Sensitivity Analysis in Electric Power Systems

Sensitivity analysis in electric power systems involves examining the impact of parameter changes on system behavior. This lecture discusses linearized sensitivity analysis, matrix notation, injection shift factors (ISFs), and more, providing insights into system reliability and security. Concepts like ISF, GSF, and LSF are explored in relation to node injections and line flows. Understanding these analytical techniques is crucial for optimizing power system operation and planning.

• Sensitivity Analysis
• Electric Power Systems
• ISF
• Matrix Notation
• Injection Shift Factors

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1. ECEN 615 Methods of Electric Power Systems Analysis Lecture 16: Sensitivity Analysis Prof. Tom Overbye Dept. of Electrical and Computer Engineering Texas A&M University overbye@tamu.edu

2. Announcements Exam average was 85.7, with a high of 100 Read Chapter 7 (the term reliability is now used instead of security) Homework 4 is assigned today, due on Thursday Nov 1 2

3. Linearized Sensitivity Analysis By using the approximations from the fast decoupled power flow we can get sensitivity values that are independent of the current state. That is, by using the B and B matrices For line flow we can approximate ( ( ) ) 2 ( ( ) ) = = V V cos = = i i j i j i j i j h s g V b V V sin i j ( ) ( ) ( ), , By using the FDPF appxomations i j ( ) ( ( ) ) = = = = i j h s b i j ( ) ( ) , , X 3

4. Linearized Sensitivity Analysis Also, for each line h h b a 0 V and so, h b b a a T A B h x L 1 = = = = L 1 0 h V 0 0 4

5. Sensitivity Analysis: Recall the Matrix Notation The series admittance of line is g +jb and we define 1 B , diag b b @ L b , , 2 We define the L N incidence matrix a T where the component j of ai is nonzero whenever line i is coincident with node j. Hence A is quite sparse, with at most two nonzeros per row 1 a T A @ 2 a T L 5

6. Linearized Active Power Flow Model Under these assumptions the change in the real power line flows are given as 1 B 0 I 1 = = = = p f BA 0 p BA B p 0 B 0 1 @ BA B The constant matrix is called the injection shift factor matrix (ISF) 6

7. Injection Shift Factors (ISFs) n The element in row and column n of is called the injection shift factor (ISF) of line with respect to the injection at node n Absorbed at the slack bus, so it is slack bus dependent Terms generation shift factor (GSF) and load shift factor (LSF) are also used (such as by NERC) Same concept, just a variation in the sign whether it is a generator or a load Sometimes the associated element is not a single line, but rather a combination of lines (an interface) Terms used in North America are defined in the NERC glossary (http://www.nerc.com/files/glossary_of_terms.pdf) 7

8. ISF Interpretation + 1 n p slack node n + + n f j i line slackbus p 1 n is the fraction of the additional 1 MW injection at node n that goes though line 8

9. ISF Properties n By definition, depends on the location of the slack bus By definition, for since the injection and withdrawal buses are identical in this case and, consequently, no flow arises on any line The magnitude of is at most 1 since 1 slackbus L 0 n n 1 Note, this is strictly true only for the linear (lossless) case. In the nonlinear case, it is possible that a transaction decreases losses. Hence a 1 MW injection could change a line flow by more than 1 MW. 9

10. Five Bus Example Reference 42 MW One Two Line 1 A 1.040 pu MVA 200 MW Line 2 A 1.050 pu MVA 260 MW Line 3 Line 4 slack 67 MW A MVA 258 MW 33 MW A 100 MW A MVA 118 MW MVA Line 5 Four 1.042 pu 1.042 pu 100 MW Line 6 Three 1.044 pu Five 118 MW 100 MW 10

11. Five Bus ISF, Line 4, Bus 2 (to Slack) 52 MW One Two Line 1 A 1.040 pu MVA 200 MW Line 2 A 1.050 pu MVA 280 MW Line 3 Line 4 slack A 63 MW 86% MVA 238 MW 37 MW A 100 MW A MVA 128 MW MVA Line 5 128 118 20 0.5 Four 1.042 pu 2 l 1.042 pu 4 100 MW Line 6 Three 1.044 pu Five = = 118 MW 100 MW 11

