrants. The feasble operatng zones of unt can be escrbe as follows: P P P mn U, k 1 U, n P P P P P P L,1 L, k max k 2,3,... n (4.13) where: = unt nex mn P = unt mnmum generaton lmt

6 82 max P = unt maxmum generaton lmt n = number of prohbte zones for unt k = nex of prohbte zones of a unt L U P,, k = lower/upper bouns of the th k prohbte zones for unt Austng the generatng unts output P must avo unt operaton n the prohbte zones. The ynamc economc spatch of power generaton of commtte unts s to be one for the gven loa eman by satsfyng the above constrants Tratonal metho of hyro an thermal power The followng formulaton s use for solvng the hyrothermal coornaton problem through PSO technque to be scusse n the next secton. Arbtrarly select a tme nterval n the scheule horzon. Let the unknown thermal generaton generaton as scusse n the prevous chapter. PT n the th tme nterval be the epenent Water Inflow X PGH PGT Hyro plant Thermal plant Water scharge P Fgure 4.1 Hyrothermal coornaton systems

7 83 The PT can be calculate by assumng that the thermal generatons of the nepenent ntervals,.e., PT for = 1, 2,..- 1, +1,..Z are known. In orer to obtan the PT, the hyro generaton n the epenent nterval PH s requre to be calculate. Soluton of PH an PT Conser the case that there are ntervals n the scheule horzon. Substtutng the equatons (4.3) an (4.2) as gven n Hota et al., we get the expresson as 2 PH PH PD PT 0 k (4.14) From the equaton (4.4) an (4.5), the water scharge rate n any nterval then becomes a functon g PD PT an the total water scharge s expresse as q T z 1 n g PD PT (4.15) an the equaton (4.15) can be rewrtten as q T z 1 n g PH (4.16) Gven the quantty of total water scharge q T an loa pattern, from equaton (4.15) the water scharge rate n the epenent nterval s obtane as g z PH qt n gph 1, n (4.17) Where n s the epenent tme nterval.

8 84 After obtanng the hyro scharges, hyro generaton can be calculate from Eq (4.4) by smple algebrac calculaton, as the scharge n the present case s a functon of hyro generaton. After calculatng the hyro generatons, by usng Eq (4.14) an the gven loa eman, the thermal generaton n the epenent nterval can be calculate as PT PD k 2 PH PH (4.18) The hyrogeneraton n the non epenent ntervals are then obtane by solvng the followng expressons. k 2 PH PD PH PT 0 for = 1, 2,,(-1), (+1),..Z (4.19) The volume of water n the reservor at the en of each nterval s then calculate usng an Equaton (4.6). All generaton level an water volumes must be checke aganst ther lmtng values accorng to the Equatons (4.7) (4.8) an (4.9). In etermnng optmal soluton for the hyrothermal coornaton accorng to the above mentone problem solvng formulaton, the man obectve s to etermne thermal generaton n the nonepenent nterval. In ths research partcle swarm optmzaton base algorthm has been apple to etermne the non-epenent thermal generatons an hence, a better optmal hyrothermal coornaton scheule s obtane. 4.3 PROPOSED SOLUTION METHODOLOGY The mnmzaton of the cost of operaton n a hyrothermal power system nvolves a proper thermal commtment as well as an approprate allocaton of hyro generaton n fferent tme pero. Hence, the soluton

9 85 process conssts of two sub processes. The frst sub-process proves the optmal spatch of hyro unts. The secon sub-process optmzes the thermal commtment prove that the outputs of hyro unts are known Overvew of PSO base hyrothermal scheulng PSO s one of the moern heurstcs algorthms. In 1995, Kenney an Eberhart frst ntrouce the PSO technque, motvate by socal behavor of organsm such as fsh schoolng an br flockng. So, as a heurstc optmzaton tool, t proves a populaton-base search proceure n whch nvuals calle partcles change ther poston (states) wth tme. In a PSO system, partcles fly aroun n a multmensonal search space. Durng flght, each partcle austs ts poston accorng to ts own experence an the neghborng experences of partcles, makng use of the best poston encountere by tself an ts neghbors. Let x an v enote a partcle poston an ts corresponng velocty n a search space respectvely. The mofe velocty an poston of each partcle can be calculate usng the current velocty an the stance from pbest to gbest as shown n the followng expressons: (t ) (t ) (t ) v =. v + c 1 * ran 1 * ( pbest - x ) + c 2 * ran 2 * ( gbest - x ) (4.20) ( t 1) where, (t ) v : Velocty of partcle at teraton t. : Inerta weght factor c 1, c 2 : Acceleraton constant ran 1, ran 2 : ranom number between 0 an 1. (t) x : Current poston of partcle I at teraton t.

