# Maintenance Management of a Transmission Substation with Optimization
## 4 Optimization Process of Maintenance Activity
### Objective and Penalty Function  
The maintenance model is linked to the optimization algorithm with objective and penalty functions. Three objective functions and one penalty function were defined to achieve the reliability-based maintenance concept. Since the objective functions are related to different quantities, they are normalized in the optimization process to enable merging of criteria. The first objective function \(f_{1}\) is related to the expected costs of outage during the maintenance for all significant states. Figure 4a shows the fuzzy function (bell-shaped membership function) used to normalize the expected costs of non-transmitted energy \(C_{\text{in},q}\). The concept of normalizing is such that the costs of non-transmitted energy, which in \"usual\" significant states lie between \(x_{1}\) = 5 and \(x_{u}\) = 300 EUR/year, represent 90% variability of the normalized value. The costs of expected outages \(C_{\text{in},q}\) that occur several times per year amount to approximately EUR 1,000,000 per year (Table 3) and have no more impact on the value of the objective function. Figure 4: (**a**) Presentation of normalization of the objective functions \(f_{1}\) (expected costs of outages) and \(f_{3}\) (\(c\)-\(i\) indices); (**b**) presentation of normalizing of the penalty function \(p_{\text{k}}\) (leaking of SF\({}_{6}\)). Figure 3: Algorithm of the optimization process of maintenance. The second objective function \(f_{2}\) in the optimization process that represents the revision costs for a certain TS element \(C_{\text{r},q}\) for all tasks that require de-energizing of an individual bay. Since the range of these costs is smaller than for the first objective function, there is no need for the use of fuzzy function-based normalization. The normalization is performed using (7)  
\[f_{2}=\sum_{q=1}^{n_{\text{SB}}}\frac{C_{\text{r},q}}{y_{q}}/\sum_{q=1}^{n_{ \text{SB}}}C_{\text{r},q};\ n_{\text{SB}}=31;\ y\in\{1,\ 2,\ 3\} \tag{7}\]  
where the objective function encompasses the total costs of all the 31 transmission substation bays and transformers, where the revision time period (1, 2, or 3 years) is taken into consideration for each bay separately. The third objective function \(f_{3}\) is related to the indices of the RCM maintenance concept. In the block diagram of input data (Figure 3), the RCM indices \(c\) and \(i\) are included in the common index \(d\). In the denotation of the vector, the weighting factor \(w_{\text{SSj}}\) needs to be taken into consideration. For the determination of the objective function \(f_{3}\), the common part \(d_{q}\) is used, which includes all the above-mentioned factors. An easier insight into the situation is enabled by the \(c-i_{\text{avg}}\) diagram, shown in Figure 2b. If the value of \(d_{q}\) is lower than 35.3 (green area), this objective function yields \(f_{3}=0\). If the value of \(d_{q}\) lies between 35.3 and 70.7 (yellow area), the value of the objective function \(f_{3}\) is linear between 0 and 1. If the value exceeds 70.7 the value of the objective function is \(f_{3}=1\). The third objective function is illustrated graphically in Figure 4a. Penalty function \(p_{\text{k}}\) is related to the variations of pressure of the greenhouse SF\({}_{6}\) gas in the circuit breakers. The penalty function describes that the leaking of SF\({}_{6}\) in the period \(\Delta t\) is the reason for more frequent maintenance interventions. The higher the change of pressure in the period \(\Delta t\), the steeper is the penalty function \(p_{\text{k}}\), i.e., the sooner it approaches the value of 1 (Figure 4b). If the pressure drops below the factory-set value \(a\), the protection disables operation of the HV device, which requires a maintenance intervention--replacement of the element. The equation of the penalty function (8) can be derived from Figure 4b. \[p_{k}=\left\{ \begin{array}{l}1;\ \left(\Delta p_{q}>a\right)\Rightarrow \text{replacement}\\ 2^{\frac{\Delta p_{q}}{a}}-1;\ \Delta p_{q}<a\end{array} \right.