VOLTAGE stability issue remains a major concern in

Transcription

1 Impacts of Mert Order Based Dspatch on Transfer Capablty and Statc Voltage Stablty Cuong P. guyen, Student Member, IEEE, and Alexander J. Flueck, Member, IEEE Abstract In ths paper, the goal s to nvestgate the mpacts of generaton mert order based dspatch on transfer capablty and statc voltage stablty. The concept of generaton mert order can be based on tradtonal costs or market bds. The new mert order feature has been ncorporated nto the exstng Contnuaton Method Trace Tool (CMTRACE). More specfcally, the wellknown contnuaton power flow problem s reformulated to take nto account the mert order effect of dspatchng generators on the statc P-V curve, gven a power system operatng pont, a load demand forecast and a table of generaton MW blocks and prces. The modelng dffcultes n dealng wth sequentally ordered generaton block dspatches, such as exact transton pont search and soluton tme, are effcently solved. The numercal results on two case studes of a 3493-bus system show that the mert order of dspatchng generaton has sgnfcant mpacts on the P-V curve and the smulaton tme. Index Terms Contnuaton power flow, dstance to collapse, statc bfurcaton, voltage stablty I. ITRODUCTIO VOLTAGE stablty ssue remans a major concern n the operaton and control of power systems nowadays. The lmt of exstng transmsson resources, the hgh growth rate of power demand and supply, together wth the ncrease n electrc power transactons are pushng power systems to operate closer to ther stablty boundary. Contnuaton power flow s an effcent method to address the quas-statc voltage stablty problem, whch has been well studed for the past 7 years [],[4],[5],[6],[7]. Typcally n ths study, the generaton/demand level of a set of partcpatng buses s consdered to vary slowly, hence quas-statcally, from a base case level to some pre-defned level. Snce all partcpatng generators are parameterzed to ncrease or decrease ther power output at the same tme, ths way of dspatchng generaton s called smultaneous dspatch. Recently n [0], an mproved CPFLOW tool consderng multple load varaton and generaton re-dspatch patterns was developed but the generators are stll dspatched n a smlar manner. The exstng CMTRACE tool [2], on whch the Mert Order Based Dspatch (MOBD) algorthm s developed, s a comprehensve tool for tracng power system steady-state behavor due to load and generaton varatons. CMTRACE s a contnuaton power flow tool wth some enhanced features such as effcent step sze selecton and an embedded algorthm to search for breakng ponts due to the lmted reactve capablty of generators. A common drawback n CMTRACE C. P. guyen (emal: cnguyen7@t.edu) and A. J. Flueck (emal: flueck@t.edu) are wth the Department of Electrcal and Computer Engneerng, Illnos Insttute of Technology, Chcago, IL, 6066 USA. as well as other avalable tools s that they can only generate P-V curves gven a power system operatng pont and a sngle drecton of load growth and generaton dspatch. As such, those tools are better suted for generalzed voltage studes. On the other hand, n specalzed voltage studes, generaton blocks wth dfferent prces are dspatched based on the mert order table. Ths table s created smply by sortng all generaton blocks n non-decreasng order of prce. The cheaper blocks wll be dspatched before the more expensve ones. Conversely when decreasng the generaton, the more expensve generators should respond before the less expensve ones. Ths paper s organzed as follows: Secton II dscusses the formulaton of the MOBD problem. In Secton III, some background on the contnuaton method s brefly ntroduced. The soluton algorthm of MOBD s presented n Secton IV. umercal results are gven n Secton V. Fnally, conclusons and contrbutons are summarzed n Secton VI. A. Smultaneous Dspatch II. PROBLEM FORMULATIO Let F (x, λ) represent the parameterzed power flow equatons. The smultaneous generaton dspatch [] has the followng formulaton: [ ] P (x) P nj F (x, λ) = Q(x) Q nj λb () where vector [ x = ( V, θ) represents ] the state varables and ˆP vector b = nj P nj ˆQ nj Q nj, whch parameterzes the total scheduled