hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. An Aaptve Energy-Conservng Strategy for Parallel Dsk Systems Mas Nm School of Computng Unversty of Southern Msssspp Hattesburg, MS 39406 mas.nm@usm.eu http://orca.st.usm.eu/~mas Aam Manzanares, Xao Qn Department of Computer Scence an Software Engneerng Auburn Unversty, Auburn, AL 36849 {acm0008,xqn}@auburn.eu http://www.eng.auburn.eu/~xqn Abstract In the past ecae parallel sk systems have been hghly scalable an able to allevate the problem of sk I/O bottleneck, thereby beng wely use to support a we range of ata- ntensve applcatons. Optmzng energy consumpton n parallel sk systems has strong mpacts on the cost of backup power-generaton an coolng equpment, because a sgnfcant fracton of the operaton cost of ata centres s ue to energy consumpton an coolng. Although a varety of parallel sk systems were evelope to acheve hgh performance an energy effcency, most exstng parallel sk systems lack an aaptve way to conserve energy n ynamcally changng workloa contons. o solve ths problem, we evelop an aaptve energy-conservng algorthm, or DCAPS, for parallel sk systems usng the ynamc voltage scalng technque that ynamcally choose the most approprate voltage supples for parallel sks whle guaranteeng specfe performance (.e., esre response tmes for sk requests. We conuct extensve experments to quanttatvely evaluate the performance of the propose energy-conservng strategy. Expermental results consstently show that DCAPS sgnfcantly reuces energy consumpton of parallel sk systems n a ynamc envronment over the same sk systems wthout usng the DCAPS strategy.. Introucton In the last ecae, parallel sk systems have been wely use to support ata-ntensve applcatons, nclung but not lmte to veo survellance [], remote-sensng atabase systems, an gtal lbrares [5], he performance of ata-ntensve applcatons eeply reles on the performance of unerlyng sk systems ue to the raply wenng gap between CPU an sk I/O spees [7]. Parallel sk systems play an mportant role n achevng hgh-performance for atantensve applcatons, because the hgh parallelsm an scalablty of parallel sk systems can allevate the sk I/O bottleneck problem. A growng number of ata centers ntrouce a momentous problem a substantal amount of energy s consume by harware resources n ata centers. For example, the power consumpton of toay s ata center ranges from 75 W/ft 2 to 50-200 W/ft 2. Snce ths tren wll unoubtely contnue n the near future [8], the energy-consumpton problem n ata centers wll become even more serous. Growng evence shows that among varous harware resources n a ata center, storage systems (e.g., parallel sk systems are one of the bggest consumers of energy. A recent nustry report reveals that storage evces account for almost 27% of the total energy consume by a ata center[4]. hs problem s exacerbate by the avalablty of faster sks wth hgher power nees. herefore, t esrable to esgn energy-effcent parallel sk systems by extensvely nvestgate energy-conservaton software technques. Moern ata-ntensve applcatons are lkely to ynamcally change ther sk I/O patterns an performance requrements. As such, t s mperatve for next-generaton parallel sk systems to flexbly an aaptvely reuce energy consumpton urng the course of the executon of a ata-ntensve applcaton. Aaptvely conservng energy n parallel sk systems becomes partcularly crtcal for ata-ntensve applcatons n whch sk requests nee to be complete wthn specfe response tmes or esre response tmes. Hence, energy-effcent parallel sk systems wll have to am at achevng two maor goals: low energy sspaton an hgh guarantee of specfe performance. Dsk scheulng algorthms play an mportant role n reucng the performance gap between processors an sk I/O [9]. he shortest seek tme frst (SSF
