Steady state load-shedding by Alliance Algorithm

Transcription

1 Steady state load-sheddng by Allance Algorthm V. Calderaro, V. Gald, V. Lattarulo, A. Pccolo, P. Sano Department of Informaton and Electrcal Engneerng, Faculty of Engneerng, Salerno Unversty, Italy Abstract-Ths paper ntroduces the Allance Algorthm (AA), a new soft-computng algorthm used to solve optmal steady state load-sheddng. The AA performance s measured n terms of response speed and post emergency state of the power system. The solutons are compared wth genetc and smulated annealng algorthms testng two case studes. Keywords- Allance Algorthm, Optmal Load-Sheddng, Soft-computng. I. ITRODUCTIO The ncreasng load demand makes power systems more vulnerable, worsenng the ablty to wthstand the mpact of sudden changes such as loss of generators or transmsson lnes [1]. So, the growth of the load must be accompaned by approprate strateges that mprove securty. One of possble solutons to prevent catastrophc mpacts s load-sheddng that forces the perturbed system to a new stable equlbrum state. Snce after a contngency the restoraton strategy for power systems plays a key part n mprovng servce relablty, there has been consderable research effort focused on load-sheddng scheme. Many challenges have been n searchng optmal restoraton strategy solutons both reducng the search space and an acceptable computng tme. Approaches proposed for ths problem are based on lnear and non-lnear programmng [2-5] or by means of soft-computng technques [6-8]. In ths paper a new soft computng algorthm for optmal steady state load-sheddng scheme s ntroduced. It s used to mnmze unserved load and prefer hgh prorty loads. The algorthm s named Allance Algorthm (AA) because t s based on the allances among trbes. The strongest allance takes possesson of the envronment thanks to the aggregatons wth other trbes/allances. Durng the evoluton of the algorthm the allances change untl a stop crteron s reached. The performance of the algorthm s measured n terms of response speed and post emergency state of the power system. The algorthm s appled to two case studes on standard test IEEE networks and the results are compared wth other algorthms. II. PROBLEM DESCRIPTIO The goal of the load-sheddng problem s to mnmze the unserved loads classfed accordng to dfferent load prortes. It s assumes that the load-sheddng s necessary followng the loss of a generator. The problem s subject to operatng constrants and the mathematcal model can be expressed as follows: ( b, p) x( b) (1) b where f1 f2 wth: f 1 ( b p ) and f 2 ( b ) (2) 1 1 In (2), s the total number of the loads, b denotes the set of the swtch status assocated to each load and =[0,1] s the nteger soluton space such as: b 1 0 f the load s connected f the load s dsconnected p s the set of the assgned load prorty so that the sngle element can assume one of the followng values: 1 load wth low prorty p 2 load wth medum prorty (4) 3 load wth hgh prorty The parameters and are the factors of load prorty and dsconnected loads, respectvely. They are useful to evaluate the mportance of the sngle functon f 1 and f 2. The electrcal and operatonal constrants are summarzed by: (3) ( b) ( b) 0 (5) where the vector functon (b) descrbes the equalty and nequalty constrants for a balanced system operaton, accordng to: b Pg Pd V V j Yj cos( j j ) 1... b j1 b Qg Qd V V j Yj sn( j j ) 1... b j1 (6) mn V V V 1... b mn Pg Pg Pg 1... g mn Qg Qg Qg 1... b I h I h h 1... l sh sh Pg ( fmn f0 ) D PL Pg ( f f0 ) D where b, l and g are the total number of the buses, lnes and generators, respectvely. I h s the current on the h-lne, P g and Q g are the generated actve and reactve power at -bus, P d and Q d are the demanded actve and reactve power at the Reference umber: W

2 -bus, V s the magntude of the voltage at -bus, Y j s the magntude of (,j) element of admttance matrx, θ j s the angle of (,j) element of the admttance matrx, s the angle sh of the voltage at -bus, P g s the trpped generator power, f 0 s the nomnal frequency, D s the dampng load constant and P L s the load power to dsconnect. The terms wth and mn ndcate the lmts of the correspondng quanttes. The last equaton of ( 6) allows steady state frequency to mantan wthn a permssble range. It has been obtaned gnorng generator droops [9]. III. ALLIACE ALGORITHM The ntroduced load-sheddng problem can be solved by usng the Allance Algorthm, a new algorthm based on the concept of the nteractons among trbes n an envronment to conquer [10]. These nteractons allow creatng allances on the bass of two dfferent characterstcs: 1. the trbe strength; 2. the trbe resources. All trbes stay n a shared envronment, n whch there s a mum amount of resources that t offers to the trbes. As the amount of resources s less than the sum of the amounts of resources that trbes need, the trbes have to become stronger to reman n the envronment. In order to mprove the probablty of survval, the trbes create allances. Fnally, a great allance