Optimal power dispatch of DGs in DC power grids: a hybrid Gauss-Seidel-Genetic-Algorithm methodology for solving the OPF problem

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1 Optmal power dspatch of DGs n power grds: a hybrd Gauss-Sedel-Genetc-Algorthm methodology for solvng the OPF problem Oscar Danlo Montoya Unversdad Tecnológca de Bolívar Km 1 vía Turbaco Cartagena, COLOMBIA omontoya@utb.edu.co Walter Gl-González Unversdad Tecnológca de Perera Carrera 27 # Barro Alamos Perera, COLOMBIA wjgl@utp.edu.co Lus F. Grsales-Noreña Insttuto Tecnológco Metropoltano Calle 73 No. 76A - 354, Vía al Volador Medelln, COLOMBIA lusgrsales@tm.edu.co Abstract: Ths paper addresses the optmal power flow (OPF) problem n drect current () power grds va a hybrd Gauss-Sedel-Genetc-Algorthm methodology through a master-slave optmzaton strategy. In the master stage, a genetc algorthm s employed to select the power dspatch for any dstrbuted generator whle the slave stage, Gauss-Sedel method s used for solvng the resultng power flow equatons wthout recurrng to matrx nversons. Ths approach s mportant snce t can be easly mplementable over any smple programmng toolbox fndng the optmal soluton of the OPF problem. Genetc-Algorthm proposed n ths paper corresponds to a contnuous varant of the conventonal bnary approaches. Computatonal results show the effcency and accuracy of the proposed optmzaton method when s compared to GAMS/CONOPT nonlnear solver. Key Words: Drect current power grds, dstrbuted generaton, Gauss-Sedel method, genetc algorthm, hybrd master-slave optmzaton strategy, optmal power flow problem. 1 Introducton 1.1 General context Electrcal power grds are an ndspensable part of the human development ncludng all technologcal advances, whch have allowed mprovng people qualty lfe [1, 2]; nevertheless, conventonal electrcal power systems have also produced harmful effects around the world manly evdenced as global warmng, whch s caused by the consumpton of fossl fuels (transportaton system and thermo-electrc plants) producng a lot of tons of greenhouse effect gases [3]. To deal wth these problems new paradgms n electrcal systems have been developed n recent decades as are the cases of smart grds and mcrogrds, based on a combnaton of renewable energy resources and energy storage technologes as can be seen n Fg These combnatons allow replacng gradually the dependence of fossl fuels for electrcty generaton [3]. To supply all power consumpton for the constantly ncreasng demand two man electrcal dstrbuton technologes based on alternatng and drect current (AC and ) have been developed as well as ther hybrd combnatons [5]. Power grds operatng under AC reference frame 1 Ths fgure was transformed from the AC confguraton presented n [4] nto an equvalent grd Load 1 Load 2 G Wnd P AC Solar AC power grd MICROGRID + ESS M motor Fgure 1: Typcal nterconnecton of dstrbuted energy resources, whch conform a power mcrogrd have been wdely explored n specalzed lterature from the pont of vew of dynamcal and statc analyss [6],.e., under transent and steady-state condtons. In case of dynamcal analyss, dfferental equaton methods are requred, whle n case of statc analyss nonlnear equatons appear n the numercal reasonng [7, 8]. In ths sense, power flow analyses n case of statc approaches correspond to one of the most studed problems n AC power grds by usng lnear and nonlnear technques [9, 10, 11]. Electrcal power grds are not the excepton E-ISSN: X 335 Volume 13, 2018

2 of these studes, manly when power electroncs have allowed real power grd mplementatons [5], for mprovng the conventonal dstrbuton system performance [12]; followng ths lne, power flow analyss corresponds to the essental technque for plannng and operaton [13, 14, 15], whch becomes ths research topc to an excellent opportunty to propose novel, effcent and easly mplementable solvng methodologes. 1.2 Motvaton When power grds are analyzed va power flow methodologes, t s necessary to know that n specalzed lterature the expresson optmal power flow corresponds to an AC smplfed power flow formulaton and t s not related wth power flow analyss n drect current power networks [14, 16]; nevertheless, the usage of the same expresson for two dfferent problems could cause confuson between non-famlarzed readers. For ths reason, we prefer to use the complete name of the problem (.e., optmal power flow analyss n power grds) to make reference to ths research area. Ths work s motvated by two man reasons. The frst corresponds to the needed of mplementng an optmal power flow n mcrogrds for obtanng the power values of the dstrbuted generators that allow reducng the actve power losses and satsfyng the techncal restrctons of the system. The second reason s the mportance of havng easly mplementable tools for solvng mportant and recurrent problems n electrcal engneerng as s the case of optmal power flow analyss avodng to recur to sophstcated software or optmzaton packages for solvng these problems. 