国家重点研发计划(2016YFB0901900)
国家自然科学基金(61573303,61503324)
[1] Xia X, Elaiw A M. Optimal dynamic economic dispatch of generation: A review. Electric Power Syst Res, 2010, 80: 975-986 CrossRef Google Scholar
[2] 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 CrossRef ADS Google Scholar
[3] Yang S, Tan S, Xu J X. Consensus Based Approach for Economic Dispatch Problem in a Smart Grid. IEEE Trans Power Syst, 2013, 28: 4416-4426 CrossRef ADS Google Scholar
[4] Xie L, Gu Y, Zhu X. Short-Term Spatio-Temporal Wind Power Forecast in Robust Look-ahead Power System Dispatch. IEEE Trans Smart Grid, 2014, 5: 511-520 CrossRef Google Scholar
[5] Fan J Y, Zhang L. Real-time economic dispatch with line flow and emission constraints using quadratic programming. IEEE Trans Power Syst, 1998, 13: 320-325 CrossRef ADS Google Scholar
[6] Guo T, Henwood M I, van Ooijen M. An algorithm for combined heat and power economic dispatch. IEEE Trans Power Syst, 1996, 11: 1778-1784 CrossRef ADS Google Scholar
[7] Lin C E, Chen S T, Huang C L. A direct Newton-Raphson economic dispatch. IEEE Trans Power Syst, 1992, 7: 1149-1154 CrossRef ADS Google Scholar
[8] Bakirtzis A. Genetic algorithm solution to the economic dispatch problem. IEE Proc Gener Transm Distrib, 1994, 141: 377-382 CrossRef Google Scholar
[9] Gaing Z L. Particle swarm optimization to solving the economic dispatch considering the generator constraints. IEEE Trans Power Syst, 2003, 18: 1187-1195 CrossRef ADS Google Scholar
[10] D'Andrea R, Dullerud G E. Distributed control design for spatially interconnected systems. IEEE Trans Automat Contr, 2003, 48: 1478-1495 CrossRef Google Scholar
[11] Yu W, Wen G, Yu X. Bridging the gap between complex networks and smart grids. J Control Decision, 2014, 1: 102-114 CrossRef Google Scholar
[12] Binetti G, Davoudi A, Lewis F L. Distributed Consensus-Based Economic Dispatch With Transmission Losses. IEEE Trans Power Syst, 2014, 29: 1711-1720 CrossRef ADS Google Scholar
[13] Zhang Y, Rahbari-Asr N, Chow M Y. A robust distributed system incremental cost estimation algorithm for smart grid economic dispatch with communications information losses. J Network Comput Appl, 2016, 59: 315-324 CrossRef Google Scholar
[14] Guo F, Wen C, Mao J. Distributed Economic Dispatch for Smart Grids With Random Wind Power. IEEE Trans Smart Grid, 2016, 7: 1572-1583 CrossRef Google Scholar
[15] Guo X, Yang Y, Zhu T. ESI: A Novel Three-Phase Inverter With Leakage Current Attenuation for Transformerless PV Systems. IEEE Trans Ind Electron, 2018, 65: 2967-2974 CrossRef Google Scholar
[16] Guo X, Yang Y, Zhang X. Advanced Control of Grid-Connected Current Source Converter Under Unbalanced Grid Voltage Conditions. IEEE Trans Ind Electron, 2018, 65: 9225-9233 CrossRef Google Scholar
[17] Guo X Q, Yang Y, Wang X. Optimal Space Vector Modulation of Current Source Converter for DC-Link Current Ripple Reduction. IEEE Trans Ind Electron, 2018, : 1-1 CrossRef Google Scholar
[18] Guo X. A Novel CH5 Inverter for Single-Phase Transformerless Photovoltaic System Applications. IEEE Trans Circuits Syst II, 2017, 64: 1197-1201 CrossRef Google Scholar
[19] Xie J, Chen K X, Yue D, et al. Distributed economic dispatch based on consensus algorithm of multi agent system for power system. Electric Power Autom Eq, 2016, 36: 112--117. Google Scholar
