Decomposed Iterative Optimal Power Flow with Automatic Regionalization
Abstract
:1. Introduction
 (i)
 Optimality: the OPF is nonconvex in both its objective function and constraints, and hence finding the global optimum is not guaranteed.
 (ii)
 Scalability: OPF is an NPhard problem and the complexity usually grows exponentially with the size of the system. It is quite possible that a proposed solution works well for small systems, but fails to converge to a solution for large systems.
 (i)
 The decomposition is achieved manually in an ad hoc manner without any guidance. To date, there has been no reported effort to utilize the topology and property of the original system to conduct better regionalization such that the distributed OPF can achieve improved computational efficiency and optimality.
 (ii)
 Limited interactions among subsystems are involved during the optimization process. The obtained solutions are ad hoc combinations of the optimal solutions for local subsystems, which compromise the optimality for the whole system.
 (iii)
 Only one decomposition is studied, which means that the optimization is made for the decomposed local parts without a global consideration of the entire system.
 Automatic: an automatic regionalization technique is proposed to decompose the original large system into smaller subsystems, which will balance the computational loads of the resultant subsystems and achieve the maximum independence of the variables in different subsystems to accelerate the convergence speed in iterative optimization.
 General: at the first level, many existing OPF problems or general optimization tools can be applied. We first adopt the two most popular OPF tools, one based on the interiorpoint method (IPM) and the other based upon ADMM.
 Global: the proposed threelayer framework, especially the added third layer of iteration among different decompositions, will greatly increase the extensiveness of the dimensions the algorithm searches and make the optimization results global in view.
2. Problem Statement
2.1. Standard OPF Formulation
2.2. Decomposed OPF Formulation
3. Automatic Regionalization Based OPF (AROPF)
3.1. Overview of the Algorithm
3.2. Automatic Regionalization via Spectral Clustering
 (1)
 Topology Based Similarity (TBS): The simplest method is utilizing the topology of the system directly to define the similarity which results in creating a weighted adjacency matrix with 1 s representing a transmission line connects the buses and 0s for otherwise.
 (2)
 MeasurementBased Similarity (MBS): The states of a bus can only affect its neighbors during operation, which means that there is an available measurement that both buses are involved simultaneously. More specifically, the similarity considers whether there is at least one power flow measurement available on a transmission line, or one power injection measurement available at either one of the buses connected to the line.
 (3)
 Weighted MeasurementBased Similarity (WMBS): In fact, no two of the buses have the same amount of measurements involved, and not all measurements have the same amount of effect in coupling those involved buses. Under certain circumstances, it will be more favorable to cluster two strongly coupled buses into one region and divide two weakly coupled buses into different regions. Therefore WMBS is considering not only the availability of the measurements, but also the amount and impact of the measurements.
3.3. CrossSubsystem Variable Update
3.3.1. Subsystem Variable Update
Algorithm 1: Decomposed OPF. 
1 Initialization: Set $n=0$ and initialize ${\mathbf{y}}_{k}^{\left(n\right)}$ and ${\mathbf{v}}^{\left(n\right)}$ 
2 Private variable update: Set ${\mathbf{y}}_{k}={\mathbf{y}}_{k}^{\left(n\right)}$ and $\mathbf{v}={\mathbf{v}}^{\left(n\right)}$. Each bus $k\in \mathcal{N}$ updates ${\mathbf{x}}_{k}$ locally, where we let $({\mathbf{z}}_{k}^{(n+1)},{\mathbf{u}}_{k}^{(n+1)})$ be the primal and dual (possibly) optimal variables achieved for the following problem:
$$\begin{array}{cccc}\hfill \phantom{\rule{1.em}{0ex}}& \mathrm{min}\phantom{\rule{1.em}{0ex}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& {f}_{k}^{G}\left({p}_{k}^{G}\right)+{\mathbf{y}}_{k}^{T}({\mathbf{v}}_{k}{\overline{\mathbf{E}}}_{k}\mathbf{v})+\frac{\rho}{2}\left\right{\mathbf{v}}_{k}{\overline{\mathbf{E}}}_{k}\mathbf{v}{\left\right}_{2}^{2},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \mathrm{s}.\phantom{\rule{4.pt}{0ex}}\mathrm{t}.\phantom{\rule{1.em}{0ex}}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& {\mathbf{z}}_{k}=({P}_{k}^{G},{Q}_{k}^{G},{P}_{k},{Q}_{k},Re\left\{{i}_{k}\right\},Im\left\{{i}_{k}\right\},Re\left\{{\mathbf{v}}_{k}\right\},Im\left\{{\mathbf{v}}_{k}\right\},Re\left\{{\overline{\mathbf{i}}}_{k}\right\},Im\left\{{\overline{\mathbf{i}}}_{k}\right\},{\overline{\mathbf{P}}}_{k},{\overline{\mathbf{Q}}}_{k})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\alpha}_{k}\left({\mathbf{z}}_{k}\right)=\mathbf{0}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\lambda}_{k}\left({\mathbf{z}}_{k}\right)=\mathbf{0}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\mu}_{k}\left({\mathbf{z}}_{k}\right)=\mathbf{0}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\beta}_{k}\left({\mathbf{z}}_{k}\right)\le \mathbf{0}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\gamma}_{k}\left({\mathbf{z}}_{k}\right)\le \mathbf{0}\hfill & \hfill \phantom{\rule{1.em}{0ex}}& \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & {\left({\mathbf{v}}_{k}^{min}\right)}_{r}^{2}\le {\left(Re\left\{{\mathbf{v}}_{k}\right\}\right)}_{r}^{2}+{\left(Im\left\{{\mathbf{v}}_{k}\right\}\right)}_{r}^{2}\le {\left({\mathbf{v}}_{k}^{max}\right)}_{r}^{2},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\hfill & & r=1,\dots ,{\mathcal{N}}_{k},\hfill \end{array}$$

