Predctve Contro of a Smart Grd: A Dstrbuted Optmzaton Agorthm wth Centrazed Performance Propertes* Phpp Braun, Lars Grüne, Chrstopher M. Keett 2, Steven R. Weer 2, and Kar Worthmann 3 Abstract The authors recenty proposed severa mode predctve contro MPC) approaches to managng resdenta eve energy generaton and storage, ncudng centrazed, dstrbuted, and decentrazed schemes. As expected, the dstrbuted and decentrazed schemes resut n a oss of performance but are scaabe and more fexbe wth regards to network topoogy. n ths paper we present a dstrbuted optmzaton approach whch asymptotcay recovers the performance of the centrazed optmzaton probem performed n MPC at each tme step. Smuatons usng data from an Austraan eectrcty dstrbuton company, Ausgrd, are provded showng the beneft of a varabe step sze n the agorthm and the mpact of an ncreasng number of partcpatng resdenta energy systems. Furthermore, when used n a recedng horzon scheme, smuatons ndcate that teratng the teratve dstrbuted optmzaton agorthm before convergence does not resut n a sgnfcant oss of performance.. NTRODUCTON Wth the proferaton of resdenta rooftop soar photovotacs and the ncreasng avaabty of cost-effectve resdenta-scae energy storage soutons e.g., batteres or fue ces), there s a need to coordnate the storage charge/dscharge schedues so as to avod arge demand peaks or troughs. n [4], [3], the authors proposed three dfferent mode predctve contro MPC) schemes to smooth the energy demand of a coecton of resdences. These MPC schemes nvoved a centrazed approach, requrng fu communcaton of a reevant system varabes, a dstrbuted approach, requrng mted communcaton of reevant system varabes, and a decentrazed approach, requrng no communcaton of system varabes. Whe a three approaches succeeded n smoothng the aggregate energy demand, unsurprsngy the centrazed approach acheved better performance when compared to the dstrbuted and decentrazed approaches, but suffered from an nabty to scae to a arge number of resdenta systems. n ths paper, we present a dstrbuted optmzaton agorthm wth the goa of recoverng the performance of the centrazed MPC scheme whst remanng scaabe. n other *C.M. Keett s supported by ARC Future Feowshp FT746. L. Grüne s supported by the Deutsche Forschungsgemenschaft, Grand GR 569/3-. P. Braun and L. Grüne are wth the Mathematca nsttute, Unverstät Bayreuth, 9544 Bayreuth, Germany, e-ma: {phpp.braun, ars.gruene}@un-bayreuth.de. 2 C. M. Keett and S. R. Weer are wth the Schoo of Eectrca Engneerng and Computer Scence at the Unversty of Newcaste, Caaghan, New South Waes 238, Austraa, e-ma: {chrs.keett, steven.weer}@newcaste.edu.au. 3 K. Worthmann s wth the nsttute for Mathematcs, Technsche Unverstät menau, 99693 menau, Germany, e-ma: kar.worthmann@tumenau.de. words, we focus on the souton of a snge, fnte tme horzon, optmzaton probem mpemented n a dstrbuted fashon. At east n the contro terature, the fed of dstrbuted optmzaton traces ts roots to the thess of Tstsks [] see aso [2]). Much of the recent work n ths fed has nvoved mut-agent systems tryng to optmze a goba objectve functon under dfferent condtons; see for exampe [5], [7], [8], [9], [5] and the references theren. A common feature n many of these references s the assumpton that the goba cost functon can be decomposed as a sum of the cost functons for each ndvdua agent. However, the cost functon naturay used to sove the probem of smoothng the