transfer, represents the change n bus njectons due to varatons n generaton dspatch and load demand. ˆP nj, ˆQ nj are target real, reactve power njecton levels and λ s the (controllng) parameter subject to varaton. The Lmt Pont (LP), ether a saddle-node bfurcaton pont or a breakng pont, corresponds to the maxmum value of the physcal parameter λ n the drecton gven by the parameterzaton vector b. In ths smultaneous parameterzaton scheme all actve generators are parameterzed n a sngle vector b, hence they are dspatched at the same tme. However, ths doesn t make sense when each generator has generaton blocks wth dfferent prces. B. Sequental Dspatch The sequental generaton dspatch algorthm consders generaton prce as a key factor n power transfer studes. Two

2 2 Pgen2 Lmt pont of sequental dspatch Gen bus 52 Sequental Dspatch Pgen Qgen ETP Target generaton schedule Base generaton Lmt pont of smultaneous dsptach MW, MVAr Qmax Pgen 0 Fg.. Real power generaton pattern of smultaneous dspatch and sequental dspatch Fg. 2. Output of generator 52 n MOBD common types of transfer are typcally of nterest n studyng statc voltage collapse: Generaton to Load Transfer: one set of pckup generators, called the source, ncreases ts generaton and another set of loads, called the snk, ncreases ts consumpton (Fg. 3). Generaton to Generaton Transfer: one set of generators, consdered as the source, ncreases ts output and another set of generators, called the snk, decreases ts output (Fg. 4). In ether case, the mert order of each generator or load partcpatng n the transfer must be determned frst based on the bddng prces. Then, a seres of bus njecton vectors b,..., b m can be bult up correspondngly. Whle Fg. 3 shows smultaneous schedulng of load, t s possble to schedule load sequentally based on prce. The sequental dspatch power flow equaton s reformulated based on (), as follows: [ ] P (x) P nj F (x, λ,..., λ m ) = Q(x) Q nj (2) λ b... λ m b m where λ s the (controllng) parameter assocated wth vector b, =,..., m. Intally, all λ = 0. The physcal parameter λ [0, ] assocated wth generators partcpatng n b s nterpreted as follows: f λ = 0 then b s nactve (generators parameterzed n b are nactve) else f λ 0 then b s actve (generators parameterzed n b are actve) else f λ = then b becomes an addtonal constant bus njecton vector n (2). ow when b s fully dspatched, the next step s to actvate b +. ote that each b represents a subtransfer, the sum of all subtransfers yelds the total scheduled transfer as seen n the smultaneous dspatch case. Fg. s an llustratve example of smultaneous dspatch versus sequental dspatch where two generators, expensve generator and cheap generator 2, are scaled up to serve the ncreased demand. There are two subtransfers n ths case, generator whch s parameterzed n b s actve before generator 2 n b 2. C. Exact Transton Pont (ETP) Search In mert order based dspatch often tmes there s more than one subtransfer. The Exact Transton Pont s the pont where b s fully dspatched,.e. λ =, so there can be as many as m transton ponts. The strategy to solve for ETP s as follows: ) Detect f the predcted parameter ˆλ >, f true then goto 2) else go to 5) 2) Rescale predctor step sze such that ˆλ = 3) Correct the predcted ETP. If the corrected λ s wthn the desred tolerance of.0, then go to 4) else go to 2) 4) Done wth b 5) Correct the predctor pont as normal It can easly be seen n Fg. that at the ETP, unt 2 generates power at ts target level whle unt stll produces power at the base case level. D. Generator Reactve Power Modelng Reactve power generaton of actve generators s a dependent varable but t can be modeled va parameterzng maxmum reactve power capablty Qmax [3]. Fg. 2 ndcates that generator 52 reactve power output for some transfer study keeps ncreasng n order to support ts termnal voltage as the system becomes more stressed. Eventually t hts Qmax at 83 MVAr before the generator tself actually becomes actve to decrease the MW and MVAr outputs down to zero. III. BACKGROUD O COTIUATIO METHOD Ths secton presents background materal on contnuaton methods for tracng soluton branches of general nonlnear algebrac systems.

