A Convex Relaxation Framework for Strategic Bidding in Electricity - PowerPoint PPT Presentation

A Convex Relaxation Framework for Strategic Bidding in Electricity Markets Mahdi Ghamkhari Department of Computer Science University of California Davis

Outline M. Ghamkhari, A. Sadeghi-Mobarakeh, H. Mohsenian-Rad, “Strategic Bidding for Producers in Nodal Electricity Markets: A Convex Relaxation Approach,” Accepted for Publication in IEEE Transactions on Power Systems , July 2016 Joint work with Ashkan Sadeghi-Mobarakeh and Hamed Mohsenian-Rad

History

History

Electricity Market 29 27 G Generators Consumers 30 G 28 25 26 S3 S4 G 24 23 19 18 15 20 G 21 17 S2 14 (Price, Quantity) 16 10 G 22 13 12 11 9 3 6 8 4 1 2 5 G G 7 S1 G Electricity Network constitutes of Generators, Consumers and Transmission Lines Strategic Generator seeks to maximizes its profit by bidding in a strategic way

MPEC T T f x Maximize x F x + 2 M athematical P rogram with E quilibrium C onstraints ( MPEC ) T p ≥ 0 ∀ p x + i i 0 i T v = 0 ∀ v x + m m 0 m T T x Q x + 2 q x = 0 ∀ z z z T q Q = d Will be z needed later z z Inherent relation between parameters

Mixed Integer Linear Program T T f x Maximize x F x + 2 T Binary Variable p ≥ 0 ∀ p x + i i 0 i T v = 0 ∀ v x + m m 0 m T T q x ≤ Binary × ( L arg eNumber ) ⇔ 0 ≤ T x Q x + 2 q x = 0 ∀ z z z z d x + 2 ≤ 1 − Binary × ( L arg eNumber ) T 0 ≤ z T q Q = d Binary Variable z z z

Solutions • MILP: gives global solution • MILP: Computation time increases Exponentially

Solutions • MILP: gives global solution • Our Approach: gives global solution with 99% Optimality • MILP: Computation time increases Exponentially • Our Approach:Computation Time increases Linearly

Our Approach Upper Bound Minimize Λ T T f x Maximize x F x + 2 T p ≥ 0 ∀ p x + i i 0 i T v = 0 ∀ v x + m m 0 m T T x Q x + 2 q x = 0 ∀ z z z

Our Approach Upper Bound Minimize Λ T T f x Maximize x F x + 2 T p ≥ 0 ∀ p x + i i 0 i T v = 0 ∀ v x + m m 0 m T T x Q x + 2 q x = 0 ∀ z z z Λ is upper bound if and only if ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Positivestellensatz Polynomials that are positive on semi Algebraic sets ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z Polynomial Semi Algebraic Set

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z Polynomial Semi Algebraic Set

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z

Schmudgen Positivestellensatz T T f x − Λ− x F x − 2 I SOS T ∑ p x + p ) − ( ) ( Polynomial i i 0 i = 1 I I SOS T T ∑ ∑ p x + p )( p x + p ) − ( Polynomial ) ( i i 0 j j 0 i = 1 j = 1 I I I SOS T T T ∑ ∑ ∑ ( p x + p )( p x + p )( p x + p ) − Polynomial ) ( Variables are polynomials i i 0 i i 0 t t 0 i = 1 j = 1 t ! M Arbitray ∑ T v x + v ) − ( ) ( Polynomial m m 0 m = 1 Z Arbitrary T 2 q x ) ≥ 0 ∀ x ∈ ∑ T n x Q x + ( ) ( " Polynomial z z z = 1 ⎧ ⎫ T p ≥ 0 ∀ p x + ⎪ ⎪ i i 0 i ⎪ ⎪ ⎪ ⎪ T T T f x ≥ 0 Λ− x F x − 2 v = 0 ∀ v x + is positive on ⎨ ⎬ m m 0 m ⎪ ⎪ T ⎪ T ⎪ x Q x + 2 q x = 0 ∀ ⎪ ⎪ ⎩ ⎭ z z z