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. algorthm s effcent n mnmzng seek tmes; SSF s starvaton-boun an unfar n nature[0]. he SCAN scheulng algorthm can solve the unfarness problem whle optmzng seek tmes [0]. Rest an Danel propose a parameterze generalzaton of the SSF an SCAN algorthms []. Most exstng sk scheulng algorthms are naequate for aaptve energy conservaton n parallel sk systems. o remey ths problem, n ths stuy we evelop an aaptve energy-conservaton scheme or DCAPS usng the ynamc voltage scalng (DVS technque for parallel sks systems. More mportantly, our scheme can proves sgnfcant energy savngs whle guaranteeng esre response tmes of sk requests by seamlessly ntegratng the DVS technque wth sk scheulng mechansms. Dsk I/O parallelsms can be prove n forms of both nter-request an ntra-request parallelsms. he nter-request parallelsm allows multple nepenent requests to be serve smultaneously by an array of parallel sks, whereas the ntra-request parallelsm enables a sngle sk request to be processe by multple sks n parallel. A parallelsm egree of a ata request s the number of sks where the requeste ata reses. he DCAPS strategy evelope n ths research are capable of ealng wth both types of parallelsms. he rest of the paper s organze as follows. We summarze relate work n the next secton. Secton 3 escrbes a system archtecture for energy-effcent parallel sk systems. In Secton 4, we propose the aaptve energy-conservaton scheme. Secton 5 evaluates the performance of the propose energysavng technque by comparng an exstng approach. Secton 6 conclues the paper wth summary an future rectons. 2. Relate Work Dsk I/O has become a performance bottleneck for ata-ntensve applcatons ue to the wenng gap between processor spees an sk access spees [3]. o help allevate the problem of sk I/O bottleneck, a large boy of work has been one on parallel sk systems. For example, Kallahalla an Varman esgne an on-lne buffer management an scheulng algorthm to mprove performance of parallel sks[4]. Scheuermann et al. aresse the problem of makng use of strpng an loa balancng to tune performance of parallel sk systems. Raasekaran an Jn evelope a practcal moel for parallel sk systems [5]. Kotz an Ells propose nvestgate several wrte back polces use n a parallel fle system mplementaton [6]. Our research s fferent from the prevous stues n that we focuse on energy savngs for parallel sk systems. Atonally, our strategy s orthogonal to the exstng technques n the sense that our scheme can be realy ntegrate nto exstng parallel sk systems to substantally mprove energy effcency an performance of the systems. Most of the prevous research regarng conservng energy focuses on sngle storage system such as laptop an moble evces to exten the battery lfe. Recently, several technques propose to conserve energy n storage systems nclue ynamc power management schemes [9], power aware cache management strateges [7], power aware perfectng schemes [8], softwarerecte power management technques [9], reunancy technques [9], an mult-spee settngs[20]. However, the research on energy-effcent parallel sk systems s stll n ts nfancy. It s mperatve to evelop new energy conservaton technques that can prove sgnfcant energy savngs for parallel sk systems whle mantanng hgh performance. he ynamc voltage scalng technque or DVS s a wely aopte approach to conservng energy n processors. he DVS technque can ynamcally reuce the voltage supples of processors to conserve energy consumpton n processors (see, for example,[2]. hus, processor voltage supples are scale own to the most approprate levels, thereby quaratcally reucng power whenever possble. Compare wth tratonal systems wth fxe voltage supply, systems wth DVS can acheve hgh energy effcency. Our approach ffers from the conventonal DVS methos, because ours s the frst technque of ts kn esgne exclusvely for energy-effcent parallel sk systems amng to guarantee specfe performance of ata-ntensve applcatons. Our aaptve energy-conservng strategy makes use of the DVS technque to acheve extremely low energy consumpton n parallel sk systems whle guaranteeng esre response tmes of sk requests. 3. System Archtecture an Moel 3. System Archtecture Frst of all, let us escrbe a framework wthn whch we can evelop an aaptve energy-conservaton technque for parallel sk systems. he framework for energy-effcent parallel sk system s elneate n Fg.. It s worth notng that the framework esgne n ths stuy s general enough to accommoate a we range of storage systems, nclung both network 2
