本文以結合樣式搜尋最佳化演算法於機率負載潮流中，計算並規劃最佳化的電容器配置以降低系統損失，在使用兩點機率估計法的負載潮流於電容器的機率配置下，能將由於變動負載及太陽能發電的不穩定電力輸出，所造成的電力系統中不確定因素加入考量；藉由負載及太陽能發電機發電量的歷史資料及預測下，本方法能在大多數案例下可得到最佳化的目標；本文使用34 bus 電力系統進行案例測試，並以機率負載曲線所得的結果和一般尖峰負載的案例進行比較，同時使用符合系統限制的隨機案例進行強韌性測試，結果證實本文所提出的機率配置法可在電力系統有不確定的因素時得到更低的系統傳輸損失。

An implementation for probabilistic allocation of capacitors is introduced with the use of Two Point Estimate Probabilistic Load Flow, this technique allows to consider uncertainties that arise in the power system due to load variation and installed PV DG solar irradiance dependent output. This uncertainty is accounted for by combining Pattern Search optimization algorithm with Probabilistic Load Flow for the planning decision evaluation of where is best to allocate capacitor banks to minimize the system losses; this allows to obtain a solution that will result to be optimal in most of the cases that could probably occur depending on the system load and PV DG’s historical and expected behavior. The formulation is tested in a 34 bus power system with a defined probabilistic load profile and the solution obtained compared against those obtained based on peak load case studies. The robustness of each solution is tested with randomly generated cases that follow the expected system behavior. The probabilistic allocation method produces a solution that yields lower losses in the power system when uncertainty is present.

Table of Contents
Acnowledgements..................................................................................................i
Abstract...............................................................................................................ii
Chinese Abstract...........................................................................................................iii
Table of Contents.........................................................................................................iv-vi
List of Figures and Tables..........................................................................................vii-ix
Chapter 1 Introduction
1.1 Introduction..................................................................................................................1
1.2 Motivation...................................................................................................................3
1.3 Literature review..........................................................................................................3
1.4 Content Organization...................................................................................................5
Chapter 2 Basic Concepts on Power Systems and Micro Grid
2.1 The power system elements..........................................................................................6
2.2 Load Flow Analysis....................................................................................................10
2.3 Complex power & Reactive Power Compensation.....................................................11
2.4 The smart grid............................................................................................................15
2.5 Distributed generation................................................................................................17
Chapter 3 Power System Planning and Probabilistic Load Flow
3.1 Power system planning...............................................................................................20
3.2 Planning in the presence of uncertainties....................................................................22
3.3 The capacitor allocation Problem in the planning process..........................................23
3.4 The distributed generation allocation Problem in the planning process......................24
3.5 The Allocation with analytical and numerical optimization methods.........................25
3.6 Integrating uncertainty in the optimization of the allocation and planning problem...28
3.7 Uncertainty in statistics.............................................................................................28
3.8 Basic statistics concepts.............................................................................................29
3.9 The need for a probabilistic power flow explained.....................................................33
3.10 Monte Carlo probabilistic power flow......................................................................34
3.11 Two point estimate probabilistic power flow............................................................37
Chapter 4 Estimation of Uncertain Parameters
4.1 Uncertainty of Distributed Generation.......................................................................47
4.2 Solar irradiance uncertainty........................................................................................47
4.3 PV panels modeling....................................................................................................49
4.4 Solar PV penetration level..........................................................................................52
4.5 Modeling of solar PV in Load Flow study.................................................................52
4.6 Uncertainty of loads...................................................................................................53
4.7 Probabilistic forecast..................................................................................................54
Chapter 5 Probabilistic Planning Procedure and Optimization Method
5.1 Probabilistic planning: Definition..............................................................................55
5.2 Pattern search optimization as part of the probabilistic planning................................56
5.3 Probabilistic planning: Capacitor and PV distributed generation allocation...............62
5.4 Probabilistic Load profile: definition and formulation...............................................63
5.5 Probabilistic objective function with weighted average: definition............................66
5.6 Integrating two point estimate load flow in probabilistic planning.............................67
5.7 Optimization Procedure considering uncertainty.......................................................68
5.8 Optimization procedure Flow Chart..........................................................................69
Chapter 6 Capacitor and DG allocation Study Cases
6.1 34 bus-study system for allocation study: data and schematics...................................71
6.2 Optimization procedure: Cases..................................................................................74
6.3 Case #1: Capacitor allocation under load uncertainty.................................................76
6.4 Case #2: PV-DG allocation under load uncertainty....................................................82
6.5 Case #3: Capacitor allocation under PV-DG and load uncertainty.............................87
6.6 Results discussion......................................................................................................92
Chapter 7 Conclusion
7.1 Conclusion.................................................................................................................93
7.2 Further Work..............................................................................................................95
References...................................................................................................................96
Appendix.....................................................................................................................99


