Power grid vulnerability analysis

Transcription

1 Power grid vulnerability analysis Daniel Bienstock Columbia University Dimacs 2010 Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

2 Background: a power grid is three systems GENERATION TRANSMISSION DISTRIBUTION Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

3 Challenges to analysis Power grids follow the laws of physics, characterized by nonlinear, nonconvex equations that make fast computation difficult. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

4 Challenges to analysis Power grids follow the laws of physics, characterized by nonlinear, nonconvex equations that make fast computation difficult. Furthermore, direct control is difficult: we cannot dictate how power will flow. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

5 Challenges to analysis Power grids follow the laws of physics, characterized by nonlinear, nonconvex equations that make fast computation difficult. Furthermore, direct control is difficult: we cannot dictate how power will flow. Power grids are subject to noise which is difficult to model accurately. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

6 Challenges to analysis Power grids follow the laws of physics, characterized by nonlinear, nonconvex equations that make fast computation difficult. Furthermore, direct control is difficult: we cannot dictate how power will flow. Power grids are subject to noise which is difficult to model accurately. Power grids can exhibit non-monotone behavior as a result of control or adversarial actions. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

7 Challenges to analysis Power grids follow the laws of physics, characterized by nonlinear, nonconvex equations that make fast computation difficult. Furthermore, direct control is difficult: we cannot dictate how power will flow. Power grids are subject to noise which is difficult to model accurately. Power grids can exhibit non-monotone behavior as a result of control or adversarial actions. Power grids can cascade. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

8 AC power flows polar coordinates Voltage at a node ( bus ) k is of the form U k e jθ k, where j = 1 Power flowing on edge ( line ) {k, m} equals p km + jq km, where p km = U 2 k g km U k U m g km cos θ km U k U m b km sin θ km q km = U 2 k (b km + b sh km ) + U k U m b km cos θ km U k U m g km sin θ km. Here, θ km = θk θ m g km, b km, bkm sh are known parameters (series conductance, series reactance, shunt susceptance) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

9 Voltage at k = U k e jθ k ; power on line {k, m} = p km + jq km, where p km = U 2 k g km U k U m g km cos θ km U k U m b km sin θ km q km = U 2 k (b km + b sh km ) + U k U m b km cos θ km U k U m g km sin θ km ( θ km. = θk θ m ) P k = Σ {k,m} p km (active power), Q k = Σ {k,m} q km (reactive power) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

10 Voltage at k = U k e jθ k ; power on line {k, m} = p km + jq km, where p km = U 2 k g km U k U m g km cos θ km U k U m b km sin θ km q km = U 2 k (b km + b sh km ) + U k U m b km cos θ km U k U m g km sin θ km ( θ km. = θk θ m ) P k = Σ {k,m} p km (active power), Q k = Σ {k,m} q km (reactive power) Power flow problem: Choose the vectors p, q, θ, P, Q so as to satisfy all equations above, and Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

11 Voltage at k = U k e jθ k ; power on line {k, m} = p km + jq km, where p km = U 2 k g km U k U m g km cos θ km U k U m b km sin θ km q km = U 2 k (b km + b sh km ) + U k U m b km cos θ km U k U m g km sin θ km ( θ km. = θk θ m ) P k = Σ {k,m} p km (active power), Q k = Σ {k,m} q km (reactive power) Power flow problem: Choose the vectors p, q, θ, P, Q so as to satisfy all equations above, and meet demand requirements and generator constraints Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

12 Voltage at k = U k e jθ k ; power on line {k, m} = p km + jq km, where p km = U 2 k g km U k U m g km cos θ km U k U m b km sin θ km q km = U 2 k (b km + b sh km ) + U k U m b km cos θ km U k U m g km sin θ km ( θ km. = θk θ m ) P k = Σ {k,m} p km (active power), Q k = Σ {k,m} q km (reactive power) Power flow problem: Choose the vectors p, q, θ, P, Q so as to satisfy all equations above, and meet demand requirements and generator constraints and, ideally, meet thermal constraints (flow limits) on the power lines Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

13 Research challenges Do we have fast and reliable algorithm for the power flow problem? Should not require human input in order to terminate. When no acceptable solution exists, should produce a certificate that this is the case. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

14 Research challenges Do we have fast and reliable algorithm for the power flow problem? Should not require human input in order to terminate. When no acceptable solution exists, should produce a certificate that this is the case. What about the cases where multiple solutions exist? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