energy demand s not decomposabe n ths way. n [4], a cosey reated probem s soved where an eectrcty retaer ams to mze the cost due to dscrepances between the power the retaer bds to use and what ts customers actuay use. Agan, ths gves rse to a dfferent cost functon to that whch we propose. The paper s organzed as foows. n Secton we ntroduce the mathematca mode of the Resdenta Energy System RES) and defne the desred performance metrcs. The centrazed MPC approach s presented n Secton and our proposed dstrbuted computaton agorthm s descrbed n Secton V-A. A bref comparson wth prma/dua decomposton s provded n Secton V-B. A smuaton study usng data from an Austraan eectrcty dstrbuton company, Ausgrd, s undertaken n Secton V. n partcuar, we demonstrate the beneft of a varyng step-sze n the dstrbuted optmzaton agorthm Secton V-A), we exae the mpact of ncreasng the number of systems Secton V-B), and the effect of eary teraton of the dstrbuted optmzaton agorthm s ustrated Secton V- C). Concudng remarks are provded n Secton V.. THE RESDENTAL ENERGY SYSTEM Let N be the number of RESs connected n the oca area under consderaton. We summarze a smpe mode of RES, {,..., }, presented n [3] x k + ) x k) + T u k), ) z k) w k) + u k) where x s the state of charge of the battery n [kwh], u s the battery charge/dscharge rate n [kw], w s the statc oad us the oca generaton n [kw], and z s the power supped by/to the grd n [kw]. Here, T represents the ength of the sampng nterva n [h] hours); e.g., T.5[h] corresponds to 3 utes. Whe the system dynamcs )
s autonomous, the performance output 2) depends on the tme varyng quantty w ). The RES network s then defned by the foowng dscrete-tme system xk + ) fxk), uk)), zk) huk), wk)) where x, u, w, z R, and the defntons of f and h are gven componentwse by ) and 2), respectvey. For each RES {,..., }, the constrants on the battery capacty and charge/dscharge rates are descrbed by the constants C, u R > and u R <,.e., x k) C and u u k) u k N. 3) Our goa s to fatten the performance output z. We ntroduce two reevant performance metrcs. To ths end, et Πk) : z k) denote the average power demand at tme k and et N denote the number of sampes comprsng a smuaton ength. The performance metrc of peak-to-peak PTP) varaton of the average demand of a RESs s gven by ) ) max Πk) k {,...,N } Πk) k {,...,N }. PTP) The second performance metrc of the root-mean-square RMS) devaton from the average s defned as N N k Πk) Υ wth the average demand Υ : N N k w k).. MODEL PREDCTVE CONTROL APPROACHES RMS) We reca a mode predctve contro MPC) agorthm for the contro of a network of RESs ntroduced n [3] and [4], respectvey. Ths approach s a centrazed MPC CMPC) scheme, n whch fu communcaton of a reevant varabes for the entre network as we as a known mode of the network are requred. n Secton V-A we present a dstrbuted optmzaton agorthm whch s based on oca optmzaton probems, keepng the fexbty of the network topoogy, whe mantanng optmaty wth respect to the CMPC approach. A correspondng proof of convergence s gven n the Appendx. MPC teratvey mzes an optmzaton crteron wth respect to predcted trajectores and mpements the frst part of the resutng optma contro sequence unt the next optmzaton s performed see, e.g., [] or [6]). To ths end, we assume that we have predctons of the resdenta oad and generaton some tme nto the future that s concdent wth the horzon of the predctve controer. n other