Harmonic Reduction via Optimal Power Flow and the Frequency Coupling Matrix

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1 27 IEEE Conference on Control Technology nd Applictions (CCTA August 27-3, 27. Kohl Cost, Hwi'i, USA Hrmonic Reduction vi Optiml Power Flow nd the Frequency Coupling Mtrix Ynhu Tin University of Toronto, Cnd N Li Hrvrd University, USA Joshu A. Tylor University of Toronto, Cnd Abstrct In this pper we propose new optiml power flow scheme tht tkes into ccount the hrmonics generted by power electronics interfced distributed genertion (DG. The objective is to minimize the cost of genertion under constrints on the totl hrmonic distortion (THD of voltge. The frequency coupling mtrix (FCM is used to model the hrmonic current injected by converter. Network current nd voltge re modeled for ech hrmonic frequency. Constrints limiting the mximum voltge THD re introduced to the threephse optiml power flow (OPF problem. We construct semidefinite relxtion, which cn be solved using commercilly vilble softwre pckges. We give numericl results for the hrmonic-constrined optiml power flow for two test systems. I. INTRODUCTION Incorporting renewble distributed genertions (DG into power network cn reduce genertion costs nd improve relibility [], [2]. Optiml power flow (OPF is used to minimize the cost of genertion subject to system constrints [3]. Though the nominl OPF is non-convex nd hence hrd to solve, semidefinte relxtion (SDR cn pproximte OPF s convex semidefinite progrm [4]. However, the stndrd OPF omits the hrmonics introduced into the network by power electronic-interfced DGs nd non-liner lods. Excessive hrmonics cn increse conductor heting loss, cuse trnsformer derting nd power qulity issues [5], [6]. The frequency coupling mtrix (FCM is populr pproch to modeling the coupling between n individul converters voltge nd current hrmonics [7], [8]. The vlue of the FCM cn be clculted either from piecewise liner differentil equtions or field mesurements [9], []. To dte, FCM-bsed pproches hve not been used to incorporte hrmonics in OPF. In this pper, we develop new frmework tht minimizes the cost of genertion while minting voltge hrmonic distortions beneth specified level. The proposed hrmonicconstrined optiml power flow (HC-OPF is novel in the following regrds: Inclusion of the FCM model for ech power electronic converter. The FCM of converter links its hrmonic current injection nd the voltge hrmonics t the point of common coupling in the network. Construction of network nd lod impednce models t hrmonic frequencies, nd the modeling of network voltge nd current hrmonics. Constrints limiting the network hrmonic voltge in terms of the totl hrmonic distortion (THD. TABLE I NOMENCLATURE N Set of nodes in the microgrid ε Set of distribution lines in the microgrid S Set of nodes feturing DG N n,φ Set of neighbouring nodes of n N tht lso contin phse φ P n Set of the phses t node n P mn Set of the phses of line (m, n n,φ Complex line-to-ground voltge t phse φ of node n I n,φ Complex current injection t phse φ of node n I mn,φ Complex current on phse φ of line (m, n P L,n,φ Active lod demnded t node n on phse φ Q L,n,φ Rective lod demnded t node n on phse φ P G,n,φ Active power supplied t node n on phse φ Q G,n,φ Rective power supplied t node n on phse φ P mn,φ Active power exiting phse φ of node m on line (m, n Q mn,φ Rective power exiting phse φ of node m on line (m, n Z mn Phse impednce mtrix of line (m, n Y s,mn Shunt dmittnce mtrix of line (m, n n, n oltge mgnitude lower (upper limit t nodes of n N P G,s,φ, P G,s,φ Minimum (Mximum ctive power supplied t phse φ of node s S Q G,s,φ, Q G,s,φ Minimum (Mximum