Departamento de Engenharia Elétrica, Universidad de Sâo Paulo, Sâo Carlos, June 30,

Transcription

1 Bieve programming appied to power system vunerabiity anaysis under mutipe contingencies José M. Arroyo E-mai: Departamento de Ingeniería Eéctrica, Eectrónica, Automática y Comunicaciones Universidad de Castia La Mancha Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

2 Contents Introduction Atacker-defender bieve programming modes Numerica resuts Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

3 Power system Physica components: wires, transformers, etc. Eectrica companies Reguators Retaiers Consumers Market Networks (transmission, distribution) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

4 Power system Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

5 Power system Operation Mainy performed by private companies Operation monitoring Government Critica infrastructure subject to new risks Risks must be managed by both groups of entities Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

6 Power system Critica infrastructure: Provision of vita services to society or government (security = high standard of ife) Cosey reated to socia wefare Avaiabiity taken for granted Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

7 Power system However, experience has shown it does fai: Recent backouts (North America, Itay, Greece, centra Europe) Recent maicious actions Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

8 Recent changes in power systems Liberaization Power markets compexity of socia network Moved from a monopoy controed by government to a competitive framework New agents and rues However, physica network unchanged Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

9 Recent changes in power systems Internationaization Different use with respect to the origina design Use of information and communication technoogies Internet Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

10 Vunerabiity of power systems Susceptibiity of attack or damage Characteristic of the design, impementation or operation of the infrastructure that makes it susceptibe of destruction by a threat Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

11 Power system Factors that make it vunerabe Asocciated with generation and transmission assets Technica weakness Market Operators driven by economic issues Pervasive use of open communication networks (cyber-attacks) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

12 Technica weaknesses Faiure of critica components (ageing, overheating, etc.) Inadequate maintenance Incorrect tripping of ine protections Incorrect tripping of generators Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

13 Technica weaknesses Insufficient oad shedding Insufficient communication and cooperation among operators Insufficient monitoring by operators Incorrect action by operators Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

14 Types of outages in power systems Impact on eectricity suppy (eectrica instaations and/or market) Unintentiona outage Uncertain random event (with probabiity distribution) Deiberate outage Uncertain non-random event Maximization of damage Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

15 Traditiona definition of risk Objective measure of risk: Risk = Probabiity Consequence Usefu for unintentiona outages Subjective aspects must aso be considered (there is risk when perceived by society) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

16 New risk definitions Risk measure under deiberate outages: Probabiity Risk = Threat Vunerabiity Consequence Probabiity Risk = Capabiity Intention Vunerabiity Consequence Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

17 New risk definitions Difficuties in risk anaysis: Estimation of probabiities and consequences of unikey but catastrophic events Need for advances in Statistics Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

18 Traditiona security assessment System designed to survive a set of credibe contingencies which are seected according to: Past events Apparent occurrence probabiity Consequences Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

19 Traditiona security assessment N-1 criterion System survives the oss of any singe component Considered components: Generators Power ines Transformers Compensation devices Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

20 Traditiona security assessment Drawbacks of N-1 criterion: Impementation depends on the country (number of credibe contingencies) Mutipe contingencies are not considered Faiures of communication and information systems are not considered Maicious intentionaity is not considered Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

21 Contents Introduction Atacker-defender bieve programming modes Numerica resuts Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

22 Vunerabiity anaysis under deiberate outages Identification of critica system components under intentiona attacks Usefu for: Network panner Terrorist (regrettaby!!) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

23 Objective of the network panner Identification of the extended contingency set the system is most vunerabe Impementation of adequate surveiance and protection actions Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

24 Objective of the terrorist Identification of an interdiction scheme to: Maximize the damage subject to imited destructive resources, or Minimize the destructive resources to reach a pre-specified eve of damage Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

25 Objective of the terrorist Interdictabe system components: Generators Substations Buses Lines Transformers Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

26 System operator After an attack, the system operator reacts: Impementation of corrective actions (generation redispatch, redirection of ine power fows, oad shedding) Objective Minimization of system damage Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

27 Measure of damage or vunerabiity System oad shedding Invountary and nonremunerated disconnection of power demanded Other measures are possibe Load shedding in particuar areas, economic cost of unserved energy, etc. Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

28 Attacker-defender bieve mode Two antagonistic agents operating sequentiay Terrorist Designs interdiction pan System operator Reacts against the attack Each agent optimizes its own objective function over a jointy dependent set Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

