slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
An Efficient Computational Method for Nonlinear Power Optimization Problems PowerPoint Presentation
Download Presentation
An Efficient Computational Method for Nonlinear Power Optimization Problems

Loading in 2 Seconds...

play fullscreen
1 / 27

An Efficient Computational Method for Nonlinear Power Optimization Problems - PowerPoint PPT Presentation


  • 93 Views
  • Uploaded on

Javad Lavaei Department of Electrical Engineering Columbia University Joint work with Somayeh Sojoudi and Ramtin Madani. An Efficient Computational Method for Nonlinear Power Optimization Problems. Power Networks. Optimizations: Optimal power flow (OPF) Security-constrained OPF

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'An Efficient Computational Method for Nonlinear Power Optimization Problems' - kenton


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

JavadLavaeiDepartment of Electrical EngineeringColumbia UniversityJoint work with SomayehSojoudiand RamtinMadani

An Efficient Computational Method for Nonlinear Power Optimization Problems

power networks
Power Networks
  • Optimizations:
    • Optimal power flow (OPF)
    • Security-constrained OPF
    • State estimation
    • Network reconfiguration
    • Unit commitment
    • Dynamic energy management
  • Issue of non-convexity:
    • Discrete parameters
    • Nonlinearity in continuous variables
  • Transition from traditional grid to smart grid:
    • More variables (10X)
    • Time constraints (100X)

Javad Lavaei, Columbia University

2

broad interest in optimal power flow
Broad Interest in Optimal Power Flow
  • OPF-based problems solved on different time scales:
    • Electricity market
    • Real-time operation
    • Security assessment
    • Transmission planning
  • Existing methods based on linearization or local search
  • Question: How to find the best solution using a scalable robust algorithm?
  • Huge literature since 1962 by power, OR and Econ people

Javad Lavaei, Columbia University

3

local solutions
Local Solutions

P1

P2

OPF

Local solution: $1502 Global solution: $338

Javad Lavaei, Columbia University

4

local solutions1
Local Solutions

Source of Difficulty: Power is quadratic in terms of complex voltages.

  • Anya Castillo et al.
  • Ian Hiskens from Umich:
  • Study of local solutions by Edinburgh’s group

Javad Lavaei, Columbia University

5

summary of results
Summary of Results

Project 1:How to solve a given OPF in polynomial time? (joint work with Steven Low)

  • A sufficient condition to globally solve OPF:
    • Numerous randomly generated systems
    • IEEE systems with 14, 30, 57, 118, 300 buses
    • European grid
  • Various theories: Itholds widely in practice

Project 2:Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)

  • Distribution networks are fine.
  • Every transmission network can be turned into a good one.

Javad Lavaei, Columbia University

6

summary of results1
Summary of Results

Project 3:How to design a distributed algorithm for solving OPF? (joint work with Stephen Boyd, Eric Chu and Matt Kranning)

  • A practical (infinitely) parallelizable algorithm
  • It solves 10,000-bus OPF in 0.85 seconds on a single core machine.

Project 4:How to do optimization for mesh networks? (joint work with RamtinMadani and Somayeh Sojoudi)

  • Developed a penalization technique
  • Verified its performance on IEEE systems with 7000 cost functions

Javad Lavaei, Columbia University

7

geometric intuition two generator network
Geometric Intuition: Two-Generator Network

Javad Lavaei, Columbia University

8

optimal power flow
Optimal Power Flow

Cost

Operation

Flow

Balance

  • Extensions:
    • Other objective (voltage support, reactive power, deviation)
    • More variables, e.g. capacitor banks, transformers
    • Preventive or corrective contingency constraints

Javad Lavaei, Columbia University

9

various relaxations
Various Relaxations

OPF

  • SDP relaxation:
    • IEEE systems
    • SC Grid
    • European grid
    • Random systems

Dual OPF

SDP

  • Exactness of SDP relaxation and zero duality gap are equivalent for OPF.

Javad Lavaei, Columbia University

10

ac transmission networks
AC Transmission Networks
  • How about AC transmission networks?
    • May not be true for every network
    • Various sufficient conditions
  • AC transmission network manipulation:
    • High performance (lower generation cost)
    • Easy optimization
    • Easy market (positive LMPs and existence of eq. pt.)

