Attribute-Preserving Optimal Network Reductions

Transcription

1 Attrbute-Preservng Optmal Network Reductons Dan Tylavsky, Yuja Zhu, Shrut Rao Arzona State Unversty wth Wllam Schulze, Ray Zmmerman, Dck Shuler, Jubo Yan Cornell Unversty Bao Mao Rensselaer Polytechnc Unversty Dan Shawhan Resources for the Future CERTS R&M Cornell Aug

2 Context Objectve: Develop reduced network equvalencng procedures that preserve certan attrbutes. Reduced network equvalents have been used: Speed executon of problems Sze problems to avalable computaton resources. E4ST Applcaton Dynamc smulatons, etc. Tradtonal network reductons only preserve certan structures Ward reducton Preserves nodal voltages, and branch flows for base case only under lnearty assumpton. The mproved Ward (e.g. PV-Ward or extended Ward) Gves better performance on matchng reactve support. REI Reactve support better modeled. Hot start method whch can preserve base case power flow solutons (bus voltage, branch flow, etc.). Inaccurate when operatng condton changes. Objectve: Targeted network reductons. Benefts: Allow more accurate smulatons of electrc power networks.

3 Scope Developng attrbute-preservng network equvalents. Topology Branch values Generator placement Load models Reduced dc equvalents that preserve branch flow values. Fndng optmal branch reactances for ac-to-dc model converson Bus aggregaton Ward-type reducton Generalzed optmzaton formulaton for dc equvalents Ths past cycle looked at: Generalzed optmzaton-based Ward-type reducton formulaton appled large dc systems. Appled optmal generator placement n reductons of large dc systems. Reductons whch preserve bus voltage values through VC n ac systems. Network reducton toolbox upgrade. Transmsson expanson corrdors.

4 Outlne Optmzaton based Ward reducton (OP-Ward) dc systems (Yuja Zhu) Optmal generator placement on ERCOT, WECC and EI (Yuja) Network reducton toolbox upgrade (Yuja) Transmsson expanson corrdors (Team) Inverse functon equvalents central dea lnear case (Shrut Rao) Applcaton of nverse functon equvalents to (nonlnear) ac systems for bus voltage preservaton (Shrut)

5 OP-Ward reducton Last year: We showed that the Ward and OP-Ward gave dentcal results for 6-bus system. Tested the method on a 9-bus and IEEE 118- bus systems wth mxed results. Identfed a fundamental ssue causng a rank defcency problem n some cases.

6 OP-Ward reducton Idea: Mnmze the branch flow errors n the retaned model porton. Formulate the problem as an unconstraned optmzaton problem: Objectve: mn Λ1 y b (1) y 2 where: Λ 1 PTDF PTDF = PTDF r full r full r full C C C T T T dag( c dag( c dag ( c 1 2 ) ) N 1 ) C s the branch-bus ncdence matrx and cc s the th column n C. bb ffff s the th column n the full model branch susceptance matrx. N-1 s number of retaned buses. b b b = b f 1 f 2 f, N 1

7 Test cases: OP-Ward reducton Case # Test system # of retaned buses # of external buses 1 9-bus IEEE 118-bus IEEE 118-bus IEEE 118-bus Error metrc: Max branch reactance error %. Large errors (>50%) occurred. The Λ 1 matrx s rank defcent.

8 OP-Ward reducton Star-mesh converson. A D A D E B C B C Λ 1 PTDF PTDF = PTDF r full r full r full C C C T T T dag( c dag( c dag ( c 1 2 ) ) N 1 ) rr PPPPPPPP ffffffff s the porton of the PTDF matrx of the full model correspondng to retaned branches n the reduced model. In the star-mesh converson, no branch s preserved thus the Λ 1 matrx n (1) can not be created.

9 OP-Ward reducton Curng the rank defcency problem. Theory: Add enough pseudo branches to full network to make the Λ 1 matrx of full rank. Remove pseudo branches from the reduced model.

10 OP-Ward reducton Pror to the reducton process add three pseudo branches (red lnes n the fgures below) parallel to the three equvalent branches. A D A D E B C B C The Λ 1 matrx based on the three pseudo branches s of full rank.

11 OP-Ward reducton Test results Case # # of rank ncrease Error (%) Problem solved? E-13 Y E-14 Y E-13 Y E-13 Y All cases yelded neglgble errors.

12 OP-Ward reducton Heurstc rules for mnmzng number of pseudo branches as follows*. 1. Every bus must have ether a pseudo or retaned branch ncdent on t. 2. The number of pseudo branches added n a network must be no less than the maxmum number of equvalent lnes ncdent on any bus. Reduced the number of pseudo branches from 338 to 21 n Test Case #4 whle retanng a small maxmum error (6.3E-11% v. 3.1E-13%). *Assumng radal buses and loops were properly handled.

