COMPUTATION AND VISUALIZATION OF REACHABLE DISTRIBUTION NETWORK SUBSTATION VOLTAGE

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1 24 th Interntionl Conferene on Eletriity Distriution Glsgow, June 2017 Pper 0615 COMPUTATION AND VISUALIZATION OF REACHABLE DISTRIBUTION NETWORK SUBSTATION VOLTAGE Mihel SANKUR Dniel ARNOLD Lun SCHECTOR Emm STEWART Lwrene Berkeley Ntionl Lortory USA Lwrene Berkeley Ntionl Lortory USA Lwrene Berkeley Ntionl Lortory USA Lwrene Berkeley Ntionl Lortory USA ABSTRACT In this work, we present two visuliztions of the ontrollility of susttion voltge in multiphse distriution network with distriuted energy resoures (DER). We utilize linerized unlned power flow model to formulte mthemtil progrm tht determines the voltge fesiility nd ost (in terms of DER ontrol effort) to move susttion voltges. The lgorithms re tested vi numeril simultions on n IEEE test feeder. INTRODUCTION The overll eletri grid network is mnged y mny disprte entities. For exmple, in Cliforni there is oth system opertor, mostly onerned out mintining the delite lne of lod nd genertion, nd utility opertors, mostly onerned out the reliility nd sfety of the lower network levels. Within utility there re lso the distriution opertors, nd within this mix we onsider diverse set of stkeholders. These stkeholders inlude the individul ustomers, nd now ggregtors of DER to the grid. DER ggregtors will in the ner future e le to id their resoures into new mrkets nd will e onsidered n integrl prt of the genertion lndspe. This in tndem with the modeling nd dt integrtion issues outlined erlier leds to onundrum of how to ount for these resoures nd distriution ehvior, while working within oth humn nd mhine limittions. Trditionlly trnsmission opertors onsider the distriution system to e lod, nd distriution opertions onsider trnsmission n infinite soure. With DER integrtion the former is no longer true, nd distriution my eome resoure to trnsmission. Numerous pprohes hve een proposed to integrte trnsmission nd distriution models into singulr pltform. In ertin ses, the rute fore pproh of using high performne omputing to simulte lrger nd lrger, nd multi-lyered, network models is pproprite. However, in the sene of rel time lrge nd multilyer network simultions, trnsmission opertors my require useful visuliztion of the performne of distriution system, nd the stte to whih they n ontrol it t the susttion us (or lower) [1 - [3. We seek to ddress this seond se providing useful nd urte dt to opertors on the ggregte performne nd ontrollility of the distriution system, in simple visuliztion environment. In this pper, we present two novel visuliztions s metris of distriution system ontrol performne. The first visuliztion shows the fesile three phse voltge mgnitude t the distriution level susttion s three dimensionl shpe. The seond visuliztion plots metri of DER ost to hieve desired susttion voltge profile. To the uthor s knowledge, no work in this re hs een pulished. This pper is orgnized s follows: We first present linerized unlned power flow model. Next, we inorporte the model into mthemtil progrm, nd develop lgorithms to visulize susttion voltge fesiility nd ost. Finlly, we disuss results from simultions performed on n IEEE test feeder nd provide onluding remrks. LINEAR THREE PHASE POWER FLOW MODEL In this setion, we outline liner three-phse power flow model. A derivtion of the model n e found in [4, whih re three-phse representtion of the LinDistFlow equtions [5. Consider rdil distriution feeder T = (N, E) where N is the set of nodes nd E is the set of edges. Edge (m, n) ε onnets nodes m nd n. A single susript of m or n denotes the node; doule susript denotes n edge (m, n). A single supersript denotes phse from the set {,, }. The omplex lod on phse t node m is given y: s m = s m (A PQ,m + A Z,m V m 2 (1) ) + w m where A PQ,m + A Z,m = 1. The omplex DER power dispth is w m = u m + jv m. The onservtion of omplex power t node m is given y: S m = s m + S n n:(m,n) ε where S m = [S m S m S m T = P m + jq m C 3 1 nd (2) CIRED /5

