依據材料特性參數來建構出多變量線性迴歸模型以預測鋰離子電池的電化學性能表現

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研究生：	鄭賀名 Ho-Ming Cheng
論文名稱：	依據材料特性參數來建構出多變量線性迴歸模型以預測鋰離子電池的電化學性能表現 Developing Multivariate Linear Regression Models to Predict the Electrochemical Performance of Lithium Ion Batteries Based on Material Property Parameters
指導教授：	朱瑾 Jinn P. Chu 王復民 Fu-Ming Wang
口試委員:	朱瑾 Jinn P. Chu 王復民 Fu-Ming Wang 蘇威年 Wei-Nien Su 劉偉仁 Wei-Ren Liu 蔡丕椿 Pi-Chuen Tsai
學位類別：	博士 Doctor
系所名稱：	工程學院 - 材料科學與工程系 Department of Materials Science and Engineering
論文出版年：	2018
畢業學年度：	106
語文別：	中文
論文頁數：	209
中文關鍵詞：	多變量線性迴歸模型、假設檢定、樣本數計算、約束最佳化、時間序列分析、主成分分析、因素分析、結構方程模型、偏最小平方法、灰色預測、熵值法、組合預測
外文關鍵詞：	multivariate linear regression model, hypothesis testing, sample size calculation, constrained optimization, time series analysis, principal component analysis, factor analysis, structural equation modelling, partial least squares, grey model, information entropy, combination forecasting
相關次數：	點閱：377 下載：6
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若能在鋰離子電池的活性材料組裝之前能對它的電化學表現作出預測，將能夠減少電池組裝的成本花費與省下充放電實驗所需耗費的時間。所以能夠建立一個基於統計學原理的數學模型來對鋰離子電池中活性材料的電化學性能作準確預測的話，會對電池材料的製作與優化有非常大的幫助。在這份研究中，我們量測了共11種由廠商製備的磷酸鐵鋰LiFePO4/C正極材料粉末的性質，並在其組裝為電池後測試它們的電化學性能。所得到的材料特性參數會透過多變量線性迴歸模型來與電化學表現的評分關聯起來。在第一個研究的部分，我們起先利用X光繞射技術、傅立葉轉換紅外線光譜儀、元素分析儀等量測技術去分別分析這11種磷酸鐵鋰材料的晶體結構、磷酸根離子PO43-官能基的振動情形、包碳含量等的材料特性參數。接下來我們將這11種LiFePO4粉末製作成電極並將其組裝為鈕扣型電池，然後我們將這些電池做了在不同充放電速率時的電容量測試以及做了速率在2C時的一千個充放電迴圈的循環壽命實驗。迴歸模型中的迴歸係數預估值是利用最小平方法去計算的，經過這樣的計算我們能夠把迴歸模型建構起來。我們期望推廣這一套統計學預測工具在材料科學中的應用以幫助未來的科研工作者最佳化他們的目標研究產物。
第二部分我們進一步依據材料特性參數來求算出能預測於循環伏安掃描量測中所分析出的極化電位的迴歸模型，並且運用小樣本數的「成對t值檢定」的方法來驗證在95 %的信心水準下，能接受預估值與實際值是一致的，接著再使用F分配檢定法來確認兩組數據在統計範圍內能有一樣的資料分散變異程度。第三部分是透過「樣本數計算」的分析來估算出若要透過假設檢定來驗證我們所推導出的迴歸模型的準確性，在滿足設定的信心水準與檢定力的要求條件下所需收集的最少樣本數是多少。第四個主題是以「約束最佳化」的分析方法來解答我們所推導出的迴歸模型能預期的在最佳情況下極化電位的極小值是多少，以及對應於此最好電池性能時的各項材料特性參數數值為何。第五個主題是運用「時間序列分析」的方法來預測正極材料循環壽命的未來趨勢，以幫助電池管理系統準確管控電池的健康狀況，另外還採用了灰色預測來對電容量的衰退建模，並以熵值法求算出權重係數以進行組合預測，以進一步提高預測的品質。第六項研究是套用「主成分分析」的方法，找出變數的合適線性組合以濾除對數據變異而言不重要的變數，幫助我們減少分析數據時所需考慮的預測變量數目並能最大程度地解釋資料的變異。第七項研究是以「因素分析」的研究方法，協助我們從變數中萃取出可能存在而未被觀察到的潛在因素，讓科研人員彙整出主要影響變異的原因並能對資料數據有更深一層的認識。在第八個主題中，「結構方程模型」的建構能夠直觀與圖像化地將預測變數、潛在變數與響應變數之間的統計學關聯給呈現出來，並讓我們能清晰地評估與比較各項變量在迴歸模型中的相對重要性。第九個主題透過「偏最小平方法」的數學技巧，進一步地解決了運用普通最小平方法所求得的迴歸模型有無法處理樣本數量少與其迴歸係數不宜直接相比較等問題的缺陷，同時此方法還有和主成分分析一樣的篩選出重要變數的功能，使得分析更加地全面與深入。在本論文的研究中我們施用了共九項的統計學方法來解析材料特性參數與電化學性能的迴歸預測模型、驗證迴歸模型的準確性、預估電化學表現的發展趨勢、找出迴歸模型所預測的最佳解答、剖析影響數據變異的主要成分等等的資料處理問題，因此我們期望本論文中所介紹的數學分析工具與方法能裨益於科研社群以在未來完成更優異的研究成果。

Predicting the electrochemical performance of active materials before their assembly in lithium ion batteries would be a path to cutting costs and time for assembling coin cells and running charging and discharging tests. Therefore, it is valuable to establish a statistical model to precisely predict the electrochemical performance of active materials in lithium ion batteries before cell assembly. In this study, we employed 11 different LiFePO4 powders prepared by manufacturers as the cathode active material and measured its properties, and then prepared cathode electrodes and ran electrochemical experiments. The acquired material property parameters and the electrochemical scores were correlated using multivariate linear regression models. We first used XRD, FTIR, and EA techniques to measure the crystal structure, vibration of PO43- functional group, and the carbon content, respectively. Next we made the cathode electrodes using these 11 LiFePO4 products and assembled them into coin cells, we then ran capacity tests at various current rates and cycleability tests at a 2 C current rate for 1,000 cycles. Estimates of the regression coefficients in the regression models were calculated by the least squares method, and thus the regression models were established. We expect to popularize this powerful material science statistical predictive strategy, to allow future researchers to predict performance of products in a cost-effective and timely manner.
