COMPARED with AC microgrids, Direct-Current microgrids

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1 1 Optmal Power Flow n Stand-alone DC Mcrogrds Ja L, Feng Lu, Member, IEEE, Zhaojan Wang, Steen H Low, Fellow, IEEE, Shengwe Me, Fellow, IEEE, arx: [math.oc] 17 Aug 2017 Abstract Drect-current mcrogrds (DC-MGs) can operate n ether grd-connected or stand-alone mode. In partcular, stand-alone DC-MG has many dstnct applcatons. Howeer, the optmal power flow problem of a stand-alone DC-MG s nherently non-conex. In ths paper, the optmal power flow (OPF) problem of DC-MG s nestgated consderng conex relaxaton based on second-order cone programmng (SOCP). Mld assumptons are proposed to guarantee the exactness of relaxaton, whch only requre unform nodal oltage upper bounds and poste network loss. Furthermore, t s reealed that the exactness of SOCP relaxaton of DC-MGs does not rely on ether topology or operatng mode of DC-MGs, and an optmal soluton must be unque f t exsts. If lne constrants are consdered, the exactness of SOCP relaxaton may not hold. In ths regard, two heurstc methods are proposed to ge approxmate solutons. Smulatons are conducted to confrm the theoretc results. Index Terms DC mcrogrd, optmal power flow, conex relaxaton. I. INTRODUCTION COMPARED wth AC mcrogrds, Drect-Current mcrogrds (DC-MGs) hae been recognzed as an attracte alternate for numerous applcatons due to hgher effcency, more natural nterface to many types of renewable energy resources and energorage systems, better complance wth consumer electroncs, etc. [1]. Addtonally, when components are coupled around a DC bus, there are no ssues wth reacte power flow and frequencablty, resultng n a notably less complex power system [2]. DC-MGs may operate n ether grd-connected or stand-alone mode. The latter has many dstnct applcatons n shpboard [3], [4], arcraft [5], [6], automote [7], as well as the electrcty supply of remote rural areas. In ths regard, t s of great mportance to nestgate the optmal power flow of DC-MGs. Second-order cone programmng (SOCP) has been extensely used n AC networks for solng the optmal power flow (OPF) problem [8] [13]. In [8], radal dstrbuton load flow s formulated as a conc quadratc optmzaton problem, and soled effcently usng nteror-pont methods. In [10], a two-step SOCP relaxaton approach s proposed, whch consst of angle relaxaton and conc relaxaton. The exactness of angle relaxaton requres the so-called cyclc condton that the sum of angle dfferences on each cycle must be zero. Wth the cyclc condton satsfed, the conc relaxaton s exact, proded there are no upper bounds on loads. The formulatons of OPF problem and ther relaxatons are summarzed n J. L, F. Lu, Z. Wang and S. Me are wth the State Key Laboratory of Power System, the Department of Electrcal Engneerng, Tsnghua Unersty, Bejng , Chna.lfeng@tsnghua.edu.cn} S. H. Low s wth the Department of Electrcal Engneerng, Calforna Insttute of Technology, Pasadena, CA, USA, (emal:slow@caltech.edu) [11], and the suffcent condtons under whch the conex relaxatons are exact are presented n [12]. Another suffcent condton for the exact SOCP relaxaton of AC OPF n radal dstrbuton networks s proposed n [13], whch requres that the allowed reerse power flow s only reacte or acte, or none. As the OPF problem of DC power network s nherently nonlnear and non-conex, SOCP s extended to realze the conexfcaton of OPF problem n DC dstrbuted networks [14]. Such a result, howeer, reles on the assumpton that there exsts a large substaton wth unlmted capablty to prode power njecton and keep ts nodal oltage constant. Unfortunately, a stand-alone DC-MG s not the case because: 1) there s no substaton wth an unconstraned power njecton, and eery dstrbuted generaton theren has a lmted capacty; 2) there s no substaton wth a fxed nodal oltage, and all nodal oltages are allowed to ary wthn a certan range. Such dfferences motate us to extend the preous work [14] to ge rse to a better understandng of DC-MG n dfferent operatng modes. To ths end, the followng steps are taken. Step 1: Equalent Transformaton. By ntroducng slack arables, the orgnal problem OPF1 s transformed equalently nto OPF2 wth non-conex rank constrants. Step 2: Seocnd-Order Conc Relaxaton. By remong the rank constrants, the non-conex problem OPF2 s relaxed nto a conex SOCP problem,.e., RL1. Moreoer, extendng the results of [14], ths paper proes the exactness of SOCP relaxaton under mld assumptons, whch only requre unform nodal oltage upper bounds and poste network loss. Step 3: Equalent Conerson. By ntroducng alternate arables, RL1 s transformed equalently nto a branch flow model, namely RLS1, for the sake of mprong numercal stablty. The soluton approach and the relaton between the assocated theoretc results are llustrated n Fg. 1. OPF1 (Non-conex) Step 1 OPF2 (Non-conex) Step 2 Step 3 RL1 (Conex) RLS1 (Conex) Fg. 1. Proposed soluton approach. Theorem 1: Equalent Transformaton Theorem 2: Exact Conc Relaxaton Theorem 4: Unque Optmal Soluton Theorem 5: Equalent Conerson Theorem 3: Topologcal Independence