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. attache storage evces (NAS an storage area networks (SAN. he framework embraces a parallel sk system, networks, an aaptve energy-conservng mechansm, a response tme estmator, an a ata parttonng mechansm. he energy-conservng mechansm, whch s at the heart of the propose system framework, s responsble for aaptvely savng energy consumpton n parallel sks wthout sgnfcantly egrang performance of the parallel sk system. hus, the energy-conservng mechansm ams to acheve the best traeoff between energy effcency an performance. Fg. he framework for aaptve energy savng technques. More specfcally, the aaptve energy-conservng mechansm reuce energy consumpton by makng use of the ynamc voltage scalng technque to ucously lower voltage supply levels of sks as long as specfe performance requrements can be met. Secton 4.3 gves much greater etal on the esgn an evelopment of the aaptve energy-conservng mechansm. he ata parttonng mechansm s geare to ve a large amount of ata nto fxe-sze of ata unts store on a number of sks. In ths stuy we conser fle strpng, whch s a generc metho for a vast varety of ata types. o etermne an optmal parallelsm egree (also known as strpe unt sze for each sk request, the ata parttonng mechansm has to leverage the response tme estmator to prect the response tme of the request. he process of ata parttonng s escrbe n suffcent etal n Secton 4.. Moreover, the response tme estmator s nspensable for the aaptve energy-conservng mechansm n the sense that estmatng response tmes make t possble to save energy by ynamcally aust voltage supply levels wthout volatng tmng requrements (.e., esre response tmes Secton 4.2 outlnes a means of estmatng response tmes of sk requests submtte to a parallel sk system. 3.2 Energy Consumpton Moel Before evelopng the aaptve energy-conservng mechansm, we frst ntrouce a power consumpton moel for parallel sk systems. We conser a sequence of sk requests R = { r, r 2,, } L submtte to a parallel sk system. Each sk request r R has an arrval tme a, a esre response tme t, an ata sze. Ieally, request r nees to be complete wthn the esre response tme t. A multple-voltage sk system has a number of screte voltages; the sk system can nstantaneously swtch from one voltage to another. Wthout loss of generalty, we assume the parallel sk system can be operate at a fnte set V = { v, v2, L, vmax } of voltage supply levels. Gven a sk voltage v, we can accorngly etermne the banwth b of the sk. Because energy sspaton n sks quaratcally proportonal to supply voltages, voltage scalng can acheve sgnfcant energy savngs for sks. hus, the energy consumpton rate P of the th sk can be expresse as below: α 2 ( v, vt P = C v,, v, V, v, vt ; C2 ( where C, C 2, an α [,2] are constants epenng on physcal characterstcs of sk evces, v, s the supply voltage, an v t s the threshol voltage. Let D enote a set of sk requests to be processe by the th sk n the parallel sk system. Gven a sk request r to be processe by the th sk, we can calculate the energy consumpton of the request as below: = P v v (2 where ( ( E,, θ, v, s the voltage supply level etermne for the sk request, P ( v, s the sk s energy consumpton rate, an θ ( v, s the processng tme of the sk request. Both P (, an θ ( v r n v, 3
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. largely rely on the supply voltage ( v, of the sk; P v, can be straghtforwarly erve from Eq. (. he energy consumpton E of sk s wrtten as a summaton of energy consumpton cause by each sk request hanle by the sk. hus, we have ( v ( v E = E = P,, r D r D, θ (3 Suppose there are m sks n the parallel sk system, the total energy consumpton E of the sk system can be expresse as: E = m = E = m = r D E, = m P = r D ( v θ ( v,, (4 We can now obtan the followng non-lnear optmzaton problem formulaton to compute the energy consumpton of a parallel sk system m Mnmze E = P ( v, θ ( v, = r D Subect to (a { v, v, L v } v, 2, max f t (5 (b where f s the response tme of the th sk request. f t n Expresson (5 sgnfes that the esre response tme constrants must be met. 4. Aaptve Energy-Conservng Strategy he propose aaptve energy-conservng strategy encompasses three components, namely, a ata parttonng technque, response tme estmaton metho, an an aaptve DVS algorthm. In ths secton, we escrbe the esgn of these three components n more etal. 4. Data Parttonng One of the maor