Linear AC Power Flow for Disaster Management

Linear AC Power Flow for Disaster Management

Carleton Coffrin

D-4

Infrastructure Analysis

Department of Computer Science

Optimization Lab

2005, 2011 2 2000, 2011 2010, 2011

LA-UR 10-03860

Power System 5 L B G - Generator - Bus - Load - Power Line B L L L L L L L L G B B B L G L

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

Restoration Plans

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Bus 1 Line 1 Bus 2 Line 2 Line 3

Bus 1 Line 2 Line 1 Bus 2

Challenges of Power Restoration Plans

• NOT close to normal operations

• Constraints:

1) Load Balance (Active / Reactive Generation Limits) 2) Line Capacities

3) Voltage Support

4) Standing Phase Angles 5) Ramp Rates

6) Control system / Transient Stability 7) Frequency Management

• Often close to operation limits

8

Disaster Recovery - Maximal Dispatch 9 P_i = n X k=1 b_ik( _{i k}) -120 0 0 0 +210 0 -30 -70 0 0 Maximal Dispatch

Inputs:

pgn - maximum active injection for bus n

pl

n - desired active load at bus n

f_nm - line load limits

Variables:

pg_{n 2} (0, pgn) - active generation at bus n

ln 2 (0,1) - percentage of load served at bus n

Maximize:_X n₂N ln (1) Subject to: pl nln = pgn + X m bnm(✓n ✓m) 8n 2 N (2) fnm  bnm(✓n ✓m)  fnm 8hn, mi 2 L (3)

Disaster Recovery - Maximal Dispatch (LDC)

What is a Restoration Plan Optimization

11

? ? ? ? ?

LP LP LP LP LP

MIP

See: Vehicle Routing for the Last Mile of Power System Restoration.

P. Van Hentenryck, C. Coffrin, and R. Bent. (PSCC'11)

Disaster Recovery - LDC Restoration Plan 12 0 10 20 30 40 50 6000 7000 8000 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LDC

LDC AC

Steady-State AC Model 13 Q_i = n X k=1 |V_i||V_k|(g_ik sin( _{i k}) + b_ik cos( _{i k})) P_i = n k=1 |V_i||V_k|(g_ik cos( _{i k}) + b_ik sin( _{i k})) AC: P_i = n X k=1 b_ik( _{i k}) LDC: ?

Disaster Recovery - LDC Restoration Plan (AC) 14 0 10 20 30 40 50 6000 7000 8000 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LDC 0 10 20 30 40 50 6000 7000 8000 AC Restoration Timeline Restoration Action A C P o w er Flo w (MW) LDC AC not converged

Solving a large AC system without a known base-point can be “maddeningly difficult” [Overbye ’04]

LA-UR 10-03860

Accuracy of LDC Under Large Contingencies

A Damaged Network Experiment

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• 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)

IEEE-30 Contingencies 17 N-9 N-11 N-12 N-13 N-15 N-16 N-17 % LDC 7436 6511 5344 6805 5931 7236 6877 66%

What’s broken?

Comparing Network Flows 18 AC Value LDC V alue _(0,0) DC Overestimation DC Underestimation

- Typical Solution - Large Angle Solution

15 _{ |}✓_i ✓_k|

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Reactive power and voltage drops are a problem.

Idea: Use the LPAC model to enforce bounds on

Variables:

pg_{n 2} (0, pgn) - active generation at bus n

q_ng ₂ ( ₁,₁) - reactive generation at bus n

ln 2 (0, 1) - percentage of load served at bus n

Variables from core LPAC Model

Maximize:_X n₂N ln (1) Subject to: pn = pl_nln + pg_n 8n 2 N (2) qn = q_nl ln + q_ng 8n 2 N (3) q_ng = 0 ₈n ₂ N _\ G (4) qn = n_X6=m m₂N ˆ q_nmt + ˆq_{nm 8}n ₂ G (5) q_ng _ qng 8n 2 G (6) 0.1 _ n  0.1 (7)

Constraints from core LPAC Model

Disaster Recovery - Maximal Dispatch (LPAC)

IEEE-30 Contingencies 25 N-9 N-11 N-12 N-13 N-15 N-16 N-17 % LDC 7436 6511 5344 6805 5931 7236 6877 66% LPAC+R 9990 9988 9996 9936 9996 9847 9386 98.8% LPAC +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 LDC 14.2 20.14 57.77 73.67 44.54 64.58 67.88 LPAC+R 35.89 30.35 57.74 62.87 57.15 66.14 64.63 LPAC +R+V 35.96 30.38 57.74 62.83 57.49 66.69 64.73

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With LPAC we have a feasible linear model!

Revisit the ROP. Can LPAC be converted to a AC power flow solution?

0 10 20 30 40 50 6000 7000 8000 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LDC 0 10 20 30 40 50 6000 7000 8000 AC Restoration Timeline Restoration Action A C P o w er Flo w (MW) LDC Legend LDC LPAC+R+V LPAC+R

Disaster Recovery - LPAC Restoration Plan (S1) 27 0 10 20 30 40 50 60 5000 6000 7000 8000 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LDC LPAC+R+V 0 10 20 30 40 50 60 5000 6000 7000 8000 AC Restoration Timeline Restoration Action A C P o w er Flo w (MW) LDC LPAC+R+V

Line Overloads (S1) 28 0 10 20 30 40 50 60 0 200 400 600 800 AC Line Overloads Restoration Action Cum ulativ e Ov er load (MV A) LDC LPAC+R+V Potential Line Failure

Reactive Injection Overloads (S1) 29 0 10 20 30 40 50 60 0 200 400 600 800 1000

Reactive Generation Violations

Restoration Action Cum ulativ e Violations (MV ar) LDC LPAC+R+V

Extreme Voltage Values (S1) 30 0 10 20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 AC Voltage Violations Restoration Action Cum ulativ e Violation (V olts p .u.) LDC LPAC+R+V

Disaster Recovery - LPAC Restoration Plan (S12) 31 0 5 10 15 20 25 30 35 6500 7000 7500 8000 8500 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LDC LPAC+R+V 0 5 10 15 20 25 30 35 6500 7000 7500 8000 8500 AC Restoration Timeline Restoration Action A C P o w er Flo w (MW) LDC LPAC+R+V

Line Overloads and Reactive Overloads (S12) 32 0 5 10 15 20 25 30 35 0 100 200 300 400 500 600 700 AC Line Overloads Restoration Action Cum ulativ e Ov er load (MV A) LDC LPAC+R+V 0 5 10 15 20 25 30 35 0 200 400 600 800 1200

Reactive Generation Violations

Restoration Action Cum ulativ e Violations (MV ar) LDC LPAC+R+V

Disaster Recovery - LPAC Restoration Plan (S16) 33 0 10 20 30 40 50 6000 7000 8000 AC Restoration Timeline Restoration Action A C P o w er Flo w (MW) LPAC+R LPAC+R+V 0 10 20 30 40 50 6000 7000 8000 DC Restoration Timeline Restoration Action DC P o w er Flo w (MW) LPAC+R LPAC+R+V

Conclusion

• Be skeptical of the LDC model under abnormal network conditions.

• The LPAC 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)

F

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[email protected]