An MINLP approach to forming secure islands in electricity networks

Transcription

1 An MINLP approach to forming secure islands in electricity networks Waqquas Bukhsh, Jacek Gondzio, Andreas Grothey, Ken McKinnon, Paul Trodden School of Mathematics University of Edinburgh GOR 2012, 4-7 September 2012, Hannover A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

2 Blackout Prevention by Islanding For a power system in the early stages of a cascading blackout, is it possible to deliberately split the system into islands to contain the fault EPSRC funded, Multi-site project: Southampton (Maths), Durham/Edinburgh (Power Systems Engineering) OR, Edinburgh: looking at Models, Solvers (Ken McKinnon, Jacek Gondzio, Andreas Grothey, Paul Trodden, Waqquas Bukhsh) Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 Island 4 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

3 USA August 2003 Happy customers A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

4 USA August 2003 Happy customers Oooops! There goes Long Island, Detroit, Ottowa, Toronto... A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

5 USA Aug 2003 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

6 USA August 2003 Inadequate system understanding situational awareness tree trimming diagnostic support A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

7 The N- Oh... NO Contingency Problem Goal Develop a tool that can create a Firewall to isolate the troubled area from the rest of the network. Leave network in a stable steady state. Could be used for off line analysis to prepare responses to faults in different areas or, if fast enough, to react in real time. A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

8 Islanding of uncertain network Network has uncertain buses/lines/generators Firewall around uncertain buses, lines, generators A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

9 Islanding of uncertain network Network has uncertain buses/lines/generators Firewall around uncertain buses, lines, generators A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

10 Islanding of uncertain network Network has uncertain buses/lines/generators Firewall around uncertain buses, lines, generators Isolate uncertain components To: achieve new safe steady state By: cutting lines, shedding loads, adjusting generation Goal: minimize load lost / at risk A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

11 Islanding of uncertain network Uncertain busesb 0 & linesl 0 Isolate by cutting lines Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

12 Islanding of uncertain network Uncertain busesb 0 & linesl 0 Isolate by cutting lines Section 0 uncertain parts, Section 1 certain parts Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

13 Islanding of uncertain network Uncertain busesb 0 & linesl 0 Isolate by cutting lines Section 0 uncertain parts, Section 1 certain parts Optimization decides what else in Sect. 0/1 Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

14 Islanding of uncertain network Uncertain busesb 0 & linesl 0 Isolate by cutting lines Section 0 uncertain parts, Section 1 certain parts Optimization decides what else in Sect. 0/1 All load in Section 0 is at risk chanceβ of surviving Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

15 Effect ofβ Growth of section 0 in 24-bus network asβ 1 Pre islanding ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

16 Effect ofβ Growth of section 0 in 24-bus network asβ β 0.56 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

17 Effect ofβ Growth of section 0 in 24-bus network asβ β 0.93 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

18 Effect ofβ Growth of section 0 in 24-bus network asβ β 0.97 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

19 Effect ofβ Growth of section 0 in 24-bus network asβ β 1.00 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

20 Effect ofβ Growth of section 0 in 24-bus network asβ β 1.00 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

21 Sectioning constraints Uncertain busesb 0 & linesl 0 γ b = section for bus b ρ bb = 0 iff line{b, b } cut Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

22 Sectioning constraints ρ bb 1 +γ b γ b {b, b } L\L 0 ρ bb 1 γ b +γ b {b, b } L\L 0 No lines between sections No lines between sections Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

23 Sectioning constraints ρ bb 1 +γ b γ b {b, b } L\L 0 No lines between sections ρ bb 1 γ b +γ b {b, b } L\L 0 No lines between sections γ b = 0 b B 0 All unsafe buses in 0 Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

24 Sectioning constraints ρ bb 1 +γ b γ b {b, b } L\L 0 No lines between sections ρ bb 1 γ b +γ b {b, b } L\L 0 No lines between sections γ b = 0 b B 0 All unsafe buses in 0 ρ bb 1 γ b {b, b } L 0 No unsafe lines in 1 ρ bb 1 γ b {b, b } L 0 No unsafe lines in 1 Section 1 Section 0 Section 1 Island 1 Island 2 Island 3 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

