TWI737562B - Container wall thickness estimation modeling method, system, computer program product, and computer-readable recording medium - Google Patents
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本發明係關於一種容器厚度估量技術,特別是關於一種基於機器學習技術建立模型以便用於估測高爐銅冷卻壁殘厚之估測容器壁厚的建模方法、系統、電腦程式產品及電腦可讀取紀錄媒體。The present invention relates to a vessel thickness estimation technology, in particular to a modeling method, system, computer program product and computer program for estimating vessel wall thickness for establishing a model based on machine learning technology for estimating the residual thickness of blast furnace copper stave Read the recording medium.
在一些需要高溫容器的應用場所(例如工廠)中,由於容器使用過程的溫度極高,導致難以即時評估容器特徵變化。In some application places (such as factories) that require high-temperature containers, it is difficult to immediately evaluate changes in container characteristics due to extremely high temperatures during the use of the containers.
以煉鋼高爐為例,習知銅冷卻壁殘厚監測方法是在定期維修時量測銅壁殘厚,但此方法並非在高溫線上作業時進行量測;又,高爐定期維修周期長,且不一定有充足時間來量測所有部位的銅壁厚度作為參考數據。Take the steelmaking blast furnace as an example. The conventional copper stave residual thickness monitoring method is to measure the residual thickness of the copper wall during regular maintenance, but this method is not to measure the residual thickness of the copper wall during high-temperature line operation; in addition, the regular maintenance cycle of the blast furnace is long, and There may not be enough time to measure the copper wall thickness of all parts as reference data.
以往雖有一些改進技術被提出,例如利用銅壁內外溫度估算銅壁厚度,但銅壁內部高溫難以取得,且估算過程過於複雜,仍需改善。Although some improvement techniques have been proposed in the past, such as using the inner and outer temperature of the copper wall to estimate the thickness of the copper wall, it is difficult to obtain the high temperature inside the copper wall, and the estimation process is too complicated and still needs to be improved.
有鑑於此,有必要提供一種有別以往的技術方案,以解決習知技術所存在的問題。In view of this, it is necessary to provide a technical solution different from the past to solve the problems existing in the conventional technology.
本發明之一目的在於提供一種估測容器壁厚的建模方法,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據。An object of the present invention is to provide a modeling method for estimating the wall thickness of a container, which builds a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of the wall thickness of the container.
本發明之次一目的在於提供一種估測容器壁厚的建模系統,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據。The second purpose of the present invention is to provide a modeling system for estimating the wall thickness of a container, which builds a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of the wall thickness of the container.
本發明之再一目的在於提供一種電腦程式產品,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據。Another object of the present invention is to provide a computer program product that builds a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of the container wall thickness.
本發明之又一目的在於提供一種內儲程式之電腦可讀取紀錄媒體,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據。Another object of the present invention is to provide a computer-readable recording medium with a program stored therein, and to establish a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of the container wall thickness.
為達上述之目的,本發明的一方面提供一種估測容器壁厚的建模方法,被配置由耦接一記憶體的一處理器執行被儲存在該記憶體中的指令,該建模方法包括步驟:讀取以離散時間方式取得自一容器壁部的數個外壁溫度及對應的數個厚度;在該數個外壁溫度與時間形成的一關係曲線中尋找溫度以大於一升溫閾值的至少一斜率陡升至數個峰值再下降所形成的數個波峰,依據該數個波峰相應的斜率進行累加運算以產生數個斜率累加值;及依據該數個厚度與由該數個外壁溫度所衍生的該峰值、該斜率及該斜率累加值產生一訓練集,依據該訓練集訓練一學習模型,直到該學習模型的訓練結果滿足一停止條件,依據該學習模型被訓練後的特徵產生一估厚模型。To achieve the above objective, one aspect of the present invention provides a modeling method for estimating the wall thickness of a container, which is configured to be executed by a processor coupled to a memory to execute instructions stored in the memory. The modeling method It includes the steps of: reading several outer wall temperatures and corresponding several thicknesses obtained from a container wall in discrete time; searching for at least one temperature greater than a temperature rise threshold in a relationship curve formed by the several outer wall temperatures and time A slope rises sharply to a few peaks and then drops to form a number of peaks, accumulate according to the corresponding slopes of the peaks to generate a number of slope cumulative values; and according to the number of thicknesses and the temperature of the outer wall The derived peak, the slope, and the cumulative value of the slope generate a training set, and a learning model is trained based on the training set until the training result of the learning model meets a stopping condition, and an estimate is generated based on the characteristics of the learning model after being trained. Thick model.
在本發明之一實施例中,估測容器壁厚的建模方法還包括步驟:基於一測試溫度依據該估厚模型產生一相應厚度。In an embodiment of the present invention, the modeling method for estimating the thickness of the container wall further includes the step of generating a corresponding thickness according to the estimated thickness model based on a test temperature.
在本發明之一實施例中,將該數個波峰相應的峰值排序並分別簡化成數個峰值等級中的一個,將該數個波峰相應的斜率排序並分別簡化成數個斜率等級中的一個,將該數個斜率累加值排序並分別簡化成數個累加等級中的一個,依據該數個厚度與由該數個外壁溫度所衍生的該數個峰值等級、該數個斜率等級及該數個累加等級產生該訓練集。In an embodiment of the present invention, the peaks corresponding to the several wave crests are sorted and simplified into one of several peak levels respectively, and the slopes corresponding to the several wave crests are sorted and simplified into one of several slope levels respectively, and The slope accumulation values are sorted and simplified into one of the accumulation levels, based on the thickness and the peak levels, the slope levels, and the accumulation levels derived from the outer wall temperatures Generate the training set.
在本發明之一實施例中,該數個峰值等級中的任一個為一25%峰值等級表徵、一50%峰值等級表徵、一75%峰值等級表徵或一最高峰值等級表徵;該數個斜率等級中的任一個為一25%斜率等級表徵、一50%斜率等級表徵、一75%斜率等級表徵或一最高斜率等級表徵;及該數個累加等級中的任一個為一25%累加等級表徵、一50%累加等級表徵、一75%累加等級表徵或一最高累加等級表徵。In an embodiment of the present invention, any one of the several peak levels is a 25% peak level characterization, a 50% peak level characterization, a 75% peak level characterization, or a highest peak level characterization; the several slopes Any one of the grades is a 25% slope grade characterization, a 50% slope grade characterization, a 75% slope grade characterization, or a highest slope grade characterization; and any one of the several cumulative grades is a 25% cumulative grade characterization , A 50% cumulative grade characterization, a 75% cumulative grade characterization, or a highest cumulative grade characterization.