1.3 Bref state-of-art In specalzed lterature exst multple research papers whch analyses the OPF problem n electrcal power grds. These nvestgatons can be dvded nto an AC and power grds, respectvely [9]. Nevertheless, t s mportant to pont out that both OPF models share the same characterstcs n terms of mathematcal complexty,.e., nonlnear, non-convex problem [14]. We focus ths revew of state of the art on power grds from hgh-voltage to low-voltage power grds. For solvng the OPF problem n power grds have been proposed equvalent convex formulatons of the problem as presented n [17] and [18]. The frst case proposes a convex reformulaton of the OPF equatons va semdefnte programmng by relaxng the non-convex constrant assocated to rank one of the matrx of varables, then, after solvng the OPF problem, the voltage profles are recoverng va egenvalues and egenvectors decomposton [19, 20]. In the second case, a second-order cone programmng model s proposed, the authors apply the same relaxng concept to solve and recover the soluton varables. Both approxmatons are compared to the exact soluton of the problem obtaned through a GAMS optmzaton package wth a hgh grade of fdelty n terms of objectve functon. A port-hamltonan approach for solvng OPF problems n power grds s proposed n [21]. Ths formulaton guarantees stablty propertes n the sense of Lyapunov for passve crcuts; nevertheless, not constant power loads are takng nto account n the formulaton, whch reduces ts applcablty to lnear crcuts [22]. On the other hand, the exstence of the power flow solutons for power grds have also studed n [13, 14, 23]. They manly focus on the convergence propertes of the power flow equatons and ther solvng regon; nevertheless, they do not analyze the OPF for power grds drectly, snce ther objectve s to analyze the structural and geometrcal propertes of the power flow equatons. Notce that the mcrogrds control theory solves the OPF problem when analyses herarchcal controllers [24] or consensus algorthms [25]; notwthstandng, they concentrate ther analyss on the control desgn from the dfferental equatons pont of vew, whch relegates the OPF for problem to a second plan snce constant power loads are ncluded nto the grd va power electronc converters, whch facltate ther manpulaton n terms of stablty propertes [26, 27, 28]. In terms of optmzaton, some approxmatons of the OPF problem for power grds have been presented, and ther correspondng OPF equatons are solved va optmzng packages and optmzaton technques [17, 29, 30] consderng the possble nterconnecton of dstrbuted energy resources, ncludng wnd and photovoltac generaton as well as battery energy storage systems [31, 32, 33, 34]. It s mportant to stand out that n the revson of state-of-the-art made n ths paper was found that the ntegraton of DGs n grds s a research topc n progress, for ths reason n the specalzed lterature exsts low documentaton and nvestgatons about of optmal szng of dstrbuted generaton n electrcal dstrbuton system. The aforementoned stuaton hghlghts the mportance of explorng ths problem and proposng new methodologes n ths research lne. Addtonally, to the best knowledge of the authors, n the specalzed lterature, the OPF problem for power grds have not addressed from hybrd E-ISSN: X 336 Volume 13, 2018

3 metaheurstc optmzaton technques (.e., Genetc- Algorthm) and conventonal numercal methods (.e., Gauss-Sedel), whch s a clear gap that ths research tres to fll. 1.4 Contrbuton and scope Ths paper presents as man contrbuton the hybrdzaton of a conventonal numercal method known as Gauss-Sedel [10] and optmzaton technque named genetc-algorthms [35] for solvng the optmal power flow problem n power grds. Addtonally, we present the possblty to adapt the classcal bnary-nteger genetc-algorthm for solvng contnuous optmzaton problems. Another mportant fact, t s that we propose the soluton of the optmal power flow problem n power grds from the pont of vew 100% algorthmc avodng the needed of usng any specalzed software to carry out ths task. The optmal power flow analyss for power grds presented n ths paper assumes that electrcal network has been desgned to support all power consumpton guaranteeng voltage stablty condtons [36], whch mples that we assume that the conventonal power flow equatons exhbt soluton for any power load and dstrbuted generaton value n the range of analyss [23]. 1.5 Document organzaton The reman of ths document s organzed as follows: Secton 2 presents of mathematcal formulaton for the optmal power flow problem n power grds by usng ts nonlnear non-convex representaton. Secton 3 shows the hybrd Gauss-Sedel Genetc-Algorthm methodology, focusng on the man aspects assocated to the evoluton strateges n the genetc algorthm as well as the recursve equatons for solvng power flow equatons va Gauss-Sedel numercal method. Secton 4 presents the man characterstcs of the test system and the proposed smulaton scenaros. Secton 5 shows computatonal results va MATLAB software and ther comparson to GAMS conventonal optmzaton package. Fnally, some concludng remarks are provded n Secton 6. 