inear AC Power Flow for Disaster Management Carleton Coffrin Department of Computer Science Optimization ab D-4 Infrastructure Analysis 1
2010, 2011 2010, 2011, 2012 2000, 2011 2005, 2011 2
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A Model of Power Restoration A-UR 10-03860
Power System G G - Generator B - Bus B - oad B - Power ine G B B 5
What do we know? G G B B G B B B G B B B 6
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
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
Disaster Recovery - Maximal Dispatch +210-120 P i = nx k=1-30 -70 0 b ik ( i k ) 0 0 0 0 Maximal Dispatch 0 9
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
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
Disaster Recovery - DC Restoration Plan DC Restoration Timeline DC Power Flow (MW) 6000 7000 8000 DC AC DC 0 10 20 30 40 50 Restoration Action 12
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
Disaster Recovery - DC Restoration Plan (AC) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) 6000 7000 8000 DC AC Power Flow (MW) 6000 7000 8000 AC not converged DC 0 10 20 30 40 50 0 10 20 30 40 50 Restoration Action Restoration Action Solving a large AC system without a known basepoint can be maddeningly difficult [Overbye 04] 14
Accuracy of DC Under arge Contingencies A simple experiment A-UR 10-03860
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
IEEE-30 Contingencies N-9 N-11 N-12 N-13 N-15 N-16 N-17 % DC 7436 6511 5344 6805 5931 7236 6877 66% What s broken? (N-13 in detail - P, Theta, Q, V ) 17
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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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
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
IEEE-30 Contingencies N-9 N-11 N-12 N-13 N-15 N-16 N-17 % DC 7436 6511 5344 6805 5931 7236 6877 66% PAC+R 9990 9988 9996 9936 9996 9847 9386 98.8% PAC +R+V 9998 10000 9996 9981 9998 10000 9911 99.8% N-9 N-11 N-12 N-13 N-15 N-16 N-17 DC 14.2 20.14 57.77 73.67 44.54 64.58 67.88 PAC+R 35.89 30.35 57.74 62.87 57.15 66.14 64.63 PAC +R+V 35.96 30.38 57.74 62.83 57.49 66.69 64.73 25
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) 6000 7000 8000 0 10 20 30 40 50 Restoration Action DC AC Power Flow (MW) 6000 7000 8000 0 10 20 30 40 50 Restoration Action DC DC PAC+R PAC+R+V 26
Disaster Recovery - PAC Restoration Plan (S1) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) 5000 6000 7000 8000 DC PAC+R+V AC Power Flow (MW) 5000 6000 7000 8000 DC PAC+R+V 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Restoration Action Restoration Action 27
ine Overloads (S1) AC ine Overloads Cumulative Overload (MVA) 0 200 400 600 800 Potential ine Failure DC PAC+R+V 0 10 20 30 40 50 60 Restoration Action 28
Reactive Injection Overloads (S1) Reactive Generation Violations Cumulative Violations (MVar) 0 200 400 600 800 1000 DC PAC+R+V 0 10 20 30 40 50 60 Restoration Action 29
Extreme Voltage Values (S1) AC Voltage Violations Cumulative Violation (Volts p.u.) 0.0 0.2 0.4 0.6 0.8 DC PAC+R+V 0 10 20 30 40 50 60 Restoration Action 30
Disaster Recovery - PAC Restoration Plan (S12) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) 6500 7000 7500 8000 8500 DC PAC+R+V AC Power Flow (MW) 6500 7000 7500 8000 8500 DC PAC+R+V 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Restoration Action Restoration Action 31
ine Overloads and Reactive Overloads (S12) AC ine Overloads Reactive Generation Violations Cumulative Overload (MVA) 0 100 200 300 400 500 600 700 DC PAC+R+V Cumulative Violations (MVar) 0 200 400 600 800 1200 DC PAC+R+V 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Restoration Action Restoration Action 32
Disaster Recovery - PAC Restoration Plan (S16) DC Restoration Timeline AC Restoration Timeline DC Power Flow (MW) 6000 7000 8000 PAC+R PAC+R+V AC Power Flow (MW) 6000 7000 8000 PAC+R PAC+R+V 0 10 20 30 40 50 0 10 20 30 40 50 Restoration Action Restoration Action 33
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
Fin cjc@cs.brown.edu 35
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