13 Generator Placement Last year: Tested three generator placement methods on small systems: Shortest Electrcal Dstance (SED) based method: place the external generator at a retaned generator bus whch s closest to ts orgnal locaton n terms of electrcal dstance. Optmzaton based Generator Placement (OGP) method: place the external generators by solvng an mxed nteger lnear programmng problem whose objectve s mnmzng generaton cost whle retanng congeston status wthn the system.

14 Generator Placement Mnmum Shft Factor Change (Mn-SF) based method: place the external generator at the retaned generator bus whch has the most smlar shft factor to the orgnal external generator bus. In the test results we showed last year on small systems, we found that the Mn-SF method s the most robust and more accurate than the OGP method. We tested the Mn-SF and the SED methods on ERCOT, WECC and EI*. * Tests on EI system n progress.

15 Generator Placement Two metrcs were used Average LMP error Error n Average Energy Cost (AEC=Total $/MWh) Error Calculaton Average LMP error ($/MWh) Err LMP = 1 N ( ) LMP LMP Average energy cost (AEC) error ($/MWh) Err = AEC AEC AEC Where: s the ndex of retaned buses NN s the number of retaned buses full full reduced reduced

16 Generator Placement Baselne LMP and AEC values taken as the dc OPF results for the unreduced model. Compared wth dc OPF results for the reduced model wth generators place by: SED method Mn-SF method

17 Generator Placement Loadng scenaros generated for large systems by unformly scalng the loads across the system. Only the scenaros n whch the unreduced model yelded feasble dc OPF results were consdered.

18 Generator Placement Statstcs of the three nterconnectons # of buses n less aggressve reduced model # of bus n full model # of branches n full model # of generators Reducton percentage (%) Full model statstcs ERCOT WECC EI Reduced models statstcs # of buses n more aggressve reduced model Reducton percentage (%) # of branches n nonaggressve reduced model Reducton percentage (%) # of branches n aggressve reduced model Reducton percentage (%) ERCOT WECC

19 LMP error ($/MWh) Generator Placement Results of WECC (6851 bus system less aggressve 50%) Comparson of average LMP error Load scale factor (%) SED MnSF 0.06 Comparson of AEC error AEC error ($/MWh) Load scale factor (%) SED MnSF

20 LMP error ($/MWh) Generator Placement Results of ERCOT (3025 bus system less aggressve 50%) Comparson of average LMP error Load scale factor (%) SED MnSF AEC error ($/MWh) Comparson of AEC error Load scale factor (%) SED MnSF

21 Generator Placement Next more aggressve reductons on ERCOT, WECC and EI systems were tested where the systems were reduced to about one tenth of ther orgnal sze.

22 5 Generator Placement Results of WECC (2000 bus system 10%) ) Comparson of average LMP error LMP error ($/MWh) AEC error ($/MWh) Load scale factor (%) SED MnSF Comparson of AEC error Load scale factor (%) SED MnSF

23 LMP error ($/MWh) Generator Placement Results of ERCOT (424 bus system 10%) ) Comparson of average LMP error Load scale factor (%) SED MnSF Comparson of AEC error AEC error ($/MWh) Load scale factor (%) SED MnSF

24 Generator Placement Concluson: The two placement methods yelded smlar results. Concluson: Systems cannot be reduced ndefntely wthout consequences to accuracy.

25 Network Reducton Toolbox Last year: Sparsty technque was not suffcently used n the beta release and resulted n two drawbacks. Hgh memory demand. Relatvely long executon tme. Major updates: Rewrote the algorthm of the partal LU factorzaton so that the reduced model can be constructed durng the factorzaton process. Improved symbolc processng of sparsty pattern of the reduced model.

26 Network Reducton Toolbox Effcency before and after the update Case # of buses Calculaton Tme Unreduced Reduced Before Update After Update ERCOT mn 25 sec WECC mn 20 sec WECC hour 2.4 mn EI Out of Memory 1.3 hour Computaton Envronment: Run on Matlab 2014a. CPU Intel Core I7 3770, 3.4 GHz. 16 GB DDR 3 memory.

27 Network Reducton Toolbox Network Reducton Toolbox Dstrbuton The toolbox s dstrbuted along wth MATPOWER 5.1. The toolbox s also avalable on the E4ST webste. The toolbox s currently used by the Ben Hobbs group to do a study on transmsson expanson n WECC system.

28 Transmsson Expanson Assstng Cornell group n dentfyng transmsson expanson projects for comparson. Proposed three canddate transmsson lnes #1 Quebec New York (Champlan-Hudson Power Express) Bll Schulze #2 Southern Calforna Arzona #3 Mantoba Mnnesota

29 Transmsson Expanson Canddate #1: Champlan Hudson Power Express Ths project s a 1000 MW HVDC lne. Currently t s beng studed by the E4ST research group. Connectng Hertel substaton n La Prare wth New York Cty.

