Probabilistic Transient Stability Assessment and On-Line Bayes Estimation

Abstract

It is a well-known fact that the increase in energy demand and the advent of the deregulated market mean that system stability limits must be considered in modern power systems reliability analysis. In this chapter, a general analytical method for the probabilistic evaluation of power system transient stability is discussed, and some of the basic contributes available in the relevant literature and previous results of the authors are reviewed. The first part of the chapter is devoted to a review of the basic methods for defining transient stability probability in terms of appropriate random variables (RVs) (e.g. system load, fault clearing time and critical clearing time) and analytical or numerical calculation. It also shows that ignoring uncertainty in the above parameters may lead to a serious underestimation of instability probability (IP). A Bayesian statistical inference approach is then proposed for probabilistic transient stability assessment; in particular, both point and interval estimation of the transient IP of a given system is discussed. The need for estimation is based on the observation that the parameters affecting transient stability probability (e.g. mean value and variances of the above RVs) are not generally known but have to be estimated. Resorting to “dynamic” Bayes estimation is based upon the availability of well-established system models for the description of load evolution in time. In the second part, the new aspect of on-line statistical estimation of transient IP is investigated in order to predict transient stability based on a typical dynamic linear model for the stochastic evolution of the system load. Then, a new Bayesian approach is proposed in order to perform this estimation: such an approach seems to be very appropriate for on-line dynamic security assessment, which is illustrated in the last part of this article, based on recursive Bayes estimation or Kalman filtering. Reported numerical application confirms that the proposed estimation technique constitutes a very fast and efficient method for “tracking” the transient stability versus time. In particular, the high relative efficiency of this method compared with traditional maximum likelihood estimation is confirmed by means of a large series of numerical simulations performed assuming typical system parameter values. The above results could be very important in a modern liberalized market in which fast and large variations are expected to have a significant effect on transient stability probability. Finally, some results on the robustness of the estimation procedure are also briefly discussed in order to demonstrate that the methodology efficiency holds irrespective of the basic probabilistic assumptions made for the system parameter distributions.

The singular and plural of names are always spelled the same; boldface characters are used for vectors; random variables (RVs) are denoted by uppercase letters.

Appendices

Appendix 1: An Analytical Study of the IP

Under the assumed hypothesis of LN pdf for both the CCT and the FCT, it was shown that the IP, q = P(CCT < FCT) = P(T _x  < T _y ), can be expressed by:

$$ q = \Uppsi (u) = \int\limits_{u}^{\infty } {{\frac{1}{{\sqrt {2\pi } }}}} \exp \left( { - {\frac{{\xi^{2} }}{2}}} \right){\text{d}}\xi $$

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where u is the “SM”:

$$ u = {\frac{{\alpha_{x} - \alpha_{y} }}{{\sqrt {\beta_{x}^{2} + \beta_{y}^{2} } }}} = {\frac{E(X) - E(Y)}{{\sqrt {V(X) + V(Y)} }}} $$

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being X = ln(T _x ); Y = ln(T _y ), and denoting by V(X) and V(Y) their variances.

The function q = q(u) is shown in Fig. 4. Since q(u) decreases very quickly towards 0, especially when u is large enough, two different curves are shown: one (left curve) is relevant to the interval (0 < u < 2.5), the other (right curve) relevant to the interval (2.5 < u < 5): the latter is the one which often occurs in practice since, in this interval, the IP typically assumes realistic small values, less than 6e−3. To appreciate the quickness with which q(u) decreases, the following values are given as examples:

Fig. 4

A curve of the IP as a function q = q(u), u being the SM

• q(2.0) = 2.28e−2;

• q(2.5) = 6.20e−3;

• q(3.0) = 1.30e−3;

• q(5.0) = 2.85e−7

In order to appreciate the variation of the IP as a function of the SM u, the following well-known asymptotic approximation of the Ψ function which may be found in many books (e.g. [36]) is given:

$$ \Uppsi (u) \approx {\frac{1}{{u\sqrt {2\pi } }}}\exp \left( { - {\frac{{u^{2} }}{2}}} \right) = {\frac{\phi (u)}{u}},\quad {\text{for }}u{\text{ large enough}} $$

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with ϕ(u) being the standard Gaussian pdf. In practice, the above approximation is satisfactory for u ≥ 3 (e.g. it yields 0.0015 for u = 3, with a relative error of less than 0.8%). From the above relation, it is readily shown (as discussed in the following) that the relative variation of q with u is in practice linear in u for typical values of u: therefore, the larger—as desired—the SM is, the more abrupt the variation (decrease) in the IP value. Indeed, a curve of the relative variation in the IP versus the argument u is shown in Fig. 5. It is apparent that such a function, which is of course negative (and decreasing), is approximately linear with u, especially for high u values.