List of Figures and Tables
Figures
Figure 2.1 Basic Power System scheme..........................................................................10
Figure 2.2 Complex Power Triangle................................................................................13
Figure 2.3 Reactive Power Compensation........................................................................14
Figure 3.1 Skewness Concept Illustration........................................................................31
Figure 3.2 Normal probability distribution.......................................................................33
Figure 3.3 Flow chart: Monte Carlo simulation probabilistic load flow...........................36
Figure 3.4 Conceptual comparison: Deterministic load flow Vs. Prob. Load flow........43
Figure 4.1 Solar PV Model in Load Flow Study.............................................................53
Figure 5.1 Pattern Search Algorithm Flow Chart.............................................................61
Figure 5.2 Probabilistic allocation routine........................................................................69
Figure 5.3 Deterministic allocation routine.....................................................................70
Figure 6.1 34 Bus Distribution System.............................................................................72
Tables
Table 2.1 Today’s Grid vs. Smart Grid.............................................................................16
Table 2.2 Distributed Generation Classification..............................................................19
Table 3.1 Point Estimate Methods Comparison..............................................................37
Table 3.2 General Prob. Case: Bus voltages and angles....................................................44
Table 3.3 General Prob. Case: Line Flows and Losses.....................................................46

Table 4.1 Higüey’s Monthly Average Insolation.............................................................48
Table 4.2 Table 4.1 Higüey’s Monthly Average Irradiance..............................................49
Table 5.1 Classical Planning vs. Probabilistic Planning...................................................56
Table 5.2 Probabilistic Load Profile.................................................................................64
Table 5.3 Solar and Non-Solar Irradiance Period Case Data............................................66
Table 5.4 General Probabilistic Case Data.......................................................................66
Table 6.1 34 Bus System Line Data..................................................................................72
Table 6.2 Probabilistic Load Profile.................................................................................73
Table 6.3 Case #1: Peak Load Case data...........................................................................77
Table 6.4 Case #1: Capacitor Allocation Peak Load Case Results....................................78
Table 6.5 Case #1: General Probabilistic Case Load Data................................................79
Table 6.6 Case #1: Probabilistic Capacitor Allocation Results........................................80
Table 6.7 Case #1: Results Test........................................................................................81
Table 6.8 Case #1: Results Comparison...........................................................................82
Table 6.9 Case #2: Peak Load Case Data..........................................................................83
Table 6.10 Case #2: Peak Load Case for PV-DG Allocation Results...............................83
Table 6.11 Case #2: Solar Irradiance Period Case Data...................................................84
Table 6.12 Case #2: PV-DG Probabilistic Allocation Results..........................................85
Table 6.13 Case #2: Results Test......................................................................................86
Table 6.14 Case #2: Results Comparison.........................................................................86
Table 6.15 Case #3: 34 Bus System-Peak Load Case Data...............................................88
Table 6.16 Case #3: Peak Load Case Capacitor Allocation Results.................................88
Table 6.17 Case #3: Probabilistic Load Data....................................................................89
Table 6.18 Case #3: Capacitor Probabilistic Allocation W/DG Results...........................90
Table 6.19 Case #3: Results Test......................................................................................91
Table 6.20 Case #3: Results Comparison.........................................................................91

