Controlling variability in power systems

Transcription

1 Daniel APAM Nov

2 A simple example:

3 A simple example: Only one solution:

4 A simple example: Only one solution: But what if the red node suddenly injects power?

5 Only one solution:

6 Only one solution: If red node suddenly injects power, offset using blue node:

7 Better example: red node is cheap but unreliable source error Base case solution: If the red node suddenly injects power, offset using blue node: D D 50 + D

8 Examples: X Y X Y capacity = 15 Any combination with X + Y = 25 works so long as Y X 18 + Y X Y Any combination with X + Y = 12 works so long as Y 3.

9 Power engineering for non-engineers conductor rotor stator steam magnetic field ω energy source current, voltage

10 Power engineering for non-power engineers

11 Power engineering for non-power engineers

12 Power engineering for non-power engineers

13 Power engineering for non-power engineers

14 AC Power Flows

15 AC Power Flows Real-time:

16 AC Power Flows Real-time: I(t) km k m V(t) k Voltage at bus k: v k (t) = V max k cos(ωt + θ V k ) Current injected at k into km: i km (t) = I max km cos(ωt + θi km ).

17 AC Power Flows Real-time: I(t) km k m V(t) k Voltage at bus k: v k (t) = V max k cos(ωt + θ V k ) Current injected at k into km: i km (t) = I max km cos(ωt + θi km ). Power injected at k into km: p km (t) = v k (t)i km (t). Averaged over period T : p km. = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km ).

18 I(t) km k m V(t) k. p km = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km )

19 I(t) km k m V(t) k. p km = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km ) v k (t) = V max k Re e j(ωt+θv k ), i km (t) = I max km Re ej(ωt+θi km )

20 I(t) km k m V(t) k. p km = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km ) v k (t) = V max k Re e j(ωt+θv k ), i km (t) = I max km Re ej(ωt+θi km ) V k. = V max k 2 e jθv k, Ikm. = I max km 2 e jθi mk

21 I(t) km k m V(t) k. p km = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km ) v k (t) = V max k Re e j(ωt+θv k ), i km (t) = I max km Re ej(ωt+θi km ) V k. = V max k 2 e jθv k, Ikm. = I max km 2 e jθi mk p km = V k I km cos(θ V k θi km ) = Re(V ki km )

22 I(t) km k m V(t) k. p km = 1 T T 0 p(t) = 1 2 V k max I max km cos(θv k θi km ) v k (t) = V max k Re e j(ωt+θv k ), i km (t) = I max km Re ej(ωt+θi km ) V k. = V max k 2 e jθv k, Ikm. = I max km 2 e jθi mk p km = V k I km cos(θ V k θi km ) = Re(V ki km ) q km. = Im(Vkm I km ) and S km. = p km + jq km

23 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (1)

24 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (1) I km = y {k,m} (V k V m ),

25 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (1) I km = y {k,m} (V k V m ), y {k,m} = admittance of km. (2)

26 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (1) I km = y {k,m} (V k V m ), y {k,m} = admittance of km. (2) Network Equations k

27 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (3) I km = y {k,m} (V k V m ), y {k,m} = admittance of km. (4) Network Equations km δ(k) p km = ˆP k, km δ(k) q km = ˆQ k k (5)

28 V k. = V max k 2 e jθv k, I km. = I max km 2 e jθi mk (voltage, current) p km = Re(V k I km ), q km = Im(V km I km ) (3) I km = y {k,m} (V k V m ), y {k,m} = admittance of km. (4) Network Equations km δ(k) p km = ˆP k, km δ(k) q km = ˆQ k k (5) Generator: ˆP k, V k given. Other buses: ˆP k, ˆQ k given.