words, gven a predcton horzon N N, we assume knowedge of w j) for j {k,..., k + N }, where k N s the current tme. A. Centrazed Mode Predctve Contro CMPC) To mpement the CMPC agorthm, we compute the network-wde average demand at every tme step k over the predcton horzon by ζk) : N k+ jk w j) 4) and then mze the jont cost functon k+ V xk); k) : ζk) w j) + û j)) û ) } {{ } jk ẑ j) 5) wth respect to the predcted contro nputs û ) û ), û 2 ),..., û )) T wth û ) û j)) k+ jk, {, 2,..., }, subject to the system dynamcs ), the current state xk) x k),..., x k)) T, and the constrants 3) for {,..., }. The vector of the predcted performance output ẑ ) s defned n the same way as the predcted contro û ). To smpfy the notaton, the current tme k s dropped when t does not dever extra nformaton. Addtonay we use the notaton uj) u j),..., u j)) T for a fxed tme j N. The same hods for the other varabes x, w and z. n Fgure the aggregated energy profe and the aggregated battery profe for a smuaton of one week N 336, T.5[h]) for RESs, nta condtons x ).5[kWh], constrants u u.3[kw] and battery capacty C 2[kWh] for a {,..., } are vsuazed. The oad and generaton data for ths smuaton was coected by an Austraan eectrcty dstrbuton company, Ausgrd, as part of ther Smart Grd, Smart Cty project. The fgures compare the uncontroed system dynamcs wth the cosed oop dynamcs of CMPC. z n [KW].8.6.4.2 Uncontroed CMPC 24 48 72 96 2 44 68 Tme n hours x n [KWh] 2.5.5 24 48 72 96 2 44 68 Tme n hours Fg.. Performance of CMPC for a smuaton ength of one week and RES. The eft fgure shows the average power demand whe the rght fgure shows the average state of charge of the batteres. V. CENTRALZED MPC WTH DSTRBUTED COMPUTATON n ths secton, we propose a herarchca dstrbuted mode predctve contro DMPC) approach where each RES can communcate wth a centra entty to acheve the performance of the CMPC agorthm,.e., a network-wde objectve whe keepng fexbty. The optma vaue returned by the dstrbuted optmzaton agorthm concdes wth the optma vaue of the mzaton probem 5) cf. the Appendx for a proof).
A. The Dstrbuted Optmzaton Agorthm The dstrbuted optmzaton agorthm s based on the cost functon 5) ntroduced n the centrazed settng. nstead of sovng one mzaton probem, severa teratons are performed at every tme step k n whch every RES mzes ony over ts own contro varabes. The centra entty communcates the aggregated performance output between the systems and computes an approprate step sze θ n every teraton. At tme step k, the agorthm s ntazed wth ζ : ζk) cf. Equaton 4)), w j) : w k + j), j,..., N, {, 2,..., }, and x) : xk). Agorthm Dstrbuted Optmzaton Agorthm nput: RES, {, 2,..., }: nta state of charge x ), predcton horzon N, energy profe w j)), and ζ. Centra Entty: Number of RESs, N, ζ, maxma teraton number max N { }, desred precson ε. ntazaton: RES, {, 2,..., }: defne and transmt ẑ j)) and ẑ j)). Centra Entty: Set the teraton counter and V, receve ẑ j)), {.2...., }. Phase Centra Entty): ncrement the teraton counter. Then, receve ẑ j)),, 2,...,. Compute the step sze θ as arg θ [,] ζ θẑ j) + θ)ẑ j) 6) Compute ẑ + j) : θ ẑ j) + θ )ẑ j) and the predcted average demand Π j) : ẑ+ j) for j {,,..., N }. Then, evauate the performance ndex V + : ζ Π j). 