2 Mathematcal formulaton For obtanng the general formulaton of the optmal power flow problem n a grd, let us consder a grd as a set of nodes represented by N = {1, 2,..., n}, a set of generator termnals G N and a set of constant power loads L N. The grd lnes are represented by a set E = {(, j)} N N and the nodal conductance matrx s defned as G bus R n n whch s a symmetrc and postve semdefnte matrx such that [G],j = G j. Notce that, any consumpton modeled as a constant resstance value s ncluded nto the conductance matrx snce ts mathematcal model s defned by a straghtforward lnear relaton (.e., Ohm s law). Besdes, t s mportant to hghlght that there s only two contnuous varables per node, ths s, the voltage profle v and the net power njected p, for the th node, respectvely [37, 13]. Optmal power flow problem for power grds can be formulated as a nonlnear-non-convex optmzaton problem [13, 14], as follows: Objectve functon: mn z = N Set of constrants: G j v v j 2 G 0 v (1) j N p g pd = j N G j v v j { N }, (2) v mn p g,mn v v max { N }, (3) p g pg,max { G}, (4) where p g s the actve power generated n the node, p g,mn and p g,max correspond to the mnmum and maxmum power generaton lmts for each generator located n the node, respectvely; v mn and v max are the mnmum and maxmum allowed voltage profles at node, whle z s the total actve power losses n the network. Notce, that G 0 represents the constant mpedance load connected at th, whch corresponds to lnear consumpton (resstve load n the network), and t can not be consdered part of the actve power losses as presented n the frst part of 1, (remember that the conductance matrx contans all resstve effects n the network ncludng the constant resstve loads). The mathematcal optmzaton model gven from (1) to (4) has the next nterpretaton. Equaton (1) determnes the total actve power losses n the grd caused by the resstve effects n all dstrbuton lnes, these actve power losses are calculated as functon of the voltage profles n the entre network; expresson (2) determnes the power balance per node,.e., ths equaton corresponds to a set of nonlnear non-convex equatons wdely well-known n specalzed lterature as power flow equatons [13, 36, 37]. On the other hand, expressons (3) and (4) are bounded by constrants assocated wth voltage regulaton polces and power capabltes n all power generators. E-ISSN: X 337 Volume 13, 2018

4 Notce that ths paper focuses on the possblty to decouple the optmzaton problem above presented n two subproblems, whch allows solvng t va numercal methods wthout approxmatng ts mathematcal model; The general OPF problem s composed frst by the optmal selecton of the total power generated by any dstrbuted generator, and second, by the calculaton of the voltage profles n the entre power grd, for ths reason, we propose a hybrd Gauss-Sedel Genetc-Algorthm (GS-GA) for decouplng ths optmzaton problem nto a generaton problem named master problem and classcal power flow problem named slave problem. The man advantage of ths approach les that for solvng the optmal power flow problem n power grds any specalzed optmzaton package or specalzed software s requred to enhance ts optmal soluton, snce t corresponds only to an algorthmc soluton (evoluton optmzaton process), as wll be evdenced n next sectons. 3 Proposed optmzaton methodology The optmal power flow problem for power grds s addressed n ths paper from the pont of vew of master-slave soluton strategy where the master problem defnes the power generaton for each dstrbuted generator through a genetc-algorthm optmzaton approach that guarantees mnmum power losses n the grd; whle the slave problem solves the conventonal power flow equatons va Gauss-Sedel numercal method [14]. 3.1 Master problem In general terms the master problem conssts to determne the power generaton n each power controlled node,.e., n all dstrbuted generators wthout the capablty to control voltage profle. In ths sense, we consder: Assumpton 1 The power grd contans at least one constant voltage node. A constant voltage node s completely necessary to avod trval soluton to the power flow equatons for electrcal AC or power grds, n ths sense, ths node has the capacty to generate (absorb) the mssng (excessng) power n the entre electrcal network, n other words, ths node s wdely used n specalzed lterature as oscllatng node or slack node [8, 17]. Consderng that, the constrant (2) can be rewrtten as follows: p g pd = j N G j v v j { N S}, (5) v k = vk c { k S}, (6) R { k S}, (7) where S represents the set of constant voltage nodes wth well-know output voltages vk c. Now, t s mportant to pont out that to solve (5),.e., to fnd all voltage profles n the remans of set of nodes, we need to know the power generaton or consumpton n all these nodes, whch does not happen yet, snce p g s an unknown varable for each generaton node. Based on the aforementoned requrement, a genetc algorthm s proposed to determne the power generaton n all dstrbuted generators. 3.2 Genetc algorthm We propose a contnuous genetc algorthm metaheurstc technque to solve the optmal power flow problem consdered that n each step the slave problem has been solved satsfactorly as wll be presented n next secton. Now, we are gong to explan the man aspects of the genetc algorthm mplementaton. Followng ths lne, a genetc algorthm corresponds to a classcal well-known optmzaton technque to solve manly bnary-nteger optmzaton problems,.e., mult-stage transmsson plannng [38] or optmal placement and szng dstrbuted generators n dstrbuton networks [39], among others; nevertheless, multple authors have prevously adapted ths optmzaton technque for contnuous optmzaton, such as, optmzaton of nonlnear contnuous functons [40], second-order boundary dfferental equatons [41] or optmal AC power dspatch [42], among others; wth satsfactory results. The genetc algorthm has fve man characterstcs to know: p g k. Generaton of the ntal populaton.. Ftness functon calculaton.. Genetc operators for generatng the descendng populaton. v. New populaton calculaton. v. Stoppng crtera. All of them are extremely mportant to solve satsfactorly any optmzaton problem va genetc algorthms, for ths reason, each one of them s gong to be explaned as follows: E-ISSN: X 338 Volume 13, 2018