components n the propose framework (see Secton 3. s the metho of ata parttonng that etermnes the optmal parallelsm egrees for sk requests. Dynamc ata parttonng s of mportance for our aaptve energy-conservng strategy, because the ata parttonng metho helps n mnmzng the response tmes of requests, thereby creatng more space to reuce energy consumpton by scalng own sk supply voltages. As such, n the frst place our strategy ams to shorten the response tmes by aaptvely etermnng the optmal parallelsm egree of each request (see Step 3 n Fg. 2. We enote the parallelsm egree an ata sze of a request r by p an, respectvely. Before proceeng to the analyss of optmal parallelsm egrees, let s frst formally erve the sk servce tme sk (, p of request r. hus, the sk servce tme can be compute as sk (, p = seek ( p + rot ( p + trans (, p, (6 where seek ( p, rot ( p, an trans (, p are the seek tme, rotaton tme, an transfer tme of the sk request he seek tme can be approxmate as below, where C s the number of cylners on a sk, a an b are two sk-type-nepenent constants, whereas e an f are sk-type-epenent constants. seek ( p = ec( a b ln( p + f (7 he value of rotaton tme can be expresse as Eq. (8, where RO s the rotaton tme of a sk. p rot ( p = RO (8 p + he transfer tme can be approxmate by Eq. (9, where B sk s the sk banwth. trans (, p p B = (9 Substtutng Eqs. (7-(9 nto Eq. (6, we obtan the value of sk servce tme as p sk (, p = ec( a bln( p + f + RO +. p + p Bsk (0 Now we are postone to calculate the optmal parallelsm egree of request r by etermnng the mnmum of the functon sk (, p. hus, we can obtan the optmal value of p by solvng Eq. (. sk (, p RO p RO ecb = = 0. 2 2 ( p p + ( p + p p Bsk ( he parallelsm egree etermne by Eq. ( can not excee m, whch s the number of sks n the system. herefore, the optmal parallelsm egree s gven by mn( p, m. sk 4.2 Response me Estmator 4
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. o aaptvely aust the voltage supply of sk requests, we nee to estmate each request s maxmum response tme, whch s efne as an nterval between the tme a request submtte an the tme the parallel sk system completes corresponng sk I/O operatons. Gven a newly ssue request r, the response tme of r s estmate by Eq. (2. ( r, σ = queue + partton p + max = { ( r, σ } proc (2 where p s the parallelsm egree etermne by the ata parttonng mechansm, v = ( v, v2, L, v p s the request s vector of the supply voltage for p strpe unts, queue s the queueng elay at the clent se, partton s the tme spent n ata parttonng, an s the system processng elay experence by the th strpe unt of the request. Wth respect to the th strpe unt of the request, the system processng elay proc can be expresse as proc proc ( r, v = network ( r, v + sk ( r, v (3 where network, an sk are the elays at the network subsystem, an parallel sk subsystems, respectvely. We assume that when the th strpe unt of a request arrves at the network queue, there are k strpe unts watng to be elvere to the parallel sk sub-system. Suppose strpe unts are transmtte n a frst-n-frst-out orer, all the strpe unts that are alreay n the queue pror to the arrval of the th strpe unt must be transmtte earler than the th strpe unt. Hence, the elay n the network subsystem r, v can be wrtten as network ( network + p r, v = B k = ( (4 network where s the ata sze of the th strpe unt n the network queue, an B network s the effectve network banwth. It s worth notng that k n Eq. (4 s the optmal parallelsm egree etermne by the ata parttonng mechansm (see Eq. n Secton 4.. Smlarly, t s assume that when the th strpe unt of the request arrves at sk, there are k sk requests must be processe by sk before hanlng the strpe unt. hus, the elay n the sk subsystem r, v s gven by the followng formula sk ( sk k sk, ( p + sk, ( l l= ( r, v = (5 where sk, ( s the sk processng tme of a request contanng bytes of ata. We can quantfy as follows, ( sk, ( seek + rot + B sk = (6 sk where seek an rot are the seek tme an rotatonal latency, an B sk s the ata transfer tme epenng on the ata sze an sk banwth B sk. 4.3 he Aaptve Energy-Conservaton Algorthm Now we are postone to esgn the aaptve energy-conservaton algorthm or DCAPS for parallel sk systems. he DCAPS algorthm ams