AN OPTIMIZATION APPROACH TO UNCERTAINTY PROPAGATION IN BOUNDARY LOAD FLOW

Size: px

Start display at page:

Download "AN OPTIMIZATION APPROACH TO UNCERTAINTY PROPAGATION IN BOUNDARY LOAD FLOW"

Michael Holmes
5 years ago
Views:

1 AN OPIMIZAION APPROACH O UNCERAINY PROPAGAION IN BOUNDARY LOAD FLOW Adrija. Sarić Northeaster Uiversity Bosto, Massahusetts asari@tf.kg.a.yu Brako Glišović Northeaster Uiversity Bosto, Massahusetts glisovi@ee.eu.edu Aleksadar M. Staković Northeaster Uiversity Bosto, Massahusetts astakov@ee.eu.edu Astrat Boudary load flow exemplifies a lass of power (load flow models that expliitly iorporates uertaities i system data. Its importae for operatio ad otrol of moder power systems is ireasig due to hages i ways the system is operated followig deregulatio ad due to the emergee of ew types of soures (e.g., distriuted geeratio. Modelig ased o iterval omputatios is a rigorous mathematial tool for worstase aalysis of uertai systems. his paper proposes a optimizatio-ased approah to miimizig uertaity spread i a oudary load flow model. he approah omies iterval omputatios with a liear programmig-ased heuristi that is effetive i avoidig the exessive oservativism ofte assoiated with diret iterval omputatios. We osider the ase of uertaity i oth measuremets (e.g., SCADA ad etwork parameters. he method is potetially appliale to large-sale power systems, ad we study its performae o a two ehmark examples: New Eglad/New York iteroetio with 68 odes ad stadard IEEE test system with 300 odes. Keywords Power flow aalysis, Uertaity, Optimizatio methods. INRODUCION Several types of uertaity affet the pratial feasiility of real-time power flow alulatios: topologial uertaities, 2 uertaity i etwork parameters, ad 3 possile errors i measuremets (e.g., SCADA. opologial uertaities are liked with the fidelity of sigalizatio ad uexpeted outages of power system elemets. hese are typially large disrepaies, ad i this paper we assume that they have ee hadled properly, so the topology of the etwork is exatly kow. O the other had, errors i etwork parameters ad i SCADA measuremets ted to e smaller i size, ut harder to detet i pratie. Aalogous prolems persist i power flow alulatios ased o foreasted data. his paper studies the effets of etwork parameter ad measuremet uertaities. he propagatio of these iput uertaities through power system alulatios is ofte very omplex, due for example to iversios of uertai matries like Jaoias. Differet approahes have ee suggested i the literature as ways to quatify the uertaity propagatio i power flow: a parametri ad sesitivity ased methods [], [2], proailisti methods [3], [4], iterval ad fuzzy alulatios [5], [6], ad d oudary load flow framework [7]. hese methods determie distriutio of power flow solutios ad/or oudig etwee extreme (pessimisti ad optimisti solutios. Utility experiee, as well as Mote Carlo simulatios (whih serve as stadard tool for verifiatio of various methods suggests that ritial situatios (with largest uertaity spread are very rare. he spread etwee extreme ad the most likely results uilds up quikly with the irease i prolem (system size. his, however, does ot dimiish the importae of uderstadig the impliatios of worst-ase searios, as reet experiees with lakouts learly show. his paper presets a power flow aalysis model that iludes uertaities i iput data (SCADA measuremets ad etwork parameters i a exat, rigorous mathematial way. he model is ased o iterval arithmeti aalysis omied with liear programmig, whih performs maximizatio of iterval widths to determie the worst ases. he origial optimizatio model with iterval values of ost futio ad of equality ostraits is trasformed ito a stadard liear programmig (LP formulatio with determiisti ost futio ad pairs of o-iterval iequality ostraits. Our methodology predits variatios i odal (magitude, agle ad lie (power flow quatities aused y model ad measuremet uertaities. While to some degree oservative, our results are muh tighter tha those otaied y diret appliatio of iterval omputatio methods. At the same time, our results are rigorous worst-ase preditios, ad as suh may prove useful i operatio ad otrol of power systems. While our urret implemetatio is iteded for off-lie aalyses, we feel that alteratios (e.g., with a state-of-the-art LP software like GAMS would make it apale of solvig systems with a few thousad variales i a olie eviromet. he rest of the paper is orgaized as follows i Setio 2 we formulate the prolem of iterval load flow i detail, followig solutio y optimizatio method desried i Setios 3 ad 4; i Setio 5 we apply proposed methods to two ehmark power systems (the first is New Eglad/New York 68 odes/86 lies, ad the seod is the IEEE ehmark with 300 odes/4 lies, followed y rief olusios i Setio 6. he Appedix otais equatios used to model the power system. 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page