5 3.2.1 Intal populaton Ths s the frst step for any optmzaton technque, n ths sense, we propose a populaton wth a sze of a rows and s columns,.e., an a s matrx, where a corresponds to the number of potental solutons named set of ndvduals and s s the number of dstrbuted generators to be dspatched (s = S ). In the case of the power flow problem, ths ntal populaton has the followng structure: p g 11 p g 12 p g 1k p g 1s p g 21 p g p g 2k p g 2s p g l1 p g l2... p g lk... p g ls p g a1 p g a2... p g ak... p g as a s where p lk represents the actve power generated by the generator k at the l soluton ndvdual. Ths value s calculated as random number contaned between ( p g,mn k and ) p g,max k,.e., p g lk = p g,mn k + p g,max k p g,mn k rand, where rand (0, 1). Notce that, ths ntal populaton fulflls generaton capabltes defned by (4), whch mples that all row n the ntal populaton s feasble n terms of power generaton Ftness functon The ftness functon n metaheurstcs theory corresponds to the performance functon assgned to any ndvdual contans n the populaton, n other words, t determnes what s the qualty of an arbtrarly soluton l. It s mportant to menton that genetc algorthms solve optmzaton problems by becomng a constrant optmzaton problem nto a condtonal problem. For ths reason, we propose the followng ftness functon. z = z + f p N ( f ( v, v mn, v max )), (8) where z represents the ftness functon, f p corresponds to the penalty factor (f p 0) and f ( ) represents a bnary functon whch s calculated as presented below: f ( v, v mn, v max ) = 1, v < v mn 1, v > v max 0, otherwse, { N } Notce that to calculate the ftness functon s requred to know the voltage profles n all nodes of the system, whch wll be solved va slave problem. Addtonally, ths penalty strategy tres to elmnate any ndvdual such that presents bad voltage performance by assgnng to ts ftness functon a hgher losses value; on the other hand, f the voltage profle s fulflled n all nodes of the system, then, the ftness functon corresponds to the real actve power losses of the power grd,.e., z = z. It s mportant to hghlght that penalty factors are commonly employed for evolutve algorthms, snce they allow explorng nfeasble regons, that would be closed to promssory solutons Descendng populaton As a genetc algorthm corresponds to an teratve optmzaton process t s necessary to generate new potental solutons to the studed problem, to replace the bad solutons contaned n the current populaton. To generate ths set of solvng ndvduals a classcal selecton, recombnaton and mutaton operators are adapted to solve contnuous optmzaton. Selecton: The descendng populaton starts selectng an arbtrary subset of ndvduals contaned n the current populaton, n ths selecton a random number r between 1 to a s chosen,.e., r = 1 + (a 1)rand. If r < a, an addtonal (a r) s matrx wth potental solutons are generated by usng the same strategy employed for the ntal populaton. The total set of selected ndvduals are conformed by the combnaton of the both aforementoned strateges. Recombnaton: Ths process alters the descendng populaton though the followng prncple. If the recombnaton probablty r p s grater than 50% (ths value has been arbtrary selected), then, two arbtrary ndvduals (randomly selected) are recombned n an arbtrary poston selected va random number between 1 to s 1. If r p s lower than 50%, then two arbtrary ndvduals (randomly chosen) are averaged to generate a new potental ndvdual; notce that, ths operaton always generates feasble ndvduals, snce the ntal populaton as well as random solutons are generate nsde of the admssblty regon of the dstrbuted generators. Ths process contnues to obtan descendng populaton wth a potental solutons. Mutaton: In ths pont the mutaton probablty m p s explored,.e., f m p s grater than 50% (ths value has been arbtrary selected), an arbtrary poston of the potental soluton l s modfed by an arbtrary power generaton value guaranteeng that (4) be satsfed. If m p s lower than 50% the potental soluton l s not modfed. Ths process contnues untl all descendng ndvduals are analyzed. E-ISSN: X 339 Volume 13, 2018