12. Five Bus Example = diag B 6.25, 12.5, 12.5, 12.5, 12.5, 10 The row of A correspond to the lines and transformers, the columns correspond to the non-slack buses (buses 2 to 5); for each line there is a 1 at one end, a -1 at the other end (hence an assumed sign convention!). Here we put a 1 for the lower numbered bus, so positive flow is assumed from the lower numbered bus to the higher number -1 0 0 0 0 -1 0 0 0 0 -1 0 = A 1 -1 0 0 0 1 -1 0 0 0 1 -1 12

13. Five Bus Example 0 0 0 18.75 12.5 0 0 12.5 37.5 12.5 0 12.5 35 10 T = = B A BA 10 10 -0.4545 -0.3636 -0.1818 0.5455 0.1818 0 -0.1818 -0.5455 -0.2727 -0.1818 0.2727 0 -0.0909 -0.2727 -0.6364 -0.0909 -0.3636 0 -0.0909 -0.2727 -0.6364 -0.0909 -0.3636 -1.0000 1 = = = = B A B With bus 1 as the slack, the buses (columns) go for 2 to 5 13

14. Five Bus Example Comments At first glance the numerically determined value of (128-118)/20=0.5 does not match closely with the analytic value of 0.5455; however, in doing the subtraction we are losing numeric accuracy Adding more digits helps (128.40 117.55)/20 = 0.5425 The previous matrix derivation isn t intended for actual computation; is a full matrix so we would seldom compute all of its values Sparse vector methods can be used if we are only interested in the ISFs for certain lines and certain buses 14

15. Distribution Factors Various additional distribution factors may be defined power transfer distribution factor (PTDF) line outage distribution factor (LODF) line addition distribution factor (LADF) outage transfer distribution factor (OTDF) These factors may be derived from the ISFs making judicious use of the superposition principle 15

16. Definition: Basic Transaction A basic transaction involves the transfer of a specified amount of power t from an injection node m to a withdrawal node n t t m n 16

17. Definition: Basic Transaction We use the notation w m n t @ , , injection node withdrawal node quantity to denote a basic transaction 17

18. Definition: PTDF NERC defines a PTDF as In the pre-contingency configuration of a system under study, a measure of the responsiveness or change in electrical loadings on transmission system Facilities due to a change in electric power transfer from one area to another, expressed in percent (up to 100%) of the change in power transfer Transaction dependent We ll use the notation to indicate the PTDF on line with respect to basic transaction w In the lossless formulation presented here (and commonly used) it is slack bus independent w ( ) 18

19. PTDFs t + t t + t m n f + + f j i line Note, the PTDF is independent of the amount t; which is often expressed as a percent f t w ( ) @ 19

20. PTDF Evaluation in Two Parts 1 1 n m w ( ) = j i line 1 1 n m m n + j j i i 0 0 1 1 20

21. PTDF Evaluation + 1 t t + 1 m n + + m f n j i line 0 1 1 = = w m n ( ) 21

22. Calculating PTDFs in PowerWorld PowerWorld provides a number of options for calculating and visualizing PTDFs Select Tools, Sensitivities, Power Transfer Distribution Factors (PTDFs) Results are shown for the five bus case for the Bus 2 to Bus 3 transaction 22

23. Five Bus PTDF Visualization 52 MW One Two Line 1 27% PTDF 1.040 pu 200 MW Line 2 18% PTDF 1.050 pu 280 MW Line 3 Line 4 slack 63 MW 73% PTDF 238 MW 37 MW 100 MW 9% PTDF 9% PTDF 128 MW Line 5 Four 1.042 pu 1.042 pu 100 MW Line 6 Three 1.044 pu Five 118 MW 100 MW PowerWorld Case: B5_DistFact_PTDF 23

24. Nine Bus PTDF Example Display shows the PTDFs for a basic transaction from Bus A to Bus I. Note that 100% of the transaction leaves Bus A and 100% arrives at Bus I 400.0 MW 400.0 MW 300.0 MW A B D 250.0 MW 43% PTDF C 10% PTDF 57% PTDF 30% PTDF 13% PTDF 20% PTDF 35% PTDF 2% PTDF F E G 150.0 MW 250.0 MW 250.0 MW 34% PTDF 32% PTDF 34% PTDF 34% PTDF I H slack 50.0 MW 200.0 MW PowerWorld Case: B9_PTDF 24

25. Eastern Interconnect Example: Wisconsin Utility to TVA PTDFs In this example multiple generators contribute for both the seller and the buyer Contours show lines that would carry at least 2% of a power transfer from Wisconsin to TVA 25

