R. MAGESHVARAN*, T. JAYABARATHI, VAIBHAV SHAH, SHIVAM SOOD

Transcription

1 Journal of Engneerng Scence and Technology Vol. 10, No.10 (2015) School of Engneerng, Taylor s Unversty OPTIMUM STEADY STATE LOAD SHEDDING USING SHUFFLED FROG LEAPING ALGORITHM TO AVERT BLACKOUT IN POWER SYSTEMS DURING OVERLOAD AND GENERATION CONTINGENCIES R. MAGESHVARAN*, T. JAYABARATHI, VAIBHAV SHAH, SHIVAM SOOD School of Electrcal Engneerng, VIT Unversty, Vellore , Taml Nadu, Inda *Correspondng Author: rmageshvaran@vt.ac.n Abstract Durng generaton and overload contngences n a power system, the system voltage and frequency wll declne due to the defcency of real and reactve powers. Consequently cascaded falures may occur whch wll lead to complete blackout of certan parts of the power system. Load sheddng s consdered as the ultmate step of emergency control acton that s necessary to prevent a blackout n the power system. Ths paper proposes a memetc meta-heurstc algorthm known as shuffled frog leapng algorthm (SFLA) to fnd a soluton for the steady state load sheddng problem presented here. The optmum steady state load sheddng problem uses squares of the dfference between the connected actve and the reactve load and the suppled actve and reactve power. The suppled actve and reactve powers are treated as dependent varables modeled as functons of bus voltages only. The proposed algorthm s tested on IEEE 14 and 30 bus test systems. The vablty of the proposed method s establshed by comparson wth the other conventonal methods presented earler n terms of soluton qualty and convergence propertes. Keywords: Optmal load sheddng, Shuffled frog leapng algorthm, Overload contngency, Generaton contngency, Voltage dependent load model. 1. Introducton The man objectve of the power utlty s to operate the power system wthout volatng the system constrants and operatonal lmts. However under certan stuatons lke unexpected outages or sudden ncrease n system demand, the system constrants and operatonal lmts are volated. Load sheddng s consdered 1239

2 1240 R. Mageshvaran et al. Nomenclatures NB Total number of buses NG Total number of generator buses P d Actve power suppled to the load P d Connected actve load P G Actve power generaton at bus P Actve power njectons at bus max P G Maxmum lmt of real power generatons P Mnmum lmt of real power generatons mn G Q d Q d Reactve power suppled to the load Connected reactve load Q Reactve power njectons at bus Q G Reactve power generaton at bus max Q G Maxmum lmt of reactve power generatons mn Q G Mnmum lmt of reactve power generatons V, V j The bus voltage magntude max V Maxmum lmt of bus voltages of the system mn V Mnmum lmt of bus voltages of the system Greek Symbols α, β Parameters related to prorty assgned to the demand at each bus δ, δ Voltage angles at bus and bus j θ j j Abbrevatons SFLA VDLM Angle of admttance of the transmsson lne between bus and bus j Shuffled frog leapng algorthm Voltage dependent load model as a last resort to avod cascaded trppng and blackout. Load sheddng s defned as coordnated sets of controls that decrease the electrc load n the system to restore the system back to ts normal operatng condton. After load sheddng the dsturbed system state can settle to a new equlbrum state. An optmal load sheddng program fnds a best steady-state stable operatng pont for a post fault system wth a mnmum amount of load shed. Reference [1] has formulated the optmal steady state load sheddng problem that uses the sum of the squares of the dfference between the connected actve and the reactve load and the suppled actve and reactve power. The optmzaton package MINOS [11], developed to solve large scale dense or sparse lnearly or nonlnearly constraned or unconstraned optmzaton problems s used to smulate the algorthm. An algorthm to mnmze the amount of load curtalment whch s based on Newton-Raphson (NR) method for solvng the power flow equatons, and Kuhn-