25 Network Flow Model v p L b,δ b bb v b,δ b ρ l bus b bus b Notation Parameters: Pd D, QD d real and reactive power demand at load d Variables: v b,θ b voltage and phase angle at bus b pbb L, ql bb real and reactive power flows into line (b, b ) from bus b pg G, qg g real and reactive power output of generator g α d proportion of load d connected α d [0, 1] A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

26 Network Flow Model v p L b,δ b bb v b,δ b ρ l bus b bus b Bounds on variables p bb + p b b H bb Thermal limit on line V b v b V + b Voltage limit at bus e.g. v b [0.95, 1.05] Q G g q G g Q G+ g ζ g Pg G pg ζ G g Pg G+ Reactive Power limits Q G g, Q G+ g independent of current operating point Real Power limits Pg G, Pg G+ dependent on current operating point ζ g = 0 if generator is switched off ζ g = 1 if generator remains on A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

27 Network Flow Model v p L b,δ b bb v b,δ b ρ l bus b bus b Kirchhoff Current Law (KCL): conservation of flow at buses pg G = α d Pd D + pbb L Real balance g G b d D b b B b q G g = α d Q D d + Reactive balance g G b d D b b B b q L bb A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

28 Network Flow Model v p L b,δ b bb v b,δ b ρ l bus b bus b Kirchhoff Current Law (KCL): conservation of flow at buses pg G = α d Pd D + pbb L Real balance g G b d D b b B b q G g = α d Q D d + Reactive balance g G b d D b b B b q L bb Kirchhoff Voltage Law pbb L = G bbvb 2 + v ) bv b ( Gbb cosδ bb + B bb sinδ bb q L bb = B bbv 2 b + v b v b ( Gbb sinδ bb B bb cosδ bb ) Cbb v 2 b Because of power losses in the lines: p L bb pl b b and ql bb ql b b. A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

29 Disconnecting Lines v p L b,δ b bb v b,δ b ρ l bus b bus b Modified KVL pbb L = G bb ˆv b (ˆv b ˆv b cosδ bb ) B bb ˆv bˆv b sinδ bb Pbb 0 (1 ρ bb ) Whenρ bb = 1, (line is connected): Force ˆv b = v b, ˆv b = v b, δ bb =θ b θ b, KVL holds Whenρ bb = 0, (line is not connected): Pbb 0 := G bb V b (V b V b cos 0) B bb V b V b sin 0 Force ˆv b = Vb, ˆv b = V b, δ bb = 0, pbb L = 0 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

30 Disconnecting Lines v p L b,δ b bb v b,δ b ρ l bus b bus b Modified KVL pbb L = G bb ˆv b (ˆv b ˆv b cosδ bb ) B bb ˆv bˆv b sinδ bb Pbb 0 (1 ρ bb ) Whenρ bb = 1, (line is connected): Force ˆv b = v b, ˆv b = v b, δ bb =θ b θ b, KVL holds Whenρ bb = 0, (line is not connected): Pbb 0 := G bb V b (V b V b cos 0) B bb V b V b sin 0 Force ˆv b = Vb, ˆv b = V b, δ bb = 0, pbb L = 0 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

31 Disconnecting Lines v p L b,δ b bb v b,δ b ρ l bus b bus b Modified KVL pbb L = G bb ˆv b (ˆv b ˆv b cosδ bb ) B bb ˆv bˆv b sinδ bb Pbb 0 (1 ρ bb ) Whenρ bb = 1, (line is connected): Force ˆv b = v b, ˆv b = v b, δ bb =θ b θ b, KVL holds Whenρ bb = 0, (line is not connected): Pbb 0 := G bb V b (V b V b cos 0) B bb V b V b sin 0 Force ˆv b = Vb, ˆv b = V b, δ bb = 0, pbb L = 0 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