In the second analysis of this study, a regression model for predicting polarization potential in CV measurements of LiFePO4/C cathodes was developed based on several material property parameters. In order to assess that whether the predicted values are about the same with the observed data with a 95 % level of confidence, a paired t-test was employed to compare the means of the 2 populations. Moreover, an F-test was applied to examine the ratio of the variance of the two datasets for confirming that the variables of the 2 populations have about the same expectation of the squared deviations from their means. Sample size calculation technique was adopted for evaluating the required minimum sample amount for achieving the predetermined power and significance level for our hypothesis tests in the third analysis. The 4th topic is to use the constrained optimization method to figure out the lowest anticipated overpotential in the CV tests and the corresponding material property parameters according to the fitted equation we established in part 3. In the 5th subject, cycle life tendencies of cathode materials were simulated based on the time series analysis. Time series analysis is beneficial for battery management system to monitor the battery health precisely, and thus would be helpful for improving the safety of LIB. Grey model was furthur employed to construct the prediction equation for assessing the degradation of the cell capacity during long-term cycling. Moreover, the idea based on information entropy also helps us develope the combination model for forecasting, and thus the precision of the resultant model can be improved even more. In the 6th work, principal component analysis was performed on the variables obtained from XRD measurements of all the samples. The original data would be transformed into uncorrelated principal components, and the unimportant vectors can thus be eliminated. Therefore, the remaining principal components can explain the variance of the original data as much as possible. The 7th topic is to perform the factor analysis on a few predictor variables we selected, as a result we can extract the unobserved latent variables which might exist. In the 8th analysis, we made use of the structure equation modelling to visually present the statistical correlations among the observed variables and the latent factors intuitively. This path analysis skill clearly manifested the relative importance of various variables in the regression model. The partial least squares regression method was performed on our experimental results in the 9th subject, consequently we are able to construct the fitted equation when the sample amount we collected is less than the variables we would like to investigate. While the variables were standardized so the regression coefficients can be compared directly, PLS regression model transforms the predictor variables into principal components as well so the variance of the data can be thoroughly accounted for as possible.
