## 3 Improved PIM under Dynamic Conditions
### Improved PIM Which Considers Past Effects
\begin{array}{l}M(0,t)=M(0,t_{1}|e_{1})+M(t_{1},t|e_{2})=\\ \left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{1})-Z_{l}(e_{0})) \right)\right)^{\frac{1}{b}}\cdot(t_{1}-t_{0})+\left(\exp\left(\sum\limits_{l =1}^{m}k_{l}\cdot(Z_{l}(e_{2})-Z_{l}(e_{0}))\right)\right)^{\frac{1}{b}}\cdot (t-t_{1})\end{array} , \tag{22}\]  
The unit time damage currently is:  
\[m(0,t|e_{2})=\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{2})-Z_{ l}(e_{0}))\right)\right)^{\frac{1}{b}}, \tag{23}\]  
Applying Equations (22) and (23) into Equation (21) leads to:  
\[ \begin{array}{l}w(t|E_{2})=\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z _{l}(e_{2})-Z_{l}(e_{0}))\right)\right)^{\frac{1}{b}}\cdot a_{0}b\cdot\\ \left(\left(\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{1})-Z_{l} (e_{0}))\right)\right)\right)^{\frac{1}{b}}\cdot(t_{1}-t_{0})+\left(\exp \left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{2})-Z_{l}(e_{0}))\right) \right)^{\frac{1}{b}}\cdot(t-t_{1})\right)\end{array} , \tag{24}\]  
The general case can then be discussed. Supposing the system runs in a dynamic WC \(E_{n}\), where the system works in \(e_{1}\) from \(t_{0}(t_{0}=0)\) to \(t_{1}\), in \(e_{2}\) from \(t_{1}\) to \(t_{2}\), in \(e_{n}\) from \(t_{n-1}\) to \(t_{n}\). When \(t_{n-1}\leq t\leq t_{n}\), the FI function can be derived as:  
\[ \begin{array}{l}w(t|E_{n})\\ =\lim\limits_{\Delta t\to 0}\frac{W(\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e _{i})+M(t_{n-1},t|e_{n})+M(t,t+\Delta t|e_{n}))-W(\sum\limits_{i=1}^{n-1}M(t_ {i-1},t_{i}|e_{i})+M(t_{n-1},t|e_{n}))}{\Delta t}\\ =\lim\limits_{\Delta t\to 0}\frac{W(\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e _{i})+M(t_{n-1},t|e_{n}))-W(\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e_{i})+M(t _{n-1},t|e_{n}))}{\Delta t\cdot M(t+\Delta t|e_{n})}\cdot\frac{M(t,t+\Delta t|e _{n})}{\Delta t}\\ =m(t|e_{n})\cdot w_{0}(\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e_{i})+M(t_{n-1 },t|e_{n}))\end{array} , \tag{25}\]  
Equation (16) is rewritten as follows:  
\[w(t|e_{n})=a_{0}b\cdot\left(\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e_{i})+M(t _{n-1},t|e_{n})\right)^{b-1}\cdot m(0,t|e_{n}), \tag{26}\]The total cumulative damage is the sum of all parts of damage, which is shown in Equation (27):  
\[ \begin{array}{l}M(0,t|E_{n})=\sum\limits_{i=1}^{n-1}M(t_{i-1},t_{i}|e_{i})+M(t_ {n-1},t|e_{n})\\ =\sum\limits_{i=1}^{n-1}\left(\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot( Z_{l}(e_{i})-Z_{l}(e_{0}))\right)\right)^{\frac{1}{p}}\cdot(t_{i}-t_{i-1})\right)+ \left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{n})-Z_{l}(e_{0})) \right)\right)^{\frac{1}{p}}\cdot(t-t_{n-1})\end{array} , \tag{27}\]  
The current unit time damage is:  
\[m(0,t|e_{i})=\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{i})-Z_{l }(e_{0}))\right)\right)^{\frac{1}{p}}, \tag{28}\]  
Applying Equations (27) and (28) into Equation (26) leads to:  
\[ \begin{array}{l}w(t|E_{n})=\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot( Z_{l}(e_{n})-Z_{l}(e_{0}))\right)\right)^{\frac{1}{p}}\cdot a_{0}b\cdot\\ \left(\left(\sum\limits_{i=1}^{n-1}\left(\left(\exp\left(\sum\limits_{l=1}^{m} k_{l}\cdot(Z_{l}(e_{i})-Z_{l}(e_{0}))\right)\right)^{\frac{1}{p}}\cdot(t_{i}-t_{i-1}) \right)+\left(\exp\left(\sum\limits_{l=1}^{m}k_{l}\cdot(Z_{l}(e_{n})-Z_{l}(e_{ 0}))\right)\right)^{\frac{1}{p}}\cdot(t-t_{n-1})\right)\right)^{b-1}\end{array} , \tag{29}\]  
where the FI in dynamic WCs is expressed by the form of that in standard WCs.