3 3 Generaton to Load Transfer ) Input the nformaton of the source: generator bus, bddng block and prce 2) Input the nformaton of the snk: load bus, real/reactve power change 3) Rank the partcpatng generators based on the prce. Generaton blocks wth the same prce are grouped together 4) Set =,.e. the cheapest generaton 5) Parameterze load buses β = Pgen / P gen Pload = β P load; Q load = β Q load ote: Q gen s parameterzed as n subsecton II.D 6) Form[ vector b ] P b = gen Pload Q gen Q load 7) Increment, f > m then go to step (9) 8) Go to step (5) 9) Stop Fg. 3. Generaton to Load transfer parameterzaton strategy A. Basc Elements of Contnuaton Method The theory of contnuaton methods has been studed extensvely and has ts roots n algebrac topology and dfferental topology [8]. A contnuaton method has four basc elements: ) parameterzaton, 2) predctor, 3) step length control and 4) corrector. B. Local Parameterzaton Local parameterzaton s chosen due to ts effcency n capturng transton ponts. Gven the prevous soluton pont (x k, λ k ) and the prevous step s tangent vector (ẋ k, λ k ), local parameterzaton tes the ewton corrector solutons trajectory to a contnuaton parameter that s determned at each contnuaton step by the tangent vector. If λ k s chosen for example, the parameterzaton equaton s λ k+ = λ k + λ (3) where λ s the step sze along the soluton path. λ can be the desred MW step sze, as n ths paper, that the user wshes to take along the soluton trajectory toward the next soluton pont. C. Corrector The ewton corrector solves the augmented set of equatons wth actve b and an ntal guess (ˆx k+ k+, ˆλ ) provded by the predctor, as follows: [ ] P (x k+ ) P nj Q(x k+ ) Q nj b... b (4) λ k+ b = 0 = λ k + λ λ k+ In the curved part of the soluton curve t s possble that some x j other than λ s selected as a contnuaton parameter. Generaton to Generaton Transfer ) Input the nformaton of the source: generator bus, bddng block and prce 2) Input the nformaton of the snk: generator bus, generaton block and prce 3) Rank the source generators n the non-decreasng order of prce and the snk generators n the nonncreasng order of prce 4) Set = source =,.e. cheapest generaton block(s) snk =,.e. most expensve generaton block(s) 5) Parameterze generator buses γ = P snk snkgen / P f γ < then Pgen = P snk snkgen + γ P source P source = ( γ) P source snk = snk + else f γ > Pgen = γ P snk snkgen + P source P snk snkgen = ( γ ) P snk snkgen source = source + else Pgen = P snk snkgen + P source snk = snk + source = source + end ote: Q gen s parameterzed as n subsecton II.D 6) Form vector b ] [ P b = gen Q gen 7) Check lmt of snk, source, f any volaton then go to step (9) 8) Increment, go to step (5) 9) Stop Fg. 4. Then x k+ j (4). Generaton to Generaton transfer parameterzaton strategy = x k j + x j wll substtute as the last equaton of IV. SOLUTIO ALGORITHM Ths paper uses a predctor-corrector contnuaton method to trace soluton paths. The tangent method serves as a predctor. The ewton method s chosen as the corrector. The soluton algorthm proposed n ths paper s llustrated n Fg. 5. In tracng the soluton curve n whch there are multple tny subtransfers, solvng for each subtransfer costs at least one contnuaton step. If an exact contnuaton trajectory wth every sngle transton pont s not of nterest, then lumpng small subtransfers would save a consderable amount of smulaton tme. One opton, newly added to CMTRACE, s the user can specfy the mnmum subtransfer level. If the th subtransfer parameterzed n b s less than the threshold, then vector b wll be combned wth b +. The process s repeated untl the total combned subtransfer s greater than or equal to the mnmum subtransfer value.

4 4 TABLE I CASE STUDIES O 3493 BUS SSTEM start Case Transfer Total scheduled transfer (MW) Area 2(Gen) Area (Gen) Area 4(Gen) Area (Load) form b(),...,b(m) solve base case power flow TABLE II SIMULTAEOUS VS SEQUETIAL DISPATCH RESULTS (CASE ) Dspatch DtoC Dto V mn 242 Tme to pass Total tme type (MW) (MW) LP (m:ss) (m:ss) Sm :28 0:38 Seq :37 0:5 Dstance to Collapse pont or equvalently total feasble transfer Dstance to the crossng pont of the soluton trajectory and V mn 242 actvate b() (= ntally) compute tangent vector pass LP = + V. UMERICAL STUDIES Ths secton presents the effectveness of ths new algorthm va two smulatons, see Table I, wth the followng power