Analysis Techniques for Large-Scale Electrical Systems Lecture 19 Sensitivity Analysis

ECE 530 Analysis Techniques for Large-Scale Electrical Systems Lecture 19: Sensitivity Analysis Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 10/30/2014 1

Announcements HW 6 is due Tuesday November 11 2

Line Outage Distribution Factors (LODFs) The LODF is the portion of the pre-outage real power line flow on line k that is redistributed to line l as a result of the outage of line k To model the outage of line k, assume a virtual transaction , , k k w i j t = = f t k d with f k k = = = = ( ( ) ) k w k ( ) i k j 1 1 k k By definition we have w ( ) f k = = = = k d w k ( ) f 1 k k 3

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 set up two transactions, wk1(with value tk1) and wk2 (with value tk2) so ?1+ ??1 ?2 ??1= 0 ??1+ ??1 ?1+ ??2 ?2 ??2= 0 ??2+ ??2 4

Multiple Line LODFs Substituting into PTDFs f + + w k w k ( ) ( ) + + = = t t 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 Equation for the change in flow on line l for the outage of lines k1 and k2 is 1 2 1 2 k d 1 f f k k 2 1 d 1 k 1 = = k k f d d k 1 k 2 5

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 6

Multiple Line LODFs 42 MW One Two Line 1 A Flow goes from 117.5 to 118.0 1.040 pu MVA 200 MW 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 7

Line Closure Distribution Factors (LCDFs) The line closure distribution factor (LCDF), LCDFl,k, for the closure of line k (or its addition if it does not already exist) is the portion of the line active power flow on line k that is distributed to line l due to the closure of line k Since line k is currently open, the obvious question is, what flow on line k? Answer (in dc sense) is the flow that will occur when the line is closed (which we do not know before closure) 8

Line Closure Distribution Factors f + + f f j i j i line line kf j i j i line k Closed line base case line k addition case f = = = = k LCDF LCDF k , f k 9

LCDF : Evaluation We can evaluate LCDF by reversing the line outage Recall how we define LODF 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 10 k

LCDF : Evaluation So the post-outage line flow ?? in LCDF is exactly the pre-outage line k flow in LODF But the change in line l flow in LCDF becomes the opposite of that in LODF So for the LCDF calculation ??= ?? Hence, we have ??? ?? ?? ??????= ? = ?? We can verify this from the earlier example on ????3 4 for the 5-bus case 11

Outage Transfer Distribution Factor The outage transfer distribution factor (OTDF) is defined as the PTDF with the line k outaged The OTDF applies only to the post-contingency configuration of the system since its evaluation explicitly considers the line k outage ( ( ) ) k w ( ) This is a quite important value since power system operation is usually contingency constrained 12

Outage Transfer Distribution Factor t + t t + t f + + f m n j i line j i line k outaged line f ( ( ) ) k w ( ) @ = t k outaged 13

OTDF : Evaluation t t m n f j = i i line j line t t m n (2) f (1) f j j + i i i kf j j i 14

OTDF : Evaluation Since and f = = w (1) ( ) f t = = w k ( ) t k = = = = k k w k (2) ( ) f d f d t then k so that = = + + = = + + w k w k (1) (2) ( ) ( ) f f f d t ( ( ) ) k = = + + w w k w k ( ) ( ) ( ) d 15

Five Bus Example Say we would like to know the PTDF on line 1 for a transaction between buses 2 and 3 with line 2 out 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 16

Five Bus Example Hence we want to calculate these values without having to explicitly outage line 2 42 MW One Two Line 1 20% PTDF 1.040 pu 200 MW The value we are looking for is 0.2 (20%) Line 2 1.050 pu 260 MW Line 3 Line 4 slack 67 MW 80% PTDF 258 MW 33 MW 100 MW 20% PTDF 20% PTDF 118 MW Line 5 Four 1.042 pu 1.042 pu 100 MW Line 6 Three 1.044 pu Five 118 MW 100 MW 17

