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Matt Kraning, Eric Chu, Javad Lavaei and Stephen Boyd (2014), "Dynamic Network Energy Management via Proximal Message Passing", Foundations and Trends® in Optimization: Vol. 1: No. 2, pp 73-126. http://dx.doi.org/10.1561/2400000002

© 2014 M. Kraning, E. Chu, J. Lavaei, and S. Boyd

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**In this article:**

1. Introduction

2. Network Model

3. Device Examples

4. Convexity

5. Proximal Message Passing

6. Numerical Examples

7. Extensions

8. Conclusions

Ackowledgments

References

We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its own
dynamic constraints and objective, connected by AC and DC lines. The problem is to minimize the total network objective subject to
the device and line constraints, over a given time horizon. This is a large optimization problem, with variables for consumption
or generation for each device, power flow for each line, and voltage phase angles at AC buses, in each time period. In this paper
we develop a decentralized method for solving this problem called *proximal message passing*. The method is iterative: At each step,
each device exchanges simple messages with its neighbors in the network and then solves its own optimization problem, minimizing its
own objective function, augmented by a term determined by the messages it has received. We show that this message passing method
converges to a solution when the device objective and constraints are convex. The method is completely decentralized, and needs no
global coordination other than synchronizing iterations; the problems to be solved by each device can typically be solved extremely
efficiently and in parallel. The method is fast enough that even a serial implementation can solve substantial problems in
reasonable time frames. We report results for several numerical experiments, demonstrating the method’s speed and scaling,
including the solution of a problem instance with over 10 million variables in under 50 minutes for a serial implementation;
with decentralized computing, the solve time would be less than one second.

1. Introduction

2. Network Model

3. Device Examples

4. Convexity

5. Proximal Message Passing

6. Numerical Examples

7. Extensions

8. Conclusions

Ackowledgments

References

*Dynamic Network Energy Management via Proximal Message Passing* presents a fully decentralized method for
dynamic network energy management based on message passing between devices. It considers a network of devices, such as generators,
fixed loads, deferrable loads, and storage devices, each with its own dynamic constraints and objective, connected by AC and DC
lines. The problem is to minimize the total network objective subject to the device and line constraints, over a given time
horizon. This is a large optimization problem, with variables for consumption or generation for each device, power flow for each
line, and voltage phase angles at AC buses, in each time period. This text develops a decentralized method for solving this problem
called *proximal message passing*. The method is iterative: At each step, each device exchanges simple messages with its neighbors
in the network and then solves its own optimization problem, minimizing its own objective function, augmented by a term determined
by the messages it has received. It is shown that this message passing method converges to a solution when the device objective
and constraints are convex. The method is completely decentralized, and needs no global coordination other than synchronizing
iterations; the problems to be solved by each device can typically be solved extremely efficiently and in parallel. The method is
fast enough that even a serial implementation can solve substantial problems in reasonable time frames. Results for several
numerical experiments are reported, demonstrating the method’s speed and scaling, including the solution of a problem instance
with over ten million variables in under fifty minutes for a serial implementation; with decentralized computing, the solve time
would be less than one second.