Transcript (3)概率應用方法(probabilistic robust design)

概率優化設計之介紹及
提升風車可靠度的應用
吳爾融博士
概率優化設計之基本觀念
優化設計的日的
(1)滿足產品要求條件
(2)提高生產良率
(3)降低成本
優化設計方法
(1)傳統的線性程式法(linear programming)
(2)田口(Taguchi)或正交距陣(orthogonal array)方法
(3)概率應用方法(probabilistic robust design)
優劣性的比較
(1)傳統的線性程式法(linear programming)
優點: 容易處理簡單問題
缺點: 不能處理設計變數之不確定性(uncertainty)
不能計算可靠度(reliability)
(2)田口(Taguchi)方法
優點:屬實驗設計法,可減少trial & error之時間, 可處統計上
之偏差(deviation).
缺點:設計變數的選擇或實驗的採樣受正,交距陣所限制, 不
能有太多變數, reliability 計算亦有困難.
(3)概率應用方法(probabilistic robust design)
•優點:無上述兩方法之缺點, 可處理uncertainty,
可計算reliability, 可用於design for six siga.
因從計算慨率為出發點.可快速找到優化穩定的
最佳變數組合.
•缺點:數學運算複雜.不能有太多constraints
設計變數間的相聯限制使運算困難
設計決策的方式:
(1) 定數分析法(Deterministic Approach)
(2) 統計分析法(Statistic Approach)
(3) 概率優化法(Probabilistic Approach)
(1)定數分析法(Deterministic Approach)
建立物理模型(physical model) 及計算公式
以定數設計元數(constant design parameters)帶入計算公式
而滿足物理模型的條件. 對於實際不確定性(realistic uncertainty)
不凊楚, 而以安全係數(safety factor) 修正設計條件.
此法對成功與失敗之邊界模糊不清
(2)統計分析法(Statistic Approach)
沒有物理模型,對生產的樣品做抽樣檢查分析(sampling process)
以微化(minimize) 偏差(deviation or variance) 而得最佳平均值
(mean value) 以做為修正設計之標的.
(3)概率優化法(Probabilistic Approach)
建立物理模型如方法(1), 並應用統計方法以平均值(μ)及標準離差(σ)
建立設計變數的概率分佈函數(probability distribution function)
而帶入物理模型去運算並聯接以成功或失敗為基礎的優化計算平台
(calculation platform) 去求得最佳設計點且微化偏差(minimum
variation)
極限函數/目標函數的觀念
極限函數
G = R (resistance) – S (strain)
G>0 代表成功 G<0 代表失敗
即有成功概率(或然率) 及可靠度(reliability) 之產生
目標函數
G(X) = G (target) – G (physical model)
G(x) → 0 and gradient G(x) minimum
即偏差最小就是最佳最穩定的設計組合
物理模型(physical model):
(1) 應用軟体, 自有軟体或商業軟体,可在輸入 點把常數(constants)
換成概率分佈函數(PDF),然後直 接與優化軟体如UNIPASS聯合
(interface)
(2) 以擬合方式產生的公式, 當應用軟体不能以(1)方式與優化軟体聯
合時或只有實驗資科, 可將所得結果以統計回歸方法求物理模型的
公式, 再與優化軟体聯合. 此方式多一運算程序, 但免去受應用軟体輸入資
料之限制.
概率(或然率)的分佈函數
或然率的分佈函數,PDF, 是或然率應用計算的基本要素, 經過數值統計
與曲線擬合(curve fitting), 已有多種可供應用於實際問題的分佈函數
存在, 例如常見的正規分佈(normal distribution)及工程上常用的韋伯
分佈 (Weibull distribution). 現簡述PDF建立的基本原理.
任何事件, 出現一次以上, 即有出現頻率, 也就有或然率的存在. 以數值
為變數, 下列圖一為頻率記錄(frequency histogram), 圖二為經過面
積正規化(normalized), 總面積等於一, 成為相對頻率記錄(relative
frequency histogram), 連接各長方塊的平圴值則得圖三, 所形成的曲
線代表或然率密度的分佈, 其曲線下之面積等於一, 圖四展示所謂或然
率密度分佈函數, PDF. 然後, 可應用恰當擬合(goodness of fitting)
的方法使所得PDF逼近(approximate) 相似的可以數值公式代表的或
然率密度分佈函數 , 如此隨機變數的PDF即可以平均值(mean value)
與偏差(deviation) 為主軸參數的公式代之. 方程式(1)的運算就可進行.
圖一 出現頻率記錄 (Frequency Histogram)
圖二 出現相對頻率記錄 (Relative Frequency Histogram)
圖三 修正出現相對頻率記錄 (Relative Frequency Histogram)
圖四 或然率分佈函數 (PDF)
結論
了解上述或然率(概率)的應用原理, 再取得運算工具及協助
資料, 即可進行產品的優化設計, 以達到提升良率降低成目標.