rective power supplied t phse φ t node s S The new frmework is bsed on three-phse unblnced OPF [], which ccommodtes with both single-phse nd three-phse power electronic converters. We use SDR to obtin semidefinite relxtion, in which power flow nd totl hrmonic distortion constrints re convex. Numericl tests demonstrte tht the proposed frmework cn find OPF solutions stisfying network hrmonic distortion limit. This pper is orgnized s follows. Section II provides the bckground bout the three phse lod flow problem nd the FCM. In section III we propose frmework tht combines the FCM with three phse lod flow to nlyze hrmonics cross the network. Section I gives the hrmonicconstrined OPF problem nd its relxtion. Section presents the numericl result for solving the problem on -bus test network nd IEEE 3-bus feeder. Section I is the conclusion. The nottion used in this pper is listed in Tble I /7/$3. 27 IEEE 25

2 A. Three-Phse Lod Flow II. BACKGROUND Distribution networks re unblnced by nture. Consequently, ll three phses should be included in the model of lod flow problem. We denote the nodes N = {,,, N} nd edges ε N N connecting pirs of nodes. DGs re connected to the network t subset of nodes S. Ech node n N is connected to subset of phses P n {, b, c}. Let n,φ nd I n,φ C be the line-to-ground voltge nd injected current of phse φ P n t node n. Active nd rective lod t node n re specified for ech phse φ s P L,n,φ nd Q L,n,φ respectively. Similrly, P G,n,φ nd Q G,n,φ re the ctive nd rective power injection of the genertor if DG is connected to node n, i.e. n S. Ech edge mn ε represents distribution line or cble connecting set of phses P mn P m P n from node m to n. The line is modeled s π network nd chrcterized by its phse impednce nd shunt dmittnce mtrices Z mn, Y s,mn P mn P mn [2]. Becuse the system is unblnced, these mtrices re typiclly not digonl. Denoting the complex current flow from node m to n s I mn = {I mn,φ φ P mn }, Ohm s Lw is given by: I mn = Z mn([ m ] Pmn [ n ] Pmn, mn ε. ( The nottion [] Pmn denotes the sub-vector of tht only contins elements of the phses in the line mn, i.e. [ m ] Pmn, [ n ] Pmn C Pmn. In ddition, Kirchoff s Current Lw (KCL holds t ech node nd phse: I n,φ = m N n,φ [( 2 Y s,mn + Z mn [ n ] Pmn Z mn [ m ] Pmn ]φ, n N, φ P n, (2 where N n,φ is the set of neighbouring bus of node n tht lso include φ in its vilble phses, nd the subscript [ ] φ denotes the elements of the brcketed vector corresponding to phse φ. KCL nd Ohm s Lw for ll nodes in the system cn be written in mtrix form s: I = Y, (3 where Y is the system dmittnce mtrix nd, I C N n= Pn re system node voltge vector nd current injection vector: = [,,,b,,c, N,b, N,c ] T I = [I,, I,b, I,c, I N,b, I N,c ] T. We follow the convention tht P G,n,φ = Q G,n,φ = if n N \ S. The totl power injection into the network t node is given by: n,φ I n,φ = (P G,n,φ P L,n,φ +j (Q G,n,φ Q L,n,φ φ P n, n N. (4 Energy Source IDC DC = [DC, DC, DC 2,, DC K ] IDC = [IDC, IDC, IDC 2,, IDC K ] Fig.. DC DC AC IAC b c Grid AC = [bc, bc, bc 2,, bc K ] IAC = [Ibc, Ibc, Ibc 2,, Ibc K ] Power Electronics Interfced DG We designte one node in the network s the slck bus nd nd fix its voltge to one per unit. Often the Point of Common Coupling (PCC where the distribution network is connected to the trnsmission network is set to be the slck[ bus nd its voltge ] is set to be [,,,b,,c ] = e j, e 2π 3 j, e 2π 3 j per unit. For given lod {P L,n,φ, Q L,n,φ } nd genertor input {P G,s,φ, Q G,s,φ }, the voltge vlue t ech bus is determined by equtions (3 nd (4. B. Frequency Coupling Mtrix Trditionlly, converter hrmonics re modeled s fixed current source with hrmonic spectrum. However, this