7) f V + V < ε or max hods, terate the agorthm. Otherwse transmt θ and Π j)) Phase 2 RES, {, 2,..., }): Receve θ and Π j)) For j,,..., N compute ẑ + j) : θ ẑ j) + θ )ẑ j) 8) Sove the oca) mzaton probem û ) ζ Π j) + ẑ+ j) w j) + û j) subject to the system dynamcs ), ˆx ) x ), and the constrants 3) to obtan the unque mzer ẑ + j)) : w j) + û + j)). Transmt ẑ + j)). Note that Π ) ony depends on ẑ + ). The ndex s chosen n such a way that n teraton, the predcted average Π ) has to be transmtted. A feasbe ntazaton of RES, {,..., } s for exampe gven by ẑ j) ẑ j) : w j), whch corresponds to the choce u ) and can be repaced by any other admssbe ntazaton. Agorthm s terated ether after a fxed number of teratons max or f the stoppng crtera V + V < ε s satsfed. The communcaton structure of Agorthm s vsuazed n Fgure 2. teraton `, Phase teraton `, Phase 2 CE Compute ` Update ẑ`+ Compute ` ẑ?` ẑ?` `, ` ẑ?`+ RES Update ẑ`+ Compute ẑ?`+ `, ` Fg. 2. Communcaton structure of Agorthm. The nput uk) s defned by the update rue of Equaton 8),.e., as a convex combnaton of the ast two computed nputs. Snce the constrants defne a convex set t s ensured that ẑ + ) corresponds to an admssbe nput sequence n every teraton. Theorem.4 n the Appendx ensures that the vaue V converges to the unque optma vaue f the teraton ndex tends to nfnty. Moreover, note that sovng the mzaton probem 6) s equvaent to a smpe functon evauaton as proven n the foowng proposton. Proposton 4.: f ẑ j)) ẑ j)), the parameter θ n teraton s gven by the projecton of θ : ζ ẑ j) )) ẑ j) ẑ j)) ẑ j) ẑ j))) to the nterva [, ],.e., θ max{, { θ, }}. Proof: n order to show the asserton, we defne the functon F θ) : ζ ζ θẑ j) + θ)ẑ j) ẑ j) θ ẑ j) ẑ j) ) Snce F s strcty convex, the asserton foows by sovng F θ) and projectng the souton on the nterva [, ]. Hence, showng that θ soves F ˆθ) competes the proof.
Ths foows by computng 2 /2 F θ): [ ) ] ζ ẑ j) θ j) j) ζ ẑ j) )) j) θ j) wth j) ẑ j) ẑ j). Remark 4.2: Aternatvey to the varabe step sze θ computed n Equaton 6), the fxed step sze θ / eads to a decrease of the optma vaue V n every teraton and convergence to the optma vaue of the CMPC mzaton probem whch s an mmedate consequence of the proof of Lemma.2. n Secton V the mpact of a fxed and a varabe step sze s ustrated by smuatons. n every teraton, the centra entty communcates N vaues the average consumpton at each tme wthn the predcton wndow) and the parameter θ to a RESs. n the reverse drecton, each RES transmts N vaues n each teraton. Hence, the amount of data transmtted by the centra entty s ndependent of the number of systems and the nformaton can be broadcast. Snce the optmzaton probems are soved by the RESs ndvduay, the compexty of the agorthm does not grow wth the number of systems. The centra entty does not make use of the constrants 3). Changng system dynamcs, constrants or addng/removng snge systems can be acheved easy on a oca eve, makng the agorthm ncey scaabe n contrast to CMPC. B. Comparson to prma and dua decomposton n ths secton we compare Agorthm wth prma and dua decomposton agorthms descrbed n [3]. Decomposton approaches descrbe methods to break a snge optmzaton probem nto severa optmzaton probems whch are easer to sove. Prma decomposton refers to the decomposton of the orgna probem whe dua decomposton manpuates the dua formuaton. Consder the mzaton probem v,y s.t. fv, y) v, y) P gven n [3]. Here f denotes a convex functon and P a poyhedron of sutabe dmenson. Assume that the functon f and the poyhedron P can be spt such that the mzaton probem 9) can be equvaenty wrtten as v,y f v, y) 9) s.t. v P,..., y P y ) wth convex functons f and poyhedra P y and P for {,..., }. Hence the objectve functon s decouped wth respect to the varabes v, and for a fxed vaue y P y, one can sove the mzaton probems v f v, y) s.t. v P ) separatey. Ths technque of rewrtng 9) as severa probems of the form ) s caed prma decomposton. To sove the probem n a dstrbuted way, ) s soved for a {,..., } and a fxed vaue y P y. Afterwards, the optmzaton varabe y s updated and the process s repeated unt an optma souton s found. n our case, the mzaton probem 5) can be wrtten as v,...,v f v,..., v ) s.t. v P,..., where v u and fv,..., v ) ξj) v j) wth constant vaues ξj). Observe that due to the square, the functon f s not separabe wth respect to the varabes v,..., v. Addtonay, an anaog of the varabe y does not exst n our settng. Nevertheess, t s possbe to fnd smartes between prma decomposton and Agorthm. We defne the vaues y j) ξj) ṽ j) j; j for gven vaues ṽ j). Then we can defne the functons fv, y ) y j) v j) and the correspondng mzaton probems v f v, y ) s.t. v P whch are separated for constant vaues y or constant vaues ṽ, respectvey. Hence, the mzaton probems can be soved n a dstrbuted manner by teratvey updatng ṽ. One way of updatng ṽ s gven by Agorthm. n contrast to prma decomposton, however, we pont out that n our case y s not an optmzaton varabe and we need an ndvdua y for every f. n dua decomposton, the mzaton probem ) s wrtten n the form v,y f v, y ) s.t. v P,..., y P y,..., y y j, j,...,. 2) nstead of fxng the parameter y, y s used as an addtona optmzaton varabe. The optmzaton probem 2) can be separated by ookng at the Lagrangan and fxng the Lagrange varabes. n dua decomposton, the mzaton probems are soved for the unknowns x, y ) and fxed Lagrange varabes for the next teraton, the Lagrange varabes are updated unt a souton s found. As emphaszed above, the varabe y does not exst n our objectve functon and hence, dua decomposton s not appcabe n our context.
V. A NUMERCAL CASE STUDY A numerca case study s presented n order to show the beneft of DMPC compared to CMPC. Ths case study s based on anonymzed oad and generaton profes of resdenta customers provded by an Austraan eectrcty dstrbuton company, Ausgrd, based n New South Waes. The numerca experments are conducted usng the nteror pont sover POPT [2] and the HSL mathematca software brary [] to sove the underyng mzaton probems and near systems of equatons, respectvey. For a numerca experments we fx the nta vaues x ).5[kWh] and the constrants C 2[kWh], u u.3[kw] for a {,..., }. A. Choce of the Step Sze θ n ths subsecton we nvestgate the roe of the step ength θ. To ths end, 2 RESs are smuated for a duraton of 3 days N 44, T.5[h]). n Fgure 3 we vsuaze the number of teratons unt a certan accuracy V k) V k), {, 2,..., 5}, s reached. a) Varabe θ b) Fxed θ / Number of teratons 2 5 5 24 48 72 96 2 44 Tme ndex k Number of teratons 2 5 5 24 48 72 96 2 44 Tme ndex k Fg. 3. Number of teratons to obtan a certan accuracy ε for,..., 5 at tme step k,.e., V k) V k) ε where V denotes the souton of the centrazed MPC agorthm. constant ne after approxmatey 2 teratons s due to the optmzaton accuracy of POPT. N N k V k) V k) 2 2 4 6 8 5 5 2 25 3 teraton Fg. 4. Average speed of convergence of the dstrbuted optmzaton agorthm wth fxed θ / back) and varabe θ bue). B. mpact of the Number of Systems Next, we anayze the dependence of the average) number of teratons on the number