Fig. 5

A curve of the relative variation of the IP as a function of the SM

Indeed, the relative variation of the function Q(u) may be analysed using the derivative of its logarithm since:

$$ {\frac{{{\text{d}}Q}}{Q}} = \left[ {{\frac{{Q^{\prime}(u)}}{Q}}} \right]{\text{d}}u = D[\ln Q(u)]{\text{d}}u. $$

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For what has been discussed above, the function: K(u) = D[ln(Qu)] tends to approach the value (−u) if u → ∞. However, K(u) is readily expressed for any finite value of u by means of available statistical functions, since:

$$ K(u) = D[\ln \Uppsi (u)] = - {\frac{\phi (u)}{\Uppsi (u) \, }} $$

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So, the availability of the standard Gaussian pdf and cdf (e.g. the functions “normpdf” and “normcdf” in MATLAB) provides an easy computations of K(u)¹⁰ and the possibility of drawing graphs such as the one in Fig. 5.

Moreover, by virtue of the above asymptotic approximation, we get the above-mentioned linear approximation of K(u), which is also confirmed by Fig. 5:

$$ K\left( u \right) \to - u,\quad {\text{as }}u \to + \infty $$

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The above relationship between the IP and the SM u can be readily expressed (see also [13]) as a function of the basic statistical parameters of the CCT—or the ones of the load on which the CCT depends—and the clearing time.

This allows a rapid sensitivity analysis of the IP for these parameters. For instance, using the already relations between the LN parameters and the mean value and the CV of the LN distributions given above, the dependence of q on μ and v (the CV value) of both the FCT and CCT is straightforward. The following curves are obtained assuming, for illustrative purposes only, a common value v of the CV, i.e. from the expression:

$$ q = \Uppsi \left( u \right);\quad u = {\frac{{\ln \mu_{x} - \ln \mu_{y} }}{{\sqrt {2\ln (1 + v^{2} )} }}} $$

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The curves in Fig. 6 describe the variation of q (in %) as a function of the mean FCT, for a fixed value of the mean CCT, chosen equal to 0.1 s. as in the numerical examples of the chapter and with 2 different values of the (common) value of the CV, i.e. v = 0.10 and v = 0.12.

Fig. 6

Curves of the IP (in %) versus mean FCT (in s) with mean CCT = 0.1 s. Each curve refers to a given (common) value of the CV for FCT and CCT, namely CV = 0.10 (below) and 0.12 (above)

These curves illustrate the high or extreme variability of the IP versus the mean FCT and also the CV. This last aspect is confirmed by the curve depicted in Fig. 7 which expresses the IP—on a logarithm scale—versus the CV, assuming mean CCT value = 0.1 s and mean FCT value = 0.145 s. All the above aspects are very important in view of the estimation process, and this is why they have been illustrated in detail.

Fig. 7

Curves of the IP (in %)—on a logarithm scale—versus the (supposed common) value of the CV of FCT and CCT, assuming a mean CCT value of 0.1 s and a mean FCT value of 0.145 s

Appendix 2: Bayes Point Estimation for the Gaussian Model

2.1 Known Variance

For the purposes of making inference in the application in this chapter, some known results [22–24] on Bayes point estimation for the Gaussian model have been applied. Indeed, the Log-Normal model assumed for both FCT and CCT can be easily converted in the Gaussian one by means of a logarithmic transformation. Some results which are specific to the Log-Normal model are given, for example, in [21].

Let us assume that X = (X ₁,…, X _n ) is a random sample of n elements generated by a Gaussian model with the same mean μ, and SD σ; let μ be an unknown to be estimated whereas σ is known. Therefore, for each k = 1,…, n, the conditional pdf of X _k , for a given value of μ is a N(μ, σ) pdf. Formally X _k |μ ~ N(μ,σ).