Within this study, we have utilized 9 statistical analyses to resolve the correlations among the measured experimental data. A number of topics researched in this work include: development of regression equations for forecasting electrochemical performances according to material properties, accuracy verifications of the regression models we established, trend anticipation of the cycle lives of the studied cathode samples, the best battery capability and the associated variable values indicated by the prediction function, and the solutions of the principal components which are able to express the data dispersion as much as possible, etc. We expect these mathematical tools can lead the scientific community to perfect their achievements in the future.

摘要........................................................................... I
Abstract .................................................................... III
誌 謝 ........................................................................ VI
目 錄 ........................................................................ XI
圖 目 錄 .................................................................... XVI
表 目 錄 ..................................................................... XX
符 號 索 引 ............................................................... XXIII
第一章 緒 論 ................................................................... 1
1 前言 ...................................................................... 1
2 鋰離子電池的實際應用 ........................................................ 6
3 鋰離子電池正極材料的發展現況 ................................................ 9
4 磷酸鐵鋰正極材料的介紹 ..................................................... 11
5 建構數學模型對鋰離子電池的電化學性能進行預測 ................................. 12
6 研究動機與目的 ............................................................ 12
第二章 文獻回顧 ............................................................... 16
1 磷酸鐵鋰正極材料的材料性質 ................................................. 16
2 多變量線性迴歸模型的介紹 ................................................... 18
3 電容量衰退趨勢的模型預測方法 ............................................... 19
第三章 實驗方法與儀器設備以及分析軟體 ........................................... 21
1 鑑定與分析正極材料特性的儀器與設備 .......................................... 21
1.1 同步輻射X光繞射分析實驗站 ................................................ 21
1.2 高解析度場發射掃描式電子顯微鏡 ........................................... 22
1.3 能量色散X射線能譜儀...................................................... 22
1.4 元素分析儀 .............................................................. 23
1.5 傅立葉轉換紅外線光譜儀 ................................................... 23
1.6 拉曼光譜儀 .............................................................. 24
1.7 振實密度儀 .............................................................. 24
1.8 固態核磁共振光譜儀 ...................................................... 24
1.9 比表面積分析儀 .......................................................... 25
1.10 超導量子干涉儀 ......................................................... 26
1.11 X射線光電子能譜儀 ...................................................... 26
1.12 同步輻射X光吸收能譜實驗站 ............................................... 27
2 電極的製備與電池的組裝 ..................................................... 27
2.1 將粉體材料製作成電極 ..................................................... 27
2.2 鈕扣電池的組裝........................................................... 28
3 鈕扣電池的電化學性能量測 ................................................... 28
3.1 鋰離子電池充放電測試儀 ................................................... 28
3.2 電化學恆電位測試儀 ...................................................... 29
4 分析數據用的工具軟體 ....................................................... 29
4.1 Microsoft® Excel 2013 .................................................. 29
4.2 OriginPro 8.5 .......................................................... 30
4.3 IBM® SPSS® Statistics 22 ............................................... 30
4.4 IBM® SPSS® Amos 24 ..................................................... 30
4.5 ZView® ................................................................. 31
4.6 FullProf Suite ......................................................... 31
4.7 G*Power 3.1 ............................................................ 31
4.8 Lingo 12 ............................................................... 31
4.9 ATHENA ................................................................. 32
第四章 統計學的理論與方法 ...................................................... 33
1 多變量線性迴歸模型 ........................................................ 33
1.1 運用最小平方法去求算迴歸係數 ............................................. 34
1.2 計算出95 %的信賴區間以剔除不重要的預測變數 ................................ 35
2 成對t值與兩族群變異數相等性的假設檢定 ........................................ 36
2.1 假設檢定的介紹 .......................................................... 37
2.2 兩群體平均值的成對t值檢驗法 .............................................. 37
2.3 成對樣本數據變異數的F檢定法 .............................................. 38
3 最小樣本數計算 ............................................................ 39
4 最佳化分析................................................................. 40
