By using the distributed consensus theory in multi-agentsystems, the strategy of economic power dispatch is studied in a smartgrid, where many generation units work cooperatively to achieve an optimal solution in a local area. The relationship between thedistributed optimization solution and consensus in multi-agentsystems is first revealed in this paper, which can serve as a general framework forfuture studies of this topic. First, without the constraints ofcapacity limitations, it is found that the total cost forall the generators in a smart grid can achieve the minimal value if theconsensus can be reached for the incremental cost of all the generation unitsand the balance between the supply and demand of powers iskept. Then, by designing a distributed consensus control protocol inmulti-agent systems with appropriateinitial conditions, incremental cost consensus can be realized and the balance for the powers can also besatisfied. Furthermore, the difficult problem for distributedoptimization of the total cost function with bounded capacitylimitations is also discussed. A reformulated barrier function isproposed to simplify the analysis, under which the total cost canreach the minimal value if consensus can be achieved for themodified incremental cost with some appropriate initial values.Thus, the distributed optimization problems for the costfunction of all generation units with and without bounded capacitylimitations can both be solved by using the idea of consensus inmulti-agent systems, whose theoretical analysis is still lackingnowadays. Finally, some simulation examples are given toverify the effectiveness of the results in this paper.
This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0800401), National Natural Science Foundation of China (Grant Nos. 61673107, 61673104, 61621003, 61532020), National Ten Thousand Talent Program for Young Top-notch Talents, Cheung Kong Scholars Programme of China for Young Scholars, Six Talent Peaks of Jiangsu Province of China (Grant No. 2014-DZXX-004), and Fundamental Research Funds for the Central Universities of China (Grant No. 2242016K41058).
[1] Ahn S J, Nam S R, Choi J H, et al. Power scheduling of distributed generators for economic and stable operation of a microgrid. IEEE Trans Smart Grid, 2013, 4: 398--405. Google Scholar
[2] Bakirtzis A, Petridis V, Kazarlis S. Genetic algorithm solution to the economic dispatch problem. Proc Gener Trans Distrib, 1994, 141: 377--382. Google Scholar
[3] Attaviriyanupap P, Kita H, Tanaka E, et al. A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function. IEEE Trans Power Syst, 2002, 17: 411--416. Google Scholar
[4] Gaing Z L. Particle swarm optimization to solving the economic dispatch considering the generator constraints. IEEE Trans Power Syst, 2003, 18: 1187--1195. Google Scholar
[5] Park J, Lee K, Shin J, et al. A particle swarm optimization for economic dispatch with nonsmooth cost functions. IEEE Trans Power Syst, 2005, 20: 34--42. Google Scholar
[6] Yu X, Cecati C, Dillon T, et al. The new frontier of smart grids: an industrial electronics perspective. IEEE Ind Electron Mag, 2011, 5: 49--63. Google Scholar
[7] Yu W W, Wen G H, Yu X H, et al. Bridging the gap between complex networks and smart grids. J Control Decis, 2014, 1: 102--114. Google Scholar
[8] Cao Y, Yu W, Ren W, et al. An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans Ind Inf, 2013, 9: 427--438. Google Scholar
[9] Ren W, Beard R W. Distributed Consensus in Multi-vehicle Cooperative Control. Berlin: Springer, 2008. Google Scholar
[10] Yu W, Wen G, Chen G, et al. Distributed Cooperative Control of Multi-agent Systems. Newark: Wiley, 2016. Google Scholar
[11] Cao M, Morse A S, Anderson B D O. Reaching a consensus in a dynamically changing environment: a graphical approach. SIAM J Control Optimiz, 2008, 47: 575--600. Google Scholar
[12] Chen Y, Lu J, Lin Z. Consensus of discrete-time multi-agent systems with transmission nonlinearity. Automatica, 2013, 49: 1768--1775. Google Scholar
[13] Saber R O, Murray R M. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Auto Control, 2004, 49: 1520--1533. Google Scholar
[14] Qin J H, Yu C B, Hirche S. Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology. IEEE Trans Ind Inf, 2012, 8: 986--994. Google Scholar
[15] Ren W, Beard R W. Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Auto Control, 2005, 50: 655--661. Google Scholar
[16] Yu W, Chen G, Cao M. Consensus in directed networks of agents with nonlinear dynamics. IEEE Trans Auto Control, 2011, 56: 1436--1441. Google Scholar