2 2 BOUNDARY INERVAL LOAD FLOW (BILF FORMULAION he geeral form of power flow equatios with iterval variales is give y: gxy (, ; ( hs( X, Y, = S, (2 X =[ XX ; ] vetor of ukow iterval state variales (voltage agles i PQ ad PV odes, voltage magitudes i PQ odes, ad reative powers i PV odes, or X= [ V Q ] ; PQ+ PV PQ PV Y = [ Y; Y ] vetor of etwork iterval rah (lies ad trasformers admittaes, or Y = G+j B ; =[ ; ] vetor of SCADA measured (or foreasted iput iterval variales (real ijetios i PQ ad PV odes, reative ijetios i PQ odes, ad voltage magitudes i PV odes, or = [ P Q V ] ; PQ+ PV PQ PV S = [ S; S ] vetor of ukow output iterval MVA omplex variales (real ad reative power flows i rahes, or S = P+j Q ; g, h S ode ijetio ad MVA rah flow futios, respetively;, lower ad upper iterval ouds, respetively; supersript deotes omplex variale. he detailed versios of ( ad (2 are give i Appedix. he flow-hart of BILF is show i Figure. Itervals of etwork parameter Y Y Load flow futio g Figure : he flow-hart of BILF. Itervals of iput variales Brah MVA flow futio h S LP-ased solutio X S LP-ased solutio X Boudaries of state variales Boudaries of output variales 3 LP-BASED SOLUION OF INERVAL EQUAIONS A geeral -dimesioal system of liear iterval equatios a e formulated as: AX =, (3 with lower/upper oud ostraits: X X X, (4 mi max A = [ AA, ; ] X = [ XX ; ] ad = [; ]. System of equatios (3 (4 a e re-formulated as a liear programmig (LP prolem, with the aim to maximize the width of a iterval solutio [8]: ( X - X, (5 max j j sujet to iterval equality ostraits: Aij Xj = i ; i, 2,, =. (6 ad lower/upper iterval oud ostraits (miimal deviatio of itervals from middle poit solutio: Xj,mi Xj Xm Xj Xj,max ; j, 2,, =. (7 A iterval equality ostrait (6 a e reformulated as the followig pair of o-iterval ostraits: Aij Xj i ; i=, 2,,, (8 Aij = Aij ; X j = X j, iff X j 0; Aij = Aij ; X j = X j, iff X j 0. Aij Xj i ; i=, 2,,, (9 Aij = Aij ; X j = X j, iff X j 0; Aij = Aij ; X j = X j, iff X j 0. he algorithmi solutio of the LP prolem (5 (9 is eumerative: we eed to solve the model twie, oe with assumptio that X j 0, ad oe with the assumptio that X j 0, ad hoose the versio yieldig the optimum of iterest. he, for j =, 2,, it will require 2 solutios for eah optimum of iterest. his umer ireases expoetially, whih is ot aept- 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 2

