Cooperative Distributed Scheduling for Storage Devices in Microgrids using Dynamic KKT Multipliers and Consensus Networks

Transcription

1 Cooperave Dsrbued Schedulng for Sorage Devces n Mcrogrds usng Dynamc KK Mulplers and Consensus Newors Navd Rahbar-Asr Yuan Zhang Mo-Yuen Chow Deparmen of Elecrcal and Compuer Engneerng Norh Carolna Sae Unversy Ralegh NC USA hs paper s acceped for presenaon a he 1 IEEE PES General Meeng July n Denver Colorado USA and wll be publshed n he proceedngs of he conference.

2 1 Cooperave Dsrbued Schedulng for Sorage Devces n Mcrogrds usng Dynamc KK Mulplers and Consensus Newors Navd Rahbar-Asr Yuan Zhang Mo-Yuen Chow Deparmen of Elecrcal and Compuer Engneerng Norh Carolna Sae Unversy Ralegh NC USA nrahbar@ncsu.edu yzhang@ncsu.edu chow@ncsu.edu Absrac Schedulng of sorage devces n mcrogrds wh mulple renewable energy resources s crucal for her opmal and relable operaon. Wh proper schedulng he sorage devces can capure he energy when he renewable generaon s hgh and uly energy prce s low and release when he demand s hgh or uly energy prce s expensve. hs schedulng s a mul-sep opmzaon problem where dfferen me-seps are dependen on each oher. Convenonally hs problem s solved cenrally. he cenral conroller should have access o he real-me saes of he sysem as well as he predced load and renewable generaon nformaon. I should also have he capably o send dspach commands o each sorage devce. However as he number of devces ncreases he cenralzed approach would no be scalable and wll be vulnerable o sngle pon of falure. Combnng he dea of dynamc KK mulplers wh consensus newors hs paper nroduces a novel algorhm ha can opmally schedule he sorage devces n a mcrogrd solely hrough peer-o-peer coordnaon of devces wh her neghbors whou usng a cenral conroller. Index erms Mcrogrds; Opmal Schedulng; Dsrbued Algorhms; Opmal Conrol; I. INRODUCION Mcrogrds are small scale power sysems wh local generaon resources sorage devces [1] and loads. hey are he val elemens o negrae dsrbued renewable energy resources o he grd []. he sorage devces n he mcrogrds can capure he elecrc energy of he renewables when generaon s more han demands or he uly elecrcy prce s low and hen release durng hgh demands or when he elecrcy s expensve. In order o operae mcrogrds effcenly he energy resources and sorage devces need o be opmally scheduled. hs schedulng has o ae real-me operang condons of he sysem as well as he predced nformaon regardng he fuure renewable generaons and demands no consderaon [3]. Because of he sorage devces he amoun of energy generaed/consumed n one me sep s no ndependen of oher me seps. herefore he schedulng becomes a mulsep decson mang process and ypcally cenralzed echnologes are proposed o solve. For example cenralzed cooperave parcle swarm algorhm n [] cenralzed mxed neger lnear programmng n [] and genec algorhms n [] are used for opmal allocaon of resources n mcrogrds. As he number of devces n mcrogrds ncreases and mcrogrds are furher combned o form smar grds he cenralzed soluons would no be effcen anymore [7]. Frs of all hey are requred o have communcaons wh all he conrollable devces n he sysem whch wll ncrease he congeson. Second hey are no scalable wh he ncreasng number of devces. And hrd hey are vulnerable o sngle pons of falure. For hs reason n recen years n he leraure of smar grds dsrbued soluons have araced aenon. For nsance dsrbued load sheddng [] dsrbued economc dspach for energy resources [9] [11] dsrbued generaon and demand response [7] and dsrbued elecrc vehcle charge opmzaon [1]. However many of he proposed dsrbued approaches only solve sngle sep opmzaon problems. In [13] [1] and [1] mul-sep dsrbued opmzaons are nroduced based on Lagrange mulplers o solve he mcrogrd schedulng problem bu he dsrbued conrollers sll need o communcae wh all he oher conrollers regardng her common