Notes o owe System Load Flow Aalysis usig a Excel Woboo Abstact These otes descibe the featues of a MS-Excel Woboo which illustates fou methods of powe system load flow aalysis. Iteative techiques ae epeseted by the Newto-Raphso ad Gauss-Seidel methods. The Woboo also icludes two seach algoithms geetic algoithms ad simulated aealig.. Itoductio Load flow studies [,] ae used to esue that electical powe tasfe fom geeatos to cosumes though the gid system is stable, eliable ad ecoomic. Covetioal techiques fo solvig the load flow poblem ae iteative, usig the Newto-Raphso o the Gauss-Seidel methods. Recetly, howeve, thee has bee much iteest i the applicatio of stochastic seach methods, such as Geetic Algoithms [,4,5], to solvig powe system poblems. The iceasig pesece of distibuted alteative eegy souces, ofte i geogaphically emote locatios, complicates load flow studies ad has tiggeed a esugece of iteest i the topic. The piciples of powe system load flow studies ae taught withi elective modules i the late yeas of udegaduate electical egieeig couses, o as essetial compoets of specialist mastes pogammes i electical powe egieeig. Fom the educatioal viewpoit, theefoe, the topic is impotat, yet a complete coveage pesets some sigificat challeges. e-equisites iclude fudametal cocepts fom a.c. cicuit aalysis, such as phaso otatio, impedace ad admittace, powe ad eactive powe, thee-phase ad pe-uit systems, all of which ae egaded as difficult by may studets. The load flow solutio techiques big exta mathematical hudles, icludig matix epesetatio (with complex umbe coefficiets), iteative methods ad pobability fuctios. To assist i the teachig of load flow aalysis techiques the Excel Woboo illustates fou diffeet methods of solvig a simple load flow poblem, which evetheless has all of the featues to be foud i a lage-scale system. The Woboo allows studets to vay both the poblem beig solved, by adjustig the powe o voltage levels ad lie impedaces, ad also the paametes of the solutio methods, such as umeical acceleatio factos. These otes peset a oveview
of the geeal powe system load flow poblem ad descibe its solutio usig fou techiques Newto-Raphso, Gauss-Seidel, Geetic Algoithm ad Simulated Aealig. Also icluded ae illustative umeical esults elatig to the paticula powe system cofiguatio aalysed i the Woboo.. Fomulatio of the Load Flow oblem Load flow studies ae based o a odal voltage aalysis of a powe system. As a example, coside the vey simple system epeseted by the sigle-lie diagam i Fig.. Hee two geeatos ( ad ) ae itecoected by oe tasmissio lie ad ae sepaately coected to a load () by two othe lies. If the phaso cuets ijected ito the system ae I, I, ad I, ad the lies ae modelled by simple seies admittaces, the it is possible to daw the equivalet cicuit fo oe epesetative phase of the balaced thee-phase system, as show i Fig.. geeato, geeato, y y y I I load, I Fig. Sigle-lie diagam of a simple example powe system Fig. Equivalet cicuit fo oe phase of the system show i Fig. Fo the cicuit i Fig. the odal voltage equatios ca be witte diectly. Fo example, at ode ( y + y ) y y I () I geeal, fo a system with odes, the at ode I + +... + +... + () whee sum of all admittaces coected to ode - (sum of all admittaces coected betwee odes ad ) I cuet ijected at ode Fo the complete system of odes