6 Once the descendng populaton has been generated ts ftness functon are calculated as gven n (8) New populaton In the new populaton wll be saved the set of best solutons found by the genetc algorthm, untl the current teraton t. To generate the new populaton, we proceed as follows: A new populaton s generated by combnng the current and descendng set of ndvduals, whch produces a populaton wth 2a potental solutons; then, two potental solutons are dentcal, then, one of them s elmnated to ths lst. Ths procedure s repeated untl guaranteeng that all potental solutons are dfferent. Now, wth the resultng potental soluton lst, we ordered n ascendant form all ndvduals as a functon of ther ftness functon, and the frst a potental solutons are selected as a new populaton to pass to the next teraton cycle t Stoppng crtera The proposed contnuous genetc algorthm fnshes ts optmzaton process, when one of the followng stoppng condtons are acheved:. The total teraton cycles has been reached.. The best potental soluton does not been mproved after m consecutve teratve cycles. Otherwse, the genetc algorthm back to the descendng populaton step. 3.3 Slave problem The soluton of the slave problem s ndspensable to carry out to determne the ftness functon of each potental soluton contaned n the populaton of the genetc algorthm. The slave algorthm resolves the conventonal power problem gven from (5) to (7) va Gauss-Sedel numercal method as we wll be presented n next secton Power flow soluton va Gauss-Sedel method Gauss-Sedel power flow method corresponds to one of the frst numercal technques reported n specalzed lterature to solve power flow equatons n power grds. Ths soluton methodology has been manly used for AC power flow problems; nevertheless, ths methodology s easly applcable on power grds snce ts structure preserves the same structure of AC grds, both formulatons only dffer n ther soluton space,.e., AC power flows are analyzed nsde of the complex number set, whle power grds are analyzed nsde of the real numbers set. In the case of the power flow problem the Gauss-Sedel method solve teratvely (5) consderng that the followng assumptons are fulflled: Assumpton 2 The graph that descrbe the grd s connected (radal or mesh grds),.e., there are not slanded nodes on the power grd. Assumpton 3 The power grd s operatng under steady state condtons,.e., there are not external perturbatons. Assumpton 4 All possble generaton-load scenaros are nsde of the admssble power flow soluton regon,.e., the grd s stable n terms of voltage. and, v b+1 k = 1 G kk pg k pd k v b k + j<k G kj v b+1 j + 1 G kj v G j, b { k {N S}} kk j>k (9) where b s the current teraton of the Gauss-Sedel method and the voltage profle and generaton n the slack node(s) s(are) gven by (6) and (7). The accuracy of any power flow soluton method (Newton-Raphson, Gauss-Sedel, lnear methods) s hghly dependent of the startng pont, n other words, of the assgned values for the frst teraton calculaton. A common practce n specalzed lterature s to start all voltage n the grd as 1 p.u; nevertheless, we employ the open voltage crcut of the network calculated as the voltage profle n all nodes of the network when constant power loads and dstrbuted generators are dsconnectng, snce ths practce allows mprovng the convergence rates of the numercal methods for power flow analyss, n terms of number of teratons and processng tmes [43] Advantages of the Gauss-Sedel method Gauss-Sedel numercal method for solvng power flow equatons n power grds has the followng advantages [10]:. ts convergence can be guaranteed through pont fxed theorems as presented n [14].. t s not requred to make nverse of the matrces to obtan the soluton vector whch contans all voltage profles of the entre network. E-ISSN: X 340 Volume 13, 2018

7 . t can be appled over mesh or radal power grds ndstnctly. It s mportant to pont out that Gauss-Sedel method was selected over other power flow soluton methods snce t s easly mplementable of any programmng language, whch mples that not specalzed software or optmzaton package s needed. 3.4 Pseudo-code for the proposed methodology Algorthm 1 (pseudo-code verson) shows the man characterstcs for solvng optmal power flow problem va hybrd GS GA by usng a master-slave strategy. Data: power grd, genetc algorthm, Gauss-Sedel parameters. for t = 1 : t max do m = 0; f t == 1 then Generate the ntal populaton; for = 1 : a do Solve the power flow problem; Evaluate the ftness functon; end else Generate the descendng popupulaton; for = 1 : a do Solve the power flow problem; Evaluate the ftness functon; end Determne the new populaton; f (m > m max t == t max ) then Result: Impress results Break; end end end Algorthm 1: Proposed pseudo-code for the hybrd Gauss-Sedel Genetc-Algorthm for solvng the optmal power flow problem 4 Test system and smulaton scenaros As test system we employ a power grd reported n [14] whch has 10 nodes operatng under radal topology. Ths power grd has constant resstve load as well as constant power loads as presented n Table 1. Notce that we assume that all values are showed AC Table 1: Electrcal parameters of the test system From To R [pu] Type of node P [pu] - R [pu] Step-node P P P R Step-node P P R DG Fgure 2: Electrcal confguraton of the low-voltage dc power grd n per-unt, consderng 1 kv and 100 kw as voltage and power bases. Besdes, the power capabltes n all dstrbuted generators are contaned from 0 p.u to 3.0 p.u, whle the maxmum and mnmum voltages n the grd are assgned from 0.9 p.u to 1.1 pu. The proposed Gauss-Sedel Genetc-Algorthm s valdated by consderng two smulaton scenaros as descrbed below: Esc. 1: The OPF problem s solved consderng the possblty to allocate a dstrbuted generator (one at tme) at the endng nodes of the test system as presented n Fg. 2. Esc. 2: The OPF problem s solved consderng the possblty to allocate two dstrbuted generators (two at tme) at the endng nodes of the test system. It s mportant to menton that the locaton of the DGs n the test system s made n arbtrary form, snce the approach of ths artcle t s to analyze the optmal szng of the DGs (OPF n grds) and t does not corresponds the optmal locaton of these. Addtonally, all smulaton results are compared n terms of objectve functon (see (8)) wth GAMS optmzaton package. 9 E-ISSN: X 341 Volume 13, 2018