在本發明之一實施例中,該升溫閾值為每秒溫度上升攝氏2度。In an embodiment of the present invention, the temperature rise threshold is 2 degrees Celsius per second.
在本發明之一實施例中,該學習模型為一迴歸模型。In an embodiment of the present invention, the learning model is a regression model.
在本發明之一實施例中,該容器壁部為一煉鋼高爐側壁或一鍋爐側壁。In an embodiment of the present invention, the container wall is a steel-making blast furnace side wall or a boiler side wall.
為達上述之目的,本發明的另一方面提供一種估測容器壁厚的建模系統,包括一處理器及一記憶體,該處理器耦接該記憶體,該記憶體儲存至少一指令,該處理器執行該指令,以執行如上所述之估測容器壁厚的建模方法。To achieve the above objective, another aspect of the present invention provides a modeling system for estimating the wall thickness of a container, which includes a processor and a memory, the processor is coupled to the memory, and the memory stores at least one instruction, The processor executes the instruction to execute the modeling method for estimating the wall thickness of the container as described above.
為達上述之目的,本發明的另一方面提供一種電腦程式產品,當電腦載入該電腦程式並執行後,該電腦能夠執行如上所述之估測容器壁厚的建模方法。To achieve the above objective, another aspect of the present invention provides a computer program product. After the computer program is loaded and executed, the computer can execute the above-mentioned modeling method for estimating the wall thickness of the container.
為達上述之目的,本發明的另一方面提供一種電腦可讀取紀錄媒體,該電腦可讀取紀錄媒體內儲程式,當電腦載入該程式並執行後,該電腦能夠完成如上所述之估測容器壁厚的建模方法。In order to achieve the above objective, another aspect of the present invention provides a computer-readable recording medium. The computer can read a program stored in the recording medium. After the computer loads and executes the program, the computer can complete the above-mentioned A modeling method for estimating the wall thickness of a container.
本發明的估測容器壁厚的建模方法、系統、電腦程式產品及電腦可讀取紀錄媒體,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據,可以有效簡化估測容器壁厚所需的參數,改善習知技術的計算過程複雜等問題,有利於即時監測容器壁厚(如高爐銅冷卻壁殘厚),便於相關人員作為容器應用、維修、設計或研究等參考。The modeling method, system, computer program product, and computer readable recording medium for estimating container wall thickness of the present invention establish a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of container wall thickness, which can be effective Simplify the parameters required for estimating the wall thickness of the vessel, and improve the complicated calculation process of the conventional technology, which is conducive to real-time monitoring of the vessel wall thickness (such as the residual thickness of the blast furnace copper stave), and is convenient for relevant personnel to use, repair, design or use the vessel. Research and other references.
為了讓本發明之上述及其他目的、特徵、優點能更明顯易懂,下文將特舉本發明較佳實施例,並配合所附圖式,作詳細說明如下。再者,本發明所提到的方向用語,例如上、下、頂、底、前、後、左、右、內、外、側面、周圍、中央、水平、橫向、垂直、縱向、軸向、徑向、最上層或最下層等,僅是參考附加圖式的方向。因此,使用的方向用語是用以說明及理解本發明,而非用以限制本發明。In order to make the above and other objectives, features, and advantages of the present invention more obvious and understandable, the preferred embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings. Furthermore, the directional terms mentioned in the present invention, such as up, down, top, bottom, front, back, left, right, inside, outside, side, surrounding, center, horizontal, horizontal, vertical, vertical, axial, The radial direction, the uppermost layer or the lowermost layer, etc., are only the direction of reference to the attached drawings. Therefore, the directional terms used are used to describe and understand the present invention, rather than to limit the present invention.
本發明的一方面提供一種估測容器壁厚的建模方法,該估測容器壁厚的建模方法可例如使用一溫度估測容器壁厚的建模系統來執行,用於建立模型以基於溫度估測容器壁厚,例如煉鋼高爐之銅冷卻壁殘厚,但不以此為限,也可適用於基於溫度估測其他容器壁厚,譬如用於加熱的鍋爐厚度等。以下舉例說明相關實施態樣,惟不以此為限。One aspect of the present invention provides a modeling method for estimating the wall thickness of a container. The modeling method for estimating the wall thickness of a container can be performed, for example, using a modeling system for estimating the wall thickness of the container by using a temperature to build a model based on Temperature estimation of vessel wall thickness, such as the residual thickness of the copper cooling wall of a steelmaking blast furnace, but not limited to this, it can also be applied to estimate the wall thickness of other vessels based on temperature, such as the thickness of a boiler used for heating. The following examples illustrate related implementation aspects, but are not limited to this.
舉例而言,請參閱第1圖所示,該溫度估測容器壁厚的建模系統例如可包含:一資料庫1、一訓練單元2及一測試單元3。該訓練單元2耦接該資料庫1與該測試單元3,該耦接方式可以是能夠用於傳遞資料載體(如電、光、磁或其組合等)的有線連接(如電性連接或網路連接等)或無線耦合(如光電耦合或電磁耦合等),使得被耦接的兩個物體間可以相互傳遞資料,以利進行在此所揭露的內容,其係所屬技術領域中具有通常知識者可以理解,不再贅述於此。For example, referring to FIG. 1, the modeling system for estimating the wall thickness of the container may include a database 1, a
舉例而言,如第1圖所示,該資料庫1可以是具有資料儲存功能的儲存載體,例如:可選自於記憶體、碟式儲存載體(如光碟或硬碟等)及資料伺服器所組成的群組,但不以此為限,也可以被配置成雲端儲存媒體(如Google Drive等),用以儲存一資料集,例如:該資料集中的資料可包含讀取一容器壁部(譬如煉鋼高爐側壁或鍋爐側壁等)以離散時間方式連續取得的數個外壁溫度(譬如溫度感測元件測量銅冷卻壁的外表溫度等)及對應的數個厚度(譬如超音波測厚數據等),作為模型訓練的依據。For example, as shown in Figure 1, the database 1 can be a storage carrier with a data storage function, for example: it can be selected from memory, disk storage carriers (such as optical disks or hard disks, etc.) and data servers The group, but not limited to this, can also be configured as a cloud storage medium (such as Google Drive, etc.) to store a data set, for example: the data in the data set can include reading a container wall (For example, steelmaking blast furnace side wall or boiler side wall, etc.)Several outer wall temperatures continuously obtained in a discrete time manner (such as temperature sensing elements to measure the surface temperature of copper staves, etc.) and corresponding thicknesses (such as ultrasonic thickness measurement data) Etc.) as the basis for model training.