I I I............. o [] I [ ][. ] () whee [] is the odal admittace matix. Fomulatio of the load flow poblem is most coveietly caied out with the tems i the odal admittace matix expessed i pola otatio θ. The Excel Woboo (Sheet ) allows uses to ete seies impedace data fo the thee lies ad the automatically calculates the tems i the odal admittace matix. Covetioal cicuit aalysis poceeds diectly fom equatio () by ivetig the odal admittace matix ad hece solvig fo the odal voltages []. Howeve, the load flow poblem is complicated by the lac of uifomity i the data about electical coditios at the odes. Thee ae thee distict types of odal data, which elate to the physical atue of the powe system a) Load odes, whee complex powe S s s +jq s tae fom o ijected ito the system is defied. Such odes may also iclude lis to othe systems. At these load odes, the voltage magitude ad phase agle must be calculated. b) Geeato odes, whee the ijected powe, s, ad the magitude of the odal voltage ae specified. These costaits eflect the geeato s opeatig chaacteistics, i which powe is cotolled by the goveo ad temial voltage is cotolled by the automatic voltage egulato. At the geeato odes the voltage phase agle must be calculated c) At least oe ode, temed the floatig bus o slac bus, whee the odal voltage magitude ad phase agle ae specified. This ode acts as the efeece ode ad is commoly chose to have a phase agle o. The powe ad eactive powe deliveed at this ode ae ot specified. I the system cofiguatio of Fig., which is aalysed i the Excel Woboo, each type of ode is epeseted with ode beig a floatig bus, ode beig a geeato ode ad ode beig a load ode. Cosequetly values must be specified fo the powe ( s ) ijected at ode, ad the powe ( s ) ad eactive
powe (Q s ) ijected at ode. All thee of these powe values may be chaged by the use, though default values ae povided ( s.; s -.5; Q s -.), with egative values idicatig that powe o eactive powe is beig daw fom the system. The magitude of the voltage at ode ca be specified, with the default value beig. pu, while the phase agle is fixed at o (. ). At the geeato ode (ode ), the voltage magitude ca be set by the use with the default value beig. pu (. ),ad the phase agle is calculated duig the load flow solutio. At the load ode (ode ) the voltage magitude ad phase agle have to be calculated ( ). So the complete load flow poblem fo this paticula powe system cofiguatio ivolves the calculatio of the voltage magitude ad the phase agles,.. Newto Raphso Method.. Geeal Appoach The Newto-Raphso method is a iteative techique fo solvig systems of simultaeous equatios i the geeal fom f ( x,.. x,.. x ) K f ( x,.. x,.. x ) K j (4) f ( x,.. x,.. x ) K whee f,...f...f ae diffeetiable fuctios of the vaiables x,...x,...x ad K,...K...K ae costats. Applied to the load flow poblem, the vaiables ae the odal voltage magitudes ad phase agles, the fuctios ae the elatioships betwee powe, eactive powe ad ode voltages, while the costats ae the specified values of powe ad eactive powe at the geeato ad load odes. owe ad eactive powe fuctios ca be deived by statig fom the geeal expessio fo ijected cuet (Eq. ) at ode I so the complex powe iput to the system at ode is S I (5) 4
5 whee the supescipt deotes the complex cojugate. Substitutig fom () with all complex vaiables witte i pola fom { } θ S (6) The powe ad eactive powe iputs at ode ae deived by taig the eal ad imagiay pats of the complex powe { } { } θ R cos S (7) { } { } Q θ I si S (8) The load flow poblem is to fid values of voltage magitude ad phase agle, which, whe substituted ito (7) ad (8), poduce values of powe ad eactive powe equal to the specified set values at that ode, s ad Q s. The fist step i the solutio is to mae iitial estimates of all the vaiables, whee the supescipt idicates the umbe of iteative cycles completed. Usig these estimates, the powe ad eactive powe iput at each ode ca be calculated fom (7) ad (8). These values ae compaed with the specified values to give a powe ad eactive powe eo. Fo ode { } s θ cos (9) { } s Q Q θ si () The powe ad eactive powe eos at each ode ae elated to the eos i the voltage magitudes ad phase agles, e.g.,, by the fist ode appoximatios + + + + + +. Q Q Q Q Q Q Q ()
