ECE 530: Lecture 22 Various Numerical Techniques used in PowerWorld Simulator

1. ECE 530: Lecture 22Various Numerical Techniques used in PowerWorld Simulator Special Guest Lecture by James Weber, Ph.D. Director of Software Development PowerWorld Corporation weber@powerworld.com

2. Introduction • Dr. Overbye is in Atlanta at a NERC conference on Geomagnetic Disturbances and their impact on Power Systems • I’m James Weber (Jamie) • Dr. Overbye’s first Ph.D. student • Defended my Ph.D. in September 1999 • Have worked at PowerWorld Corporation since the company’s inception in 1996 • Oversee the software development

3. Topics for Today • Not one topic for today, but just a smattering of topics related to this class • Always more to it than you cover in class! • You learn this stuff when you sit down to code it and your system doesn’t match the • Variations on Linear Analysis • Generator MW control in a power flow solution • Remote Generator Voltage Control and Mvar sharing • Line Drop Compensation and Reactive Current Compensation • Solving a Power Flow • Contouring Algorithms Introduction

4. AC Power Flow Equations • Full AC Power Flow Equations • Solution requires iteration of equations • Note: the large matrix (J) is called the Jacobian

5. Full AC Derivatives • Real Power derivative equations are • Reactive Power derivative equations are

6. Decoupled Power Flow Equations • Make the following assumptions • Derivatives simplify to • Note: If assumption is insteadthen replace all with • Option in Simulator Options under Power Flow Solution, DC Options sub-tab

7. Other Typical Assumptions about Transformers • Ignore Transformer Impedance Correction Tables • If we do not ignore, then as tap or phase changes the series branch impedances change • Ignore Phase Shift Angle Effects • If we do not ignore, then as the phase angle changes the series impedance seen between the buses will vary • Both of these are normally done because if they are not ignored, then the solution matrices become a function of the system state (i.e. impedance will vary with tap or phase) • This defeats the purpose of using the Decoupled equations. Recommendation: Leave these checked

8. B’ and B’’ Matrices • Define • Now Iterate the “decoupled” equations • What are B’ and B’’? • B’ is the imaginary part of the Y-Bus with all the “shunt terms” removed • B’’ is the imaginary part of the Y-Bus with all the “shunt terms” double counted

9. “DC Power Flow” • The “DC Power Flow” equations are simply the real part of the decoupled power flow equations • Voltages and reactive power are ignored • Only angles and real power are solved for by iterating

10. Bus Voltage and Angle Sensitivities to a Transfer • Power flow was solved by iterating • Model the transfer as a change in the injections DP • Buyer: • Seller: • Then solve for the voltage and angle sensitivities by solving • These are the sensitivities of the Buyer and Seller “sending power to the slack bus”

11. What about Losses? • If we assume the total sensitivity to the transfer is the seller minus the buyer sensitivity, then and • This makes assumption that ALL the change in losses shows up at the slack bus. • Simulator assigns the change to the BUYER by defining • Then

12. Why does “k” assign the losses to the Buyer?

13. Lossless DC Voltage and Angle Sensitivities • Use the DC Power Flow Equations • Then determine angle sensitivities • The DC Power Flow ignores losses, thus

14. Lossless DC Sensitivities with Phase Shifters Included • DC Power Flow equations • Augmented to include an equation that describes the change in flow on a phase-shifter controlled branch as being zero. • Thus instead of DC power flow equations we use • Otherwise process is the same.

15. 230 kV Phase Shifter Canada 500 kV Path BPA 115 kV Phase Shifter 115 kV Phase Shifter California Lossless DC with Phase Shifters • Phase Shifters are often on lower voltage paths (230 kV or less) with relatively small limits • They are put there in order to manage/prevent the flow on a path that would commonly see overloads • Thus, they constantly show up as “overloaded” when using linear analysis if they are not accounted for • Example: Border of Canada with Northwestern United States • PTDFs between Canada and US without Phase-Shifters • 85% on 500 kV Path • 15% on Eastern Path • PTDF With Phase-Shifters • 100% goes on 500 kV Path • 0% on Eastern Path • This better reflects real system