29 Attacker-defender bieve mode INTERDICTION PLAN DESIGN DAMAGE MAXIMIZATION OR MINIMIZATION OF DESTRUCTIVE RESOURCES SELECTION OF CORRECTIVE ACTIONS DAMAGE MINIMIZATION (ATTACK PLAN) TERRORIST SYSTEM OPERATOR NETWORK PLANNER IDENTIFICATION OF VULNERABLE COMPONENTS Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

30 Bieve programming 2-payer game of non-zero sum and perfect information Sequentia, non cooperative, 1-round game Feasibiity region impicity characterized by 2 optimization probems which are soved in a pre-determined sequence Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

31 2 payers Leader, upper-eve agent or outer-eve agent Decision variabes x X Optimizes its objective function by seecting a strategy (variabes x) that anticipates the reactions of the other payer The other payer s objective function is known Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

32 2 payers Foower, ower-eve agent or inner-eve agent Decision variabes y Y Reacts against the eader s strategy (variabes x) by seecting its strategy (variabes y) to optimize its objective function without considering the externa consequencies of its actions Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

33 Genera bieve formuation MinF( x,y) x,y Subject to: G ( x,y) 0 y argmin f y' ( x,y' ) Subject to: g ( x,y' ) 0 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

34 Mathematica characterization Optimization probem Even with inear constraints Non-convex probem Loca optima Deveopment of soution procedures Difficut task Existing methods of imited appication Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

35 Soution approaches Under certain convexity and differentiabiity conditions Conversion to a standard mathematica program KKT optimaity conditions Linear ower-eve probem Dua probem + Strong duaity theorem Max-min probems Dua probem Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

36 Bieve programming modes for vunerabiity anaysis Two antagonistic agents acting sequentiay: Leader Terrorist Foower System operator Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

37 Leader Terrorist Decision variabes x {0,1} x = 0 System component is destroyed/attacked x = 1 System component is not destroyed/attacked Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

38 Foower System operator Decision variabes Network operation (simpified DC mode): Power fows Noda phase anges Generation eves Noda oad shedding Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

39 Bieve programming modes for vunerabiity anaysis Two modes: Minimum vunerabiity mode Maximum vunerabiity mode Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

40 δ Gen, δ Line Minimum vunerabiity mode, δ Bus, δ Min Sub Subject to:,b,v, γ,p Gen,P Line,S c, θ Rec_dest ( Gen Line Bus Sub δ, δ, δ, δ ) Feasibiity of ( Gen Line Bus Sub δ, δ, δ, δ,b,v) γ γ γ = P Gen min,p Line,S Subject to: c, θ c S c Feasibiity of ( Gen Line P,P,S,θ) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, c Terrorist System operator

41 δ Gen, δ Maximum vunerabiity mode Max Line, δ Bus, δ Sub Subject to:,b,v γ Feasibiity of ( Gen Line Bus Sub δ, δ, δ, δ,b,v) ( Gen Line Bus Sub Rec_dest δ, δ, δ, δ ) M γ = P Gen min,p Line,S Subject to: c, θ c S c Feasibiity of ( Gen Line P,P,S,θ) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, c Terrorist System operator

42 Gen g Terrorist s constraints { 0,1, } g δ Generator attack δ Line { 0,1, } Line attack Bus i { 0,1, } i δ Bus attack Sub δ { 0,1, } s Substation attack s v { 0,1, } Line operation b g { 0,1, } g Generator operation Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

43 Terrorist s constraints v = ( Line) ( Bus) ( Bus δ ) ( ) δ ( ) δ Sub 1 1 o 1 d( ) 1 δs ( Line 1 δ ) ', ' ' L Par s L Sub s b g = ( Bus)( Gen 1 δ 1 δ ), g i ( g) g Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

44 Terrorist s constraints Destructive resources Rec_dest ( Gen Line Bus Sub δ, δ, δ, δ ) = g M Gen g δ Gen g + M Line δ Line + i M Bus i δ Bus i + s M Sub s δ Sub s Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

45 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, System operator s constraints: OPF(v,b) ( ) ( ) ( ) µ θ θ =, : x v P d o Line ( ) ( ) i, : d P P S P i C c c i d Line i o Line C c c G g Gen g i i i λ = + + = = ( ) ϕ φ,, : P P P Line Line Line Line power fows Power baances Line capacities

46 System operator s constraints: OPF(v,b) θ θ i θ ( χ, ε ), i : i i Ange imits Gen Gen 0 Pg bgpg : γg, g Generation imits 0 Sc dc : αc, c Load shedding imits Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

47 Probem characterization Bieve programming probem Non-inear (products of variabes) Mixed-integer Large scae Soution based on duaity theory Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