15 Research challenges Do we have fast and reliable algorithm for the power flow problem? Should not require human input in order to terminate. When no acceptable solution exists, should produce a certificate that this is the case. What about the cases where multiple solutions exist? After a contingency has take place, or a control has been applied: which solution should be instantiated? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

16 Research challenges Do we have fast and reliable algorithm for the power flow problem? Should not require human input in order to terminate. When no acceptable solution exists, should produce a certificate that this is the case. What about the cases where multiple solutions exist? After a contingency has take place, or a control has been applied: which solution should be instantiated? What if all solutions are bad? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

17 Solution methodologies Newton-Raphson (iterative) algorithms to solve system of equations Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

18 Solution methodologies Newton-Raphson (iterative) algorithms to solve system of equations The claim: this always works fast. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

19 Solution methodologies Newton-Raphson (iterative) algorithms to solve system of equations The claim: this always works fast. normal system. At least in the case of a Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

20 Solution methodologies Newton-Raphson (iterative) algorithms to solve system of equations The claim: this always works fast. normal system. At least in the case of a New result: Low et al (2010). Some (many?) optimal power flow problems can be solved using semidefinite programming. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

21 Linearized ( DC ) model: sin(θ i θ j ) (θ i θ j ) for θ i θ j A power flow is a solution f, θ to: ij f ij ij f ji = b i, for all i, where b i > 0 for each generator i, b i < 0 for demand node i, Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

22 Linearized ( DC ) model: sin(θ i θ j ) (θ i θ j ) for θ i θ j A power flow is a solution f, θ to: ij f ij ij f ji = b i, for all i, where b i > 0 for each generator i, b i < 0 for demand node i, x ij f ij θ i + θ j = 0 for all (i, j). ( x ij = reactance ) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

23 Linearized ( DC ) model: sin(θ i θ j ) (θ i θ j ) for θ i θ j A power flow is a solution f, θ to: ij f ij ij f ji = b i, for all i, where b i > 0 for each generator i, b i < 0 for demand node i, x ij f ij θ i + θ j = 0 for all (i, j). ( x ij = reactance ) Lemma: Given a choice for b with i b i = 0 (a requirement), Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

24 Linearized ( DC ) model: sin(θ i θ j ) (θ i θ j ) for θ i θ j A power flow is a solution f, θ to: ij f ij ij f ji = b i, for all i, where b i > 0 for each generator i, b i < 0 for demand node i, x ij f ij θ i + θ j = 0 for all (i, j). ( x ij = reactance ) Lemma: Given a choice for b with i b i = 0 (a requirement), the system has a unique (in f) solution. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

25 A quote from: Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations, U.S.-Canada Power System Outage Task Force, April 5, ( Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

26 A quote from: Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations, U.S.-Canada Power System Outage Task Force, April 5, ( Cause 1 was inadequate system understanding Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

27 A quote from: Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations, U.S.-Canada Power System Outage Task Force, April 5, ( Cause 1 was inadequate system understanding stated 20 times Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

28 A quote from: Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations, U.S.-Canada Power System Outage Task Force, April 5, ( Cause 1 was inadequate system understanding stated 20 times Cause 2 was inadequate situational awareness stated 14 times Cause 3 was inadequate tree trimming stated 4 times Cause 4 was inadequate RC diagnostic support stated 5 times Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

29 Cascades Load (demand) Generator Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

30 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

31 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

32 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

33 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

34 Cascades = lost demand Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

35 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

36 Cascades Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

37 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

38 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

39 Islanding supply > demand Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

40 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

41 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

42 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3. The next set of faults takes place. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

43 Formal cascade model (Dobson et al) Initial fault event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3. The next set of faults takes place. (Stochastic or history-dependent criterion) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

44 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

45 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

46 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) f e r 1, where 0 α e 1 is given. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

47 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) f e r 1, where 0 α e 1 is given. f r e = running average of f e. r = round (time). Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

48 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) f e r 1, where 0 α e 1 is given. f r e = running average of f e. r = round (time). e fails if f e > u e. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

49 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) f e r 1, where 0 α e 1 is given. f r e = running average of f e. r = round (time). e fails if f e > u e. or: e fails if f e u e Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

50 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) fe r 1, where 0 α e 1 is given. f r e = two-round average of f e. r = round (time). e fails if f e > u e, Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

