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#### The Transactions of the Korean Institute of Electrical Engineers

##### ISO Journal TitleTrans. Korean. Inst. Elect. Eng.
• SCOPUS
• KCI Accredited Journal

Distributed optimization, economic dispatch (ED), consensus ADMM, Newton-Raphson method.

## 1. INTRODUCTION

The economic dispatch (ED) is one of the most important problems in power system operation, where generator outputs are decided by minimizing system generation cost subject to power balance constraints. Sun et al., Yang et al., and Chen et al. introduced interconnected systems for economic cooperation. Due to the expansion of the electricity market, the models and data within the region of the system became a commercial secret, which makes the data exchange between regions difficult if not impossible. From these reasons, system operators need to solve sub-problems with limited information(1-3).

These changes converted the traditional centralized system operation decentralized and distributed. To solve the ED problem, Boyd et al. suggested simple and powerful alternating direction method of multipliers (ADMM) suited for distributed convex optimization(4). Yang et al. applied it to the optimization of large-scale power system; a fully distributed and robust algorithm for alternating current optimal power flow (AC OPF) is proposed. The algorithm is based upon ADMM which is customized as a region-based optimization procedure(5).

Erseghe, Peng et al., and Ma et al. have applied ADMM to solve ED and OPF problems(6-8). In the distributed AC OPF, entire power system is separated into multiple areas and each area solves its own ED or OPF problem with the minimal information of the other areas. Especially consensus ADMM and proximal ADMM were suggested for the ED and AC OPF with second-order conic programming (SOCP) relaxation. Also, Yang et al. proposed a distributed consensus and a supply–demand balance approach based on quadratic cost functions to solve the ED under switching topologies and guaranteed the global feasibility of the algorithm(9). Chen et al. solved ED problem with general convex cost functions using an ADMM-based distributed algorithm(10). Moreover, the traditional centralized ADMM is extended to a distributed implementation problem by using the center-free algorithm and projection method.

This paper presents an approach for an ED problem based on consensus ADMM algorithm that does not require any form of central coordination. The solution of a local or regional optimization is exchanged only between neighboring areas. The convergence speed depends upon the number of sub-problems and decision variables, hence in order to improve the convergence speed and number of iterations, the optimal solution of a consensus area and its average are calculated using the decision variables at every iteration. As a result, the solution of the overall system can be found more efficiently based on sub-problems.

The proposed techniques have been applied to a 10-bus system to demonstrate their effectiveness and compared to the Newton-Raphson method.

## 2. PROBLEM FORMULATION

This section defines the formulation for the economic dispatch (ED) which is used by Saadat (1999), and modifies it so that each area containing the overlapping buses can be optimized through consensus ADMM which is used by Ma et al. (2016)(8).

### 2.1 Objective Function

The objective function for the conventional generating plants consists of quadratic cost functions as follows:

##### (1)
$\min . C_{G}=\Sigma_{i=1}^{n}\left(a_{i}+b_{i}P_{G,\:i}+c_{i}P_{G,\:i}^{2}\right)$

where

### References

1
A. X. Sun, D. T. Phan, S. Ghosh, July 2013, Fully decentralized AC optimal power flow algorithms, IEEE Power and Energy Society General Meeting, pp. 1-5
2
H. Yang, D. Yi, J. Zhao, Z. Dong, 2013, Distributed optimal dispatch of virtual power plant via limited communication, IEEE Transactions on Power Systems, Vol. 28, No. 3, pp. 3511-3512
3
G. Chen, C. Li, Z. Dong, March 2016, Parallel and distributed computation for dynamical economic dispatch, IEEE Transactions on Smart Grid, Vol. 8, No. 2, pp. 1026-1027
4
S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, January 2010, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, Vol. 3, No. 1, pp. 1-122
5
L. Yang, T. Zhang, G. Chen, Z. Zhang, J. Luo, S. Pan, may 2018, Fully distributed economic dispatching methods based on alternating direction multiplier method, Journal of Electrical Engineering Technology, Vol. 13, No. 5, pp. 1778-1790
6
T. Erseghe, may 2014, Distributed optimal power flow using admm, IEEE transactions on power system, Vol. 29, No. 5, pp. 2370-2380
7
Q. Peng, S. H. Low, 2014, Distributed algorithm for optimal power flow on a radial network, IEEE 53rd Annual Conference in Decision and Control, pp. 167-172
8
M. Ma, L. Fan, Z. Miao, 2016, Consensus admm and proximal admm for economic dispatch and ac opf with socp relaxation, 2016 IEEE North American Power Symposium, pp. 1-6
9
Z. Yang, J. Xiang, Y. Li, February 2016, Distributed consensus-based supply–demand balance algorithm for economic dispatch problem in a smart grid with switching graph, IEEE Transactions on Industry Electronics, Vol. 64, No. 2, pp. 1600-1610
10
G. Chen, Q. Yang, September 2017, An admm-based distributed algorithm for economic dispatch in islanded microgrids, IEEE Transactions on Industrial Informatics, Vol. 14, No. 9, pp. 3892-3903
11
R. S. Kar, Z. Miao, M. Zhang, L. Fan, 2017, ADMM for nonconvex AC optimal power flow, IEEE North American Power Symposium, pp. 1-6
12
M. Grant, S. Boyd, The CVX Users’ Guide Release 2.2
13
H. Saadat, Power system analysis, McGraw-Hill, chapter 7
14
MATLAB, R2020a, 2020, The MathWorks Inc., Natick, MA, USA

## 저자소개

##### 김규호(Kyu-Ho Kim)

Kyu-Ho Kim received the B.S., M.S. and Ph.D. degrees from Hanyang University, South Korea, in 1988, 1990 and 1996, respectively.

He is a Professor in the Department of Electrical Engineering at Hankyong National University, South Korea.

He was a Visiting Scholar at Baylor University for 2011-2012 and Unversity of Colorado Denver for 2020-2021.

His research interests include power system control and operation, optimal power flow and the development of control techniques for wind power plants.