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研究生： 陳長煌
Chang-Huang Chen
論文名稱： 精英導向粒子群優法及其在經濟調度上之應用
Personal best oriented particle swarm optimizer and its application to economic dispatch problems
指導教授： 葉勝年
Sheng-Nian Yeh
黃仲欽
Jonq-Chin Hwang
口試委員: 潘晴財
Ching-Tsai Pan
劉志文
Chih-Wen Liu
陳建富
Jiann-Fuh Chen
謝冠群
Guan-Chyun Hsieh
蘇慶宗
Ching-Tzong Su
劉添華
Tian-Hua Liu
吳瑞南
Ruay-Nan Wu
學位類別： 博士
Doctor
系所名稱： 電資學院 - 電機工程系
Department of Electrical Engineering
論文出版年： 2008
畢業學年度： 96
語文別： 英文
論文頁數： 166
中文關鍵詞： 差分方程式 、經濟調度 、粒子群優法
外文關鍵詞： difference equation, economic dispatch, particle swarm optimization
相關次數： 點閱：290  下載：5 
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    • 粒子群優法(particle swarm optimization-PSO)是近年來發展出來的一種解最佳化問題的演算法，並已成功應用在許多工程與科學問題上。雖然它有許多的優點，但並不保證都能求得全域最佳解。許多的研究者不斷提出各種策略，試圖改善其性能，但都免不了要加入一些邏輯判斷與.數學運算步驟，這不僅增加演算法的複雜性，也增加執行時間。

    本論文提出二種新型粒子群優法，分別稱為精英粒子群優法(personal best oriented particle swarm optimizer-PPSO)與簡化精英粒子群優法(simplified personal best oriented particle swarm optimizer-SPPSO)。經驗顯示，前者對大部分測試的驗證函數(benchmark functions)，都能獲得比傳統粒子群優法更好的解，收斂速度也比較快。雖然後者用於求解單極點問題時，解的品質沒有PSO或PPSO好，但應用於求解多極點(multimodal)函數時，卻有極佳的表現。這兩種演算法都沒有加入額外的數學或邏輯運算，不會增加額外的計算負擔，因此具有與傳統粒子群優法一樣容易實現的優點。此外，SPPSO使用較少的參數，可以減輕參數調校的問題。

    為了解精英粒子群優法的內在特性及動態行為，本文提出以一階差分方程式(first-order difference equation)來描述精英粒子群優法的數學模型，並據以決定參數的設定。理論分析顯示精英粒子群優法與傳統粒子群優法都以環繞某一隨機加權中心的方式，搜尋最佳解，此一隨機加權中心係搜尋過程中所發現的全域最佳解(global best solution)與個別最佳解(personal best solution)或稱精英解的線性組合，但兩者的組合方式有明顯的不同。透過搜尋軌跡的觀察，可以證實此一現象。

    同時，為了進一步驗證本文所提出方法的可行性與能力，除使用高維度的驗證函數來測試精英粒子群優法外，並應用於電力系統的經濟調度問題(economic power dispatch problems)，結果顯示精英粒子群優法確實能獲得比傳統粒子群優法更低的調度成本。

    The particle swarm optimization (PSO) is a newly developed optimization problem solver, which has been successfully applied to various engineering optimization problems. Although it has been shown that it performs well on many optimization problems, it is occasionally trapped in local optimum. Many attempts have been tried to make it more robust by using new operations or hybridizing with other algorithms to avoid premature. However, they inevitably introduce extra mathematical and/or logical computations which complicate original simplicity of PSO and increase computation time.
    Two variants of particle swarm optimization, called personal best oriented particle swarm optimizer (PPSO) and simplified PPSO (SPPSO), are proposed in this dissertation. Empirical studies demonstrate that, in general, PPSO outperforms PSO both in computation quality and convergent speed. As for SPPSO, it performs extremely well for multimodal benchmark functions, although it is not as good as PSO and PPSO in unimodal functions. Moreover, without incurring any new computational and logical operations, PPSO and SPPSO are as simple as PSO, reserving the merit of easy implementation. Meanwhile, SPPSO use fewer parameters which will release some burden in choosing and tuning parameters.
    To reveal the intrinsic behavior of PPSO and SPPSO, mathematical models to characterize PPSO and SPPSO are developed in this dissertation. It is shown that a first-order difference equation is sufficient to describe the behavior of PPSO and SPPSO. Theoretical analysis depicted that a particle stochastically moves within a region in real space. On each dimension, the center of the region approximately equals to a random weighted mean of the best positions found by an individual and its neighbors. This phenomenon is observed and verified from the real trajectory profiles demonstrated in this dissertation.
    In order to verify the feasibility and capability of PPSO and SPPSO, in addition to testing on several high dimensional benchmark functions, they are also applied to economic power dispatch problems with various kinds of cost functions as well as different constraints. Experimental results demonstrate that, for most of the problems, PPSO and SPPSO indeed can obtain better solutions than PSO.