29 Optimal power flow (economic dispatch, tertiary control) Used periodically to handle the next time window (e.g. 15 minutes, one hour) Choose generator outputs Minimize cost (quadratic) Satisfy demands, meet generator and network constraints Constant load (demand) estimates for the time window

30 DC-OPF: min c(p) (convex piecewise-linear or quadratic) s.t. Bθ = p d (6) y ij (θ i θ j ) u ij for each line ij (7) P min g p g P max g for each bus g (8) Notation: p = vector of generations R n, d = vector of loads R n B R n n, (bus susceptance matrix) y ij, ij E (set of lines) i, j : B ij = k;{k,j} E y kj, i = j 0, otherwise

31 Managing changing demands

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34 What happens when there is a generation/load mismatch conductor rotor stator steam magnetic field ω energy source current, voltage

35 What happens when there is a generation/load mismatch conductor rotor stator steam magnetic field ω energy source current, voltage Frequency response:

36 What happens when there is a generation/load mismatch conductor rotor stator steam magnetic field ω energy source current, voltage Frequency response: mismatch P

37 What happens when there is a generation/load mismatch conductor rotor stator steam magnetic field ω energy source current, voltage Frequency response: mismatch P frequency change ω c P

38 The swing equation H ω = p m (t) p e (t) Dω ω = ω(t) = frequency p m (t) = mechanical power supplied by motor p e (t) = electrical power supplied by motor D > 0 (tamping)

39 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes.

40 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics.

41 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds.

42 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds. Agent: algorithms, pre-set controls.

43 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds. Agent: algorithms, pre-set controls. 3 Tertiary control: OPF (Optimal power flow). Manages longer lasting changes. Run every few minutes. Goal: economic generation that meets demands while maintaining feasibility (stability).

44 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds. Agent: algorithms, pre-set controls. 3 Tertiary control: OPF (Optimal power flow). Manages longer lasting changes. Run every few minutes. Goal: economic generation that meets demands while maintaining feasibility (stability). Agent: algorithmic computations, humans.

45 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds. Agent: algorithms, pre-set controls. 3 Tertiary control: OPF (Optimal power flow). Manages longer lasting changes. Run every few minutes. Goal: economic generation that meets demands while maintaining feasibility (stability). Agent: algorithmic computations, humans. 4 Once (?) a day: unit commitment problem. Chooses which generators will operate in the next day or half-day.

46 Managing changing demands 1 Primary frequency control. Handles instantaneous (small) changes. Agent: physics. 2 Secondary control. Handles changes that span more than a few seconds. Agent: algorithms, pre-set controls. 3 Tertiary control: OPF (Optimal power flow). Manages longer lasting changes. Run every few minutes. Goal: economic generation that meets demands while maintaining feasibility (stability). Agent: algorithmic computations, humans. 4 Once (?) a day: unit commitment problem. Chooses which generators will operate in the next day or half-day. Agent: algorithms, humans.

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48 CIGRE -International Conference on Large High Voltage Electric Systems 09 Large unexpected fluctuations in wind power can cause additional flows through the transmission system (grid) Large power deviations in renewables must be balanced by other sources, which may be far away Flow reversals may be observed control difficult A solution expand transmission capacity! Difficult (expensive), takes a long time Problems already observed when renewable penetration high

49 CIGRE -International Conference on Large High Voltage Electric Systems 09 Fluctuations 15-minute timespan Due to turbulence ( storm cut-off ) Variation of the same order of magnitude as mean Most problematic when renewable penetration starts to exceed 20 30% Many countries are getting into this regime

50 Control model 1 w i + w i = output of renewable at bus i. w i = forecast, w i = error (uncertain). 2 δ j = response at bus j. Generic linear control: δ j = i α ji w i A: matrix of all values α ji ; (A,..., A) K

51 Control model 1 w i + w i = output of renewable at bus i. w i = forecast, w i = error (uncertain). 2 δ j = response at bus j. Generic linear control: δ j = i α ji w i A: matrix of all values α ji ; (A,..., A) K e.g. j (1 α ji ) = 0 i for full-dimensional uncertainty set.

