## 2 Estimation of Interruption Costs Arising from Transmission System Originated Failures
### Integrated Interruption Cost Model  
This section describes the methodology to develop the interruption cost model for each customer sector that can be integrated with the SPD model described in Section 2.1. The interruption cost incurred to a customer sector due to bulk component failure depends on the magnitude of the load curtailed and the duration of the interruption. As the magnitude of the load varies with time with a characteristic specific to a customer sector, it will be necessary to determine the interruption cost as a function of time of day, time of year, and the interruption duration. This information can be extracted from a customer survey [18] of all the customer sectors within the power system. The outage cost estimation from the customer survey assumes that the customers are in the best position to assess the cost of their outage that may occur at different times and scenarios. An inaccurate response from a customer can deviate from the survey results, and therefore, careful screening is required to identify and filter out outliers from the survey sample. It should be noted that the outage cost derived is the best possible estimate, as intangible and indirect costs associated with emotional and other social values are very difficult to quantify. The statistical estimation method [14; 15; 16] is applied to obtain the interruption cost \(IC_{p}\) at each of the 'N + 1' periods \(p\) of the SPD model for a range of outage durations as shown in (11). \[IC_{p}=\big{\{}ic_{p,t}\mid t=1,2,\ldots,T\big{\}} \tag{11}\]  
where, \(ic_{p,t}\) is the interruption cost in CAD/interruption for an outage of duration \(t\) during the load period p. The outage duration, \(t\) = 1, 2, \(\ldots\), \(T\), representing 1 min, 20 min, 1 h, 2 h, 4 h, 8 h, 24 h [14; 15; 16; 41; 42] are usually considered in these studies. The interruption cost is then normalized by the representative load for each period of the SPD model. The demand-normalized cost \(DNC_{p,t}\) in CAD/kW is obtained from (12),  
\[DNC_{p,t}=\big{\{}ic_{p,t}/L_{p}\big{\}}_{for\ p=1\ to\ N+1\ and\ t=1\ to\ T} \tag{12}\]  
The expected demand-normalized cost \(EDNC_{t}\) for an outage duration \(t\) is next obtained from (13) by taking the weighted sum of all the DNC calculated for all the N + 1 demand periods of the SPD model. \[EDNC_{t}=\sum_{p=1}^{\text{N+1}}\big{(}\text{Prob}_{p,t}\times DNC_{p,t}\big{)} \tag{13}\]  
The customer damage function (CDF) is the expected demand-normalized cost as a function of the interruption duration. The CDF for a sector, SCDF, is given by (14):  
\[SCDF=\big{\{}EDNC_{1},EDNC_{2},\ldots,EDNC_{t},\ldots,EDNC_{T}\big{\}} \tag{14}\]  
The composite customer damage function (CCDF) at a bulk load point is calculated by weighting the SCDF from (14) by load compositions (peak load and energy consumption percentages) of customer sectors as given by (15). \[CCDF{=}\left\{\sum_{m=1}^{\text{M}}\Big{(}EDNC_{m,t}\times wf_{m,t}\Big{)} \right\}_{for\ t=1\ to\ T} \tag{15}\]  
where \(m=1,2,\ldots,M\) denotes the participating customer sectors, and \(wf_{m,t}\) is the weighting factor for a customer sector at the load point such that [41]\[wf_{m,t}=\left\{ \begin{array}{ll}L_{peak,m}/L_{peak}&for\ t<1\ \text{hour}\\ E_{m}/E&for\ t\ \geq\ 1\ \text{hour}\end{array} \right.\]  
where \(L_{peak,m}\) and \(E_{m}\) are the annual peak load and energy consumption of Sector \(m\), respectively, and \(L_{peak}\) and \(E\) are the annual peak load and energy consumption of the overall system. Finally, the monetary equivalence of power loss is expressed as the expected cost of customer interruptions (ECOST) as given by (16). \[ECOST_{k}=\sum_{i=1}^{NC}L_{ki}\times F_{i}\times C_{i}(D_{i})\frac{MW}{kW} \$/\text{yr} \tag{16}\]  
where \(L_{ki}\), \(F_{i}\), and \(C_{i}(D_{i})\) represent the magnitude (MW), frequency (occ/year), and CCDF (CAD/kW) of an outage of duration \(D_{i}\) from contingency \(i\) at load point \(k\), respectively. Also, \(NC\) represents the total number of outages leading to the power interruptions at load point \(k\).