bndng consrans. hs paper provdes a novel cooperave dsrbued algorhm for opmal schedulng of sorage devces n a mcrogrd. he algorhm combnes he conceps of dynamc KK mulplers and consensus newors. In hs approach each energy devce n he mcrogrd s equpped wh a dsrbued conroller. he dsrbued conrollers have he ably o exchange nformaon wh her neghbors. Provded ha he communcaons newor among dsrbued conrollers s conneced each dsrbued conroller can fnd he opmal schedule for s conneced devce n an erave procedure. hree man feaures of he proposed algorhm are as follows: 1) Solves he mul-sep opmzaon problem wh emporally and spaally coupled consrans n a fully dsrbued way whou requrng any cenral conroller/coordnaor/leader. ) he nodes are only requred o communcae wh her neghbors. Unle some dsrbued schedulng approaches n he leraure he devces do no have o communcae wh all he devces ha share he same bndng consrans. 3) Devces do no have o dsclose her consumpon/generaon nformaon o oher devces. he nformaon beng exchanged s only he esmaons of global prmal/dual varables. Organzaon of he paper s as follows. Secon II formulaes he opmal schedulng of he sorage devces n he mcrogrd as a dscree opmal conrol problem. Secon III explans he proposed dsrbued schedulng algorhm. Secon IV

3 demonsraes he performance of he algorhm on he Fuure Renewable Elecrc Energy Delvery and Managemen (FREEDM) Sysem [1] as a realzaon of a fuure smar grd. Fnally secon V summarzes he paper and brngs he concludng remars. II. PROBLEM FORMULAION he mcrogrd of neres n hs paper consss of dsrbued renewable resources sorage devces and local loads. Schedulng of sorage devces n hs ype of mcrogrd can be formulaed as a dscree-me opmal conrol problem. o hs end we defne he conrollable/unconrollable npus saes objecve funcon and consrans. In he remander of he paper s assumed ha all he devces n he mcrogrd are assgned wh a unque ndex and he se of ndces of all devces s denoed by I. A. Conrollable Inpus he vecor of conrollable npus s denoed by u() whch consss of: 1) Power commands o sorage devces: he power command o he sorage devce wh ndex a me sep s denoed by PB() where and B s he se of ndces of sorage uns. A posve value denoes njecng power no he mcrogrd. ) Power command o he grd nerface: he power o be drawn from he grd a me sep s denoed by Pgrd () where s he ndex assgned o he grd nerface and he se conanng hs ndex s denoed by grd. B. Unconrollable Inpus he vecor of unconrollable npus s denoed by w() whch consss of: 1) Renewable generaon: he power produced by he renewable generaon un wh ndex a me sep s denoed by PR() where and R s he se of ndces of renewable uns. ) Demands: he power consumed by he load un wh ndex a me sep s denoed by PD() where and D s he se of ndces of demand uns. 3) Energy prce: he prce of energy a he ouer grd a me sep s denoed by p(). C. Sysem Saes Mcrogrd saes are he values of sored energy n sorage devces. he energy sored a energy sorage devce wh ndex a me s denoed by x() whch has he followng dynamcs: B : x ( 1) x P B (1) (1) where Δ s he me nerval beween wo me seps (for nsance 1 hour). he vecor of saes s denoed by = [ : ]. D. Objecve Funcon he objecve s o mnmze he operaonal cos of he mcrogrd over a specfed me horzon 1 mn J C x u w () u( ) where ( ) s he operaonal cos of he mcrogrd a me-sep. Here <γ 1 s a dscoun facor o pu less wegh on fuure values of he coss. In hs paper we are no consderng any hermal plans nsalled n he mcrogrd so he only operaonal cos s due o he energy bough from he uly. herefore C x u w p P. (3) grd E. Consrans hree classes of consrans should be sasfed: 1) Power Balance Consran: A all mes he amoun of generaon should equal o he amoun of load: 1 : P B P R P grd P D. () B R D Noe ha n hs paper we do no consder losses. ) Energy Consran for Sorage Devces: A all mes he energy sored n he sorage devces canno be more han her capacy