Let the prior information about the unknown parameter μ be described by a prior Normal distribution with known parameters (μ₀, σ₀), i.e. μ ~ N(μ₀, σ₀),¹¹ so that the prior pdf is:

$$ g(\mu ) = {\frac{1}{{\sigma_{0} \sqrt {2\pi } }}}\exp \left[ { - {\frac{{(\mu - \mu_{0} )^{2} }}{{2\sigma_{0}^{2} }}}} \right],\quad \mu \in \Re \, $$

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The prior parameters (μ₀, σ₀) are also denoted “hyper-parameters” and are assumed to be known. Before observing data, the “best” estimator of μ cannot be that its prior mean: E[μ] = μ₀, with prior variance:

$$ {\text{Var}}[\mu ] = \sigma_{0}^{2} + {\frac{{\sigma^{2} }}{n}} $$

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The LF of the observed sample X = (X ₁,…, X _n ), conditional to μ, is expressed by:

$$ L\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} |\mu } \right) = \left( {2\pi \sigma^{2} } \right)^{{{\frac{ - n}{2}}}} {\text{e}}^{{ - \sum {\left( {x_{i} - \mu } \right)}^{2} /2\sigma^{2} }} $$

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Then, multiplying the above two functions for applying the Bayes theorem, after some algebra, the following well-known result is obtained for μ°, the Bayes estimator of μ, i.e. the posterior mean:

$$ \mu^{ \circ } = E[\mu |X] = {\frac{{\sigma^{2} \mu_{0} + n\sigma_{0}^{2} M_{n} }}{{\sigma^{2} + n\mu_{0}^{2} }}} = {\frac{{{\frac{{\sigma^{2} }}{n}}\mu_{0} + \sigma_{0}^{2} M_{n} }}{{{\frac{{\sigma^{2} }}{n}} + \sigma_{0}^{2} }}} $$

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where M _n is the sample mean (which is equal in this case to the classical ML estimator of μ):

$$ M_{n} = (1/N)\sum\limits_{k = 1}^{N} {X_{k} } $$

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In the final equation expressing μ°, the prior variance (σ ²₀ ) and the one of Mn (σ ²/n) are clearly indicated. In this form, the above relationship shows the known property that the Bayesian estimator of μ can be, in a suggestive way, expressed as the weighted mean (a linear convex combination, in fact) of the prior estimator and the sample mean. The posterior variance is given by:

$$ {\text{Var}}\left[ {\mu |X} \right] = {\frac{{\sigma_{0}^{2} \sigma^{2} }}{{n\sigma_{0}^{2} + \sigma^{2} }}} = {\frac{{{\frac{{\sigma^{2} }}{n}}\sigma_{0}^{2} }}{{{\frac{{\sigma^{2} }}{n}} + \sigma_{0}^{2} }}} $$

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2.2 Unknown Variance

Although unknown variance is not considered in the application of this chapter, it seems opportune to mention it, even if very briefly.

Let us first consider known mean, μ = m. The conjugate prior pdf for the variance, here denoted by V, is the so-called “Inverted Gamma” model, characterized by the following pdf, with argument v (a realization, of course positive, of the RV V) and (positive) parameters r and ϕ:

$$ g(v;r,\phi ) = {\frac{{\phi^{r} }}{{v^{(r + 1)} \Upgamma (r)}}}\exp ( - \phi /v),\quad v > 0 $$

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in which Γ(·) is the Euler–Gamma special function. The “Inverted Gamma” pdf is so denoted since it can be deduced as the pdf of the reciprocal of a Gamma RV. It is not difficult to deduce that the posterior pdf of V is again an Inverted Gamma pdf. This result derives from expressing the LF of the observed sample X = (X ₁,…, X _n ) as above—Eq. 79—but conditioning to V = ν:

$$ L\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X}|v} \right) = \left( {2\pi v} \right)^{{{\frac{ - n}{2}}}} {\text{e}}^{{ - \sum {\left( {x_{i} - m} \right)}^{2} /2v}} $$

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(here, the mean m is assumed to be a known constant whereas the variance v is the argument under investigation). By multiplying the above prior pdf and LF, it is apparent that the posterior pdf of V is again an Inverted Gamma pdf and the updated values of r and ϕ are obvious.

Then, let us also consider the general case of unknown mean μ, i.e. both mean μ and variance V are unknown. Here, the most adopted prior model for μ is again described—conditionally to the variance V—by a Gaussian prior pdf. This prior model, multiplied by the above Inverted Gamma pdf for V, constitutes the so-called “Normal Inverted Gamma” prior pdf. This is indeed the conjugate prior model, as the joint posterior pdf of μ and V is again a Normal Inverted Gamma pdf [23, 24]. A similar model can also be developed in the dynamic framework [29].

