1. PS ERC 1 Reactive Power Considerations in Linear Load Flow with Applications to Available Transfer Capability Pete Sauer (With a lot of help from Santiago Grijalva) University of Illinois at Urbana-Champaign PS ERC Internet Seminar December 3, 2002 © 2002 University of Illinois

2. PS ERC 2 Overview Linear load flow methods Linear load flow methods Linear transfer capability calculations Linear transfer capability calculations Reactive linear ATC calculations Reactive linear ATC calculations Examples Examples Conclusions Conclusions

3. PS ERC 3 Linear Load Flow An approximation used to estimate the result of a change in operating conditions from some base case An approximation used to estimate the result of a change in operating conditions from some base case Small change sensitivities Small change sensitivities Large-change sensitivities Large-change sensitivities

4. PS ERC 4 PTDFs Power Transfer Distribution Factors (PTDFs) indicate how “injection power” flows in the lines Power Transfer Distribution Factors (PTDFs) indicate how “injection power” flows in the lines A 5% PTDF for a given injection set and line means that 5% of the injection flows in that line A 5% PTDF for a given injection set and line means that 5% of the injection flows in that line Principle is simple current division Principle is simple current division

5. Base Case 5

6. P12 = 10 MW 6

7. PS ERC 7 Numerical PTDFs Real power distribution factors for real power transfer from area 1 to 2 (area 2 reduced by 10 MW) Real power distribution factors for real power transfer from area 1 to 2 (area 2 reduced by 10 MW)  jk,sr =  P jk /  P sr  12,12 = 0.4,  13,12 = 0.6,  32,12 = 0.6  12,12 = 0.4,  13,12 = 0.6,  32,12 = 0.6 (40% of the transfer goes down line 12, 60% goes down line 13, and 60% goes down line 32) (40% of the transfer goes down line 12, 60% goes down line 13, and 60% goes down line 32)

8. PS ERC 8 Use these to predict the result of a 100 MW transaction from 1 to 2 P 12 = 0+0.4*100 = 40 MW P 13 = 0+0.6*100 = 60 MW P 32 = 0+0.6*100 = 60 MW

9. P12 = 100 MW OK, but look at the VARS 9

10. PS ERC 10 How about 200 MW? P 12 = 0+0.4*200 = 80 MW P 13 = 0+0.6*200 = 120 MW P 32 = 0+0.6*200 = 120 MW

11. P12 = 200 MW OK, but look at the VARS 11

12. PS ERC 12 Behavior of Distribution Factors Close to Collapse p p* p 0 1.0

13. PS ERC 13 Analytical Distribution Factors These PTDFs can be analytically constructed from line admittances only, or line admittances plus base case operating point values. These PTDFs can be analytically constructed from line admittances only, or line admittances plus base case operating point values. There are also “Line Outage Distribution Factors” that estimate the change in flows due to a line outage. There are also “Line Outage Distribution Factors” that estimate the change in flows due to a line outage.

14. PS ERC 14 Total Transfer Capability (TTC) Specify sending point or points Specify sending point or points Specify receiving point or points Specify receiving point or points Increase sending power injection Increase sending power injection Decrease receiving power injection Decrease receiving power injection Monitor security limits Monitor security limits Stop when limit reached Stop when limit reached

15. PS ERC 15 Computation Linear algebraic Linear algebraic Non-linear algebraic Non-linear algebraic Time domain simulation Time domain simulation

16. PS ERC 16 1996 NERC definitions Transmission Reliability Margin (TRM) is supposed to account for uncertainty in conditions and model Transmission Reliability Margin (TRM) is supposed to account for uncertainty in conditions and model Capacity Benefit Margin (CBM) is supposed to account for reliability criteria (neighboring reserve etc.) Capacity Benefit Margin (CBM) is supposed to account for reliability criteria (neighboring reserve etc.)