16. Power Transfer Distribution Factors (PTDFs) • PTDF: measures the sensitivity of line MW flows to a MW transfer. • Line flows are simply a function of the voltages and angles at its terminal buses • Thus, the PTDF is simply a function of these voltage and angle sensitivities. • This is the “Chain Rule” from Calculus • Pkm is the flow from bus k to bus m Voltage and Angle Sensitivities that were just determined

17. Pkm Derivative Calculations • Full AC equations • Lossless DC Approximations yields

18. What do Flow Sensitivities (PTDFs, GSFs, TLRs, …) give us? • Give the ability to model a change in power injection without actually doing a new power flow solution • Transfer of power between two places • Generator outage • Load outage • Still can’t model a line outage or line closure yet, but that is what LODFs will give us

19. Line Outage Distribution Factors (LODFs) • LODFl,k: percent of the pre-outage flow on Line K will show up on Line L after the outage of Line K • Linear impact of an outage is determined by modeling the outage as a “transfer” between the terminals of the line

20. Modeling an LODF as a Transfer Create a transfer defined by Assume Then the flow on the Switches is ZERO, thus Opening Line K is equivalent to the “transfer”

21. Modeling an LODF as a Transfer • Thus, setting up a transfer of MW from Bus n to Bus m is equivalent to outaging the transmission line • Let’s assume we know what is equal to, then we can calculate the values relevant to the LODF.

22. Calculation of LODF • Estimate of post-outage flow on Line L • Estimate of flow on Line L after transfer • Thus we can write • We have a simple function of PTDF values

23. Line Closure Distribution Factors (LCDFs) • LCDFl,k: percent of the post-closure flow on Line K will show up on Line L after the closure of Line K • Linear impact of an closure is determined by modeling the closure as a “transfer” between the terminals of the line

24. Modeling the LCDF as a Transfer Create a “transfer” defined by Assume Then the net flow to and from the rest of the system are both zero, thus closing line k is equivalent the “transfer”

25. Modeling an LCDF as a Transfer • Thus, setting up a transfer of MW from Bus n to Bus m is linearly equivalent to outaging the transmission line • Let’s assume we know what is equal to, then we can calculate the values relevant to the LODF. • Note: The negative sign is used so that the notation is consistent with LODF “transfer”

26. Calculation of LCDF • Estimate of post-closure flow on Line L • Thus we can write • Thus the LCDF, is exactly equal to the PTDF for a transfer between the terminals of the line

27. We have a Problem! • We don’t know what the flow on the branch after closing the line will be! • The line is presently OPEN, so the present flow is zero! • Instead define Pk as the flow on the presently open branch using the present voltage and angle at the line terminals. • Calculate PTDFk based on the bus voltages and sensitivities (again even though the branch is presently open) • Think a while and you’ll realize you can write

28. Numerical Trick to calculate MLCDF • In a piece of software we won’t actually store the “LCDF” factor but instead MLCDF • Instead we’ll define what I like to call the “Modified LCDF” which is based on the percentage of Pk instead. • This is a bit weird to a user because Pk represents the flow on an open branch  this is just a numerical trick

29. LODF and LCDF • LODF (LCDF) – Gives the ability to model a single line outage (closure) event • OTDF – incremental impact on a particular branch while also considering a single line outage • OMW – estimate of the flow on a particular branch after a single line outage • Still can’t model multiple line outages simultaneously • Can not just sum because the lines that are being outaged also interact with each other

30. Linear Impact of a Contingency with Multiple Outages • Outage Transfer Distribution Factors (OTDFs) • The percent of a transfer that will flow on a branch M after the contingency occurs • Outage Flows (OMWs) • The estimated flow on a branch M after the contingency occurs

31. OTDFs and OMWs • Single Contingency • Multiple Contingencies • What are and ?

32. Determining NetPTDFKand NetMWK • Each NetPTDFK is a function of all the other NetPTDFs because the change in status of a line affects all other lines. • Assume we know all NetPTDFs except for NetPTDF1. Then we can write: • In general for each Contingent Line N, write

