SCIENTIA SINICA Informationis, Volume 48, Issue 10: 1364-1380(2018) https://doi.org/10.1360/N112018-00055

Distributed algorithm for economic dispatch based on gradient descent and consensus in power grid

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  • ReceivedMay 25, 2018
  • AcceptedJun 11, 2018
  • PublishedOct 9, 2018


This study aims to propose, a distributed optimization algorithm to solve the economic dispatch problem encountered in power grids with the ultimate objective of minimizing the total generation cost. The proposed approach is based on the gradient descent method and the consensus protocol. No central unit was required to broadcast the global information to each bus, and only local information was exchanged between the neighboring buses to balance power supply and demand. Theoretical analysis revealed that the proposed algorithm can converge to the optimal solution of the primal problem by selecting the appropriate step size and initial values. Simulation studies on the IEEE 9-bus system were conducted to show the validity of the proposed algorithm.

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[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


    Algorithm 1 基于梯度下降和一致性的分布式经济调度算法

    Require:令$k=0$, 设定初始的局部变量$\hat{\lambda}_{i}(0)$, $i\in~\mathbb{N}$; $~~\text{设定一致性计算次数}~\varphi;$



    $\text{基于局部变量}~\hat{\lambda}_{i}(k),~~\text{各条母线分别计算第}~k~\text{次迭代的发电机出力}$: $x_{i}(k)={\left[{C'_{i}}^{-1}(\hat{\lambda}_{i}(k))\right]}^{x_{i}^{\rm max}}_{x_{i}^{\rm min}},~i\in \mathbb{N};\tag{12}$


    $\text{相互连通的母线之间对各自的局部信息进行交换,~~对}~u(k)~\text{进行}~\varphi~\text{次的一致性更新}$:$v_{i}^{1}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}u_{j}(k),~i\in \mathbb{N},\tag{14}$ $v_{i}^{2}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{1}(k),~i\in \mathbb{N},\tag{15}$ $v_{i}^{\varphi}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{\varphi-1}(k),~i\in \mathbb{N};\tag{16}$

    $\text{计算下一阶段的局部变量值}~\hat{\lambda}_{i}(k+1),~ i\in \mathbb{N}$: $\hat{\lambda}_{i}(k+1)=v_{i}^{\varphi}(k),~i\in \mathbb{N},\tag{17}$ $\text{令}~k=k+1\text{, 转1.}$

  • 1   Table 1Parameters of the cost functions
    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
  • 2   Table 2Optimal generator output under different step sizes $\alpha$
    $\alpha$ $\widetilde{x}_{1}^{*}$ (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
  • 3   Table 3Parameters of the cost functions
    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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