51 Outage mechanism f e = flow on line e u e = flow limit (threshold) on e Prob( e fails) = F ( f e /u e ), where F (x) 1 as x +. Set f r e = α e fe r + (1 α e) fe r 1, where 0 α e 1 is given. f r e = two-round average of f e. r = round (time). e fails if f e > u e, (or, e fails if f e u e ) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

52 Stochastic faults e fails if u e < f r e, Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

53 Stochastic faults e fails if u e < f r e, e does not fail if (1 γ)u e > f e r, (γ = tolerance) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

54 Stochastic faults e fails if u e < f r e, e does not fail if (1 γ)u e > f e r, (γ = tolerance) if (1 γ)u e f r e u e then e fails with probability 1/2 Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

55 Formal cascade model (Dobson et al) Initial outage event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3. The next set of outages takes place. (Stochastic or history-dependent criterion) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

56 Formal cascade model (Dobson et al) Initial outage event takes place (an act of God ). For r = 1, 2,..., 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3. The next set of outages takes place. (Stochastic or history-dependent criterion) If no more faults occur or too much demand has been lost, STOP Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

57 Online control Initial outage event takes place. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

58 Online control Initial outage event takes place. Compute control algorithm. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

59 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

60 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

61 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3a. Take measurements and apply control to shed demand. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

62 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3a. Take measurements and apply control to shed demand. 3b. Reconfigure generator outputs; Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

63 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3a. Take measurements and apply control to shed demand. 3b. Reconfigure generator outputs; get new power flows. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

64 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3a. Take measurements and apply control to shed demand. 3b. Reconfigure generator outputs; get new power flows. 4. The next set of outages takes place. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

65 Online control Initial outage event takes place. Compute control algorithm. For r = 1, 2,..., R 1 1. Reconfigure demands and generator output levels. 2. New power flows are instantiated. 3a. Take measurements and apply control to shed demand. 3b. Reconfigure generator outputs; get new power flows. 4. The next set of outages takes place. At round R, reduce demands so as to remove any line overloads. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

66 Deterministic, no history model Optimal control via integer programming formulation Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

67 Deterministic, no history model Optimal control via integer programming formulation? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

68 Deterministic, no history model Optimal control via integer programming formulation? f r j y r j d r j = flow on arc j at round r = 1, if arc j fails in round r, 0 otherwise = demand at node i in round r and many other variables Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

69 Subject to: max X d R i i D 8 X f r j X f r < j = j δ + (i) j δ : (i) si r i G di r i D 0 otherwise 1 r R (1) f r j = π r j ν r j j A and 1 r R (2) π r j Dp r j, νr j Dn r j, j A and 1 r R (3) p r j + n r j r 1 X = 1 y h j, j A and 1 r R (4) h=1 π r j + ν r j u j π r j + ν r j Dy r j j A and 1 r R (5) u j y r j j A and 1 r R 1 (6) π R j + ν R j u j j A (7) φ r i φ r j x j f r r 1 X j M j y h j j A (8) h=1 0 s r i s i i G, 0 d r i d i i D, (9) p r j, nr ij, yr j = 0 or 1, j A and 1 r R (10) 0 π r j, 0 νr j, j A and 1 r R. (11) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

70 What s bad about the formulation probably can t solve it for medium to large networks stochastic variant probably needed, harder Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

71 What s bad about the formulation probably can t solve it for medium to large networks stochastic variant probably needed, harder optimal solutions = complex policies Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

72 Adaptive affine controls For each demand v, and round r, control cv r, br v, sr v computed to be Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

73 Adaptive affine controls For each demand v, and round r, control cv r, br v, sr v computed to be Parameterized by integer r > 0. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

74 Adaptive affine controls For each demand v, and round r, control cv r, br v, sr v computed to be Parameterized by integer r > 0. At round r, Let κ = maximum overload of any line within radius r of v Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

75 Adaptive affine controls For each demand v, and round r, control cv r, br v, sr v computed to be Parameterized by integer r > 0. At round r, Let κ = maximum overload of any line within radius r of v If κ > cv r, demand at v reduced (scaled) by a factor max { 1, s r v (cr v κ) + br v}. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

76 Adaptive affine controls For each demand v, and round r, control cv r, br v, sr v computed to be Parameterized by integer r > 0. At round r, Let κ = maximum overload of any line within radius r of v If κ > cv r, demand at v reduced (scaled) by a factor max { 1, s r v (cr v κ) + br v}. The goal: pick control to maximize demand being served at the end of round R. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