[20] Binetti G, Davoudi A, Naso D. A Distributed Auction-Based Algorithm for the Nonconvex Economic Dispatch Problem. IEEE Trans Ind Inf, 2014, 10: 1124-1132 CrossRef Google Scholar
[21] Zhang W, Liu W, Wang X. Online Optimal Generation Control Based on Constrained Distributed Gradient Algorithm. IEEE Trans Power Syst, 2015, 30: 35-45 CrossRef ADS Google Scholar
[22] Li C J, Yu X H, Yu W W. Optimal economic dispatch by fast distributed gradient. In: Proceedings of International Conference on Control Automation Robotics and Vision, Singapore, 2014. 571--576. Google Scholar
[23] Xu T, Wu W, Sun H. Fully distributed multi-area dynamic economic dispatch method with second-order convergence for active distribution networks. IET Generation Transmission Distribution, 2017, 11: 3955-3965 CrossRef Google Scholar
[24] Chen G, Zhao Z. Delay Effects on Consensus-Based Distributed Economic Dispatch Algorithm in Microgrid. IEEE Trans Power Syst, 2018, 33: 602-612 CrossRef ADS Google Scholar
[25] Bai L, Ye M, Sun C. Distributed Economic Dispatch Control via Saddle Point Dynamics and Consensus Algorithms. IEEE Trans Contr Syst Technol, 2017, : 1-8 CrossRef Google Scholar
[26] Yang S, Li X L, Jia S J, et al. Distributed economic dispatch strategy for smart grids considering environmental factors. Electr Energ Manage Technol, 2017, 16: 57--65. Google Scholar
[27] Yi P, Hong Y, Liu F. Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems. Automatica, 2016, 74: 259-269 CrossRef Google Scholar
[28] Zhang X, Li N, Papachristodoulou A. Achieving real-time economic dispatch in power networks via a saddle point design approach. In: Proceedings of IEEE Power and Energy Society General Meeting, Denver, 2015. Google Scholar
[29] Wood A J, Wollenberg B F, Sheble G B. Power Gneration, Operation, and Control. New York: Wiley, 2012. Google Scholar
[30] Hug G, Kar S, Wu C. Consensus + Innovations Approach for Distributed Multiagent Coordination in a Microgrid. IEEE Trans Smart Grid, 2015, 6: 1893-1903 CrossRef Google Scholar
[31] Bertsekas D P. Nonlinear Programming. Nashua: Athena Scientific, 1995. Google Scholar
[32] Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004. Google Scholar
[33] Yang B, Johansson M. Distributed optimization and games: a tutorial overview. Netw Control Syst, 2011, 406: 109--148. Google Scholar
[34] Boyd S, Diaconis P, Xiao L. Fastest Mixing Markov Chain on a Graph. SIAM Rev, 2004, 46: 667-689 CrossRef ADS Google Scholar
[35] Hastings W K. Monte Carlo Sampling Methods using Markov Chains and their Applications. Biometrika, 1970, 57: 97-109 CrossRef ADS Google Scholar
[36] Johansson B, Keviczky T, Johansson M, et al. Subgradient methods and consensus algorithms for solving convex optimization problems. In: Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, 2008. 4185--4190. Google Scholar
[37] NediÄ A. Subgradient methods for convex minimization. Dissertation for Ph.D. Degree. Cambridge: Massachusetts Institute of Technology, 2002. Google Scholar
[38] Rudin W. Principles of Mathematical Analysis. 3rd ed. New York: McGraw-Hill, 1976. Google Scholar
Figure 1
(Color online) The structure of a power system consisting of multiple buses
Figure 2
(Color online) The equivalent topological structure
Figure 3
Simplified illustration of the IEEE 9-bus system