2 24 th Interntionl Conferene on Eletriity Distriution Glsgow, June 2017 Pper 0615 s m = [s m s m s m T C 3 1. The vetor Y m R 3 1 is omprised of the squred voltge mgnitudes of the voltge phsors t node m. The omplex slr V m represents the voltge phsor for phse phi t node m. If phse does not exist t node m, then Y m = V m = 0. Y m V m 2 Y m = [ Y m = [ V m 2 (3) Y m V m 2 The reltion etween voltge mgnitudes etween nodes m nd n is: (4) Y m Y n M P n N Q n with M nd N defined s: M 2r = [ r + 3x r 3x N = [ x x 2x 3r + 3r r 3x 2r r + 3x x x + 3r 2x 3r r + 3x r 3x x x where z ψ = r ψ ψ + jx etween phses nd ψ on edge (m,n). VISUALIZATION ALGORITHMS 2r 3r + 3r 2x (5) (6) is the omplex impedne In this setion, we present optimiztion progrms nd lgorithms designed to visulize the performne nd ontrollility of distriution network. Susripts for voltges (e.g. Y 650 ) refer to nodes of the test feeder studied in simultions shown in Figure 1. Algorithm for Finding Distriution Susttion Fesile Voltge Mgnitude Our first ojetive is to determine the set of fesile voltges t the network susttion tht re rehle given ville DER. We formulte onvex optimiztion progrm (7) tht seeks to minimize the voltge mgnitude of phse t the susttion, sujet to the liner three-phse power flow model, voltge onstrints, nd ounds on DER power. = min Y 650 Y,P,Q,u,v Y 650 sujet to: (1) (6) 0.95 V n 1.05 Y 650 = mx Y Y,P,Q,u,v 650 sujet to: (1) (6) 0.95 V n 1.05 (7) (8) Similrly, (8) seeks to mximize the sme voltge mgnitude, sujet to the sme onstrints. The lgorithm to find the fesile susttion voltge mgnitudes is outlined s follows: Algorithm 1: Determintion of Fesile Voltges 1. Solve 7 nd 8 for Y 650 nd Y 650, respetively 2. Grid over phse mgnitude from Y 650 to Y At eh phse grid point A k, solve 7 nd 8 for nd Y 650, respetively, with the dditionl Y onstrint tht Y 650 = A k 4. At the urrent phse grid point A k, grid over phse from Y 650 to Y At eh phse grid point B k, solve 7 nd 8 for Y 650 nd Y 650, respetively, with the dditionl onstrints tht Y 650 = A 2 2 k nd Y 650 = B k 6. After ll grid points re exhusted, find the onvex hull of mgnitude vlues. The onvex nture of 7 nd 8 ensure tht n optiml solution is otined for every instne. The DER mgnitude onstrints n lso e pproximted s set of liner equtions, thus mking (7) nd (8) liner progrms. Algorithm for Determining Susttion Voltge Mgnitude DER Cost Our seond ojetive is to determine the pproximte ost to hieve ertin voltge mgnitude profile t the susttion. We formulte third optimiztion progrm in whih the ost funtion is the sum of the mgnitudes of ll DER output. L( V n, V n, V n ) = min w n 2 sujet to: (1) (6) Y,P,Q,u,v n N,ε{,,} 0.95 V n 1.05 We formulte 9 with generi ojetive funtion of DER power dispth in lieu of n eonomi ost. The lgorithm to find the optiml ost of hieving susttion voltge mgnitude profile is outlined s follows: Algorithm 2: Optiml Voltge Profile DER Cost 1. Fix phse voltge mgnitude to desired vlue A D 2. Solve 7 nd 8 for Y 650 nd Y 650, respetively, with 2 the dditionl onstrint tht Y 650 = A D 3. Grid over phse from Y 650 to Y At eh phse grid point B k, solve 7 nd 8 for Y 650 nd Y 650, respetively, with the dditionl onstrints (9) CIRED /5

3 24 th Interntionl Conferene on Eletriity Distriution Glsgow, June 2017 Pper tht Y 650 = A D nd Y 650 = B k 5. At the urrent phse grid point B k, Grid over phse from Y 650 to Y At eh phse grid point C k, solve 9 with the 2 2 dditionl onstrints tht Y 650 = A D nd Y 650 = B k 2 nd Y 650 = C k This lgorithm is designed to rete ost visuliztion for desired phse voltge mgnitude, s we onsider olor mp within three dimensionl shpe to e poor presenttion of dt. Without loss of generlity, this n e done for desired voltge mgnitudes on other phses. Remrks on Algorithms The lgorithms presented grid over the fesile voltge mgnitude spe. Thus lrge numer of onvex optimiztion progrm lls must e ompleted, whih my result in high omputtion time for lrger networks. We re investigting wys to redue this omputtion effort. The fesile susttion voltge mgnitude n e mthemtilly defined using polytope projetions. The liner power flow model, (1) - (6), n e formulted s the liner mtrix eqution Ax. The voltge mgnitude onstrins of 0.95 V n 1.05 re liner. The DER dispth mgnitude ound n e pproximted y set of liner equtions, therefore oth sets of onstrins from (7) nd (8) n e inorported into Ax. SIMULATION RESULTS We perform simultions using modified version of the IEEE 13 node test feeder. Feeder topology n e seen in Figure 1. The voltge regultor etween nodes 650 nd 632 nd the trnsformer etween nodes 633 nd 634 re omitted. The swith etween nodes 671 nd 692 is ssumed losed. susttion nd trnsmission line re onneted y the trnsformer speified in the test feeder doumenttion. We ssume the ross phse impednes of the trnsformer re one fifth of the sme phse impedne. Spot lods re multiplied y ftor of to produe phse voltge violtions in zero DER dispth. For simpliity ll lods, hve prmeters A PQ,m = 0.85 nd A Z,m = 0.15 for simpliity. DERs re pled t nodes 632, 671, 692, nd 684, on ll existing phses t eh node. For simpliity, DERs re ssumed to e single phse four qudrnt opertion ple inverters. DER pprent power dispth is limited to p.u. for ll inverters. We ssume tht the voltges on the trnsmission line re 1.0 p.u. for phses,, nd. Solving power flow, ssuming ll DER power to e zero, susttion voltge mgnitudes for phses,, nd re , , nd , respetively. Simultions results of the first visuliztion, showing rehle susttion voltge mgnitude, re given in Figure 2 - Figure 5. Figure 2 shows the rehle voltge mgnitude s three-dimensionl polygon. Figure 3 shows this shpe projeted onto the phse plne, Figure 4 shows this shpe projeted onto the phse plne, nd Figure 5 shows the rehle voltge projeted onto the phse plne. Figure 2 shows tht voltge mgnitude profile of Y 650 = [1 1 1 T is hievle, showing tht with proper DER ontrol, lning of voltges with limited loss of power is effetive with this strtegy. It n lerly e seen in Figure 3, Figure 4, nd Figure 5 tht the phse mximum voltge mgnitude is more onstrined thn tht of nd. This my indite higher loding onditions or less DER vilility or penetrtion. Figure 1: IEEE 13 node test feeder topology Figure 2: Fesile voltge mgnitude t susttion We ssume the susttion, node 650, is onneted to trnsmission line whih ts s n infinite soure. The CIRED /5