[17] Yu W, Chen G, Cao M, et al. Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics. IEEE Trans Syst Man Cybern, 2010, 40: 881--891. Google Scholar
[18] Yu W, Zhou L, Yu X, et al. Consensus in multi-agent systems with second-order dynamics and sampled data. IEEE Trans Ind Inf, 2013, 9: 2137--2146. Google Scholar
[19] Nedic A, Ozdaglar A. Distributed subgradient methods for multiagent optimization. IEEE Trans Auto Control, 2009, 54: 48--61. Google Scholar
[20] Yuan D, Ho D W C, Xu S. Regularized primal-dual subgradient method for distributed constrained optimization. IEEE Trans Cybern, 2016, 46: 2109--2118. Google Scholar
[21] Liu Q, Wang J. $L_1$-minimization algorithms for sparse signal reconstruction based on a projection neural network. IEEE Trans Neural Netw Learn Syst, 2016, 27: 698--707. Google Scholar
[22] Zhang Z, Ying X C, Chow M Y. Decentralizing the economic dispatch problem using a two-level incremental cost consensus algorithm in a smart grid environment. In: Proceedings of North American Power Symposium, Boston, 2011. 1--7. Google Scholar
[23] Zhang Z, Chow M Y. Convergence analysis of the incremental cost consensus algorithm under different communication network topologies in a smart grid. IEEE Trans Power Syst, 2012, 27: 1761--1768. Google Scholar
[24] Zhang Z, Chow M Y. The Influence of Time Delays on Decentralized Economic Dispatch by Using Incremental Cost Consensus Algorithm. Berlin: Springer, 2012. 313--326. Google Scholar
[25] Dominguez-Garcia A D, Cady S T, Hadjicostis C N. Decentralized optimal dispatch of distributed energy resources. In: Proceedings of IEEE 51st Annual Conference on Decision and Control, Maui Hawaii, 2012. 3688--3693. Google Scholar
[26] Kar S, Hug G. Distributed robust economic dispatch in power systems: a consensus+innovation approach. Power Energy Soc Gen Meet, 2012. Google Scholar
[27] Yang S, Tan S, Xu J. Consensus based approach for economic dispatch problem in a smart grid. IEEE Trans Power Syst, 2013, 28: 4416--4426. Google Scholar
[28] Cherukuri A, Cortes J. Distributed generator coordination for initialization and anytime optimization in economic dispatch. IEEE Trans Control Netw Syst, 2015, 2: 226--237. Google Scholar
[29] Horn R A, Johnson C R. Matrix Analysis. Cambridge: Cambridge University Press, 1985. Google Scholar
[30] Chen G, Wang X, Li X. Introduction to Complex Networks: Models, Structures and Dynamics. Beijing: High Education Press, 2012. Google Scholar
[31] Godsil C, Royle G. Algebraic Graph Theory. Berlin: Springer, 2001. Google Scholar
[32] Hale J, Lunel S V. Introduction to Functional Differential Equations. Berlin: Springer, 1993. Google Scholar
[33] Barabasi A L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: 509--512. Google Scholar
Figure 1
Illustration of networks with pinning control.
Figure 2
(Color online) (a) The incremental cost of all three generation units as well as (b) their corresponding output powers in Table
Figure 3
(Color online) (a) The incremental cost of all three generation units as well as (b) their corresponding output powers at power demands 650 MWh and 850 MWh.
Figure 4
(Color online) (a) The incremental cost of all generation units as well as (b) their corresponding output powers in a scale-free network structure.
Figure 5
(Color online) (a) The modified incremental cost of all three generation units with capacity limitations as well as (b) their corresponding output powers in Table
Figure 6
The diagram of IEEE 30-bus.
Figure 7
The communication topology of IEEE 30-bus.
Figure 8
(Color online) (a) The states for the modified incremental cost of all generation units with capacity limitations and (b) their corresponding output powers in Table
Unit | $\alpha_i$ | $\beta_i$ | $\gamma_i$ | $P_{Gi}(0)$ |
1 | 561 | 7.92 | 0.001562 | 300 |
2 | 310 | 7.85 | 0.00194 | 250 |
3 | 78 | 7.8 | 0.00482 | 100 |
Unit | $\alpha_i$ | $\beta_i$ | $\gamma_i$ | $P_{Gi}(0)$ |
$i$ | [100,~500] | [7.5,~8] | [0.001,~0.004] | [175,~225] |
Unit | $P_i^m$ | $P_i^M$ | $P_{Gi}(0)$ |
1 | 250 | 300 | 260 |
2 | 200 | 300 | 220 |
3 | 150 | 200 | 170 |
Unit (Generator No.) | $\alpha_i$ | $\beta_i$ | $\gamma_i$ | $P_{\min}$ | $P_{\max}$ |
1 (1) | 0 | 2.00 | 0.00375 | 50 | 200 |
2 (2) | 0 | 1.75 | 0.01750 | 20 | 80 |
3 (5) | 0 | 1.00 | 0.06250 | 15 | 50 |
4 (8) | 0 | 3.25 | 0.00834 | 10 | 35 |
5 (11) | 0 | 3.00 | 0.02500 | 10 | 30 |
6 (13) | 0 | 3.00 | 0.02500 | 12 | 40 |
Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有
京ICP备18024590号-1