Appendix 3: A BCI for the IP Using the Beta Distribution

In order to establish a BCI, a numerical procedure derived from a similar one, proposed in [27] and already proved satisfactory by the authors in [16], is illustrated. In [16], it was used in a different context (the one of classical statistic estimation) whereas here it is revised in the Bayesian framework. As discussed above, in the Bayesian approach the IP Q is an RV in (0, 1), depending on the four random parameters (α _x ,α _y ,β _x ,β _y ). The need for a numerical procedure is based on the fact that an analytical expression of such a pdf is impossible to find. A reasonable choice for its characterization is the approximation of its true pdf with a suitable distribution such as the Beta which is very flexible for describing RV in (0, 1) and is capable of producing a large variety of shapes. The Beta is in fact the most commonly used distribution for describing random probabilities because it is also a conjugate pdf under a Binomial sampling [22–24, 36]. The analytical expression of the Beta pdf, as a function of the values q assumed by the RV Q in (0, 1), is [19, 36, 37]:

$$ f\left( {q;\omega ,\xi } \right) = \left( {\begin{array}{*{20}c} {\Upgamma (\omega + \xi ){\frac{{q^{\omega - 1} (1 - q)^{\xi - 1} }}{\Upgamma (\omega )\Upgamma (\xi )}}} & {(0 < q < 1)} \\ 0 & {\text{elsewhere}} \\ \end{array} } \right. $$

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where Γ(·) is the already introduced Gamma special function, and ω and ξ are positive shape parameters. Mean value and variance of the Beta distribution are given by:

$$ \mu_{\text{B}} = {\frac{\omega }{\omega + \xi }};\quad \sigma_{\text{B}}^{2} = \mu_{\text{B}}^{2} \left\{ {{\frac{\xi }{\omega (\omega + \xi + 1)}}} \right\} $$

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In order to choose the approximating Beta pdf for Q, an adequate choice of the two parameters (ω, ξ) must be made, for instance—as proposed in [27]—by equating the above Beta statistical parameters (μ _B, σ ²_B ) to opportune (as explained in the following) values of the mean value M and variance V, thus obtaining the following equations which give the Beta parameters as functions of the mean M and variance:

$$ \omega = M{\frac{{\left( {M - M^{2} - V} \right)}}{V}};\quad \xi = {\frac{\omega (1 - M)}{M}} $$

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The above mean value M and variance V of the RV Q cannot be of course obtained from its (unknown) distribution, but an excellent approximation for them is obtained: the resulting approximate values are, respectively, denoted as (M′, V′). They are obtained, still following [27], by considering an expansion of Q = Ψ(U) expressed as a function, say G = G(α _x ,α _y ,β _x ,β _y ), of the four variables (α _x ,α _y ,β _x ,β _y ) in a Taylor series about the point Π₀ = (A _x ,A _y ,B _x ,B _y ), being (A _x ,A _y ,B _x ,B _y ) the a.m. ML estimators (see Sect. 4.4) of the random parameters (α _x ,α _y ,β _x ,β _y ). In particular, expanding Q in a Taylor series about Π₀ up to second-order terms, the following values (M′, V′) are obtained by the well-known “Delta method” or the “statistical differentials” method [19, 36]:

$$ M^{\prime} = \Upphi (U) - 0.5\Upphi (U)\left[ {{\frac{{A_{y} - A_{x} }}{{B^{3} }}}} \right]\chi $$

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$$ V^{\prime} = \Upphi (U)^{2} \left[ {{\frac{{B_{x}^{2} }}{{nB^{2} }}} + {\frac{{B_{y}^{2} }}{{mB^{2} }}} + 0.5\left( {A_{y} - A_{x} } \right)^{2} {\frac{{B_{x}^{4} }}{{n_{1} B^{6} }}} + 0.5\left( {A_{y} - A_{x} } \right)^{2} {\frac{{B_{y}^{4} }}{{m_{1} B^{6} }}}} \right] $$

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being: m ₁ = m − 1, n ₁ = n − 1;

$$ U = {\frac{{A_{x} - A_{y} }}{{\sqrt {B_{x}^{2} + B_{y}^{2} } }}}. $$

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$$ B = \sqrt {\left( {B_{x}^{2} + B_{y}^{2} } \right)} ; $$