4 24 th Interntionl Conferene on Eletriity Distriution Glsgow, June 2017 Pper 0615 Plots suh s Figure 2 - Figure 9 my enefit system opertors in understnding the stte of the network. Additionlly, this informtion my inform opertors in their DER ontrol deisions, suh how ostly it is to lne voltge mgnitude t the susttion. Figure 3: Fesile susttion voltge mgnitude plotted on AB phse plne Figure 6: DER p.u. power ost plot for hievle susttion voltge mgnitude with phse voltge mgnitude of 0.98 Figure 4: Fesile susttion voltge mgnitude plotted on AC phse plne Figure 7: DER p.u. power ost plot for hievle susttion voltge mgnitude with phse voltge mgnitude of 0.99 Figure 5: Fesile susttion voltge mgnitude plotted on BC phse plne Figure 6 through Figure 9 show the p.u. DER power ost for hieving susttion voltge mgnitude profiles for four fixed phse mgnitudes. The differenes in rehle voltge in the phse plne is lerly visile, s phse mgnitude is inresed. The DER power dispth ost is lso lerly shown for different desired susttion voltges s is the different osts to lne susttion voltge t different mgnitudes. Figure 8: DER p.u. power ost plot for hievle susttion voltge mgnitude with phse voltge mgnitude of 1.00 CIRED /5

5 24 th Interntionl Conferene on Eletriity Distriution Glsgow, June 2017 Pper 0615 to investigte temporl rehility with hnging lod onditions nd energy storge. REFERENCES [1 S. Chtzivsileidis, M. Bonvini, J. Mtnz, 2014, "Cyer-Physil Modeling of Distriuted Resoures for Distriution System Opertors", Proeedings of the IEEE vol. 104, [2 U.S. Deprtment of Energy, 2015, Grid Moderniztion Multi-Yer Progrm Pln, U.S. Deprtment of Energy, Wshington, DC, USA, Figure 9: DER p.u. power ost plot for hievle susttion voltge mgnitude with phse voltge mgnitude of 1.01 CONCLUSIONS In this work, we present mthemtil progrms nd n lgorithm to ompute the rehle voltge mgnitude of unlned rdil distriution network susttion. We lso present n lgorithm to lulte the optiml ost of hieving desired susttion voltge profile. Simultion results re presented s visul tools for system nd utility opertors. We pln to further develop this work in three key res. The first is to develop more relisti single nd three phse inverter models. The seond is to improve omputtion time y exploring polytope representtions of network mode, nd grid model redution. The third is [3 J. Dily, et l., 2014, Frmework for Network Co- Simultion t the 3rd Workshop on Next-Genertion Anlytis for the Future Power Grid, Pifi Northwest Ntionl Lortory, Rihlnd, Wshington, USA, 4-5. [4 L. Gn, 2014, "Convex Relxtions nd liner Approximtion for Optiml power Flow in Multiphse Rdil Networks", Proeedings IEEE Power Systems Computtion Conferene (PSCC). [5 M. Brn, F.F. Wu, 1989, Optiml Sizing of Cpitors Pled on Rdil Distriution System, IEEE Trnstions on Power Delivery, vol. 4, CIRED /5