# Life Cycle Cost Estimation and Analysis of Transformers Based on Failure Rate
## 2 Materials and Methods
### Life Cycle Cost Analysis
at 6.2% [22] of the equipment purchase cost, while other costs mainly include equipment debugging cost, which can be calculated at 11.8% [22] of the equipment purchase cost. 2. Operating cost C2  
The operating cost refers to the cost of maintaining the normal operation of the transformer during its lifespan from operation to scrapping, including energy consumption cost and maintenance cost. For the cost of operating maintenance, factors such as labor compensation, maintenance costs, and insurance premiums have been considered as CPI increases, and the model has been improved. The operating cost model is as follows [22]:  
\[C_{2}=C_{\text{ol}}+C_{\text{om}} \tag{2}\]  
where \(C_{\text{ol}}\) is the operation loss cost, and \(C_{\text{om}}\) is operation and maintenance cost. The specific mathematical model of the machine operation and maintenance cost is as follows [21]:  
\[C_{\text{ol}}=\left[P_{0}+\beta^{2}(P_{\text{w}}+P_{\text{f}})\right]\cdot \eta\cdot 8760\cdot P_{\text{r}} \tag{3}\]  
\[C_{\text{om}}=\sum_{t=1}^{N}C_{\text{om}}(t)\bigg{(}\frac{1+r}{1+R}\bigg{)}^{ t-1} \tag{4}\]  
where \(P_{0}\) is the no-load loss; \(P_{\text{w}}\) is the load loss; \(P_{\text{f}}\) is the auxiliary loss; \(\beta\) is the average load rate, which can take a value of 70%; \(P_{\text{r}}\) is the comprehensive electricity prices, the general value of which is CNY 0.56/kW\(\cdot\)h; \(\eta\) is the annual load loss rate, with a value of 0.608; \(C_{\text{om}}(t)\) is the cost of equipment operation and maintenance in year; \(R\) is the social discount rate; and \(r\) is the rate of inflation. 3. Repair cost C3  
The repair cost includes both major and minor repairs. The cost of each inspection and maintenance includes the equipment, material fees, and service fees that the supplier needs to provide for the activity. This also includes the owner's equipment, material costs, and labor costs in this activity [21]. \[C_{3}=\sum_{t-1}^{N}\lambda(t)\big{[}C_{\text{maj}}(t)p(t)+C_{\text{min}}(t)( 1-p(t))\big{]}\times\bigg{(}\frac{1+r}{1+R}\bigg{)}^{t-1} \tag{5}\]  
\[p(t)=\frac{(1+\mu)\big{[}\frac{t-1}{t_{\text{maj}}}\big{]}}{t_{\text{maj}}} \tag{6}\]  
where \(\lambda(t)\) is the failure probability of the power transformer and \(p(t)\) is the probability of power transformer major repair; then, \(1-p(t)\) is the probability of minor repair. \(C_{\text{maj}}(t)\) represents the average cost of a major repair; \(C_{\text{min}}(t)\) represents the average cost of a minor repair; N indicates the operating years; \(t_{\text{maj}}\) is the power transformer overhaul cycle; \(\mu\) is the overhaul ratio rate of increase in cases; and \(\big{[}\frac{t-1}{t_{\text{maj}}}\big{]}\) indicates taking not greater than \(\frac{t-1}{t_{\text{maj}}}\), the largest integer. 4. Failure cost C4  
The cost of transformer failure refers to the cost of losses caused by power shortage or power interruption due to transformer failure, including failure recovery costs and failure loss costs. The cost of failure recovery includes the cost of removing failure equipment, transporting failure transformers, and installing new equipment. The cost of failure losses is an indirect loss caused by power outages, damage to the company's reputation, and social impact and responsibility, which can be converted based on multiple electricity prices [21]. \[C_{4}=\sum_{t=1}^{N}\left[\alpha WT+\lambda(t)C_{r}\cdot MTTR+(T-T_{0})C_{4} \right]\times\bigg{(}\frac{1+r}{1+R}\bigg{)}^{t-1} \tag{7}\]where \(\alpha\) is the value of the average interrupted power supply of the user, which varies with the region where the user is located; \(W\) indicates that equipment failure interrupts the power supply; \(T\) is the annual failure interruption power supply time of the equipment; \(\lambda(t)\) is the annual average number of equipment failures; \(C_{\rm r}\) is the average repair cost of failure equipment; \(MTTR\) is the mean time to repair; and \(C_{\rm d}\) represents the compensation cost for unit power outage time.