8 5 Computatonal results The computatonal mplementaton was carred-out through MATLAB 2017a software n a desk computer wth 8 Gb RAM, 3.6 GHz, wndows 10 Home Sngle Language, 64 bts. For comparson purposes, the actve power losses to the base case corresponds to kw, whch has been calculated by solvng the power flow problem va GS method consderng null the power njecton at all dstrbuted generators. Addtonally, for the GA, the populaton sze, number of teraton and convergence s error are selected n 10, 2000 and , respectvely; whle the GS numercal method allows 1000 teratons and the convergence s error s fxed at Frst smulaton scenaro Table 2 presents the power losses after applyng the optmzaton process (see Esc. 1.). It s mportant to hghlght that the proposed GS GA and the CONOPT solver fnd the same objectve functon, whch mples that the proposed method converges 100% to the global optma; nevertheless, the power generaton per node suffers small varatons, whch may be attrbuted to the numercal precson of GAMS and MATLAB. On the other hand, Fg. 4 shows the percentage of power losses reducton for each possble locaton of a dstrbuted generator analyzed n the Esc. 1. Table 2: Power generaton and actve power losses for Esc. 1. [kw] GAMS/CONOPT GS GA Node p g z p g z Notce that the locaton of the dstrbuted generator affects sgnfcantly the objectve functon performance, n ths sense, when one dstrbuted generator s consdered, the node 8 s less attractve for power losses reducton (49.08%), whle the node 10 s the most attractve node wth 79.01% of power losses reducton. Nonetheless, ther dfference (29.93%) mples around of kw addtonal of power njecton, whch may be not effcent from the economcal pont of vew. 5.2 Second smulaton scenaro In ths smulaton scenaro, we present the possblty to solve the OPF problem for a combnaton of two dstrbuted generators n four nodes, whch produces sx dfferent alternatves. Table 3 presents the results obtaned when GS-GA as well as CONOPT solver are employed to solve ths problem. Notce that the GS GA and GAMS optmzng package fnd exactly the same optmal soluton, whch guarantees the 100% of convergence of the proposed GS GA when s compared to a commercal wdelyknown solver. In Fg. 4 are presented the percentage of power losses reducton as functon of the dstrbuted generators allocaton. In ths snce, t s possble to observe that the combnaton between nodes 9 and 10 represent 91.59% of power losses reducton, whch s the most mportant reducton obtaned n the Esc. 2., whle the combnaton between 8 and 9 reduces the actve power losses around 62.40%, whch corresponds to the lower reducton n ths scenaro. Nevertheless, the combnaton between 9 and 10 nodes requres kw, whle 8 and 9 nodes requres only kw, whch mples that for mprovng the power losses reducton from 62.40% to 91.59% are needed kw addtonal, whch may be noneconomcal sustanable by the grd operator. Power losses reducton [%] Node Node Node Node 10 Fgure 3: Total reducton of power losses for for Esc. 1. Power losses reducton [%] Nodes 5-8 Nodes 5-9 Nodes Nodes 8-9 Nodes 8-10 Nodes 9-10 Fgure 4: Total reducton of power losses for Esc. 2. E-ISSN: X 342 Volume 13, 2018

9 Table 3: Power generaton and actve power losses for Esc. 2. [kw] Nodes GAMS/CONOPT GS GA p g p g z p g p g z Addtonal analyss and commentares For the sake of completng, n Fg. 5 s presented the voltage profle performance for the best power reducton mpact,.e., node 10 for Esc. 1. and nodes 9 and 10 for Esc. 2. as well as the base case. Recall that the performance of the voltage profle s hghly dependent of the total power njecton, nevertheless, t s not possble to affrm that the voltage profle ncreases lnearly wth the power njecton, snce the power flow equatons have nonlnear ntrnsc relatons between both varables that complcates ther analyss. On the other hand, t s mportant to menton that, when the Esc. 2. s observed, the voltage profle evdences a constant tendency for all nodes, ths behavor occurs because the njecton of actve power n dfferent nodes reduce sgnfcantly the current through the dstrbuton lnes, whch reduces the voltages drops between neghborhood nodes whch tend to equlbrate ther voltage profles. Nevertheless, n the Esc. 2. ths stuaton s less evdent snce the total power njecton s concentrated n a unque pont, whch has local and not global consequences n the voltage profle. We consder that the results presented n ths research are mportant for optmzaton as well as control ssues snce the GS GA presents a straghtforward form to solve complex nonlnear problems wth classcal and well-known optmzaton and numercal Voltage profles [pu] Base case Esc. 1. Esc Load ncreasng Fgure 5: Voltage profle at load nodes when for dfferent ncrements of the capacty of generaton and consumpton technques, wthout recurrng to specalzed software, whch s manly attractve for free-software developers and researchers. Fnally, t s mportant to menton that for future comparson purposes the averaged tme employed for the Gauss-Sedel method for each power flow evaluaton s 2.8 ms; wth 439 teratons. 