10 86 Pbest : pbest of partcle. gbest : gbest of the group. Smlar to other evolutonary algorthms, PSO must also have a ftness functon that takes the partcle s poston an assgns to t a ftness value. For consstency, the ftness functon s the same as for the other algorthms. The poston wth mnmum ftness value n the entre run s calle the global best (gbest). Also each partcle keep trackng ts mnmum ftness value, calle as local best (l best or pbest). Each partcle s ntalze wth a ranom poston an velocty. The velocty v t of the th partcle, each of n mensons, accelerate towars the global best an ts own personal best. PSO has a well balance mechansm to enhance both global an local exploraton abltes. Ths s realze by nerta weght w an s calculate by the followng expresson: max mn ter ter max (4.21) max where, max,mn s the ntal an fnal weght, termax s the maxmum teraton count, an ter s the current teraton number. From the above equaton certan velocty can be calculate, whch graually gets close to pbest an gbest. The current poston (searchng pont n the soluton space) can be mofe by the followng expresson: x ( t 1) x v ( t ) ( t 1), = 1,2,3,,n, an =1,2,3,..,m (4.22)

11 87 Crazness functon: The man rawbacks of the PSO are ts premature convergence, especally whle hanlng problems wth more local optma an heavly constrane. To solve ths, the concept of crazness, wth the partcles havng a preetermne probablty of crazness, s ncorporate, whle n general ncreases the probablty of fnng a better soluton n the complex oman. Thus, crazy agents are ntate, when they fn a premature convergence of the proceure. In ths thess, the probablty of crazness cr (entfcaton of partcles an ranomzng ts velocty) s expresse as a functon of nerta weght, to ensure the control of nerta weght urng the search. t cr mn exp (4.23) max Thus V t ran (0, v max), f cr ran (0,1) t V, otherwse (4.24) where, t s the nerta weght at the tth teraton of the run, an ran(0,1) s the ranom number between 0 an 1. It s obvous that, f the PSO proceure gets stuck n the begnnng of the run, a hgh value of cr wll be use to generate crazy partcles. Whle the run progresses, a comparatvely low value of cr wll be use to generate Crazy partcles. Thus the sgnfcance of control of nerta weght n the PSO algorthm s also retane. Partcle veloctes on each menson are clampe to maxmum allowable velocty v max f the sum of acceleratons excees the lmt. Vmax s

12 88 an mportant parameter that etermnes the resoluton wth whch regons between the present poston an the target postons are searche. If v max s too hgh, agents may fly past goo regons. If t s low, agents may not explore suffcently beyon locally goo regons. To enhance the performance of the PSO, v max s set to the value of the ynamc range of each control varable n the problem Optmzaton scheme of PSO ) Evaluaton of Each Partcle Each Partcle s evaluate usng the ftness functon of the problem to mnmze the fuel cost functon gven by (4.1). The best ftness value of each partcle up to the current teraton s set to that f the local best of that partcle p,. Lbest ) Mofcaton of Each Searchng Pont: Usng the global best an the local best of each partcle up to the current teraton, the searchng pont of each agent has to be mofe accorng to the followng expresson: p (4.25) ( t ) v ( t ) p ( t 1) where v ( t) w( t) v ( t 1) c ran ( p p ( t 1)) 1 1 L best, c ran ( p p ( t 1)) (4.26) 2 2 G best where ran 1&2 are ranom numbers between 0 an 1, an C 1, C 2 are acceleraton factors. Smlar to nerta weght, acceleraton factors also controls the exploraton of the PSO. These are the stochastc acceleraton

13 89 terms that pull each partcle towars P best an G best postons. Thus mprove performance of PSO can be obtane carefully selectng sutable values for nerta weght C 1 an C 2. Thus, new searchng ponts were explore for the next teraton to further explot the search. The elements of the new searchng pont matrx P (t) shoul be force to satsfy the real power generaton lmts gven n Fan an Mconal (1994). Once the new searchng ponts were etermne, nerta weght ha to be mofe usng (4.21). ) Mofcaton of the Global an the Local bests: Each partcle shoul be evaluate usng the ftness functon of the ynamc economc spatch, as was one n pont () n ths secton. P Gbest an P Lbest, have to be mofe accorng to the present ftness functon values evaluate usng the new searchng ponts of the partcle. If the best ftness value of all the ftness functon values s better than the O,best (t-1), then change P Gbest to ths value of the searchng pont of the corresponng partcle contrbute for ths best ftness value. Smlarly, the local best of other partcle n the populaton shoul be change accorngly f the present ftness functon value s better than the prevous one. v) Termnaton Crtera: Repeat from () untl the maxmum number of teratons s reache Implementaton of HTC problem usng PSO (C) algorthm To apply a PSO algorthm for optmzaton problem, some essental components nee to be esgne. The mplementaton of these PSO components for solvng the hyrothermal coornaton problem as escrbe below.