3 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng Tucker theorem for the optmzaton has been descrbed n [2]. Here, the load sheddng polcy, defnng the prorty schedules s obtaned frst, and then the mnmum load to be shed at each bus s calculated. System reactve power and losses are not ncluded n the problem formulaton and the actve and reactve power of loads s assumed to be ndependent of bus voltages. A method for optmal load curtalment, that consders generator control effects, voltage and frequency characterstcs of the load, has been proposed n [3]. The resultng nonlnear optmzaton problem s solved usng Second-order gradent technque. Inablty to curtal system loads durng an emergency s the major drawback of the formulaton n [3]. Another drawback s the algorthm s nablty to converge durng emergency condton due to over defnton of system generaton through the governor acton and the poor reconclaton of the frequency varatons appled to the load model equatons. Ref. [4] has presented a reformulated optmal load sheddng polcy that takes nto account generator control effects and voltage and frequency characterstcs of loads. Ref. [5] has presented another work of optmal load sheddng problem that uses the sum of squares of the dfference between the connected actve and reactve load and the suppled actve and reactve power. In the formulaton, the suppled actve and reactve power s consdered as dependent varables modeled as a functon of bus voltages only. A senstvty based approach to solve the load sheddng problems and to mnmze the loss of loads has been proposed n [6]. In order to lmt the sze of the load beng dropped, dfferent prortes to loads are assgned usng a weghted error crteron. Ref. [7] and [8] has formulated a non-lnear optmzaton problem for the optmal load sheddng and reschedulng of generators durng an emergency state. The non-lnear problem has been approxmated by an accurate senstvty model whch takes nto account the real and reactve nodal njectons, voltage magntudes and angles and loads senstvty to voltage magntudes. The accuracy of the soluton n both these papers s affected by the lnearzaton approach. In [9] and [10] two dfferent methods for generaton reschedulng and load sheddng to allevate lne overloads, based on the senstvty of lne overloads to bus power ncrements have been developed. In [11], a mesh approach has been developed for the formulaton of the network equatons n the load flow analyss. A hybrd approach usng a combnaton of an mpedance matrx method and a nodal-admttance matrx method whch explots the salent characterstcs of the mpedance and admttance method s developed. A new power flow model for the steady state behavor of large complex power system that allows the study of power flow under normal and abnormal operatng condtons has been developed n [12]. In [13], dfferental evoluton algorthm has been mplemented for optmal allocaton of repar tmes and falure rates n meshed dstrbuton system. An optmal under-voltage load sheddng scheme to provde long term voltage stablty usng a new hybrd partcle swarm based smulated annealng optmzaton technque has been presented n [14]. The techncal and economc aspects of each load are consdered by ncludng the senstvtes of voltage stablty margn nto the cost functon. In [15], a new voltage stablty margn ndex consderng load characterstcs has been ntroduced n under-voltage centralzed load sheddng scheme. Quantum nspred evolutonary programmng

4 1242 R. Mageshvaran et al. has been mplemented n [16] for the optmal locaton and szng of dstrbuted generatons (DGs) n radal dstrbuton system. In [17], an optmal load sheddng scheme have been proposed to montor the load-generaton unbalance n the plants wth nternal co-generaton and to quckly ntate sheddng of an optmal amount of load durng a contngency. DC optmal load shed recoveres wth transmsson swtchng model have been presented n [18]. Ths model reduces the amount of load shed requred durng generaton and/or transmsson lne contngences, by modfyng the bulk power system topology. An approach based on parallel-dfferental evoluton has been proposed n [19] for the optmal load sheddng aganst voltage collapse. The non-lnearty of the problem s fully consdered n ths approach and thereby able to escape from local optma and not lmted to system modelng. Bascally, the optmal load sheddng strateges are classfed nto two types, namely, centralzed load sheddng and de-centralzed or dstrbuted load sheddng. Centralzed load sheddng strateges are solved based on stablty margn senstvtes. These methods are based on the assumptons of lnearty and constancy of the senstvtes [21], and depend on lnear programmng technques to solve the comprehensve optmzaton problem. In actual practce, these assumptons are not realstc [22], partcularly when the non-lnear characterstcs of the system components, such as, reactve power generaton lmts, actons of swtched shunt devces load-tap changers and so on are consdered. A mult-stage method to solve the non-lnear optmal load sheddng problem stage by stage has been presented n [22]. Here, each stage corresponds to a lnearzed sub-problem based on senstvty analyss. Usually these methods do not consder prortes for the loads to be shed, whereas, n dstrbuted load sheddng schemes prortes for the loads are beng consdered. Moreover, n the mathematcal formulaton of optmal load sheddng schemes, reactve power of loads to be shed are not consdered [13-23]. Also, the loads are consdered to be ndependent of the system voltage, but n actual practce, the real and reactve power of the loads depends on the system voltage [1]. In the present paper the optmal load sheddng problem to mnmze the sum of squares of the dfference between the connected loads and the suppled power has been formulated. The Shuffled frog leapng (SFL) algorthm s mplemented to solve the formulated optmzaton problem. The performance of the proposed algorthm for generaton defcency and overload contngences are analyzed. Testng s done usng IEEE 14 and 30, representng small power system. The optmal solutons are compared wth those reported n [1, 3, 5]. 2. Problem Formulaton Durng emergency condtons an operatonal objectve s to mnmze the dfference between the connected load and the suppled power [1]. System frequency devaton must be ncluded n the problem formulaton f the system consdered s under transent state. In ths paper, system under steady state condton s consdered for the problem formulaton and thus the frequency s treated as constant. The objectve functon can be expressed as