or negave: 1 B : x E full () where Efull s he oal amoun of energy ha can be sored n he sorage devce wh ndex. Combnng (1) and (): s 1 s B : x E full P B x () where x s he nal value of he sored energy a he sorage devce wh ndex. 3) Power Rang Consrans: A all mes he power commands sen o he sorage devces or o he grd nerface should be whn he maxmum and mnmum lms: P B mn P B P B max 1 : (7) Pgrd mn P grd Pgrd max where PB mn PB max are he mnmum and maxmum power lms of he sorage devce wh ndex and Pgrd mn Pgrdmax are he mnmum and maxmum power ha can be drawn from he grd. 1 III. DISRIBUED PREDICIVE MICROGRID SCHEDULING ALGORIHM In hs secon he seps o derve he algorhm o solve he formulaed opmal conrol problem n a dsrbued approach are explaned. A. Lagrangan of he problem he augmened Lagrangan of he problem by addng KK mulplers consrans and penaly erms o he objecve funcon would be: 1 J p P grd 1 P D P B P R P grd 1 D B R x E full P B 1 B s1 P x B 1 B s1 ()

4 3 D B R grd P P P P 1 D B R x E full P B 1 B s1 P B x 1 B s1 where λ() μ() and ξ() are he KK mulplers for he consrans and [ ] projecs s argumen o he posve values. he las hree erms are penaly erms for he consrans and mprove he convergence properes of he erave algorhms [17] and ρ s he penaly facor. B. Updae Equaons based on Graden Descen/Ascen he soluon o he opmal conrol problem can be found n an erave procedure by movng he prmal varables (conrollable npus) n he oppose drecon and dual varables (KK mulplers) n he drecon of he graden of he Lagrangan [1]. herefore he updae equaons would be: {1... }: 1 1 P P p P (9) grd grd {1... } B : P ( l) ( l) l l 1 l (1) P B P B x E full P B (s) l s1 l P B (s) x l s1 {1... }: 1 P (11) {1... } B : j 1 (1) x E full P B s1 {1... } B : j 1 (13) P B x s1 where {1... }: P P P P P. (1) D B R grd D B R By choosng small enough posve values for and he updae equaons (9)-(13) would converge o he saddle pon of he Lagrangan whch s he opmal pon of he problem [1]. However usng equaons (9)-(11) requres each node havng access o ceran global nformaon of he mcrogrd whch are he forecased load of all he demand uns forecased generaon of all he renewable generaon uns and conrollable npus decded for all he dspachable nodes. C. Consensus Newors o Esmae Global Informaon o mae he algorhm presened by (9)-(13) fully dsrbued nsead of usng global nformaon we use local esmaons of global nformaon a each node and allow he nodes o coordnae her esmaons wh her neghbors hrough consensus newors [19]. hus equaons (9) and (1) become: {1... }: 1 1 ˆ ˆ (1) P P p P grd grd {1... } B : ˆ ( ) ˆ P l ( l ) l l 1 l P B P B (1) x E full P B l s1 l P B x l s1 where Δ and are he esmaons of node a eraon from he average of he power mbalance n he newor Δ and he dual varable respecvely. he esmaon varables are updaed n all he nodes as follows: {1... } B D R grd j { B D R grd}: ˆ 1 ˆ ˆ ˆ ˆ j j jn w P Pˆ Pˆ w Pˆ Pˆ P P 1 1 j j j j jn (17) where N s he se of neghbors of node n he communcaons newor and wj = wj s he connecvy srengh beween node and node j and s chosen such ha < < max.. o sasfy he convergence crera for he consensus newor [19]. Devces such as ransformers/converers n he mcrogrd do no acvely generae/consume power. However from he communcaons pon of vew hey mgh parcpae n he nformaon exchange process. herefore for all hose devces he esmaon varables are updaed as: {1... } I \ B R D grd : ˆ 1 ˆ ˆ ˆ ˆ j j jn w P P P w P P. ˆ 1 ˆ ˆ ˆ j j jn IV. NUMERICAL SIMULAIONS (1) o valdae he proposed approach we consder applcaon of he algorhm on a mcrogrd confguraon shown n Fg 1. he fgure shows a ypcal confguraon of he FREEDM Sysem whch s a realzaon of he smar mcrogrds [1]. I consss of hree Dsrbued Energy Sorage Devces (DESDs) a wnd urbne a PV panel wo load nodes as well as Sold Sae ransformers (SSs) whch are power elecroncs converers. On op of he physcal sysem here s he communcaons newor hrough whch dfferen devces can exchange nformaon. he nodes are shown wh crcles and he communcaon lns are shown wh doed lnes. he exsence