52 Chance-constrained problems (one period) Optimization Problem min P g,a c k (P g k ) k s.t. the following system is feasible: Flow balance: B θ = P g + renewables {}}{ w + w Line limits: P( f km > fkm max ɛ km km (ignoring e.g. generator constraints) linear control {}}{ A w P d

53 Safety-constrained problems (one period) Optimization Problem min P g,a c k (P g k ) k s.t. the following system is feasible: Flow balance: B θ = P g + Line limits: renewables {}}{ w + w linear control {}}{ A w E(f km ) + ν km Std(f km ) f max km km P d

54 Optimization Problem min P g,a c k (P g k ) k s.t. the following system is feasible: Flow balance: B θ = P g + Line limits, renewables {}}{ w + w km linear control {}}{ A w P d πkm T y km E(P g + w + w A w P d ) + ν km Std(f km ) fkm max, = shift factors from row-differences of pseudo-inverse of B π T km

55 Optimization Problem min P g,a c k (P g k ) k s.t. the following system is feasible: Flow balance: B θ = P g + Line limits, renewables {}}{ w + w km linear control {}}{ A w P d π T km y km E(P g + w + w A w P d ) + ν km Std(f km ) f max km, from row-differences of pseudo-inverse of B

56 Optimization Problem min P g,a c k (P g k ) k s.t. the following system is feasible: Flow balance: B θ = P g + Line limits, renewables {}}{ w + w km linear control {}}{ A w P d πkm T y km E(P g + w + w A w P d ) + ν km Std(f km ) fkm max, = shift factors from row-differences of pseudo-inverse of B π T km

57 Optimization Problem min P g,a c k (P g k ) k s.t. Flow balance: (1 α ji ) = 0 i; j Line limits, km (P g i + w i Pi d ) = 0 i π T km y km E(P g + w + w A w P d ) + ν km Std(f km ) f max km

58 E(P g + w + w A w P d ) = P g + w P d Var(f ij ) = y 2 ij πt ij (I A)Ω(I AT )π ij ;

59 E(P g + w + w A w P d ) = P g + w P d Var(f ij ) = y 2 ij πt ij (I A)Ω(I AT )π ij ; Ω = covariance of w.

60 E(P g + w + w A w P d ) = P g + w P d Var(f ij ) = y 2 ij πt ij (I A)Ω(I AT )π ij ; Ω = covariance of w. Yields SOCP formulation for safety-constrained problem.

61 E(P g + w + w A w P d ) = P g + w P d Var(f ij ) = y 2 ij πt ij (I A)Ω(I AT )π ij ; Ω = covariance of w. Yields SOCP formulation for safety-constrained problem. Caution!

62 E(P g + w + w A w P d ) = P g + w P d Var(f ij ) = y 2 ij πt ij (I A)Ω(I AT )π ij ; Ω = covariance of w. Yields SOCP formulation for safety-constrained problem. Caution! SOCP, but not easy in larger cases. Should use sparse formulation. Should use first-order or outer-envelope method.

63 Previous work on chance-constrained OPF (review), Chertkov, Harnett Roald, Andersson, several coauthors

64 Previous work on chance-constrained OPF (review), Chertkov, Harnett Roald, Andersson, several coauthors 1 Chance-constrained DC-OPF with linear control δ j = i α ji w i can be implemented as convex optimization problem under suitable assumptions

65 Previous work on chance-constrained OPF (review), Chertkov, Harnett Roald, Andersson, several coauthors 1 Chance-constrained DC-OPF with linear control δ j = i α ji w i can be implemented as convex optimization problem under suitable assumptions 2 However such convex problems (SOCPs) beyond solvers But first-order methods fast and accurate

66 An extreme example of variability 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Quantity k is large. Bus b has a load of L units. Stochastic injection at bus b = ω, E(ω) = µ, Var(ω) = σ 2.

67 An extreme example of variability 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Quantity k is large. Bus b has a load of L units. Stochastic injection at bus b = ω, E(ω) = µ, Var(ω) = σ 2. 2σ > µ. Linear generation cost function at i (0 i k + 1): c i p i. c 0 < c 1 = c 2... = c k < c k+1. Safety parameters equal to 3. (specify line limits later)

68 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0.