Five Bus Example Evaluating: the PTDF for the bus 2 to 3 transaction on line 1 is 0.2727; it is 0.1818 on line 2 (from buses 1 to 3); the LODF is on line 1 for the outage of line 2 is -0.4 Hence ( ( 0.2727 ( 0.4) (0.1818) + + ) ) k = = + + w w k w k ( ) ( ) ( ) d = = 0.200 For line 4 (buses 2 to 3) the value is 0.7273 + + (0.4) (0.1818) = = 0.800 18

UTC Revisited We can now revisit the uncommitted transfer capability (UTC) calculation using PTDFs and LODFs Recall trying to determine maximum transfer between two areas (or buses in our example) For base case maximums are quickly determined with PTDFs = = ( ( ) ) 0 max f f ( ( ) ) 0 m n u min ( ( ) ) w , ( ( ) ) w 0 19

UTC Revisited For the contingencies we use max ( ) 0 k ( ) 0 k f f d f ( ( ) ) 1 m n = = u min ( ( ) ) , k ( ( ) ) k w ( ) w ( ) 0 Then as before = = ( ) 0 m n (1) m n u min u u , m n , , , 20

Five Bus Example t ( ( ) ) 0 T = = = = w 2, 3, f 42, 34,67 ,118, 33,100 T max= = f 150 , 400 ,150 ,150 ,150 ,1,000 42 MW One Two Line 1 A 1.040 pu MVA 200 MW Line 2 A 1.050 pu 34 MW 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 21

Five Bus Example Therefore, for the base case ( ( ) ) 0 max f f ( ( ) ) 2,2 0 = = u min ( ( ) ) w ( ( ) ) w 0 150 42 400 , 34 150 , 67 150 118 150 , 0.0909 0.7273 33 = = min , 0.2727 0.1818 0.0909 = = 44.0 22

Five Bus Example For the contingency case corresponding to the outage of the line 2 max f u min ( ) 0 0 2 ( ) 2 f d f = = (1) 2,3 ( ( ) ) 2 ( ( ) ) 2 w ( ) w ( ) 0 The limiting value is line 4 max ( ) 0 0 2 ( ) 2 f f d f 150 118 0.4 34 = = ( ( ) ) 2 0.8 w ( ) Hence the UTC is limited by the contingency to 23.0 23

Contingency Considerations Traditionally contingencies consisted of single element failures (N-1), though utilities have long considered multiple element contingencies Some can be quite complex N-2 involves considering all double element contingencies N-1-1 is used to describe contingencies in which a single element contingency occurs, the system is re- dispatched, then a second contingency occurs The number of contingencies considered following the first contingency can be quite large, coming close to N-2 24

Additional Comments Distribution factors are defined as small signal sensitivities, but in practice, they are also used for simulating large signal cases Distribution factors are widely applied in the operation of the electricity markets where the rapid evaluation of the impacts of each transaction on the line flows is required Applications to actual system show that the distribution factors provide satisfactory results in terms of accuracy For multiple applications that require fast turn around time, distribution factors are used very widely, particularly, in the market environment 25

Additional References M. Liu and G. Gross, Effectiveness of the Distribution Factor Approximations Used in Congestion Modeling , in Proceedings of the 14th Power Systems Computation Conference, Seville, 24- 28 June, 2002 R. Baldick, Variation of Distribution Factors with Loading , available: http://www.ucei.org/pdf/csemwp104.pdf T. Guler and G. Gross, Detection of Island Formation and Identification of Causal Factors under Multiple Line Outages, " IEEE Transactions on Power Systems, vol. 22, no. 2, May, 2007. 26

Additional References T. Guler, G. Gross and M. Liu, "Generalized Line Outage Distribution Factors," IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 879 881, May 2007 C.M. Davis, T.J. Overbye, "Multiple Element Contingency Screening," IEEE Trans. Power Systems, vol. 26, pp. 1294-1301, August 2011 27