system: Buses: 3493; Generators: 83; Loads: 2565; Fxed shunts: 957; Swtchable shunts: 25; Lnes: 5953; Fxed transformers: 654; Fxed phase shfter: ; ULTC transformers: 82; ULTC phase shfters: 8; Areas: 30. The test bed used n ths paper s a SU 420R Worksever wth four Processors and 4 GB of Ram, runnng the Solars operatng system. solve for LP select contnuaton parameter predct pass ETP A. Case Study : Generaton to Generaton Transfer Ths case smulates a total scheduled transfer of MW from nexpensve generaton n Area 9 to expensve generaton n Area. The output of all generators n Area 9 s scaled up by 36.0% whle that n Area s decreased by 00%. A step sze of 200 MW and no mnmum subtransfer level are selected n ths study. The voltage magntude trajectores of generator 242 n the mportng Area are presented n Fg. 6 and Fg. 8. In smultaneous dspatch, all MW and MVAr generaton n the snk area s decreased at the same tme regardless of prce. The voltage at bus 242 decreases gradually, see Fg. 6. Ths trend s also observed n other buses n Area. In the sequental case however, snce generator 242 s ranked as the second most expensve generator after generator 243 n Area, t s parameterzed n the second subtransfer. The voltage magntude shown n Fg. 8 drops quckly to 72 pu when generator 242 decreases ts output to zero. Ths s ntutve snce generator 242 s the only one n the mportng area partcpatng n the second subtransfer. The sold horzontal lnes n these fgures represent the mnmum allowed voltage level, whch s 6 pu, at the measured bus. Table II ndcates that the maxmum transfers or dstances to collapse for smultaneous and sequental methods are 2958 MW and 3346 MW respectvely. The mnmum voltage constrant further shortens the maxmum transfers to MW and MW respectvely. The sequental method clearly shows more negatve mpact from the voltage pont of vew for ths example. Fg. 5. correct Done wth b() = m stop reset state back to ETP Flowchart of mert order based dspatch algorthm In terms of power output, the expensve generator 242 s brought out of servce n the sequental dspatch (see Fg. 9) whle t stll onlne wth an output of about 45 MW just before encounterng the lmt pont n the smultaneous case (see Fg. 7). Tme statstcs of the two dspatches are depcted n Table II. Obvously, the sequental process s more tme consumng than the smultaneous counterpart, an ncrease of 34% n total CPU tme. Ths s expected, as smultaneous dspatch has only one sngle transfer whereas sequental dspatch has sequental subtransfers. Interestngly, t happens that sequental dspatch has a longer dstance to collapse compared to smultaneous

5 5..05 Bus 242 Smultaneous Dspatch pred corr..05 Bus 242 Sequental Dspatch pred corr tran 5 5 Fg. 6. Voltage magntude at bus 242 n smultaneous dspatch Fg. 8. Voltage magntude at bus 242 n sequental dspatch Gen bus 242 Smultaneous Dspatch Pgen Qgen Gen bus 242 Sequental Dspatch Pgen Qgen MW, MVAr MW, MVAr Fg. 7. Output of generator 242 n smultaneous dspatch Fg. 9. Output of generator 242 n sequental dspatch dspatch, about 3% ncrease. B. Case Study 2: Generaton to Load Transfer Ths case smulates a total scheduled transfer of MW from nexpensve generaton n Area 4 to the ncreased load n Area. The output of all generators n Area 4 s scaled up by 25.3% and the load growth rate n Area s assumed to be 94.2%. Fg. 0 shows that there are many transton ponts,.e. many tny subtransfers, n the range of 500 MW to 500 MW. Therefore the total soluton tme wthout combnng any subtransfers n sequental dspatch s 56% longer than smultaneous dspatch, see Table III. When the mnmum subtransfer threshold s ncreased, the soluton tme decreases, see Table IV. However there sn t any tme mprovement between the 00 MW and 200 MW mnmum lmts because the total contnuaton steps reman the same. Moreover, for mnmum subtransfers larger than 400 MW, the smulaton tme ncreases a bt as the step sze s stll fxed at 50 MW. Therefore, further ncreasng the threshold beyond 400 MW wll not yeld any tme mpovement. Fg. depcts the P-V curve at bus 6 wth the mnmum transfer set at 200 MW. VI. COCLUSIOS Investgatng transfer capablty and statc voltage stablty s of great mportance n power system operaton. However usng the exstng statc voltage stablty analyss tools to obtan P-V curves n the case of mert order based dspatch can be very tedous, even mpossble. Wth the support of the enhanced CMTRACE tool ths procedure can be effcently automated. The numercal studes prove that MOBD has sgnfcant mpacts on system voltage profle, dstance to collapse as well as smulaton tme compared to the smultaneous dspatch. The contrbutons of ths paper are:

6 6. Bus 6 - Sequental Dspatch tran. Bus 6 Sequental Dspatch tran Tny subtransfers 5 Fg. 0. Voltage magntude at bus 6 n sequental dspatch wthout mn subtransfer constrant TABLE III SIMULTAEOUS VS SEQUETIAL DISPATCH RESULTS (CASE 2, O MIIMUM SUBTRASFER COSTRAIT, STEP SIZE = 50 MW) Dspatch DtoC Dto V mn 242 Tme to pass Total tme type (MW) (MW) LP (m:ss) (m:ss) Sm :29 0:32 Seq :46 0:50 TABLE IV SEQUETIAL DISPATCH WITH DIFFERET MIIMUM SUBTRASFER COSTRAITS (CASE 2, STEP SIZE = 50 MW) Mn subtransfer (MW) Tme to pass LP (m:ss) 0:46 0:37 0:37 0:27 0:29 Total tme (m:ss) 0:50 0:44 0:44 0:30 0:3 Reformulatng the exstng contnuaton power flow problem to take nto account the mert order based dspatch of generaton or load bddng blocks. Handlng effcently the transton ponts between any two subtransfers. The newly mproved CMTRACE tool fnds ts applcaton n near term power transacton studes as well as long term plannng n modern power systems n whch power transfers are ncreasng both n magntude and multtude. VII. REFERECES [] H. D. Chang, A. J. Flueck, K. S. Shah and. Balu, CPFLOW: A Practcal Tool for Tracng Power System Steady-state Statonary Behavor Due to Load and Generaton Varatons, IEEE Trans. on Power Systems, vol. 0, no. 2, pp , May 995. [2] A. J. Flueck, J. R. Dondet, A ew Contnuaton Power Flow Tool for Investgatng the onlnear Effects of Transmsson Branch Parameter Varatons, IEEE Trans. on Power Systems, vol. 5, no., pp , Feb [3] A. J. Flueck, H. D. Chang, K. S. Shah, Investgatng The Installed Real Power Transfer Capablty of a Large Scale Power System Under a Proposed Multarea Interchange Schedule Usng CPFLOW, IEEE Trans. on Power Systems, vol., no. 2, pp , May 996. [4] V. A. Ajjarapu and C. Chrsty, The Contnuaton Power Flow: A Tool for Steady State Voltage Stablty Analyss, IEEE Trans. on Power Systems, vol. 7, no., pp , Feb Fg.. Voltage magntude at bus 6 n sequental dspatch wth mn subtransfer constrant of 200 MW [5] C. A. Cañzares and F. L. Alvarado, Pont of Collapse and Contnuaton Methods for Large AC/DC System, IEEE Trans. on Power Systems, vol. 8, no., pp. -8, Feb [6] S. Greene, I. Dobson, F. L. Alvarado, Senstvty of the Loadng Margn to Voltage Collapse wth respect to Arbtrary Parameters, IEEE Trans. on Power Systems, vol. 2, no., pp , Feb [7] A. Sode-ome,. Mthulananthan, K.. Lee, A maxmum loadng margn method for statc voltage stablty n power systems, IEEE Trans. on Power Systems, vol. 2, no. 2, pp , May [8] R. Seydel, From Equlbrum to Chaos: Practcal Bfurcaton and Stablty Analyss, ew ork: Elsever, 988. [9] H. B. Keller, umercal Soluton of Bfurcaton and onlnear Egenvalue Problems, Applcatons of Bfurcaton Theory, ew ork, Academc Press, pp , 977. [0] S. H. L, H. D. Chang, Contnuaton Power Flow wth Multple Load Varaton and Generaton Re-Dspatch Patterns, IEEE Power Engneerng Socety General Meetng [] A. J. Flueck, Advances n umercal Analyss of onlnear Dynamcal Systems and The Applcaton to Transfer Capablty of Power Systems, Ph.D. thess, Cornell Unversty, Ithaca,, Aug VIII. BIOGRAPHIES Cuong P. guyen receved the B.S. degree (2003) n power system engneerng from Hano Unversty of Technology, Vetnam and the M.S. degree (2005) n electrcal engneerng from Southern Illnos Unversty at Carbondale, USA. He s currently workng toward the Ph.D. degree wth Prof. A. J. Flueck n the Department of Electrcal and Computer Engneerng, Illnos Insttute of Technology, Chcago. Hs research nterests nclude modelng of power system components, contnuaton methods for parameterzed nonlnear systems, and contngency screenng. Alexander J. Flueck receved the B.S. degree (99), the M.Eng. degree (992) and the Ph.D. degree (996) n electrcal engneerng from Cornell Unversty. He s currently an Assocate Professor at Illnos Insttute of Technology n Chcago. Hs research nterests nclude transfer capablty of large-scale electrc power systems, contngency screenng of multple branch or generator outages wth respect to voltage collapse, contnuaton methods for parameterzed nonlnear systems, transent stablty and parallel smulaton of power systems va message-passng technques on dstrbuted-memory computer clusters.