at ucously lower the parallel sk system voltage usng ynamc voltage scalng technque or DVS, thereby reucng the energy consumpton experence by sk requests runnng on parallel sk systems. he processng algorthm separately repeats the process of controllng the energy by specfyng the most approprate voltage for each sk request. hus, the algorthm s geare to aaptvely choose the most approprate voltage for strpe unts of a sk request whle warrantng the esre response tme of the request. Specfcally, the algorthm s carre out n three phases: ynamc ata parttonng (see Secton 4., response tme estmaton (see Secton 4.2, an aaptve energy consumpton controller. o mnsh the energy consumpton of the sk systems, DCAPS eneavors to mnmze the supply voltage of a request. Hence, the frst phase ynamcally calculates the optmal parallelsm egree of the request, thereby reucng elays at the parallel sk subsystems (see Eq. 8. Durng the secon phase of the algorthm, the response tme of each strpe unt s estmate (see Eqs. 5 an 6. Phase three, gue by the estmate response tme obtane an esre response tme, aaptvely reuce the supply voltage for each strpe unt prove that the request s response tme oes not excee the request s esre response tme. he complete algorthm of DCAPS s outlne n Fg. 2., 5
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. When a sk request s ssue to the system, the DCAPS strategy nserts the newly arrve requests nto the watng queue base on the earlest esre response tme frst polcy (see Step. After the ata portonng of each request n the queue, DCAPS ntalzes the voltage of all the strpe unts of request r to the maxmum supply voltage v max (see Step 6. In ong so, DCAPS are more lkely to guarantee esre response tmes uner heavly loae contons. Usng the ynamc voltage scalng technque, the DCAPS strategy aaptvely makes sks operate at low voltage supply levels for all the strpe unts to conserve the total energy consumpton of the parallel sk system. Assume that the maxmum supply voltage v max. s 3.3 Volts, the supply voltage can be reuce as long as the sk request can be accomplshe wthn ts esre response tme or the supply voltage reach the mnmum voltage v mn. In ths stuy, we assume that the threshol voltage s 0.8 [onlne]. In an effort to stealy reuce the voltage of strpe unts, DCAPS guarantees that all requests wll be complete before ther esre response tmes. hus, the followng property nees to be satsfe n DCAPS.. he reuce supply voltage v s greater than the mnmum voltage v mn ; 2. ( r, p, σ t, where s the response tme of the th stpe unt, t s the esre response tme of the request, an r, p, σ = + + ( r, p, σ ( queue partton proc 3. he reuce supply voltage v s greater than the mnmum voltage v mn ; 4. ( r, p, σ t, where s the response tme of the th stpe unt, t s the esre response tme of the request, an ( r, p, σ = queue + partton + proc ( r, p, σ Steps 0- are repeately performe to scale own sk voltage untl a request s esre response tme cannot be guarantee (see Step 2 or the supply voltage are approachng the threshol voltage. Consequently, DCAPS aaptvely reuce the supply voltage whle makng the best effort to complete all the sk requests before ther esre response tme. 5. Expermental Results o evaluate the performance of the DCAPS strategy n an effcent way, we smulate a parallel sk system wth all the functons that are necessary to mplement our system. able summarzes mportant parameters use to resemble real worl sks. In aton, we mplemente a ata-parttonng algorthm to optmze parallelsm egrees of large sk I/O requests. We wll frst compare the performance of a parallel sk system wth DCAPS wth that of another system wthout employng DCAPS. We wll then stuy effects of varyng arrval rates, ata sze, an sk banwth on the performance of the two sk systems. Next, we wll compare an evaluate the two sk systems base on varyng the voltage. Fnally, we wll also analyse the performance mpacts of parallelsm egrees on the parallel sk systems. able. Dsk parameters of the smulate parallel sk system In our smulaton experments, we mae use of the followng three performance metrcs to emonstrate the effectveness of the DCAPS scheme. ( Satsfe rato s a fracton of total arrve sk requests that are foun to be fnshe wthn ther esre