国家重点研发计划(2016YFB0901900)
国家自然科学基金(61573303,61503324)
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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
$\text{基于局部变量}~\hat{\lambda}_{i}(k),~~\text{各条母线分别计算第}~k~\text{次迭代的发电机出力}$: $x_{i}(k)={\left[{C'_{i}}^{-1}(\hat{\lambda}_{i}(k))\right]}^{x_{i}^{\rm max}}_{x_{i}^{\rm min}},~i\in \mathbb{N};\tag{12}$ |
$\text{对于}~\hat{\lambda}_{i}(k),~~i\in~\mathbb{N},~~\text{采用梯度下降法进行计算}$:$u_{i}(k)=\hat{\lambda}_{i}(k)-\alpha(x_{i}(k)-q_{i}),~~i\in~\mathbb{N};\tag{13}$ |
$\text{相互连通的母线之间对各自的局部信息进行交换,~~对}~u(k)~\text{进行}~\varphi~\text{次的一致性更新}$:$v_{i}^{1}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}u_{j}(k),~i\in \mathbb{N},\tag{14}$ $v_{i}^{2}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{1}(k),~i\in \mathbb{N},\tag{15}$ $v_{i}^{\varphi}(k)=\sum\limits_{j\in\mathcal{N}_i}[W]_{ij}v_{j}^{\varphi-1}(k),~i\in \mathbb{N};\tag{16}$ |
$\text{计算下一阶段的局部变量值}~\hat{\lambda}_{i}(k+1),~ i\in \mathbb{N}$: $\hat{\lambda}_{i}(k+1)=v_{i}^{\varphi}(k),~i\in \mathbb{N},\tag{17}$ $\text{令}~k=k+1\text{, 转1.}$ |
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 |