model fils to cpture the dditionl hrmonic currents generted by hrmonic voltge present t the PCC. From network perspective, the hrmonic current injected by n individul converter distorts the voltge profile cross the network, nd in turn impcts the hrmonic current injection of other converters. This coupling between hrmonic voltge nd current cnnot be cptured by fixed current source model. The frequency coupling mtrix of converter models its dmittnce t hrmonic frequencies nd the cross-coupling of different hrmonics. Fig. shows DG connected to the grid vi threephse converter. The FCM of converter linerly reltes its AC-side current with its AC-side voltge nd DC-side current, where ech voltge nd current vector contins elements of both the fundmentl nd hrmonic frequencies up to ny desired order. The FCM of voltge source converter (SC using Pulse Width Modultion (PWM cn be mthemticlly derived from its prmeters s in [9], or experimentlly mesured s in []. In [9], ech time-domin signl is decomposed into its Fourier series nd represented by its fundmentl nd hrmonic phsors, which exist in conjugte pirs for ech hrmonic order k nd k ccording to Euler s eqution: f(t = f + = f + n cos(nωt + b n sin(nωt n= c n e jnωt + c ne jnωt n= Since the hrmonic components of the converter DC-side current I dc re negligible [9], the FCM for converter cn 25

3 be pproximted s: In (5, I c = [ K,, K] T I c = [ I K c = [ K I c = [ FCM ] [ ] c. (5 I dc [ I K,, I K] T nd c = for single-phse converter.,, Ib K, ] I K c,, Ic K T nd ], K,, K T,, I K, I K b,, K b b, K c,, K c for three-phse converter. The size of the FCM depends on the mximum hrmonic order K, which is design prmeter of the FCM. Also I n = I n nd n = n. The FCM in the bove equtions cn be clculted when the prmeters of the converter re known [9]. Our objective is to use the FCM to link the hrmonics with fundmentl voltge nd power quntities. To fcilitte this, we mke the following modifictions to eqution (5. A converter s FCM over its operting rnge is pproximted s constnt, nd clculted when the converter is injecting its rted power into the network vi nominl PCC voltge. Since converters in power systems re intended to primrily generte fundmentl frequency voltge nd current, the impct of higher order hrmonics on the fundmentl frequency current is neglected. The FCMs re reformulted into three prts tht re ssocited with the fundmentl frequency voltge, hrmonic frequency voltge nd DC current respectively. If the DC-side voltge DC of SC is well regulted[3], nd the conversion losses of the converter re negligible, the DC-side current I DC cn be replced with its AC-side power injection by dividing the ssocited entries of FCM by DC. Since the hrmonic voltge nd current re regulted to be smller thn the fundmentl frequency components, the totl power consumed by the converter is pproximted by its power injection t the fundmentl frequency, P. With these modifictions, equtions (5 cn be re-formulted s: I h = [ ] [ ] K + [ ] K P P + [ ] [ ] K h h (6 I h I bh = [ ] K I ch b c b c h + [ ] K P P + [ ] K h h bh ch h bh ch, where the vectors with subscript h contin hrmonic components, nd for three-phse converter P is the sum of the (7 Fundmentls Hrmonics Fig. 2. I = Y P + jq = I * FCM, FCM2, I = Y I 2 = Y 2 2 : I K = Y K K, FCMs Frmework for Network Hrmonics Anlysis fundmentl frequency power injection of ll three phses: I h = [ I I 2 I K] T I h = [ I I 2 I K P = P + Pb + Pc. The modified hrmonic model (6 nd (7 cpture the dependence of the hrmonic current injection I h on the fundmentl frequency power flow solution ( nd P, s well s voltge hrmonics h present t the converter terminl. ] T III. NETWORK HARMONICS