of RESs. To ths end, the number of RESs,, s vared wthn the set {, 2,..., 3}. Then, the number of teratons s counted unt the accuracy V V 2 s obtaned both for varabe and fxed step sze θ. n Fgure 5, we observe a near growth n the number of teratons for fxed θ whe ths number s sgnfcanty smaer and seems to grow subneary n the case of varabe θ. n concuson, the number of teratons stays moderate for varabe θ whe t may become too arge for θ / to make the agorthm appcabe for a very) arge number of RESs. f a fxed step sze θ / s used nstead of a varabe θ accordng to Proposton 4. the requred number of teratons s, on average, twce as arge, see Tabe. Accuracy average no. maxmum no. mum no. θ / varabe / varabe / varabe ε 8.6 3.8 2 6 6 3 ε 2 23.9 5.5 42 24 6 ε 3 59.33 33.4 86 46 3 ε 4 99.85 5.44 3 67 4 6 ε 5 42.69 65.89 76 89 6 9 TABLE Average, mum, and maxmum number of teratons to acheve a certan accuracy for varabe and fxed θ. Average number of teratons 45 4 35 3 25 2 5 5 5 5 2 25 3 Number of RES n Fgure 4 the average devaton n teraton from the N benchmark CMPC souton,.e., N k V k) V k), s vsuazed. The average s taken wth respect to each sampng nstant k wth smuaton ength N 44. Hence, the convergence speed of the dstrbuted optmzaton agorthm wth step sze θ n accordance wth Proposton 4. ceary outperforms ts counterpart usng constant θ /. The Fg. 5. Average number of teratons needed to ensure the accuracy V V 2 n dependence of the number of RES wth fxed θ / red) and wth varabe θ bue). The dashed nes show the maxma and ma number of teratons. C. mperfect Optmzaton Agorthm needs about 42 teratons on average to obtan an accuracy of 2 n the settng of RESs and
varabe θ, cf. Fgure 5. However, n practce, t may be necessary to terate the agorthm after a fxed number of teratons; e.g., due to a fxed aowabe computaton tme. We exae two ssues. The frst s merey the performance of Agorthm wth a fxed number of teratons. The second s the cosed oop performance of Agorthm wth a fxed number of teratons when used n a recedng horzon fashon. We frst compute the devaton V k) V k) at each tme nstant k wthn the smuaton wndow and, then, we anayze the MPC cosed oop performance. f the step sze θ s chosen such that 6) s soved n each teraton the tota devaton s st arge after teratons, but the cosed oop performance aready ooks convncng, see Fgure 6. V corresponds to a arge sma) devaton from the average ζ. Therefore, we use the absoute error nstead of the reatve error V V ɛ V V ɛ V as a quatatve measure of the resuts. f V s sma the performance wth respect to our metrcs s good even f the reatve error mght st be arge. The choce ε 2 for most of the numerca smuatons seems to be reasonabe for our appcaton, but can be repaced by any other vaue. V k) V k).2.8.6.4.2 3 teratons 5 teratons teratons 48 96 44 92 24 288 336 Tme ndex k z n [KW].8.7.6.5.4.3 CMPC DMPC.2 24 48 72 96 2 44 68 Tme n hours Fg. 6. Devaton and MPC cosed oop evouton for RESs usng varabe θ and ncompete optmzaton teratons). On the contrary, the cosed oop performance s not satsfactory for fxed θ as seen n Fgure 7. V k) V k).2.8.6.4.2 48 96 44 92 24 288 336 Tme ndex k z n [KW].8.7.6.5.4.3.2 24 48 72 96 2 44 68 Tme n hours Fg. 7. Devaton and MPC cosed oop evouton for RESs usng fxed θ and ncompete optmzaton teratons). The same concusons can be drawn for even smaer teraton numbers see Tabe ). Number of teratons varabe θ θ / DMPC - CMPC PTP RMS PTP RMS 3..22.294.33 