69 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0. Stochastic flow on ab = 3σ µ ω, with variance σ 2.

70 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0. Stochastic flow on ab = 3σ µ ω, with variance σ 2.

71 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0. Stochastic flow on ab = 3σ µ ω, with variance σ 2. Suppose we want to reduce variance on ab by 50%. Then α k+1 =

72 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0. Stochastic flow on ab = 3σ µ ω, with variance σ 2. Suppose we want to reduce variance on ab by 50%. Then α k+1 = sum of line flow variances > (.5 + (.293) 2 (D + 1))σ 2.

73 0 L µ, σ 2 b k+d a k+2 k+1 1 k 0,1,...,k+1 = generators 1,...,k+1 = participating generators b = load b = stochastic node Unique optimal safety-constrained solution: P g 0 = L µ 3σ. For 1 i k: α i = 1/k and P g i = 3σ/k. α k+1 = P g k+1 = 0. Stochastic flow on ab = 3σ µ ω, with variance σ 2. Suppose we want to reduce variance on ab by 50%. Then α k+1 = D = 10, sum of line flow variances 1.35 σ 2.

74 Variance-aware SCOPF min P g,a s.t. j c k (P g k ) + (v 2 ) k (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 π T km y km E(f km ) + ν km Std(f km ) f max km km (9) i v 2. = vector with entries Var(fij ). = variance metric.

75 Variance-aware SCOPF min P g,a s.t. j c k (P g k ) + (v 2 ) k (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 π T km y km E(f km ) + ν km Std(f km ) f max km km (9) i v 2. = vector with entries Var(fij ). = variance metric. Special case: (Var(f )) = ij F ij (Var(f ij )) ij convex nondecreasing,

76 Variance-aware SCOPF min P g,a s.t. j c k (P g k ) + (v 2 ) k (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 π T km y km E(f km ) + ν km Std(f km ) f max km km (9) i v 2. = vector with entries Var(fij ). = variance metric. Special case: (Var(f )) = ij F ij (Var(f ij )) ij convex nondecreasing, but F could depend on solution

77 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem.

78 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50.

79 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50. Convex problem?

80 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50. Convex problem? Convex.

81 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50. Convex problem? Convex. 3 F = set of N lines with largest flow magnitude, N fixed.

82 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50. Convex problem? Convex. 3 F = set of N lines with largest flow magnitude, N fixed. Convex?

83 Variance metric ij F ij(var(f ij )) 1 F = all lines. Var-aware SCOPF is a convex optimization problem. 2 F = set of N lines with largest flow variance, N fixed. E.g. N = 50. Convex problem? Convex. 3 F = set of N lines with largest flow magnitude, N fixed. Convex? 4 Logarithmic barrier function. ij where ρ > 0 = ρ log(f max km E(f km) ν km Std(f km ))

84 Variance-aware SCOPF min P g,a s.t. j c k (P g k ) + ij F ij (Var(f ij )) k (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 π T km y km E(f km ) + ν km Std(f km ) f max km km (10) i min P g,a s.t. j k c k (P g max k ) ρ log(fkm E(f km) ν km Std(f km )) (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 i

85 Variance-aware SCOPF min P g,a s.t. j c k (P g k ) + (v 2 ) k (1 α ji ) = 0 i; (P g i + w i Pi d ) = 0 π T km y km E(f km ) + ν km Std(f km ) f max km i km

86 Correction vs Formal Variance-Aware Optimization Several groups of authors: 1 Outer-approximation algorithm for chance-constrained DC-OPF converges in few iterations. 2 Only a few lines are at risk in realistic cases.

87 Correction vs Formal Variance-Aware Optimization Several groups of authors: 1 Outer-approximation algorithm for chance-constrained DC-OPF converges in few iterations. 2 Only a few lines are at risk in realistic cases. 3 Low-hanging fruit:

88 Correction vs Formal Variance-Aware Optimization Several groups of authors: 1 Outer-approximation algorithm for chance-constrained DC-OPF converges in few iterations. 2 Only a few lines are at risk in realistic cases. 3 Low-hanging fruit: instead of variance-aware optimization, first solve SC-OPF problem, and then correct or adjust to reduce variability without increasing cost (by much).

