## 4 Materials and Methods
### Extenics-Based Identification Primitive Modeling of Coil Manufacturing Process Information
#### 4.4.1 All Elements Information Fusion Based on Extensible Petri Net Model
\((u_{1},1)\) transitions to \((u_{2},\) 1) through a traceability scenario (\(s_{1},\) 1). During this transition, the following traceability information transformation relationship exists:  
\[u_{2},1.\text{Info }=u_{1},1.\text{Info }\oplus s_{1},1.\text{Info} \tag{11}\]  
\[P\left(u_{2},1\right)=\left(\Gamma(0,\ u_{2,1})\right)^{-1}(\mathop{\cup}_{j= 1}^{n}u_{2,j})=\mathop{\cup}_{i=1}^{m}u_{1,j}=\left\{u_{1},1\right\} \tag{12}\]  
\[F(u_{1},1)=\Gamma(1,\ u_{1},1)\left(\mathop{\cup}_{j=1}^{m}u_{1,j}\right)= \mathop{\cup}_{i=1}^{n}u_{2,i}=\left\{u_{2},1\right\} \tag{13}\]  
(2) Disassembly Type  
The fundamental type represented by a traceability scenario \"\(s\)\" with a fan-in of 1 and a fan-out greater than 1 is referred to as a disassembly traceability scenario. As depicted in Figure 8, \((u_{1},1)\), \((u_{2},1)\), \((u_{2},2)\), \(\ldots\), \((u_{2},n)\) are n + 1 traceability units. \((u_{1},\) 1) transitions to \((u_{2},\) 1), \((u_{2},2)\), \(\ldots\), \((u_{2},n)\) through a traceability scenario (\(s_{1},\) 1). During this transition, the following traceability information transformation relationship exists:  
\[u_{2},i.Info=u_{1},1.Info\oplus s_{1},1.Info(\ i)\wedge\forall i\in\left\{1,2, \ldots,n\right\} \tag{14}\]  
\[P\left(u_{2},i\right)\overset{\forall i\in\left\{1,2,\ldots,n\right\}}{=}s_{1,1}^{-1}(\mathop{\cup}_{j=1}^{n}u_{2,j})=\mathop{\cup}_{i=1}^{m}u_{1,i} \tag{15}\]  
\[F(u_{1},1)=s_{1},1\left(\mathop{\cup}_{j=1}^{m}u_{1,j}\right)=\mathop{\cup }_{i=1}^{n}u_{2,i} \tag{16}\]  
Figure 7: Linear Traceability Scenario. (3) Merging Type  
Merging Type: The fundamental type represented by a traceability scenario \"\(s\)\" with a fan-in greater than 1 and a fan-out of 1, referred to as a merging traceability scenario. As shown in Figure 9, (\(u_{1}\), 1), (\(u_{1}\), 2), \(\ldots\), (\(u_{1}\), \(n\)), and (\(u_{2}\), 1) are \(n+1\) traceability units. (\(u_{1}\), 1), (\(u_{1}\), 2), \(\ldots\), (\(u_{1}\), \(n\)) transition to (\(u_{2}\), 1) through a traceability scenario (\(s_{1}\), 1) to form a single traceability unit. During this transition, the following traceability information transformation relationship exists as the following:  
\[u_{2},1.Info=u_{1},1.Info\oplus u_{1},2.Info\oplus\ldots\oplus s_{1},n.Info \tag{17}\]  
Figure 7.0  
Figure 7.1  
Figure 7.2  
Taking the traceability of quality information in the coil manufacturing process as an example, coils flow continuously along the process path, with states and information constantly changing and migrating. The paths of information association are extremely complex, making it exceptionally difficult to trace quality issues; the main characteristic elements of quality traceability information are shown in Tables 9 and 10. To address this challenge, integrate the above mathematical model with Petri nets; a traceability path is constructed based on the physical element model corresponding to each place P and the event element model corresponding to each transition t in the transformer coil manufacturing process. Taking the influencing factors of coil manufacturing quality as nodes, a traceability and control path for quality anomalies has been established. As shown in the Figure 10, \(P_{\text{q1}}\)-\(P_{\text{q10}}\) represent ten key control stations for the quality of coils (traceability scenario); during the coil production process, the tasks mainly include mold inspection, raw material inspection such as insulation paper, full-process inspection of the winding process, casting and curing inspection, coil polishing, finished product appearance inspection. Each \(P_{qi\text{-}j}\) associated with \(P_{qi\text{-}j}\) represents all the traceability units included in that traceability scenario. For example, from \(P_{\text{q10-i}}\) to \(P_{\text{q10}}\), it includes traceability units such as mold outer diameter measurement, circumference measurement, perpendicularity measurement, and overall appearance cleanliness inspection in the \"Mold Quality Inspection\" scenario.