事實上, 此優化設計方法因限制的絛件少,可應用於例如機械
電子,化工,化學,土木,生物科技及含有隨機變數的事件之預測,
下列一些作者所發表的應用例子:
• 汽車工業------車輪(金屬輪圈)的優化設計[1]
• 潤滑工程------特殊軸承的研發[2]
• 中草藥的研發-------有效成分之優化設計[3]
• 一般藥物的研發-----找出有效成分[4]
• 選舉選票分析---------成功可能性之預測[5]
風車系統簡介與優化設計之應用
風車是一部能量轉換機器
Kinetic Energy
→
風的動能
Wind
Mechanical
Energy
→
風車
→
Wind Turbine
Windmill
Electrical Energy
發電機
→
Generator
Dutch windmill
Dutch Windmill interior Structure
Modern Wind Turbine 450kw
Ocean Tidal Turbine
Modern Wind Turbine Sketch
Basic Mechanical Transmission and Power Generation
Wind
Turbine
Grid frequency
Europe 50hz
USA
60hz
→
Gearbox
Transmission
Generator has
to match grid
frequency,
required rpm
→ Generator
→ grid
Gearbox
converts
turbine
speed to
generator
speed
Fixed transmission: turbine rotor is controlled to feed
gearbox with a constant speed.
Variable transmission turbine rotor is free to provide its
variable speeds to the gearbox or add a frequency
inverter aft the generator to feed the grid network.
Basic Mechanical Transmission and Power Generation
Wind
Turbine
Grid frequency
Europe 50hz
USA
60hz
→
Gearbox
Transmission
Generator has
to match grid
frequency,
required rpm
→ Generator
→ grid
Gearbox
converts
turbine
speed to
generator
speed
Fixed transmission: turbine rotor is controlled to feed
gearbox with a constant speed.
Variable transmission turbine rotor is free to provide its
variable speeds to the gearbox or add a frequency
inverter aft the generator to feed the grid network.
Ideal and realistic wind turbine power extraction
Ideal turbine power extracted from free stream air
P= Cp Po
Cp= 0.593 momentum theory
P0 = ½ ρV03A
free stream power
Realistic turbine power considering friction and rotor characteristics
P=Cp ½ ρVw3A
Cp= λCQ
M= CQ ½ ρVw2AR λ=Vt / Vw
Rotor power coefficient vs. turbine tip speed
Whole wind turbine structure and generator
Gearbox changes low speed rotor to high speed generator
Gearbox close-up
SKF Roller Bearing for Wind Turbine
Roller Bearing Housing
Large Roller Bearing
Wind Turbine Loading Situation
Causes of Wind Turbine Failure
(1) Material Fatigue
(2) Vibration—Rotor-Bearing Dynamics
(3) Natural Disasters----Cyclone, Tornado, Tsunami and
Earthquake
(1) Fatigue: Tower, Blade, Hub and Nacelle
(2) Vibration: Rotor, Bearing and Gearbox “Failure”
(3) Disaster: Foundation and Control Mechanism
Possible Solutions:
(1) Design (good tool)
(2) Tests and Data Acquisition (field operation)
(3) Experiments (models)
(4) Research and Development (new tool & components)
Wind Turbine Damage
Wind Turbine Gearbox Damage
A Sample of Tilting-Pad Bearing
Tilting-Pad Bearing Sketch
Robust Design and Reliability
Three Methods:
(1) (Deterministic) Physical Modeling with Safety Factor:
Structure Dynamics, Computational Fluid Dynamics,
Rotor Bearing Dynamics, Structure-Fluid interaction
Fatigue Analysis, Test Data Acquisition etc.
After computation, add a safety factor for design.
(2) Statistic Method:
Effective sampling, Searching for Expectation (mean) Value
and Minimizing Variance. Find µ with minimum σ
(3) Probabilistic Method (Statistic + Physical Modeling):
Define a
Limit state function= Target Value – Results of Physical Model
In side the physical model, convert design parameters into
probability distribution functions (random numbers for uncertainty).
Analytic or Simulation Methods to satisfy limit-state function and
minimize variance. Design numbers are expectation numbers µ’s.
Example of Probabilistic Robust Design of Wind Blade
Wind Power equation:
P= Cpρ/2 Vw3 A
A= πR2
Let’s say to build a 1.5 MW turbine with maximum
power coefficient Cp=0.48 (3 blades) from the figure
Let Vw and R be random variables. We want to find
best values of Vw and R (expectation numbers) for
steady power output of 1.5 MW.
We assign normal distribution pdf’s for Vw and R with
mean values in the ranges:
Vw : 4 meter/sec-----12 meter/sec
R : 50 meters ------ 100 meters
UNIPASS calculation results:
Vw = 10.5 meter/sec
R = 37.4 meter
Let us look at GE 1.5 MW series Wind Turbines
Rotor power coefficient vs. turbine tip speed
GE 1.5MW series Wind Turbines
Operation and Maintenance
“Tracking”, “Tracking”, “Tracking”
“Location”, “Location”, “Location”
Maintenance Cost:
1.0-1.6 cents per kwhr.
Turbine Cost:
5 cents per kw.
Maintenance Difficulties:
(1) Component Accessibility
(2) Lubrication Challenge
(3) Taiwan Special Situation: Earthquake and Typhon
• The whole system is influenced by its uncertain environment.
• 整個系統受不確定的周遭環境所影響
• Engineering ingenuity and probabilistic approach may serve as
technical drivers for future developments.
• 對未來的發展, 工程創新能力及概率應用方法是主要的
推動力.
簡報完畢，敬請指教