Customer interruption cost data normalization and customer damage functions

CIC estimates retrieved from survey respondents are absolute cost values. To consistently apply the cost data in power system reliability planning or operation, it is necessary to transform the CIC data by normalization with an appropriate factor. This allows for the calculation of aggregate or average cost of different electricity customers with similar cost characteristics [28].

Normalization also reduces the magnitude of the cost data and speeds up computational procedures. The normalized CIC for a respondent in a electricity interruption scenario at a time reference time t may be represented as:

𝐶_𝑛,𝑖(𝑑, 𝑡) =^𝐶𝑖_𝑁𝐹^(𝑑,𝑡)

𝑖 (𝑅/𝑘𝑊ℎ 𝑜𝑟 𝑘𝑊) (2.1)

𝐶_{𝑛,  𝑖}(𝑑, 𝑡) is the normalized CIC estimate for respondent i for an electricity interruption of duration d, occurring at time t. 𝐶𝑛,  𝑖(𝑑, 𝑡) can be also called the individual customer damage function (ICDF).

𝐶_𝑖(𝑑, 𝑡) is the CIC estimate for respondent i for an electricity interruption of duration d, occurring at time t.

N𝐹_𝑖 is the chosen normalization factor for respondent i.

The customer damage function for sector j (i.e. 𝐶_𝑛,𝑗(𝑑, 𝑡)) consisting of N electricity customers may be calculated by:

I. Averaging the ICDF for the electricity customers in that sector:

𝐶_𝑛,𝑗(𝑑, 𝑡)  =  _𝑁¹∑^𝑁_𝑖=1𝑐_{𝑛,  𝑖}(𝑑, 𝑡) (2.2)

II. Aggregating the CIC estimates for sector j and dividing the aggregate by a chosen normalization factor for the sector.

 𝐶_𝑛,𝑗(𝑑, 𝑡)  =  ^{∑𝑁𝑖=1}_𝑁𝐹^{𝐶 𝑖(𝑑,𝑡)}

𝑗 (2.3)

III. Fitting a suitable probability distribution function (PDF) to the N data points of individual normalized cost of the electricity customers in sector j. Normalized CIC data for a sector

37

having a finite range may exhibit significant skewness. Thus, the chosen PDF must be versatile enough to accommodate these characteristics of the data. The effectiveness of the beta PDF in characterizing CIC data probabilistically has been demonstrated and validated by past researches [28, 79, 109, 110]. A concise mathematical and graphical description of the beta distribution is presented in Appendix A. Describing CIC probabilistically allows for a probabilistic evaluation of a network’s performance in terms of value-at-risk (e.g. Rands@Risk).

Normalized CIC estimates for a reference electricity interruption scenario may be temporally differentiated using time-weight factors [42, 78]. A time-element matrix may also be developed that characterizes the activity levels of electricity customers into distinct time-season cells [79].

Each time-season cell may represent periods where the power system is susceptible to certain types of risk [6].

Choosing normalization factors

The intended application of CIC data and availability of ancillary data during a study or project period influences the choice of a normalization factor. Dzobo [28] summarizes various normalization factors based on electrical energy or load and their data requirements (Table 2.7).

The predominant CIC normalization factors in the literature include annual electricity consumption (kWh/MWh) and peak load [34]. In some cases, electricity customers may be able to provide data on their monthly or annual electricity consumption. Where electric utilities are involved or interested in the CIC study, they can also provide such data. Otherwise, annual electricity consumption may be deduced from tariff information. Information on peak load at a reference electricity interruption time are not usually publicly available but can be estimated from load curves [28].

Several opinions have been aired by different authors on the most suitable normalization factor for specific customer types or electricity interruption scenarios. For instance, Ghajar and Billinton [111] claim that the effect of peak load on CIC is more significant for short electricity interruption duration, while the effect of annual electrical energy consumption on CIC is more significant for longer duration. Thus, peak load should be used for normalizing CIC for very short electricity

38

interruption duration, while annual electrical energy consumption should be used for normalizing CIC for longer duration. However, Sullivan et al [38] opine that normalization based on peak load and that based on annual electrical energy consumption do not yield similar reliability worth evaluation results. Alternatively, the unsupplied energy for a sector may be used to normalize the estimated CIC for the sector [112]. Normalizing CIC by unsupplied energy requires estimation of the unsupplied energy for the reference electricity interruption scenario from load curves.

However, it might be difficult to ascertain whether the variation in CIC results directly from a variation in unsupplied energy or from assumptions and approximations made during the estimation of unsupplied energy [113]. Some other authors [114] opine that a more appropriate normalization factor for large industries is their annual turnover. They argue that annual turnover has a more prominent effect on CIC than electricity consumption, because the annual turnover for such large industries is significantly higher than their annual electrical energy consumption.

However, annual turnover is seldom used for normalizing CIC.

Table 2.7: Normalization factors based on electrical energy demand or load [28]

Factor Definition Data requirement

Annual electricity consumption (kWh or MWh)

Total annual electricity units consumed.

Total annual electricity consumption monitored as input to the electricity bill.

Average load (kW) Annual electricity consumption / 8760

Total annual electricity consumption monitored as input to the electricity bill.

Peak load (kW) Maximum hourly load in a year Load data: 8760 hourly loads based on hourly metering or general load curves.

Interrupted load (kW) The estimated power that would have been supplied at the time of the electricity interruption (or voltage disturbance) if the

Energy not supplied The estimated energy that would have been supplied if an electricity interruption did not occur.

Load data: 8760 hourly loads based on hourly metering or general load curves.

Monthly energy cost The total amount of money paid by the electricity customer to buy electricity for a month.

Total monthly electricity bill.

39

Academic researchers may not be able to obtain accurate information from survey respondents on their annual or monthly electrical energy consumption [70, 115] and electric utilities might be unwilling to divulge such data or may not have it at a disaggregated level. In such cases, average monthly electricity cost may be used for CIC normalization. High correlation has been observed between average monthly electricity bill and CIC for electricity customers with similar economic activity [47, 70].