77 For each demand v, and round r, control c r v, br v, sr v At round r, if κ > cv r, demand at v reduced (scaled) by a factor min { 1, [ s r v (cr v κ) + br v ]+ }. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

78 For each demand v, and round r, control c r v, br v, sr v At round r, if κ > cv r, demand at v reduced (scaled) by a factor min { 1, [ s r v (cr v κ) + br v ]+ }. This talk: r = n (number of nodes) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

79 For each demand v, and round r, control c r v, br v, sr v At round r, if κ > cv r, demand at v reduced (scaled) by a factor min { 1, [ s r v (cr v κ) + br v ]+ }. This talk: r = n (number of nodes) Special case: (optimal scaling problem) Insist that for each r, (c r v, br v, sr v ) = (cr, b r, s r ) for every v Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

80 For each demand v, and round r, control c r v, br v, sr v At round r, if κ > cv r, demand at v reduced (scaled) by a factor min { 1, [ s r v (cr v κ) + br v ]+ }. This talk: r = n (number of nodes) Special case: (optimal scaling problem) Insist that for each r, (c r v, br v, sr v ) = (cr, b r, s r ) for every v Then, equivalent problem: In round r, let α r (K ) 1 be chosen for each component of the network in round r If node v component K, then its demand is scaled by α r (K ) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

81 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

82 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

83 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Theorem: Θ R (t, ˆβ) is nondecreasing piecewise-linear with at most m R /R! + O(m R 1 ) breakpoints. m = no. of arcs Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

84 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Theorem: Θ R (t, ˆβ) is nondecreasing piecewise-linear with at most m R /R! + O(m R 1 ) breakpoints. m = no. of arcs In round 1, the optimal scale is equal to u j /(t ˆf j ) for some arc j. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

85 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Theorem: Θ R (t, ˆβ) is nondecreasing piecewise-linear with at most m R /R! + O(m R 1 ) breakpoints. m = no. of arcs In round 1, the optimal scale is equal to u j /(t ˆf j ) for some arc j. So arc j will become critical Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

86 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Theorem: Θ R (t, ˆβ) is nondecreasing piecewise-linear with at most m R /R! + O(m R 1 ) breakpoints. m = no. of arcs In round 1, the optimal scale is equal to u j /(t ˆf j ) for some arc j. So arc j will become critical And recursively... Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

87 Notation: ˆβ = supply/demand vector at time 0 ˆf = corresponding power flows at time 0 Θ R (t, β) : R + R + = total demand, at the end of round R, using optimal control, if the supply/demand vector is t β Note: supply/demand = tβ means flow = t ˆf Theorem: Θ R (t, ˆβ) is nondecreasing piecewise-linear with at most m R /R! + O(m R 1 ) breakpoints. m = no. of arcs In round 1, the optimal scale is equal to u j /(t ˆf j ) for some arc j. So arc j will become critical And recursively... Robust/stochastic version? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

88 General case: simulation-based optimization Given a control vector ũ = (cv r, br v, sr v ) (over all v and r), Θ(ũ) = throughput (total demand) satisfied at end of cascade Maximization of Θ(ũ) should be (very?) fast Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

89 General case: simulation-based optimization Given a control vector ũ = (cv r, br v, sr v ) (over all v and r), Θ(ũ) = throughput (total demand) satisfied at end of cascade Maximization of Θ(ũ) should be (very?) fast Optimization should be robust (noisy process) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

90 General case: simulation-based optimization Given a control vector ũ = (cv r, br v, sr v ) (over all v and r), Θ(ũ) = throughput (total demand) satisfied at end of cascade Maximization of Θ(ũ) should be (very?) fast Optimization should be robust (noisy process) From a strict perspective, Θ(ũ) is not even continuous Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

91 General case: simulation-based optimization Given a control vector ũ = (cv r, br v, sr v ) (over all v and r), Θ(ũ) = throughput (total demand) satisfied at end of cascade Maximization of Θ(ũ) should be (very?) fast Optimization should be robust (noisy process) From a strict perspective, Θ(ũ) is not even continuous Θ(ũ) is obtained through a simulation Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

92 Derivative-free optimization Conn, Scheinberg, Vicente, others Rough description: Sample a number of control vectors ũ Use the sample points to construct a convex approximation to Θ Optimize this approximation; this yields a new sample point Scalability to large dimensionality? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

93 First order method Given a control vector ũ 1 Estimate the gradient g = Θ(ũ) through finite differences. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