5 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng F 2 2 ( P Pd) + β ( Q Q ) α (1) d d d = 1 = NB where NB s the number of buses n a system, P d, and Q d are the actve and reactve powers suppled to the load. P d, and Q d are the connected actve and reactve load. The weghtng factors α and β are the parameters related to prorty assgned to the demand at each bus. Flat values are assgned to the prortes of the loads. The objectve functon gven n Eq. (1) s subjected to both network and system lmts constrants resultng n a non-lnear optmzaton problem. The equalty constrants are the power flow equatons of the networks and nequalty constrants are characterzed by the lmts on the system varables Equalty constrants The transmsson network s represented by a steady-state AC power flow model. In ths model every node s characterzed by two nonlnear algebrac equatons. In these network equatons the actve and reactve power njected at a node s related to the power mported through the branches ncdent at that node. These network equatons are equalty constrants whch are to be satsfed n order to obtan the balance of actve and reactve power at each node. For a network wth NB number of nodes, a total of 2 NB equatons can be wrtten as P Q ( V) = P P ( V) P( V, δ) = 0 G d (2) ( V) = Q Q ( V) Q ( V, δ) = 0 G d (3) where P G and Q G represents the respectve actve and reactve power generatons at bus. P and Q represents the actve and reactve power njectons at bus. The actve and reactve power njectons at bus n terms of bus voltage magntude and phase angle s expressed as P Q NB ( V, ) = V V Y cos( δ δ θ ) δ j j j j (4) = 1 NB ( V, ) = V V Y sn( δ δ θ ) δ j j j j (5) = Inequalty constrants The nequalty constrants consdered are the lmts of real and reactve power generatons, bus voltage magntudes and angles, and lne flows. The lmts of real and reactve power generatons of the system are mn max PG PG PG = 1,... NG (6) mn max QG QG QG = 1,... NB (7)

6 1244 R. Mageshvaran et al. where P mn G and mn Q G are the mnmum real and reactve power generatons and max PG and Q max G are the maxmum avalable real and reactve power generatons. The lmts of bus voltages of the system are mn max V V V = 1,... NB (8) mn max wherev and V are the mnmum and maxmum lmts of bus voltages of the system. Ether current magntude constrant due to thermal consderatons or electrcal angle (dfference n voltage angle across a lne) constrant due to stablty consderatons can be consdered for transmsson lne loadng lmts. In the present formulaton the electrcal angle nequalty constrant s used, whch can be expressed as LF= δ δ j j = 1,... NB 1; j = + 1,... NB (9) whereδ and voltage phase angle dfference between and j. δ j are the voltage angles at bus and bus j, and j s the maxmum 2.3. Load model The system actve and reactve power demands can be expressed usng dfferent load models n terms of bus voltage and system frequency. A polynomal functon of the bus voltage s used n ths formulaton to express the actve and reactve power demands at any gven bus as N1 N 2 V + V P d = P d Pp + Pc P (10) z V V N 3 N 4 V + V Q d = Q d Qq + Qc Q (11) z V V where P p, P c, P z, Q q, Q c and Qz are constants assocated wth ths voltage dependent load model and N1, N2, N3 and N4 are the powers of polynomal. 3. Implementaton of Shuffled Frog Leapng Algorthm for the Optmal Load Sheddng Problem The SFL algorthm s a meta-heurstc optmzaton method based on observng, mtatng and modelng the behavor of groups of frogs when searchng for the locaton that has maxmum amount of avalable food [23] s mplemented n followng steps. The mplementaton of SFL algorthm for the optmal load sheddng problem s shown n Fgs. 1(a) and (b).