30 Transmsson Expanson Canddate #2: Southern Calforna - Arzona Facts: Wthn natonal congeston corrdor defned by the 2006 and 2009 Natonal Electrc Transmsson Congeston Study.

31 Transmsson Expanson Canddate #2: Southern Calforna Arzona Southern Calforna Edson (SCE) n Apr proposed 500 kv ac transmsson lne project (DPV2) the Devers-Palo Verde No. 2. The project was approved on Calforna sde by Calforna Publc Utltes Commsson (CPUC).

32 Transmsson Expanson Canddate #2: Southern Calforna Arzona On Arzona sde, the project was dened by Arzona Corporaton Commsson (ACC) n June The major concern s that the ACC beleved that the proposed transmsson lne wll lower the rate on Calforna sde however rase the rate n Arzona.

33 Transmsson Expanson Canddate #2: Southern Calforna Arzona Current status: The constructon of Calforna porton s completed. Calforna porton of DPV2 project

34 Transmsson Expanson Canddate #3: Mantoba Mnnesota Facts: The Great Northern Transmsson Lne (between Mantoba Hydro and Mnnesota Power) a 500 kv ac transmsson lne between provnce of Mantoba n Canada and Blackberry Sub. n Itasca County.

35 Transmsson Expanson Canddate #3: Mantoba Mnnesota Status The project was proposed n 2012 and s currently under federal and state revew. On June 30, 2015 the Mnnesota Publc Utltes Commsson (PUC) ssued a wrtten order for a Certfcate of Need for the Great Northern Transmsson Lne. More capablty to delver clean power. Hydro power to be delvered from Mantoba. Wnd power to be delvered from Mnnesota. Improve system relablty.

36 Inverse Functon Network Reducton Tradtonal (e.g., Ward-type and REI) reducton methods: Lnearze nonlnear (PQ) loads at external buses: Impedances Current Injectons Dstrbute lnear loads va reducton rules. Convert lnear to equvalent nonlnear (PQ) loads at base case loadng. Do not handle nonlnear (PQ) loads accurately because of complexty of nonlnear reducton. Examne whether retanng a nonlnear model n the reducton process was mportant for ac bus voltage preservaton.

37 Inverse Functon Network Reducton Consder a three-bus network as shown below. Ward reducton: Convert PQ load at bus 1 to current njectons Elmnate bus 1 usng Ward reducton method splt I 1 between buses 0 and 2. Convert current njectons to equvalent S=PjQ loads at buses 0 and 2. Accuracy Test: Scale loads unformly to the voltage collapse pont.

38 Inverse Functon Network Reducton Statc voltage collapse pont: Unreduced Network: VC=7.63 Base_Load Inverse Functon Approach: VC=7.61 Ward Reducton: VC=7.17 Bus 2 voltage plot. Voltage magntude on bus Full model Inverse functon Ward reducton Voltage (pu) Load scale factor

39 Inverse Functon Network Reducton Bus 2 voltage error plot Dfference of bus 2 voltage Inverse functon Ward Reducton Error (pu) Load scale factor

40 Inverse Functon Network Reducton Lnear case. Ax=b b(a,x) (A=admttance matrx, x=voltage, b=current njecton.) Inverse functon: x(a, b) (Voltage as a functon of current njectons.) Network Reducton: A(x,b) (Admttance matrx as a functon of loads.) Ax = b ( I D) x = b x = Dx b Holomorphcally embed the recurson relaton wth parameter. ( ) = Dx( ) b x Represent x() as a power seres n. 2 N x( ) = x[0] x[1] x[2] x[ N T T ] Equate correspondng powers of on both sdes of the equaton. x[0] = b x[1] = Dx[0] x[ N ] = Dx[ T N T 1]

41 Inverse Functon Network Reducton Use Padé approxmate to represent x() as ratonal approxmant. N T N T x x x x x ] [ [2] [1] [0] ) ( 2 = Equate correspondng powers of on both sdes of the equaton. 1] [ ] [ [0] [1] [0] = = = T N T Dx N x Dx x b x Last step s to get: Trcker for a nonlnear problem. ), ( ) ( ) ( ) ( ] [ [2] [1] [0] ) ( D b x b a b b b b a a a a O M L x x x x x M M L L M L M L = = = = ), ( b x D Holomorphc Seres Method (HSM)

42 Voltage-Preservng Network Equvalents usng the HSM In the past, we have developed network equvalents that preserve branch flows for dc network power flow formulatons. Preserve the bus voltage magntude and angle n ac network reductons usng ths holomorphc seres method (HSM). Ths s of partcular nterest for studes nvolvng voltage stablty.