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$$ \chi = \left[ {{\frac{{B_{x}^{2} }}{n}} + {\frac{{B_{y}^{2} }}{m}} + \frac{1}{2}\left( {A_{y} - A_{x} } \right)^{2} {\frac{{B_{x}^{4} }}{{n_{1} B^{4} }}} - 1.5{\frac{{B_{x}^{4} }}{{n_{1} B^{2} }}} + \frac{1}{2}\left( {A_{y} - A_{x} } \right)^{2} {\frac{{B_{y}^{4} }}{{m_{1} B^{4} }}} + 1.5{\frac{{B_{y}^{4} }}{{m_{1} B^{2} }}}} \right] $$

Φ(x) and ϕ(x), respectively, the already introduced standard Normal cdf and pdf.

Indeed, let us denote by F _B (q; ω, ξ) the generic Beta cdf of the RV Q, evaluated in q, with parameters (ω, ξ), i.e.

$$ F_{\text{B}} (q;\omega ,\xi ) = P\left( {Q < q} \right) $$

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This Beta distribution is used for inference on the BCI. For example, the estimated η-quantile of the above IP is given—still denoting by τ′ the value of a true parameter τ estimated by this procedure—by:

$$ Q^{\prime} = F_{\text{B}}^{ - 1} (\eta \cdot ;\omega^{\prime},\xi^{\prime}) $$

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i.e. by the inverse function of the above Beta cdf F _B(x; ω′, ξ′) evaluated in η, namely the solution, q*, of: η = F _B(q*; ω′, ξ′). So, any “upper confidence bound” for the IP mentioned in Sect. 4 (see Eq. 34) can be computed easily, since the Beta quantiles are largely available in most software packages (e.g. using the function “Betainv” of MATLAB). The above equation is equivalent indeed to the following:

$$ P(Q < Q^{\prime} \eta ) = \eta $$

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And thus Q’_η = F ⁻¹_B (η; ω′, ξ′) coincides with the upper confidence bound of probability (degree of belief) η. Of course, this procedure also allows easy estimation of the whole distribution of Q and establishes any desired confidence interval for the IP, also a bilateral one.

A simple practical numerical example is given to evaluate the BCI. In this example the FCT T _y is therefore assumed to be deterministic, with value t _y  = 0.1 s, and the LCCT mean α _x  = E[ln(T _y )] is assumed to follow a prior Gaussian distribution with a mean value equal to 0.145 s and an SD equal to 1% of the mean value (i.e. a CV value equal to 0.01). The values (0.10 s, 0.145 s) of the CCT and of the mean FCT, respectively, are typical values, equal to those used in the computations already performed in the VST application of the present chapter. A CV value equal to 0.01 for the mean FCT may be also a reasonable value for describing uncertainty in such kind of on-line applications.¹² The following values of parameters (α _x ,α _y ,β _x ,β _y ) correspond to the above CCT and FCT values: α _y  = ln(0.1) = −2.3026, β _y  = 0, β _x  = 0.0998 whereas α _x is an RV with the above pdf.¹³

For illustrative purposes, a simulated sample of N = 10⁴ values of M = α _x was generated and the corresponding empirical pdf of the IP has been evaluated and compared with the approximated theoretical Beta pdf obtained as mentioned above. The goodness of fit of this Beta pdf to the random sample has been validated through the Kolmogorov–Smirnov test of hypothesis [28]. For graphical evidence, this is also confirmed by histograms such as the one in Fig. 8 in which the frequency histograms of the sampled IP values (measured in per cent) and the corresponding hypothetical frequency distribution obtained by the a.m. Beta pdf are superimposed. Also the “Q–Q plots” [28] confirmed this adequacy, as also shown in [16].

Fig. 8

Frequency histograms of the sampled IP values (in %) and the corresponding hypothetical frequency distribution obtained by the theoretical approximating Beta pdf

To be more specific, the values M′ and V′ of the above mean and variance approximations resulted equal, for such an example, to: M′ = 0.0155%; V′ = (0.0155)². A value of 0.0155 is therefore obtained for the SD S′ = √V, with a corresponding CV equal to 0.8326, much higher than the CV = 0.1 of the basic RV M, thus confirming the already discussed high variability of the IP. This is confirmed by the values assumed by the 5th and 95th percentiles of the IP sample, i.e. 0.0032 and 0.0399, respectively, with an increase of 1147% from the former to the latter.