33. Determining NetPTDFKand NetMWK • Thus we have a set of nc equations and nc unknowns (nc= number of contingent lines) • Thus • Same type of derivation shows Known Values

34. Putting all these Linear Steps together • Ultimate goal in some tools is to solve for a contingency solution without doing a power flow • What if you wanted to run 10,000,000 contingencies and only had a few minutes to do it? • PTDFs • Model a group of injection changes as just a “transfer” from the positive injection changes to the negative changes • Generator outage • DC line setpoint change • Load outage • Line outages that isolate generation or load • LODF and LCDF combinations • Multiple outage LODF “matrices” as Dr. Overbye showed earlier

35. Trick with Multiple Line Outages Resulting in an Island Every Row of matrix sums to zero This means it’s SINGULAR!

36. Trick with Multiple Line Outages Resulting in an Island • Theoretical Choice • Don’t model outage of every line. Just model outage of all but one of the lines • Lazy/Clever coder • During the LU decomposition done for matrix inversion, if you come across a zero on the diagonal, then just stick a giant number in it • Same as forcing NetMWnc = 0 which essentially eliminates the last row and column but doesn’t require code to figure out when to do this

37. Generator MW Control in a Power Flow • Just jump over to the advanced training topic on Advanced Power Flow • http://www.powerworld.com/files/S02AdvancedPowerFlow.pdf

38. Generator Remote Bus Voltage Regulation and Sharing • I’ll just write this on the board generally • Didn’t have time to make slides for this! • It’s rather clever really

39. Line Drop Compensation • Power Flow cases often model have remote bus voltage regulation configured • In reality this isn’t done in the very short term control systems (dozens of seconds and faster) • Results in unstable operation if a generator’s exciter is chasing around every 10 milliseconds trying to keep this voltage pegged • Long ago the generator only controlled it’s terminal bus voltage in the exciter to a set-point • Line Drop Compensation is a compromise • Allows you to regulate of fictitious point a certain impedance out into the system • Typically values of approximately 50% of the impedance of the step-up transformer are used • BPA research in the 1980s showed this worked best to improve voltage stability but not cause instability (Carson Taylor)

40. Generator S, I, V definitions

41. Fictitious Impedance “Xcomp”

42. Voltage-Dependent Reactive Power Output

43. Voltage-Dependent Reactive Power Output • Generator is modeled as a PQ bus in the power flow, but the reactive power is voltage-dependent • Modeled very similarly to a constant current load which is also voltage dependent • Requires the calculation of dQ/dV for an extra term on the diagonal of the Jacobian matrix

44. Mvar Limits • This is always the hardest part of the power flow equations! • Hitting a generator generator hit its min/max limit • This is pretty simple. If it’s outside limits, just change to a constant PQ injection • Backing off a generator Mvar limit • For a normal generator we calculate dV/dQ. If sensitivity says that backing off the limit pushes us toward meeting the generator voltage setpoint, then back-off the limit • Trickier here, because we need to calculate derivate of the voltage at a fictitious bus to a change in reactive power injection. • Do some algebra!

45. Use Chain Rule to calculate dV/dQ • Basic Chain Rule • Pieces are calculated as • Last piece is calculate in customary way using spare vector methods on factored Jacobian

46. More tricks to handle bad data • What if • We have a problem as we’re taking square root of a negative value • This represents the maximum power transfer point • This happens in software when a typo occurs • User enters Xcomp = 0.5 instead of 0.05 • This in never possible in a real system, but software inputs can be messed up so you have to handle it! • Some special trick also employed to ensure that the following is always positive

47. Reactive Current Compensation • Two generators on the same terminal bus • This happens all the time at Hydro Electric Power Plants • Generators at the same bus can not have their control systems set up to regulate their terminal buses to the same voltage • This is inherently unstable because their exciters end up fighting with each other • Solution: Reactive Current Compensation • Generator regulate a fictitious voltage “behind” their terminal bus (X < 0) • These are different electrical points so the control system doesn’t fight

48. Combination of the Line Drop and Reactive Current • It’s also possible to have a combination of Line Drop and Reactive Current Compensation! • Just draw on board and generally explain