43 Holomorphc Seres Method (HSM) Use HSM to obtan the voltages as a functon of the current and/or complex power njectons,.e., fnd the nverse functon. The power balance eq. (PBE) for a PQ bus can be wrtten as: S N * Y kvk = * k = 1 V To use the HSM, frst the above equaton can be holomorphcally embedded as follows: N * S Y kvk ( ) = * * k = 1 V ( ) Wth ths embeddng, scales complex load, S. Next V() s represented as ts Maclaurn seres expressed 2 N as: V ( ) = V[0] V[1] V[2] V[ N T T ] wth N T number of terms n the seres.

44 Holomorphc Seres Method (HSM) The nverse voltage functon on the RHS of the holomorphcally embedded equaton can be expressed as an nverse seres W() where Thus the PBE s represented as: The soluton at =0 (germ) and s obtaned by equatng the constant terms: Subsequent seres terms obtaned through a recurrence relaton obtaned by equatng lke powers of on both sdes. ) ( 1 ) ( V W = ) ] [ [2] [1] [0] ( ) ] [ [2] [1] [0] ( * 2 * * * * 1 2 T T N T N k N T k k k k k N W W W W S N V V V V Y = = 0 [0] 1 = = N k Y k V k 1] [ ] [ * * 1 = = n W S n V Y N k k k

45 Holomorphc Seres Method (HSM) Smlarly the equatons for PV buses can be embedded as follows: N P jq ( ) * YkVk ( ) = * sp 2 * * V ( ) V ( ) = V V ( ) k = 1 where P s the known power njected nto the bus and V sp s the specfed voltage for the PV bus. The embedded equaton for the slack bus s gven by: sp V slack () = V The terms of the voltage seres for the PV buses can be obtaned n a smlar manner as that for PQ buses: N k = 1 = Y k PW V * k [ n] j [ n 1] * * ( Q [ n] W [0] Q [0] W [ n] ) j n 1 Q [ k] W * [ n k]

46 Holomorphc Seres Method (HSM) The voltage magntude constrant ultmately leads to: V [0] V = ( V * [ n] V [1] V * [ n [ n] V * [0] 1]... V [ n 1] V [1]) Combnng the slack, PQ and PV bus equatons, the PBE s of a power system can be solved recursvely to obtan the terms of the voltage power seres. Challenge: The voltage power seres may not always converge. Padé approxmants are used to obtan a converged soluton, f t exsts. *

47 Padé approxmants Stahl s Padé convergence theory- For an analytc functon wth fnte sngulartes, the sequence of near-dagonal Padé approxmant converges to the functon... [1] Padé approxmants are ratonal approxmants to the gven power seres gven by: [1] H. Stahl, On the Convergence of Generalzed Padé Approxmants, Constructve Approxmaton, 1989, vol. 5, pp ) ( ) ( ) ( ] [ [2] [1] [0] ) ( b a b b b b a a a a O M L V V V V V M M L L M L M L = = =

48 Estmatng Voltage Collapse Pont (VCP) from Roots of the Padé Approxmant Need to know the lmts over whch the Pade approxmant s vald. VCP estmate s the smallest real zero of the numerator or denomnator polynomals of the Padé approxmants of any bus voltage 1.[2] 1. A formulaton such that the soluton at dfferent values of represents the soluton at dfferent loadng levels of the system, must be used. [2] George A. Baker, Jr., Peter Graves-Morrs, Padé approxmants, Cambrdge Unversty Press,

49 Inverse Functon Network Equvalents Once the voltage seres for a gven power flow problem are obtaned, can develop reduced radal networks whose branch admttances are represented as a power seres. Let the reduced system nclude the slack bus and any two buses from a large system. (Note that the topology s arbtrary.) 49

50 Inverse Functon Network Equvalents To fnd branch admttances as functons of, Y k (), for the reduced network, use the voltage seres of the retaned buses n the PBEs. N * S Y k ( ) Vk ( ) = * * V ( ) k = 1 N k = 1 = S The admttance and voltage varables n the above equaton are expanded nto power seres as: 2 NT 2 (( Y [0] Y [1] Y [2] Y [ N ] ))( V [0] V [1] V [2] V * k ( W * k [0] W * k [1] W * 2 [2] k T W * [ N T ] N T k ) k k k [ N T ] N T ) Equate the same powers of on both sdes of the equaton, to fnd the Y seres. Ths reduced network more fathfully preserves the voltages. 50

51 Results of the HSM generated network equvalent Tested the approach for systems wth PQ buses only. For the IEEE 14 bus system, the four PV buses (2,3,6 and 8) were converted to PQ buses and a reduced radal network was generated wth the slack bus connected to bus 2, 2 to 3 and 3 to 4. 51

52 Results of the HSM generated network equvalent Plot: log of voltage error v. load scalng factor. Voltage collapse pont scalng factor for the orgnal network =

53 Lunchtme