2 2 Under the proposed mld assumptons, the exactness of SOCP relaxaton and the unqueness of optmal soluton are reealed, whch are ndependent of topologes and operatng modes of DC-MGs. These propertes of DC-MG are ery helpful for optmal control and market desgn. When lne constrants are consdered, the stuaton turns to be much more complcated, snce the rank constrant cannot be guaranteed. In ths context, suffcent condtons are explored to justfy the exactness of SOCP relaxaton and two heurstc methods are suggested to construct feasble solutons when the rank constrant s not satsfed. The rest of ths paper s organzed as follows. The OPF problem of a stand-alone DC-MG s formulated and equalently transformed n Secton II. The SOCP relaxaton s gen n Secton III. Numercal studes are proded n Secton IV. The nfluence of lne constrants are dscussed n Secton V. Secton VI draws the conclusons. A. Basc Formulaton II. OPF PROBLEM OF DC-MGS Consder a graph G := (N, E), where N := 1,, n} denotes the set of all buses and E denotes the set of all lnes n the network of a DC-MG. G s assumed to be connected. Index the buses by 1,, n and abbreate, j} E as j. Denote ( j & < j) by j. For each bus N, denote V as ts oltage, and p as ts power njecton. For each lne j, y j denotes ts conductance. A letter wthout subscrpts denotes a ector of the correspondng quanttes, e.g., V = [V ] N. The notatons are summarzed n Fg.2. In p Bus V Fg. 2. Summary of notatons. y j Bus j a DC-MG, V, p, y j are real numbers, and y j > 0. Then the OPF problem of a stand-alone DC-MG reads V j... OPF1: mn h(p) f (p ) N oer: p, V ; s.t. p V (V V j ) y j, N ; p p p, N ; V V V, N. p j (1a) (1b) (1c) Here, f (p ) s strctly ncreasng n p. Eq. (1a) s the power njecton equaton for bus. The nodal oltages are constraned by (1c) wth the lower bound V > 0 and upper bound V. The power njectons are constraned by (1b) wth the lower bound p and upper bound p. It s assumed that p 0, whch ndcates the followng two cases: 1) Bus s a pure generaton bus wthout load. In ths case, p = 0 as the generators n a DC-MG can be turned off; 2) Bus s a pure load bus wthout generaton or a mxture power njecton of both load and generaton. In ths case, there s p < 0. In fact, the model of a grd-connected DC-MG [14] can be ewed as a specal case of OPF1 ncludng a substaton bus wth an unconstraned power njecton (p 0 =, p 0 = ) and a fxed nodal oltage (V 0 = V0 ref). Note that, n a DC-MG, lne constrants usually do not bnd n the normal operaton condton, snce the network s oerprosoned. Therefore, n ths paper, we frst consder the OPF problem of DC-MGs wthout lne constrants. Then, we dscuss the nfluence of lne constrants on the exactness of SOCP relaxaton. Throughout the paper, we do not assume any specfc topology of the power network. B. Equalent Transformaton The proposed OPF problem (1) s a non-lnear non-conex problem. By ntroducng slack arables, t can be transformed nto an equalent counterpart, where the non-conex power njecton equaton (1a) s conerted nto a rank constrant. Introduce slack arables to formulate a map f such that = V 2, N ; (2a) f := W j = V V j, j; (2b) and defne a matrx [ ] W R j := j (3) W j j for eery j. Then OPF1 (1) s transformed nto OPF2: mn h(p) oer: p,, W s.t. p p p p, N ( W j ) y j, N (4a) V 2 V 2, N W j 0, j W j = W j, j R j 0, j rank(r j ) = 1, j (4b) (4c) (4d) (4e) (4f) (4g) where the non-conexty n (1a) (n OPF1) s conerted nto the non-conexty n the rank constrant (4g) (n OPF2). R j s poste semdefnte as shown n (4f). Theorem 1: OPF1 and OPF2 are equalent. To proe Theorem 1, we frst ge the followng lemma. Lemma 1: Gen > 0 for N and W j 0 for j, let W j = W j for j. If rank(r j ) = 1 for j, then there exsts a unque V satsfyng V > 0 for N and (2). Moreoer, V s determned by V = for N. The proof of Lemma 1 can be found n Appendx A. Lemma 1 mples that for each (, W ), there exsts a unque V that satsfes the map gen by (2). Dfferng from [14], we

3 3 do not hae a substaton bus wth a fxed oltage. Instead, we assume V > 0 for all N. Wth Lemma 1, Theorem 1 can be proen. The proof can be found n Appendx B. It s worth notng that Theorem 1 does not rely on the topology of network. III. EXACTNESS OF CONIC RELAXATION A. SOCP Relaxaton of OPF n DC-MGs By remong (4g), the orgnal non-conex problem OPF2 s transformed to an SOCP problem (named as RL1) as below. RL1: mn h(p) oer: p,, W s.t. (4a) (4f) The only dfference between RL1 and OPF2 s that RL1 has no constrant (4g). Therefore, RL1 s exact, proded that ts eery optmal soluton satsfes (4g). To ensure the exactness of conc relaxaton, addtonal assumptons are requred. B. Assumptons Throughout the paper, we make the followng assumptons. Assumpton 1: V 1 = V 2 = V n > 0. Assumpton 2: N p > 0. Assumpton 1 requres all the nodal oltages hae the same upper bounds, whch s reasonable n DC-MGs, snce the scale of system s usually small. Assumpton 2 s tral as t means the total network loss s poste. Such assumptons relax those n [14], whch requre negate power njecton lower bounds and an unconstraned power njecton, to admt the features of stand-alone DC-MGs. C. Exactness of Conc Relaxton Wth Assumptons 1 and 2 mentoned aboe, we hae the followng man theorem: Theorem 2: RL1 s exact f Assumptons 1 and 2 hold for OPF1 (equalently OPF2). Theorem 2 clams that RL1 s an exact SOCP relaxaton of OPF2 (equalently OPF1) under Assumptons 1 and 2. To proe ths theorem, we ntroduce the followng lemmas. Lemma 2: Assume Assumpton 1 holds. Let (p,, W ) be feasble for RL1 but olate the rank constrant (4g) on a certan lne (s t) E. If p s = p s, then s < V 2 s. Meanwhle, f p t = p t, then t < V 2 t. The proof of Lemma 2 can be found n Appendx C. Lemma 2 mples that, for each bus, the power njecton s lower bound and the nodal oltage s upper bound cannot bnd at the same tme. Therefore, for any feasble soluton (p,, W ) of RL1, f t olates the rank constrant (4g) for a certan lne (s t) E and the constrant p s p s (or p t p t ) s bndng, then s V 2 s (or t V 2 t ) cannot bnd. Lemma 3: Assume Assumptons 1 and 2 hold for OPF1 and let (p,, W ) be a feasble soluton to RL1. If 1) (p,, W ) olates the rank constrant (4g) on a certan lne (s t) E; 2) p s > p