17. PS ERC 17 TRM Components Forecasting errorForecasting error Data uncertaintyData uncertainty – Impedances – Ratings – Measurements

18. PS ERC 18 TRM Alternatives Fixed MW amount Fixed MW amount Fixed % Fixed % Resolve with limits reduced by some amount (I.e. 4%) Resolve with limits reduced by some amount (I.e. 4%)

19. PS ERC 19 CBM Issues Loss of the biggest unit will result in import from neighbors. There must be capability to allow this import. Loss of the biggest unit will result in import from neighbors. There must be capability to allow this import. Some companies use CBM = 0 and include the loss of units in the contingency list. Some companies use CBM = 0 and include the loss of units in the contingency list.

20. PS ERC 20 Available Transfer Capability ATC = TTC - TRM - CBM - ETC ATC = TTC - TRM - CBM - ETC Available = Total - Margins - Existing Available = Total - Margins - Existing

21. PS ERC 21 TTC Computation Errors Linear vs nonlinear flow calculations Linear vs nonlinear flow calculations MW vs MVA limits MW vs MVA limits Neglecting voltage constraints Neglecting voltage constraints Neglecting stability constraints Neglecting stability constraints

22. PS ERC 22 Estimating Maximum Power Transfers Recall the 3-bus example: Recall the 3-bus example: Real power distribution factors for real power transfer from area 1 to 2 Real power distribution factors for real power transfer from area 1 to 2  12,12 = 0.4,  13,12 = 0.6,  32,12 = 0.6  12,12 = 0.4,  13,12 = 0.6,  32,12 = 0.6 Maximum transfer 1-2 is minimum of: Maximum transfer 1-2 is minimum of: 100/.4 = 250, 130/.6 = 217, 140/.6 = 233 100/.4 = 250, 130/.6 = 217, 140/.6 = 233

23. P12 = 217 MW 23

24. P12 = 203 MW 24

25. P12 = 212 MW 25

26. PS ERC 26 P12 error Linear vs nonlinear error plus MW vs MVA error = 217 MW vs 203 MW (7%) Linear vs nonlinear error plus MW vs MVA error = 217 MW vs 203 MW (7%) Linear vs nonlinear error only = 217 MW vs 212 MW (2%) Linear vs nonlinear error only = 217 MW vs 212 MW (2%)

27. LIMITING CIRCLE Q jk P jk (P jk 0, Q jk 0 )  P jk * Linear ATC  P sr * =  P jk * /  jk,sr 27

28. PS ERC 28 Line power flow relations jk j X jk P + j Q jk P kj + jQ kj P jk = V j V k B jk sin (  j -  k ) P jk = V j V k B jk sin (  j -  k ) Q jk = V j 2 B jk - V j V k B jk cos (  j -  k ) P jk 2 + (V j 2 B jk - Q jk ) 2 = (V j V k B jk ) 2 kk VkVk jj VjVj

29. jk Bus j j X P jk + j Q jk P kj + jQ kj jj VjVj kk VkVk Bus k R jk + -j(1/B jj ) -j(1/B kk ) P jk = + V j 2 G jk - V j V k Y jk cos (  j -  k +  jk ) Q jk = - V j 2 B jj - V j 2 B jk - V j V k Y jk sin (  j -  k +  jk )  (P jk -V j 2 G jk ) 2 +(Q jk +V j 2 B jj +V j 2 B jk ) 2 = (V j V k Y jk ) 2 Line power flow relations 29

30. Limiting Circle: P jk 2 + Q jk 2 = (S jk max ) 2 P jk Operating and Limiting Circles Operating Circle: (P jk -V j 2 G jk ) 2 +(Q jk +V j 2 B jj +V j 2 B jk ) 2 = (V j V k Y jk ) 2 r = S jk max (P jk0, Q jk0 ) Q jk r =S jk0 (P jk *, Q jk *) 30