wll conquer the envronment. It wll consst of some trbes that have globally a lesser amount of resources than those offered by the envronment and the mum strength. All the others trbes and allances leave the competton because they are weaker than the great allance, so the other allances and trbes go away from the envronment. So, at the end, only the allance that can use all possble resources remans. Ths allance s composed by trbes that represent the soluton of the problem. Analytcally, let T the number of trbes (1,2, T), where the progressve numbers are the IDs of the trbes. Every trbe have a strength s, needs of resources r and stays n an allance a, whch s empty when the trbe enters n another allance, otherwse a contans at least the trbe. The allance can also contan other trbes but wth an ID superor than. So, a generc trbe s composed by ( s, r, a mn ), where mn s the mnmum trbe s ID of the trbe s allance. The problem conssts to mze the functon: T H s b (7) 1 consderng the constrant: T r b R (8) 1 where b 0,1 and R s the amount of the envronment resources. IV. COMPUTATIOAL PROCEDURE The AA can be mplemented consderng the followng ten steps: 1) ntalzaton, 2) token phase, 3) choosng of an ally, 4) verfcaton, 5) new members become alled, 6) remove some group s members, 7) update remanng Data Structures, 8) control state, 9) search the strongest allance, 10) soluton 1/0 of the strongest allance. 1. Intalzaton In ths frst phase, the T trbes are ntalzed and placed n the envronment. Ths ntalzaton conssts of assgnng an ID, a strength and an amount of resources needed for each trbe. Also the amount of the resources R that the envronment can offer s set. 2. Token Phase The Token Phase conssts of usng a token to choose whch allance has to go to the thrd step. So, n ths step a token moves among the allances; f an allance has the token can ask to other trbes f they want jon n the allance. 3. Choosng of an ally In ths phase the allance chooses the trbe that can become a new ally. The allance can choose only trbes that are not already alled; there are two prncpal ways to choose an ally: 1. by scannng all the trbes; 2. by usng a random way. 4. Verfcaton Wthout loss of generalty to explan ths step t s possble to suppose that there are two allances wth a generc number of trbes. In fact, the nteracton also n a complex system, wth more than two allances, s always between two allances at the tme. The trbes of the frst allance a 1 are: [(s 1,r 1,a 1 ),,(s p,r p,a 1 )] and the trbes of the second allance a t are: [(s t,r t,a t ),,(s T,r T,a t )] where 1 s the mnmum ID n the frst group and t s the mnmum ID n the second group. The a 1 asks to the trbe x of the other allance to jon n, durng ths phase the trbe x has to answer consderng four possbltes: 1. f the sum of the amount of resources requred by the two allances s less than the amount of resources that the envronment can offer then there s a unfcaton of the two allances. Analytcally, consderng only two allances 1 and t, gven the sum of the amount of resources necessary for the frst allance: r 1, 1 where 1 s the number of trbes present n the frst allance, and gven the sum of the amount of resources necessary for the second allance: Reference umber: W

3 r t 1, where s the number of trbes present n the second allance, n order to create a new allance t s necessary to satsfy: r r t R 1, 1 1, (9) So, the fnal result s the followng aggregaton: [(s 1,r 1,a 1 ),(s 2,r 2,a 1 ),,(s T-1,r T-1,a 1 ),(s T,r T,a 1 )] 2. If the sum of the amount of resources requred by the two allances s more than the amount of avalable resources: r r t R 1, 1 1, (10) but f the sum of the amount of resources of a 1 and the amount of resources of trbe x s less than the amount of resources that the envronment can offer such as: r rx( R 1, 1 (11) the trbe x changes ts allance only f the sum of the a 1 s strength and the trbe x s strength s more than a t s strength: s sx t s t 1, 1 ( ) 1, (12) The subscrpt x( ndcates that the trbe x s changng allance: t s leavng the t allance and enters nto the 1 allance; the fnal result s the followng aggregaton: [(s 1,r 1,a 1 ),..,(s x,r x,a 1 ),..,(s p,r p,a 1 ),(s t,r t,a t ),,(s T,r T,a t )] where p s the mum ID of the allance If the sum of the amount of resources of a 1 and trbe x s greater than the amount of resources that the envronment can offer such as: r rx( R 1, 1 (13) and f the strength of the trbe x s greater than the mnmum strength of at least one trbe n a 1 : s x( >mn(a 1 ) (14) where mn(a 1 ) s the weakest trbe of the allance a 1, a 1 tres to remove one or more trbes to ntegrate the trbe x. Analytcally: r,1 rx( r removed R 1 1, (15) Obvously, the sum of the strength of the trbes that the allance removes has to be less than the strength of the trbe x such as: s,1 sx( s 1 1, removed s 1, 1 (16) where the terms wth the word removed n (1 5)-(16) represent the amount of resources and the strengths, respectvely, of the group of the removed trbes from the allance 1 for the entrance of the new trbe. So, for example, f the trbe y s removed by the a 1 and the trbe x enters n the a 1 the fnal result s: [(s 1,r 1,a 1 ),,(s x,r x,a 1 ),,(s p,r p,a 1 ), (s y,r y,a y ),(s t,r t,a t ),,(s T,r T,a t )] On the other hand, f the strength of the trbe x s less than the mnmum strength of at least one trbe n a 1 : s x mn(a 1 ) (17) the trbe x cannot jon the a 1 and the fnal result does not change from the prevous aggregaton. 