MODEL In this section, we propose frmework for nlyzing the hrmonics cross the network by combining the fundmentl power flow nlysis nd converter FCM models. In this nlysis, genertor power injection nd lod vlues re tken s input nd nodl voltge nd current injection t fundmentl nd hrmonic frequencies re solved s output. We lso provide numericl exmple on three-bus test system. A. Modeling Our model contins two prts: for ech hrmonic order new impednce network is constructed to model hrmonic current flow t tht frequency; nd for ech converter in the network, its FCM is employed to couple its hrmonic current injection with its terminl hrmonic voltge nd the fundmentl power flow solution. First, dditionl nottions nd vribles re defined. Let K be the mximum order hrmonic in the model. This is lso the dimension of the FCM specified for ech converter. Hrmonic voltge vector h C K N n= Pn is defined in terms of the vector n,φh C K Pn,φ t individul node n 252

4 nd phse φ s follows:,h,bh,ch h =. N,bh N,ch n,φ 2 n,φ n,φh = n,φ 3.. n,φ K Let S Φ {(n, φ n N, φ P n } be the set of node nd ssocited phse pirs where single-phse DGs re connected into the distribution network. Let S 3Φ be the sets of nodes where three-phse DGs re instlled in the distribution network. By definition ech node in S 3Φ contins ll three phses. The hrmonic current injection nd terminl voltge for ech DG in the network cn be modeled vi its FCM s follows: I G,n,Φh = [ ] [ ] K n,φ,nφ + [ ] K P,nΦ PG,n,φ n,φ + [ ] [ ] K n,φh h,nφ (n, φ S Φ (8 n,φh I G,n,h I G,n,bh = [ ] K,n I G,n,ch n, n,b n,c n, n,b + [ ] K h,n + [ ] K P,n PG,n,bc n,c n,h n,bh n,ch n,h n,bh n,ch n S 3Φ (9 Similr to ( nd (2, Ohm s Lw for ech line nd KCL t ech bus cn be used to form model t ech hrmonic frequency: I k mn = (Z k mn ( [ k m] Pmn [ k n] Pmn I k n,φ = m N n,φ (m, n E k = {, 2,, K} ( [ ( 2 Yk s,mn + (Z k mn [n] k Pmn (Z k mn [ k m] Pmn ] φ n N, φ P n, k = {, 2,, K} ( In the bove equtions, Z k mn nd Y k s,mn indictes the mtrix t hrmonic frequency k. The exct vlues of these mtrices re not vilble from the stndrd specifictions. We extrpolte these vlues by scling the rective components bsed on the new frequency: X L (ω = jωl nd X C (ω = jωc. oltge vrible k n C Pn nd m k C Pm re constructed by collecting the k-th order hrmonic voltge terms t buses n nd m. Equivlently, KCL of ll the nodes in the system t hrmonic order k cn be compctly written in mtrix form s: I k = Y k k, (2 where Y k is the network dmittnce mtrix t hrmonic frequency kω. k nd I k re the system node voltge vector nd current injection vector for hrmonic order k respectively. Unlike the cse for fundmentl frequency, no lods re specified t hrmonic frequencies, which elimintes the needs for dditionl power flow constrints similr to (4. Lod modelling t hrmonic frequencies hs been studied by mny reserchers [4] [6]. Two types of lod re considered: liner lod nd non-liner lod. Liner lod re pssitive nd do not generte dditionl hrmonics while non-liner lod similr to converters re hrmonic sources. The behvior of vrious types of liner lod (constnt power, constnt impednce nd constnt current t hrmonic frequencies re different. However, in ech cse n equivlent RLC circuit model is typiclly constructed bsed on the lod decomposition nd hrmonic frequency. In this study, we model liner lod of power P L,n,φ nd Q L,n,φ connected t bus n of phse φ s n RL impednce pproximted by: Z L,n,φ = 2 P L,n,φ +Q L,n,φ j = R L,n,φ + jx L,n,φ is the nominl operting voltge for the lod, i.e. = per unit. The lod impednce lso vries with frequency nd cn be pproximted s: Z k L,n,φ = R L,n,φ + jkx L,n,φ For ech hrmonic frequency order k, construct the digonl lod dmittnce mtrix YL