s, p t > p t ; then there exsts another feasble soluton (p,, W ) that 1) satsfes (4a) (4f); 2) satsfes h(p ) < h(p). The proof of Lemma 3 can be found n Appendx D. Lemma 3 says that f a feasble pont (p,, W ) olates the rank constrant (4g) for a certan lne s t, whle both p s and p t are not bndng, then we can always fnd another feasble pont (p,, W ) wth a better objecte alue. It mples that f the optmal soluton (p,, W ) to RL1 olates the rank constrant (4g) for a certan lne s t, then at least one of p s and p t must hae reached ts lower bound. Lemma 4: Assume Assumptons 1 and 2 hold for OPF1 and let (p,, W ) be a feasble soluton to RL1. If 1) (p,, W ) olates the rank constrant (4g) on a certan lne (s t) E; 2) ether (p s = p s & p t > p t ) or (p s > p s & p t = p t ); then there exsts another feasble soluton (p,, W ) that 1) satsfes (4a) (4f); 2) satsfes h(p ) < h(p). The proof of Lemma 4 can be found n Appendx E. Lemma 4 means that f a feasble pont (p,, W ) olates the rank constrant (4g) for a certan lne s t, whle ether p s or p t s bndng, then we can always fnd another feasble pont (p,, W ) wth a better objecte alue. Lemma 3 and 4 mply that f the optmal soluton (p,, W ) to RL1 olates the rank constrant (4g) for a certan lne s t, then t must satsfy p s = p s and p t = p t. Otherwse, we can always fnd a better soluton. Lemma 5: Assume Assumptons 1 and 2 hold for OPF1 and let (p,, W ) be a feasble soluton to RL1. If 1) (p,, W ) olates the rank constrant (4g) on a certan lne (s t) E; 2) p s = p s and p t = p t ; then there always exsts (p,, W ) that 1) satsfes (4a) (4f); 2) olates rank constrant (4g) for all the neghbourng lnes of s t,.e. j wth }, j}} s}, t}}. The proof of Lemma 5 can be found n Appendx F. Lemma 5 says that f a feasble pont (p,, W ) olates the rank constrant (4g) for a certan lne s t, whle the correspondng power njectons lower bounds are bndng, we can always fnd another feasble pont (p,, W ) whch olates the rank constrant (4g) for s t and all ts neghborng lnes. It should be noted that n Lemma 5, the constructon of feasble pont (p,, W ) does not change p. Snce the network G s connected, Lemma 5 mples that we can contnue such propagaton to obtan a feasble pont that olates the rank constrant (4g) for all the lnes, wthout changng p. As mentoned before, f the optmal soluton (p,, W ) to RL1 olates the rank constrant (4g) for a certan lne s t, Lemma 3 and 4 ensure that t must satsfy p s = p s and p t = p t. In ths case, accordng to Lemma 5, we can always fnd a feasble (p,, W ) whch satsfes p = p and olates the rank constrant (4g) for all the lnes. Followng ths dea, we can proe Theorem 2 based on Lemma 2 5. The proof of Theorem 2 can be found n Appendx G.

4 4 D. Topologcal Independence Recall the SOCP relaxaton n AC networks [10]. The approach conssts of two relaxaton steps: angle relaxaton and conc relaxaton. Smlarly, our method also contans two steps: equalent transformaton and conc relaxaton. Dfferng from AC networks, DC networks do not nole oltage angles. The frst step n our method (.e., equalent transformaton) s exact, as Theorem 1 states. Howeer, n the second step, drectly remong the rank constrants may result n nexactness. Thus, addtonal assumptons (Assumptons 1, 2) are made to ensure the exactness of conc relaxaton. By notng that none of the two steps depends on specfc network topologes, we drectly hae the followng theorem: Theorem 3: Assume Assumptons 1 and 2 hold. Then the exactness of RL1 s ndependent of network topologes. Remark 1: In terms of a grd-connected DC-MG, t has also been demonstrated that the SOCP relaxaton s topologyndependent [14]. Actually, when a DC-MG works n grdconnected state, one can smply assgn a substaton bus n RL1 by lettng p 0 =, p 0 = + and 0 = [V ref 0 ]2. Hence, combnng the results n [14] and Theorem 3 mmedately concludes that the SOCP relaxaton of the OPF problem of DC-MGs does not rely on the topology of network, whether the DC-MG works n grd-connected or stand-alone mode. E. Unqueness of the Optmal Soluton Next we show the the optmal soluton to RL1 s unque. Theorem 4: Assume Assumptons 1 and 2 hold for RL1. Then the optmal soluton to RL1 s unque. The proof of Theorem 4 can be found n Appendx H. In a grd-connected DC-MG, t has also been proen that the SOCP problem has at most one optmal soluton [14]. Thus, the unqueness of optmal soluton to the SOCP problem does not depend on the operatng mode of DC-MG. F. Branch Flow Model In an optmal soluton to RL1, and W j may be numercally close to each other, snce the range of nodal oltage s small (usually p.u.) and Rank(R j ) = 1 s satsfed, whch mples that j = W j W j. Thus, RL1 s ll-condtoned snce equaton (4a) requres the subtractons of and W j. Howeer, such subtractons can be aoded by conertng RL1 nto a branch flow model, so that the numercal stablty s mproed. In lght of [14], by defnng z j := 1/y j and adoptng alternate arables P, l, RL1 can be conerted nto a branch flow model a the map g : (, W ) (, P, l) defned as below. =, N ;(5a) g := P j = ( W j ) y j, j; (5b) l j = yj 2 ( W j W j + j ), j. (5c) Wth g, RL1 s conerted nto the followng optmzaton problem wth branch flow model (BFM): RLS1: mn h(p) oer: p, P,, l; s.t. p P j, N ; p p p, N ; V 2 V 2, N ; P j + P j = z j l j, j; j = z j (P j P j ), j; (6a) (6b) (6c) (6d) (6e) l j P j 2, j (6f) where P j denote the power flow through lne j, and l j denote the magntude square of the current through lne j. Theorem 5: RL1 and RLS1 are equalent. The proof of Theorem 5 can be found n Appendx I. Snce RL1 s a conex relaxaton of the non-conex OPF2 by remong the rank constrant (4g), Theorem 5 