32 Disconnecting Lines v p L b,δ b bb v b,δ b ρ l bus b bus b Modified KVL pbb L = G bb ˆv b (ˆv b ˆv b cosδ bb ) B bb ˆv bˆv b sinδ bb Pbb 0 (1 ρ bb ) Whenρ bb = 1, (line is connected): Force ˆv b = v b, ˆv b = v b, δ bb =θ b θ b, KVL holds Whenρ bb = 0, (line is not connected): Pbb 0 := G bb V b (V b V b cos 0) B bb V b V b sin 0 Force ˆv b = Vb, ˆv b = V b, δ bb = 0, pbb L = 0 Voltage and angle relations enforced by standard MILP constraints. A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

33 Objective Minimize load shed / at risk, Maximize expected delivery Planned supply pd D =α { dpd D β if in section 0 Supply probability = 1 if in section 1 Objective maxβ α d Pd D + α d Pd D d S 0 d S 1 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

34 Objective Minimize load shed / at risk, Maximize expected delivery Planned supply pd D =α { dpd D β if in section 0 Supply probability = 1 if in section 1 Objective maxβ α d Pd D + α d Pd D d S 0 d S 1 d S 0 /S 2 is a decision new formulation Formulation Splitα d α d =α 0d +α 1d, 0 α 1d γ b Objective max d P D d (βα 0d +α 1d ) A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

35 Solving the problem Islanding formulation with AC power flow is MINLP Procedure 1 Solve to optimum with approximated power flow model (MILP) 2 Cut lines 3 Solve AC optimal load shedding opt. (NLP) A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

36 Solving the problem Islanding formulation with AC power flow is MINLP Procedure 1 Solve to optimum with approximated power flow model (MILP) 2 Cut lines 3 Solve AC optimal load shedding opt. (NLP) We consider 2 approximations to Kirchhoff s Laws: DC Constant Loss: Drop the reactive power constraints Set all voltages to 1 Assume line loss p Loss bb = its pre-islanding value A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

37 Solving the problem Islanding formulation with AC power flow is MINLP Procedure 1 Solve to optimum with approximated power flow model (MILP) 2 Cut lines 3 Solve AC optimal load shedding opt. (NLP) We consider 2 approximations to Kirchhoff s Laws: DC Constant Loss: Drop the reactive power constraints Set all voltages to 1 Assume line loss p Loss bb = its pre-islanding value AC PWL: Include reactive power constraints Linearize KVL Include PWL approximation of cosine terms A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

38 Approximated Power flow models: DC v p L b,δ b bb v b,δ b ρ l bus b bus b AC power flow ˆp bb L = G bbvb 2 ) + v b v b ( Gbb cosδ bb + B bb sinδ bb ˆq bb L = B bbvb 2 + v ) bv b ( Gbb sinδ bb B bb cosδ bb A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

39 Approximated Power flow models: DC v p L b,δ b bb v b,δ b ρ l bus b bus b DC power flow B bb G bb v b 1 sinδ bb δ bb (susceptance conductance) A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

40 Approximated Power flow models: DC v p L b,δ b bb v b,δ b ρ l bus b bus b DC power flow B bb G bb v b 1 sinδ bb δ bb Then p bb = B bb δ bb q bb = 0 (susceptance conductance) A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

41 Approximated Power flow models: DC v p L b,δ b bb v b,δ b ρ l bus b bus b DC power flow B bb G bb v b 1 sinδ bb δ bb Then No losses (susceptance conductance) p bb = B bb δ bb q bb = 0 p bb + p b b = 0 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

42 Computational tests Steps 1 Solve OPF: for pre-islanded state using the full AC equations. 2 Remember predicted line-loss 3 Based on this solution solve the DC -MILP problem to decide the best islands and generator shut downs. 4 With the islands and generator shutdowns fixed, solve an optimal load shedding problem using the full AC equations. A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

43 Computational tests Steps 1 Solve OPF: for pre-islanded state using the full AC equations. 2 Remember predicted line-loss 3 Based on this solution solve the DC -MILP problem to decide the best islands and generator shut downs. 4 With the islands and generator shutdowns fixed, solve an optimal load shedding problem using the full AC equations. Software and examples 1 and 4 use IPOPT (COIN-OR). 3 uses CPLEX 12.3 (using 8 cores) IEEE test cases up to 300 buses. A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

44 AC infeasibility (DC constant loss) Setup: all single lines/generators marked as uncertain in turn: Some AC solutions were infeasible because voltage limits were violated. Percentage that were AC infeasible: n B AC infeasible A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