請再參閱第1圖所示,該訓練單元2及該測試單元3可分別用於進行模型訓練及測試作業。舉例而言,該訓練單元2及該測試單元3可被配置成軟硬體協同運作裝置,例如:採用伺服器、處理器或特殊應用積體電路等。該訓練單元2可依據該訓練集中的資料訓練一學習模型,例如可預測一特定數值(如高爐銅冷卻壁殘厚等)的迴歸模型,用以產生一估厚模型,以便基於溫度估測容器壁厚;該測試單元3可依據該測試集中的資料測試該訓練後的估厚模型。Please refer to Figure 1 again. The
示例地,該溫度估測容器壁厚的建模方法可被配置為由一處理器耦接一記憶體來執行,該處理器執行被儲存在該記憶體中的指令,例如該記憶體可被配置成具備上述資料庫的功能,該處理器可被配置成具備上述訓練單元及測試單元的功能,用於執行該估測容器壁厚的建模方法,以下舉例說明本發明上述實施例的實施態樣,惟不以此為限。For example, the method for modeling the wall thickness of the temperature estimation container can be configured to be executed by a processor coupled to a memory, and the processor executes instructions stored in the memory. For example, the memory can be Configured to have the function of the aforementioned database, the processor can be configured to have the functions of the aforementioned training unit and test unit for executing the modeling method for estimating the wall thickness of the container. The following examples illustrate the implementation of the aforementioned embodiment of the present invention State, but not limited by this.
舉例而言,如第2圖所示,該估測容器壁厚的建模方法可包括步驟:一獲取步驟S1、一抽取步驟S2及一訓練步驟S3,以下僅以煉鋼高爐之銅冷卻壁殘厚為例進行說明,但不以此為限,也可適用於基於溫度估測其他容器壁厚,譬如加熱用的鍋爐厚度等。For example, as shown in Figure 2, the modeling method for estimating the wall thickness of a vessel may include steps: an acquisition step S1, an extraction step S2, and a training step S3. The following only uses the copper stave of a steelmaking blast furnace The residual thickness is described as an example, but it is not limited to this, and it can also be applied to estimate the wall thickness of other vessels based on temperature, such as the thickness of a boiler for heating.
該獲取步驟S1,可讀取以離散時間方式取得自一容器壁部的數個外壁溫度及對應的數個厚度;該抽取步驟S2,可在該數個外壁溫度與時間形成的一關係曲線中尋找溫度以大於一升溫閾值的至少一斜率陡升至數個峰值再下降所形成的數個波峰,依據該數個波峰相應的斜率進行累加運算以產生數個斜率累加值;該訓練步驟S3,可依據該數個厚度與由該數個外壁溫度所衍生的該峰值、該斜率及該斜率累加值產生一訓練集,依據該訓練集訓練一學習模型,直到該學習模型的訓練結果滿足一停止條件,依據該學習模型被訓練後的特徵產生一估厚模型。The obtaining step S1 can read several outer wall temperatures and corresponding thicknesses obtained from a container wall in a discrete time manner; the extracting step S2 can be in a relationship curve formed by the several outer wall temperatures and time Looking for a number of peaks formed by the temperature rising sharply to a number of peaks and then falling with at least a slope greater than a temperature rise threshold, and accumulating operations are performed according to the corresponding slopes of the peaks to generate a number of cumulative slope values; the training step S3, A training set can be generated according to the thickness and the peak value derived from the outer wall temperature, the slope, and the slope cumulative value, and a learning model is trained according to the training set until the training result of the learning model satisfies a stop Condition, based on the characteristics of the learning model after being trained, an estimation model is generated.
舉例來說,該資料集中的一部分可被當成產生該訓練集的依據,例如該處理器可獲取來自該容器壁部(如煉鋼高爐側壁或鍋爐側壁等)的以離散時間方式取得的數個外壁溫度及其對應的數個厚度,如多次對外壁進行量測所獲取的多組溫度與厚度,作為後續訓練模型之依據。For example, a part of the data set can be used as the basis for generating the training set. For example, the processor can obtain several discrete-time data from the vessel wall (such as the side wall of a steelmaking blast furnace or a boiler side wall). The temperature of the outer wall and its corresponding thicknesses, such as multiple sets of temperatures and thicknesses obtained by multiple measurements of the outer wall, are used as the basis for subsequent training models.
在一示例中,以該容器壁部為一煉鋼高爐側壁為例。首先,可被探討的是熱傳導效應。如第3圖所示,一高爐側壁可被簡化為包括一高爐容物H、一銅冷卻壁B及一冷卻水層W,該高爐容物譬如高溫達攝氏千度等級的鐵水等,該銅冷卻壁B位於該高爐容物H與該冷卻水層W之間。假設相關熱傳導參數諸如該高爐容物H的溫度為T h,該銅冷卻壁B的溫度及厚度分別為T b及d,該冷卻水層W的溫度為T w,可知熱傳導關係如下公式(1)及(2)所示: (1) (2) 其中,T h為該高爐容物H的溫度;T w為該冷卻水層W的溫度;q為單位熱傳量;d為該銅冷卻壁B的厚度;k為該銅冷卻壁B的熱傳導係數;T b為該銅冷卻壁B的溫度,h為該銅冷卻壁B的冷端熱對流係數。從而,藉由如上公式(1)及(2)可解出該銅冷卻壁B的厚度d。 In an example, the container wall is taken as the side wall of a steelmaking blast furnace. The first thing that can be explored is the effect of heat conduction. As shown in Figure 3, a blast furnace side wall can be simplified to include a blast furnace content H, a copper stave B, and a cooling water layer W. The blast furnace content is, for example, molten iron with a high temperature of 1,000 degrees Celsius. The copper stave B is located between the blast furnace content H and the cooling water layer W. Assuming that the relevant heat conduction parameters such as the temperature of the blast furnace content H is Th , the temperature and thickness of the copper stave B are T b and d, respectively, and the temperature of the cooling water layer W is T w , it can be seen that the heat conduction relationship is as follows (1 ) And (2) show: (1) (2) Among them, T h is the temperature of the blast furnace content H; T w is the temperature of the cooling water layer W; q is the unit heat transfer; d is the thickness of the copper stave B; k is the copper stave The thermal conductivity of B; T b is the temperature of the copper stave B, and h is the heat convection coefficient of the cold end of the copper stave B. Therefore, the thickness d of the copper stave B can be solved by the above formulas (1) and (2).
