WO2012129561A1 - Analyse dynamique de risques utilisant des bases de données d'alarme - Google Patents

Analyse dynamique de risques utilisant des bases de données d'alarme Download PDF

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WO2012129561A1
WO2012129561A1 PCT/US2012/030559 US2012030559W WO2012129561A1 WO 2012129561 A1 WO2012129561 A1 WO 2012129561A1 US 2012030559 W US2012030559 W US 2012030559W WO 2012129561 A1 WO2012129561 A1 WO 2012129561A1
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variables
quality
abnormal
copula
abnormal events
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PCT/US2012/030559
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Ankur PARIYANI
Warren D. SEIDER
Ulku G. OKTEM
Masoud Soroush
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Pariyani Ankur
Seider Warren D
Oktem Ulku G
Masoud Soroush
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Publication of WO2012129561A1 publication Critical patent/WO2012129561A1/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F11/00—Error detection; Error correction; Monitoring
    • G06F11/008—Reliability or availability analysis
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B23/00—Testing or monitoring of control systems or parts thereof
    • G05B23/02—Electric testing or monitoring
    • G05B23/0205—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults
    • G05B23/0218—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults
    • G05B23/0224—Process history based detection method, e.g. whereby history implies the availability of large amounts of data
    • G05B23/024—Quantitative history assessment, e.g. mathematical relationships between available data; Functions therefor; Principal component analysis [PCA]; Partial least square [PLS]; Statistical classifiers, e.g. Bayesian networks, linear regression or correlation analysis; Neural networks

Definitions

  • the present invention relates to methods and systems for dynamic risk analysis for improving process safety and/or product quality.
  • Incidents can be broadly classified as accidents or near-misses. Every accident is typically preceded by several near-misses which are less severe events/conditions/consequences, having the potential to lead to accidents. 3,4
  • CPIs Chemical Process Industries
  • a variable that is measured frequently online and describes the process dynamics (for example, temperatures, pressures, flow rates and their rates of change, etc.) is called a process variable.
  • a variable that is related to the quality of the product (for example, viscosity, density, average molecular weight, etc.) and is often estimated/inferred using mechanistic and/or statistical models is called a quality variable
  • a primary or secondary variable can be a process or quality variable.
  • primary process variables are denoted by pPs and primary quality variables by pQs.
  • the primary variables are selected during the design and commissioning of plants by carrying-out analyses of tradeoffs between the safety and profitability of the plant. De-dimensionalization (or scaling to obtain more meaningful quantities; e.g., the Damk5hler number and reactant conversion) and principal-component analyses can be used to identify primary variables systematically that should be monitored along with individual process variables to improve the tracking of process dynamics.
  • Figure 1 shows a typical control chart for a primary variable.
  • the chart is divided into four zones, beginning with its green-belt zone (normal operation), during which the variable lies within acceptable limits. When the variable moves beyond these limits, into its yellow-belt zones, high/low alarms are triggered. When it moves beyond the limits of its yellow-belt zones, into its orange-belt zones, high-high/low- low alarms are triggered. The borders between its orange- and red-belt zones are the threshold limits for the triggering of the ESD system. For secondary variables, similar control charts do not have red-belt zones.
  • abnormal events are classified in three categories: least-critical abnormal events that cross the high/low alarm thresholds, but do not cross the high-high/low- low alarm thresholds; moderately-critical abnormal events that cross the high-high/low-low alarm thresholds, but do not cross the ESD thresholds; and most-critical abnormal events that cross the ESD thresholds. Because secondary variables do not have red-belt zones, most-critical abnormal events are not associated with them.
  • DCS databases contain abnormal event data, which include alarm identity tags for the variables, alarm types (low, high, high-high, etc.), times at which the variables cross their alarm thresholds (in both directions), and variable priorities.
  • Their associated ESD databases contain trip event data, timer-alert data, etc.
  • a screenshot of a typical DCS database for a brief period is shown in Figure 2. Every row represents a new entry, associated with either a process or quality variable. Column A displays the times of events in chronological order, with each entry displaying the 'Year-Month-Day Hour: Minute AM/PM'. Note that the second entry is provided, although not shown in Figure 2.
  • Column B indicates the entry type: alarm, change in controller settings, etc.
  • Column C shows the alarm tag of the variable (defined during the commissioning of the unit).
  • Column D shows the alarm type [LO (low), LL (low- low), HI (high), etc.].