Figure 4
(Color online) Convergence of the generator output
Figure 5
(Color online) Convergence of the local information $\hat{\lambda}_{i}$
Figure 6
(Color online) Convergence of the power supply
Figure 7
(Color online) Convergence behavior of the local information with different times of consensus updates.protect łinebreak (a) $\varphi=10$; (b) $\varphi=20$; (c) $\varphi=30$; (d) $\varphi=40$
Figure 8
(Color online) Convergence behavior of the generator output with different times of consensus updates.protect łinebreak (a) $\varphi=10$; (b) $\varphi=20$; (c) $\varphi=30$; (d) $\varphi=40$
Figure 9
(Color online) Convergence behavior of the generator output with a diminishing step size. (a) $\alpha=0.02$;protect łinebreak (b) $\alpha=0.005$; (c) $\alpha=0.02\rightarrow0$
Figure 10
(Color online) Convergence of the power supply tested on the IEEE 9-bus and IEEE 39-bus system. (a) IEEE 9-bus; (b) IEEE 39-bus
$\text{基于局部变量}~\hat{\lambda}_{i}(k),~~\text{各条母线分别计算第}~k~\text{次迭代的发电机出力}$: $x_{i}(k)={\left[{C'_{i}}^{-1}(\hat{\lambda}_{i}(k))\right]}^{x_{i}^{\rm max}}_{x_{i}^{\rm min}},~i\in \mathbb{N};\tag{12}$ |
$\text{对于}~\hat{\lambda}_{i}(k),~~i\in~\mathbb{N},~~\text{采用梯度下降法进行计算}$:$u_{i}(k)=\hat{\lambda}_{i}(k)-\alpha(x_{i}(k)-q_{i}),~~i\in~\mathbb{N};\tag{13}$ |
$\text{相互连通的母线之间对各自的局部信息进行交换,~~对}~u(k)~\text{进行}~\varphi~\text{次的一致性更新}$:$v_{i}^{1}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}u_{j}(k),~i\in \mathbb{N},\tag{14}$ $v_{i}^{2}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{1}(k),~i\in \mathbb{N},\tag{15}$ $v_{i}^{\varphi}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{\varphi-1}(k),~i\in \mathbb{N};\tag{16}$ |
$\text{计算下一阶段的局部变量值}~\hat{\lambda}_{i}(k+1),~ i\in \mathbb{N}$: $\hat{\lambda}_{i}(k+1)=v_{i}^{\varphi}(k),~i\in \mathbb{N},\tag{17}$ $\text{令}~k=k+1\text{, 转1.}$ |
| G | $a_{i}$ | $b_{i}$ | $c_{i}$ | $x_{i}^{\rm~min}$ (MW) | $x_{i}^{\rm~max}$ (MW) |
| G1 | 0.001562 | 7.92 | 561 | 150 | 600 |
| G2 | 0.00194 | 7.85 | 310 | 100 | 400 |
| G3 | 0.00482 | 7.97 | 78 | 20 | 200 |
| $\alpha$ | $\widetilde{x}_{1}^{*}$ (MW) | $\mid\widetilde{x}_{1}^{*}-x_{1}^{*}\mid$ (MW) | $\widetilde{x}_{2}^{*}$ (MW) | $\mid\widetilde{x}_{2}^{*}-x_{2}^{*}\mid$ (MW) | $\widetilde{x}_{3}^{*}$ (MW) | $\mid\widetilde{x}_{3}^{*}-x_{3}^{*}\mid$ (MW) |
| 0.005 | 393.1844 | 0.0146 | 334.4777 | 0.1261 | 122.3379 | 0.1115 |
| 0.010 | 393.1989 | 0.0291 | 334.3519 | 0.2519 | 122.4492 | 0.2228 |
| 0.015 | 393.2133 | 0.0435 | 334.2263 | 0.3775 | 122.5604 | 0.3340 |
| 0.020 | 393.2276 | 0.0578 | 334.1008 | 0.5030 | 122.6715 | 0.4451 |
| 0.025 | 393.2418 | 0.0720 | 333.9756 | 0.6282 | 122.7825 | 0.5561 |
| 0.02$\rightarrow$0 | 393.1971 | 0.0273 | 334.6201 | 0.0163 | 122.2396 | 0.0132 |
| G | $a_{i}$ | $b_{i}$ | $c_{i}$ | $x_{i}^{\rm~min}$ (MW) | $x_{i}^{\rm~max}$ (MW) |
| G1 | 0.0046 | 7.065 | 135.88 | 135 | 500 |
| G2 | 0.00111 | 3.53 | 214.92 | 214 | 400 |
| G3 | 0.0029 | 7.58 | 78 | 108 | 400 |
| G4 | 0.0045 | 2.24 | 127.69 | 127 | 500 |
| G5 | 0.00104 | 8.53 | 232.56 | 100 | 600 |
| G6 | 0.0029 | 7.85 | 240 | 200 | 500 |
| G7 | 0.0021 | 3.375 | 44.628 | 44 | 300 |
| G8 | 0.0032 | 9.435 | 234.48 | 234 | 500 |
| G9 | 0.0047 | 6.45 | 74.6 | 74 | 400 |
| G10 | 0.0048 | 8.71 | 172 | 172 | 600 |
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