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

SLIDE 1

A Convex Relaxation Framework for Strategic Bidding in Electricity Markets

Mahdi Ghamkhari Department of Computer Science University of California Davis

SLIDE 2

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

SLIDE 3

History

SLIDE 4

History

SLIDE 5

Electricity Market

1 3 4 2 5 7 6 8 9 11 12 13 10 14 15 16 17 18 23 19 20 21 22 24 26 25 28 27 29 30

S1 S2 S3 S4 G G G G G G G G

Generators Consumers (Price, Quantity) Strategic Generator seeks to maximizes its profit by bidding in a strategic way Electricity Network constitutes of Generators, Consumers and Transmission Lines

SLIDE 6

MPEC

z

Q =

z

d

z T

q

Inherent relation between parameters Will be needed later

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

Mathematical Program with Equilibrium Constraints (MPEC)

SLIDE 7

Mixed Integer Linear Program

⇔

Binary Variable Binary Variable

0 ≤

z T

q x ≤ Binary × (LargeNumber) 0 ≤

z T

d x + 2 ≤ 1−Binary × (LargeNumber)

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

z

Q =

z

d

z T

q

SLIDE 8

Solutions

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

SLIDE 9

Solutions

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

SLIDE 10

Our Approach

Minimize Λ

Upper Bound

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

SLIDE 11

Our Approach

Minimize Λ

Upper Bound

Λ is upper bound if and only if

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

SLIDE 12

Positivestellensatz

Polynomials that are positive on semi Algebraic sets Semi Algebraic Set Polynomial

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

SLIDE 13

Schmudgen Positivestellensatz

Semi Algebraic Set Polynomial

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 14

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

SLIDE 15

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 16

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 17

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 18

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 19

Schmudgen Positivestellensatz

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 20

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

Schmudgen Positivestellensatz

Variables are polynomials

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 21

( SOS Polynomial )=

2

(Polynomial) ∑

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

Schmudgen Positivestellensatz

Minimize Λ

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 22

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

Schmudgen Positivestellensatz

Arbitrary Polynomials

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 23

( SOS Polynomial )=

2

(Polynomial) ∑

T

Λ−x F x −2

T

f x ≥ 0

is positive on

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪

Schmudgen Positivestellensatz

x is not variables x is an index for infinite constraints

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 24

A Convex Optimization Problem

Minimize Λ

T

Λ−x F x −2

T

f x −

( SOS Polynomial )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

i=1 I

∑

t I

∑ (

SOS Polynomial )

j=1 I

∑

(

i T

p x +

i0

p )(

i T

p x +

i0

p )(

t T

p x +

t0

p )−

! ( Arbitray Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Polynomial )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

"

SLIDE 25

T

Λ−x F x −2

T

f x −

( Positive Scalar )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( Positive Scaler )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

( Linear Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Scalar )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

!

Relaxation of Polynomials in Psatz

Quadratic Expression

! " # # $ ##

SLIDE 26

Computationally Tractable Reformulation

T

1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γ 1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥0

γ =

Λ

T

f − f F

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ − ( Positive Scalar )

i=1 I

∑

i0

p

i T

p

2

i

p

2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −

i=1 I

∑

( Positive Scaler )

j=1 I

∑

j0

p

j

p 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

T

j0

p

j

p 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

− Scalar

m=1 M

∑ 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

T

m0

v

m

v 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

−

m=1 M

∑

Scalar

l=1 n

∑

l

e ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

T

m0

v

m

v 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

( Positive Scalar )

z=1 Z

∑

i T

q

i

q

z

Q

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

Quadratic Expression

! " # # $ ##

T

Λ−x F x −2

T

f x −

( Positive Scalar )

i=1 I

∑

(

i T

p x +

i0

p )−

i=1 I

∑

( Positive Scaler )

j=1 I

∑

(

i T

p x +

i0

p )(

j T

p x +

j0

p )−

( Linear Polynomial )

m=1 M

∑

(

m T

v x +

m0

v

)− ( Arbitrary Scalar )

z=1 Z

∑

(

z

T

x Q x +

z T

2 q x ) ≥ 0 ∀ x ∈

n

!

SLIDE 27

Computationally Tractable

γ ≻0

Semi Definite

T

1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γ 1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥0

SLIDE 28

Computationally Tractable

Λ

is upper bound if

γ ≻0 γ ≻0

Semi Definite

T

1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γ 1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥0

SLIDE 29

Computationally Tractable

Λ

is upper bound if

γ ≻0 γ ≻0

Semi Definite

T

1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γ 1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥0

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

Best customized upper bound

Minimize Λ γ ≻0

SLIDE 30

Computationally Tractable

is 97% optimal

Λ

Minimize Λ γ ≻0

SLIDE 31

Computationally Tractable

is 97% optimal

Λ

Minimize Λ γ ≻0

x: Optimal optimization variables in MPEC ? What happened to x?