6 Conclusons and future works A hybrd GS GA for solvng the optmal power flow problem n power grds was presented. The man advantage of the proposed methodology was that t dd not requre any specalzed software or optmzaton packages to the determne the optmal power generaton n each dstrbuted generator for mnmzng the total power losses. Besdes, the proposed methodology avoded makng nverse matrces snce GS worked drectly on the power flow equatons by recursvely solvng the convergence under normal operatng condtons of the network. A modfcaton of the conventonal bnary GA was proposed to solve nonlnear contnuous optmzaton problems through transformng the constraned optmzaton problem nto an equvalent nonconstraned problem; n addton, most of the constrants were drectly fulflled by the GA codfcaton makng easer to resolve the optmzng problem under analyss. Smulaton results allowed valdatng the applcablty and the qualty of the results regardng the values of the objectve functon snce these were the same found by the GAMS commercal optmzaton package. Addtonally, the numercal results presented n ths paper showed that for mnmzng power losses on power grds, not only the optmal power flow problem was mportant, snce the locaton of the dstrbuted generators, as well as the quantty of them, affects the total power losses of the grd sgnfcantly. As future nvestgaton works, the optmal locaton and dmensonng of dstrbuted energy resources such a renewable generaton and energy storage systems can be explored by usng hybrd algorthms, such E-ISSN: X 343 Volume 13, 2018

10 as bnary-contnuous genetc algorthms as well as conventonal power flow solutons lke Gauss-Sedel or Gauss-Jacob or lnear approxmatons. In addton, the hybrd GS GA proposed n ths paper can be used for mcrogrd control applcaton to determne the set pont of the controllers under any possble operatng condton (combnaton of loads and dstrbuted energy resources.) Fnancal support Ths work was partally supported by the Natonal Scholarshp Program Doctorates of the Admnstratve Department of Scence, Technology and Innovaton of Colomba (COLCIENCIAS), by callng contest References: [1] T. Slough, J. Urpelanen, and J. Yang, Lght for all? Evaluatng Brazls rural electrfcaton progress , Energy Polcy, vol. 86, pp , [Onlne]. Avalable: scence/artcle/p/s [2] D. R. Thomas and J. Urpelanen, Early electrfcaton and the qualty of servce: Evdence from rural nda, Energy for Sustanable Development, vol. 44, pp , [Onlne]. Avalable: com/scence/artcle/p/s [3] O. D. Montoya, A. Grajales, A. Garces, and C. A. Castro, Dstrbuton systems operaton consderng energy storage devces and dstrbuted generaton, IEEE Latn Amerca Transactons, vol. 15, no. 5, pp , May [4] O. D. Montoya, A. Garcs, and G. Espnosa-Prez, A generalzed passvty-based control approach for power compensaton n dstrbuton systems usng electrcal energy storage systems, Journal of Energy Storage, vol. 16, pp , [Onlne]. Avalable: com/scence/artcle/p/s x [5] S. Parhz, H. Lotf, A. Khodae, and S. Bahramrad, State of the art n research on mcrogrds: A revew, IEEE Access, vol. 3, pp , [6] Chapter 2 - uhv ac grd and system stablty, n UHV Transmsson Technology. Oxford: Academc Press, 2018, pp [Onlne]. Avalable: scence/artcle/p/b [7] A.-F. Atta, R. A. E. Sehemy, and H. M. Hasanen, Optmal power flow soluton n power systems usng a novel Sne-Cosne algorthm, Internatonal Journal of Electrcal Power & Energy Systems, vol. 99, pp , [Onlne]. Avalable: artcle/p/s [8] A. Garces, A lnear three-phase load flow for power dstrbuton systems, IEEE Transactons on Power Systems, vol. 31, no. 1, pp , Jan [9] H. Abd, S. D. Begvand, and M. L. Scala, A revew of optmal power flow studes appled to smart grds and mcrogrds, Renewable and Sustanable Energy Revews, vol. 71, pp , [Onlne]. Avalable: scence/artcle/p/s [10] P. Murty, Chapter 10 - power flow studes, n Power Systems Analyss (Second Edton), second edton ed., P. Murty, Ed. Boston: Butterworth-Henemann, 2017, pp [Onlne]. Avalable: artcle/p/b [11] O. D. Montoya-Graldo, W. J. Gl-González, and A. Garcés-Ruíz, Optmal power flow for radal and mesh grds usng semdefnte programmng, Tecno Lógcas, vol. 20, no. 40, pp , [12] D. Gandn and A. T. de Almeda, Drect current mcrogrds based on solar power systems and storage optmzaton, as a tool for costeffectve rural electrfcaton, Renewable Energy, vol. 111, no. Supplement C, pp , [13] A. Garces, On Convergence of Newtons Method n Power Flow Study for Mcrogrds, IEEE Transactons on Power Systems, pp. 1 1, [14], Unqueness of the power flow solutons n low voltage drect current grds, Electrc Power Systems Research, vol. 151, no. Supplement C, pp , E-ISSN: X 344 Volume 13, 2018