26. Line Outage Distribution Factors (LODFs) Power system operation is practically always limited by contingencies, with line outages comprising a large number of the contingencies Desire is to determine the impact of a line outage (either a transmission line or a transformer) on other system real power flows without having to explicitly solve the power flow for the contingency These values are provided by the LODFs The LODF is the portion of the pre-outage real power line flow on line k that is redistributed to line as a result of the outage of line k k d 26

27. LODFs f f + + f j j i i line line kf j i j i line k line k outaged base case outage case f k = = = = d d ,k f k Best reference is Chapter 7 of the course book 27

28. LODF Evaluation We simulate the impact of the outage of line k by adding the basic transaction k , , i j = = w t k and selecting tk in such a way that the flows on the dashed lines become exactly zero + + f f j i line j i In general this tk is not equal to the original line flow + + f f k k kt kt line k 28

29. LODF Evaluation We select tk to be such that f f + + = = t 0 k k k where k is the active power flow change on the line k due to the transaction wk The line k flow from wk depends on its PTDF t = = f t w k ( ) kf k k f k k = = = = it follows that ( ( ) ) k w k ( ) i k j 1 1 k k 29

30. LODF Evaluation For the rest of the network, the impacts of the outage of line k are the same as the impacts of the additional basic transaction wk w ( ) k w ( ) = = = = f t f k k k w k ( ) 1 k Therefore, by definition the LODF is w ( ) f k = = = = k d w k ( ) f 1 k k 30

31. Five Bus Example Assume we wish to calculate the values for the outage of line 4 (between buses 2 and 3); this is line k 52 MW One Two Say we wish to know the change in flow on the line 3 (Buses 3 to 4). PTDFs for a transaction from 2 to 3 are 0.7273 on line 4 and 0.0909 on line 3 Line 1 A 1.040 pu MVA 200 MW Line 2 A 1.050 pu MVA 280 MW Line 3 Line 4 slack A 63 MW 86% MVA 238 MW 37 MW A 100 MW A MVA 128 MW MVA Line 5 Four 1.042 pu 1.042 pu 100 MW Line 6 Three 1.044 pu Five 118 MW 100 MW 31

32. Five Bus Example Hence we get f 128 k = = = = = = t 469.4 k w k ( ) 1 0.7273 1 k w ( ) f 0.0909 1 0.7273 4 3 3 = = = = = = = = 4 3 d 0.333 w ( ) f 1 4 4 4 ( ( ) ) = = = = = = f f 0.333 0.333 128 42.66MW 3 4 32

33. Five Bus Example Compensated Here is the system with the compensation added to bus 2 and removed at bus 3; we are canceling the impact of the line 4 flow for the reset of the network. 184 MW A One Two Line 1 124% MVA 1.040 pu 200 MW Line 2 A 1.050 pu MVA 749 MW Line 3 Line 4 A slack 108 MW 312% MVA 238 MW 8 MW A 100 MW A MVA 465 MW MVA Line 5 Four 1.027 pu 1.016 pu 100 MW Line 6 Three 1.029 pu Five 587 MW 100 MW 33

34. Five Bus Example Below we see the network with the line actually outaged 180 MW A One Two The line 3 flow changed from 63 MW to 106 MW, an increase of 43 MW, matching the LODF value Line 1 121% MVA 1.040 pu 200 MW Line 2 A 1.050 pu MVA 280 MW Line 3 Line 4 slack 106 MW 238 MW 6 MW A 100 MW A MVA 0 MW MVA Line 5 Four 1.042 pu 1.044 pu 100 MW Line 6 Three 1.044 pu Five 118 MW 100 MW 34

35. Developing a Critical Eye In looking at the below formula you need to be thinking about what conditions will cause the formula to fail ( ) 1 w ( ) k w = = = = f t f k k k w k ( ) k Here the obvious situation is when the denominator is zero That corresponds to a situation in which the contingency causes system islanding An example is line 6 (between buses 4 and 5) Impact modeled by injections at the buses within each viable island 35

36. Calculating LODFs in PowerWorld Select Tools, Sensitivities, Line Outage Distribution Factors Select the Line using dialogs on right, and click Calculate LODFS; below example shows values for line 4 36

37. Blackout Case LODFs One of the issues associated with the 8/14/03 blackout was the LODF associated with the loss of the Hanna- Juniper 345 kV line (21350-22163) that was being used in a flow gate calculation was not correct because the Chamberlin-Harding 345 kV line outage was missed With the Chamberlin-Harding line assumed in-service the value was 0.362 With this line assumed out-of-service (which indeed it was) the value increased to 0.464 37