[1] Neagle & Samson, “Loss Reduction from Capacitors Installed on Primary Feeders”, in presentation at the AIEE North Eastern District Meeting, Rochester, N. Y, 1956.
[2]Y.G Bae, “Analytical Method of Capacitor Allocation on Distribution Primary Feeders” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, No. 4, 1978.
[3]J.J. Grainger, S.H. Lee, “Optimum Size and Location of Shunt Capacitors for Reduction of Losses on Distribution”, IEEE Transactions on Power Apparatus and Systems, Vol. Pas-100, No. 3, 1981.
[4]Suat Ertem & James R. Tudor, “Optimal Shunt Capacitor Allocation By Nonlinear Programming”, IEEE Transactions on Power Delivery, Vol. PWRD-2, No. 4, 1987.
[5] R. Srinivasa Rao and S. V. L. Narasimham, “Optimal Capacitor Placement in a Radial Distribution System using Plant Growth Simulation Algorithm”, International Science Index 21, 2008 waset.org/publications/11813
[6] Hong-Chan Chin, Whei-Min Lin, “Capacitor Placements for Distribution Systems with Fuzzy Algorithm”. TENCON '94. IEEE Region 10's Ninth Annual International Conference. Theme: Frontiers of Computer Technology. Proceedings of 1994
[7]Hadi saadat, “Power System Analysis”, Chapter 1 & Chapter 2, Mc Graw Hill, 2nd edition 2002.
[8] James Momoh, “SMART GRID Fundamentals of Design and Analysis”, John Wiley & Sons, 2012.
[9] Thomas Ackermann et al. “Distributed generation: a definition”, Electric Power Systems Research, Vol. 57, Issue 3, 20 April 2001, Pages 195–204.
[10] Hossein Seifi, Mohammad Sadegh Sepasian. “Electric Power System Planning: Issues”, Algorithms and Solutions, Springer, 2011.
[11] Mohammad Masoum, “Chapter 10: Optimal Placement and Sizing of Shunt Capacitor Banks in the Presence of Harmonics”, Power Quality in Power Systems and Electrical Machines, Elsevier, 2008.
[12] J. V. Schmill, “Optimum Size and Location of Shunt Capacitors on Distribution Feeders”, IEEE Trans. on Power Apparatus and Systems, vol. 84, pages 825-832, 1965.
[13] Chun-Lien Su, “Probabilistic Load-Flow Computation Using Point Estimate Method”, IEEE transaction on power systems, Vol. 20, No. 4, 2005.
[14] Lund Research Ltd. "Descriptive and Inferential Statistics". Statistics.laerd.com. Retrieved 23/01/2015.
[15] Dr. David C. Stone and Jon Ellis, “Statistics Course Notes” Chemistry, University of Toronto: chem.utoronto.ca, 2006.
[16] Susan Dean, Barbara Illowsky "Descriptive Statistics: Skewness and the Mean, Median, and Mode", Connexions website.
[17] William Feller, "An Introduction to Probability Theory and Its Applications", (Vol.1), Wiley, 3rd Ed, (1968).
[18] B. S. Everitt, “The Cambridge Dictionary of Statistics”, Cambridge University Press, Cambridge, 2006.
[19] Leonard J. Kazmier, “Schaum's Outline of Business Statistics”, McGraw Hill Professional, 2003.
[20] P. Jmgensen and J.O. Tande, “Probabilistic Load Flow Calculation using Monte Carlo Techniques for Distribution Network with Wind Turbines”, 8th International Conference On Harmonics and Quality of Power Proceedings, 1998.
[21] Juan M. Morales and Juan Perez Ruiz, “Point Estimate Schemes to Solve the Probabilistic Power Flow”, IEEE Transaction on Power Systems, Vol. 22, No. 4, 2007.
[22] Stattrek.com: Linear Transformations of Random Variables, Retrieved 20/01/2015.
[23] Statlect.com: Linear Combinations of Normal Random Variables, Retrieved 22/01/2015.
[24] Heiko Hahn et al, “Electric load forecasting methods: Tools for decision making”, European Journal of Operational Research, Volume 199, Issue 3 , 2009, Pages 902–907.
[25] Torczon, V.J. "On the convergence of pattern search algorithms". Society for Industrial and Applied Mathematics, Journal on Optimization, 1997.
[26]Matlab R2015-Global Optimization User´s Guide of Matlab: Pattern Search, pages 163-170, 2015.
[27] Yadolah Dodge,The Concise Encyclopedia of Statistics, Pooled Variance ,pp 427-428, 2008.
[28] Zhe Zhang et al. “Stochastic Evaluation of Voltage in Distribution Networks Considering the Characteristic of Distributed Generators”, 4th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), 2011.
[29] NASA, Surface meteorology and Solar Energy, at Site: eosweb.larc.nasa.gov/sse, retrieved: 25/02/2015.