response tmes. (2 Energy consumpton s the total energy consume by the parallel sk systems. An (3 Energy conservaton rato 5. Impact of Arrval Rate hs experment s focuse on comparng a parallel sk system wth the DCAPS strategy aganst a stanar parallel sk system wth a fxe voltage supply level. We stuy the mpacts of sk request arrval rate on the satsfe rato an normalze energy consumpton. o acheve ths goal, we ncrease the arrval rate of sk requests from 0. to 0.5 No./Sec wth an ncrement of 0. No./Sec. Fgs. 3 an 4 plot the satsfe ratos, normalze energy consumpton, an energy conservaton rato of the parallel sk systems wth an wthout DCAPS. Fgs 3(a reveals that the DCAPS scheme yels satsfe ratos that are very close to those of the parallel sk system wthout employng DCAPS. hs 6
hs paper appeare n the Proceengs of the 2th IEEE Internatonal Symposum on Dstrbute Smulaton an Real me Applcatons (DS-R 08, Vancouver, Brtsh Columba, Canaa, Oct. 2008. s essentally because DCAPS eneavors to save energy consumpton at the margnal cost of satsfe rato. More mportantly, Fgs. 3(b an 4 show that DCAPS sgnfcantly reuces the energy sspaton n the parallel sk system by up to 7% wth an average of 52.6%. he mprovement n energy effcency can be attrbute to the fact that DCAPS reuces the sk supply voltages n the parallel sk system whle makng the best effort to guarantee esre response tmes of the sk requests. Furthermore, t s observe that as the sk request arrval rate ncreases, the energy consumpton of the both parallel sk systems soars. Fg. 4 shows that as the loa ncreases, the energy conservaton rato tens to ecrease. hs result s not surprsng because hgh arrval rates lea to heavly utlze sks, forcng the DCAPS to boos sk voltages to process larger number of requests wthn ther corresponng esre response tmes. Increasng number of sk request an scale-up voltages n turn gve rse to the ncrease energy sspatons n the parallel sk systems. ths paper, we focuse on the esgn of novel parallel sk systems that can acheve both great energy effcency an hgh guarantee of specfe performance. Specfcally, we evelope an aaptve energyconservng strategy, whch ynamcally scale own sk voltage supples to the most approprate levels, thereby sgnfcantly reuce energy sspaton n parallel sk systems. he expermental results have confrme that our scheme can acheve up to 70% energy savngs compare wth stanar parallel sk systems wth fxe supply voltage. Our approach s the frst technque of ts kn esgne exclusvely for energy-effcent parallel sk systems amng to guarantee specfe performance of ata-ntensve applcatons. As a future recton, we wll propose a ynamc voltage scalng technque at the level of ata-ntensve applcatons. Further, we plan to exten our approach by conserng overhea of scalng sk supply voltages. Fg. 4. Impact of request arrval rate on energy conservaton rato. 6. Conclusons an Future Work Parallel sk systems play an mportant role n achevng hgh-performance for ata-ntensve applcatons, because the hgh parallelsm an scalablty of parallel sk systems can allevate the sk I/O bottleneck problem. However, growng evence show that a substantal amount of energy s consume by parallel sk systems n ata centers. It s therefore hghly esrable to esgn energy-effcent parallel sk systems by extensvely nvestgate energyconservaton software technques. Aaptvely conservng energy n parallel sk systems becomes partcularly crtcal for ata-ntensve applcatons n whch sk requests nee to be complete wthn specfe response tmes or esre response tmes. In Fg. 3. Impact of request arrval rate on satsfe rato an normalze energy consumpton when sk banwth s 30MB/sec. Acknowlegments: he work reporte n ths paper was supporte by the US Natonal Scence Founaton uner Grants No. CF- 074287, No. CNS-0757778, No. CNS-083502, No. OCI-0753305, an No. DUE-062307, an Auburn Unversty uner a startup grant. References [] Avtzour, Novel scene calbraton proceure for veo survellance systems, IEEE rans. Aerospace an Electronc Systems, Vol. 40, No. 3, pp. 05-0, July 2004. [2] Al Manzak an Chatal Chakrabart, varable Voltage ask Scheulng Algorthms for Mnmzaton Energy/Power, IEEE rans very Large Scale Integraton Sys, vol., no. 2, Aprl 7
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