k s follows: YL k = dig { (ZL,, k, (ZL,,b k,, (ZL,N,c k } Then Y k L k will be the vector of k-th order hrmonic current bsorbed by liner lod in the network: I k LL, = Y k L k (3 Denote the current vector I k NLL s the k-th order hrmonic current generted by the non-liner lod in the system, it cn be modeled in two wys. The simplest wy is considering it s constnt hrmonic current source I NLL,n,Φh = [ T INLL,n,φ, I2 NLL,n,φ NLL,n,φ],, IK. If the hrmonic current generted by the non-liner lod is vrible, its hrmonic chrcteristics is similr to tht of DG nd cn be modeled by its FCM. In tht cse the coupling between I NLL,n,Φh nd n,φh cn be described in the sme form s eqution (8 nd (9 for single-phse nd three-phse non-liner lod respectively. When combining current injected by DGs, liner lod nd liner lod, eqution (2 cn be re-written s: 253

5 Slck Bus P, Q,, I h, Ih Fig bc Network Admittnce: Y, Y, Y 2, Lod P, Q,, I h, Ih bc, Y K Network Hrmonics Anlysis Test System DG P2, Q2, 2, I2 2h, I2h Mximum oltge THD Power Injection of DG (kw or equivlently: I k G I k LL I k NLL = Y k k, (4 I k G I k NLL = ( Y k + YL k k (5 In summry, the power flow solution long with the converter FCMs provide boundry condition t the point where the converter is connected. They lso provide the coupling from the fundmentl to ll the hrmonics. In ddition, the current flow cross the network is modeled for every hrmonic frequency bsed on the line dmittnce nd lod impednce t tht frequency. Any non-liner lod is considered fixed hrmonic current source or modeled s converter with its own FCM. The structure described bove is illustrted in Fig. 2. The fundmentl frequency lod nd genertor power vribles P nd Q re considered s input to our problem nd fundmentl nd hrmonic voltge nd current vribles, I, h nd I h re solved s output. B. An exmple of hrmonic nlysis In this section, we pply our frmework on three-bus test system. Fig. 3 shows the schemtic of the network with vribles ssocited with ech bus mrked beside it. This is rdil network with three buses rted t 4.6 k. Ech bus contins ll three phses. The line impednces re ll set to Z mn = [ j.8,.56 + j.52,.58 + j.424;.56 + j.52, j.48,.54 + j.385;.58 + j.424,.54 + j.385,.34 + j.35]. Bus # is the slck bus. A lod of kw+8 kar is connected t ech phse of bus #. A DG is connected to bus #2 vi converter whose prmeters re listed in Tble II. The FCM of the converter is computed ccording to [9] with mximum hrmonic frequency order K =. To quntify the mount of hrmonic present in the network, we use the voltge totl hrmonic distortion (THD s metric. THD is typiclly defined s the sum of the squred hrmonic mgnitudes normlized by the mgnitude of the fundmentl frequency: k=,2,,k THD n,φ = n,φ k2 n,φ 2 n N, φ P n In this exmple we vry the mount of power generted by the DG from kw to kw, nd solve the three-phse Fig. 4. Network Hrmonics Anlysis Result: Mximum oltge THD lue vs. Incresing DG Power Input lod flow for ech cse using the Newton-Rphson method. The solution is then used s input for the network hrmonics nlysis. The mximum voltge THD in the network is plotted with respect to the DG genertion in Fig. 4. The plot illustrtes the impct of DG genertion on network hrmonics. As shown in the figure the mximum voltge THD increses when more power is generted by the power electronic-interfced DG. I. HARMONIC-CONSTRAINED OPTIMAL POWER FLOW In this section, we describe the hrmonic-constrined OPF problem nd its SDP relxtion. First the non-convex problem is formed by combining the network hrmonic model with regulr OPF problem; then the problem is relxed to semidefinite progrm. A. Problem Formultion For given lod {P L,n,φ, Q L,n,φ }, the OPF determines