7 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng Start Intalze parameters Populaton sze Number of memeplexes Number of teratons wthn each memeplex Generate random populaton of solutons (frogs) Calculate ftness of each ndvdual frog Sort populaton n descendng order of ftness Dvde P solutons nto memeplexes Local Search ( shown n Fg 1-b) Shuffle evolved memeplexes No Termnaton = true Yes Determne the best soluton Stop Fg. 1. (a) Flowchart of SFLA. Step-1: Intalze the populaton sze ( P ), number of memeplexes (m) and number of teraton wthn each memeplex. Generate random populaton of ' P ' feasble solutons by generatng random values of the n varables (amount of load to be shed n each bus) wth n ther search space,.e., T X ( 0) = [ x 1, x 2, x 3,... x 1], = 1,2,... P (12) These ' P ' solutons are known as frogs. Step-2: Usng these generated solutons obtan the value of objectve functon gven n Eq. (1) and calculate the ftness of each ndvdual soluton,.e., frogs.

8 1246 R. Mageshvaran et al. Start k=1 k=k+1 =1 = +1 Determne and for k th memeplex and Evaluate worst frog poston usng (13) and (14) Yes Is the new frog poston better than the worst poston? No Replace X b wth X g and evaluate worst frog poston usng (13) and (14) Is the new frog poston better than the worst poston? Generate a new frog randomly No Yes Replace the worst frog wth the new frog If = no. of teratons Yes If k= no. of memeplexes No No End Yes Fg. 1. (b) Local search of SFLA. Step-3: The frogs are sorted n descendng order accordng to ther ftness. Then the entre populaton s parttoned nto m subsets referred to as memeplexes, each contanng n frogs (.e., P = m n ). The strategy of the parttonng s as follows: the frst frog goes to the frst memeplex, the second frog goes to the second memeplex, the th m frog goes to the th ( m + 1) frog goes back to the frst memeplex, and so forth. th m memeplex, the Step-4: A local search for the best soluton from each memeplexes should be carred out. Select each memeplex and set the teraton count = 1. Determne,

9 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng X w, X b & X g for each memeplexes, where X w X b & X g frogs wth worst, best and global best ftness respectvely., were the postons of Step-5: Evaluate the new frog poston to replace the worst frog usng the followng equatons D = Rand ( X X ) (13) X D new w mn = X * b w current w < D < D max + D where D max and D mn are the maxmum and mnmum step szes allowed for a frog s poston respectvely. Step-6: If the new frog poston s better compared to the worst frog poston then replace the worst frog wth new frog else replace X by X and evaluate the new frog poston usng Eq. (13) and Eq. (14). In ths case also f the new frog s not better than the worst frog then generate new frog and replace the worst frog wth ths new generated frog. The evaluatons wll contnue for a specfc number of teratons. Therefore, SFLA smultaneously performs an ndependent local search n each memeplex usng a process smlar to the PSO algorthm. After a predetermned number of memetc evolutonary steps wthn each memeplexes, the solutons of evolved memeplexes ( X1,..., X P ) are shuffled for the global nformaton exchange among the frogs. b g (14) Step-7: If the constrants are satsfed, obtan the best soluton, f not go to step3 and contnue. 4. Smulaton Results and Analyss The predcton of the aerodynamc coeffcents of the nvestgated projectles The proposed SFL algorthm has been verfed on two small systems IEEE 14-bus and 30-bus. The results obtaned by the proposed approach are compared wth those obtaned by usng conventonal methods reported earler, such as projected augmented Lagrangan method mplemented usng MINOS an optmzaton package [1], gradent technque based on Kuhn-Tucker theorem [2] and second order gradent technque [3]. The sngle lne dagram and the detaled data of IEEE-14 and 30 bus systems are gven n [11]. The software was wrtten n Matlab and executed on 2.4 GHz, Intel 3 processor wth 2 GB RAM PC. The constants and the powers of the polynomal assocated wth the load model gven n Eq. (10) and (11) were assumed as follows: P p =0.2, and 4 N =2. P c =0.3, P z =0.5, Q q =0.2, Q c =0.3 and Q z =0.5. N 1=1, N 2 =2, N 3 = Applcaton to small sze systems IEEE 14, 30-bus test systems [11] are consdered here. The two cases of contngences analyzed are loss of generaton and overload. The populaton sze used for the proposed SFL algorthm for these test systems s 50.

