Brief Overview of Applications of Importance Sampling in the

in the Existing Literature

The previous section demonstrated how there is a great deal of variation of constraint costs in each simulator evaluation for a given power system background, both on an annual and snapshot basis. Section 3.4.3 stated that an average of many annual simu- lator evaluations is used to acquire an estimate of mean annual constraint costs. This is to remain consistent with National Grid methodology of working with mean constraint costs. However, acquiring an estimate of annual constraint costs which evaluates all 17520 snapshots of the year is quite expensive. In turn, this means taking an average of many such annual evaluations is very expensive.

Figure 4.3 shows how the mean constraint costs of a snapshot can vary greatly with snapshot, and suggests that when estimating mean constraint costs for a year, certain snapshots contain more relevant information to the estimate of mean annual constraint costs than others. Therefore, when aiming to acquire an accurate estimate of mean annual constraint costs it may be more important to have accurate estimates of costs in certain snapshots more than others. This is the fundamental principle behind im- portance sampling, where a careful selection of sampling density can reduce the work necessary to achieve a given level of accuracy in an estimate [29, 6, 31, 77].

In order to detail how importance sampling is commonly applied in the existing lit- erature, consider a problem where the expected value of a function, f (x), of a set of discrete random variables, x, is of interest. The expectation can be calculated as

µ =X

x

f (x)p(x) (4.2.1)

where p(x) is the probability density function of x.

However, if the function f (x) is expensive to evaluate it will not be feasible to calculate

µ exactly via Equation 4.2.1. Therefore, Monte Carlo simulation is commonly used to

approximate Equation 4.2.1 by randomly drawing values of x from the distribution of

4.2. Importance Sampling 65

distribution p(x) such that xi is the ith value sampled, an estimate of µ via Monte

Carlo simulation can be acquired as ˆ µp = 1 n n X i=1 f (xi) (4.2.2)

However, certain values of x may be more relevant to the estimate of µ than others. This is particularly true when estimating the probability of a rare event occurring (i.e.

µ is the probability of a rare event occurring with f (x) acting as an indicator function,

taking the value 1 if the rare event occurs for a particular value of x and 0 otherwise) as is considered in the examples of [9, 36, 104, 96]. In such examples, sampling values of x from the distribution p(x) will be very inefficient, as the rare event will not be observed for the majority of values of x. In particular, [96] notes that if µ corresponds

to a rare event with a probability of 10−6 of occurring it would require a Monte Carlo

sample size of the order of hundreds of millions to acquire an estimate of µ with an error of 10%.

Therefore, importance sampling can be applied by instead sampling values of xi from

some alternative distribution q(x), where q(x) gives more weight to values of x more relevant to the estimate of µ. If n random drawings of x are sampled from the distri-

bution q(x) such that xi is the ith value sampled, an estimate of µ using importance

sampling can be calculated as [77, 36, 16, 96] ˆ µq = 1 n n X i=1 f (xi) p(xi) q(xi) (4.2.3) It can be shown that such a method has the same expectation as the estimate from Monte Carlo simulation as

E(ˆµq) = Eq(x) f (x)p(x) q(x) ! =X x f (x)p(x) q(x) q(x) = X x f (x)p(x) = Ep(x)(f (x)) = E(ˆµp) (4.2.4) When applying importance sampling to estimate the probability of a rare event it is desirable that the alternative density, q(x), results in an increased probability of the rare event occurring [104, 16, 82], with there being several potential methods of achieving this such as exponential twisting [9, 36] or iterative schemes such as the cross- entropy method [104, 96] or adaptive importance sampling [47]. An iterative scheme suitable for estimating mean annual constraint costs (as considered in this thesis) via

importance sampling will be proposed in Section 4.4.2.

There are a wide variety of applications of importance sampling to various examples in the existing literature. For example, [47] consider an application to fishery stock assessment, where importance sampling is used to estimate the biomass stock of the orange roughy (a type of fish), as well as to estimate the probability that the stock exceeds a particular value. Further, these estimates are then used to make policy decisions for the fisheries, with consideration also given to how the policy maker’s attitude to risk affects the decision made.

The safety of lane change events of automated vehicles is considered in [104], where im- portance sampling is used to estimate the probabilities of conflict, crashes and injuries during the lane change events. A comparison is also given to show the benefit of using importance sampling over Monte Carlo sampling without importance sampling. An application related to power systems is presented in [96], where importance sampling is used to estimate the loss of load probability (the probability that a power system will not be able to satisfy all demand) and expected power not served (the expected demand, in MW, that will not be satisfied) for a variety of power system backgrounds.

Many other applications of importance sampling to a wide variety of topics exist, in- cluding estimating the probability of failure of the Data Communication System (DCS) of large scale passenger rail systems [82]; predicting the yield of an integration circuit (i.e. the probability of failure of a complex circuit) under variability due to the man- ufacturing process [16]; biological/medical applications (such as testing for distinctive charge clusters in proteins or identifying cancer indicators in mammograms) [68] and path tracing using the photon map [43].

As importance sampling is not the main focus of this thesis, but rather a tool used within the simulation process to estimate mean annual constraint costs, this section is not intended to be taken as a full literature review, but rather a brief overview of applications of importance sampling in the existing literature.

4.2. Importance Sampling 67