whee the matix of patial diffeetials is called the Jacobia matix, [J]. The elemets of the Jacobia ae calculated by diffeetiatig the powe ad eactive powe expessios (7,8) ad substitutig the estimated values of voltage magitude ad phase agle. At the ext stage of the Newto-Raphso solutio, the Jacobia is iveted. Matix ivesio is a computatioally-complex tas with the esouces of time ad stoage iceasig apidly with the ode of [J]. This equiemet fo matix ivesio is a majo dawbac of the Newto-Raphso method of load flow aalysis fo lage-scale powe systems. Howeve, with the ivesio completed, the appoximate eos i voltage magitudes ad phase agles ca be calculated by pe-multiplyig both sides of () + + J. Q () The appoximate eos fom () ae added to the iitial estimates to poduce ew estimated values of ode voltage magitude ad agle. Fo ode + () + (4) Because fist-ode appoximatios ae used i () the ew estimates (deoted by the supescipt ) ae ot exact solutios to the poblem. Howeve, they ca be used i aothe iteative cycle, ivolvig the solutio of Equatios (9-4). The pocess is epeated util the diffeeces betwee successive estimates ae withi a acceptable toleace bad. The desciptio above elates specifically to a load ode, whee thee ae two uows (the voltage magitude ad agle) ad two equatios elatig to the specified powe ad eactive powe. Fo a geeato ode the voltage magitude ad powe ae specified, but the eactive powe is ot specified. The ode of the calculatio ca be educed by. Thee is o eed to esue that the eactive powe is at a set value ad oly the agle of the ode voltage eeds to be calculated, so 6
oe ow ad colum ae emoved fom the Jacobia. Fo the floatig bus, both voltage magitude ad agle ae specified, so thee is o eed to calculate these quatities... Applicatio of the Newto-Raphso Method to the Specific oblem Fo the example system show i Fig. ad aalysed i the Excel Woboo, thee ae thee uows (,, ) ad thee set values of powe ad eactive powe ( s, s, Q s ). Geeal expessios fo powe ad eactive powe iput at odes ad ca be deived fom (7) ad (8) { θ } + cos{ θ } + { θ } (5) cos cos { θ } + cos{ θ } + { θ } (6) cos cos { θ } + si{ θ } + { θ } Q (7) si si I the iteative solutio pocess, (5-7) ae used to calculate the powe ad eactive powe iputs fom latest estimates of ode voltages ad the usig (9,) to calculate the powe eos. The tems i the Jacobia ae obtaied by patial diffeetiatio of (5-7). Fo example cos { θ } + cos{ θ } + cos{ θ } (8) si { θ } (9) Ivesio of the x Jacobia with scala elemets is easily accomplished i Excel. () ca be used to deive the appoximate eos i the thee vaiables ad ew estimates fomed fom (,4). These ew estimates ae applied i the subsequet iteative cycle... Sample Results fo the Newto-Raphso Method Fo the default system iputs, defied i Sectio, the Newto-Raphso method, implemeted i the d sheet of the Excel Woboo, poduces the esults show i Table. I this simple example, oly 5 iteatios ae eeded to geeate esults which ae stable to thee decimal places, eve though the iitial estimates of all thee uows ae fa fom the coect value. Whe maig compaisos with othe methods of solutio, it is impotat to ealise that each iteative cycle of the Newto- 7
Raphso method ivolves cosideable computatioal effot, otably to ivet the Jacobia. 4. Gauss-Seidel Method 4.. Geeal Appoach The Gauss-Seidel Method is aothe iteative techique fo solvig the load flow poblem, by successive estimatio of the ode voltages. Equatio () ca be eaaged to give a expessio fo the complex cojugate of the cuet iput at ode + I + + () Substitutig fo S I fom () ito (5) ad e-aagig + + + () + S + () The ode voltage appeas o both sides of (), which caot, theefoe, be used to give a diect solutio. Howeve, this equatio is used i the Gauss-Seidel method as the basis fo a iteative solutio. If p ad at ode afte p ad p+ iteatio cycles, () ca be witte (p+ ) (p+ ) + (p) S + p+ deote the values of the voltage Note that i evaluatig the th ode voltage, the latest estimates of the othe ode voltages ae used. I the p+ th iteatio cycle whe pefomig the calculatio fo ode, the voltages at the odes - ae available, but fo the othe ode voltages the values fom the pevious (p th ) cycle have to be used. The foegoig discussio is appopiate to load odes. At the floatig bus the voltage is ow ad so does ot eed to be calculated. Geeato odes ae paticulaly poblematical fo the Gauss-Seidel Method. At these odes the powe ad voltage ( p) () magitude ae specified, so i () p+ caot be calculated because Q (the 8