48 Foower s dua constraints λ λ i i ( g) +γg 0 g ( c) +αc 1 c λo( ) +λd( ) +µ +φ +ϕ = 0 o ( ) v µ x + ( ) = i d = i vµ x +χ i +ε i = 0, i χ 0, i ε 0, i γ 0, g i i φ 0, ϕ 0, α 0, c g c Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

49 Strong duaity theorem c S c = ( ) ( ) Line ϕ φ P + αc +λi( c) c d c + g γ g b g P Gen g + ( ε χ) i i i θ Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

50 Conversion to a singe eve: MINLP Optimize terrorist s objective function Subject to: Terrorist s non-inear constraints SO s non-inear prima constraints SO s non-inear dua constraints SO s non-inear strong duaity equaity Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

51 Linearization of product of 0/1 variabes MILP mode: z 0 z, i= 1, K, n z x i n x i i= 1 xi n+ 1 { } 0,1 z, x { 0,1} i = n i= 1 x i Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

52 v v v v Linearization of constraints Leader ( Line 1 δ ), ( Line) Par 1 δ,, ' L ' ( Bus 1 δ ), o ( ) ( Bus 1 δ ), d ( ) v ( Sub) Sub 1 δ, s, L s Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, s

53 Linearization of constraints Leader v ( Line) ( Bus) ( Bus 1 δ ) ( ) + 1 δ ( ) + 1 δ ( ) + Sub o d 1 δs + ( Line) ( Par) ({ Sub 1 δ }) ' 3+ card L + card s : Ls ' L Par + 1, s L Sub s, v 0, Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

54 Linearization of constraints Leader b b g g ( Bus 1 δ ), g i ( g) ( Gen 1 δ ), g g b g 0, g b g ( Bus) ( Gen 1 δ + 1 δ ) 1, g i ( g) g Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

55 Linearization of product of 0/1 variabe and continuous variabe z= xp { }, x 0,1 [, MILP mode: z= p r xp min z xp max ( ) min ( ) max 1 x p r 1 x p p p min, p max ] x { 0,1} Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

56 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, Linearization of constraints Foower ( ) ( ) ( ) [ ] θ θ θ θ =, ~ ~ x 1 P d d o o Line ( ) θ θ θ θ, v ~ v o o ( ) θ θ θ θ, v ~ v d d ( ) ( ) θ θ θ, v 1 ~ v v 1 o ( ) ( ) θ θ θ, v 1 ~ v v 1 d

57 Linearization of constraints Foower o ( ) µ ~ x ( ) ( ) µ + ( ) = i d = i µ µ ~ x +χ i +ε i = 0, i v µ µ µ ~ v µ, v ( 1 µ ) µ v ( 1 µ ), ~ Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

58 Linearization of constraints Foower c S c = + ( ) ( ) Line ϕ φ P + αc +λi( c) g ~ ( ) Gen γ γ + ( ε χ) g g Pg i i c i θ d c b g γ g γ g γ ~ g 0, g ~ ( b ) γ γ 0, g 1 g g g Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

59 Resuting MILP probem Optimize terrorist s objective function Subject to: Terrorist s inear constraints SO s inear prima constraints SO s inear dua constraints SO s inear strong duaity equaity Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

60 Contents Introduction Atacker-defender bieve programming modes Numerica resuts Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

61 Minimum vunerabiity mode 5-bus exampe and One-Area IEEE RTS Scenario of peak demand Ony destruction of ines Resuts parameterized as a function of the minimum eve of system oad shed γ Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

62 Minimum vunerabiity mode GAMS and CPLEX 9.0 De PowerEdge 6600, 2 processors, 1.6 GHz, 2 GB of RAM Computing time for optimaity 10 seconds Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

63 5-bus exampe 0 MW P g1 150 MW 0 MW P g2 150 MW ~ ~ 50 MW 170 MW Bus pu Bus pu pu pu Bus 3 0 MW P g3 150 MW 90 MW ~ pu Bus pu Bus 5 ~ ~ 30 MW 300 MW 0 MW P g4 150 MW 0 MW P g5 150 MW Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

64 5-bus exampe # destroyed ines System oad shed (MW) Worst combination of destroyed ines , , 3-5, , 3-5, , 3-5, , 3-5, , 2-3, 3-5, , 1-3, 2-3, 3-5, , 1-4, 2-3, 3-5, , 1-3, 1-4, 2-3, 3-5, 4-5 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

65 5-bus exampe Range of γ (MW) Destroyed ines 0< γ < γ , < γ , 2-3, 3-5, 4-5 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