3 ale for realisti power systems. For this reaso, we use the followig heuristi: alulate o-iterval (middlepoit solutio, ad the with alulated ad fixed sigs for X j evaluate their orrespodig itervals. Give our desire to alulate o-oservative, tight itervals widths, it is ritial that alulatio oth leftside matrix A ad right-side vetor e ased o realisti ouds. I the power system ase, their variatios are additioally ostraied. For example, i sumatrix PPQ+PV i (A all diagoal elemets are approximately equal to the egative sum of off-diagoal ele- PQ+PV mets. his meas that various etries should ot e treated as mutually idepedet, as ommoly assumed i iterval solvers [9], [0]. he additioal ostraits a e oveietly hadled with the row-ased LPapproah from (5 (9. 4 MAXIMIZED BILF MODEL he solutio of load flow equatios ( y stadard iterval aalysis tools (suh as iterval matrix iversio [9], differet otrators [0], diret appliatio of iterval omputatios [5], et. gives very wide iterval variales, rederig iosequetial for idustrial appliatios. Startig with liearized power flow equatios ( we a reformulate the prolem i the form: or: Ji X= Ji( X- Xm = - m =, (0a JiX=, (0 where J = [; JJ ] is iterval Jaoia matrix defied o asis eq. (A, = - m+ JiX m ad susript m deotes variales i the middle poit of iterval. he iterval Jaoia matrix J is alulated for itervals of etwork parameters Y ad middle poit values ( X m of ukow iterval variale vetor X = [ XX. ; ] For algorithm desried i Setio 3, formulatio (0 is preferred over stadard formulatio, as it makes the sig of iterval variales equal to the sig of middle poit variales. For (0 we a apply LPased formulatio for solutio of iterval equatios: max X j - X j k j X m sujet to iterval equality ostraits:, (a Jij Xj = i ; i, 2,, =, ( ad lower ad upper oudary ostraits for PQ-ode voltages ad PV-ode reative powers, respetively: V V ; ( VPQ,mi V PQ ; PQ PQ,max Q Q, (d QPV,mi Q PV ; PV PV,max where susripts mi ad max deote miimum ad maximum, respetively. he oeffiiet k j i (a aims to equalize otriutios to the riterio futio of various types of etries (voltage agles ad magitudes, et. i the state vetor X= [ PQ+ PV VPQ QPV ]. We foud empirially that the followig seletio works well i simulatios: = κ V = N NV = NPQ + NPV NPV κ, ( ad κ Q = N NQ = ( NPQ + NPV NPQ. As expeted, if the deoupled model (e.g., Stott-Alsa is used, the κ = κv = κ Q =. I a similar way, the iitial iterval rah flows S (2 are alulated (e.g., for real powers as: Pi = J X P, (2 where JP = [ JP; J P] is iterval rah flow sesitivity matrix (equatio (A4 i Appedix ad P = P- Pm+ JPiX m. he iterval sesitivity matrix J P is alulated for itervals of etwork parameters Y ad middle poit of itervals of system ode variales Xm = [ m Vm ]. However, rah flow iter- vals alulated y (2 ted to e very wide, ad sometimes eve otraditory with the odal power alae. After (2 has ee solved, the real power rah flows are additioally ostraied with the LP-ased approah: L s = ( P - P max, (3a sujet to iterval iequality (rah real power flow iremets with perturatio ad equality (ode real load alae ostraits, respetively: JP, j Xj P - Pm + J PXm ; =, 2,, Ls k P k, =.(3 + P = 0 ; k =, 2,, us π SL. (3 Ls = L- LSL ( L SL is set of rahes oeted with slak (SL ode(s; P k real ijetios (geeratio, loads, ompesatio, shut rah admittaes i the k-th ode; k umer of rahes oeted with k-th ode. 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 3