應被理解的是,在實際高爐應用過程中,上述公式所需的各項參數如欲完整取得極為困難。例如:該冷卻水層W的溫度T w僅能從入水口與出水口量測溫度,但無法對高爐中的諸多銅冷卻壁中的每一片銅冷卻壁量測溫度;另,該銅冷卻壁B的冷端熱對流係數h則有計算複雜,以及,尚須實驗進行驗證方可求得正確數據等問題;另,高爐於高達千度溫度下使用時,高爐內部如同一黑盒子,無法直接量測該高爐容物H的溫度T h。實際上,便於取得的數據僅有該銅冷卻壁B的溫度T b及可定期量測的超音波測厚數據,如該銅冷卻壁B的厚度d。因此,可行的相關熱傳導求解方式乃為利用該銅冷卻壁B的溫度T b求得該銅冷卻壁B的厚度d。 It should be understood that in the actual blast furnace application process, it is extremely difficult to obtain all the parameters required by the above formula completely. For example: the temperature T w of the cooling water layer W can only be measured from the water inlet and the water outlet, but the temperature cannot be measured for each of the many copper staves in the blast furnace; in addition, the copper stave B’s cold-end thermal convection coefficient h is complicated to calculate, and it needs to be verified by experiments to obtain the correct data. In addition, when the blast furnace is used at a temperature of up to 1,000 degrees, the inside of the blast furnace is like a black box and cannot be directly Measure the temperature Th of the content H of the blast furnace. In fact, the only data that can be easily obtained are the temperature T b of the copper stave B and the ultrasonic thickness measurement data that can be measured regularly, such as the thickness d of the copper stave B. Therefore, a feasible way to solve the related heat conduction is to use the temperature T b of the copper stave B to obtain the thickness d of the copper stave B.
示例地,由上式(1)及(2)可知,該銅冷卻壁B的熱傳導係數k及該銅冷卻壁B的冷端熱對流係數h可視為常數,其餘未知數尚有該冷卻水層W的溫度T w及該高爐容物H的溫度T h。其中,由於該冷卻水層W的溫度通常會被設定維持在攝氏30至40度(℃)之間,相較於該高爐容物H的溫度通常高達上千度,該冷卻水層W的溫度T w可假設被視為一常數;另,由於高爐中的鐵水是反映高爐內部溫度的主要材質,因此,高爐鐵水溫度可被類比成該高爐容物H的溫度T h。 For example, from the above equations (1) and (2), it can be seen that the thermal conductivity k of the copper stave B and the cold end heat convection coefficient h of the copper stave B can be regarded as constants, and the remaining unknowns still have the cooling water layer W The temperature T w of the blast furnace and the temperature T h of the contents of the blast furnace H. Wherein, since the temperature of the cooling water layer W is usually set to maintain between 30 and 40 degrees Celsius (°C), compared to the temperature of the blast furnace content H, which is usually as high as thousands of degrees, the temperature of the cooling water layer W T w can be assumed to be regarded as a constant; in addition, since the molten iron in the blast furnace is the main material reflecting the internal temperature of the blast furnace, the temperature of the molten iron in the blast furnace can be compared to the temperature T h of the content H of the blast furnace.
如第4圖所示,其係在一示例時段(如某一天的0時0分0秒到隔天的0時0分0秒)的高爐鐵水溫度(如單位為℃)的示意圖。由圖可知,高爐內部存在一區間變動的溫度變化,因此,該高爐容物H的溫度T h可被視為在一區間內變動的常數。為了減少其變動造成對模型訓練的影響,還可選擇在數據的收集過程中將其簡化為數個區間代表值,例如:以四分位法為例,可將諸多數據排序后分成四個等分級數,譬如25%等級、50%等級、75%等級及最高等級。藉此,可將變動範圍簡化成四種等級中的一種,以便降低數值變動造成的影響(如運算複雜度等)。因此,可將厚度簡化為d=(K×T b),其中K可被視為一組模型特徵(如數個參數等),可進一步藉由機器學習和深度學習方式,使用諸多超音波測厚數據及該銅冷卻壁的溫度數據,進行模型訓練與求解過程。 As shown in Figure 4, it is a schematic diagram of the temperature of molten iron in a blast furnace (for example, the unit is °C) during an exemplary time period (for example, 0:00:00 on a certain day to 0:00:00 on the next day). It can be seen from the figure that there is a range of temperature changes inside the blast furnace. Therefore, the temperature Th of the contents of the blast furnace H can be regarded as a constant that changes within a range. In order to reduce the impact on model training caused by its changes, you can also choose to simplify it into several interval representative values during the data collection process. For example, taking the quartile method as an example, many data can be sorted and divided into four equal parts. Levels, such as 25% level, 50% level, 75% level and the highest level. In this way, the variation range can be simplified to one of four levels, so as to reduce the impact caused by the numerical value variation (such as computational complexity, etc.). Therefore, the thickness can be simplified as d=(K×T b ), where K can be regarded as a set of model features (such as several parameters, etc.), and further machine learning and deep learning methods can be used to measure the thickness by ultrasound. The data and the temperature data of the copper stave are used for model training and solution process.