10 1248 R. Mageshvaran et al IEEE 14-bus system Ths system conssts of 20 lnes, two generators, three synchronous condensers, three transformers and one statc capactor. The generated reactve power lmts are: 150 Q G 0 Q G 2 140, 0 Q G Q 140, G6 0 Q G8 140 The generated actve power lmts are: 0 P 200, G1 0 P G2 200 The connected load for ths test system s 259 MW. Under normal operatng condtons the suppled power to the connected load usng NR method wth VDLM s MW. The reactve power suppled by the method used here s MVAR. The suppled powers n Refs. [2], [3] and [5] are MW, MW and MW respectvely for a connected load of 259 MW. For the same connected load the suppled power In Ref [1] s MW. The reactve power suppled by the methods reported n Refs [2], [3] and [5] are MVAR, MVAR and MVAR respectvely. The defct n the suppled power obtaned n ths paper and n Refs. [1],[3] and [5] represents the effect of usng a voltage dependent load model (VDLM) to express the actve power. The total actve and reactve power generaton obtaned under normal operatng condton by the NR method wth VDLM used n ths paper s MW and MVAR respectvely. The total real power loss obtaned here s MW. The total actve power generaton obtaned by the method reported n Refs [2], [3] and [5] for the normal operatng condtons were MW, 270 MW and MW respectvely. The correspondng reactve power generaton reported n these Refs [2], [3] and [5] were 8.56 MVAR, MVAR and MVAR respectvely. The total actve power loss presented n the Refs [2], [3] and [5] were MW, MW and MW respectvely. The bus voltages vary between 1.01 pu and 1.08 pu n the NR method wth VDLM, whereas the voltages vary from 0.98 pu to 1.07 pu n Ref. [1,5], 0.93 pu to pu n Ref. [2] and pu to pu n Ref. [3]. The am of optmal load sheddng s to restore normal operatng condtons followng loss of generaton and overload contngences by sheddng mnmum load. a) Loss of Generaton Contngences An abnormal operatng condton representng the loss of generatng unt #2 generatng 72.0 MW or 26% of normal generaton s consdered here. The amount of load shed usng the proposed SFLA for ths contngency s MW and the suppled actve power demand s MW.Whereas the load shed and the suppled actve power demand reported by Mostafa n Refs [1] and [5] are MW and MW respectvely. It can be observed that the proposed approach has yelded lower amount of load shed and hgher suppled actve power demand when compared wth other methods. The total actve power

11 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng demand presented n Refs [2] and [3] for these contngences were 192 MW and MW respectvely. Reactve powers suppled by the proposed method and by the methods used n the Refs [2], [3] and [5] are MVAR, MVAR, 65.2 MVAR and MVAR respectvely. The total actve and reactve power generaton obtaned by the proposed SFL approach for ths abnormal condton s 200 MW and 82.5 MVAR respectvely. The correspondng loss obtaned by ths approach s MW. Whereas the total actve power generaton obtaned by the method reported n Refs [2], [3] and [5] for ths loss of generaton contngency were 200 MW, MW and MW respectvely. The correspondng reactve power generaton reported n these Refs [2], [3] and [5] were MVAR, MVAR and MVAR respectvely. The correspondng actve power losses reported n the Refs [2], [3] and [5] are MW, MW and MW respectvely. The bus voltages vary between 1.01 pu and 1.09 pu n the proposed SFL algorthm. Whereas the voltages vary from pu to pu n Refs. [1,5] and pu to 1.1pu n Ref. [2]. The proposed approach yelds better bus voltage profle as compared wth other approaches. Fgure 2 shows the convergence characterstcs of the proposed SFL algorthm for the test system operated under the generaton contngency consdered here. The maxmum number of teratons to converge for the proposed approach s Objectve functon Number of Iteratons Fg. 2. Convergence Characterstcs of SFL Algorthm for IEEE 14-Bus System Under Loss of Generatng Unt Contngency. b) Range of generaton defcts contngency The test system s also subjected to contngences characterzed by generaton defcts. The range of generaton s vared from 260 MW to 160 MW, wth a connected load of 259 MW, whch means, the resultng generaton defct vares from 0 to 99 MW. Fgure 3 shows that the mnmum teraton to converge for the proposed SFL algorthm s 19 correspondng to 160 MW generaton. Fgure 4(a) shows the total suppled power obtaned by the proposed SFL approach decreases from MW at 260 MW generatons to MW at 160 MW generatons. Fgure 4(b) shows the correspondng actve power loss decrease from MW to MW.