mples that RLS1 s the conex relaxaton of another non-conex problem, namely, OPF3, whch s obtaned by conertng OPF2 usng the one-to-one map g (5). OPF3: mn h(p) oer: p, P,, l s.t. (6a) (6e) l j = P 2 j, j Under Assumptons 1 and 2, let F denote a feasble set of a certan optmzaton problem, whch s ndcated by the subscrpt. Then the relatonshp between dfferent feasble sets s shown as below: F OP F 2 = f(f OP F 1 ) F RL1 F RLS1 = g(f RL1 ) where f s the one-to-one map gen n Theorem 1, and g s the one-to-one map gen n Theorem 5. The relatonshp between dfferent feasble sets s depcted n Fg. 3. F OP F 1 s a non-conex regon and (p, V ) denotes the optmal soluton to OPF1. F OP F 1 s transformed equalently nto another non-conex regon F OP F 2 by the map f. F OP F 2 s conexfed by remong the rank constrant (4g), yeldng F RL1, whch s larger than F OP F 2. The exactness of RL1 ensures that the optmal soluton to RL1,.e., (p,, W ) s not n the shaded regon of F RL1. Snce the objecte functon of OPF2 and RL1 are the same, (p,, W ) s also the optmal soluton to OPF2. Applyng the one-to-one map f, (p,, W ) s transformed nto (p, V ). The one-to-one map g conerts F RL1 nto an equalent conex regon F RLS1, and conerts F OP F 2 nto an equalent non-conex regon F OP F 3. (p, P,, l ) s the optmal soluton to RLS1, whch s also the optmal soluton to OPF3, snce RLS1 s exact. Moreoer, (p, P,, l ) can be conerted equalently nto (p,, W ) usng the map g. Extendng the work n [14], we hae shown that under the proposed assumptons, the OPF problem n DC-MGs can be conerted equalently nto a conex SOCP problem, regardless of topologes or operatng modes. Thereby solng the relaxed conex SOCP can obtan the global optmum wth theoretc guarantee. Furthermore, we hae shown that the SOCP problem has at most one optmal soluton, whether the

5 5 F OPF1 (p*,v*) f F OPF2 (p*,*,w*) g F OPF3 (p*,p*,*,l*) The oltage lower and upper bounds are set as 0.95 p.u. and 1.05 p.u., respectely. The objecte s to mnmze total network loss and the problems are soled usng MOSEK. The results are lsted n Table I. F RL1 Fg. 3. Relatonshp between dfferent feasble sets. DC-MG s workng n grd-connected or stand-alone mode. Remark 2: In OPF1 OPF3, lne flow constrants are not consdered. Whereas we hae not found any olaton of lne constrants n all our tests, we cannot theoretcally exclude the possblty of such olaton. In such a crcumstance, approxmate solutons can be heurstcally constructed. We wll dscuss the stuaton ths ssue later on n Secton V. A. 16-bus System IV. CASE STUDIES Tests are conducted on the modfed 16-bus system [15] to demonstrate the effcacy of the proposed method when the system works n dfferent operatng modes and network topologes. The three-feeder system s shown n Fg. 4 where 6 dstrbuted generators (DGs) are added wth the same capacty of 5MW. The dashed lnes represent te lnes wth te breakers. The system s able to work n two modes: F RLS1 Grd-connected: the system s connected to a power grd a all the three feeders (Feeders A, B and C). Stand-alone: all the three feeders are dsconnected from the grd. Addtonally, the network can swtch between tree and mesh topologes by openng or closng the te breakers. Tree topology: all the te breakers are swtched off. Mesh topology: all the te breakers are swtched on. Thus, the followng four cases are studed. 1) GT: grd-connected mode wth a tree topology. 2) GM: grd-connected mode wth a mesh topology. 3) ST: stand-alone mode wth a tree topology. 4) SM: stand-alone mode wth a mesh topology. G Feeder A Feeder B Feeder C G 5 G G 11 Fg bus system wth DGs G G TABLE I RESULTS OF 16-BUS SYSTEM GT GM ST SM Exactness 5.73E E E E-10 Objecte Value (p.u.) Computaton Tme (s) At a numercal RLS1 soluton (p,, W ), R j can be obtaned for each j. If RLS1 s exact, then for all j, we should hae rank(r j ) = 1,.e., the dfference D j := j W j W j = 0. Hence the smaller D j ndcates, the closer R j s to rank one. The row Exactness lsts the maxmum alue of D j for all j, ndcatng RLS1 s exact for all the test networks. Moreoer, t ndcates that the proposed SOCP relaxaton does not depend on the operatng mode or network topology. The objecte alues ndcate that n ths system, mesh topology reduces network loss n both grd-connected and stand-alone modes, snce mesh topology s able to support hgher nodal oltages. Addtonally, workng n the same network topology, grd-connected mode experences less network loss than stand-alone mode, snce more power s njected from the feeders to support hgher nodal oltages. The results of power njectons and nodal oltages are lsted n Table II. TABLE II RESULTS OF POWER INJECTIONS AND NODAL VOLTAGES Power Injecton (p.u.) Nodal Voltage (p.u.) Bus GT GM ST SM GT GM ST SM B. Exactness and Comparson More tests are conducted on both mesh and tree networks to check the exactness of the SOCP relaxaton. In addton, the objecte alue and computaton tme of the relaxed model (.e., RLS1) and the orgnal model (.e., OPF1) are compared to show the effcacy of the relaxaton. The objecte of both models s to mnmze total network loss. The mesh networks [14] are modfed from MATPOWER by gnorng lne reactances. All the lne resstances are reduced to 10% of the orgnal alues to smulate the DC-MG condton. Partcularly, the zero resstances are reset as 10 3 p.u.. Addtonally, the IEEE 118-bus system s appled to show the scalablty of