應被注意的是,在銅冷卻壁的溫度數據背後隱含的物理意義,例如:鐵水會磨損銅冷卻壁,導致銅冷卻壁的厚度變薄,更加重要的是,銅冷卻壁上會附著有渣皮,導致熱傳導係數改變,造成附著有渣皮的銅冷卻壁的溫度偏低。為了避免銅冷卻壁的溫度過於失真,還可觀察渣皮脫落前後的銅冷卻壁的溫度變化,作為訓練更能適合用於銅冷卻壁特性模型的依據。It should be noted that the hidden physical meaning behind the temperature data of the copper stave, for example: molten iron will wear the copper stave, causing the thickness of the copper stave to become thinner, and more importantly, there will be adhesions on the copper stave There is a slag skin, which causes the thermal conductivity to change, causing the temperature of the copper stave with the slag skin to be lower. In order to avoid excessive distortion of the temperature of the copper stave, the temperature change of the copper stave before and after the slag skin falls off can also be observed, which can be used as a basis for training to be more suitable for the characteristic model of the copper stave.
舉例而言,如第5圖所示,其係銅冷卻壁附著的渣皮脫落過程之爐壁溫度示意圖。其中,銅冷卻壁在渣皮脫落時,由於爐壁局部的銅冷卻壁突然少了渣皮卻仍承受內部高溫能量,導致爐壁溫度會在渣皮脫落後快速提高,譬如一溫度曲線C的溫度從最左方的一時間起點(標示為0)處快速陡升,約在第0.9分處,到達一局部高點(其被視為可信度較高的實際爐壁溫度),之後,溫度再逐步下降,直到第23.5分處,溫度呈現緩和。在此過程中,可觀察到在銅冷卻壁附著的渣皮脫落前後,由該銅冷卻壁測得的溫度曲線C會出現以大於一升溫閾值的一斜率陡升至一峰值再下降所形成的一波峰。在一高爐示例中,該升溫閾值可為每秒溫度上升攝氏2度,但不以此為限,該升溫閾值可依實際情況進行微調;此外,如果渣皮隨著時間重複性地附著又脫落,則可推知該溫度曲線C會出現以大於該升溫閾值的至少一斜率陡升至數個峰值再下降所形成的數個波峰。For example, as shown in Figure 5, it is a schematic diagram of the furnace wall temperature during the process of peeling off the slag skin attached to the copper cooling stave. Among them, when the slag skin falls off the copper stave, because the part of the copper stave on the furnace wall suddenly loses the slag skin but still bears the internal high temperature energy, the temperature of the furnace wall will increase rapidly after the slag skin falls off, such as a temperature curve C The temperature rises rapidly from the leftmost time starting point (marked as 0), and reaches a local high point (which is regarded as the actual furnace wall temperature with higher reliability) at about the 0.9th minute. After that, The temperature gradually decreased until the 23.5 minute point, the temperature appeared to ease. During this process, it can be observed that before and after the slag attached to the copper stave falls off, the temperature curve C measured by the copper stave will rise sharply to a peak and then drop at a slope greater than a temperature rise threshold. A crest. In an example of a blast furnace, the temperature rise threshold can be 2 degrees Celsius per second, but it is not limited to this. The temperature rise threshold can be fine-tuned according to the actual situation; in addition, if the slag crust repeatedly adheres and falls off over time , It can be inferred that the temperature curve C will have several peaks formed by steeply increasing to several peaks and then decreasing with at least one slope greater than the temperature rising threshold.
因此,為了訓練更加適用於煉鋼高爐的銅冷卻壁的特性模型,還可加入上述渣皮脫落過程觀測到的溫度曲線的波峰特徵,以便模型訓練結果更能貼近真實的銅冷卻壁特性。Therefore, in order to train the characteristic model of copper staves more suitable for steelmaking blast furnaces, the peak characteristics of the temperature curve observed during the slag peeling process can also be added, so that the model training results can be closer to the real copper stave characteristics.
舉例而言,由於該溫度曲線會出現以大於該升溫閾值的斜率陡升至該峰值再下降所形成的波峰,因此,為了精簡構成該波峰的數據量,在具有數個波峰的溫度曲線波形中,可取該溫度曲線的數個波峰相應的峰值及斜率,並依據該數個波峰相應的斜率產生數個斜率累加值,分別表示渣皮脫落過程的溫度特徵,例如該峰值與渣皮確實脫落與否有關,該斜率與渣皮脫落範圍大小有關,該斜率累加值與溫度變化趨勢有關。For example, since the temperature curve will have a peak formed by a steep rise to the peak and then drop with a slope greater than the temperature rise threshold, in order to simplify the amount of data that constitutes the peak, in the temperature curve waveform with several peaks , You can take the corresponding peak values and slopes of the several peaks of the temperature curve, and generate several cumulative slope values according to the corresponding slopes of the several peaks, which respectively represent the temperature characteristics of the slag peeling process. For example, the peak and the slag peeling are indeed No, the slope is related to the size of the slag peeling range, and the cumulative value of the slope is related to the temperature change trend.
可選地,在一實施例中,可將該數個波峰相應的峰值排序並分別簡化成數個峰值等級中的一個,將該數個波峰相應的斜率排序並分別簡化成數個斜率等級中的一個,將該數個斜率累加值排序並分別簡化成數個累加等級中的一個,依據該數個厚度與由該數個外壁溫度所衍生的該數個峰值等級、該數個斜率等級及該數個累加等級產生該訓練集。藉此,可有效適用於訓練模型貼近銅冷卻壁真實溫度特性,還可進一步精簡資料量,以便降低數值變動造成的影響(如運算複雜度等)。Optionally, in an embodiment, the peaks corresponding to the several wave crests can be sorted and simplified into one of several peak levels respectively, and the slopes corresponding to the several wave crests can be sorted and simplified into one of several slope levels respectively. , The several slope accumulation values are sorted and simplified into one of several accumulation levels respectively, based on the several thicknesses and the several peak levels derived from the several outer wall temperatures, the several slope levels and the several Accumulate the levels to generate the training set. In this way, it can be effectively applied to the training model close to the true temperature characteristics of the copper stave, and the amount of data can be further streamlined to reduce the impact of numerical changes (such as computational complexity, etc.).
舉例而言,為了進一步精簡資料量,還可將該溫度曲線的數個波峰衍生的數個峰值、數個斜率及數個斜率累加值簡化為數個區間代表值,例如:以四分位法為例,可將諸多數據排序后分成四個等分級數,譬如25%等級、50%等級、75%等級及最高等級,惟不以此為限。For example, in order to further simplify the amount of data, the number of peaks, the slopes, and the cumulative values of the slopes derived from the peaks of the temperature curve can be simplified into representative values of intervals, for example, the quartile method is For example, a lot of data can be sorted and divided into four equal grades, such as 25% grade, 50% grade, 75% grade and the highest grade, but it is not limited to this.