the power injection of ech genertor {P G,s,φ, Q G,s,φ } in order to minimize the operting cost of the system. Typicl choices of this function re the cost of genertion or the totl loss in the network. For this study, the cost of power genertion is used. In ddition to equtions (3 nd (4, which describe the physicl lws governing current nd power flow, sfety nd relibility constrints re imposed to restrict the voltge nd DG power injection vlues within the cceptble limits: n n,φ n φ P n, n N (6 P G,s,φ P G,s,φ P G,s,φ φ P s, s S (7 Q G,s,φ Q G,s,φ Q G,s,φ φ P s, s S. (8 Next, we dd hrmonic constrints to mitigte the dverse effects of hrmonics cross the network. From Section III, equtions (8, (9 nd (5 re included to model the hrmonic current flow cross the network. To restrict the hrmonic effect, the voltge totl hrmonic distortion is constrined by n upper limit α, or equivlently: n,φ k2 αn,φ 2 n N, φ P n (9 k=,2,,k In summry, the hrmonic-constrined OPF problem is s follows: 254

6 (P Minimize: C = c s P G,s,φ {P G, Q G,, h } s S φ P s Subject to: (3, (4, (6, (7, (8 (8, (9, (5, (9 This problem is non-convex becuse of constrints (4, (6 nd (9. In the next section, we will pply semidefinite relxtion to mke this problem convex. B. Semidefinite Relxtion Semidefinite relxtion is powerful technique for pproximting non-convex qudrticlly constrined problems like OPF [4], [7]. We now pply this relxtion to the hrmonicconstrined OPF. Consider rnk positive-semidefinite mtrix W =. Element W i,j is relted to s W i,j = i j. For ech node n nd phse φ denote the following dmittnce mtrix: where: e n,φ = Y n,φ = e n,φ e n,φy Φ n,φ = e n,φ e n,φ, [ ] T T P, T P,, T P, n eφ,t P, n T P,, n+ T P N nd e φ P n denotes the cnonicl bsis of R Pn. Then the non-convex constrints (4, (6 nd (9 cn be re-written s liner constrints of W s follows: T r(y n,φ W = (P G,n,φ P L,n,φ j (Q G,n,φ Q L,n,φ n 2 T r(φ n,φ W n 2 k=,2,,k φ P n, n N (2 φ P n, n N (2 k2 n,φ αφ n,φ W n N, φ P n (22 For the stndrd OPF problem, ll constrints cn be rewritten in terms of W. However, in the hrmonic ugmented problem the voltge vrible is present in FCM constrints (8 nd (9. The voltge vrible cn be pproximted from the vlues of the first coloum of W, due to the slck bus convention, = nd, n,φ = n,φ. With these modifictions, nd by relxing the rnk constrint of W, the semidefinite relxtion of hrmonic-constrined OPF P is: (P2 Minimize: C = c s P G,s,φ {P G, Q G, W,, h, IG} s S φ P s Subject to: (2, (2, (7, (8 (8, (9, (5, (22 = We W P2 is semidefinite progrm nd cn be effectively solved using commercil solvers.. NUMERICAL RESULT In this section, we solve the hrmonics-constrined OPF using CX, pckge for specifying nd solving convex progrms [8], [9], nd MOSEK solver [2] for the following two networks: -bus unblnced system s shown in Fig. 5; nd the IEEE 3-bus test feeder [2] s shown in Fig. 8. In both networks three-phse converters with prmeters listed in Tble II re included s distributed resources. The converter rel power lower nd upper limits re set to be from kw nd 3 kw, nd the rective power is regulted t kar. The FCM is clculted ccording to [9] with mximum hrmonic frequency order K =. The line nd lod dt for the IEEE 3-bus test feeder is dopted from [2] with the following modifictions to remove the disconnect switch nd other equipment not considered in the OPF formultion []: The trnsformer between bus 4 nd 5 is replced with 5 ft. overhed line of Config. #62 overhed line; The disconnect switch between bus 2 nd 8 is replced with ft. underground line of Config. #66; Only the spot lod re included, nd ll spot lod re modeled s P Q type; the voltge supporing cpcitors re not included. For the 4.6 