# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 1 Introduction  
Society is dependent on a reliable supply of electricity for its normal operation, and thus power outages can have severe consequences. Major blackouts are often due to multiple component outages, caused by environmental factors [1, 2, 3, 4, 5], and/or dependent failures such as protection system failures [6]. The increased probability of failure of one or more components in a short period of time due to extreme weather has been termed failure bunching, and models have been developed to capture such effects in power system reliability studies by using both analytical methods and Monte Carlo simulation methods [7, 8, 9, 10]. Large blackouts are often a consequence of complex series of events, such as cascading failures [11]. A distinction in the structure of cascading blackouts is made in [12] between a triggering event, which can be simple component failure(s), and generation of propagating events, caused by preceding events and a change in the power system state.  
One of the causes of propagating events is the misoperation of protection systems [4, 11, 13]. Overlapping outages of highly reliable components, whose probability is the product of the individual component outage probabilities, can be rather small even if their individual probabilities are relatively high. Dependent failures, such as protection system failures, can however lead to the same set of simultaneous component outages with a much higher probability thanwhat the simultaneous independent outages might suggest [14]. This will in turn increase the risk of the scenario, as a function of its probability and consequence [15]. Previous studies have shown how protection system failures can have a significant impact on system reliability evaluation [16; 17]. Combining the effects of failure bunching with protection system failures may give more realistic picture of the power system reliability assessment.  
The hypothesis in this paper is that the combination of failure bunching effects and protection system failures will increase the risk of blackout events. To the best of our knowledge, this hypothesis was not investigated before in a quantifiable manner with respect to system reliability. The goal of this paper is to carry out investigations to verify the proposed hypothesis. The unique contributions made in this paper are as follows: A generalized compact system of equations using graph theory has been formulated to quantify the impact of protection system failures on power system reliability, based on [17]. Graph theory helps establish a systematic identification of propagation of transmission line failures due to misoperation of protection systems. These equations are then used to incorporate the effects of failure bunching due to harsh weather on the system reliability. Subsequently, these equations are applied to time-series of failure probability of transmission lines specifically due to wind conditions. It should be noted that the procedure is not only applicable to wind-dependent threats but can be used with any time-varying threats, e.g., due to other weather conditions.  
The paper is structured as follows: In Section 2, previous work on failure bunching effects and protection system reliability is presented, together with a short review of graph theory and its use in this study of power systems reliability. In Section 3, a general and compact method for calculating reliability indices including protection system misoperations is presented, using both approximate methods and a time-series method. Section 4 presents a case study where the method is applied, before the paper is concluded in Section 5. This paper is an extension of the work presented in [18], incorporating an extended literature review, updated equations, calculation of unavailability for the time-series method, additional illustrations as well as an extended case study and results.