89 GENERIC CORRECTION TEMPLATE Input: an instance of the safety-constrained problem and a variance metric. Step I. Solve safety-constrained OPF problem, with solution ( P g, Ā). Step II. Perform a small number of adjustment iterations which shift ( P g, Ā) to a new feasible solution that attains an improved value of the variance metric, while at the same time increasing generation cost in a moderate manner.

90 Reroute(τ, Â) Â : a given control matrix

91 Reroute(τ, Â) Â : a given control matrix So Var(f km ) = ykm 2 πt km (I Â)Ω(I ÂT )π km. = ˆV km is fixed for every 0 < τ < 1.

92 Reroute(τ, Â)  : a given control matrix So Var(f km ) = ykm 2 πt km (I Â)Ω(I ÂT )π km. = ˆV km is fixed for every 0 < τ < 1. min P g s.t. c k (P g k ) k (P g i + w i Pi d ) = 0 i π T km y km E(f km ) + ν km ˆV km (1 τ )f max km Here, f km is an implicit function of P g and  km

93 VShift( P g ) P g : generation vector. f : corresponding flow vector. min A km F( f ) constraints on A: km ( y 2 km πt km (I A)Ω(I AT )π km ) (1 α ji ) = 0 i j Keep power flows fixed, improve on control

94 Correction procedure Input: Feasible solution (P g,0, A 0 ), 0 < τ < 1. For k = 1, 2,..., K perform iteration k: 1. Run Reroute(A k 1, τ ). If infeasible STOP. Else let P g,k be optimal. 2. Solve VShift(P g,k ), with solution  k. 3. Set A k (1 λ)a k 1 + λâ k. 0 < λ < 1 chosen so that P g,k, A k feasible 4. If (A k ) (A k 1 ), STOP.

95 Correction procedure Input: Feasible solution (P g,0, A 0 ), 0 < τ < 1. For k = 1, 2,..., K perform iteration k: 1. Run Reroute(A k 1, τ ). If infeasible STOP. Else let P g,k be optimal. 2. Solve VShift(P g,k ), with solution  k. 3. Set A k (1 λ)a k 1 + λâ k. 0 < λ < 1 chosen so that P g,k, A k feasible 4. If (A k ) (A k 1 ), STOP. Thm.: In convex case if we stop in Step 4, then minimum.

96 Example on case2746wp 2746 buses, 3514 branches, 520 generators 22 stochastic injection sites, sum of mean injections (penetration 18.5% penetration) average ratio of standard deviation to mean 30%.

97 Example on case2746wp 2746 buses, 3514 branches, 520 generators 22 stochastic injection sites, sum of mean injections (penetration 18.5% penetration) average ratio of standard deviation to mean 30%. Variance metric: ij F Var(f ij ), where F is the union of

98 Example on case2746wp 2746 buses, 3514 branches, 520 generators 22 stochastic injection sites, sum of mean injections (penetration 18.5% penetration) average ratio of standard deviation to mean 30%. Variance metric: ij F Var(f ij ), where F is the union of The 100 lines with largest flow magnitude Lines ij where f ij + ν ij Std(f ij ) (1 τ )f max ij

99 Two iterations Initial variance metric:

100 Two iterations Initial variance metric: After one iteration:

101 Two iterations Initial variance metric: After one iteration: After two iterations: Approx. 40% reduction relative to original.

102 Two iterations Initial variance metric: After one iteration: After two iterations: Approx. 40% reduction relative to original. Cost nearly constant after two iterations

103 Two iterations Initial variance metric: After one iteration: After two iterations: Approx. 40% reduction relative to original. Cost nearly constant after two iterations Variance-shifting SOCP has approximately variables and constraints and nonzeros Solution times of a few seconds ThThu.Nov @blacknwhite