**Javad LavaeiDepartment of Electrical EngineeringColumbia** University Convex Relaxation for Polynomial Optimization: Application to Power Systems and Decentralized Control

**Outline** • Convex relaxation for highly structured optimization • (Joint work with: Somayeh Sojoudi) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 2

**Penalized Semidefinite Programming (SDP) Relaxation** • Exactness of SDP relaxation: • Existence of a rank-1 solution • Implies finding a global solution • How to study the exactness of relaxation? Javad Lavaei, Columbia University 3

**Highly Structured Optimization** • Abstract optimizations are NP-hard in the worst case. • Real-world optimizations are highly structured: • Sparsity: • Non-trivial structure: • Question: How do structures affect tractability of an optimization? Javad Lavaei, Columbia University 4

**Example** • Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph: Javad Lavaei, Columbia University 5

**Example** Optimization: Javad Lavaei, Columbia University 6

**Real-Valued Optimization** Edge Cycle Javad Lavaei, Columbia University 7

**Complex-Valued Optimization** • 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 8

**Complex-Valued Optimization** • Consider a real matrix M: • Polynomial-time solvable for weakly-cyclic bipartite graphs. Javad Lavaei, Columbia University 9

**Outline** • Convex relaxation for highly structured optimization • (Joint work with: Somayeh Sojoudi) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 10

**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 11

**Resource Allocation: Optimal Power Flow (OPF)** Voltage V Current I Complex power = VI*=P + Q i • OPF: Given constant-power loads, find optimal P’s subject to: • Demand constraints • Constraints on V’s, P’s, and Q’s. Javad Lavaei, Columbia University 12

**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 13

**Optimal Power Flow** Cost Operation Flow Balance Javad Lavaei, Columbia University 14

**Project 1** 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 Javad Lavaei, Columbia University 15

**Project 2** Project 2:Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang) • Distribution networks are fine due to a sign definite property: • Transmission networks may need phase shifters: PS Javad Lavaei, Columbia University 16

**Project 3** 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 using ADMM. • It solves 10,000-bus OPF in 0.85 seconds on a single core machine. Javad Lavaei, Columbia University 17

**Project 4** Project 4:How to do optimization for mesh networks? (joint work with RamtinMadani and Somayeh Sojoudi) • Observed that equivalent formulations might be different after relaxation. • Upper bounded the rank based on the network topology. • Developed a penalization technique. • Verified its performance on IEEE systems with 7000 cost functions. Javad Lavaei, Columbia University 18

**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 19

**Low-Rank Solution** Javad Lavaei, Columbia University 20

**Penalized SDP Relaxation ** • Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix: • Modified 118-bus system with 3 local solutions (Bukhsh et al.) • IEEE systems with 7000 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 21

**Outline** • Convex relaxation for highly structured optimization • (Joint work with: Somayeh Sojoudi) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 22

**Distributed Control** • Computational challenges arising in the control of real-world systems: • Communication networks • Electrical power systems • Aerospace systems • Large-space flexible structures • Traffic systems • Wireless sensor networks • Various multi-agent systems Distributed control Decentralized control Javad Lavaei, Columbia University 23

**Optimal Decentralized Control Problem** • Optimal centralized control: Easy(LQR, LQG, etc.) • Optimal distributed control (ODC): NP-hard (Witsenhausen’s example) • Consider the time-varying system: • The goal is to design a structured controller to minimize Javad Lavaei, Columbia University 24

**Two Quadratic Formulations in Static Case** • Formulation in time domain: • Stack the free parameters of K in a vector h. • Define v as: • Formulation in Lypunov domain: • Consider the BMI constraint: • Define v as: Javad Lavaei, Columbia University 25

**Graph of ODC for Time-Domain Formulation** Javad Lavaei, Columbia University 26

**Recovery of Rank-3 Solution** • How to find such a low-rank solution? • Nuclear norm technique fails. • Add edges in the controller cloud to make it a tree: • Add a new vertex and connect it to all existing nodes. • Perform an optimization over the new graph. Javad Lavaei, Columbia University 27

**Numerical Example (Decentralized Case)** Penalized SDP Relaxation SDP Relaxation • Rank: 1 or 2 • Exactness in 59 trials • Rank: 1 • 99.8% optimality for 84 trials Javad Lavaei, Columbia University 28

**Distributed Case** • 100 random systems with standard deviation 3 for A and B, and 2 for X0. • Each communication link exists with probability 0.9. Javad Lavaei, Columbia University 29

**Outline** • Convex relaxation for highly structured optimization • (Joint work with: Somayeh Sojoudi) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi, and Ghazal Fazelnia) Javad Lavaei, Columbia University 29

**Polynomial Optimization** • Consider an arbitrary polynomial optimization: • Fact 1: • Fact 2: Javad Lavaei, Columbia University 31

**Polynomial Optimization** • Technique: distributed computation • This gives rise to a sparse QCQP with a sparse graph. • Question: Given a sparse LMI, find a low-rank solution in polynomial time? • We have a theory for relating the rank of W to the topology of its graph. Javad Lavaei, Columbia University 32

**Conclusions** • Convex relaxation for highly structured optimization: • Complexity of SDP relaxation can be related to properties of a generalized graph. • Optimization over power networks: • Optimization over power networks becomes mostly easy due to their structures. • Optimal decentralized control: • ODC is a highly sparse nonlinear optimization so its relaxation has a rank 1-3 solution. • General theory for polynomial optimization: • Every polynomial optimization has an SDP relaxation with a rank 1-3 solution. Javad Lavaei, Columbia University 33