14 90 ) Representaton of tral soluton vector: generaton Accorng to the problem soluton methoology, a epenent thermal PT s ranomly selecte. The non-epenent thermal generaton PT for = 1, 2, Z,, are together taken as (Z 1) mensonal tral vector. Let P PT PT,... PT, PT,..., PT th component of a partcle. be the tral vector of the 1,, 2 ( 1) ( 1) z ) Intalzaton of Partcle of tral vectors: Let the partcle sze be N p. Each ntal parent partcle tral vector P, 1,2,..., N, s selecte ranomly from a feasble range n each p menson. Ths s one by settng the th components of each parent partcle as PT ran PT, PT mn max, for = 1, 2,.,( -1), ( + 1),.Z (4.27) where, ran PT mn, PT max enotes a unform ranom varable rangng over PT mn, PT max Fgure The above proceure s elneate n the flowchart shown n 4.4 NUMERICAL SIMULATION In ths secton two fferent hyrothermal test systems are solve to valate the propose technque. Frst system conssts of one hyro an one thermal unt, secon system comprses sngle hyro unt an ten thermal unts. Also, the secon system s solve for two fferent loa eman patterns. The avalablty of maxmum hyro unts are taken nto account an eucte from the total power eman. The balance loa eman s met by the thermal unts, such that the thermal unts to be commtte for generaton s efne by the hybr SA base technque. The PSO (C) has to be provng the optmal hyro scheule for 24 hrs base on the volume of the water reservor an scharge rate. The thermal unt commtment scheule has been ncorporate n the DEDP sub-problem an solve by the propose hybr SA-EP-TPSO technque, whch mnmzes the total generaton cost.

15 91 Input ata on loa pattern, fuel cost curve of equvalent thermal unt, operatng lmts on thermal an hyro unts, an reservor water volume lmts Input control varables of PSO Generate at ranom ntal partcle vectors t = 1 Estmate the thermal generaton an ts respectve generaton cost usng SA-EP-TPSO metho. t = t + 1 Calculate total water scharge Calculate epenent thermal generaton Are all generaton levels an water volumes wth ther lmtng values? No No Is t == Np? Yes Is stoppng rule satsfe? Yes No Yes Evaluate obectve functons corresponng to each partcle Output the hyrothermal scheulng results Upate the partcles velocty & poston Upate Inerta weght Fgure 4.2. Flowchart for the HTC problem by PSO (C) technque

16 92 Test System 1: In ths secton, the frst system s teste to valate the propose hybr PSO(C) technque. It comprses a hyro an an equalent thermal plant. The scheule horzon s 3 ays an there are sx 12-hr ntervals gven n Appenx A1.3. The test ata are taken from the Woo an Woolenberg (1984).Ths unt has got the prohbte operatng zones as gven n Table 4.1. Table 4.1 Prohbte Operatng Zones for the 1-unt DEDP system Unt Zone 1 Zone 2 Zone 3 Zone 4 1 [870, 910] [790, 810] [ ] [ ] To valate the feasblty of the propose algorthm, the test system was also solve usng the graent search algorthm an the PSO algorthm wthout nclung the crazy functon. The smulaton was conucte for 100 tral runs to stuy the robustness of the evelope PSO (C) base HTC algorthm. The followng smulaton parameters are selecte for the PSO algorthms to solve the HTC problem. The selecton proceure has been aopte from Aruloss an Ebenezer (2005). These smulaton parameters are foun to be most sutable for the test case aopte to emonstrate the feasblty of the propose algorthms. Partcle sze = 100, Maxmum nerta weght = 1.3, Mnmum nerta weght = 0.7, C1 = C2 = 2, maxmum velocty = P, k max = 30. For PSO the penalty parameters are taken as Termnaton of the PSO algorthms s one when there s no mprovement n the soluton for a pre-specfe number of teratons.