γ ≻0

T

1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ γ 1 x ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥0

SLIDE 32

Recovery

Minimize Λ γ ≻0

maximize trace

T

f f F

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

trace

i0

p

i T

p

2

i

p

2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i trace

l

e

T

m0

v

m

v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟

= 0 ∀ m ,l trace

i0

p

i

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

T

j0

p

j

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i j trace

z T

q

z

q

z

Q

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ z

11

X = 1 X ≻ 0

Duality Dual Form PrimalForm

SLIDE 33

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

maximize trace

T

f f F

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

trace

i0

p

i T

p

2

i

p

2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i trace

l

e

T

m0

v

m

v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟

= 0 ∀ m ,l trace

i0

p

i

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

T

j0

p

j

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i j trace

z T

q

z

q

z

Q

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ z

11

X = 1 X ≻ 0

First column of X

Approximated solution for variables in MPEC

SLIDE 34

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

maximize trace

T

f f F

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

trace

i0

p

i T

p

2

i

p

2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i trace

l

e

T

m0

v

m

v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟

= 0 ∀ m ,l trace

i0

p

i

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

T

j0

p

j

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i j trace

z T

q

z

q

z

Q

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ z

11

X = 1 X ≻ 0

First column of X

Approximated solution for variables in MPEC is not feasible in MPEC

*

x

SLIDE 35

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

to produce a feasible solution to MPEC

*

x

we use

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

z

Q =

z

d

z T

q

Approximated solution for variables in MPEC

SLIDE 36

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

to produce a feasible solution to MPEC

*

x

we use

z

Q =

z

d

z T

q

z T

d x + 2

( )

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

Approximated solution for variables in MPEC

SLIDE 37

z T

d x + 2

( )= 0

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

to produce a feasible solution to MPEC

*

x

we use

z T

d x + 2

( )

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

z T

d

*

x + 2 ≤ ε

z T

q

*

x ≥ Δ

and

Approximated solution for variables in MPEC

SLIDE 38

z T

d x + 2

( )= 0

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

to produce a feasible solution to MPEC

*

x

we use

z T

d x + 2

( )

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0 and

z T

d

*

x + 2 ≥ Δ

z T

q

*

x ≤ ε

Approximated solution for variables in MPEC

SLIDE 39

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

Recovery

to produce a feasible solution to MPEC

*

x

we use

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

Approximated solution for variables in MPEC

SLIDE 40

Algorithm

• 1. Solve the dual of customized Psatz Relaxation
• 2. Obtain an approximated solution for MPEC
• 3. Eliminate some non-convex constraints
• 4. Solve a reduced complexity MILP

maximize trace

T

f f F

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟

trace

i0

p

i T

p

2 i

p

2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i trace

l

e

T m0

v

m

v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟

= 0 ∀ m ,l trace

i0

p

i

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

T j0

p

j

p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ i j trace

z T

q

z

q

z

Q

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

X

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

≥ 0 ∀ z

11

X = 1 X ≻ 0

1

*

x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =

*

X

T

1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

z T

d x + 2

( )

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

z T

d x + 2

( )= 0

z T

q x

⎛ ⎝ ⎜ ⎞ ⎠ ⎟= 0

Maximize

T

x F x +2

T

f x

i T

p x +

i0

p ≥ 0

i

∀

m T

v x +

m0

v = 0

m

∀

z

T

x Q x +

z T

2 q x = 0

z

∀

SLIDE 41

Simulation Results

1 2 3 4 5 6 7 8 9 10

Average Compution Time (minutes)

100 200 300 400 500 600 700

(a)

MILP Approach in [1] Proposed Approach

2 4 6 8 10 4 8 12 16

Number of Scenarios

The impact of increasing the number of random scenarios on the computation time of the proposed approach and the MILP approach

1 3 4 2 5 7 6 8 9 11 12 13 10 14 15 16 17 18 23 19 20 21 22 24 26 25 28 27 29 30 S1 S2 S3 S4 G G G G G G G G

IEEE 30-Bus Network

SLIDE 42

Simulation Results

The impact of increasing the number of random scenarios on the

• ptimality of the proposed approach.

Number of Scenarios

1 2 3 4 5 6 7 8 9 10

Average Optimality

0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1

(b)

Proposed Approach

1 3 4 2 5 7 6 8 9 11 12 13 10 14 15 16 17 18 23 19 20 21 22 24 26 25 28 27 29 30 S1 S2 S3 S4 G G G G G G G G

IEEE 30-Bus Network

SLIDE 43

Thank You