94 First order method Given a control vector ũ 1 Estimate the gradient g = Θ(ũ) through finite differences. Requires O(1) simulations per demand node. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

95 First order method Given a control vector ũ 1 Estimate the gradient g = Θ(ũ) through finite differences. Requires O(1) simulations per demand node. 2 Estimate step size argmax Θ(ũ + σg) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

96 First order method Given a control vector ũ 1 Estimate the gradient g = Θ(ũ) through finite differences. Requires O(1) simulations per demand node. 2 Estimate step size argmax Θ(ũ + σg) Easily parallelizable Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

97 Line searches 0.91 ls ls ls Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

98 Current parallel implementation: boss-nerd Boss carries out search algorithm Nerds simulate cascades with given control Communication using Unix sockets Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

99 Scaling Example: nodes, lines 5 gradient steps 8-core i7 CPUs (3 machines total) cores wall-clock sec Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

100 Initial experiments with Eastern Interconnect nodes, lines generator nodes, 6261 demand nodes Equivalent DC flow version Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

101 Initial experiments with Eastern Interconnect nodes, lines generator nodes, 6261 demand nodes Equivalent DC flow version Methodology for experiments 1 Generate an interdiction of the grid ( initial event ) 2 Compute control and simulate Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

102 Initial experiments with Eastern Interconnect nodes, lines generator nodes, 6261 demand nodes Equivalent DC flow version Methodology for experiments 1 Generate an interdiction of the grid ( initial event ) 2 Compute control and simulate 3 At least three rounds of cascade after initial event Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

103 Computing a control (1) Solve scaling problem let (c, b, s ) be optimal Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

104 Computing a control (1) Solve scaling problem let (c, b, s ) be optimal (2) Partition demand nodes into small number of segments Σ 1,..., Σ k. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

105 Computing a control (1) Solve scaling problem let (c, b, s ) be optimal (2) Partition demand nodes into small number of segments Σ 1,..., Σ k. Example = demand quantiles. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

106 Computing a control (1) Solve scaling problem let (c, b, s ) be optimal (2) Partition demand nodes into small number of segments Σ 1,..., Σ k. Example = demand quantiles. Perform segmented gradient search starting from (c, b, s ). Look for a control with (cv r, br v, sr v ) = constant for each given r and all v in a common Σ i. Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

107 Computing a control (1) Solve scaling problem let (c, b, s ) be optimal (2) Partition demand nodes into small number of segments Σ 1,..., Σ k. Example = demand quantiles. Perform segmented gradient search starting from (c, b, s ). Look for a control with (cv r, br v, sr v ) = constant for each given r and all v in a common Σ i. (3) Perform full gradient search starting from the output in (2). Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

108 Experiments K random lines taken out highly loaded lines more likely to be taken out; connectivity preserved Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

109 Experiments K random lines taken out highly loaded lines more likely to be taken out; connectivity preserved K yield, (%) yield, wallclock no control control (sec) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

110 Conjectures It is best to stop the cascade in the first round It is best to apply control in the first round only, and ride out the cascade Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

111 Conjectures It is best to stop the cascade in the first round It is best to apply control in the first round only, and ride out the cascade (Answer: both wrong) Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

112 Details: cascade with 50 (highly loaded) random lines taken out No control yield = 0% Optimal round 1 only constant control yield = 38% Optimal scaling control yield = 45% Plus segmented gradient seach yield = 50% Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

113 Load distribution at time zero (load of arc j = f j u j ) load no. of arcs Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

114 Load distribution at time zero (load of arc j = f j u j ) Optimal round 1 scale = 0.51, load no. of arcs Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

115 Load distribution at time zero (load of arc j = f j u j ) load no. of arcs Optimal round 1 scale = 0.51, so 44 faults Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

116 Out-of-sample testing: use stochastic faults at round r, e fails if u e < f r e, e does not fail if (1 γ)u e > f e r, (γ = tolerance) if (1 γ)u e f r e u e then e fails with probability 1/2 Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

117 Out-of-sample testing: use stochastic faults at round r, e fails if u e < f r e, e does not fail if (1 γ)u e > f e r, (γ = tolerance) if (1 γ)u e f r e u e then e fails with probability 1/2 What is the impact of γ? Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs

118 γ = 0.03, 0.10, 0.20, runs percent 10 percent 20 percent COUNT YIELD Daniel Bienstock (Columbia University) Power grid vulnerability analysis Dimacs