1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 10/28/2014 1 Lecture 18: Sensitivity Analysis

2. Injection Shift Factors 2

3. Five Bus Example: Base Case

4. Five Bus ISF, Line 4, Bus 2 (to Slack) 4

5. Five Bus Example The rows 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 5 ~ -

6. Five Bus Example 6

7. Five Bus Example Comments 7

8. Distribution Factors Various additional distribution factors may be defined – power transfer distribution factor (PTDF) – line outage distribution factor (LODF) – line closure distribution factor (LCDF) – outage transfer distribution factor (OTDF) These factors may be derived from the ISFs making judicious use of the superposition principle 8

9. 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 9

10. We use the following notation to denote a basic transaction Definition: Basic Transaction injection node withdrawal node quantity 10

11. 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 l with respect to the basic transaction w In the lossless formulation presented here (and commonly used) it is slack bus independent 11

12. PTDFs Note, the PTDF is independent of the amount Δt; often expressed as a percent 12

13. PTDF : Evaluation = + 13

14. PTDF : Evaluation 14

15. Calculating PTDFs in PowerWorld PowerWorld provides a number of options for calculating and visualizng PTDFs – Select Tools, Sensitivities, Power Transfer Distribution Factors (PTDFs) 15 Results are shown for the five bus case for the Bus 2 to Bus 3 transaction

16. PowerWorld PTDF Visualization To visualize the PTDFs, right click on the one-line background and, Animated Flows – Select Oneline Display Options and then set the Base Flow Scaling on PTDF Percentage Flows – Select Pie Chart/Gauges and change Pie Chart Style to PTDF, set the Show Percent to a small value, like 1% 16

17. Five Bus PTDF Visualization 17 PowerWorld Case: B5_DistFact

18. Nine Bus PTDF Example 18 PowerWorld Case: B9 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

19. Eastern Interconnect Exmaple: Wisconsin Utility to TVA PTDFs 19 Contours show lines that would carry at least 2% of a power transfer from Wisconsin to TVA In this example multiple generators contribute for both the seller and the buyer

20. 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 l as a result of the outage of line k 20

21. LODFs outaged base case outage case A good reference is Power Generation, Operation and Control by Wood and Wollenberg; there is now a 3 rd edition 21

22. LODF : Evaluation We simulate the impact of the outage of line k by considering the basic transaction and selecting  t k in such a way that the flows on the dashed lines become exactly zero 22

23. LODF : Evaluation We select  t k to be such that where  f k is the active power flow change on the line k due to the transaction w k As the transaction w' results in a flow on line k of it follows that 23

24. 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 w k Therefore, by definition 24

25. 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 25 Say we wish to know the change in flow on the line 3 (Buses 3 to 4). PTDFs for the transaction from 2 to 3 are 0.7273 on line 4 and 0.0909 on line 3

26. Five Bus Example Hence we get 26

27. Five Bus Example Compensated 27 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.

28. Five Bus Example Here is the system with line 4 actually outaged 28 The line 3 flow changed from 63 MW to 106 MW, an increase of 43 MW, matching the LODF value

29. Developing a Critical Eye In looking at the below formula we need to think about what conditions will cause this formula to fail Here an obvious case 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 29

30. 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 30