  • Column E shows the drift status of the alarms, either ALM (alarm) or RTN (return); that is, whether the variable drifts beyond or returns within the alarm thresholds. Therefore, given the occurrence and return times of the variables, the durations of abnormal events (or recovery times) can be calculated.
  • Column F shows the alarm priority, and column G presents a brief description of the alarm. Similar data entries exist for the ESD database.
  • a safety and quality management structure responds to abnormal events with typically six safety, quality, and operability systems (SQOSs), which are components of the DCS and ESD system, and involve human operators. Unlike independent protection layers, these systems are interdependent to reduce the risk- levels of the process. Furthermore, they are usually activated sequentially and their actions are interdependent.
  • SQLOSs safety, quality, and operability systems
  • Figure 1 depicts various operating belt zones and alarm thresholds for a primary variable.
  • Figure 2 is a screenshot from a typical DCS database showing data entries for a few minutes.
  • Figures 3A-3B shows the steps to prepare compact likelihood data for Bayesian analysis.
  • Figure 4 is a dynamic risk assessment pyramid showing different stages.
  • Figure 5 is an event-tree for primary process/quality variables in their yellow- and/or orange-belt zones.
  • Figure 6 is an event-tree for primary process variables (pPs) in their red-belt zones.
  • Figure 7 is an event-tree for primary quality variables (pQs) in their red-belt zones.
  • Figure 8 is an event-tree for secondary process/quality variables in their yellow-/orange-belt zones.
  • Figure 9 is a schematic of the end-state function for a general abnormal events history, showing several process records and individual paths traced by the process and quality variables.
  • Figure 10 is a schematic representation of the Bayesian analysis framework.
  • Figures 12A-12B depicts the marginal posterior distributions for ⁇ using a multivariate normal copula.
  • Figure 13 shows probability distributions for the occurrence of an ESD and an accident per period, calculated using a multivariate normal copula.
  • Figure 14 shows probability distributions for the occurrence of an ESD and an accident per abnormal event, calculated using a multivariate normal copula.
  • Figures 15A-15B shows Box and Whisker Plots for probabilities of an ESD and an accident per period, for cumulative periods (outliers are not shown).
  • Figures 16A-16B shows Box and Whisker Plots for probabilities of an ESD and an accident per abnormal event, for cumulative periods (outliers are not shown).
  • Figures 17A-17B shows marginal posterior distributions for the elements of W calculated using a multivariate normal copula.
  • a three-step dynamic risk analysis methodology uses massive alarm databases to improve process safety and product quality.
  • the three steps are: (i) tracking of abnormal events over an extended period of time, (ii) event-tree and set-theoretic formulations to compact the abnormal event data, and (iii) Bayesian analysis to calculate the likelihood of the occurrence of incidents.
  • Large alarm databases including many near- misses associated with process and quality variables, are used for the projection of unsafe plant conditions as well as quality problems.
  • the event trees and set theoretic formulations allow for compaction of massive numbers (millions) of abnormal events, making it possible to carry out the Bayesian analysis in real time.
  • the invention relates to a method for risk analysis using alarm data.
  • one or more event trees representing actions of safety, quality and
  • Basic Process Control System (BPCS) - SQOS 1 - refers to an automated basic control system within the DCS, designed to take control actions (that is, adjust the manipulating variables) to keep the process and quality variables within their normal operating ranges.
  • BPCS Basic Process Control System
  • SQOS 1 - refers to an automated basic control system within the DCS, designed to take control actions (that is, adjust the manipulating variables) to keep the process and quality variables within their normal operating ranges.
  • Level I - SQOS 2 - refers to human operator-assisted control to keep the variables within their high/low alarm thresholds and to return them to normal operating conditions. When unsuccessful, variables enter into their orange- and red-belt zones.
  • Level II- SQOS 3 - refers to human operator-assisted control to keep the variables within their high-high/low-low alarm thresholds and to return them to normal operating conditions.
  • Override controller - SQOS 4 - refers to an automatic controller that takes radical actions when select primary variables enter their red-belt zones. When successful, no tripping occurs. Typically, it is a safety system, associated with select primary process variables. However, herein, it is treated as a SQOS.
  • ESD Automatic Emergency Shutdown
  • ESD Manual Emergency Shutdown
  • a framework involving event-trees and multisets provides a compact representation of vast DCS and ESD databases to facilitate statistical analysis using Bayesian theory.
  • the combined framework accounts for the complex interactions that occur between the DCS, human operators, and the ESD system - yielding enhanced estimates and predictions of the failure probabilities of the safety, quality, and operability systems and, more importantly, the probabilities of the occurrence of shutdowns and accidents.