5..2.2592.942..4.228.759 TABLE Devaton of Dstrbuted MPC wth ncompete optmzaton and CMPC for RES n dependence of the step sze θ. Remark 5.: For the consdered data set n ths secton,.e., the 44 sampes and a varabe number of RESs, the vaues of V are n the nterva [.54,.85]. A arge sma) V. CONCLUSON n ths paper we have presented a dstrbuted optmzaton agorthm for the appcaton to the probem of smoothng energy consumpton n a resdenta eectrcty network where resdences have sma scae generaton e.g., rooftop soar photovotac panes) and storage e.g., a battery). Ths teratve message-passng agorthm asymptotcay recovers the optma vaue of the centrazed optmzaton probem. Va a smuaton study, the dstrbuted optmzaton agorthm has been shown to scae we wth the number of systems and, when used n an MPC scheme, to retan good performance when the agorthm s terated after a fxed number of teratons. Furthermore, we have demonstrated the beneft of mpementng a varabe step sze. APPENDX n ths secton, we prove convergence of Agorthm to the optma vaue of 5),.e., we show that the mt V : m V correspondng to Agorthm concdes wth the optma vaue V of the mzaton probem ẑ ),...,ẑ ) s.t. ζ ẑj) x) ˆx) x j + ) x j) + T u j) z j) w j) + u j) u u j) u x j + ) C, j) {,..., } {,..., N } 3) whch has to be soved n every tme step of CMPC. To ths end, we defne the functons v ẑ ); ) : ζ Π j) + ẑ j) ẑ j) ) 4) and rewrte the oca mzaton probem from Phase 2 of
Agorthm ẑ ) vẑ ); ) s.t. x ) ˆx ) x j + ) x j) + T u j) z j) w j) + u j) u u j) u x j + ) C j {,..., N } 5) for {,..., }. The constrants of 5) defne a convex and compact set. The functon v s strcty convex and contnuous n ẑ ) and n the parameters ζ Π j)+ẑ j)/, j {,..., N }. Hence the optma vaue v ẑ ); ), where ẑ ) denotes the unque mzer of the oca mzaton probem, depends contnuousy on the parameters ζ Π j) + ẑ j)/, j {,..., N }. Snce we w use ths resut n the foowng we w state t n a Lemma. Lemma.: The optma vaue v ẑ ); ) of the oca mzaton probem 5) of RES {,..., } s contnuous wth respect to the parameters ζ Π j) + ẑ j)/, j {,..., N }. Before we can prove the convergence of the sequence V ) N we show the weaker resut of monotoncty. Lemma.2: The sequence V ) N generated by Agorthm s monotoncay decreasng,.e., V + V hods for a N. f, addtonay, ẑ ) ẑ ), then V + < V hods. Hence, the sequence V ) N s strcty monotoncay decreasng unt Agorthm stops. Proof: Snce θ [, ] s chosen such that F θ) attans ts mum, see Remark 4., repacng θ by yeds a arger vaue V + V. ζ Π j) ζ ẑ j) + ζ Π j) + θ θ ζ Π j) + ẑ j) ẑ ẑ j) ẑ ẑ j) ẑ ) j) ) j) ) ) j) ζ Π j) + ) ẑ j) ẑ j) } {{ } v ẑ ); ) v ẑ );) ζ Π j) The frst nequaty foows wth θ /. The second nequaty foows from the defnton of convex functons or Jensen s nequaty),.e., M ) M M f α x α fx ), α, α m m m apped to fx) x 2. The thrd nequaty s a drect consequence of the optmaty of ẑ ). Snce v ; ) s strcty convex we obtan v ẑ ); ) < v ẑ ); ) f there exsts an ndex, j) {, 2,..., } {,,..., } such that ẑ j) ẑ j) hods. The proof of Lemma.2 shows that / s a possbe, fxed, choce for θ n Agorthm. Hence, the convergence aso hods f the optma) step sze n Agorthm s repaced by the step sze /. Coroary.3: For the sequence V ) N R of Agorthm converges,.e., m V V R. Proof: Snce V and V ) N s monotoncay decreasng by Lemma.3, V ) N converges to ts nfmum V. n Lemma.2 and Coroary.3 we have shown that the sequence V ) N s convergng. What s eft to show, s the convergence