6 6 the proposed method. Four radal dstrbuton networks n the lterature are also used to erfy the topologcal ndependence of the relaxaton. In these systems, all the lne resstances are also reduced to 10% of the orgnal alues to smulate the DC- MG condton. In all the test systems, the oltage lower and upper bounds are set as 0.95 p.u. and 1.05 p.u., respectely. RLS1 s soled usng MOSEK, whle OPF1 s soled usng IPOPT. The results are lsted n Table III. TABLE III EXACTNESS OF RLS1 AND COMPARISON OF TWO MODELS Exactness Objecte Value (p.u.) Tme (s) Topology System of RLS1 RLS1 OPF1 RLS1 OPF1 case6ww 1.24E E E case9 7.17E E E Mesh case eee E E E case E E E case E E E bus [16] a 1.28E E E Tree 70-bus [17] b 5.35E E E bus [18] c 3.09E E E a 6 DGs are added at bus 5, 10, 15, 20, 25, 30, wth the capacty of 50kW. b 13 DGs are added at bus 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, wth the capacty of 50kW. c 13 DGs are added at bus 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, wth the capacty of 50kW. It s obsered that RLS1 s exact for both mesh and tree topologes. The scale of a DC-MG s small, howeer, more and more DC-MGs may be bult n the future to ntegrate dstrbuted renewable energy generaton and electrc ehcles. The results of computaton tme mples that the proposed method may be appled n large scale systems, for example, the cluster of DC-MGs. We also tested the model wth lne constrants (.e., RLS2), on the aboe systems. Howeer, we hae not found any case where the optmal soluton olates the rank constrants (4g). In ths regard, we conjecture that the RLS2 s almost always exact n practce. Snce we cannot theoretcally exclude the possblty of nexactness of OPF n ths case, we ge some dscussons and nsghts n the next secton. V. DISCUSSION ON LINE CONSTRAINTS A. OPF wth Lne Constrants Let I j denote the threshold of current through lne j, then the lne constrant can be formulated as whch s correspondng to y 2 j( W j W j + j ) I 2 j (7) y 2 j(v V j ) 2 I 2 j (8) n the orgnal problem. By addng (7) nto RL1, we hae the followng model. RL2: mn h(p) oer: p,, W s.t. (4a) (4f), (7) Usng the transformaton n Theorem 5, the lne constrant (7) can be be added nto RLS1, yeldng the followng model. RLS2: mn h(p) oer: p, P,, l s.t. (6a) (6f) l j I 2 j (9) B. Exactness Condtons When Consderng Lne Constrants We frst ntroduce the followng theorem to ge the condtons that guarantee the exactness of RL2. Theorem 6: Assume Assumptons 1 and 2 hold for RL2. Then RL2 s exact under ether of the followng condtons. Condton 1: ( j) E, constrant (7) s not bndng ; Condton 2: For each lne (s t) E, f constrant (7) s bndng for ths lne, then the correspondng lower bounds on p s and p t are not bndng. The proof of Theorem 6 can be found n Appendx J. Furthermore, f the soluton to RL2 s not exact, t s possble to fnd an approxmate soluton by relaxng the bounds on the power njectons, as the followng theorem ndcates. Theorem 7: Assume Assumptons 1 and 2 hold for RL2 and let (p,, W ) be a feasble soluton to RL2. If 1) (p,, W ) olates the rank constrant (4g) for a certan lne (s t) E where the lne constrant (7) s bndng; 2) at least one of p s = p s and p t = p t s satsfed; then there must exst another soluton (p,, W ) such that 1) satsfes all the constrants of RL2 except for (4b) (.e., (4a), (4c) (4f) and (7)); 2) satsfes the rank constrant (4g). 3) has a lower objecte alue than (p,, W ). The proof of Theorem 7 can be found n Appendx K. Theorem 7 ndcates that f the adjustment of power njecton ɛ s allowed at bus s and bus t, then a soluton whch satsfes the rank constrant (4g) can be constructed. In a DC- MG, such adjustment may be acheed by employng demand response or energorage. C. Constructng Approxmate Optmal Solutons Snce RL2 s not always exact, we can check the soluton after solng RL2. If the soluton satsfes the rank constrant (4g), then t s global optmal for the orgnal problem OPF1 (1). Otherwse, nspred by the recoery methods of semdefnte program (SDP) relaxaton [19], we propose two heurstc methods to construct a nearly optmal soluton. Theorem 6 ndcates that f (p,, W ) s the optmal soluton to RL2, and olates the rank constrant (4g) for a certan lne (s t) E where the lne constrant (7) s bndng, then at least one of p s and p t must reach ts lower bound. Otherwse, we can always fnd another feasble pont, whch has a smaller objecte alue than (p,, W ). Therefore, we only need to dscuss the cases that at least one of p s and p t reaches ts lower bound. 1) Drect constructon method: In [19], a drect recoery method s proposed for AC networks, usng the frst column of the optmal soluton matrx W to recoer a nearly optmal soluton. It nspres a drect constructon method for DC-MGs. Assume (p,, W ) s the optmal soluton to RL2, whch olates the rank constrant (4g) for some lnes. Instead of usng the frst column of W (see n [19]), we use to construct an approxmate soluton (p, V ) to the orgnal problem

7 7 OPF1. Frst, V ( N ) s dered by lettng V =. Then, p ( N ) s dered by substtutng V nto the power balance equaton (1a). Next we show the approxmate soluton (p, V ): 1) satsfes power balance equaton (1a), the oltage constrant (1c), lne constrant (8); 2) may olate (1b) for some N, but the olatons hae lmted bounds. Accordng to the constructon of (p, V ), t s straghtforward to check that (p, V ) satsfes (1a) and (1c). Hence we only need to examne constrant (8) to justfy the frst asserton. For any N, let C denote the set of buses mmedately connected to bus. Let B C denote the subset such that for any j B, lne j olates the rank constrant (4g). If all the neghborng lnes of bus do not olate the rank constrant (4g), we hae B =. It follows that for all ( j) E, j > W j for j B, whle j = W j for j / B. Snce (p,, W ) satsfes (7), for any j(j B ), t follows that yj(v 2 V j ) 2 = yj(v 2 2 2V V j + Vj 2 ) = yj( 2 2 j + j ) and for any j(j / B ) < y 2 j( 2W j + j ) I 2 j yj(v 2 V j ) 2 = yj(v 2 2 2V V j + Vj 2 ) = yj( 2 2 j + j ) = y 2 j( 2W j + j ) I 2 j. Therefore, (p, V ) satsfes (8). Next, we check constrant (1b). It follows from (1a) that p V (V V j ) y j ( ) j y j,j / B +,j B,j / B = p. ( ) j y j ( ) j y j ( ) j y j ( W j) yj +,j B ( Wj ) yj Hence, p may olate (1b). And the olaton s bounded by p p ( ) j + W j y j.,j B 2) Slack arable method: Accordng to Theorem 6, RL2 s exact f the lower bounds of power njectons are not bndng. Thus, after solng RL2, f the soluton olates the rank constrant (4g) for a certan lne (s t) E, and any lower bound of p s and p t s bndng, for example, p s = p s, then a correspondng slack arable ε s (ε s > 0) can be added nto (4b) to reformulate the constrant as p s p s ε s p s (10) so that p s wll not reach ts lower bound n the next teraton to sole RL2. Addtonally, n order to mnmze ε s, t s also added nto the objecte functon as mn h(p) + ˆN where ˆN s the set of buses where the rank constrant (4g) s olated and the power njecton lower bound s bndng at the same tme. The procedure s shown n Fg. 5. No Sole RL2 Rank(R j )=1 for all the lnes? No e has been added for all the buses? Yes Add e for the lne where Rank(R j ) 1 Add e n the objecte functon Sole RL2 Rank(R j )=1 for all the lnes? No Yes Feasble Soluton Fg. 5. Flowchart of slack arable method. Yes ε VI. CONCLUSIONS Optmal Soluton Feasble Soluton In ths paper, we hae proposed an SOCP relaxaton of the OPF problem n stand-alone DC-MGs, whch do not consst of any substaton wth an unconstraned power njecton and a fxed nodal oltage. We hae also proposed two mld assumptons whch only requre unform oltage upper bounds and poste network losses. Under such assumptons, we hae proen the exactness of the SOCP relaxaton by extendng the results n [14]. Combnng the results n [14] and those n ths paper, we hae a more comprehense understandng on the OPF problem n DC-MGs: 1) Under the proposed assumptons, the exactness of SOCP relaxaton n DC-MGs does not rely on the operatng mode of grd. Ths property facltates the optmal operaton of DC-MG, snce t allows the operator to achee optmal operaton whether the DC-MG s workng n grdconnected or stand-alone mode. 2) Unlke AC systems, the exactness of SOCP relaxaton n DC-MGs does not rely on the topology of networks. It mples that when the topology s changed due to

8 8 lne swtchng, the OPF problem can stll be conerted equalently nto a conex SOCP problem. 3) The unqueness of the global optmal soluton to the SOCP problem s ndependent of the topology and operatng mode of DC-MGs. Ths property s especally useful for settng a target n optmal control of DC-MGs. REFERENCES [1] T. Dragčeć, J. C. Vasquez, J. M. Guerrero, and D. Škrlec, Adanced LVDC electrcal power archtectures and mcrogrds, IEEE Electrf. Mag., ol. 2, no. 1, pp , [2] T. Dragčeć, X. Lu, J. C. Vasquez, and J. M. Guerrero, DC Mcrogrds - Part I: A Reew of Control Strateges and Stablzaton Technques, IEEE Trans. Power Electron., ol. 31, no. 7, pp , [3] G. F. Reed, B. M. Granger, A. R. Sparacno, and Z.-H. Mao, Shp to grd: Medum-oltage dc concepts n theory and practce, IEEE Power Energy Mag., ol. 10, no. 6, pp , [4] Z. Jn, G. Sullgo, R. Cuzner, L. Meng, J. C. Vasquez, and J. M. Guerrero, Next-generaton shpboard dc power system: Introducton smart grd and dc mcrogrd technologes nto martme electrcal netowrks, IEEE Electrf. Mag., ol. 4, no. 2, pp , [5] P. Magne, B. Nahd-Mobarakeh, and S. Perfederc, General acte global stablzaton of multloads DC-power networks, IEEE Trans. Power Electron., ol. 27, no. 4, pp , [6], Acte stablzaton of dc mcrogrds wthout remote sensors for more electrc arcraft, IEEE Trans. Ind. Appl., ol. 49, no. 5, pp , [7] A. Emad, A. Khalgh, C. H. Retta, and G. A. Wllamson, Constant power loads and negate mpedance nstablty n automote systems: Defnton, modelng, stablty, and control of power electronc conerters and motor dres, IEEE Trans. Veh. Technol., ol. 55, no. 4, pp , [8] R. A. Jabr, Radal dstrbuton load flow usng conc programmng, IEEE Trans. Power Syst., ol. 21, no. 3, pp , [9] S. Sojoud and J. Laae, Physcs of power networks makes hard optmzaton problems easy to sole, IEEE Power Energy Soc. Gen. Meet., pp. 1 8, [10] M. Farar and S. H. Low, Branch flow model: Relaxatons and conexfcaton Part I, IEEE Trans. Power Syst., ol. 28, no. 3, pp , [11] S. H. Low, Conex relaxaton of optmal power flow Part I: Formulaton and equalence, IEEE Trans. Control Netw. Syst., ol. 1, no. 1, pp , [12], Conex relaxaton of optmal power flow Part II: Exactness, IEEE Trans. Control Netw. Syst., ol. 1, no. 2, pp , [13] S. Huang, Q. Wu, J. Wang, and H. Zhao, A suffcent condton on conex relaxaton of AC optmal power flow n dstrbuton networks, IEEE Trans. Power Syst., ol. 32, no. 2, pp , [14] L. Gan and S. H. Low, Optmal power flow n drect current networks, IEEE Trans. Power Syst., ol. 29, no. 6, pp , [15] S. Canlar, J. J. Granger, H. Yn, and S. S. H. Lee, Dstrbuton feeder reconfguraton for loss reducton, IEEE Trans. Power Del., ol. 3, no. 3, pp , [16] M. E. Baran and F. F. Wu, Network reconfguraton n dstrbuton systems for loss reducton and load balancng, IEEE Trans. Power Del., ol. 4, no. 2, pp , [17] D. Das, A fuzzy multobjecte approach for network reconfguraton of dstrbuton systems, IEEE Trans. Power Del., ol. 21, no. 1, pp , [18] C.-T. Su and C.-S. Lee, Network reconfguraton of dstrbuton systems usng mproed mxed-nteger hybrd dfferental eoluton, IEEE Trans. Power Del., ol. 18, no. 3, pp , [19] G. Fazelna, R. Madan, A. Kalbat, and J. Laae, Conex relaxaton for optmal dstrbuted control problems, IEEE Trans. Automat. Contr., ol. 62, no. 1, pp , 2017.

