Formulation of the runaway prediction problem

The problem of identifying a suitable runaway criterion can be considered as optimisation of the structure and the parameters p of the critical equation f(x(k),p) that indicates the runaway at the k-th instance of time as a binary classifier:

̂( ) ( ( ) ) (6.1) ̂( ) ( ( ) ) (6.2) where the variables of the function are the subsets of the state variables and parameters of the studied process, e.g. ( ) { ( ) ( ) ( ) ( ) }, where T(k) denotes the process temperature, Tw(k) stands for the wall temperature and qgen(k) and qrem(k) are the generated and extracted heat at the k-th instant of time, respectively.

Thanks to the utilised genetic programming algorithm the user does not have to put too much effort into the selection of the informative variables, as the most informative subset will be

selected during the optimisation procedure. This method is based on the analysis i=1,…,N operations of the reactor with k=1,…,ti operating lengths. The proposed voting system integrates nc number of different criteria. In this work, seven criteria were analysed; hence, nc is seven in the case studies. The investigated criteria are: PD, SZ, Maxi, VH, LPP, MSC and MDC (see Section 2.4). The criteria were chosen because, they are relatively commonly applied and their conservativeness is different from each other.

For each reactor states all the investigated criteria are evaluated, so the criteria are represented by a set of characteristic functions, Ic(k), as Ic(k)=1 when the c=1,…,nc-th criterion shows runaway at the k-th time instant. These criteria indications can be summarized to evaluate the given time instance: runaway, a z_i characteristic variable has been introduced. To define a robust and reliable sign of the runaway we formed a voting system that indicates the runaway when at least half of the investigated criteria indicate runaway at i-th operation and the maximum process temperature exceeded the MAT, that is: investigated operation was classified as runaway (z(i) =1). This logical connection reflects that if the maximum process temperature does not exceed MAT, then the time series of reaction states are not dangerous. Also, the reactors are designed for safe operations, which mean that at normal conditions the process temperature should be far away from the MAT, so the process temperature cannot exceed MAT without the development of thermal runaway.

The second important task is to indicate runaway early. As the analysed nc criteria indicate runaway at different states, hence at different time instances, it is crucial to take into consideration each runaway states which are classified as runaway by the criteria. Therefore, the given state at k-th time instance in i-th operation, y⁽ⁱ⁾(k), is considered as runaway, if the

analysed i-th operation is runaway (z(i) = 1) and at least one criterion indicates runaway (I⁽ⁱ⁾(k) ≥ 1).

( )( ) { ^{( )}( ) ^{( )}

(6.5)

Runaway criteria can be evaluated and developed concerning the following aims:

1. to indicate the development of a reactor runaway reliably;

2. to indicate early the development of a reactor runaway;

3. to indicate the development of a reactor runaway early and reliably.

In the first task, the aim is to generate a criterion for the reliable indication of runaway development, so the model can be evaluated as a binary classifier. The same method is applied for the reliability analysis as in Section 5. In TP true positive cases the developed runaway was indicated, so and ̂ . We denote TN as the number of true negative cases when a runaway did not develop, and was not indicated, and ̂ . The FP number of false positive cases are the number of operations when a runaway did not develop but was indicated, as and ̂ . In FN false negative cases runaway developed but where N represents the number of studied operations, z and ̂ are the real and predicted values respectively.

As the second measure should evaluate how accurately a criterion can provide early warning, the evaluation of the criteria should be based on the actual state of the reactor. For this

When a given model is used to estimate the runaway at every k time instant, based on the ̂^{( )}( ) predictions the start and end of runaways can be estimated similarly, so ̂^{( )} ( ̂^{( )}( ) ) and ̂^{( )} ( ̂^{( )}( ) ) stand for the instances of time when the runaway occurs and the reactor returns into normal operation according to the model, respectively. Based on these variables four types of false indications can be evaluated for every k time instant as it is also illustrated in Figure 6.1:

^{( )} { ^{( ) ( )} ̂^{( )}

^{( )} ̂^{( ) ( )} (6.7)

^{( )} { ^{( )} ̂^{( ) ( )}

^{( ) ( )} ̂^{( )} (6.8)

Figure 6.1 The interpretation of different failed indications (green= ( ), red= ̂( )) These indications of misclassifications can be summed to evaluate a given operation over the ti time lengths, e.g:

^{( )} ∑ ^{( )}

(6.9) and the whole set of i=1,…,N operations:

∑ ^{( )}

(6.10) Similarly to the summation of the misclassified states, the number of correct indications is the sum of the ∑ ∑ ^{( )} number of true positive states when the developed runaway was indicated, ^{( )}( ) and ̂^{( )}( ) , and true negative states ∑ ∑ ^{( )}, when a thermal runaway not developed, and was not indicated, ^{( )}( ) and ̂^{( )}( ) . The objective function of early indication is considered to be the ratio of correct indications penalized to different types of failed indications, where all terms are weighted by a w weighting factor.

∑ ∑

∑ (6.11)

The earliness and the reliability of the indications of thermal runaway development are the two most essential requirements of criterion development. As these goals are competitive we can decide what the importance of the requirements is. As a high-performing runaway criterion should be reliable and able to determine any early indications, we also propose third objective function based on the linear combination of Eqs. (6.10) and (6.11):