12 1250 R. Mageshvaran et al Objectve Functon Number of Iteratons Fg. 3. Convergence Characterstcs of SFL Algorthm for IEEE 14-Bus System Under Generaton Defct Contngency 260 Suppled power to the system n MW Generated power n MW 160 Fg. 4 (a). Optmal Suppled Load for IEEE 14-Bus System under Generaton Defct Contngency usng SFL Algorthm. 12 System Losses n MW Generated Power n MW 160 Fg. 4 (b). System Losses of IEEE 14-Bus System under Generaton Defct Contngency usng SFL Algorthm. Whereas the total suppled power n Refs. [1] and [5] decreases from MW at 260 MW generaton to MW at 160 MW generaton wth correspondng actve power loss decrease from MW to 6.22 MW and n Ref

13 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng [2] the suppled load decreases from MW to MW wth the correspondng actve power loss decrease from 7.08 MW to 2.78 MW for the same range of generaton defcts. For ths loss of generaton contngency the maxmum voltage obtaned by the proposed shuffled frog leapng approach remans constant at 1.025pu and the mnmum voltage vares between 1.01 pu and 1.03 pu. Whereas n Refs. [1] and [5] the maxmum voltage decreases from pu to pu and the mnmum voltage magntude decreases from pu to 0.77 pu. For IEEE 14 bus system, bus 3 s the bus wth heavest load and bus 4 s the bus wth second heavest load. The suppled powers at bus 3 and bus 4 by the proposed SFLA are MW and MW respectvely. In Refs [1] and [5] the suppled powers at bus 3 and bus 4 are MW and MW respectvely. The suppled powers at bus 3 n Ref [2] s MW and at bus 4 t s MW. The proposed approach supples more power at the heavest loaded buses - bus 3 and bus 4- as compared to [1, 2, 5]. c) Overload contngency: In ths contngency t s consdered overloadng up to 55.4% above the maxmum avalable generaton of 400 MW. Ths corresponds to a connected load between MW (150% of the nomnal connected load) to MW (240% of the orgnal connected load). Fgure 5 shows the mnmum teratons to converge for the proposed SFL algorthm s 22 teratons correspondng to connected load of MW. For ths contngency the suppled load obtaned usng the proposed method decreases from MW at 150% overload to MW at 240 % overload. Whereas for the same range of overload contngences, n refs [1,5] and ref [2] the suppled power decreases from MW to MW and from 383.3MW to MW respectvely. Fgure 6(a) shows the decrease n suppled power from MW at 150% overload to MW at 240% overload for the proposed SFL algorthm. The correspondng ncrease n the system losses s shown n Fg. 6(b) for the correspondng connected load. For ths range of overload contngences the system losses obtaned by the proposed method ncreases from MW to MW. Whereas the system losses for ths condton reported n Refs[1,5] and Ref[2] ncreases from MW to MW and from 16.7 MW to MW respectvely Objectve Functon Number of Iteratons

14 1252 R. Mageshvaran et al. Fg. 5. Convergence Characterstcs of SFL Algorthm for IEEE 14-Bus System under Overload Condtons. The proposed approach supples MW to bus 3 and MW to bus 4. Whereas Ref [1,5] supples to bus 3 MW and MW to bus 4.Ref [2] supples MW and MW to buses 3 and 4 respectvely. A better voltage profle s obtaned by the proposed method durng ths contngency. Maxmum voltage obtaned usng the proposed approach s constant at pu and the mnmum voltage ncreases from to pu Suppled Power to the System n MW Connected Load n MW Fg. 6 (a). Optmal Suppled Power for IEEE14-Bus System under Overload Condtons usng SFL Algorthm System Losses n MW Connected Load n MW Fg. 6 (b). System Losses of IEEE14-Bus System under Overload Condtons usng SFL Algorthm. In Refs [1,5] the maxmum voltages vary between pu and pu. and the mnmum voltage ncreases from 0.77 pu to 0.95 pu. The maxmum voltage reported n Ref [2] s constant at 1.2 pu and the mnmum voltage ncrease from pu to pu.