9 9 A. Proof of Lemma 1 APPENDIX Proof: Exstence: Let V = for N. It suffces to show that V satsfes V > 0 for N and (2). Snce > 0 for N, t follows that V = > 0 for N. It s straghtforward to check that V satsfes (2a). Snce R j s not full rank, we hae j W j W j = 0, j Snce W j 0, we hae W j = Wj 2 = W j W j = j = V V j for j,.e., V, satsfes (2b). Ths completes the proof of exstence. Unqueness: Let Ṽ denote an arbtrary soluton to Ṽ > 0 for N and (2). It suffces to show that Ṽ = for N. Assume Ṽ for some N, then t follows from (2a) that Ṽ = < 0, whch contradcts wth Ṽ > 0 for N. Thus, Ṽ = for N. Ths completes the proof of unqueness. B. Proof of Theorem 1 Proof: Let F OP F 1 and F OP F 2 denote the feasble sets of OPF1 and OPF2, respectely. Snce OPF1 and OPF2 hae the same objecte functon, t suffces to show that there exsts a one-to-one map between F OP F 1 and F OP F 2. Specfcally, we show the map f : (V ) (, W ) gen by (2) s one-to-one, snce p s unquely determned by V or (, W ). To ths end, t suffces to show the map s both nto and onto. On the one hand, t s straghtforward that for any V 1 V 2 F OP F 1, f(v 1 ) f(v 2 ) F OP F 2. On the other hand, as Lemma 1 says, for each (, W ) F OP F 2, there exsts a unque f 1 (, W ) = V F OP F 1. Ths completes the proof. C. Proof of Lemma 2 Proof: For brety, we only proe the case of bus s as the proof of bus t s the same. Wth regard to the alue of p s, we only need to dscuss two cases: p s = p s < 0 and p s = p s = 0. In the frst case, we hae p s ( s W s ) y s < 0 : s accordng to (4a). Therefore, ( s W s ) y s < 0 for some N. It follows from (4d) (4f) that W s s. Accordng to Assumpton 1, V s = V. Thus, we hae s < W s 2 2 s V s V = V 2 s. In the second case, we hae p s ( s W s ) y s = 0 (11) : s accordng to (4a). Then we dscuss the followng two cases: 1) When s W s = 0 for all s, we hae s = W st for = t. Snce (p,, W ) olates the rank constrant (4g) for s t, R st s non-sngular. Hence, W st s t W st < s t due to (4d) (4f). Accordng to Assumpton 1, V s = V t. It follows that s = W st < 2 2 s t V s V t = V 2 s. 2) When s W s = 0 does not hold for some of s, there must exst at least one s such that s W s < 0 (and accordngly there exsts at least another s such that s W s > 0 n order to satsfy (11) ). Hence we hae s < W s 2 2 s V s V = V 2 s due to V s = V. Ths completes the proof. D. Proof of Lemma 3 Proof: Snce (p,, W ) satsfes (4d) (4f), we hae 0 W j j for j. Snce (p,, W ) olates the rank constrant (4g) for s t, we hae W st s t. Thus, W st < s t. We can always choose a small enough number ɛ > 0 such that ɛ < mn ps p s, p t p t, } s t W st. Followng the lne of the proof of Lemma 7 n [14], we can use ɛ to construct W as W j W j + ɛ f, j} = s, t}; = otherwse; W j and construct p as p ( W j ) yj, N. The pont (p,, W ) satsfes (4a) due to the constructon of p. When s, t, we hae p ( W j) yj ( W j ) y j = p. When = s, t, we hae p ( W j)y j ( W j )y j ɛ = p ɛ (p, p ). Hence, (p,, W ) satsfes (4b) and p < p f = s, t; = p otherwse. It follows that h(p ) < h(p) snce f s strctly ncreasng n p for N. The pont (p,, W ) satsfes (4c) snce remans the same as the feasble pont (p,, W ). The pont (p,, W ) satsfes (4d) due to the constructon of W. The pont (p,, W ) satsfes (4e) snce W j W j = W j W j = 0 for j. Addtonally, the pont (p,, W ) satsfes (4f) snce when, j} s, t} and W j = W j [0, j ] W j = W j + ɛ [0, j ) when, j} = s, t}. Ths completes the proof.

10 10 E. Proof of Lemma 4 Proof: We present the proof for the case (p s = p s & p t > p t ). The proof for the case (p s > p s & p t = p t ) s smlar and omtted for brety. Smlar to the proof of Lemma 3, we hae W st < s t. Addtonally, t follows from Lemma 2 that s < V 2 s, so we can always choose a small enough number ɛ > 0 such that pt p ɛ < mn t, j:j s s t W st, y ( ) } sj V 2 s s, then p t ɛ > p t, W st + ɛ < s t, s + j:j s y sj ɛ < V 2 s. Followng the lne of the proof of Lemma 8 n [14], we can use ɛ to construct W as W j W j + ɛ f, j} = s, t}; = otherwse; construct as = W j + ɛ f = s; yj otherwse; and construct p as p ( W j)y j, N. The pont (p,, W ) satsfes (4a) accordng to the constructon of p. When s, t, we hae p ( W j) yj ( W j ) y j = p. When = s, we hae p s ( s W sj j:j s ) ysj ( s W sj ) y sj + ( s s ) y sj ɛ j:j s ( s W sj ) y sj = p s. j:j s When = t, we hae p t ( t W tj)y tj j:j t j:j s ( t W tj )y tj ɛ j:j t = p t ɛ (p t, p t ). Thus, (p,, W ) satsfes (4b) and p < p f = t; = p otherwse. It follows that h(p ) < h(p). Snce = f s and < < V 2 f = s, the pont (p,, W ) satsfes (4c). Smlar to the proof of Lemma 3, t s straghtforward to check that the pont (p,, W ) satsfes (4d), (4e) and (4f). Ths completes the proof. F. Proof of Lemma 5 Proof: Smlar to the proof of Lemma 3, we hae W st < s t. Accordng to Lemma 2, we hae s < V 2 s, t < V 2 t. Therefore, we can always choose a small enough number ɛ > 0 so that s j:j s ɛ < mn t W st, y ( ) sj V 2 s s, y st j:j t y ( ) } tj V 2 t t, then W st + ɛ < s t, s + j:j s y sj ɛ < V 2 s, t + j:j t y ɛ < V 2 t. tj Followng the lne of the proof of Lemma 9 n [14], we can use ɛ to construct W as W j = W j + ɛ W j f, j} = s, t}; otherwse; and construct as + ɛ = s, t; = yj otherwse. The pont (p,, W ) satsfes (4a) snce when = s, t, ( W j ) yj ( W j ) y j + ( W j ) y