k network of Fig. 5, the the lines re ll set to be of Config. #6 from [2], nd.2 mile in length. The lods t ech bus re of three-phse type with vlue connected to ech phse listed in Tble III. The cost of power supplied by the trnsmission network from Node is set to $4/MW, nd the cost of power supplied by the distributed genertions re set to be $/MW. The lower nd upper limit for ech bus voltge is set to be 95% nd 5% of the nominl level. For ech network, we solve sequence of the problem with incresing vlues of the α prmeter tht constrints the mximum llowble voltge THD t ech nodes. The simultion result for the -Bus network is shown in Fig. 6 nd 7. The result for the IEEE 3-Bus feeder is shown in Fig. 9 nd. Even the rnk constrint is relxed in the SDP, the solution W for both study cse ppers to hve single dominntly lrge eigenvlue. For the -bus network, when the THD llownce α = %, the two lrgest eigenvlues of solution W re λ = 58.7 nd λ 2 =.9. When α = 2%, the two lrgest eigenvlues of W re λ = nd λ 2 =.28. For the IEEE 3-bus feeder, when the THD llownce α = %, the two lrgest eigenvlues of solution W re λ = nd λ 2 =.7. When α = %, the two lrgest eigenvlue of W re λ = nd λ 2 =.9. The THD for ech voltge is plotted with respect to the mximum llowble vlue α for the two test networks in Fig. 6 nd 9. As the llownce for hrmonic increses, the THD increses cross the network for the -Bus feeder system but 255

7 TABLE II CONERTER PARAMETER LIST R =.2 Ω R b =.2 Ω R c =.2 Ω L =.2 H L b =.2 H L =.2 H C =.5 F Ω = 377 rd/sec TABLE III -BUS TEST NETOWRK LOAD LIST Lod Dt Bus P(kW Q(kAR oltge THD(% α Allownce(% Fig. 6. oltge THD vs. Incresing Allownce for -Bus Test System Ech curve is the THD of one bus voltge remins under the llowble α vlue. For the IEEE 3-Bus feeder, it ppers tht the trend is less uniform, where the THD for some buses decrese with higher THD llownce. This implies the voltge THD t some buses cn be reduced by llowing higher THD t other bus in the network. The totl cost of genertion with respect to the incresing hrmonics llownce α is shown s blue curve for both networks in Fig. 7 nd respectively. In ddition, the power injected by the trnsmission network t the slck bus nd the power generted by the distributed genertors in the distribution network is lso shown for ech α vlue. As shown by these result, the reduction in cost is due to the incresing mount of cheper DG genertion tht is enbled by llowing more hrmonics. Cost of Genertion Cost DG Utility α Allownce (% Fig. 7. Power Genertion Cost nd Portfolio vs. Incresing Allownce for -Bus Test System Power Injection by Utility nd DGs (MW I. CONCLUSION In this pper, we hve formulted hrmonic-constrined OPF. The frequency coupling mtrix is used to model the hrmonics chrcteristic of individul converters in the network. The network voltge nd current re modeled using b c bc bc bc bc b c c c bc bc b c 2 3 Fig. 8. IEEE 3-Bus Feeder Network Fig. 5. A -Bus Test Network 256

8 oltge THD(% α Allownce(% Fig. 9. oltge THD vs. Incresing Allownce for IEEE 3-Bus Feeder Ech curve is the THD of one bus voltge Cost of Genertion Cost DG Utility α Allownce (% Fig.. Power Genertion Cost nd Portfolio vs. Incresing Allownce for IEEE 3-Bus Feeder the modified system impednce t ech hrmonic frequency. The effect of hrmonics is constrined by limiting the totl hrmonic distortion of ech nodl voltge in the network. The non-convex optimiztion problem is relxed using semidefinite relxtion, nd implemented for two test networks. The numericl results show tht the method successfully cpture the hrmonics behvior in the network, nd the OPF solution is influenced by restricting the mximum hrmonics distortion in the