Linear AC Power Flow for Disaster Management

Transcription

1 inear AC Power Flow for Disaster Management Carleton Coffrin Department of Computer Science Optimization ab D-4 Infrastructure Analysis 1

2 2010, , 2011, , ,

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4 A Model of Power Restoration A-UR

5 Power System G G - Generator B - Bus B - oad B - Power ine G B B 5

6 What do we know? G G B B G B B B G B B B 6

7 Restoration Plans Bus 1 ine 1 Bus 2 ine 2 ine 3 Which restoration plan is best? ine 3 Bus 1 ine 2 ine 1 Bus 2 7

8 Challenges of Power Restoration Plans NOT close to normal operations Constraints: 1) oad Balance (Active / Reactive Generation imits) 2) ine Capacities 3) Voltage Support 4) Standing Phase Angles Steady State 5) Ramp Rates 6) Control system / Transient Stability 7) Frequency Management Often close to operation limits 8

9 Disaster Recovery - Maximal Dispatch P i = nx k= b ik ( i k ) Maximal Dispatch 0 9

10 Disaster Recovery - Maximal Dispatch (DC) Inputs: p g n - maximum active injection for bus n p l n - desired active load at bus n f nm - line load limits Variables: p g n 2 (0, p g n) - active generation at bus n l n 2 (0, 1) - percentage of load served at bus n Maximize: X l n (1) n2n Subject to: p l nl n = p g n + X b nm ( n m ) 8n 2 N (2) m f nm apple b nm ( n m ) apple f nm 8hn, mi 2 (3) 10

11 What is a Restoration Plan Optimization????? P P P P P MIP [Bus 1, ine 1, Bus 2, ine 2, ine 3] See: Vehicle Routing for the ast Mile of Power System Restoration. P. Van Hentenryck, C. Coffrin, and R. Bent. (PSCC'11) 11

12 Disaster Recovery - DC Restoration Plan DC Restoration Timeline DC Power Flow (MW) DC AC DC Restoration Action 12

13 Steady-State AC Model DC: P i = nx k=1 b ik ( i k ) AC: P i = Q i = n k=1 nx V i V k (g ik sin( i k )+b ik cos( i k )) k=1? V i V k (g ik cos( i k )+b ik sin( i k )) 13

14 Disaster Recovery - DC Restoration Plan (AC) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) DC AC Power Flow (MW) AC not converged DC Restoration Action Restoration Action Solving a large AC system without a known basepoint can be maddeningly difficult [Overbye 04] 14

15 Accuracy of DC Under arge Contingencies A simple experiment A-UR

16 A Damaged Network Experiment Start with a simple and well understood network (IEEE 30) Remove some lines and see what happens N-3 (10000 samples) N-4 (10000 samples)... N-20 (10000 samples) How many models converge to an AC solution? 16

17 IEEE-30 Contingencies N-9 N-11 N-12 N-13 N-15 N-16 N-17 % DC % What s broken? (N-13 in detail - P, Theta, Q, V ) 17

18 Comparing Network Flows P Theta Q V DC Value (0,0) DC Underestimation AC Value 15 apple i k - Typical Solution - arge Angle Solution 18

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23 Reactive power and voltage drops are a problem. q g n apple q g n e V apple e V apple e V Idea: Use the PAC model to enforce bounds on reactive injection and voltage deviation. 23

24 Disaster Recovery - Maximal Dispatch (PAC) Variables: p g n 2 (0, p g n) - active generation at bus n qn g 2 ( 1, 1) - reactive generation at bus n l n 2 (0, 1) - percentage of load served at bus n Variables from core PAC Model Maximize: X l n (1) n2n Subject to: p n = p l nl n + p g n 8n 2 N (2) q n = qnl l n + qn g 8n 2 N (3) qn g =0 8n 2 N \ G (4) n6=m X q n = ˆq nm t +ˆq nm 8n 2 G (5) m2n qn g apple qn g 8n 2 G (6) 0.1 apple n apple 0.1 (7) Constraints from core PAC Model 24

25 IEEE-30 Contingencies N-9 N-11 N-12 N-13 N-15 N-16 N-17 % DC % PAC+R % PAC +R+V % N-9 N-11 N-12 N-13 N-15 N-16 N-17 DC PAC+R PAC +R+V

26 With PAC we have a feasible linear model! Revisit the ROP. Can PAC be converted to a AC power flow solution? DC Restoration Timeline AC Restoration Timeline egend DC Power Flow (MW) Restoration Action DC AC Power Flow (MW) Restoration Action DC DC PAC+R PAC+R+V 26

27 Disaster Recovery - PAC Restoration Plan (S1) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) DC PAC+R+V AC Power Flow (MW) DC PAC+R+V Restoration Action Restoration Action 27

28 ine Overloads (S1) AC ine Overloads Cumulative Overload (MVA) Potential ine Failure DC PAC+R+V Restoration Action 28

29 Reactive Injection Overloads (S1) Reactive Generation Violations Cumulative Violations (MVar) DC PAC+R+V Restoration Action 29

30 Extreme Voltage Values (S1) AC Voltage Violations Cumulative Violation (Volts p.u.) DC PAC+R+V Restoration Action 30

31 Disaster Recovery - PAC Restoration Plan (S12) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) DC PAC+R+V AC Power Flow (MW) DC PAC+R+V Restoration Action Restoration Action 31

32 ine Overloads and Reactive Overloads (S12) AC ine Overloads Reactive Generation Violations Cumulative Overload (MVA) DC PAC+R+V Cumulative Violations (MVar) DC PAC+R+V Restoration Action Restoration Action 32

33 Disaster Recovery - PAC Restoration Plan (S16) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) PAC+R PAC+R+V AC Power Flow (MW) PAC+R PAC+R+V Restoration Action Restoration Action 33

34 Conclusion Be skeptical of the DC model under abnormal network conditions. The PAC model enables constraints voltage and reactive power, leading to feasible AC power flows. Validated on, A simple N-k experiment on the IEEE-30 benchmark. Restoration Order Problems arising in real-world disaster recovery data. Next steps! (transient stability, frequency management) 34

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