Transient-state Feasibility Set Approximation of Power Networks - PowerPoint PPT Presentation

Transient-state Feasibility Set Approximation of Power Networks Against Disturbances Yifu Zhang and Jorge Cort´ es Mechanical and Aerospace Engineering University of California, San Diego Power Systems II 2017 American Control Conference Seattle, Washington May 25, 2017

Power Network: Efficiency & Robustness Efficiency 1 Economic Dispatch 2 Optimal Power Flow ... Robustness 1 Voltage Collapse 2 Cascading Failure ... Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 2 / 24

Power Network: Efficiency & Robustness Efficiency 1 Economic Dispatch 2 Optimal Power Flow ... Robustness 1 Voltage Collapse 2 Cascading Failure ... How to identify the disturbances under which (a) frequencies of buses stay within safe bounds, and (b) power flows of transmission lines stay within safe bounds? Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 2 / 24

Outline Problem Statement 1 Linearized Power Network Dynamics Disturbance Modeling 2 Equivalent Transformation Time Domain Solution Set Decomposition 3 Approximation of the Feasibility Set Outer Approximations Inner Approximations 4 Simulations Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 3 / 24

Linearized Power Network Dynamics � ˙ � � �� Λ( t ) � � � Λ( t ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 P ( t ) Ω( t ) Ω( t ) Λ = [ λ 1 , λ 2 , . . . λ m ] T ∈ R m — angle difference vector Ω = [ ω 1 , ω 2 , . . . ω n ] T ∈ R n — frequency vector M ∈ R n × n — inertia matrix E ∈ R n × n — damping/droop parameter matrix Y b ∈ R m × m — susceptance matrix P = [ p 1 , p 2 , . . . p n ] T ∈ R n — power injection vector ( Y b Λ = [ f 1 , f 2 , . . . f m ] T ∈ R m — power flow vector ) Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 4 / 24

Disturbance Modeling Power network dynamics � ˙ � � �� Λ( t ) � � � Λ( t ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 P ( t ) Ω( t ) Ω( t ) Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 5 / 24

Disturbance Modeling Power network dynamics � ˙ � � �� Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 P ( t, K ) Ω( t, K ) Ω( t, K ) P ( t, K ) = P 0 ( t ) + ¯ P ( t, K ) P 0 ( t ) ∈ R n : scheduled power injection ¯ P ( t, K ) ∈ R n : power disturbance Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 5 / 24

Disturbance Modeling Power network dynamics � ˙ � � �� Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 P ( t, K ) Ω( t, K ) Ω( t, K ) Disturbance model P ( t, K ) = P 0 ( t ) + ¯ P ( t, K ) P 0 ( t ) ∈ R n : scheduled power injection ¯ P ( t, K ) ∈ R n : power disturbance ¯ P ( t, K ) = B K D ζ ( t ) K Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 5 / 24

Example 1 1( ) t K 1  t 1( ) t e K 2  1( t 0.5) K 3 2 3 Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 6 / 24

Example   1 1( t ) K 1 ¯ 0 P ( t, K ) =   1( ) t K 1 1( t ) e − t K 2 + 1( t − 0 . 5) K 3       1 0 0 1( t ) 0 0 K 1  1( t ) e − t t = 0 0 0 0 0 K 2 1( ) t e K 2       0 1 1 0 0 1( t − 0 . 5) K 3  1( t 0.5) K 3 1( t ) 1( t ) e − t 1( t − 0 . 5) � � = B K diag K = B K D ζ ( t ) K 2 3 Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 6 / 24

Example   1 1( t ) K 1 ¯ 0 P ( t, K ) =   1( ) t K 1 1( t ) e − t K 2 + 1( t − 0 . 5) K 3       1 0 0 1( t ) 0 0 K 1  1( t ) e − t t = 0 0 0 0 0 K 2 1( ) t e K 2       0 1 1 0 0 1( t − 0 . 5) K 3  1( t 0.5) K 3 1( t ) 1( t ) e − t 1( t − 0 . 5) � � = B K diag K = B K D ζ ( t ) K 2 3 ¯ P ( t, K ) = “location × trajectory form × amplitude” Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 6 / 24

Problem Statement Power network dynamics � ˙ � � �� Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 P ( t, K ) Ω( t, K ) Ω( t, K ) Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 7 / 24