The Beta pdf corresponding to values of M = M′ and V = V′, which is shown in Fig. 8, has the following values of the two parameters (ω, ξ), obtained from (M ₀, V′) as described above: ω = ω′ = 1.4047, ξ = ξ′ = 89.2197. This computation closes the procedure of finding a BCI.

For instance, the 0.95 upper confidence bound of the above IP is given by:

$$ Q_{0.95} = F_{\text{B}}^{ - 1} (0.95;\omega^{\prime},\xi^{\prime}) = 0.041\% $$

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which is very close to the sample value above reported (0.0399), with a relative difference of less than 3%. So, under this approximation:

$$ P\left( {Q < 0.041\% } \right) = 0.95 $$

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In other words, we can be confident, with a subjective probability equal to 0.95, that the IP is less than 0.041%. As apparent, and as already discussed in  Sect. 4, this information has a greater meaning than a simple point estimate, and is consistent with the establishment of possible standards.

Of course, approximately the same values of the statistical parameters of the IP and of its BCI could be obtained by performing a Monte Carlo simulation instead of computing a Beta cdf but the procedure should be repeated—in the dynamical framework here focussed on—at each time step, which is at least tedious if not time-consuming. A much more important advantage of the Beta approximation over Monte Carlo simulation is that the former allows an analytical sensitivity analysis which is cumbersome when done by simulation.

Of course, other pdf approximations can be devised for the above purposes. In our studies, an LN approximation also seemed to be adequate for describing IP randomness. However, the Beta pdf has the advantage of being theoretically limited to the interval (0, 1). Another possible adequate model in this interval is the “Negative Log-Gamma” Distribution; it was introduced in [37] and already discussed and satisfactorily adopted by the authors in studies on uncertainty characterization in reliability analyses [38], and is worth being studied also in the present context.

Appendix 4: Recursive Application of Bayes Estimation

Recursive Bayes estimation is based on repeated application of the Bayes theorem which shows the coherence of the updating process and its adequacy for a “dynamical” estimation, i.e. an estimation procedure involving stochastic processes. Let us perform a statistical inference for an unknown parameter θ, characterized by a prior pdf g(θ). Once a set of data D is observed, let the posterior pdf be g(θ|D):

$$ g(\theta |D) = g(\theta )L(D|\theta )/P(D) $$

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In view of dynamical applications, we can imagine acquiring data D in a two-stage process so that D consists of two sets of data—denoted by D ₁ and D ₂—observed in succession. Then, by repeatedly using the Bayes theorem and the “chain rule” for joint probabilities, the above posterior pdf for θ may be alternatively obtained by expanding the previous equation in the following “two stage” process:

$$ \begin{gathered} g(\theta |D_{1} \cap D_{2} ) = g(\theta )L(D_{1} \cap D_{2} |\theta )/P(D_{1} \cap D_{2} ) = g(\theta \left| {D_{1} )L(D_{2} } \right|\theta \cap D_{1} )/P\left( {D_{2} |D_{1} } \right) \hfill \\ = g_{1} (\theta )L_{1} (D_{2} |\theta )/P_{1} \left( {D_{2} } \right) \hfill \\ \end{gathered} $$

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having denoted with the suffix “1” every probability (or pdf) conditional to data D ₁, i.e.

$$ g_{1} (\theta ) = g(\theta \left| {D_{1} );\quad L_{1} (D_{2} } \right|\theta ) = L(D_{2} |\theta \cap D_{1} );\quad P_{1} \left( {D_{2} } \right) = P\left( {D_{2} |D_{1} } \right) $$

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As can be seen, the posterior pdf g(θ|D ₁ ∩ D ₂) can be obtained by applying the Bayes theorem “starting” with a prior pdf g(θ|D ₁), which is the posterior pdf after observation of D ₁, then applying the same conditioning to the LF L(D ₂|θ) and the probability of data D ₂. The updating process may be indefinitely continued in this way through successive stages, transforming every posterior information gained at the end of stage k into prior information for the next stage:

$$ g(\theta |D1 \cap D2 \cdots \cap D_{k + 1} ) = \, g(\theta |D_{1} \cap D_{2} \cdots \cap D_{k} )L(D_{k + 1} |\theta \cap D_{1} \cdots \cap D_{k} )/C $$

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where C is the “constant” (with respect to θ):

$$ C = L(D_{k + 1} |D_{1} \cdots \cap D_{k} ) $$

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