    Contents Abstract in English ------------------------------------I Abstract in Chinese ------------------------------------II Acknowledgement ----------------------------------------III Contents -----------------------------------------------V Notations ----------------------------------------------IV Lists of Figures ---------------------------------------XII Lists of Tables ----------------------------------------XV Chapter 1 Introduction ---------------------------------1 1.1 Motivation -----------------------------------------1 1.2 Objectives -----------------------------------------2 1.3 Methodology ----------------------------------------2 1.4 Contribution ---------------------------------------3 1.5 Organization of the dissertation -------------------4 Chapter 2 Background -----------------------------------5 2.1 Formulation of Optimization problems----------------5 2.2 Population based optimization algorithms------------6 2.2.1 Genetic algorithm---------------------------------7 2.2.2 Evolutionary programming--------------------------8 2.2.3 Evolution strategy--------------------------------9 2.2.4 Differential evolution ---------------------------10 2.3 Particle swarm optimization ------------------------11 2.4 Economic power dispatch problems -------------------15 2.5 Summary ------------------------------------------- 17 Chapter 3 Particle Swarm Optimization ------------------19 3.1 Original particle swarm optimization ---------------19 3.2 Inertia weight PSO ---------------------------------22 3.3 Constriction type PSO ----------------------------- 23 3.4 Analysis of PSO ----------------------------------- 24 3.4.1 Preliminary ------------------------------------- 24 3.4.1.1 Definition of convergence ----------------------24 3.4.1.2 Expectation value of uniform distributed random number -------------------------------------------------25 3.4.2 Analysis deterministic model of PSO --------------25 3.4.3 Convergent property of PSO ---------------------- 27 3.4.4 Trajectory of PSO ------------------------------- 33 3.5 Summary ------------------------------------------- 38 Chapter 4 Personal Best Oriented Particle Swarm Optimization (PPSO) ----------------------------------- 40 4.1 Constant inertia PPSO ----------------------------- 40 4.2 Simplified PPSO ----------------------------------- 42 4.3 Constriction type PPSO ---------------------------- 43 4.4 Linear decreasing inertia weight PPSO ------------- 43 4.5 Trajectory -----------------------------------------44 4.6 A sample PSO evolution experiment ----------------- 55 4.7 Summary ------------------------------------------- 62 Chapter 5 Theoretical Analysis of PPSO ---------------- 63 5.1 Simple PPSO: one particle and one dimension case ---63 5.2 Deterministic model of PPSO ------------------------64 5.2.1 Convergence property of the deterministic PPSO ---65 5.2.2 Step size ----------------------------------------67 5.2.3 Parameters selection -----------------------------70 5.2.4 Stability ----------------------------------------73 5.2.5 Equilibrium point---------------------------------73 5.3 Stochastic PPSO ------------------------------------75 5.3.1 Convergent property of the stochastic PPSO -------75 5.3.2 Step size ----------------------------------------77 5.3.3 Parameter selection for stochastic PPSO ----------78 5.3.3.1 Inertia weight ---------------------------------78 5.3.3.2 Acceleration coefficient -----------------------80 5.3.4 Equilibrium point --------------------------------81 5.4 The Simplified PPSO --------------------------------81 5.5 Testing on benchmark functions----------------------82 5.5.1 Benchmark functions ------------------------------82 5.5.2 Methodology --------------------------------------83 5.5.3 Performance comparison ---------------------------84 5.5.3.1 Constant inertia weight PPSO -------------------84 5.5.3.2 Constriction type PPSO -------------------------92 5.5.3.3 Linear decreasing inertia weight PPSO ----------94 5.6 Summary --------------------------------------------96 Chapter 6 Application of PPSO to Economic Dispatch Problems -----------------------------------------------97 6.1 Introduction ---------------------------------------97 6.2 Basic formulation of economic dispatch problems ----97 6.2.1 Cost function ------------------------------------98 6.2.1.1 Multiple-fuel options --------------------------98 6.2.1.2 Valve-point effect -----------------------------100 6.2.2 Transmission loss---------------------------------101 6.2.3 Ramp rate limits----------------------------------101 6.2.4 Prohibited zone-----------------------------------102 6.2.5 Other constraints --------------------------------102 6.3 Adapting PPSO and SPPSO to comply with ED problems -102 6.4 Case studies ---------------------------------------105 6.4.1 Case 1: 40 generator system-----------------------106 6.4.2 Case 2 : Ten generators with multiple fuel options107 6.4.3 Case 3 Thirteen units with valve-point effects ---112 6.4.4 Case 4 Ten units with multiple fuel options and valve-point effects ------------------------------------115 6.4.5 Case 5 Twenty generators accounting for transmission loss ---------------------------------------------------117 6.4.6 Case 6 Six generators with prohibited zone and ramp rate constraints ---------------------------------------120 6.4.7 Case 7 Fifteen generators with prohibited zone and ramp rate constraints ----------------------------------121 6.5 Summary --------------------------------------------124 Chapter 7 Conclusions ----------------------------------125 7.1 Summary of chapters --------------------------------125 7.2 Conclusions ----------------------------------------126 7.2 Future research ------------------------------------127 Reference --------------------------------------------- 129 Appendices ---------------------------------------------137 A Landscape view of benchmark functions ----------------137 A.1 Landscape view of Sphere function ------------------137 A.2 Landscape view of Rosenbrock function---------------137 A.3 Landscape view of Ackley function-------------------138 A.4 Landscape view of Griewank function-----------------138 A.5 Landscape view of Rastrigin function----------------139 A.6 Landscape view of Schwefel function ----------------139 B Generators data and transmission loss coefficients ---140 B.1 Unit data and cost coefficient for a 40-unit case of 40-unit system------------------------------------------140 B.2 Unit data and cost coefficients for multiple fuel options of a 10-unit case ------------------------------141 B.3 Unit data and cot coefficients of a 13-unit case ---142 B.4 Unit data of a 10-unit system with multiple-fuel and valve-point effects-------------------------------------143 B.5 Coefficients of transmission loss of a 20-unit case 144 B.6 Coefficients of transmission loss of a 6-unit case -144 B.7 Coefficients of transmission loss of a 15-unit case 145

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