## Power supply interruption costs: Models and methods incorporating time dependent patterns

##### Doctoral thesis

##### Date

1996##### Metadata

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- Institutt for elkraftteknikk [2415]

##### Abstract

The main results of this thesis are improved methods for the estimation of annual interruption costs for delivery points from electricity utilities to the end users. The improvements consist of a combined representation of time variation and uncertainties in the input variables. The time variation is only partly handled in methods reported in the literature. A particular result in this work is a unified representation of time variation in the input variables. This will enable the socio-economic costs of power supply interruptions to be determined more correctly. Consequently more credible estimates of this cost element can be provided as a basis for the optimization of the power system.
Recent customer surveys by the electric power industry on interruption costs provide better estimates of costs per interruption and more information on the characteristics of these costs, which can stimulate further studies of the annual costs. This is relevant as there is increased interest in the quality of power supply from both the customers and the regulation authorities. This calls for improved methods for assessment of interruption costs for delivery points at any system level.
Time variation and uncertainties
The annual costs from unexpected interruptions are determined by four variables: Failure rate, repair time, load and specific interruption cost. The customer surveys show that the cost per interruption has considerable variation depending on the time of occurrence. On average for the industrial and commercial sectors in Norway there is for instance a cost decrease of 40 % from working hours till midnight as well as from working days till weekends, while the monthly variation is up to 20 %. Examples from the failure statistics for distribution systems show that the probability of failures is three times higher in January than in May and about three times higher in working hours than at night. The repair times however, are about twice as high during night than during working hours and twice as high in winter as in summer. The failure statistics for the higher system levels in the transmission system show for instance that the probability of failures is three times higher in January than in summer and 60 % higher in working hours than at night, while the repair time is about 20 % higher in summer than winter and at night compared to day time.
These time dependent patterns indicate that there is a time dependent correlation between the input variables that might influence the annual costs. There are also stochastic variations in the input variables as well as other types of uncertainties, termed fuzziness in this thesis. A further result in this thesis is combining the representation of time variations and the additional uncertainties in the variables to show how these mechanisms may affect the annual interruption costs.
Models and methods
The time varying failure rate is represented by average cyclic variations based on observations of all types of failures, i.e., failures caused by climatical, technical and other causes (such ashuman). A description of these accumulated effects registered in the failure statistics is primarily suitable for the determination of expected variations in the long run. This makes use of the total number of failures observed for different types of components.
Both analytical methods and a Monte Carlo simulation method are developed using the same basic representation of time variation. The methods start with a list of outage events which may lead to interruptions at the delivery point. The annual interruption costs are thus found by summation of the contributions from the individual outage events. It is assumed that these outage events are predetermined by appropriate methods for load flow and contingency analyses. This approach allows the reliability assessment to be decoupled from timeconsuming load flow analyses, and thereby simplifies the process of determining the annual interruption costs.
The uncertainties in input variables can be handled either by a Monte Carlo simulation giving probability distributions and confidence intervals for the reliability indices or by a fuzzy description giving the degree of fuzziness in the indices, represented by fuzzy memberships and intervals at a level of confidence. Both representations give valuable additional information.
Practical applications and case studies
The methods developed in this thesis are designed for practical applications in radial and meshed systems, based on available data from failure statistics, load registrations and customer surveys. The models and methods are illustrated for case studies ranging from simple examples to real cases from the transmission and distribution system.
One of the by-products from the methods is the calculation of traditional reliability indices, such as annual interruption time, and power- and energy not supplied. Depending on the method, all indices include the time variations and uncertainties in the input variables.
The methods are presented as algorithms and/or procedures which are available as prototypes. The algorithms can be implemented in existing tools for reliability assessment with the necessary extensions of models and data bases needed for different purposes.
Significance of time variation and correlation
The case studies show that the time dependent correlation may be significant for certain combinations of input variables. The correlation is particularly significant on a weekly and daily basis. Based on the failure statistics for distribution system, there is a strong positive correlation between the number of failures and cost per interruption, shown by a correlation factor about 0.8 both on weekly and daily basis for the industrial sector. This is counteracted by a negative correlation between number of failures and repair time. The correlation factors are -0.6 and -0.7 on a weekly and daily basis respectively. In these examples the resulting correlation from the four input variables is not found significant for the annual interruption costs. Compared to the traditional method, the annual costs are reduced by 0 - 5 % while the energy not supplied is increased by about 10 %.
In radial systems each failure leads to an interruption, while in meshed systems interruptions occur only when the available capacity to supply the load is significantly decreased. This happens for a limited number of failures or outage events. If the probability of failures is high in periods when the load is high, the time varying failure rate may have a significant impact on annual reliability indices in transmission systems. This is illustrated for the transmission system case using two different relative time variations, the first with no characteristic pattern and the second with a strong positive correlation between number of failures and load. The correlation factors are about 0.5 and 0.8 on a weekly and daily basis respectively. In the last case energy not supplied is increased by 44 % and annual interruption costs by 24 % compared to the first case.
These conclusions are based on limited data and more studies are needed for radial and meshed systems to investigate the influence of the time variation on the annual indices. The methods developed in this work can be used to study this influence as well as the combined effect of time variations and uncertainties. A description which incorporates uncertainties in input variables will in most cases not influence the expectation values, but primarily give additional information. However, there are exceptions. The specific cost is a function of interruption duration. If this function is significantly nonlinear, the expected annual costs may be influenced. An example is included using two different probability distributions for the repair time: An exponential with variance 6.0 and a lognormal with variance 1.5. With a nonlinear cost function the exponential distribution gives about 6 % higher expected annual costs than the lognormal. The two distributions yield equal expected costs when the cost function is linear.
Application of specific interruption cost
Careful modelling of the data basis is necessary in the assessment of annual interruption costs. This work has shown that the application of a normalized interruption cost at a reference time may lead to significant underestimation of annual costs, i.e., when the normalization factor is energy not supplied. The absolute cost per interruption is divided by the energy not supplied providing the specific (normalized) cost. Thus, the time variation in the specific cost depends on the time variation in both the cost per interruption and the load. This yields for instance average specific costs on an annual basis which are 20 % and 57 % higher than the reference cost for an industrial load and a commercial load respectively. If a detailed time variation in the variables is not represented, the annual average cost function should be applied. Using the specific cost at reference time leads to an underestimation of the annual interruption costs of about 20-30 % in the transmission and distribution cases compared to not considering the time variation.

##### Has parts

Appendix 1: Kjølle, Gerd Hovin; Sand, Kjell. RELRAD - an analytical approach for distribution system reliability assessment in Proceedings of the 1991 IEEE Power Engineering Society Transmission and Distribution Conference - Is not included due to copyright avaialable at https:/doi.org/10.1109/TDC.1991.169586Appendix 2: Estimation of covariance and correlation

Appendix 3: Expectation method for radial systems

Appendix 4: Kjølle, Gerd H.; Holen, Arne T. Delivery point interruption costs: Probabilistic modelling, time correlations and calculation methods PSCC 1996, Dresden, Germany

Appendix 5: Cost description: Customer costs and utility costs

Appendix 6 Data and results for Chapter 7

Appendix 7 Data and results for Chapter 8