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

• AcceptedJun 11, 2018
• PublishedOct 9, 2018
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Abstract

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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• 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$\hat{\lambda}_{i}(0)$, $i\in~\mathbb{N}$; $~~\text{设定一致性计算次数}~\varphi;$

$~~\text{设定步长}~\alpha.$

Output:

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

$\text{对于}~\hat{\lambda}_{i}(k),~~i\in~\mathbb{N},~~\text{采用梯度下降法进行计算}$:$u_{i}(k)=\hat{\lambda}_{i}(k)-\alpha(x_{i}(k)-q_{i}),~~i\in~\mathbb{N};\tag{13}$

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

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

• Table 1   Parameters 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
• Table 2   Optimal 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
• Table 3   Parameters 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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