3 Net variable update: We denote ${\mathbf{v}}^{(n+1)}$ as the solution to the problem minimize
$$min\sum _{k\in \mathcal{N}}{\mathbf{y}}_{k}^{T}({\mathbf{v}}_{k}^{(n+1)}{\overline{\mathbf{E}}}_{k}\mathbf{v})+\frac{\rho}{2}\left\right{\mathbf{v}}_{k}^{(n+1)}{\overline{\mathbf{E}}}_{k}\mathbf{v}{\left\right}_{2}^{2}$$

4 Dual variable update: Each bus $k\in \mathcal{N}$ updates its dual variable ${\mathbf{y}}_{k}$ as
$${\mathbf{y}}_{k}^{(n+1)}={\mathbf{y}}_{k}^{\left(n\right)}+\rho ({\mathbf{v}}_{k}^{(n+1)}{\overline{\mathbf{E}}}_{k}{\mathbf{v}}^{(n+1)}).$$

5 Stopping criterion check: set $n:=n+1$. If the stopping criterion is not reached, go to step 2, otherwise stop and return $\left({\mathbf{z}}^{\left(n\right)},{\mathbf{v}}^{\left(n\right)},{\mathbf{u}}^{\left(n\right)},{\mathbf{y}}^{\left(n\right)}\right)=\left({\left({\mathbf{z}}_{k}^{\left(n\right)}\right)}_{k\in \mathcal{N}},{\mathbf{v}}^{\left(n\right)},{\left({\mathbf{u}}_{k}^{\left(n\right)}\right)}_{k\in \mathcal{N}},{\left({\mathbf{y}}_{k}^{\left(n\right)}\right)}_{k\in \mathcal{N}}\right)$. 
3.3.2. CrossSubsystem
4. Simulation and Results
4.1. Simulation Result Analysis
4.2. Time Efficiency and Scalability
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Regionalization  Region 1  Region 2  Region 3  Region 4 

Manual Case 1  1, 2, 5  3, 4, 7, 8, 9  10, 11, 14  6, 12, 13 
Manual Case 2  1, 2, 5  3, 4, 7, 8  9, 10, 14  6, 11, 12, 13 
TBSAR (A)  1, 2, 3, 4, 5  7, 8, 9  10, 11  6, 12, 13, 14 
MBSAR (B)  1, 2,3,5  4, 7, 8, 9  10, 11  6, 12, 13, 14 
WMBSAR (C)  1, 2, 3, 5  4, 7, 8, 9, 14  10, 11  6, 12, 13 
Results  Original  Strategy A  Strategy B  Strategy C 

Converge Time (s)  0.41  0.26  0.29  0.27 
Iteration numbers  12  54  60  56 
Objfcn Value ($/h)  8081.53  8081.53  8081.53  8081.53 
Total Gen P (MW)  268.29  268.29  268.29  268.29 
Total Gen Q (MVar)  67.63  67.63  67.63  67.63 
Branch Loss P (MW)  9.287  9.287  9.287  9.287 
Branch Loss Q (MVar)  39.16  39.16  39.16  39.16 
Results  Original  AROPF (A+B)  AROPF (A+C) 

Converge Time (s)  0.41  0.29  0.29 
Iteration numbers  12  40 (22+18)  36 (22+14) 
Objfcn Value ($/h)  8081.53  8081.53  8081.53 
Total Gen P (MW)  268.29  268.29  268.29 
Total Gen Q (MVar)  67.63  67.63  67.63 
Branch Loss P (MW)  9.287  9.287  9.287 
Branch Loss Q (MVar)  39.16  39.16  39.16 
Results  Original  AROPF (A+B)  AROPF (A+C) 

Converge Time (s)  0.54  0.35  0.35 
Iteration numbers  15  66 (33+33)  59 (33+26) 
Objfcn Value ($/h)  576.89  577.63  577.39 
Total Gen P (MW)  192.06  192.81  192.34 
Total Gen Q (MVar)  105.08  106.58  106.21 
Branch Loss P (MW)  2.860  2.772  2.825 
Branch Loss Q (MVar)  13.33  12.66  12.75 
Results  Original  AROPF (A+B)  AROPF (A+C) 

Converge Time (s)  0.83  0.48  0.46 
Iteration numbers  19  85 (49+36)  82 (49+33) 
Objfcn Value ($/h)  719,725.11  719,790.85  719,727.16 
Total Gen P (MW)  32,678.4  32,682.1  32,680.7 
Total Gen Q (MVar)  −9240.1 to 14,090.2  −9239.7 to 14,093.1  −9239.9 to 14,091.2 
Branch Loss P (MW)  302.78  303.16  302.88 
Branch Loss Q (MVar)  4599.97  4600.89  4600.11 
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Zheng, X.; Duan, D.; Yang, L.; Wang, H. Decomposed Iterative Optimal Power Flow with Automatic Regionalization. Energies 2020, 13, 4987. https://doi.org/10.3390/en13184987
Zheng X, Duan D, Yang L, Wang H. Decomposed Iterative Optimal Power Flow with Automatic Regionalization. Energies. 2020; 13(18):4987. https://doi.org/10.3390/en13184987
Chicago/Turabian StyleZheng, Xinhu, Dongliang Duan, Liuqing Yang, and Haonan Wang. 2020. "Decomposed Iterative Optimal Power Flow with Automatic Regionalization" Energies 13, no. 18: 4987. https://doi.org/10.3390/en13184987