11 [15] O. D. Montoya, L. F. Grsales-Noreña, D. González-Montoya, C. Ramos-Paja, and A. Garces, Lnear power flow formulaton for low-voltage power grds, Electr. Power Syst. Res., vol. 163, pp , [16] K. S. H. Beagam, R. Jayashree, and M. A. Khan, A new dc power flow model for q flow analyss for use n reactve power market, Engneerng Scence and Technology, an Internatonal Journal, vol. 20, no. 2, pp , [Onlne]. Avalable: com/scence/artcle/p/s [17] A. Garces, D. Montoya, and R. Torres, Optmal power flow n multtermnal hvdc systems consderng dc/dc converters, n 2016 IEEE 25th Internatonal Symposum on Industral Electroncs (ISIE), June 2016, pp [18] J. L, F. Lu, Z. Wang, S. Low, and S. Me, Optmal Power Flow n Stand-alone Mcrogrds, IEEE Transactons on Power Systems, pp. 1 1, [19] S. Boyd and L. Vandenberghe, Convex optmzaton. Cambrdge unversty press, [20] Y. Nesterov, Lectures on Convex Optmzaton, ser. Sprnger Optmzaton and Its Applcatons. Sprnger Internatonal Publshng, [Onlne]. Avalable: books?d=jsyntqeacaaj [21] E. Benedto, D. del Puerto-Flores, A. Dra- Cerezo, and J. M. Scherpen, Optmal Power Flow for resstve Networks: a Port- Hamltonan approach, IFAC-PapersOnLne, vol. 50, no. 1, pp , 2017, 20th IFAC World Congress. [Onlne]. Avalable: artcle/p/s [22] C. Alexander and M. Sadku, Fundamentals of Electrc Crcuts. McGraw-Hll Hgher Educaton, [Onlne]. Avalable: google.com.co/books?d=dm6vpwaacaaj [23] N. Barabanov, R. Ortega, R. Gr, and B. Polyak, On exstence and stablty of equlbra of lnear tme-nvarant systems wth constant power loads, IEEE Transactons on Crcuts and Systems I: Regular Papers, vol. 63, no. 1, pp , Jan [24] Z. Shua, J. Fang, F. Nng, and Z. J. Shen, Herarchcal structure and bus voltage control of dc mcrogrd, Renewable and Sustanable Energy Revews, vol. 82, pp , [Onlne]. Avalable: com/scence/artcle/p/s [25] C. DePerss, E. R. Wetenberg, and F. Drfler, A power consensus algorthm for dc mcrogrds, Automatca, vol. 89, pp , [Onlne]. Avalable: com/scence/artcle/p/s [26] Z. Lu, M. Su, Y. Sun, H. Han, X. Hou, and J. M. Guerrero, Stablty analyss of dc mcrogrds wth constant power load under dstrbuted control methods, Automatca, vol. 90, pp , [Onlne]. Avalable: scence/artcle/p/s [27] Shvam and R. Dahya, Stablty analyss of slanded mcrogrd for the proposed dstrbuted control strategy wth constant power loads, Computers & Electrcal Engneerng, [Onlne]. Avalable: artcle/p/s [28] P. G. Khorasan, M. Jooraban, and S. G. Sefossadat, Smart grd realzaton wth ntroducng unfed power qualty condtoner ntegrated wth dc mcrogrd, Electrc Power Systems Research, vol. 151, pp , [Onlne]. Avalable: com/scence/artcle/p/s [29] H. Wu, X. Lu, and M. Dng, Dynamc economc dspatch of a mcrogrd: Mathematcal models and soluton algorthm, Internatonal Journal of Electrcal Power & Energy Systems, vol. 63, pp , [Onlne]. Avalable: scence/artcle/p/s [30] C. Phuralatpam, B. S. Rajpuroht, and L. Wang, Plannng and optmzaton of autonomous mcrogrds for rural and urban applcatons n Inda, Renewable and Sustanable Energy Revews, vol. 82, pp , [Onlne]. Avalable: com/scence/artcle/p/s [31] A. A. Hamad and E. F. El-Saadany, Multagent supervsory control for optmal economc dspatch n mcrogrds, Sustanable Ctes and Socety, vol. 27, pp , [Onlne]. Avalable: com/scence/artcle/p/s E-ISSN: X 345 Volume 13, 2018

12 [32] V. Bhattacharjee and I. Khan, A non-lnear convex cost model for economc dspatch n mcrogrds, Appled Energy, vol. 222, pp , [Onlne]. Avalable: artcle/p/s [33] O. D. Montoya, A. Grajales, L. F. Grsales, and C. A. Castro., Optmal Locaton and Operaton of Energy Storage Devces n Mcrogrds n Presence of Dstrbuted Generaton (n Spansh), Revsta CINTEX, vol. 22, no. 1, pp , jun [34] J. Yue, Z. Hu, C. L, J. C. Vasquez, and J. M. Guerrero, Economc Power Schedule and Transactve Energy through an Intellgent Centralzed Energy Management System for a Resdental Dstrbuton System, Energes, vol. 10, no. 7, [Onlne]. Avalable: [35] M. T. Bhoskar, M. O. K. Kulkarn, M. N. K. Kulkarn, M. S. L. Patekar, G. Kakandkar, and V. Nandedkar, Genetc algorthm and ts applcatons to mechancal engneerng: A revew, Materals Today: Proceedngs, vol. 2, no. 4, pp , 2015, 4th Internatonal Conference on Materals Processng and Characterzaton. [Onlne]. Avalable: com/scence/artcle/p/s [40] M. Esmaelan, M. Tavana, F. J. Santos- Arteaga, and M. Val, A novel genetc algorthm based method for solvng contnuous nonlnear optmzaton problems through subdvdng and labelng, Measurement, vol. 115, pp , [Onlne]. Avalable: artcle/p/s [41] O. A. Arqub and Z. Abo-Hammour, Numercal soluton of systems of second-order boundary value problems usng contnuous genetc algorthm, Informaton Scences, vol. 279, pp , [Onlne]. Avalable: artcle/p/s [42] M. Todorovsk and D. Rajcc, An ntalzaton procedure n solvng optmal power flow by genetc algorthm, IEEE Transactons on Power Systems, vol. 21, no. 2, pp , May [43] M. Todescato, power flow feasblty: Postve vs. negatve loads, n 2017 IEEE 56th Annual Conference on Decson and Control (C), Dec 2017, pp [36] O. D. Montoya, Numercal approxmaton of the maxmum power consumpton n dc-mgs wth cpls va an sdp model, IEEE Transactons on Crcuts and Systems II: Express Brefs, pp. 1 1, [37] J. E. Machado, R. Grñó, N. Barabanov, R. Ortega, and B. Polyak, On exstence of equlbra of mult-port lnear ac networks wth constantpower loads, IEEE Transactons on Crcuts and Systems I: Regular Papers, vol. 64, no. 10, pp , Oct [38] R. A. Gallego, A. Montcell, and R. Romero, Transmsson system expanson plannng by an extended genetc algorthm, IEE Proceedngs - Generaton, Transmsson and Dstrbuton, vol. 145, no. 3, pp , May [39] W. Sheng, K. Y. Lu, Y. Lu, X. Meng, and Y. L, Optmal placement and szng of dstrbuted generaton va an mproved nondomnated sortng genetc algorthm, IEEE Transactons on Power Delvery, vol. 30, no. 2, pp , Aprl E-ISSN: X 346 Volume 13, 2018