imagiay pat of S ) is uow. This difficulty is addessed by fist calculatig Q, as follows (p+ ) (p+ ) # the voltage compoet ( ) + ( p) + (p) (4) ca be calculated immediately ad substituted ito () (p+ ) (p+ ) # jq ( ) + ( ) p (5) but, fo a geeato ode, the magitude is ow, so cosideig the magitudes i (5) (p+ ) # jq (p+ ) # jq R ( ) + + I ( ) + ( ) (6) p ( p) which ca be solved fo Q (by iteatio if ecessay). The calculated value of Q is substituted bac ito (5) ad the ew estimate of geeato ode voltage is foud. Whe compaed to the Newto-Raphso Method, the Gauss-Seidel Method ivolves simple calculatios, but is slow to covege. Theefoe, it is commo pactice to acceleate the iteative pocess, by addig to the ewly-calculated value of each vaiable a exta tem popotioal to the diffeece betwee the ew ad pevious values. Fo example (p+ ) acceleated (p+ ) + α. (p+ ) (p) { } whee α is a acceleatio facto, which has a typical value of.6. 4.. Applicatio of the Gauss-Seidel Method to the Specific oblem Fo the paticula -ode poblem, itoduced i Sectio, Equatio () ca be used diectly to calculate the voltage at the load ode (ode ) evey iteative cycle. To deal with the geeato ode (ode ), multiply (5), with, by ( p) (p+ ) ( p) ) (p+ # ( ) + jq ( p) (7) (8) ad expessig the phaso voltages i pola fom + (9) p p+ p # # θ θ jq the tem jq is imagiay ad ca be elimiated by equatig eal pats i (9) p p+ # p # cos{ θ } cos{ + θ } () p p+ # p # cos{ θ } cos{ + θ } () 9
# p + p p # θ accos cos{ + θ } () 4.. Sample Results fo the Gauss-Seidel Method The Gauss-Seidel Method, defied by (,4,5,7,), fo solvig the load flow poblem outlied i Sectio is implemeted i Sheet 4 of the Excel Woboo. Sample esults fo the default iput data ad usig a acceleatio facto α.6 ae show i Table. Compaig the esults fom Table with those fom Table, the slowe covegece of the Gauss-Seidel Method is evidet eve afte iteatio cycles the values of phase agle ae stabilised oly to a sigle decimal place. Howeve the computatioal effot ivolved i each cycle is much educed i the Gauss-Seidel Method. Nevetheless both iteative methods equie some complicated mathematical opeatios ivesio of the lage Jacobia matix i the Newto-Raphso Method ad itemediate calculatio of eactive powe iput at geeato odes i the Gauss- Seidel Method. 5. Stochastic Seach Techiques 5.. Geeal iciples Recet developmets i load flow aalysis have moved attetio away fom the iteative methods ad towads so-called stochastic seach methods. Two such methods Geetic Algoithms ad Simulated Aealig ae descibed hee ad ae implemeted i the Excel Woboo. Both appoaches use a seies of tial solutios to the poblem ad develop bette solutios i the light of expeiece gaied fom these tials. The computatioal effot fo each tial is ept as low as possible, so a vey lage umbe of tials ca be coducted. Fo the example poblem beig cosideed thoughout this pape, thee ae thee vaiables the voltage magitude ad the phase agles,. I ay oe tial some appopiate values ae chose fo these vaiables. The choice may be a etiely adom selectio acoss the etie possible age of values (temed the seach space ) o the choice may be ifomed by pevious expeiece. Oce these tial values ae chose, the phaso voltages at all thee odes ae defied, because all of the othe voltage magitudes ad phase agles ae fixed. Theefoe the
cuets ijected at each ode ca