aganst the vaue correspondng to the mzaton probem 3) whch w be done next. Theorem.4: The mt V of the sequence V ) N generated by Agorthm concdes wth the optma vaue V of the mzaton probem 3). Proof: Let z ) denote the souton of Probem 3). Snce the cost functon s contnuous and defned on a compact set, there exsts an admssbe) accumuaton pont z ) of the sequence ẑ )) satsfyng the equaty ζ 2 z j)) V. We frst assume that the mt ẑ ) s obtaned n fntey many teratons,.e., there exsts a j N such that ẑ j ) ẑ ). We defne the functon F : [, ] R as F θ) : ζ ζ θ )z j) + θ z j) ) z j) To show the asserton, we assume θ z j) z j)) F ) V < V F ). 6) Snce F ) s convex, ts drectona dervatve n wth respect to θ s ess than zero,.e., > grad F ), F θ ). 7) nequaty 6) mpes the exstence of an ndex {,..., } such that z ) z F ) and, thus, > θ ) hods. However, then the -th RES updates ẑ ), cf. 4) a contradcton to the assumpton that V s the mt of V ) N accordng to Lemma.2 snce the update ẑ j + ) j + eads to a better vaue V < V. f the accumuaton pont ẑ ) s not reached n fntey many steps then there exsts a subsequence j k ) k N such that m k ẑ j k ) ẑ ). Then, due to the contnuty of the optma vaue functon c.f. Lemma.) there exsts a k N such that
V j k + < V whch agan contradcts the propertes of V. REFERENCES [] HSL Mathematca Software Lbrary. A coecton of Fortran codes for arge-scae scentfc computaton, 24. http://hs.r.ac.uk/. [2] D. P. Bertsekas and J. N. Tstsks. Parae and Dstrbuted Computaton: Numerca Methods. Athena Scentfc, Bemont, MA, USA, 989. [3] S. Boyd, L. Xao, A. Mutapcc, and J. Mattngey. Notes on decomposton methods. Technca report, Stanford Unversty, 27. [4] T.-H. Chang, A. Nedć, and A. Scagone. Dstrbuted constraned optmzaton by consensus-based prma-dua perturbaton method. EEE Transactons on Automatc Contro, 596):524 538, 24. [5] J. C. Duch, A. Agarwa, and M. J. Wanwrght. Dua averagng for dstrbuted optmzaton: Convergence anayss and network scang. EEE Transactons on Automatc Contro, 573):592 66, 22. [6] L. Grüne and J. Pannek. Nonnear Mode Predctve Contro. Theory and Agorthms. Sprnger London, 2. [7] D. Jakovetcć, J. Xaver, and J. M. F. Moura. Fast dstrbuted gradent methods. EEE Transactons on Automatc Contro, 595):3 46, 24. [8] A. Nedć and A. Ozdagar. Dstrbuted subgradent methods for mut-agent optmzaton. EEE Transactons on Automatc Contro, 54):48 6, 29. [9] A. Nedć, A. Ozdagar, and P. A. Paro. Constraned consensus and optmzaton n mut-agent networks. EEE Transactons on Automatc Contro, 554):922 938, 2. [] J. B. Rawngs and D. Q. Mayne. Mode Predctve Contro: Theory and Desgn. Nob H Pubshng, 29. [] J. N. Tstsks. Probems n Decentrazed Decson Makng and Computaton. PhD thess, MT, Cambrdge, MA, USA, 984. [2] A. Wächter and L. T. Beger. On the mpementaton of a prmadua nteror pont fter ne search agorthm for arge-scae nonnear programg. Mathematca Programg, 6):25 57, 26. [3] K. Worthmann, C. M. Keett, P. Braun, L. Grüne, and S. R. Weer. Dstrbuted and decentrazed contro of resdenta energy systems ncorporatng battery storage. EEE Transactons on Smart Grd, 25. Do:.9/TSG.25.23928. [4] K. Worthmann, C. M. Keett, L. Grüne, and S. R. Weer. Dstrbuted contro of resdenta energy systems usng a market maker. n 9th FAC Word Congress, South Afrca, pages 64 646, 24. [5] M. Zhu and S. Martínez. On dstrbuted convex optmzaton under nequaty and equaty constrants. EEE Transactons on Automatc Contro, 57):5 64, 22.