4 Fially, we must alulate power flows for set of rahes oeted with slak ode(s L SL, ijetio(s i slak ode(s SL ad system losses o a diret way (without optimizatio. he proposed formulatio is geeral for full (e.g., Newto-Raphso or deoupled (e.g., Stott-Alsa power flow methods. I our BILF model we assume that uertaities i etwork parameters ad i SCADA measuremets (i advae speified iputs for power flow alulatios are small eough, so that a sigle Jaoia a apture the ehavior of the power flow solutio. I partiular, we use a (arefully seleted see (0 liearizatio to evaluate effets of uertaities. If the magitude of uertaity is so large that this assumptio is violated, the a iterative proedure may offer a improved solutio; we did ot pursue this path further. 5 APPLICAION he maximized BILF model for alulatio of ode voltages ad power flows with uertaities i etwork parameters ad measuremet data set has ee implemeted i Matla eviromet ad tested o two stadard test examples. 5. New Eglad/New-York iteroetio with 68 odes he first test system omprises 68 uses, 6 geerators ad 86 lies. hree of the geerators (4, 5 ad 6 are very large equivalets of eighorig systems. he values for SCADA measuremet vetor (real loads i (PQ + PV-odes, reative loads i PQ-odes, real geeratios i PV-odes ad voltage magitudes i PV-odes ad etwork rah parameter vetor ( Y are uertai i rage = ±0.5 %, refereed to middle poit values (omial, o-pertured solutio. ale shows a sample of detailed results for iterval of state variales (ode voltage magitudes ad agles, while ale 2 shows the same for iterval of output variales (real ad reative rah power flows, otaied y maximized BILF model. he sums of maximum iterval widths for differet types of state variales are show i ale 3 for uertaities i rage = ±0.5 ad ± %. I additio, we ompare our optimizatio results with a simple, ut iformative heuristi a approximate lower oud for a ode voltage magitudes ad agles is otaied y osiderig maximal (PQ + PV- ode real ad PQ-ode reative ijetios, maximal etwork impedaes, ad miimal values of measured PV-ode voltage magitudes; the dual ase is used for otaiig the approximate upper oud (referred to as lower/upper ouds later. Our results show that maximized BILF model alulates worst-ases of iterval results for state variales ad other output variales quite effetively. I Figure 2, the results for = ±0.5 ad ±.0 % are ompared with those otaied y 20,000 radom (Mote Carlo uiformly distriuted, zero-mea variatio of iput uertai variales ad with heuristi lower/upper ouds approah. he otaied results suggest that: Maximized BILF model gives useful worst-ase ouds. 2 Radom (uiformly distriuted variatio of etwork parameters ad SCADA measuremets (with zero-mea teds to uderestimate the worst-ase variatios. 3 he variatio of iput uertaities has ee amplified several times i state ad other output variales (as summarized i ale 3. Distriutio desity Distriutio desity Radom approah (±.0% Lower/upper ouds (±.0% Radom approah (±0.5% Lower/upper ouds (±0.5% Maximized BILF (±0.5% Maximized BILF (±.0% Voltage magitude i ode 50 (p.u Voltage agle i ode 50 (degrees a. Radom approah (±.0% Lower/upper ouds (±.0% Radom approah (±0.5% Lower/upper ouds (±0.5% Maximized BILF (±0.5% Maximized BILF (±.0%. Figure 2: Proaility desities i hage of solutios (with respet to omial, o-pertured ase as a futio of radom perturatios i SCADA measuremets ad etwork rah parameters, ompared with maximized BILF solutio from ales ad 2 ad heuristi lower/upper ouds solutio for uertaty: ±0.5 % ad 2 ±.0 %: a. Voltage Magitude i Node 50.. Voltage Agle i Node 50. 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 4

Node/ ype Voltage magitude Voltage agle [degrees] Node/ ype Voltage magitude Voltage agle [degrees] /PQ [0.977;.004;.030] [6.02; 7.433; 8.876] 35/PQ [0.969;.00;.052] [2.388; 3.39; 3.93] 2/PQ [0.