如第6圖所示,其係在一示例時段(如某一天)的溫度與溫差(如單位為℃)的示意圖。其中,上方表示一溫度曲線,下方表示一溫差曲線,在上方的溫度曲線中,還標示出25%等級、50%等級、75%等級及最高等級等四個峰值等級表徵,惟不以此為限。As shown in Fig. 6, it is a schematic diagram of temperature and temperature difference (for example, the unit is °C) during an example period (for example, a certain day). Among them, the upper part represents a temperature curve, and the lower part represents a temperature difference curve. In the upper temperature curve, there are also four peak level representations such as 25% level, 50% level, 75% level and the highest level. limit.
可選地,在一實施例中,該數個峰值等級中的任一個可被表示為一25%峰值等級表徵、一50%峰值等級表徵、一75%峰值等級表徵或一最高峰值等級表徵。藉此,可進一步精簡資料量,並可將變動範圍簡化成四種等級中的一種,以便進一步降低受鐵水溫度數值變動造成的影響(如運算複雜度等)。Optionally, in an embodiment, any one of the several peak levels may be represented as a 25% peak level characterization, a 50% peak level characterization, a 75% peak level characterization, or a highest peak level characterization. In this way, the amount of data can be further simplified, and the range of variation can be simplified to one of four levels, so as to further reduce the influence caused by the variation of the temperature of the molten iron (such as computational complexity).
可選地,在一實施例中,該數個斜率等級中的任一個可為一25%斜率等級表徵、一50%斜率等級表徵、一75%斜率等級表徵或一最高斜率等級表徵。藉此,可進一步精簡資料量,並可將變動範圍簡化成四種等級中的一種,以便進一步降低受鐵水溫度數值變動造成的影響(如運算複雜度等)。Optionally, in an embodiment, any one of the several slope levels may be a 25% slope level characterization, a 50% slope level characterization, a 75% slope level characterization, or a highest slope level characterization. In this way, the amount of data can be further simplified, and the range of variation can be simplified to one of four levels, so as to further reduce the influence caused by the variation of the temperature of the molten iron (such as computational complexity).
可選地,在一實施例中,該數個累加等級中的任一個可為一25%累加等級表徵、一50%累加等級表徵、一75%累加等級表徵或一最高累加等級表徵。藉此,可進一步精簡資料量,並可將變動範圍簡化成四種等級中的一種,以便進一步降低受鐵水溫度數值變動造成的影響(如運算複雜度等)。Optionally, in an embodiment, any one of the several accumulation levels may be a 25% accumulation level characterization, a 50% accumulation level characterization, a 75% accumulation level characterization, or a highest accumulation level characterization. In this way, the amount of data can be further simplified, and the range of variation can be simplified to one of four levels, so as to further reduce the influence caused by the variation of the temperature of the molten iron (such as computational complexity).
以上說明在模型訓練前的數據前處理的示例,後續,討論利用數據進行模型訓練的示例,其係作為示例性說明可用的實施過程,並非作為本發明的必要限制。The above describes the example of data pre-processing before model training, and then discusses the example of using data for model training, which is used as an example to illustrate the available implementation process and not as a necessary limitation of the present invention.
舉例而言,可以採用一迴歸模型作為該學習模型,以便預測該容器壁厚(譬如銅冷卻壁殘厚等),例如可採基於支持向量機(support vector machine,SVM)或梯度提振迴歸(Gradient Boosting regression,GBR)等迴歸模型。For example, a regression model can be used as the learning model to predict the wall thickness of the container (such as the residual thickness of the copper stave, etc.), for example, support vector machine (support vector machine, SVM) or gradient boost regression ( Gradient Boosting regression, GBR) and other regression models.
示例地,以SVM為例,其為一種監督式學習模型,可用統計風險最小化原則估計一個用於分類的超平面(hyperplane),使得一個超平面可將兩類不同數據完美區隔,譬如找出一直線,使得該直線與該兩類數據的距離最大;但不以此為限,基於SVM的變形,可為找到一直線,該直線可包括所有兩類數據,計算出此直線的公式,及可計算不同x對應的y。例如:假設資料被表示為 (x 1, y 1), . . . ,(x i, y i) ∈ R d×R,其中,x 1…i表示輸入的製程參數;y 1…i表示所對應的迴歸值,譬如銅冷卻壁的鋅層厚度,R為實數集合,R d為實數向量集合。令f(x) = w · x + b,w ∈ R d,b ∈ R,如果一停止條件為f(x)≈y,則滿足該停止條件,可知f(x)能從x準確地預測y, 這個 w 即是 SVR 所要找的超平面。因此,可知模型訓練過程可由收集來的銅冷卻壁溫度和相對應的銅冷卻壁厚度,加上一定的誤差範圍找出w,即可訓練出銅冷卻壁溫度和銅冷卻壁厚度相對應的銅冷卻壁厚度預測模型,作為該估厚模型,其運算過程係所屬技術領域中具有通常知識者可以理解,不再贅述。 For example, taking SVM as an example, it is a supervised learning model. The principle of statistical risk minimization can be used to estimate a hyperplane for classification, so that a hyperplane can perfectly separate two types of different data, such as finding Draw a straight line to maximize the distance between the straight line and the two types of data; but not limited to this. Based on the deformation of SVM, it can be used to find a straight line. The straight line can include all two types of data. The formula for this straight line can be calculated, and Calculate y corresponding to different x. For example: Suppose the data is expressed as (x 1 , y 1 ),..., (X i , y i ) ∈ R d ×R, where x 1...i represents the input process parameters; y 1...i represents the The corresponding regression value, such as the thickness of the zinc layer of the copper stave, R is the set of real numbers, and Rd is the set of real vectors. Let f(x) = w · x + b, w ∈ R d , b ∈ R, if a stopping condition is f(x)≈y, then the stopping condition is satisfied, we know that f(x) can be accurately predicted from x y, this w is the hyperplane that SVR is looking for. Therefore, it can be seen that the model training process can be based on the collected copper stave temperature and the corresponding copper stave thickness, plus a certain error range to find out w, then the copper stave temperature and the copper stave thickness corresponding to the copper cooling can be trained The wall thickness prediction model, as the thickness estimation model, the calculation process of which can be understood by those with ordinary knowledge in the technical field, and will not be repeated.
