#### Reliability Evaluation by Non-sequential MCS

The calculation of the reliability indices in nonsequential MCS can be summarized by evaluating Eq. (1):

where N is the number of simulated states, F is the test function for calculating the indices for each system state xi and EðFÞ is the estimate of the annual reliability indices.

The system states are obtained by combining the states of all system components. For a system with m components, a system state can be represented by the random vector x = [x1, x2, …, xk,…,xm] where xk represents the state of the kth component.

Sampling the component state is done by applying the Inverse Transformation (IT) method to the cumulative distribution function (CDF) of the component operating states. Considering a component modeled by n states, this is done by generating a random number Uk uniformly distributed between [0, 1] and then identifying its state xk by (2), where Pi = P(xk B i):

The models usually used by nonsequential MCS assume that the states of the components are statistically independent and therefore, the probability of occurrence of each system state is given by the product of the probability of each component state. Defining P(xk) as the probability of occurrence associated with the kth component, the probability of the ith system state P(xi) is calculated by (3):

However, as will be discussed in the next section, this approach is incorrect when there are statistically dependent components, as is the case of multiple correlated wind generations, for example.