15 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng IEEE 30-bus system: Ths system conssts of 41 lnes, three generators, three synchronous condensers, two statc capactor and three transformers. The generated reactve power lmts are: 20 Q 43, 1 10 Q 30 8, G 20 Q 2 43, 10 Q 11 45, G 20 Q 50, 5 10 Q 50 13, G The generated actve power lmts are: 0 P 175, G1 0 P G P G5 75 G G G The suppled power by the NR method wth VDLM used n ths paper under normal operatng condtons s MW for a connected load of MW. The reactve power suppled by the method used n ths paper s MVAR. The suppled powers n Refs. [2], [3] and [5] are MW, MW and MW respectvely for a connected load of MW. In Ref [1] the suppled power s MW for the same connected load. The reactve power suppled by the methods reported n Refs [2], [3] and [5] are MVAR, MVAR and MVAR respectvely. The defct n the suppled power obtaned n ths paper and n Refs. [1],[3] and [5] represents the effect of usng a VDLM to express the actve power. The total actve and reactve power generaton obtaned under normal operatng condton by the NR method wth VDLM used n ths paper s MW and MVAR respectvely. The total real power loss obtaned here s 9.44 MW. Whereas the total actve power generaton obtaned by the method reported n Refs [2], [3] and [5] for the normal operatng condtons were MW, MW and MW respectvely. The correspondng reactve power generaton reported n these Refs [2], [3] and [5] were MVAR, MVAR and MVAR respectvely MW, MW and MW are the respectve total actve power losses reported n the Refs [2], [3] and [5]. The bus voltages vary between pu and pu n the proposed approach whereas the voltages vary from pu to 1.10 pu n Ref. [1,5], pu to 1.10 pu n Ref. [2] and pu to pu n Ref. [3]. a) Loss of generaton contngences An abnormal operatng condton representng the loss of 60 MW or 20.35% of normal generaton s consdered here. The amount of load shed and the suppled actve power demand obtaned usng the proposed SFLA s MW or % of the nomnal load and MW respectvely. Whereas the load shed and the suppled actve power demand n Refs [1] and [5] are MW or % of the nomnal load and MW respectvely. For the same generaton loss, the amount of load shed and the suppled power n Ref. [2] are MW or 14.38% of the nomnal load and MW respectvely. It can be observed that the proposed approach has yelded lower amount of load shed when compared wth the Refs [1] and [5]. Reactve powers suppled by the proposed SFL

16 1254 R. Mageshvaran et al. algorthm and by the methods used n the Refs [2], [3] and [5] are MVAR, MVAR, MVAR and MVAR respectvely. The total actve and reactve power generaton obtaned by the proposed SFL algorthm for ths case s 250 MW and MVAR respectvely. The correspondng actve power loss obtaned by ths approach s MW. Whereas the total actve power generaton obtaned by the method reported n Refs [2], [3] and [5] for ths loss of generaton contngency were 250 MW, MW and 250 MW respectvely. The correspondng reactve power generaton reported n these Refs [2], [3] and [5] were MVAR, MVAR and MVAR respectvely. The correspondng actve power losses reported n the Refs [2], [3] and [5] are MW, MW and MW respectvely. Fgure 7 shows the convergence characterstcs of the proposed SFL algorthm for the test system operated under the abnormal operatng condton representng loss of generaton of 60 MW. The maxmum number of teratons to converge for the proposed approach s Objectve Functon Number of Iteratons Fg. 7. Convergence Characterstcs of SFL Algorthm for IEEE 30-Bus System under Loss of Generaton of 60 MW. The bus voltages vary between pu and 1.082pu n the proposed SFLA.Whereas the voltages vary from pu to pu n Ref. [1,5], pu to 1.10 pu n Ref. [2]. b) Range of generaton defcts contngency The test system s also subjected to contngences characterzed by generaton defcts. The range of generaton s vared from 300 MW to 190 MW, wth a connected load of MW, whch means, the resultng generaton defct vares from 0 to 93.3 MW. Fgure 8 shows the convergence characterstcs of the proposed SFL algorthm for the generaton of 190 MW and the mnmum teratons beng 27. Fgure 9(a) shows the total suppled power obtaned by the proposed SFL algorthm decreases from MW at 290 MW generatons to MW at 190 MW generatons and Fg. 9(b) shows the correspondng actve power loss

17 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng decreases from 9.25 MW to 3.12 MW. Whereas the total suppled power n Refs. [1] and [5] decreases from MW at 300 MW generaton to MW at 190 MW generaton wth correspondng actve power losses decrease from MW to 4.24 MW and n Ref [2] the suppled power decreases from MW at 300 MW generaton to MW at 190 MW generaton wth the correspondng actve power losses decrease from 9.87 MW to 3.94 MW for the same range of generaton defcts Objectve Functon Number of Iteratons Fg. 8. Convergence Characterstcs of SFL Algorthm for IEEE 30-Bus System under Generaton Defcts Contngency Suppled Power n MW Generated Power n MW Fg. 9 (a). Optmal Suppled Load for IEEE 30-Bus System under Generaton Defct Contngency usng SFL Algorthm. The maxmum bus voltage obtaned by the proposed SFL algorthm for ths generaton contngency remans constant at pu and the mnmum voltage vares between 1.06pu and 0.984pu.Whereas n Refs. [1] and [5] the maxmum voltage decreases from 1.1 pu to pu and the mnmum voltage magntude vares from pu to 0.77 pu and n Ref [3] the maxmum voltage s constant at 1.1 pu and mnmum voltage magntude ncreases from pu to pu.