17 93 Table 4.2: Best Hyrothermal Scheules obtane by propose Technque Technque Interval Thermal Generaton Hyro Generaton Volume (acre ft) Dscharge (acre-ft h -1 ) Cost (Rs) 1 st ay 0:00-12: :00-0: Propose 2 n ay 0:00-12: :00-0: r ay 0:00-12: :00-0: Table 4.3 Best Hyrothermal Scheules obtane by Graent technque Thermal Hyro Technque Interval Generaton Generaton Volume (acre ft) Dscharge (acre-ft h -1 ) Cost (Rs) 1 st ay 0:00-12: :00-0: Graent 2 n ay 0:00-12: search 12:00-0: r ay 0:00-12: :00-0: Table 4.4: Best Hyrothermal Scheules obtane by PSO technque Thermal Hyro Technque Interval Generaton Generaton Volume (acre ft) Dscharge (acre-ft h -1 ) Cost (Rs) 1 st ay 0:00-12: :00-0: n ay 0:00-12: PSO 12:00-0: r ay 0:00-12: :00-0:

18 x 10 5 PSO(C) PSO Fgure 4.3 Best soluton obtane for 100 tral runs Best results obtane usng the propose metho an the graent search algorthms are shown n Table 4.2 an 4.3. Ths result was obtane for 71 tral runs. The average cost obtane by propose PSO base HTC algorthm s Rs , an even ths cost s comparatvely less than the best cost obtane usng the graent search algorthm an the PSO algorthm wthout the crazy agents. Durng the smulaton the crazy agents are generate at an average of 12 tmes when the nerta weght reuces below an average of The best soluton obtane by the propose hybr metho an PSO for 100 tral runs s plotte n Fgure 4.3. Test System 2: In ths case, the secon test system was analyze wth two fferent loa emans for the best hyro an thermal generators scheule for twenty-four hours. The optmal hyrothermal coornaton scheule for the propose hybr PSO(C)-SA-EP-TPSO metho s gven n the table 4.5 an 4.7. In the secon test system the reserve power also consere by satsfyng the mnmum an maxmum reservor volume wth other constrants. The numercal results are valate wth optmal scheule of the hyro an thermal generators. The test ata an the unts havng prohbte operatng zones are consere as same as n the chapter 2.

19 95 Table 4.5 Fnal optmal hyrothermal scheule usng the propose metho for the secon system (LD1). S.No Loa Deman Thermal Power Generaton PT Hyro Power Generaton PH Reservor Volume (acre ft) Water Dscharge (acre ft h- 1 ) Total Cost (Rs) =

20 96 Table 4.6 Fnal commtment scheule usng the Hybr SA-EP-TPSO technque for the secon system (Frst loa eman) Hour Thermal Power Unt Status From the above tables, the generaton cost obtane from the secon test system for the frst loa eman (LD1) usng the propose hybr technque s foun to be superor to that of the cost obtane usng graent search an other methos. After etermnng the hyroelectrc power the fnal commtte unts for the thermal generators usng the hybr SA-EP-TPSO technque are scheule an gven n table 4.6.

21 97 Table 4.7 Fnal optmal hyrothermal scheule usng the propose Hybr metho for the secon system (LD2). S.No Loa Deman Thermal Power Generaton PT Hyro Power Generaton PH Volume of the Reservor (acre ft) Water Dscharge (acre ft h- 1 ) Total Cost (Rs) = The total cost of power generaton obtane by usng the propose hybr SA-EP-TPSO s gven n table 4.7. In ths system some of the unts consere POZ as constrants an the propose technque coul overcome the nonconvexty n the soluton oman an coul prouce a qualty soluton.

22 98 The fnal generaton cost obtane from the secon test system for the secon loa eman (LD2) s foun to be superor n terms of the performance of the heavly constrane hyrothermal coornaton system nclues POZ n certan thermal unts n aton to the securty constrants. The fnal commtment scheule for the 10 unt thermal system s gven n table 4.8. To valate the propose hybr algorthm several stanar an IEEE-39 bus test systems are taken an solve. Table 4.8 Fnal commtment scheule usng the hybr SA-EP-TPSO technque for the secon system (secon loa eman) Hour Thermal Power Unt Status

23 CONCLUSION A hybr PSO(C)-SA-EP-TPSO base algorthm to solve the HTC problem s propose. In ths problem the prohbte operatng zone as an nequalty constrant, consere n the thermal generatng unts. All the thermal generatng unts are represente as sngle equvalent thermal plant. In the secon case a 10-unt thermal system s solve to scheule the commtte unts. The POZ complcates the soluton oman an makes the soluton algorthm to easly trap nto a local mnmum. To solve ths heavly constrane HTC problem, the PSO algorthm s mprove by nclung the crazness functon. The UCP an DEDP part were solve usng SA-EP-TPSO technques. The effcency of ths newly propose algorthm s llustrate usng a stanar test system. Qute mpressve results have been obtane by the propose technque n terms of fuel cost savngs. Also, Numercal results emonstrate that the propose technque proves a feasble scheule compare wth the graent search technque n terms of qualty an relablty. The smulaton tme can be sgnfcantly reuce by mplementng the propose algorthm n parallel processng machnes. (Ttus an Ebenezer Jeyakumar 2007).