Problem Statement Power network dynamics � ˙ � � �� Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 � � Ω( t, K ) P 0 ( t ) + B K D ζ ( t ) K Ω( t, K ) For a given 0 � t 1 < t 2 , find all K ’s that guarantee: 1 Transient-state frequency bound: Ω min � Ω( t, K ) � Ω max , ∀ t ∈ [ t 1 , t 2 ] 2 Transient-state power flow bound: F min � Y b Λ( t, K ) � F max , ∀ t ∈ [ t 1 , t 2 ] Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 7 / 24

Problem Statement Power network dynamics � ˙ � � �� Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 � � Ω( t, K ) P 0 ( t ) + B K D ζ ( t ) K Ω( t, K ) For a given 0 � t 1 < t 2 , find all K ’s that guarantee: 1 Transient-state frequency bound: Ω min � Ω( t, K ) � Ω max , ∀ t ∈ [ t 1 , t 2 ] 2 Transient-state power flow bound: F min � Y b Λ( t, K ) � F max , ∀ t ∈ [ t 1 , t 2 ] K ∈ R s | Ω min � Ω( t, K ) � Ω max , F min � Y b Λ( t, K ) � F max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ � Ψ:( transient-state ) feasibility set Goal: Characterize Ψ! Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 7 / 24

Outline Problem Statement 1 Linearized Power Network Dynamics Disturbance Modeling 2 Equivalent Transformation Time Domain Solution Set Decomposition 3 Approximation of the Feasibility Set Outer Approximations Inner Approximations 4 Simulations Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 8 / 24

Time Domain Solution � ˙ � � � � Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 � � Ω( t, K ) P 0 ( t ) + B K D ζ ( t ) K Ω( t, K ) Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 9 / 24

Time Domain Solution � ˙ � � � � Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 � � Ω( t, K ) P 0 ( t ) + B K D ζ ( t ) K Ω( t, K ) � � � 0 m x ( t, K ) = Ax ( t, K ) + ˙ M − 1 � � P 0 ( t ) + B K D ζ ( t ) K Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 9 / 24

Time Domain Solution � ˙ � � � � Λ( t, K ) � � � Λ( t, K ) 0 m × m D 0 m = + − M − 1 D T Y b ˙ − M − 1 E M − 1 � � Ω( t, K ) P 0 ( t ) + B K D ζ ( t ) K Ω( t, K ) � � � 0 m x ( t, K ) = Ax ( t, K ) + ˙ M − 1 � � P 0 ( t ) + B K D ζ ( t ) K � Solve first-order ODE x ( t, K ) = S ( t ) + V ( t ) K where � t � t � � � � 0 m 0 m S ( t ) � e At x 0 + e A ( t − τ ) e A ( t − τ ) � d τ, V ( t ) d τ M − 1 P 0 ( τ ) M − 1 B K D ζ ( τ ) 0 0 Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 9 / 24

Equivalent Transformation K ∈ R s | Ω min � Ω( t, K ) � Ω max , F min � Y b Λ( t, K ) � F max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ � Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 10 / 24

Equivalent Transformation K ∈ R s | Ω min � Ω( t, K ) � Ω max , F min � Y b Λ( t, K ) � F max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ � � K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = where Ω max Ω min � � � � x max � , x min � Y − 1 Y − 1 F max F min b b Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 10 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 11 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = K ∈ R s | x min � S ( t ) + V ( t ) K � x max � � � = t 1 � t � t 2 Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 11 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = K ∈ R s | x min � S ( t ) + V ( t ) K � x max � � � = t 1 � t � t 2 ⇒ Ψ contains infinitely many constraints ⇒ Approximation Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 11 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 12 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = K ∈ R s | x min � � [ S ( t )] i + [ V ( t )] i K � x max � � = , ∀ t ∈ [ t 1 , t 2 ] i i i =1 , 2 ,...n + m � � Ψ i i =1 , 2 ,...n + m Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 12 / 24

Set Decomposition K ∈ R s | x min � S ( t ) + V ( t ) K � x max , ∀ t ∈ [ t 1 , t 2 ] � � Ψ = K ∈ R s | x min � � [ S ( t )] i + [ V ( t )] i K � x max � � = , ∀ t ∈ [ t 1 , t 2 ] i i i =1 , 2 ,...n + m � � Ψ i i =1 , 2 ,...n + m Approximation of Ψ i ⇒ Approximation of Ψ Y. Zhang & J. Cort´ es (UCSD) May 25, 2017 12 / 24