45 AC infeasibility (DC constant loss) Setup: all single lines/generators marked as uncertain in turn: Some AC solutions were infeasible because voltage limits were violated. Percentage that were AC infeasible: n B AC infeasible What can be done Relaxing voltage limits in the contingency 0.9 v 1.1 cures some cases. Alter the tap setting post-islanding Relax the post contingency line A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

46 AC infeasibility (DC constant loss) Setup: all single lines/generators marked as uncertain in turn: Some AC solutions were infeasible because voltage limits were violated. Percentage that were AC infeasible: n B AC infeasible What can be done Relaxing voltage limits in the contingency 0.9 v 1.1 cures some cases. Alter the tap setting post-islanding Relax the post contingency line Use AC PWL in MILP for islanding A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

47 Power flow models: Derivation of AC-PWL model v p L b,δ b bb v b,δ b ρ l bus b bus b AC power flow p bb = G bb vb 2 ) + v b v b ( Gbb cosδ bb + B bb sinδ bb q bb = B bb vb 2 + v bv b ( Gbb sinδ bb B bb cosδ bb ) Cbb vb 2 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

48 Power flow models: Derivation of AC-PWL model v p L b,δ b bb v b,δ b ρ l bus b bus b Piecewise linear AC power flow Define p ave bb = (p bb p b b)/2 p loss bb = p bb + p b b q ave bb = (q bb q b b)/2 q loss bb = q bb + q b b A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

49 Power flow models: Derivation of AC-PWL model v p L b,δ b bb v b,δ b ρ l bus b bus b Piecewise linear AC power flow Define p ave bb = (p bb p b b)/2 p loss bb = p bb + p b b q ave bb = (q bb q b b)/2 q loss bb = q bb + q b b Then p bb = pbb loss /2 + pave bb p b b = pbb loss /2 pave bb q bb = qloss bb /2 + qave bb q b b = qbb loss /2 qave bb A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

50 AC-PWL approximation: real power flow p ave bb = 1 2 (pl bb pl b b) = G bb 2 (v 2 b v 2 b ) B bb v bv b sinδ bb = G(v b v b ) + G 2 ((v b 1) 2 (v b 1) 2 ) Bδ + B(δ sinδ) + b sinδ(1 v b v b ) p loss bb = pl bb + pl b b ) = G bb (v 2 b + v 2 b ) 2G bb v bv b cosδ bb = G(v b v b ) 2 + 2G(1 cosδ) + 2G(v b v b 1)(1 cosδ) Piecewise-linear (PWL) approximation A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

51 AC-PWL approximation: real power flow p ave bb = 1 2 (pl bb pl b b) = G bb 2 (v 2 b v 2 b ) B bb v bv b sinδ bb = G(v b v b ) + G 2 ((v b 1) 2 (v b 1) 2 ) Bδ + B(δ sinδ) + b sinδ(1 v b v b ) p loss bb = pl bb + pl b b ) = G bb (v 2 b + v 2 b ) 2G bb v bv b cosδ bb = G(v b v b ) 2 + 2G(1 cosδ) + 2G(v b v b 1)(1 cosδ) Piecewise-linear (PWL) approximation Keep linear terms in v,δ A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

52 AC-PWL approximation: real power flow p ave bb = 1 2 (pl bb pl b b) = G bb 2 (v 2 b v 2 b ) B bb v bv b sinδ bb = G(v b v b ) + G 2 ((v b 1) 2 (v b 1) 2 ) Bδ + B(δ sinδ) + b sinδ(1 v b v b ) p loss bb = pl bb + pl b b ) = G bb (v 2 b + v 2 b ) 2G bb v bv b cosδ bb = G(v b v b ) 2 + 2G(1 cosδ) + 2G(v b v b 1)(1 cosδ) Piecewise-linear (PWL) approximation Keep linear terms in v,δ Use piecewise linear (SOS2 set) approximation of cos A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