network. REFERENCES [] M. Shhidehpour nd J. F. Clir, A functionl microgrid for enhncing relibility, sustinbility, nd energy efficiency, The Electricity Journl, vol. 25, no. 8, pp. 2 28, 22. [Online]. Avilble: [2] N. Htzirgyriou, H. Asno, R. Irvni, nd C. Mrny, Microgrids, IEEE Power nd Energy Mgzine, vol. 5, no. 4, pp , Jul. 27. [3] M. Huneult nd F. D. Glin, A survey of the optiml power flow literture, IEEE Trnsctions on Power Systems, vol. 6, no. 2, pp , My 99. [4] S. H. Low, Convex Relxtion of Optiml Power Flow Prt I: Formultions nd Equivlence, IEEE Trnsctions on Control of Network Systems, vol., no., pp. 5 27, Mr. 24. [Online]. Avilble: [5] H. Shrm, M. Rylnder, nd D. Dorr, Grid Impcts Due to Incresed Penetrtion of Newer Hrmonic Sources, IEEE Trnsctions on Industry Applictions, vol. 52, no., pp. 99 4, Jn. 26. [6] M. S. Rd, M. Kzerooni, M. J. Ghorbny, nd H. Mokhtri, Anlysis of the grid hrmonics nd their impcts on distribution trnsformers, in 22 IEEE Power nd Energy Conference t Illinois, Feb. 22, pp. 5. [7] Y. Sun, G. Zhng, W. Xu, nd J. G. Myordomo, A Hrmoniclly Coupled Admittnce Mtrix Model for AC/DC Converters, IEEE Trnsctions on Power Systems, vol. 22, no. 4, pp , Nov Power Injection by Utility nd DGs (MW [8] F. Yhyie nd P. W. Lehn, Using Frequency Coupling Mtrix Techniques for the Anlysis of Hrmonic Interctions, IEEE Trnsctions on Power Delivery, vol. 3, no., pp. 2 2, Feb. 26. [Online]. Avilble: [9] P. W. Lehn nd K. L. Lin, Frequency Coupling Mtrix of oltge-source Converter Derived From Piecewise Liner Differentil Equtions, IEEE Trnsctions on Power Delivery, vol. 22, no. 3, pp , Jul. 27. [Online]. Avilble: [] X. Zong, P. A. Gry, nd P. W. Lehn, New Metric Recommended for IEEE Stndrd 547 to Limit Hrmonics Injected Into Distorted Grids, IEEE Trnsctions on Power Delivery, vol. 3, no. 3, pp , Jun. 26. [Online]. Avilble: [] E. Dll Anese, Ho Zhu, nd G. B. Ginnkis, Distributed Optiml Power Flow for Smrt Microgrids, IEEE Trnsctions on Smrt Grid, vol. 4, no. 3, pp , Sep. 23. [Online]. Avilble: [2] W. H. Kersting, Distribution System Modeling nd Anlysis, Third Edition. Abingdon; Abingdon: CRC Press Tylor & Frncis Group [distributor, 22, oclc: [Online]. Avilble: [3] R. W. Erickson nd D. Mksimovic, Fundmentls of power electronics. New York: Springer Science + Business Medi, 24, oclc: [4] M. Au nd J. Milnovic, Estblishment of lod composition in ggregte hrmonic lod model t L buses bsed on field mesurements, vol. 25. IEEE, 25, pp. v2 29 v2 29. [Online]. Avilble: [5] R. Burch, G. Chng, C. Htzidoniu, M. Grdy, Y. Liu, M. Mrz, T. Ortmeyer, S. Rnde, P. Ribeiro, nd W. Xu, Impct of ggregte liner lod modeling on hrmonic nlysis: comprison of common prctice nd nlyticl models, IEEE Trnsctions on Power Delivery, vol. 8, no. 2, pp , Apr. 23. [Online]. Avilble: [6] K. R. Krishnnnd, J. Moirngthem, S. Bhndri, nd S. K. Pnd, Hrmonic lod modeling for smrt microgrids. IEEE, Aug. 25, pp [Online]. Avilble: [7] S. H. Low, Convex Relxtion of Optiml Power Flow Prt II: Exctness, IEEE Trnsctions on Control of Network Systems, vol., no. 2, pp , Jun. 24. [Online]. Avilble: [8] M. Grnt nd S. Boyd, CX: Mtlb softwre for disciplined convex progrmming, version 2., Mr. 24. [9], Grph implementtions for nonsmooth convex progrms, in Recent Advnces in Lerning nd Control, ser. Lecture Notes in Control nd Informtion Sciences,. Blondel, S. Boyd, nd H. Kimur, Eds. Springer-erlg Limited, 28, pp. 95, boyd/grph dcp.html. [2] MOSEK Aps, The MOSEK optimiztion toolbox for MATLAB mnul. ersion 7. (Revision 4., 25. [Online]. Avilble: [2] W. Kersting, Rdil distribution test feeders, vol. 2. IEEE, 2, pp [Online]. Avilble: 257

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