5 時間序列分析 .............................................................. 43
5.1 時間序列模型的介紹 ...................................................... 44
5.2 灰色預測模型 ............................................................ 47
5.3 運用熵值法來將多種預測模型通過加權係數加以組合 ............................. 51
6 主成分分析 ................................................................ 52
7 因素分析 ................................................................. 55
8 結構方程模型 .............................................................. 58
9 偏最小平方迴歸法 .......................................................... 60
第五章 實驗與分析的結果和討論 .................................................. 63
1 根據材料特性建立能預測電化學性能的迴歸模型 .................................. 63
1.1 運用XRD作材料結構的分析 .................................................. 67
1.2 運用FTIR技術解析材料的官能基振動特徵 ...................................... 69
1.3 運用元素分析儀量測材料中的碳含量 .......................................... 70
1.4 倍率性能與循環壽命等電化學性能的量測結果 .................................. 71
1.5 運用最小平方法求算出迴歸係數以建立預測方程式 ............................... 75
1.6 通過信賴區間的篩選以剔除可忽略的預測變量 .................................. 76
1.7 深入的研究與討論 ........................................................ 78
2 迴歸模型預測準確性的檢驗 ................................................... 80
2.1 假設檢定的概述與在鋰電池方面的應用方向 .................................... 80
2.2 材料結構方面的特性參數分析 ............................................... 81
2.3 運用EDX量測作材料中的元素計量比分析 ....................................... 81
2.4 材料樣品的振實密度量測 ................................................... 85
2.5 運用拉曼光譜儀分析包覆碳的石墨化程度 ...................................... 87
2.6 固態核磁共振光譜儀的實驗結果 ............................................. 92
2.7 運用超導量子干涉儀分析材料的磁學性質 ...................................... 93
2.8 電池的循環伏安法掃描結果與參數解析 ........................................ 95
2.9 建立能預測過電位的迴歸模型以及實際數值的驗證 ............................... 98
2.10 運用成對t值檢驗法進行兩組樣本群體的平均值比較 ............................. 99
2.11 以F檢定法比較兩組樣本數據的變異數大小 ................................... 100
3 最小樣本數計算 ........................................................... 101
3.1 設想樣品電容量的提升為範例來探索樣本數計算的應用 .......................... 101
3.2 常態分配檢定 ........................................................... 102
3.3 樣本數計算的結果與討論 .................................................. 104
4 利用線性規劃的求算軟體找出預測模型的最佳解 ................................. 105
4.1 目標函數的設定與變數範圍的語法編輯 ....................................... 106
4.2 LINGO軟體所解出的最佳化解答 ............................................. 107
5 運用時間序列分析法對正極材料的循環壽命進行預測 .............................. 108
5.1 運用ARIMA模型預測樣品A的電容量衰退趨勢 ................................... 109
5.2 運用指數平滑法預測樣品C的電容量衰退趨勢 .................................. 111
5.3 額外的補充討論 ......................................................... 112
5.4 運用灰色模型來分析數據以建立電極循環壽命的預測方程式 ...................... 113
5.5 運用熵值法來求算單項預測法的權重係數以建立組合預測模型 ..................... 113
6 對與X光繞射相關的材料變數進行主成分分析 .................................... 116
6.1 擷取與材料結構特性有關的重要的主成分 ..................................... 117
6.2 綜合主成分分析以及樣品的評分排序 ......................................... 119
6.3 建立評估電極過電位的主成分迴歸模型 ....................................... 120
7 綜合變數的因素分析 ....................................................... 122
7.1 因素負荷量的計算以及潛在變數的推論 ....................................... 122
8 結構方程模型的建立 ....................................................... 128
8.1 AMOS程式的操作與路徑係數的解讀 .......................................... 128
9 運用偏最小平方法迴歸來進行小樣本數的預測模型建立 ............................ 131
9.1 材料樣品的BET比表面積量測結果 ........................................... 131
9.2 運用偏最小平方法處理小樣本數的預測模型建立問題 ............................ 134
9.3 偏最小平方迴歸與相關係數的結果比較 ....................................... 136
10 材料的X光吸收光譜與電極的交流阻抗分析結果 ................................. 138
10.1 X光吸收能譜的數據 ..................................................... 138
10.2 交流阻抗頻譜的圖形與等效電路的模擬 ...................................... 140
第六章 結論與展望 ............................................................ 147
參 考 文 獻 ................................................................. 152
附 錄 ....................................................................... 165
FullProf程式碼 .............................................................. 165
口試委員的提問與回答 .......................................................... 169
朱 瑾 老師 ............................................................... 169
王復民 老師 ............................................................... 169
蘇威年 老師 ............................................................... 170
劉偉仁 老師 ............................................................... 172
蔡丕椿 老師 ............................................................... 176
個 人 簡 歷 ................................................................. 183
學 歷 ....................................................................... 183
獲得獎項 .................................................................... 183
著 作 目 錄 ................................................................. 184
                                

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