PS

Javad Lavaei, Columbia University

11

phase shifters
Phase Shifters
  • Blue: Feasible set (PG1,PG2)
  • Green: Effect of phase shifter
  • Red: Effect of convexification
  • Minimization over green = Minimization over green and red (even with box constraints)

Javad Lavaei, Columbia University

12

phase shifters1
Phase Shifters
  • Simulations:
    • Zero duality gap for IEEE 30-bus system
    • Guarantee zero duality gap for all possible load profiles?
    • Theoretical side: Add 12 phase shifters
    • Practical side: 2 phase shifters are enough
    • IEEE 118-bus system needs no phase shifters (power loss case)

Phase shifters speed up the computation:

Javad Lavaei, Columbia University

13

response of sdp to equivalent formulations
Response of SDP to Equivalent Formulations
  • Capacity constraint: active power, apparent power, angle difference, voltage difference, current?

P2

P1

Equivalent formulations behave differently after relaxation.

SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows.

Correct solution

Javad Lavaei, Columbia University

14

low rank solution
Low-Rank Solution

Javad Lavaei, Columbia University

15

penalized sdp relaxation
Penalized SDP Relaxation
  • How to turn a low-rank solution into a rank-1 solution?
  • Extensive simulations show that reactive power needs to be corrected.
  • Penalized SDP relaxation:
  • Penalized SDP relaxation aims to find a near-optimal solution.
  • It worked for IEEE systems with over 7000 different cost functions.
  • Near-optimal solution coincided with the IPM’s solution in 100%, 96.6% and 95.8% of cases for IEEE 14, 30 and 57-bus systems.

Javad Lavaei, Columbia University

16

penalized sdp relaxation1
Penalized SDP Relaxation
  • Let λ1 and λ2 denote the two largest eigenvalues of W.
  • Correction of active powers is negligible but reactive powers change noticeably.
  • There is a wide range of values for ε giving rise to a nearly-global local solution.

Javad Lavaei, Columbia University

17

penalized sdp relaxation2
Penalized SDP Relaxation

Javad Lavaei, Columbia University

18

problem of interest
Problem of Interest
  • Abstract optimizations are NP-hard in the worst case.
  • Real-world optimizations are highly structured:
  • Sparsity:
  • Non-trivial structure:
  • Question: How does the physical structure affect tractability of an optimization?

Javad Lavaei, Columbia University

19

example 1
Example 1

Trick:

SDP relaxation:

  • Guaranteed rank-1 solution!

Javad Lavaei, Columbia University

20

example 11
Example 1

Opt:

  • Sufficient condition for exactness: Sign definite sets.
  • What if the condition is not satisfied?
      • Rank-2 W(but hidden)
      • NP-hard

Javad Lavaei, Columbia University

21

sign definite set
Sign Definite Set
  • Real-valued case: “T “ is sign definite if its elements are all negative or all positive.
  • Complex-valued case: “T “ is sign definite if T and –T are separable in R2:

Javad Lavaei, Columbia University

22

exact convex relaxation
Exact Convex Relaxation
  • Each weight set has about 10 elements.
  • Due to passivity, they are all in the left-half plane.
  • Coefficients: Modes of a stable system.
  • Weight sets are sign definite.

Javad Lavaei, Stanford University

Javad Lavaei, Columbia University

17

23

formal definition optimization over graph
Formal Definition: Optimization over Graph

Optimization of interest:

(real or complex)

Define:

  • SDP relaxation for y and z (replace xx* with W).
  • f (y , z) is increasing in z (no convexity assumption).
  • Generalized weighted graph: weight set for edge (i,j).

Javad Lavaei, Columbia University

24

real valued optimization
Real-Valued Optimization

Edge

Cycle

Javad Lavaei, Columbia University

25

complex valued optimization
Complex-Valued Optimization
  • Main requirement in complex case: Sign definite weight sets
  • SDP relaxation for acyclic graphs:
    • real coefficients
    • 1-2 element sets (power grid: ~10 elements)

Javad Lavaei, Columbia University

26

conclusions
Conclusions
  • Focus: OPF with a 50-year history
  • Goal: Find a global solution efficiently
  • Obtained provably global solutions for many practical OPFs
  • Developed various theories for distribution and transmission networks
  • Still some open problems to be addressed

Javad Lavaei, Columbia University

27