38. 2000 Bus LODF Example LODF is for line between 3048 and 5120; values will be proportional to the PTDF values 38

39. 2000 Bus LODF Example NOCONA TELEPHONE 0% PTDF HENRIETTA CROSBYTON 0% PTDF DENISON 2 0% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF WHITESBORO SUMNER 0% PTDF SAINT JO SADLER MUENSTER 1 DENISON 1 MATADOR 0% PTDF 0% PTDF BONHAM 0% PTDF 0% PTDF PARIS 2 PARIS 1 MUENSTER 2 0% PTDF PARIS 3 0% PTDF DODD CITY 0% PTDF 0% PTDF 0% PTDF RALLS 1 GAINESVILLE RALLS 2 0% PTDF 0% PTDF SAVOY MONTAGUE SHERMAN 3 0% PTDF BELLS SEYMOUR LINDSAY HONEY GROVE 0% PTDF SHERMAN 2 SHERMAN 1 0% PTDF ROXTON 0% PTDF ECTOR 0% PTDF 0% PTDF 0% PTDF COLLINSVILLE 0% PTDF ARCHER CITY BOWIE 0% PTDF ARCHER 1 WHITEWRIGHT 0% PTDF ARCHER 2 0% PTDF FORESTBURG COOPER SUNSET 0% PTDF KNOX CITY 0% PTDF LEONARD TRENTON 0% PTDF WOLFE CITY 0% PTDF PATTONVILLE 0% PTDF OLNEY 1 0% PTDF ALVORD 0% PTDF 0% PTDF 0% PTDF 0% PTDF SPUR 0% PTDF COMMERCE 0% PTDF CHICO ROCHESTER 0% PTDF 0% PTDF 1% PTDF 0% PTDF 0% PTDF BRYSON 2 DECATUR 0% PTDF DIKE OLNEY 2 3% PTDF 3% PTDF 2% PTDF 0% PTDF JACKSBORO 2 BRYSON 1 0% PTDF ASPERMONT 0% PTDF BRIDGEPORT DENTON 1 2% PTDF THROCKMORTON 2% PTDF 0% PTDF 0% PTDF MOUNT VERNON 0% PTDF MCKINNEY 3 MCKINNEY 1 GREENVILLE 1 SULPHUR SPRINGS 0% PTDF 2% PTDF 0% PTDF 3% PTDF GRAHAM HASKELL 0% PTDF 0% PTDF ALLEN 1 0% PTDF 0% PTDF CUMBY 1% PTDF PARADISE 6% PTDF JACKSBORO 1 GREENVILLE 2 POOLVILLE 1% PTDF 0% PTDF 2% PTDF 0% PTDF 0% PTDF PERRIN COMO 0% PTDF PTDF RICHARDSON 2 0% PTDF KELLER 2 0% PTDF 1% POINT 0% PTDF JAYTON 0% PTDF 0% PTDF FLUVANNA 2 STAMFORD LAMESA O DONNELL 2 O DONNELL 1 ODONNELL FLUVANNA 1 QUINLAN YANTIS 0% PTDF 0% PTDF GRAFORD 2% PTDF 0% PTDF 0% PTDF 4% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF ROTAN 1% PTDF 0% PTDF 0% PTDF EMORY COAHOMA LUEDERS HAMLIN GARLAND 1 1% PTDF 0% PTDF 0% PTDF WEATHERFORD 3 0% PTDF 0% PTDF 0% PTDF MINEOLA 0% PTDF 1% PTDF 0% PTDF ALBANY 2 BRECKENRIDGE MINERAL WELLS 0% PTDF ARLINGTON 1 0% PTDF ALBANY 1 0% PTDF ANSON PALO PINTO 2 1% PTDF STRAWN 0% PTDF SNYDER 2 SYNDER TERRELL 1 0% PTDF 1% PTDF 0% 2% PTDF PTDF SNYDER 1 2% PTDF ALEDO 1 EDGEWOOD MILLSAP 0% PTDF WILLS POINT 1% PTDF 0% PTDF 0% PTDF 0% PTDF ABILENE 2 0% PTDF 1% PTDF 3% PTDF 0% PTDF 0% PTDF TRENT 2 GRAND PRAIRIE 3 PALO PINTO 1 DALLAS 1 GRAND SALINE 0% PTDF 0% PTDF HAWLEY 0% PTDF 0% PTDF 0% PTDF SANTO 1% PTDF 0% PTDF HERMLEIGH GORDON 0% PTDF 7% PTDF 1% PTDF ABILENE 3 1% PTDF 0% PTDF KAUFMAN 0% PTDF CISCO 0% PTDF 3% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF LINDALE VAN LIPAN CANTON ABILENE 1 0% PTDF ROSCOE 2 0% PTDF 0% PTDF 0% PTDF 0% PTDF MANSFIELD 1% PTDF ROSCOE 1 EASTLAND STANTON 0% PTDF 0% PTDF 0% PTDF MERKEL 3 0% PTDF ABILENE 5 0% PTDF MINGUS 0% PTDF 0% PTDF LORAINE 1 MABANK 1 TRENT 1 MIDLOTHIAN 1 BEN WHEELER 0% PTDF DYESS AFB TYLER 9 SWEETWATER 1 0% PTDF GRANBURY 2 ROSCOE 3 0% PTDF 0% PTDF KEMP 0% PTDF ABILENE 6 TOLAR 2% PTDF GLEN ROSE 1 0% PTDF 1% PTDF SWEETWATER 3 0% PTDF BAIRD 0% PTDF DUBLIN 1 0% PTDF ABILENE 4 ABILENE 7 GRANBURY 1 0% PTDF