5 Fg 1. ypcal confguraon of he FREEDM sysem of a communcaon ln beween wo nodes denoes ha hey are neghbors. All he smulaons are performed n MALAB 13a on a PC wh. GHz processor and GB RAM. he local nformaon regardng power generaon and consumpon of each devce s only accessble o he conroller conneced o ha devce. For nsance only he dsrbued conroller locaed a node 1 has access o he load forecas nformaon of he load conneced o node 1. Smlarly only node has access o he generaon forecas daa of he PV panel conneced o ha node. he objecve s ha he devces collaboravely coordnae wh each oher and fnd he opmal schedule for he nex hours. As he case sudy we have used onlne daa for prce profle 1 and scaled profles for PV wnd and loads 3 as shown n able 1. he nformaon regardng he specfcaons of DESDs and her nal sored energy s shown n able. he proposed algorhm s appled. he value of s chosen as.3 he dscoun facor s chosen as 1 and all he connecvy srenghs are chosen as wj =.17 o sasfy < < max = 1/... ABLE 1. PROFILE USED FOR HE CASE SUDY Prce Wnd Solar Load1 Load Hour (cens/wh) (W) (W) (W) (W) 1: : : : : ABLE. DESD SPECIFICAIONS AND INIIAL CONDIIONS Cap (Wh) PBmax (W) x (Wh) DESD 1 1 DESD 1 1. DESD 3. Fg shows he evoluon of he esmaons of he dual varables. Fg (a) shows he evoluon of dual varable esmaons for dfferen me seps a node 1. Fg (b) - Fg (f) show he evoluon of dual varable esmaons for each me sep by all oher nodes. I can be seen ha he esmaons of dfferen nodes of he sysem for he dual varable of each me sep converge o he same value. However he dual varables of dfferen me seps (as shown n Fg (a)) do no have o be equal as hey correspond o dfferen consrans governng each 1 a node Ieraon (a) (c) me Sep1 me Sep me Sep3 me Sep me Sep () 1 1 Node Node Node Node Ieraon 1 1 () Node Node Node Node Ieraon (1) 1 Node 1 Node Node Node eraon (e) (f) Fg. Evoluon of esmaons of dual varable λ me sep. he resulan schedule for he DESDs along wh he load wnd and PV generaon profles are shown n Fg 3. I can be seen ha durng he me seps when he prce s low (me seps and 3) and he renewable generaon s relavely hgh he DESDs are chargng and when he prce goes hgh and renewable generaon decreases he DESDs dscharge o suppor he demand and avod hgh operaonal coss. o demonsrae he opmaly of he resulan schedule we found he global opmum usng cenralzed Lnear Programmng (LP). able 3 shows he objecve value and convergence me of he proposed algorhm benchmared agans LP. As he proposed algorhm s an erave process he convergence me depends on he soppng crera. he summaon of L norms of all consran volaons as well as he relave change n he decson varables s defned as he Convergence Index (CI). he algorhm s sopped once CI < ԑ. able 3 shows he convergence me and he resulan objecve value for dfferen values of ԑ. We can see ha he smaller he value of ԑ s he closer he soluon s o he global opmum bu aes longer me. For hs small scale problem as he cenralzed approach has access o he enre nformaon can fnd he soluon faser han he dsrbued algorhm n whch each node has only paral nformaon. o see how he scale of he problem affecs he compuaonal complexy of he algorhm we ncreased he number of devces from 1 o 1 and for each case performed 3 random scenaros. Fg shows he medan of he eraons requred for convergence (CI < (b) (3) 1 1 Node Node Node Node Ieraon (d) () 1 1 Node Node Node Node Ieraon 1 hp:// hp:// 3 hp://pjm.com/mares-and-operaons/energy/real-me/hourly-prelmloads.aspx

