An Efficient Computational Method for Nonlinear Power
Optimization Problems
Javad Lavaei
Department of Electrical Engineering
Columbia University
Joint work with Somayeh Sojoudi and Ramtin Madani
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)
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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
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Local Solutions
P1
P2
OPF
Local solution: $1502
Global solution: $338
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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
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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: It holds 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.
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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 Ramtin Madani and
Somayeh Sojoudi)
 Developed a penalization technique
 Verified its performance on IEEE systems with 7000 cost functions
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Geometric Intuition: Two-Generator Network
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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
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Various Relaxations
OPF
Dual OPF
SDP
 SDP relaxation:
 IEEE systems
 SC Grid
 European grid
 Random systems
 Exactness of SDP relaxation and zero duality gap are equivalent for OPF.
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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.)
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PS
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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)
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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:
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Response of SDP to Equivalent Formulations
 Capacity constraint: active power, apparent
power, angle difference, voltage difference, current?
P1
P2
1.
Equivalent formulations behave
differently after relaxation.
2.
SDP works for weakly-cyclic networks
with cycles of size 3 if voltage
difference is used to restrict flows.
Correct solution
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Low-Rank Solution
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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.
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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.
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Penalized SDP Relaxation
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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?
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Example 1
Trick:
SDP relaxation:
 Guaranteed rank-1 solution!
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Example 1
Opt:
 Sufficient condition for exactness: Sign definite sets.
 What if the condition is not satisfied?
 Rank-2 W (but hidden)
 NP-hard
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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:
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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
JavadLavaei,
Lavaei,Columbia
Stanford University
University
17
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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
Javad Lavaei, Columbia University
for edge (i,j).
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Real-Valued Optimization
Edge
Cycle
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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)
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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
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