be evaluated diectly usig () ad the coespodig complex powe iput is calculated usig (5). The success of the tial eeds to be judged by some quatitative citeio. The tial values of ode voltage lead to values of iput powe ad eactive powe (, Q) that do ot exactly match the pe-defied values ( s, Q s ). The extet of the mis-match ca be quatified coveietly, fo this paticula poblem, with the eo fuctio E + + () ( s ) ( s ) ( Q Qs ) The stochastic seach techiques use the eo fuctio to ifom the selectio of ew potetial solutios fo the subsequet oud of tials. It is this selectio pocess which is defied by the paticula seach techique. 5.. Geetic Algoithms Geetic Algoithms imitate the pocess of evolutio, whee the fittest idividuals ae liely to suvive i a competig eviomet. A geetic algoithm [,4] stats with a adom populatio of potetial idividuals, o chomosomes, each epesetig oe possible solutio to a poblem. The chomosomes ae simply a collectio of gees, each gee beig oe of the solutio vaiables. The chomosomes ae the evolved though successive geeatios. Duig each geeatio, all the chomosomes ae evaluated, accodig to a defied fitess citeio, ad the best chomosomes ae selected to mate ad geeate offspig. The least fit chomosomes of each populatio ae the eplaced by the offspig so that the populatio size emais costat. Afte seveal geeatios, the algoithm coveges to the best chomosome which epesets a optimal solutio to the poblem. A futhe efiemet of the evolutio pocess, agai mioig atue, is that ay chomosome i ay geeatio has a fiite pobability of suffeig mutatio, i which some of the gees ae adomly petubed. It is this pocess which esues that the geetic algoithm does ot covege to a local miimum whe seachig fo a global poblem solutio. Whe applied to the load flow poblem, the gees ae the odal voltage magitude ad phase agle values ad each chomosome cotais a complete set of the gees eeded to defie uiquely a tial solutio. The fitess of each chomosome is evaluated usig the eo citeio (), which is used as the basis of selectio fo the
chomosomes i the ext geeatio. Fo the example poblem, a geetic algoithm solutio is implemeted i Sheet 5 of the Excel Woboo. Results fom the commo example poblem duig the ealy stages of the algoithm ae show i Table. A iitial populatio of chomosomes foms the fist geeatio. Chomosome is placed at the cete of the seach space, chomosomes -9 ae located at the extemes of the seach space ad chomosome has adom values. The values of the gees, voltage magitude ad the phase agles,, fo each chomosome ae show i ows - of the Table. Each chomosome, i associatio with the pe-defied values of, ad, defies a set of tial odal voltages, which ae used to calculate the iput cuets (ows 4-6) ad the powe / eactive powe iputs (ows 7-9). The set values fo the powe iputs ae ( s.; s -.5; Q s -.), ad the eo fuctio defied i () is evaluated fo each chomosome at ow with a low value of eo fuctio idicatig a close match betwee the calculated ad set powe values. The chomosomes ae the aed (ow ) by eo. Rows - epeset oe complete geeatio of the geetic algoithm s evolutioay pocess. Chomosomes fo the secod geeatio ae deived fom the pevious geeatio s chomosomes accodig to thei aig. Row idicates how each chomosome i the secod geeatio has bee fomed. The fist two chomosomes ae copies of the two highest-aed chomosomes of the pevious geeatio. Chomosomes -5 ae obtaied by beedig betwee chomosomes ad i each case oe gee i chomosome is eplaced by the coespodig gee fom chomosome. Each gee i chomosome 6 taes the aveage value of the gees i chomosomes ad. Mutatio taes place i chomosomes 7-9, with oe gee i tu fom chomosome beig eplaced by a adom value. Fially, chomosome of the secod geeatio has gees which ae etiely adomly-selected. So the ew geeatio has chomosomes, most of which ae elated to the best chomosomes fom the pevious geeatio.