18 1256 R. Mageshvaran et al System Losses n MW Generated Power n MW Fg. 9 (b). System Losses for IEEE 30-Bus System under Generaton Defct Contngency usng SFL Algorthm. For IEEE 30 bus system, bus 5 s the bus wth heavest load and bus 8 s the bus wth second heavest load. The suppled powers by the proposed SFL approach at bus 5 and bus 8 are MW and MW respectvely at a generaton of 250 MW. In Refs [1] and [5] the suppled powers at bus 5 and at bus 8 are MW and MW respectvely. The suppled power at bus 5 n Ref [2] s MW and at bus 8 t s MW. The proposed approach supples more power at the heavest loaded buses- bus 5 as compared to Refs.[2]. c) Overload contngency We consder overloadng up to % above the maxmum avalable generaton of 320 MW. Ths corresponds to a connected load between MW (110% of the nomnal connected load ) to MW (160 % of the orgnal connected load). Fgure 10 shows the mnmum teratons to converge s 31 correspondng to connected load of MW. For ths contngency the suppled load obtaned usng the proposed method decreases from MW at 110% overload to MW at 160 % overload. Whereas for the same range of overload contngences, the suppled power decreases from MW to MW and from MW to MW n Refs [1,5] and Ref [2] respectvely. Fgure 11(a) shows the decrease n suppled power from MW at 110% overload to MW at 160 % overload for the proposed SFL approach. For ths range of overload contngences the system losses obtaned by the proposed method ncreases from MW to MW whch s shown n Fg. 11(b). Whereas the system losses for ths condton reported n Refs[1,5] and Ref[2] ncreases from MW to MW and from MW to MW respectvely.

19 Optmum Steady State Load Sheddng Usng Shuffled Frog Leapng Objectve Functon Number of Iteratons Fg. 10. Convergence Graph of SFL Algorthm for 30-Bus System under Overload Condtons. The proposed approach supples MW to bus 5 and MW to bus 8. Whereas Ref [1, 5] supples to bus 5 MW and 34.6 MWMW to bus 8. Ref [2] supples MW and MW to buses 5 and 8 respectvely. So, durng the overload contngency consdered the proposed method supples more power to these heavest load buses as compared to the power suppled by the method reported n Ref [2]. A better voltage profle s obtaned by the proposed method durng ths contngency. Maxmum voltage obtaned usng the proposed approach s constant at 1.082pu and the mnmum voltage ncreases from to 0.963pu Suppled Power to the System n MW Connected Load n MW Fg. 11(a). Optmal Suppled Power for IEEE 30-Bus System under Overload Condtons usng SFL Algorthm.

20 1258 R. Mageshvaran et al System Losses n MW Connected Load n MW Fg. 11(b). System Losses of IEEE 30-Bus System under Overload Condtons usng SFL Algorthm. In Refs [1,5] the maxmum voltages vary between 1.094pu and 0.943pu. and the mnmum voltage ncreases from 0.77 pu to 0.93pu. The maxmum voltage reported n Ref[2] s constant at 1.1pu and the mnmum voltage ncrease from pu to pu. 5. Conclusons In ths paper a steady state optmal load sheddng strategy usng a memetc meta-heurstc algorthm known as shuffled frog leapng algorthm has been presented. The proposed approach has been tested on IEEE 14 and 30bus test systems. The results obtaned by the proposed approach are compared wth those obtaned by conventonal methods. The comparson s done on the bass of suppled power, system losses, total load shed and the mnmum and maxmum bus voltages. The results presented show that the proposed approach provdes more suppled power and better voltage profle as compared wth those of other methods. Also, the proposed method supples more power to the heavest load buses n the case of IEEE 14 and 30 bus test systems, as compared wth the power suppled by the other methods. The SFLA consst of both local search and global nformaton exchange. The algorthm conssts of a set of nteractng vrtual populaton of frogs parttoned n to dfferent memeplexes. An ndependent local search n each memeplex s carred out smultaneously, whch mproves the explotaton characterstcs. The exploraton characterstc of the algorthm s enhanced by the perodcal shufflng of the memeplexes and the random generaton of vrtual frogs. These characterstcs have mproved the effectveness and effcency of the proposed algorthm n order to obtan better solutons n terms of qualty and accuracy. Based on these outcomes, t can be concluded that the proposed shuffled frog leapng algorthm can be consdered as an effectve alternatve approach for optmal load sheddng. References

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