5 Node/ ype Voltage magitude Voltage agle [degrees] Node/ ype Voltage magitude Voltage agle [degrees] /PQ [0.977;.004;.030] [6.02; 7.433; 8.876] 35/PQ [0.969;.00;.052] [2.388; 3.39; 3.93] 2/PQ [0.974; 0.986; 0.999] [6.907; 9.57; 2.240] 36/PQ [0.929; 0.987;.046] [-0.799;-0.532; ] 3/PQ [0.954; 0.967; 0.979] [3.53; 6.6; 8.789] 37/PQ [0.946; 0.977;.008] [-7.508; ; -6.59] 4/PQ [0.929;0.943; 0.958] [2.654; 4.906; 7.55] 38/PQ [0.980;.005;.030] [8.529; 9.695; 0.887] 5/PQ [0.932; 0.949; 0.965] [4.002; 5.988; 7.97] 39/PQ [0.944; 0.975;.007] [-9.690; ; ] 6/PQ [0.936; 0.952;0.968] [4.776; 6.75; 8.725] 40/PQ [.04;.030;.046] [3.444; 6.004; 8.649] 7/PQ [0.923; 0.94; 0.960] [2.349; 4.2; 6.07] 4/PQ [0.989; 0.999;.009] [39.979; ; 50.54] 8/PQ [0.920; 0.940; 0.960] [.804; 3.607; 5.4] 42/PQ [0.989; 0.999;.009] [32.784; 39.77;46.932] 9/PQ [0.942; 0.983;.025] [2.350; 3.33; 3.924] 43/PQ [0.945; 0.976;.007] [-8.645; ; ] 0/PQ [0.948; 0.96; 0.975] [7.448; 9.570;.687] 44/PQ [0.945; 0.976;.007] [-8.690;-7.88; ] /PQ [0.973;.003;.035] [7.850; 8.546; 9.259] 67/PV [0.995;.000;.005] [33.486; ; ] 34/PQ [0.963;.008;.053] [2.556; 3.75; 3.8] 68/PV [0.995;.000;.005] [43.586; ; ] ale : Node voltage magitudes ad agles otaied y maximized BILF model for New Eglad/New York system. Brah Real power flow Reative power flow Brah Real power flow Reative power flow -2 [-.085; ; ] [-0.032; 0.55; 0.384] [-.742;-.742;-.742] [-0.446; -0.29; -0.00] -30 [0.895;.302;.797] [-0.00; 0.293; 0.594] [-3.308;-3.308;-3.308] [0.060; 0.58; 0.268] 2-3 [3.68; 3.828; 3.989] [0.940; 0.940; 0.940] 29-6 [-7.909;-7.909;-7.909] [0.098; 0.489; 0.932] 2-25 [-2.95;-2.95;-2.95] [0.237; 0.552; ] 9-30 [-3.547;-3.547;-3.547] [-.042; -0.49; 0.34] 2-53 [-2.488;-2.488;-2.488] [-.059; ; ] 9-36 [2.472; 3.098; 3.839] [-0.623;-0.623;-0.623] 3-4 [0.696; 0.994;.33] [0.760; 0.893; 0.893] 9-36 [2.472; 3.098; 3.839] [-0.623;-0.623;-0.623] [.778; 2.50; 3.27] [0.704; 0.704; 0.704] [-39.80;-39.80; ] [-.960; ;.520] [-.25;-.25;-.25] -0.39; ; ] -27 [-0.033; 0.024; 0.02] [-0.25; -0.23; -0.07] ale 2: Brah real ad reative power flows otaied y maximized BILF model for New Eglad/New York system. No-pertured (omial ase Iput mi V mj Qmk uertaity ( iœn PQ + PV ( jœnpq [%] [rad] Pertured (uertai ase Iput i V j Q k uertaity ( iœn PQ + PV ( jœnpq [%] [rad] ± ± Relatives Iput i V j Q k uertaity mi V mj Qmk [%] [%] [%] [%] ± ± ale 3: System-wide ompariso of iterval widths of power flow solutios for variatios i iput uertaities, New Eglad/New York system. Our quatifiatio of the oservativism of the BILF methodology is illustrated i Figure 3, where we display the rage of voltage agle variatios i ode 50 otaied y extesive Mote Carlo simulatios (iludig o-zero mea parameter variatios ad ompares it with BILF preditios. From this ad umerous other examples we are led to elieve that BILF methodology is ot exessively oservative. Figure 3: Mote Carlo versus LP preditios, us 50, 0.5 % perturatio. Please ote that due to the very high order of the prolem (approximately 220 ukows i this example, it is diffiult to guaratee that the parameter spae 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 5

6 has ee adequately overed i Mote Carlo simulatios; thus, the atual oservativism may well e smaller tha the oe show i Figure Stadard IEEE test system with 300 odes he seod test system omprises 300 uses, 69 geerators ad 4 lies. he umer of state variales is 2NPQ + 2NPV = 598. he followig aalyses for differet iput uertaities are desried i ale 4: Case : Iput uertaities i etwork data ad i measuremets: PPQ+PV = QPQ = VPV = Y = ±0.5 %. Case 2: Measuremet uertaities oly: P = Q = V = ±0.5 %. PQ+PV PQ PV Figure 4: Sum of iterval widths of various quatities as a futio of the level of iput uertaity (Case i ale 4. Case No-pertured (omial ase V mi ( iœn PQ + PV ( jœnpq [rad] Pertured (uertai ase V Q i mj j ( iœn PQ + PV ( jœnpq [rad] Q mk k ale 4: System-wide ompariso of iterval widths of power flow solutios for variatios i iput uertaities, IEEE test system with 300 odes. As expeted, the iterval widths are smaller i the seod ase (whe etwork parameters are fixed. his is partiularly true for us voltage magitudes. O the other had, the variatios i reative ijetios i PV uses are largely determied y measuremet uertaities. he variatio of the (system-wide sum of iterval widths of various quatities (Case i ale 4 as a futio of the level of iput uertaity is displayed i Figure 4. 6 CONCLUSIONS he paper proposes a roust (iterval arithmeti ad liear programmig (LP-ased method for alulatio of worst-ases i power flow aalyses with etwork ad measuremet data uertaity. he omputatioal requiremets of the proposed method sustatially exeed those of stadard methods for power flow alulatio. However, the iformatio otaied from the algorithm is also muh riher, ad iludes sesitivities to hages i model parameters ad i quality of SCADA measuremets. I its urret state, our implemetatio is suitale for off-lie aalyses. We feel that a more streamlied implemetatio (e.g., with a stateof-the-art LP software like GAMS, ad with deoupled power flow ould e apale of solvig systems with a few thousads variales, thus potetially opeig ew appliatio domais. APPENDIX BASIC POWER SYSEM EQUAIONS [] he load flow futio g ( writte i detailed form for iremets i all variales is: È PPQ+PV PPQ+PV PPQ+PV È PPQ+PV Í È PQ+PV Í Í PQ+PV VPQ QPV Í Í Í QPQ QPQ Q Í Í PQ Q PQ = Í Í V Í Í PQ PQ+PV VPQ QPV Í Í Í Í Í VPV VPV VPV ÍÎ VPV ÍÎ QPV Í PQ+PV VPQ Q Î PV,(A where elemets of Jaoia matrix are defied for asi ijetio equatios (ad alulated for opertured, omial oditio: 2 Pi = Vi Gii + Vi ÈÎ Vk Gik ik + Bik ik kœα i ( os si ; (A2 5th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 6