The cycle of chomosome evaluatio ad beedig cotiues though may geeatios. Evaluatio of the eo fuctio fo each chomosome equies elatively simple calculatios, so the geetic algoithm ca cotiue though may geeatios util the eo has falle to acceptable levels. A typical vaiatio of eo with geeatio umbe is show i Fig.. The geetic algoithm quicly educes the eo, but a vey lage umbe of geeatios is eeded to big the eo close to zeo. I the Excel Woboo, acceleated covegece is obtaied afte geeatios, by edefiig the seach space so that it is ceted o the best available solutio at that stage ad is educed i size. The effect of this edefiitio is appaet i Fig., whee the eo eductio eceives fesh impetus afte the th geeatio.. eo.5..5. 5 5 umbe of geeatios Fig. Typical eo associated with the best chomosome as a fuctio of the umbe of geeatios 5. Simulated Aealig Simulated aealig is a global seach techique i which a adomly-geeated potetial solutio,, to a poblem is compaed to a existig solutio, X. The pobability of beig accepted fo ivestigatio depeds o the poximity of to X ad the extet to which the solutio has bee developed, as epeseted by a tempeatue paamete, T, which educes thoughout the aealig pocess. Both potetial solutios ae ivestigated ad is chose to eplace X as the existig
solutio accodig to a pobability fuctio, which agai depeds o the tempeatue T. To apply this cocept to load flow studies i geeal, it is assumed that the solutios X ad,, epeset ifomatio about possible odal voltage values. I the paticula illustative example defied i Sectio, thee ae thee uow voltage values, so the solutios ca be witte [,, ] [, ] X X X X, (4) whee these solutios must lie withi the pe-defied seach space. Table 4 pesets sample esults fom the simulated aealig techique, which is implemeted i Sheet 6 of the Excel Woboo. The solutio commeces o ow with a iitial best value, which is placed at the cete of the seach space ad its eo E X evaluated usig (). The age idicates the extet of the seach space ad is defied as ( ) + ( ) + ( ) MAX MIN MAX MIN MAX MIN age (5) A ew set of voltage values,, ae selected at adom. The displacemet is the distace betwee X ad i the seach space displacemet ( ) + ( ) + ( ) (6) X X X A acceptace pobability, A, is the calculated displacemet Ts A exp (7) age T whee T is the istataeous tempeatue ad Ts is the iitial tempeatue. A is compaed to a adom value i the age [ ]. If > A, the potetial solutio is ejected ad the pocess is epeated. If < A the is accepted fo evaluatio. Note that fo small values of displacemet ad high values of tempeatue, the acceptace pobability is close to uity, so potetial solutios ae moe liely to be accepted fo evaluatio if they ae close to the existig solutio o if the tempeatue is high. Fo example, i the fist ow of Table 4, the displacemet betwee X ad is small (.) ad the tempeatue is equal to the iitial tempeatue. Theefoe the 4