6 1 Energy Prce a Grd (cens/wh) 3 PV generaon (W) assumpons such as sorage devce effcency degradaon losses ec. would be he fuure wor of he auhors me Seps Wnd Generaon (W) 1 3 me Seps Grd (W) 1 3 me Seps DESD (W) 1 3 me Seps Fg 3. Forecased profles and resulan schedule usng he proposed algorhm Fg. Medan of convergence eraons vs. number of devces for ԑ =.1 ABLE 3. BENCHMARK AGAINS CENRALIZED ALGORIHM Algorhm Objecve Value (cens) Convergence me (sec) LP Proposed Algorhm (ԑ=1e-1 ) Proposed Algorhm (ԑ=1e- ) Proposed Algorhm (ԑ=1e-3 ) Proposed Algorhm (ԑ=1e- ) ) as a funcon of number of devces whch ndcaes a lnear ncrease n he compuaonal complexy. V. CONCLUSION 1 3 me Seps hs paper nroduced a dsrbued predcve algorhm for opmal schedulng of mcrogrds conssng of renewable generaons and sorage devces. Consderng more realsc Demand (W) 1 3 me Seps DESD1 (W) 1 3 me Seps DESD3 (W) 1 3 me Seps REFERENCES [1] K. C. Dvya and J. Osergaard Baery energy sorage echnology for power sysems An overvew Elecrc Power Sysems Research vol. 79 no. pp. 11 Apr. 9. [] J. M. Guerrero J. C. Vasquez J. Maas L. G. De Vcuña and M. Caslla Herarchcal Conrol of Droop-Conrolled AC and DC Mcrogrds A General Approach oward Sandardzaon IEEE ransacons on Indusral Elecroncs vol. no. 1 pp [3] H. Kanchev D. Lu F. Colas V. Lazarov and B. Francos Energy Managemen and Operaonal Plannng of a Mcrogrd Wh a PV-Based Acve Generaor for Smar Grd Applcaons IEEE ransacons on Indusral Elecroncs vol. no. 1 pp [] J. Soares M. Slva. Sousa Z. Vale and H. Moras Dsrbued energy resource shor-erm schedulng usng Sgnaled Parcle Swarm Opmzaon Energy vol. no. 1 pp. 7 Jun. 1. [] H. Moras P. Kádár P. Fara Z. a. Vale and H. M. Khodr Opmal schedulng of a renewable mcro-grd n an solaed load area usng mxedneger lnear programmng Renewable Energy vol. 3 no. 1 pp Jan. 1. [] C. Chen S. Duan. Ca B. Lu and G. Hu Opmal Allocaon and Economc Analyss of Energy Sorage Sysem n Mcrogrds IEEE ransacons on Power Elecroncs vol. no. 1 pp Oc. 11. [7] N. Rahbar-Asr U. Ojha Z. Zhang and M.-Y. Chow Incremenal Welfare Consensus Algorhm for Cooperave Dsrbued Generaon/Demand Response n Smar Grd IEEE ransacons on Smar Grd vol. no. pp [] Y. Xu and W. Lu Sable Mul-Agen-Based Load Sheddng Algorhm for Power Sysems IEEE ransacons on Power Sysems vol. no. pp [9] S. Kar and G. Hug Dsrbued Robus Economc Dspach n Power Sysems : A Consensus + Innovaons Approach n Power and Energy Socey General Meeng 1 pp. 1. [1] S. Yang S. an and J.-X. Xu Consensus Based Approach for Economc Dspach Problem n a Smar Grd IEEE ransacons on Power Sysems vol. no. pp. 1 Nov. 13. [11] G. Bne A. Davoud D. Naso B. urchano and F. L. Lews A Dsrbued Aucon-Based Algorhm for he Nonconvex Economc Dspach Problem IEEE ransacons on Indusral Elecroncs vol. 1 no. pp [1] N. Rahbar-Asr and M.-Y. Chow Cooperave Dsrbued Demand Managemen for Communy Chargng of PHEV/PEVs Based on KK Condons and Consensus Newors IEEE ransacons on Indusral Informacs vol. 1 no. 3 pp [13] A. J.del Real A. Arce and C. Bordons Combned envronmenal and economc dspach of smar grds usng dsrbued model predcve conrol Inernaonal Journal of Elecrcal Power & Energy Sysems vol. pp. 7 Jan. 1. [1] R. Deng Z. Yang J. Chen N. Rahbar-Asr and M.-Y. Chow Resdenal Energy Consumpon Schedulng: A Coupled-Consran Game Approach IEEE ransacons on Smar Grd vol. no. 3 pp [1] A. J. del Real A. Arce and C. Bordons An Inegraed Framewor for Dsrbued Model Predcve Conrol of Large-Scale Power Newors IEEE ransacons on Indusral Informacs vol. 1 no. 1 pp Feb. 1. [1] A. Q. Huang M. L. Crow G.. Heyd J. P. Zheng and S. J. Dale he Fuure Renewable Elecrc Energy Delvery and Managemen (FREEDM) Sysem: he Energy Inerne Proceedngs of he IEEE vol. 99 no. 1 pp [17] S. Boyd P. Neal E. Chu B. Paleao and J. Ecsen Dsrbued Opmzaon and Sascal Learnng va he Alernang Drecon Mehod of Mulplers Foundaons and rends n Machne Learnng vol. 3 no. 1 pp [1] D. Fejer and F. Pagann Sably of prmal dual graden dynamcs and applcaons o newor opmzaon Auomaca vol. no. 1 pp [19] R. Olfa-Saber and J. Fax Consensus and cooperaon n newored mul-agen sysems Proceedngs of he IEEE vol. 9 no. 1 pp