示例地,以GBR為例,可透過許多的弱分類器分類,最後將所有弱分類器分類結果集合成答案,其中,弱分類器是一種簡單分類器,可先以不同條件區分不同類別作為輸出,最後集合所有弱分類器的輸出,投票選出最佳輸出結果。例如:假設資料為(X 1, Y 1), . . . ,(X n, Y n) ∈ R d×R,X 1…n表示輸入的製程參數;Y 1…n表示所對應的迴歸值,R為實數集合,R d為實數向量集合。先以一簡單的公式(可為線性或多項式等公式)計算出Y pred1,目標是Y pred1- Y n= 0,若不為0,則繼續將Y pred1- Y n帶入第二個弱分類器計算求得Y pred2,直到滿足一停止條件為Y n-∑Y pred1…n≈0,將Y pred1…n總和運算即訓練完成所需之模型。因此,可知將預測模型加上參數輸入形式的公式,即可建立銅冷卻壁厚度預測模型,作為該估厚模型,其運算過程係所屬技術領域中具有通常知識者可以理解,不再贅述。 For example, taking GBR as an example, it can be classified by many weak classifiers, and finally the classification results of all weak classifiers are aggregated into answers. Among them, the weak classifier is a simple classifier that can first distinguish different categories with different conditions as the output , And finally gather the output of all weak classifiers, and vote for the best output result. For example: assuming the data is (X 1 , Y 1 ),..., (X n , Y n ) ∈ R d ×R, X 1...n represents the input process parameter; Y 1...n represents the corresponding regression value , R is the set of real numbers, and R d is the set of real number vectors. First calculate Y pred1 with a simple formula (which can be a linear or polynomial formula), the goal is Y pred1 -Y n = 0, if it is not 0, continue to bring Y pred1 -Y n into the second weak category The device calculates and obtains Y pred2 until a stopping condition of Y n -∑Y pred1...n ≈0 is met, and the sum of Y pred1...n is calculated to complete the training required model. Therefore, it can be known that the copper stave thickness prediction model can be established by adding the prediction model to the formula in the form of parameter input. As the thickness estimation model, the calculation process can be understood by those with ordinary knowledge in the technical field and will not be repeated.
此外,附加地,如第2圖所示,該建模方法還可包括步驟:一測試步驟S4,基於一測試溫度依據該估厚模型產生一相應厚度。例如:該測試溫度可為銅冷卻壁溫度,該相應厚度可為銅冷卻壁厚度,依據該估厚模型可基於該測試溫度產生該相應厚度,該相應厚度除可作為進一步優化該估厚模型的依據,還可作為測試該估厚模型用於輸出經過估測的銅冷卻壁厚度,該銅冷卻壁厚度可被當成高爐銅冷卻壁殘厚的一高可信度參考值,供相關人員作為容器(如高爐)應用、維修、設計或研究等。In addition, as shown in Figure 2, the modeling method may further include the following steps: a test step S4, generating a corresponding thickness according to the estimated thickness model based on a test temperature. For example: the test temperature can be the temperature of the copper stave, the corresponding thickness can be the thickness of the copper stave, the corresponding thickness can be generated based on the test temperature according to the estimated thickness model, and the corresponding thickness can be used as a further optimization of the estimated thickness model. It can also be used as a basis for testing the estimated thickness model to output the estimated copper stave thickness. The copper stave thickness can be used as a high-confidence reference value for the residual thickness of the blast furnace copper stave for relevant personnel as a container (Such as blast furnace) application, maintenance, design or research, etc.
示例地,如第7及8圖所示,其係兩種銅冷卻壁厚度估測示意圖,譬如可表示兩處銅冷卻壁的厚度估測結果。其中,◆表示實際量測的超音波測厚數據,虛線與實線表示兩種估測結果。由於超音波測厚數據會有些微誤差,導致實際量測數據不穩定的情況,因此,實線表示根據每次量測數據進行厚度更新的估測結果,虛線表示由初始厚度為130毫米(mm)起算,不考慮以量測數據更新厚度的估測結果。由圖可知,該些預測結果與實際量測結果之間的預測誤差皆小於2 mm。從而,本發明實施例之估測容器壁厚的建模方法如被應用於煉鋼高爐時,該估厚模型產生的相應厚度確實可以被當成一可信度高的銅冷卻壁厚度估測結果。For example, as shown in Figures 7 and 8, which are schematic diagrams of two types of copper stave thickness estimation, for example, it can represent the thickness estimation results of two copper staves. Among them, ◆ represents the ultrasonic thickness measurement data actually measured, and the dashed line and the solid line represent the two estimation results. Because the ultrasonic thickness measurement data will have some slight errors, which will cause the actual measurement data to be unstable. Therefore, the solid line represents the estimated result of the thickness update based on each measurement data, and the dashed line represents the initial thickness from 130 millimeters (mm ), regardless of the measurement data to update the thickness estimation result. It can be seen from the figure that the prediction errors between the prediction results and the actual measurement results are all less than 2 mm. Therefore, when the modeling method for estimating the thickness of the vessel wall of the embodiment of the present invention is applied to a steelmaking blast furnace, the corresponding thickness generated by the estimating model can indeed be regarded as a highly reliable copper stave thickness estimation result. .
另一方面,本發明還提供一種估測容器壁厚的建模系統,包括一處理器及一記憶體,該處理器耦接該記憶體,該記憶體儲存至少一指令,該處理器執行該指令,以執行如上所述之估測容器壁厚的建模方法。In another aspect, the present invention also provides a modeling system for estimating the wall thickness of a container, including a processor and a memory, the processor is coupled to the memory, the memory stores at least one instruction, and the processor executes the Command to execute the modeling method for estimating the wall thickness of the container as described above.