31. Solutions Solve: (P jk -V j 2 G jk ) 2 +(Q jk +V j 2 B jj +V j 2 B jk ) 2 = (V j V k Y jk ) 2 P jk 2 + Q jk 2 = (S jk max ) 2 Define: - M 0 2 = P jk0 2 +Q jk0 2 -S jk0 2 A = (P jk0 2 + Q jk0 2 ) B = - P jk0 ((S jk max ) 2 -M 0 2 ) C = [(S jk max ) 2 -M 0 2 ] 2 /4 - Q jk0 2 (S jk max ) 2 Then: P jk * = [ - B  (B 2 -4AC) 1/2 ]/2A Q jk *= [(S jk max ) 2 - P jk * 2 ] 1 / 2 31

32. LIMITING CIRCLE OPERATING CIRCLE (P jk *, Q jk * ) #1 r = S k max Q jk P jk r =S jk0 (P jk0,Q jk0 ) (P jk *, Q jk *) #2  P jk *  P sr * =  P jk * /  jk,sr Reactive Linear ATC 32

33. Feasibility in Reactive Linear ATC Computation Q jk P jk OPERATING CIRCLE (0, V j 2 Y jk ) B C A j B C D A P jk Q jk Limiting circle I Limiting circle II (0, V j 2 Y jk ) A to B (Thermal limit) A to B to C (Feasibility limit) 33

34. PS ERC 34 Test for infeasible cases in reactive linear ATC computation [(±V j 2 G jk +V j V k Y jk ) 2 + (-V j 2 B jj -V j 2 B jk ) 2 ] 1/2 < S jk max [(±V j 2 G jk +V j V k Y jk ) 2 + (-V j 2 B jj -V j 2 B jk ) 2 ] 1/2 < S jk max  An estimate of voltage collapse limits.

35. Estimation of Reactive Power Support Consider: Q jk = Q jk0 + [S jk0 2 - (P jk - P jk0 ) 2 ] 1/2 Valid for line complex flow if voltages ~ constant. Then, for a variation in the injection at bus i:  Q jk = Q jk0 -Q jk 0 +[S jk0 2 -(  jk,sr  P sr +P jk 0 -P jk0 ) 2 ] 1/2 Therefore, the new reactive power at bus j: Q j = Q j 0 +  k {Q jk0 -Q jk 0 +[S jk0 2 -(  jk,sr  P sr +  P jk 0 -P jk0 ) 2 ] 1/2 } All terms are known except  P sr which is independent.  A way to estimate the reactive power support required for large variations in active power transactions. 35

36. Base Case 36

38. 7-bus system 38

39. 7- bus system: P 6 - 4 39

40. PS ERC 40 WSCC Summer Case Forty-two transfers across the BC Hydro, BPA, and PG&E control areas were simulated. The simulation did not include contingency sets. The model had 7,119 buses, and 9,630 lines and transformers. Total generation was 120GW. Simulations run by PowerWorld Corp.

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42. PS ERC 42 NYISO Summer case Fifty transfers across different control areas in the NYISO The simulation did not include contingency sets. The model had about 40,000 buses and included more than 6,000 generating units and 139 control areas. Simulations run by PowerWorld Corp.

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44. PS ERC 44 Transmission Loading Relief (TLR) Based on the PTDF concept Based on the PTDF concept Could benefit from consideration of reactive power in loading Could benefit from consideration of reactive power in loading

45. PS ERC 45 The future of ATC The ATC concept has other problems The ATC concept has other problems – chaining does not work – updates are difficult What will the new Standard Market Design rules use? What will the new Standard Market Design rules use?

46. PS ERC 46 Conclusions The inclusion of reactive power considerations in a linear ATC calculation can reduce error in ATC. The inclusion of reactive power considerations in a linear ATC calculation can reduce error in ATC. It may provide a way to estimate the proximity to voltage collapse limits due to a transaction. It may provide a way to estimate the proximity to voltage collapse limits due to a transaction. The inclusion of reactive power considerations in a linear ATC calculation is easy. The inclusion of reactive power considerations in a linear ATC calculation is easy.