7 2 Qj =- Vj Bjj + V È j Vk ( Gjk sijk Bjk os jk Î - kœα j, (A3 V i, i ( ik = i - k voltage ad phase agle i the i-th ode, respetively; G ik, B ik elemets of us admittae matrix Yus = Gus + jb us i positio ik, respetively; defiig, for example, the suseptae part as: Ï sh Bij, + Bi ; i= k; Ô B j ik =Ì Œα i Ô- Ó Bik, ; i π k; α i set of odes oeted with i-th ode; sh B i sum of shut suseptaes i the i-th ode; i=, 2,, ( PQ + PV ; j =, 2,, PQ. he rah power flow futio h (2 otais L- dimesioal vetors of real ( P ad reative power flows ( Q as parts: ÏÈ P P È P = diag ÌÍ Í ÓÎ V Î V ÏÈ Q Q È Q = diag ÌÍ Í ÓÎ V Î V ; (A4, (A5 where the derivatives (alulated for o-pertured, omial oditio are otaied from followig asi rah flow equatios: ( os si 2, ij, ij i i j, ij ij, ij ij P = G V - VV G + B ; (A6 2 ( sh ( si os Qij, Vi Bij, Bi VV i j Gij, ij Bij, ij = (A7 REFERENCES [] K. Almeida, F. P. Galiaa ad S. Soures, A geeral parametri optimal power flow, IEEE ras. o Power Systems, vol. 9, pp , Fe [2] P. R. Griik, D. Shirmohammadi, S. Hao ad C. L. omas, Optimal power flow sesitivity aalysis, IEEE ras. o Power Systems, vol. 5, pp , Aug [3] P. Zhag ad S.. Lee, Proailisti load flow omputatio usig the method of omied umulats ad Gram-Charlier expasio, IEEE ras. o Power Systems, vol. 9, pp , Fe [4] L. Xiaomig, C. Xiaohui, Y. Xiagge, X. ieyua ad L. Huagag, he algorithm of proailisti load flow retaiig oliearity, Pro Power System ehology Coferee - PowerCo 2002, vol. 4, Ot [5] Z. Wag ad F. L. Alvarado, Iterval arithmeti i power flow aalysis, IEEE ras. o Power Systems, vol. 7, pp , Aug [6] V. Mirada ad J. P. Saraiva, Fuzzy modelig of power system optimal load flow, IEEE ras. o Power Systems, vol. 7, pp , May 992. [7] A. Dimitrovski ad K. omsovi, Boudary load flow solutios, IEEE ras. o Power Systems, vol. 9, pp , Fe [8] J. W. Chiek ad K. Ramada, Liear programmig with iterval oeffiiets, Joural of the Operatioal Researh Soiety, vol. 5, pp , Fe [9] J. Roh, Systems of liear iterval equatios, Liear Algera ad its Appliatios, vol. 26, pp , 989. [0]L. Jauli, M. Kieffer, O. Didrit ad É. Walter, Applied iterval aalysis. Lodo, Berli, Heidelerg: Spriger-Verlag, 200. []A. J. Wood ad B. F. Wolleerg, Power Geeratio, Operatio, ad Cotrol. 2d ed., New York: Wiley, th PSCC, Liege, August 2005 Sessio 4, Paper 4, Page 7

Observer Design with Reduced Measurement Information

Observer Design with Reduced Measurement Information Observer Desig with Redued Measuremet Iformatio I pratie all the states aot be measured so that SVF aot be used Istead oly a redued set of measuremets give by y = x + Du p is available where y( R We assume