j = p. y j ɛy j ɛ When s, t, we hae ( W j) yj ( W j ) y j = p. (12) Snce p does not change, the pont (p,, W ) satsfes (4b). It follows from (12) that = f s, t and < < V 2 f = s, t. Hence, the pont (p,, W ) satsfes (4c). Smlar to the proof of Lemma 3, t s straghtforward to check that the pont (p,, W ) satsfes (4d), (4e) and (4f). Snce G s connected, there always exsts some j such that, j} s, t} and }, j}} s}, t}}. Then we hae W j = W j j < j. Partcularly, when, j} = s, t}, W j = W j + ɛ < j < j. It means (p,, W ) olates the rank constrant (4g) for s t and all ts neghborng lnes. Ths completes the proof. G. Proof of Theorem 2 Proof: To proe that RL1 s exact, t suffces to show that any optmal soluton to RL2 satsfes the rank constrant (4g). We frst assume RL1 s not exact for the sake of

11 11 contradcton. Then there must exst at least one optmal soluton (p,, W ) of RL, whch olates the rank constrant (4g) for a certan lne (s t) E. We dscuss from the followng three aspects of the power njectons p s and p t correspondng to s t. (1) p s > p s & p t > p t. In ths case, as Lemma 3 ponts out, there must exst a feasble (p,, W ) whch has a smaller objecte alue than (p,, W ). In ths stuaton, (p,, W ) cannot be optmal for RL. (2) (p s = p s & p t > p t ) or (p s > p s & p t = p t ). In ths case, Lemma 4 states that there must exst a feasble (p,, W ) that has a smaller objecte alue than (p,, W ). Hence, (p,, W ) cannot be optmal for RL. (3) p s = p s & p t = p t. In ths case, we show such an optmal soluton (p,, W ) does not exst. Accordng to Lemma 5, there always exsts a feasble (p,, W ) whch olates the rank constrant (4g) for s t and all ts neghborng lnes. Snce G s connected and p = p, we can further propagate such constructon to obtan another feasble (p,, W ) of RL, whch olates the rank constrant (4g) for all the lnes. Hence, p satsfes p = p for all N. It follows that p p 0 N N Ths contradcts Assumpton 2 whch states N p > 0. Thus, (p,, W ) does not exst. Thus, eery optmal soluton must satsfy the rank constrant (4g),.e., RL1 s exact. Ths completes the proof. H. Proof of Theorem 4 Proof: Let x 1 := (p 1, 1, W 1 ) and x 2 := (p 2, 2, W 2 ) be two optmal solutons of RL1, then we hae f (p 1 ) f (p 2 ) (13) N N Followng the lne of the proof of Theorem 3 n [14], we know 1 2 = 1 j j 2 W 1 j = = η, j; 1 1 j It follows from (4a) that p 1 = η ( 1 ( η j Wj 1 ) yj ηwj 2 ) yj = ηw 2 j. = ηp 2. Snce f (p ) s strctly ncreasng, we must hae η = 1. Otherwse, t contradcts (13). Thus, x 1 = x 2,.e., RL1 has at most one optmal soluton. Ths completes the proof. I. Proof of Theorem 5 Proof: Let F RL1 and F RLS1 denote the feasble sets of RL1 and RLS1, respectely. It suffces to show that the map g (5) s one-to-one between F RL1 and F RLS1, snce p s determned by (, W ) n F RL1 and determned by P n F RLS1. On the one hand, t s straghtforward that for any ( 1, W 1 ) ( 2, W 2 ) F RL1, g( 1, W 1 ) g( 2, W 2 ) F RLS1. On the other hand, for any (, P, l) F RLS1, t can be erfed that there exsts a g 1 (, P, l) = (, W ) F RL1. Ths completes the proof. J. Proof of Theorem 6 Proof: When the lne constrant (7) s not bndng, RL2 s equalent to RL1, whch s exact under Assumptons 1 and 2 accordng to Theorem 2. Howeer, when constrant (7) s bndng, Theorem 2 does not hold. In ths stuaton, assume (p,, W ) s the optmal soluton to RL2. Assume (p,, W ) olates the rank constrant (4g) for a certan lne (s t) E where constrant (7) s bndng, and the lower bounds of p s and p t are not bndng. Snce (p,, W ) satsfes (4d) (4f), we hae 0 Wj j for j. Snce (p,, W ) olates the rank constrant (4g) for s t, we hae Wst s t. Thus, Wst < s t. We can always choose a small enough number ɛ > 0 such that ɛ < mn p s p s, p t p t, y s t Wst st Then we can use ɛ to construct (p,, W ) where W j Wj := + ɛ f, j} = s, t}; Wj otherwse; p : ( W j ) yj, N. }. It s easy to erfy that (p,, W ) satsfes (4a) (4f) and h(p ) < h(p ). Addtonally, when, j} s, t}, we hae y 2 j( W j W j + j ) = y 2 j( W j W j + j ) I 2 j. When, j} = s, t}, snce the lne constrant (7) s bndng, y 2 j( W j W j+ j ) = y 2 j( W j W j+ j 2ɛ) < I 2 j. Thus, the pont (p,, W ) satsfes (7). It means that the pont (p,, W ) s feasble for RL2 and has a smaller objecte alue than (p,, W ). Thus, (p,, W ) cannot be the optmal soluton. In contrast, the optmal soluton to RL2 must satsfy the rank constrant (4g). Ths completes the proof. K. Proof of Theorem 7 Proof: Notng that (p,, W ) satsfes (4d) (4f), we hae 0 W j j for j. Snce (p,, W ) olates the rank constrant (4g) for s t, W st s t. Hence W st < s t. Then we can always choose ɛ := s t W st > 0 to construct another soluton (p,, W ) as below: W j W j + ɛ f, j} = s, t}; := W j otherwse; p : ( W j ) yj, N. It s easy to erfy that (p,, W ) satsfes (4a), (4c) (4f) and (7). Hence the frst asserton s proen.

12 12 In terms of the second asserton, (p,, W ) satsfes (4g) by notng and W j = W j = j for, j} s, t} W j = W j + ɛ = j for, j} = s, t}. Next we consder the thrd asserton. Accordng to the constructon of (p,, W ), for any s, t, p satsfes p ( W j) yj ( W j ) y j = p and for any when = s, t, p ( W j)y j ( W j )y j ɛ = p ɛ < p Then (p,, W ) has a lower objecte alue than (p,, W ) because the objecte functon s strctly ncreasng n p. Ths completes the proof.