53 AC-PWL approximation: reactive power flow q ave bb = 1 2 (q bb q b b) = B bb 2 (v 2 b v 2 b ) G bb v bv b sinδ bb C bb (v 2 b v 2 b ) = (B+C)(v b v b ) B+C 2 ((v b 1) 2 (v b 1) 2 ) Gδ + G(δ sinδ) + G(1 v b v b ) sinδ q loss bb = q bb + q b b = 2B bb v b v b cosδ bb B bb (v 2 b + v 2 b ) C bb (v 2 b + v 2 b ) = 2B(1 cosδ) B(v b v b ) 2 + 2B(v b v b )(1 cosδ) Piecewise-linear (PWL) approximation Keep linear terms in v,δ 2C(v b v b 1) C((v b 1) 2 + (v b 1) 2 ) Use piecewise linear (SOS set) approximation of cos Also model reactive power flow A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

54 AC-PWL approximation: reactive power flow q ave bb = 1 2 (q bb q b b) = B bb 2 (v 2 b v 2 b ) G bb v bv b sinδ bb C bb (v 2 b v 2 b ) = (B+C)(v b v b ) B+C 2 ((v b 1) 2 (v b 1) 2 ) Gδ + G(δ sinδ) + G(1 v b v b ) sinδ q loss bb = q bb + q b b = 2B bb v b v b cosδ bb B bb (v 2 b + v 2 b ) C bb (v 2 b + v 2 b ) = 2B(1 cosδ) B(v b v b ) 2 + 2B(v b v b )(1 cosδ) Piecewise-linear (PWL) approximation Keep linear terms in v,δ 2C(v b v b 1) C((v b 1) 2 + (v b 1) 2 ) Use piecewise linear (SOS set) approximation of cos Also model reactive power flow A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

55 Case: B = 5, G = 1, C = 0.5, δ <= 57, 0.95 v max DC AC PWL p Av = Bδ G(v b v b ) G/2 ((v b 1) 2 (v b 1) 2 ) B(1 v b v b ) sinδ B(δ sinδ) p Loss = 2G(1 cosδ) G(v b v b ) G(v b v b 1)(1 cosδ) q Av = Gδ (B + C)(v b v b ) (B + C)/2 ((v b 1) 2 (v b 1) 2 ) G(δ sinδ) G(1 v b v b ) sinδ q Loss = 2C(v b v b 1) B(1 cosδ) B(v b v b ) C((v b 1) 2 + (v b 1) 2 ) B(v b v b 1)(1 cosδ) A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

56 AC feasibility problems 17 ~ DC constant loss ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Infeasible voltages v 2 = 1.15, v 6 = 0.85 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

57 AC feasibility problems ~ DC constant loss ~ ~ ~ AC PWL ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Infeasible voltages v 2 = 1.15, v 6 = ~ ~ ~ Feasible voltages v 2 = 1.02, v 6 = 0.99 A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

58 l Obtaining AC-feasible solutions 24-bus network: voltages & flows (deg) δ L l PWL-AC AC-OLS Line l (a) Angle differences (MW) p L,fr PWL-AC AC-OLS Line l (b) Real flows A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

59 l Obtaining AC-feasible solutions 24-bus network: voltages & flows v b (p.u.) (MVAr) q L,fr PWL-AC AC-OLS Bus b (c) Voltages 100 PWL-AC AC-OLS Line l (d) Reactive flows A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

60 Obtaining AC-feasible solutions Table: % AC infeasible n B DC A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

61 Obtaining AC-feasible solutions Table: % AC infeasible n B DC PWL-AC A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

62 Obtaining AC-feasible solutions Table: % AC infeasible n B DC PWL-AC Table: PWL-AC median solve times n B DC Optimality A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

63 Obtaining AC-feasible solutions Table: % AC infeasible n B DC PWL-AC Table: PWL-AC median solve times n B DC Optimality PWL-AC Optimality A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

64 Obtaining AC-feasible solutions Table: % AC infeasible n B DC PWL-AC Table: PWL-AC median solve times n B DC Optimality PWL-AC Optimality PWL-AC 10% gap A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

65 Current and future research Larger networks Computation Heuristics Aggregation/decomposition Bus/node splitting Dynamic stability A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26

66 Thank you A. Grothey (Edinburgh) MINLP approach to islanding GOR / 26