calculated acceptace pobability is high (.8). The adom umbe.5 ad so is accepted fo evaluatio. Whe is accepted fo evaluatio the coespodig odal voltages ae used to calculate the iput cuets {fom ()} ad the complex powe iputs {fom (5)}. Hece the eo fuctio E ca be foud fom (). This eo is compaed to the eo E X obtaied fo the solutio X i the swap pobability fuctio, S S E + exp E E X X Ts T S is compaed to a adom value i the age [ ]. If > S, the the oigial solutio X is etaied, ad, if < S, the ew solutio, is accepted, ad X is eplaced by. Substitutio of fo X is most liely to occu if the eo Ey is small, i which case the swap pobability is high. Howeve the tempeatue T also iflueces the lielihood of swappig. I ow of Table 4 the values of E ad E X ae almost equal ad the swap pobability is.5. Howeve the adom umbe is.49, so a swap of fo X does occu. Theefoe i the secod ow of Table 4 the cuet best values ae the values fom ow. Looig moe geeally at Table 4, i ow the ext set of adom values ae accepted fo evaluatio, but poduce a eo E (.9) which is substatially lage tha E X (.9), esultig i a low swap pobability of.. The value of geeated is.68, so a swap does ot occu. Note, howeve, that thee is a fiite pobability of a wose value beig substituted. The pobability of such a evet happeig dimiishes as the tempeatue educes. Wheeve a adom value is too fa displaced fom the cuet best value, it may ot be accepted fo evaluatio, as happes i ow. The cuet ad powe calculatios do ot eed to be made i these cicumstaces. May alteatives fo the decemet fuctio of the tempeatue, T, ae available. I this wo, the decemet fuctio poposed by Xi [6] is used, with the tempeatue beig ivesely popotioal to the umbe of potetial solutios ivestigated. Covegece is assisted by e-defiig the seach space ad esettig the tempeatue to its iitial value afte evey values have bee cosideed fo acceptace. Fig. 4 shows how the eo fom the best values teds to educe with the umbe of potetial solutios ivestigated. Howeve thee ae istaces whee the (8) 5
eo iceases. Eve though the eo poduced by a ew set of tial values may be lage tha the best value eo, thee is a fiite pobability that a swap will occu, esultig i a iceased best value eo. I Fig. 4 the iceases i eo ted to occu whe the tempeatue is high. Compaig the two seach methods, geetic algoithms ad simulated aealig, the umbe of calculatios equied to poduce a acceptable eo is simila fo the default powe system paametes the geetic algoithm opeates ove geeatios, with 8 ew chomosomes to be evaluated i each geeatio, givig a total of 6 evaluatios. Simulated aealig equies i the ode of 7 potetial solutios to be ivestigated. If a adom seach was coducted acoss the etie seach space the to obtai compaable esolutio i the solutio would equie the evaluatio of appoximately voltage magitude values ad values of each phase agle, givig a total of xx.x 7 evaluatios.. eo.5..5..5 4 5 6 7 umbe of ivestigated solutios Fig. 4 aiatio of best value eo E X with the umbe of solutios accepted fo evaluatio 6. Summay of Results ad System Calculatios Sheet 7 of the Excel Woboo summaises the esults of the odal voltage calculatios fom the fou methods. The eo fom the fial best value fom the two seach methods is also peseted, so the use is aleted to ay failue to aive at a solutio with the appopiate accuacy. If ecessay, the calculatio cycle ca be epeated by pessig the (F9) ey. 6
