近年來，由於尖峰用電的需求不斷攀升，以及環保意識抬頭，再生能源
的比例逐漸提高，使得電力網路如何能夠更有智慧. 更有效率的運轉. 調度
之需求已迫在眉睫。自從1990 年代起，相量量測單元(phasor measurement
unit，PMU) 的問世，其高速的資料取樣速率以及透過GPS 同步，使得即
時監控電力網路成為可能。這些相量量測單元帶來的大數據啟發了許多廣
域量測. 保護與控制(wide area monitoring protection and control，WAMPAC)
等題目的探討; 而其中，電力網路的可觀性(observability) 更是實現智慧電
網的重要基石。本論文將探討如何以最符合經濟效益的可觀性配置，並
且同時滿足通訊上. 拓樸上等等的限制條件，如在電力領域當中被廣泛討
論的最佳化相量量測單元佈建問題(optimal PMU placement，OPP)，此一
NPcomplete
的組合最佳化問題。
在本論文中，將探討過去研究最佳化相量量測單元佈建問題的三種主流
演算法:(1) 以圖形理論文基礎的演算法，(2) 人工智慧演算法以及(3) 數學
規劃法。這三種演算法分別各有其優缺點與適合求解的最佳化相量量測單
元佈建問題模型，本論文除了介紹這三種演算法以外，更將結合圖形理論
文基礎的演算法與人工智慧演算法的優點，提出一混合型的二階段演算法，
利用圖形法求得的近似解作為人工智慧演算法的初始解，可大幅節省計算
時間與疊代次數；此外，本論文提出一理論分析以佐證此演算法確實可以
有效得到最佳解。在模型的開發上，過去絕大多數的模型中忽略了無限制
可觀性傳遞的問題，在這些可觀性相依的特性之下，無限制的可觀性傳遞，
將使得系統高度暴露於不可觀的風險之下; 因此，本論文提出了一個全新的
模型，於此模型當中，一樣是求解最符合經濟效益的可觀性配置問題，但同時加入了有限的可觀性傳遞的條件，實驗結果顯示可以有效的提升系統
的可觀性。在分散式能源是未來智慧電網的趨勢的潮流下，我們也探討如
何以樹分解(tree decomposition) 電力網路，並且應用動態規劃演算法於此樹
狀結構的樹分解來求解最佳化智慧電網可觀性配置問題。

With increasing global concerns regarding energy management, the concept of
the smart grid has become a particularly important interdisciplinary research topic.
In order to continually monitor a power utility system and efficiently observe all
of the states of electric nodes and branches on a smart grid, placing phasor measurement
units (PMUs) at selected nodes on the grid can monitor the operation
conditions of the entire power grid. The problem of monitoring a power grid with
minimum installation costs of PMUs can be transformed into the optimal PMU
placement (OPP) problem where the system is monitored under power observation
rules. The combinatorial optimization problem has been shown to be NPcomplete.
In this work, three algorithmic approaches: graphtheoretic
approaches,
metaheuristic
algorithms and mathematical programming (MP) are proposed and
discussed.
A new hybrid twophase
algorithm combining both merits of graphtheoretic
approaches and metaheuristic
algorithms is first proposed for the classical OPP
problem, in which the objective is to simultaneously minimize the number of PMUs
and ensure the complete observability of the whole power grid. The numerical
studies on various IEEE power test systems demonstrate the superior performance
of the proposed algorithm in regard to computational time and solution quality.
When the power grid encounters unexpected contingencies, conventional OPP formulations
with unlimited observability propagations may increase the risk of losing
the observability. In order to overcome this difficulty, a new observabilityenhanced
OPP formulation with considering bounded observability propagation constraints is proposed in by introducing the concept of observability propagation
depth (OPD). The OPD can characterize the depth of observability propagations applied
to observe a bus from its nearest PMU measurements. To further explore the
robustness of measurement systems under severe contingencies, a bilevel optimization
framework with a linearized probabilistic observability model is proposed. In
order to accommodate the distributed control architecture for the Distributed energy
resource (DER), another twophase
solution algorithm is also proposed by
exploring sparsity of largescale
power grids and observability rules derived from
power engineering practices. In the first phase, a tree decomposition techniques is
proposed to decompose the original graph topology of the power grid into a treelike
structure and a dynamic programming (DP) approach is applied to the tree in a
bottomup
manner in the second phase. One advantage of the proposed algorithm
is that it seems to be more flexible since some buses can be recommended to install
PMUs from topology characteristics of the power grid. These proposed algorithms
have shown their effectiveness from both the theoretical and practical perspectives.

Abstract ii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 The classic OPP problem . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Observabilityenhanced
OPP problems . . . . . . . . . . . . . . . . . 5
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Preliminary 11
2.1 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Topological Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Classic OPP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Direct PMU Measurements . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Observability Propagations . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Graphtheoretic
Approaches 16
3.1 Observation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The Power Domination Problem . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The Ideal OPP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Graphbased
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5.1 Case Studies of the Practical Model . . . . . . . . . . . . . . . . . . . 21
3.5.2 Case Studies of the Ideal Model . . . . . . . . . . . . . . . . . . . . . 21
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 MetaHeuristic
Algorithms 24
4.1 Artificial Bee Colony Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 The ABC Algorithm for the Classic OPP problem . . . . . . . . . . . . . . . . 25
4.2.1 Solution Representations . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.2 Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.3 Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Results of the Hybrid twophase
Algorithm . . . . . . . . . . . . . . . 28
4.3.2 Theoretical Analysis of Lower Bounds . . . . . . . . . . . . . . . . . 32
4.3.3 Results of the Singlephase
with ABC algorithm only . . . . . . . . . . 34
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Mathematical Programming 37
5.1 The Classic OPP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.1 Direct PMU measurements . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.2 PMU Channel Capacity Limits . . . . . . . . . . . . . . . . . . . . . . 37
5.1.3 Observability Propagations . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 ObservabilityEnhancing
OPP Problems . . . . . . . . . . . . . . . . . . . . . 40
5.3 The OPP Problem with limited Observability Propagations . . . . . . . . . . . 41
5.3.1 The Concept of Observability Propagation Depth . . . . . . . . . . . . 42
5.3.2 Observability Enhancement with Predefined
Bounded OPDs . . . . . 46
5.4 Bilevel OPP Problem under a Probabilistic Framework . . . . . . . . . . . . . 46
5.4.1 Probabilistic Formulation of the Contingency Constraints . . . . . . . . 46
5.4.2 Bilevel Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4.3 SingleLevel
Transformation . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5.1 Results of the OPDs under given PMU configurations . . . . . . . . . 53
5.5.2 Results of the OPDbased
OPP on IEEE Test Systems . . . . . . . . . 54
5.5.3 Results of the OPDbased
OPP with the Probabilistic Framework . . . 59
6 Observability Configurations in decentralized architecture 72
6.1 Reformulation by GraphTheoretic
Approaches . . . . . . . . . . . . . . . . . 73
6.2 Decomposition on a Power Network . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Tree Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.2 Nice Tree Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Dynamic Programming on a Nice Tree Decomposition . . . . . . . . . . . . . 80
6.3.1 Leaf Bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.2 Introduce Bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.3 Forget Bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.4 Join Bag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4.1 Results of the Proposed Decomposition Technique . . . . . . . . . . . 83
6.4.2 Results of OPP problems in the Proposed DP Approach . . . . . . . . 84
7 Conclusions 89
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