66 IEEE RTS 24 buses 38 ines 32 generators 17 oads Peak demand scenario (2850 MW) A ines destroyed 1607 MW shed ~ Bus 18 ~ ~ Bus 17 Bus 21 Bus 22 Bus 23 ~ ~ Bus 19 Bus 20 Bus 16 Bus 14 Bus 15 ~ Bus 13 ~ ~ Bus 24 Bus 11 Bus 12 Bus 3 Bus 9 Bus 4 Bus 5 Bus 10 Bus 6 cabe Bus 8 cabe Bus 1 Bus 2 ~ ~ ~ Bus 7 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

67 IEEE RTS γ (MW) Destroyed ines Load shed (MW) , 20-23A, 20-23B , 9-12, 11-13, , 11-14, 12-13, 12-23, , 12-13, 12-23, 14-16, , 11-13, 12-13, 12-23, 14-16, Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

68 IEEE RTS ~ Bus 18 ~ ~ Bus 17 Bus 21 Bus 22 Bus 23 ~ ~ Bus 19 Bus 20 Bus 16 Bus 14 Bus 15 ~ ~ Bus 13 ~ Bus 24 Bus 11 Bus 12 Bus 3 Bus 4 Bus 9 Bus 10 Bus 6 cabe Bus 5 Bus 8 cabe Bus 1 Bus 2 ~ ~ ~ Bus 7 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

69 IEEE RTS Bus P g (MW) P d (MW) S c (MW) Bus P g (MW) P d (MW) S c (MW) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

70 Maximum vunerabiity mode One-Area IEEE RTS and Two-Area IEEE RTS Maximum demand scenario Resuts parameterized as a function of M Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

71 Maximum vunerabiity mode Destruction of a ine or severa parae ines 1 person Destruction of a transformer 2 persons Destruction of a bus or substation 3 persons Destruction of a generator or an underground cabe Impossibe Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

72 Maximum vunerabiity mode GAMS and CPLEX 8.1 Pentium IV, 2.66 GHz, 512 MB of RAM Computing time for optimaity 180 seconds Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

73 One-Area IEEE RTS M Load shed (MW) CPU time (s) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30, M Load shed (MW) CPU time (s)

74 One-Area IEEE RTS. M = 20 ~ Bus 18 ~ ~ Bus 17 Bus 21 Bus 22 Bus 23 ~ ~ Bus 19 Bus 20 Bus 16 Bus 14 Bus 15 ~ ~ Bus 13 ~ Bus 24 Bus 11 Bus 12 Bus 3 Bus 4 Bus 9 Bus 10 Bus 6 cabe Bus 5 Bus 8 cabe Bus 1 Bus 2 ~ ~ ~ Bus 7 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

75 Two-Area IEEE RTS M Load shed (MW) CPU time (s) M Load shed (MW) CPU time (s) Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

76 Two-Area IEEE RTS. M = 12 Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

77 Future and ongoing work More precise power fow mode (AC) Aternative corrective actions (disconnection of generators and/or ines) Anaysis of power vs. energy shed Use of game theory Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

78 References A. V. Gheorghe, M. Masera, M. Weijnen, L. de Vries. Critica Infrastructures at Risk. Securing the European Eectric Power System. Springer. Dordrecht, The Netherands IEEE/CIGRÉ Joint Task Force on Stabiity Terms and Definitions. Definitions and cassification of power system stabiity. IEEE Trans. Power Syst. Vo. 19, no. 3, pp , Aug Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

79 References J. Samerón, K. Wood, R. Badick. Anaysis of Eectric Grid Security under Terrorist Threat. IEEE Transactions on Power Systems, vo. 19, no. 2, pp , May J. M. Arroyo, F. D. Gaiana. On the Soution of the Bieve Programming Formuation of the Terrorist Threat Probem. IEEE Transactions on Power Systems, vo. 20, no. 2, pp , May Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

80 References A. L. Motto, J. M. Arroyo, F. D. Gaiana. A Mixed- Integer LP Procedure for the Anaysis of Eectric Grid Security under Disruptive Threat. IEEE Transactions on Power Systems, vo. 20, no. 3, pp , August M. Carrión, J. M. Arroyo, N. Aguaci. Vunerabiity- Constrained Transmission Expansion Panning: A Stochastic Programming Approach. IEEE Transactions on Power Systems, vo. 22, no. 4, pp , November Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,

81 Thanks for your attention! GSEE: Departamento de Engenharia Eétrica, Universidad de Sâo Pauo, Sâo Caros, June 30,