舉例而言,該估測容器壁厚的建模系統可被配置成具有資料處理功能的電子裝置,例如:雲端平台機器、伺服器、桌上型電腦、筆記型電腦、平板電腦或智慧型手機等,惟不以此為限,用於執行如上所述之估測容器壁厚的建模方法,其實施方式已說明如上,不再贅述。For example, the modeling system for estimating container wall thickness can be configured as an electronic device with data processing functions, such as cloud platform machines, servers, desktop computers, laptops, tablets, or smartphones Etc., but not limited to this, it is used to implement the modeling method for estimating the wall thickness of the container as described above. The implementation mode has been described above and will not be repeated.
另一方面,本發明還提供一種電腦程式產品,當電腦載入該電腦程式並執行後,該電腦能夠執行如上所述之估測容器壁厚的建模方法。例如:該電腦程式產品可包含數個程式指令,該程式指令可利用現有的程式語言實現,以便用於執行如上所述之估測容器壁厚的建模方法,例如:可採用C、C++、Labview、Python、R等程式語言或其組合,惟不以此為限。On the other hand, the present invention also provides a computer program product. After the computer program is loaded and executed, the computer can execute the modeling method for estimating the wall thickness of the container as described above. For example: the computer program product may contain several program instructions, which can be implemented using existing programming languages to execute the modeling method for estimating the wall thickness of the container as described above, for example: C, C++, Programming languages such as Labview, Python, R, or combinations thereof, but not limited to these.
另一方面,本發明還提供一種電腦可讀取紀錄媒體,例如:光碟、隨身碟或硬碟等,該電腦可讀取紀錄媒體內儲程式(如上述電腦程式),當電腦載入該程式並執行後,該電腦能夠完成如上所述之估測容器壁厚的建模方法。On the other hand, the present invention also provides a computer-readable recording medium, such as an optical disc, a flash drive, or a hard disk. The computer can read a program stored in the recording medium (such as the above-mentioned computer program). After execution, the computer can complete the modeling method for estimating the wall thickness of the container as described above.
本發明上述實施例的估測容器壁厚的建模方法、系統、電腦程式產品及電腦可讀取紀錄媒體,基於容器外部溫度及相應厚度建立模型,以利作為即時估測容器壁厚的依據,可以有效簡化估測容器壁厚所需的參數,改善習知技術的計算過程複雜等問題,有利於即時監測容器壁厚(如高爐銅冷卻壁殘厚),便於相關人員作為容器應用、維修、設計或研究等參考。The modeling method, system, computer program product, and computer readable recording medium for estimating container wall thickness in the above embodiments of the present invention establish a model based on the external temperature of the container and the corresponding thickness, so as to be used as a basis for real-time estimation of container wall thickness , It can effectively simplify the parameters needed to estimate the wall thickness of the vessel, improve the complicated calculation process of the conventional technology and other problems, which is conducive to the real-time monitoring of the vessel wall thickness (such as the residual thickness of the blast furnace copper cooling stave), and is convenient for relevant personnel to use and maintain the vessel. , Design or research, etc.
雖然本發明已以較佳實施例揭露,然其並非用以限制本發明,任何熟習此項技藝之人士,在不脫離本發明之精神和範圍內,當可作各種更動與修飾,因此本發明之保護範圍當視後附之申請專利範圍所界定者為準。Although the present invention has been disclosed in preferred embodiments, it is not intended to limit the present invention. Anyone familiar with the art can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, the present invention The scope of protection shall be subject to the scope of the attached patent application.
1:資料庫 2:訓練單元 3:測試單元 B:銅冷卻壁 C:溫度曲線 H:高爐容物 S1:獲取步驟 S2:抽取步驟 S3:訓練步驟 S4:測試步驟 T b:銅冷卻壁的溫度 T h:高爐容物的溫度 T w:冷卻水層的溫度 W:冷卻水層 d:銅冷卻壁的厚度1: Database 2: Training Unit 3: Test Unit B: Copper Stave C: Temperature Curve H: Blast Furnace Content S1: Acquisition Step S2: Extraction Step S3: Training Step S4: Test Step T b : Copper Stave Temperature T h : the temperature of the blast furnace contents T w : the temperature of the cooling water layer W: the cooling water layer d: the thickness of the copper stave
[第1圖]:本發明實施例之估測容器壁厚的建模系統的方塊示意圖。 [第2圖]:本發明實施例之估測容器壁厚的建模方法的流程示意圖。 [第3圖]:本發明實施例被應用於煉鋼高爐之銅冷卻壁的熱傳導參數示意圖。 [第4圖]:本發明實施例被應用於煉鋼高爐之高爐鐵水溫度示意圖。 [第5圖]:本發明實施例被應用於煉鋼高爐之渣皮脫落過程之爐壁溫度示意圖。 [第6圖]:本發明實施例被應用於煉鋼高爐之銅冷卻壁單日溫度與溫差示意圖。 [第7圖]:本發明實施例被應用於煉鋼高爐之一種銅冷卻壁厚度估測示意圖。 [第8圖]:本發明實施例被應用於煉鋼高爐之另一種銅冷卻壁厚度估測示意圖。 [Figure 1]: A block diagram of a modeling system for estimating the wall thickness of a container according to an embodiment of the present invention. [Figure 2]: A schematic flow diagram of a modeling method for estimating the wall thickness of a container according to an embodiment of the present invention. [Figure 3]: Schematic diagram of the heat conduction parameters of the copper stave of the steel-making blast furnace applied to the embodiment of the present invention. [Figure 4]: Schematic diagram of the temperature of molten iron in a blast furnace where the embodiment of the present invention is applied to a steelmaking blast furnace. [Figure 5]: Schematic diagram of the furnace wall temperature of the slag skin shedding process of the steelmaking blast furnace that the embodiment of the present invention is applied to. [Figure 6]: A schematic diagram of the daily temperature and temperature difference of the copper stave of the steel-making blast furnace applied to the embodiment of the present invention. [Figure 7]: The embodiment of the present invention is applied to a schematic diagram of the thickness estimation of a copper stave in a steelmaking blast furnace. [Figure 8]: The embodiment of the present invention is applied to another schematic diagram of estimating the thickness of copper stave in a steelmaking blast furnace.
S1:獲取步驟 S1: Obtaining steps
S2:抽取步驟 S2: extraction step
S3:訓練步驟 S3: training steps
S4:測試步驟 S4: Test steps
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US20200241078A1 (en) * | 2016-09-26 | 2020-07-30 | KW Associates LLC | Estimation of arc location in three dimensions |
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