COLORADO CITY 2% PTDF 1% PTDF 0% PTDF ENNIS MIDLAND 2 0% PTDF MERKEL 1 1% PTDF 6% PTDF ROSCOE 5 MERKEL 2 TYLER 8 19% PTDF 19% PTDF DE LEON SWEETWATER 2 NOLAN 0% PTDF 0% PTDF STEPHENVILLE SWEETWATER 5 CLEBURNE 2 MABANK 2 BIG SPRING 6 TUSCOLA CHANDLER CLYDE NEMO 0% PTDF 0% PTDF 0% PTDF 0% PTDF LENORAH 0% PTDF 1% PTDF 0% PTDF 0% PTDF ATHENS 1 0% PTDF RICE 1% PTDF 0% PTDF 0% PTDF 0% PTDF ROSCOE 4 FORRESTON 2% PTDF 0% PTDF FLINT WINGATE BARDWELL BIG SPRING 5 GLEN ROSE 2 3% PTDF 0% PTDF BLACKWELL 0% PTDF 0% PTDF ITALY TRINIDAD 2 0% PTDF TRINIDAD 1 0% PTDF KERENS MURCHISON 1% PTDF FORSAN 0% PTDF MILFORD 1% PTDF MALAKOFF 0% PTDF BLOOMING GROVE OVALO 0% PTDF GORMAN KOPPERL 0% PTDF 0% PTDF BULLARD RIO VISTA 0% PTDF ATHENS 2 BIG SPRING 3 0% PTDF WALNUT SPRINGS 49% PTDF DUBLIN 2 5% PTDF CORSICANA 2 MIDLAND 5 GOLDSBORO SILVER 0% PTDF MERTENS CORSICANA 1 BIG SPRINGS FROST 0% PTDF HILLSBORO FRANKSTON 0% PTDF 0% PTDF 0% PTDF 1% PTDF 0% PTDF MIDLAND 3 CROSS PLAINS 31% PTDF 0% PTDF 0% PTDF MORGAN ODESSA 3 HICO 0% PTDF 1% PTDF BIG SPRING 4 BIG SPRING 1 0% PTDF 2% PTDF 0% PTDF WHITNEY 1% PTDF STERLING CITY 1 0% PTDF 0% PTDF 1% PTDF 0% PTDF STERLING CITY 2 0% PTDF 0% PTDF PALESTINE 2 BIG SPRING 2 TENNESSEE COLONY 0% PTDF MERIDIAN MAY MALONE 0% PTDF 0% PTDF 0% PTDF ODESSA 2 0% PTDF ABBOTT 0% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF COMANCHE WORTHAM 0% PTDF FAIRFIELD 1 FAIRFIELD 2 0% PTDF ODESSA 1 0% PTDF 0% PTDF 0% PTDF CLIFTON 1 DAWSON 0% PTDF PALESTINE 1 0% PTDF WINTERS 0% PTDF 0% PTDF HUBBARD EARLY 0% PTDF 0% PTDF 0% PTDF CLIFTON 2 WEST 0% PTDF SAN ANGELO 3 5% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF JONESBORO FAIRFIELD 3 GARDEN CITY STREETMAN 0% PTDF MOUNT CALM BROWNWOOD 0% PTDF BALLINGER COLEMAN 0% PTDF ODESSA 4 1% PTDF 0% PTDF 1% PTDF 0% PTDF 0% PTDF BANGS 0% PTDF CHINA SPRING ELM MOTT 0% PTDF ZEPHYR HAMILTON 1% PTDF VALLEY MILLS AXTELL COOLIDGE MEXIA 0% PTDF GOLDTHWAITE 2 WACO 6 0% PTDF 0% PTDF WACO 2 0% PTDF MILES ELKHART 1% PTDF 0% PTDF WACO 3 MART OAKWOOD EVANT 0% PTDF 17% PTDF WOODWAY 0% PTDF RICHLAND SPRINGS MIDLAND 1 SANTA ANNA 0% PTDF WACO 4 SAN ANGELO 1 0% PTDF 0% PTDF GROESBECK 0% PTDF GATESVILLE 1 0% PTDF TEAGUE RIESEL 1 WACO 1 1% PTDF 0% PTDF GATESVILLE 2 BUFFALO 1% PTDF 1% PTDF 1% JEWETT 1 HEWITT 0% PTDF 0% PTDF 0% PTDF 0% PTDF 0% PTDF MC GREGOR 0% PTDF GRAPELAND GATESVILLE 4 ROWENA 11% PTDF PTDF GOLDTHWAITE 1 0% PTDF 1% PTDF PTDF RIESEL 2 5% 2% PTDF 6% PTDF 0% PTDF JEWETT 2 0% PTDF 4% PTDF 0% PTDF THORNTON GOODFELLOW AFB 0% PTDF MOODY 0% PTDF LORENA 0% PTDF BIG LAKE 1% PTDF SAN ANGELO 2 LOMETA 0% PTDF 0% PTDF MARQUEZ 0% PTDF 0% PTDF EDEN 0% PTDF COPPERAS COVE 0% PTDF FRANKLIN 0% PTDF MCCAMEY 2 MCCAMEY 1 0% PTDF 0% PTDF MARLIN 0% PTDF TROY 0% PTDF BRADY 1% PTDF SAN SABA 0% PTDF 0% PTDF KILLEEN 1 FORT HOOD TEMPLE 3 CENTERVILLE NORMANGEE 1% PTDF 4% PTDF KILLEEN 2 0% PTDF LOVELADY 0% PTDF LAMPASAS 0% PTDF 0% PTDF BREMOND KEMPNER CHRISTOVAL ROSEBUD NOLANVILLE TEMPLE 2 0% PTDF 0% PTDF 3% PTDF 4% PTDF 0% PTDF HARKER HEIGHTS BELTON 39