4. Ths last case happens when () the amount of resources of the two allance are more than the resources offered by the envronment, accordng to (1 0); () the amount of resources of a 1 wth the amount of resources of trbe x s less than the amount of resources that the envronment can offer accordng to (11); () a t s stronger than the a 1 such as: s sx s t 1, 1 1, (18) So, also n ths case nothng happen from the prevous aggregaton. 1. ew members become alled Ths step depends on the step 4 because f n the verfcaton step there s a stuaton as presented n the case 4, the trbe/allance can t become alles. In ths step the trbe/allance physcally jons n the allance that has asked for t. The strength of the new member (trbe/allance) s added to that of the allance that t jons. 2. Remove some group s members Some members are removed from an allance n two dfferent ways: () t he members receve an nvte to become alles n another allance. Ths possblty happens n the cases 1 and 2 of the verfcaton step; () the members are removed by the allance because t substtutes them wth a better member. Ths possblty happens n the case 3 of the verfcaton step. In order to choose whch allance members have to leave a smple algorthm s used: a. nsert all the members wth a strength less than the strength of the member that has to jon n the allance; b. take an unused member by usng leas densty: d =s /r (19) c. control the possblty of stoppng the process by (15); d. f t s not possble to stop return to the step b; e. control f the fnal value of the allance s greater than the prevous (1 6), f yes and also (12) s verfed take the soluton proposed, otherwse t s mpossble to nsert the new member n the allance. Reference umber: W

4 3. Update remanng Data Structures In ths step, n order to make permanent the jonng of a trbe/allance n the allance wth the token, the updatng of the new aggregatons between trbes s carred out. Moreover, are also updated the possble choces that the actual allance could do. 4. Control State In ths step the AA scans and controls f there are the condtons for the endng of the algorthm. The algorthm ends when each allance has asked every trbes and remans unchanged. 5. Search the strongest Allance When the teratons between the trbes are fnshed many allances are formed but only one s the strongest allance. So n ths step the algorthm calculates the strength of each allance and takes the strongest allance. 6. Soluton 1/0 of the strongest allance Ths s the last step and after that the algorthm has chosen the allance outputs the soluton n a T dmenson vector where the -element s 0 f the trbe s not n the allance and 1 otherwse. So, n ths step there s the converson of the strongest allance to a mathematcal soluton. V. APPLICATIO OF AA TO LOAD-SHEDDIG PROBLEM In order to solve the load-sheddng problem by AA, the objectve functon (1) must be consdered. Eqn. ( 7) can be generalzed as follow: H [ ( b ( p ))] (20) 1 It s an mplct multobjectve functon where the parameters and have been ntroduced n the Secton II. It s possble to notce that (7) s a partcularzaton of (20) wth =1 and =0. Consderng that the prortes represent the strengths of the trbes, (20) can be rewrtten as (22): x Sx ( s ) s x (21) 1 1 where x s the ID of the generc allance x and the generc s s the strength of the trbe n the allance x. Moreover, (12) and (16) become (22) and (23), respectvely: 1 ( t s 1, 1 sx) ( 1 1) s 1, t t (22) ( sx s removed 1, ) (1 ) 0 (23) A. Procedure The load-sheddng problem s solved ntegratng AA nto a load flow solver based on ewton-raphson method. In partcular, the soluton proposed by AA s passed by load flow solver n order to verfy the constrants (6). If not all the constrants are respected the soluton s dscarded and archved. The next soluton wll take account of those saved. Ths cycle contnue untl the algorthm fnds a soluton that respects all the constrants. B. Case studes In order to llustrate the effcency of the AA two IEEE standard test cases have been used: the 14 bus and the 30 bus IEEE. The results obtaned by the AA have been compared wth the results obtaned by the Genetc Algorthms and Smulated Annealng. The test have been made wth =1 and =0 by usng AA, the populaton sze of GA was 20, the mum teraton of SA was Each test has been repeated 100 tmes n order to estmate the best and the average values. The smulatons have been carred out by Matlab rev.7, by usng a processor Intel Core 2 Duo P GHz and 2.27 GHz. C. s and comparson The frst case study has been run on 14 bus IEEE test system. The network