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Introduction to Reliable Computing

Many power-engineering computations, especially those related to system analysis, are affected by large and complex uncertainties. These uncertainties make it difficult to compute the "exact" problem solution, and the analyst is required to identify a proper approximated solution, which is as close as possible to the "exact" one. The difference between the "exact" and the approximated solution is commonly referred to as the solution "error."

The sources of uncertainties affecting power system analysis are multiple and heterogeneous, and can be both external and internal to the computing process. External uncertainties include measurement errors, missing data, and simplified models; while internal uncertainties are mainly related to the computing errors induced by digital processing, which frequently requires replacing rigorous mathematical models with discrete approximations (e.g. time discretization, truncation, and round-off errors).

Hence, when integrating the results of numerical analysis in power system operation tools, the impact of these uncertainties must be assessed and a formal error analysis should be considered an important part of the development process. The objective of such formal error analysis is to comprehensively assess the accuracy of all the computations involved, define the magnitude of the solution errors as a function of the values and the errors of the input data (i.e. variables and parameters).

Unfortunately, defining a formal mathematical process for specifying the accuracy of numerical computing algorithms is extremely complex, since estimating the propagation of both the external and internal errors for all the basic operations composing the computing process is often unfeasible, even for simple algorithms.

Moreover, accuracy specification requires the compliance of the input data with a set of strict prerequisites (e.g. well-conditioned matrices, no overflow, and functions with bounded derivatives). Guaranteeing or even checking these prerequisites for all the possible combinations of the input data represents another challenging issue to address.

Hence, power system analysts frequently deploy numerical algorithms without formal accuracy specifications and rigorous error analysis, checking the consistency of the obtained results on the basis of their own experience or by crude tests. This practice could hinder the integration of approximate numerical computing in modern power systems tools, which are characterized by the presence of large data uncertainties, stemming from multiple and heterogeneous sources.

To try and overcome this limitation, the power system research community started adopting reliable computing-based models, which allow the accuracy of the computed quantities to be automatically estimated as part of the process of computing them. The application of these models in numerical computations is also referred to as self-validated computing, since it can estimate "a posteriori" the error magnitude of the entire computing process (Stolfi and De Figueiredo, 1997). This feature is extremely important, especially when the data uncertainties are induced by external causes. In this case, if the output errors computed by the self-validated model become too large, i.e. overcoming a fixed acceptable threshold, then specific remedial actions can be automatically triggered in order to enhance the model accuracy (e.g. acquire more data, re-adjourn the input parameters, and use more accurate models).

1.1 Elements of Reliable Computing

Let be a continuous mathematical function, and suppose we need to compute for . For this, we should implement a discrete numerical computation , where and are discrete mathematical objects approximating the corresponding continuous variables and .

To solve this issue different reliable computing models can be adopted, including probability distributions and range-based methods.

In particular, the adoption of probability distributions can approximate, in a statistical sense, the computed result by considering each component of the vector as a real-value random variable, whose probability distribution function is frequently assumed to follow a Gaussian distribution. In this case, a reliable computing model should specify the statistical moments of each component , and the corresponding covariance matrix describing the joint Gaussian probability distribution of the random vector , given those characterizing the random input variables .

The application of this probabilistic-based reliable computing model is often limited to specific application domains, which are characterized by Gaussian uncertainties, linear mappings, and negligible truncation errors. The lack of these conditions makes the statistical characterization of the computed result extremely complex or even mathematically intractable (Stolfi and De Figueiredo, 1997).

To try and overcome this limitation, most reliable computing models approximate the computed results by ranges, rather than by probability distributions.

According to these models, the approximated solution is described by means of its range , which is a compact set containing the "exact" solutions for all the input variables lying in the range .

This important feature, which is usually referred to as the fundamental invariant of range analysis, guarantees that the range contains the true solution set, provided that the input variables vary in a fixed range.

The simplest model of range analysis is Interval Arithmetic (Moore, 1966), which defines the range of each component of the computed result by a real interval, which is a set of real numbers lying between its upper and lower bounds. Since no constraints relating these intervals are assumed, meaning that all the uncertainties are assumed to be statistically independent, the range of the computed result is the Cartesian product of the ranges of its components , since all combinations of in the box are allowed.

In more advanced reliable computing-based models, such as those based on affine arithmetic (AA), the computed result also integrates useful information about the partial correlations between the output vector components ; hence, identifying the statistical dependencies between the input variables and the computed result. In this case, the range of the computed result is a proper subset of the Cartesian product of the individual ranges.

One of the most important features which characterizes all range-based models is their capability of computing, for every function , a range extension , which is characterized by the fundamental invariant of range analysis:

• If the input vector lies in the range jointly determined by the given approximate values , then the quantities are guaranteed to lie in the range jointly defined by the approximate values .

This property is extremely useful in reliable computing, since the joint range determined by the approximated values is an outer estimation of the real solution set, namely:

hence, introducing a conservative factor in approximating the solution vectors. This conservativism is a valuable feature of range analysis-based methods, which is extremely important in reliable power system analysis, since it allows the Analyst to bound all internal and external uncertainties in numerical computing. However, obtaining a suitable (not too large) conservativism level is a relevant problem in range-based computation, since the naive application of range-based models often results in extremely conservative solution ranges, which are too wide, and hence, not useful in realistic application domains.

Therefore, in order to assess and compare the conservativism of range-based reliable computing models, proper metrics should be defined. For this purpose, we introduce the relative accuracy of the range-based approximation , which is defined as:

where and are the norms of the real solution range and the computed range , respectively.

This index is a reliable measure of the conservativism introduced by range-based computing, indicating the outer estimation accuracy by a number, which could vary between zero (i.e. the computed range is much wider than the real one) and one (i.e. no conservativism is introduced in range computing).

Obviously, this index requires that the input vector should vary in the range jointly determined by the given approximate values , as stated by the fundamental invariant of range analysis. This hypothesis cannot be verified in real application domains, where some of the input data are acquired by measurements, which can be affected by unbounded errors. In this context, the assumption related to the inclusion of the output variables in the range jointly determined by the approximate values should be revised by introducing probabilistic information about the range of the possible values . This requires determining, for the computed range [Z], the corresponding confidence range , which is the probability of to be included in [Z]. This approach differs from the Gaussian-based probabilistic models as it does not assume any hypothesis about the probability distribution of the error inside the range , but only that its integral over the computed range is at least .

Hence, the application of range-based methods in this context requires computing, for each mathematical operation, not only the joint range, but also a conservative...