The ultimate pupose of load flow studies is to calculate the flow of powe ad eactive powe though the system. Theefoe, icluded i the Sheet is the calculatio of the powe ad eactive powe flow though each of the thee lies ad a calculatio of the total powe ad eactive powe cosumed by the tasmissio system. (Note that the default system paametes defie a lossless system, so the powe cosumed is zeo.) 7. Coclusios The Woboo ca be used fo two diffeet aspects of egieeig educatio i) as a itoductio to load flow studies fo powe systems studets, i which case the focus of attetio is the system data (Sheet ) ad the load flow esults o Sheet 7, without beig coceed with the calculatio methods; ii) as a example poblem fo couses itoducig umeical methods of poblem solvig, usig both iteative methods (Sheets ad 4) ad stochastic seach methods (sheets 5 ad 6). 8. Refeeces [] A.E. Guile ad W.D. ateso, Electical powe systems, ol., (egamo ess, d editio, 977). [] W.D. Steveso J., Elemets of powe system aalysis, (McGaw-Hill, 4 th editio, 98). [] K.F. Ma, K.S. Tag, ad S. Kwog, Geetic algoithms cocepts ad applicatios, IEEE Tasactios o Idustial Electoics, 4 (996), 5, pp. 59-5. [4] M. Ge, ad R. Cheg, Geetic algoithms ad egieeig desig, (Joh Wiley & Sos, Ic., 997). [5] J.X. Xu, C.S. Chag, ad X.W. Wag, Costaied multiobjective global optimisatio of logitudial itecoected powe system by geetic algoithm, IEE oceedigs, Geeatio, Tasmissio & Distibutio, 4 (996), 5, pp 45-446. [6] Xi ao, A ew simulated aealig algoithm, Iteatioal Joual of Compute Mathematics, 56 (995), pp 6-68. 7
Iteatio o o..9 -. -.8.4 4.77 -.46.9 -.8 -.6.5-5. 4 -.4.5-5.78 5 -.4.5-5.78 Table Results fom the Newto-Raphso method Iteatio o o..9 -. -6.56.7-9.985 -.75.46-7.85 -.5.48-5.95 4 -.77.5-5.65 5 -.55.49-5.49 6 -..5-5. 7 -.54.5-5.79 8 -.5.5-5.5 9 -.9.49-5.7 -.77.49-5.6 Table Results fom the Gauss-Seidel Method 8
st Geeatio Chomosome No. 4 5 6 7 8 9 (adias)..75.75.75.75 -.75 -.75 -.75 -.75.7....9.9...9.9.97 (adias)..75 -.75.75 -.75.75 -.75.75 -.75.6 4 I 5 I 6 I.+.4j. +.4j. +.j -.7 +.75j -.7 +.75j.95 -.4j.9 +.75j.9 +.75j -4.78 -.4j -.55 -.4j -.55 -.4j.4 +.54j.7 -.4j.7 -.4j -4.5 +.54j -.9 +.75j -.9 +.75j 4.78 -.4j.7 +.75j.7 +.75j -.96 -.4j -.7 -.4j -.7 -.4j 4.5 +.58j.55 -.4j.55 -.4j -.4 +.54j -.97 +.4j.9 +.7j.8 +.57j 7..764 4.9.764 4.5-4.9 -.764-4.5 -.764 -.5 8..955-5.9.78-4.67 5.9 -.955 4.67 -.78.5 9 Q -..6.6 -.8 -.585.6.6 -.585 -.8 -.46 E.97.6 5.5. 4.54 8.986.6 7.78.748.546 aig 8 5 7 9 4 6 best d best d Geeatio Chomosome No. beed fom best beed fom best beed fom best beed fom best mutate best mutate best mutate best 4 5 6 7 8 9 4 (adias). -.75.. -.75 -.87...8.99 5......5..977..97 adom 6 (adias). -.75 -.75.. -.87 -.4...75 Table Opeatio of the Geetic Algoithm duig the fist geeatio 9
Cuet best values New adom values Acceptace Calculate Cuets Calculate owe Swap Cout x (adias) x x (adias) E x T s /T (adias) (adias) displacemet A adom, accept? I I Q E S adom, swap?....97......8.5....9. -..9..8.69.49. -.4j -.6 -.8j.6 -.45j.6 +.7j....9. -.6.98 -...6.7 N.... 4....9. -.. -.4.9.66.6 5 -.. -.4.75...9...46.4 6 -.. -.4.75.4 -.. -.8.6.86.9.9 -.45j.8 -.6j.5-.5j -.76 +.8j. +.j -.9 +.94 7 -.. -.4.75.5 -.7.96..7.47.5 N.... 8 -.. -.4.75.5 -.8.6..6.48.5 N.... 9 -.. -.4.75.5 -..9.6..55.46 -.8 -.j. +.j Table 4 Typical iitial esults fom Simulated Aealig..6.5.9.5.49 -.6.59 -.99.9..68 N..8 4.5 7.69 4.7 -.9 -.8.75.68.4.6.5 -..74.. N.4 -.98 -.86..8.66 N...8 4.8 4. 6. 6.54 4.94 7 -.86.4 -.99.6..5 N R E J R E J R E J