presents two generators wth a total power generaton of MW, the system s perturbed for the loss of a generator. The post fault system supples MW. So, the power of the trpped generator s 40 MW. The total power demand s 259 MW and the system s composed by 11 loads. In the table I are shown the load power wth ther prortes. TABLE I: Load power and prortes case 1 Loads Actve Power (MW) Prorty TABLE II: s for the case study IEEE 14 bus Alg. ame AA s 0.55s GA s 4.75s SA s 9.91s In the table II are presented the results by usng AA and the other two alternatve algorthms. In the result table the term Cycle ndcates the number of tmes that t s necessary to execute the ewton-raphson; s the value of the objectve-functon wth the found soluton; s tme consumng from the start of the algorthm to the end of the load flow untl a soluton respects all the constrants. For the best soluton 10 loads are connected and the power demand s MW and the power loss s MW, thus the power that the generator has to supply s MW. By lookng the table II, t s possble to notce that all the algorthms fnd Reference umber: W

5 a feasble soluton n only one cycle. The best and the average results are the same for all the algorthms; ths means that for all the 100 tmes the found result was 17. In the last two columns, t s hghlghted as the AA s speed to fnd the same soluton s more than the other algorthms. Table III and IV present the data and the results of the second case study, respectvely. In partcular, the IEEE test system has 30 buses wth fve generators wth a total generaton power of MW. TABLE III: Load power and prortes case 2 Loads Actve Power (MW) Prorty TABLE IV: s for the case study IEEE 30 bus Alg. ame AA s 3.08s GA s 4.65s SA s 4.83s The system s perturbed for the loss of a generator determnng a new generaton power of MW. The power of the trpped generator s MW. The total power demand s MW. In the table III are reported only the buses wth a power demand dfferent from zero. For the best soluton 16 loads are connected, the power demand s MW and the power loss s 1.26 MW, thus the power that the generator has to supply s MW. Also n ths case all algorthms fnd a feasble soluton n only one cycle. Even though the best results are the same for all algorthms, only the AA found the best average result. Furthermore, n terms of tme consumng, the AA presents the best result. So, the number of more cycles used by AA allows obtanng very low tme solutons. VI. COCLUSIO Ths work solves a steady state load-sheddng problem n order to mnmze the unserved power demand consderng load prorty. The load-sheddng s necessary followng the loss of a generator. The problem has been formulated as an mplct multobjectve mxed non nteger programmng and the soluton algorthm has been obtaned comparng three dfferent algorthms. In partcular, the best soluton has been found by usng a new ntroduced algorthm, named Allance Algorthm. It s based on the allances among trbes and the allance whch gets the hghest number of resources s consdered stronger and takes possesson of the envronment (soluton space). The nterestng results are compared wth a Genetc Algorthm and a Smulated Annealng. The performances have been evaluated n terms of response speed and post emergency state of the power system. The effectveness of the method has been valdated by means of two case studes, consderng two IEEE standard case test systems. REFERECES [1] P.M. Mahadev, R.D. Chrste, Envsonng power system data: vulnerablty and severty representatons for statc securty assessment, IEEE Trans. on Power Systems, v. 9, n. 4, ov [2] M.A. Mostafa, M.E. El-Hawary, G.A.. Mbamalu, M.M. Mansour, K.M. El-agar, A.M. El-Arabaty, A computatonal comparson of steady state loadsheddng approaches n electrc power systems, IEEE Trans. on Power Systems, v. 12, n. 1, Febr [3] u Dng, A.A. Grgs, Optmal load-sheddng strategy n power systems wth dstrbuted generaton, IEEE Wnter Meetng, v. 2, [4] Aponte, E.E., elson, J.K., Optmal Loadsheddng for Dstrbuted Power Systems, IEEE Trans. on Power Systems, v. 21, n. 1, Febr [5] Wang, P., Dng, Y., Goel, L., Relablty assessment of restructured power systems usng optmal loadsheddng technque, Generaton, Transmsson & Dstrbuton, IET, v. 3, n. 7, July [6] Luan, W.P., Irvng, M.R., Danel, J.S., Genetc algorthm for supply restoraton and optmal loadsheddng n power system dstrbuton networks, IEE Proceedngs Generaton, Transmsson and Dstrbuton, v. 149, n. 2, March [7] Sanaye-Pasand, M., Davarpanah, M., A new adaptve multdmensonal load-sheddng scheme usng genetc algorthm, Canadan Conference on Electrcal and Computer Engneerng, May [8] akawro, W., Erlch, I., Optmal Load-sheddng for Voltage Stablty Enhancement by Ant Colony Optmzaton, 15 th Int. Conf. on Intellgent System Applcatons to Power Systems, ov [9] P. Kundur, Power System Stablty and Control, EPRI Power System Engneerng, McGraw-Hll, [10] V.Lattarulo, Applcaton of an nnovatve optmzaton algorthm for the management of Energy resources, Bsc Thess, Reference umber: W