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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

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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


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


local solutions
Local Solutions




Local solution: $1502 Global solution: $338

Javad Lavaei, Columbia University


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


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


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


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

Javad Lavaei, Columbia University


optimal power flow
Optimal Power Flow





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

Javad Lavaei, Columbia University


various relaxations
Various Relaxations


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

Dual OPF


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

Javad Lavaei, Columbia University


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.)


Javad Lavaei, Columbia University


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


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


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



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


low rank solution
Low-Rank Solution

Javad Lavaei, Columbia University


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


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


penalized sdp relaxation2
Penalized SDP Relaxation

Javad Lavaei, Columbia University


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


example 1
Example 1


SDP relaxation:

  • Guaranteed rank-1 solution!

Javad Lavaei, Columbia University


example 11
Example 1


  • 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


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


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



formal definition optimization over graph
Formal Definition: Optimization over Graph

Optimization of interest:

(real or complex)


  • 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


real valued optimization
Real-Valued Optimization



Javad Lavaei, Columbia University


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


  • 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