An MINLP approach to forming secure islands in electricity networks

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 2012 1 / 26

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 2012 2 / 26

USA August 2003 Happy customers A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 3 / 26

USA August 2003 Happy customers Oooops! There goes Long Island, Detroit, Ottowa, Toronto... A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 3 / 26

USA Aug 2003 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 4 / 26

USA August 2003 Inadequate system understanding situational awareness tree trimming diagnostic support A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 5 / 26

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 2012 6 / 26

Islanding of uncertain network Network has uncertain buses/lines/generators Firewall around uncertain buses, lines, generators A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 7 / 26

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 2012 7 / 26

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 2012 8 / 26

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 2012 8 / 26

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 2012 8 / 26

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 2012 8 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 Pre islanding ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 0.00 β 0.56 ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 0.56 β 0.93 ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 0.93 β 0.97 ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 0.97 β 1.00 ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

Effect ofβ Growth of section 0 in 24-bus network asβ 1 0.00 β 1.00 ~ ~ ~ 18 21 22 17 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 9 / 26

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 2012 10 / 26

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 2012 10 / 26

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 2012 10 / 26

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 2012 10 / 26

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 2012 11 / 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 2012 11 / 26

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 2012 11 / 26

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 2012 12 / 26

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 2012 12 / 26

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 2012 13 / 26

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 2012 14 / 26

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 2012 14 / 26

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 2012 15 / 26

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 2012 15 / 26

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 2012 15 / 26

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 2012 15 / 26

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 2012 16 / 26

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 2012 16 / 26

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 9 14 24 30 39 57 118 300 AC infeasible 0 13 21 23 17 10 12 18 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 2012 17 / 26

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 9 14 24 30 39 57 118 300 AC infeasible 0 13 21 23 17 10 12 18 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 2012 17 / 26

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 2012 18 / 26

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 2012 18 / 26

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 2012 18 / 26

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 2012 19 / 26

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 2012 19 / 26

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 2012 19 / 26

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 2012 20 / 26

Case: B = 5, G = 1, C = 0.5, δ <= 57, 0.95 v 1.05. max DC AC PWL p Av = Bδ 2.500 + G(v b v b ) 0.100 + G/2 ((v b 1) 2 (v b 1) 2 ) 0.001 + B(1 v b v b ) sinδ 0.245 + B(δ sinδ) 0.103 p Loss = 2G(1 cosδ) 0.245 + G(v b v b ) 2 0.001 + 2G(v b v b 1)(1 cosδ) 0.023 q Av = Gδ 0.500 (B + C)(v b v b ) 0.495 (B + C)/2 ((v b 1) 2 (v b 1) 2 ) 0.001 + G(δ sinδ) 0.021 + G(1 v b v b ) sinδ 0.051 q Loss = 2C(v b v b 1) 0.550 2B(1 cosδ) 1.224 B(v b v b ) 2 0.050 C((v b 1) 2 + (v b 1) 2 )+ 0.000 + 2B(v b v b 1)(1 cosδ) 0.125 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 21 / 26

AC feasibility problems 17 ~ DC constant loss ~ ~ 18 21 22 ~ 16 19 ~ ~ ~ ~ 20 23 15 14 13 24 11 12 3 9 10 8 4 5 6 1 2 7 ~ ~ ~ Infeasible voltages v 2 = 1.15, v 6 = 0.85 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 22 / 26

AC feasibility problems ~ DC constant loss ~ ~ ~ AC PWL ~ ~ 17 18 21 22 17 18 21 22 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 ~ 16 19 ~ ~ ~ ~ 20 15 14 13 23 24 11 12 24 11 12 3 9 10 8 3 9 10 8 4 5 6 4 5 6 1 2 7 ~ ~ ~ Infeasible voltages v 2 = 1.15, v 6 = 0.85 1 2 7 ~ ~ ~ Feasible voltages v 2 = 1.02, v 6 = 0.99 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 22 / 26

l Obtaining AC-feasible solutions 24-bus network: voltages & flows 10 300 (deg) δ L l 5 0 5 10 15 PWL-AC AC-OLS 20 0 5 10 15 20 25 30 Line l (a) Angle differences (MW) p L,fr 200 100 0 100 200 300 400 PWL-AC AC-OLS 500 0 5 10 15 20 25 30 Line l (b) Real flows A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 23 / 26

l Obtaining AC-feasible solutions 24-bus network: voltages & flows 1.06 1.04 100 50 v b (p.u.) 1.02 1 0.98 (MVAr) q L,fr 0 50 0.96 PWL-AC AC-OLS 0.94 0 5 10 15 20 25 Bus b (c) Voltages 100 PWL-AC AC-OLS 150 0 5 10 15 20 25 30 Line l (d) Reactive flows A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 23 / 26

Obtaining AC-feasible solutions Table: % AC infeasible n B 9 14 24 30 39 57 118 300 DC 0 13 21 23 17 10 12 18 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 24 / 26

Obtaining AC-feasible solutions Table: % AC infeasible n B 9 14 24 30 39 57 118 300 DC 0 13 21 23 17 10 12 18 PWL-AC 0 0 0 0 0 0 0 2 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 24 / 26

Obtaining AC-feasible solutions Table: % AC infeasible n B 9 14 24 30 39 57 118 300 DC 0 13 21 23 17 10 12 18 PWL-AC 0 0 0 0 0 0 0 2 Table: PWL-AC median solve times n B 9 14 24 30 39 57 118 300 DC Optimality 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.6 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 24 / 26

Obtaining AC-feasible solutions Table: % AC infeasible n B 9 14 24 30 39 57 118 300 DC 0 13 21 23 17 10 12 18 PWL-AC 0 0 0 0 0 0 0 2 Table: PWL-AC median solve times n B 9 14 24 30 39 57 118 300 DC Optimality 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.6 PWL-AC Optimality 0.0 0.9 1.4 3.4 2.6 15.4 261.9 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 24 / 26

Obtaining AC-feasible solutions Table: % AC infeasible n B 9 14 24 30 39 57 118 300 DC 0 13 21 23 17 10 12 18 PWL-AC 0 0 0 0 0 0 0 2 Table: PWL-AC median solve times n B 9 14 24 30 39 57 118 300 DC Optimality 0.0 0.1 0.1 0.1 0.2 0.2 0.3 0.6 PWL-AC Optimality 0.0 0.9 1.4 3.4 2.6 15.4 261.9 PWL-AC 10% gap 0.0 0.9 0.8 3.1 1.0 11.6 59.4 128.3 A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 24 / 26

Current and future research Larger networks Computation Heuristics Aggregation/decomposition Bus/node splitting Dynamic stability A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 25 / 26

Thank you A. Grothey (Edinburgh) MINLP approach to islanding GOR 2012 26 / 26