40. Multiple Line LODFs LODFs can also be used to represent multiple device contingencies, but it is usually more involved than just adding the effects of the single device LODFs Assume a simultaneous outage of lines k1 and k2 Now setup two transactions, wk1 (with value tk1)and wk2 (with value tk2) so f f f t f f f t + + + + f t + + + + + + + + = = = = t 0 0 k k k 1 1 2 k 1 k k k 2 1 2 k ) 2 w k w k ( ) ( = = t 0 k k 1 2 k 1 k k k 1 1 1 2 1 w k w k ( ) ( ) + + + + = = f t t t 0 k k 1 2 k 2 k k k 2 1 2 1 2 40

41. Multiple Line LODFs Hence we can calculate the simultaneous impact of multiple outages; details for the derivation are given in C.Davis, T.J. Overbye, "Linear Analysis of Multiple Outage Interaction," Proc. 42nd HICSS, 2009 Equation for the change in flow on line for the outage of lines k1 and k2 is 1 f f k k 2 1 d 1 d k 1 = = k k 1 2 f d d k k 1 2 1 k 2 41

42. Multiple Line LODFs Example: Five bus case, outage of lines 2 and 5 to flow on line 4. 1 f f k k 2 1 d 1 d k 1 = = k k 1 2 f d d k k 1 2 1 k 2 1 1 0.6 0.75 1 0.336 0.331 = = = = f 0.4 0.25 0.005 42

43. Multiple Line LODFs 42 MW One Two Line 1 A 1.040 pu MVA 200 MW Flow goes from 117.5 to 118.0 Line 2 1.050 pu 260 MW Line 3 Line 4 slack 100 MW A MVA 258 MW 0 MW A 100 MW MVA 118 MW Line 5 Four 1.040 pu 1.036 pu 100 MW Line 6 Three 1.042 pu Five 118 MW 100 MW 43