  • This causative relationship between the SQOSs is modeled using copulas (multivariate functions that represent the dependencies among the systems using correlation coefficients).
  • a process is said to be in an upset state when process or quality variables move out of their green-belt zones, indicating "out-of-control" or “perturbed” operation. Upset states lead to deterioration in operability, safety, and/or quality performances of the process. Equations to estimate the operability and safety performances have been proposed.
  • the upset states are:
  • Operability-Only Upset State where at least one of the secondary process variables lies outside of its green-belt zone, but all of the quality variables and the primary process variables lie within their green-belt zones. In this case, only the operability performance deteriorates, whereas safety and product quality are maintained. This occurs, for example, when the flow rate of a stream (a secondary process variable) moves just above its green-belt zone, but not sufficiently far to move the product quality or primary process variables out of their green-belt zones.
  • Safety Upset State where at least one of the primary process variables lies outside of its green-belt zone, but all of the quality variables lie within their green-belt zones. In this case, both safety and operability performances are affected and a safety problem is likely to occur.
  • the dynamic risk assessment method herein consists of three steps, shown as three regions of the pyramid in Figure 4.
  • the steps are: (1) Near-miss Tracking, (2) Event-tree and Set-theoretic Formulation, and (3) Bayesian Analysis.
  • Near-Miss Tracking refers to identification and tracking of near-misses over an extended period of time (weeks, months, etc.).
  • the abnormal events experienced by the process and quality variables are recognized as near-misses.
  • Pareto charts and alarm frequency diagrams variables that experience excessive numbers of abnormal events can be identified.
  • Event-tree and Set-theoretic Formulation permits the transformation of near- miss data to information on the performances of the safety, quality, and operability systems.
  • the near-miss data that tracks: (a) special-causes, (b) abnormal events, (c) the propagation of abnormal events, and (d) the attainment of end-states, is stored in set-theoretic formulations to represent the branches of event-trees.
  • SQOSs As special-causes arise in processes, they are handled by the SQOSs, whose actions guide the process/quality variables through their green-/yellow-/orange-/red- belt zones, resulting in either continued normal operation (variables in their green-belt zones) or upset states (OOUS, SUS, QUS, S+QUS).
  • the sequences of responses (that is, successes or failures) of the SQOSs are the paths followed by the process/quality variables and are described by the branches of the event-trees. They track abnormal events to their end-states (i.e., normal operation, plant shut-down, accident, etc.).
  • these paths are represented in a condensed format - to facilitate Bayesian analysis.
  • near-miss data from the DCS and ESD system databases show how variables move among their green-, yellow-, orange-, and red-belt zones to their end-states. From these, event- trees are created and represented with new set-theoretic notations. This permits the systematic utilization of the historical alarm databases in real time Bayesian calculations to estimate failure probabilities, the probabilities of accidents, and the like.
  • Bayesian Analysis refers to the utilization of the transformed data to obtain knowledge (that is, statistical estimates) of the performances (in terms of failure probabilities) and pair- wise interaction coefficients of the safety, quality, and operability systems. Also, the probabilities of incidents are estimated. These estimates help to identify the root-causes in the process, for example, the SQOSs with high failure probabilities or variables experiencing high abnormal event rate. In particular, consider a case when the failure probabilities of the operator corrective actions (SQOS 2 and SOQS 3 ) are high - giving operators and managers incentives to identify their root-causes - possibly due to insufficient operator training, stress factors, etc.
  • the performance of a SQOS is likely to influence the performances of the other SQOSs, because of: (a) nonlinear relationships between the variables, and (b) human behavior-based factors. For example, deterioration in the performance of the BPCS is likely to distract the operators and impede their performance. This causal relationship between the two SQOSs is accounted for using copulas, which are multivariate functions that model dependences.
  • SQOS is represented by a node, with the success or failure of each system denoted by S or F, respectively, along two branches leaving each node.
  • Figures 5, 6, and 7 show the event-trees for primary process/quality variables (pPs or pQs) that enter their yellow-/orange-/red-belt zones.
  • the trees are illustrated for the six typical SQOSs discussed earlier.
  • the first three systems are shown across the top in Figure 5 and are denoted as SQOS 1 , SQOS 2 and SQOS 3 , whereas the remaining three systems are shown across the top in Figure 6 and are denoted as SQOS*, SQOS 5 and SQOS 6 .
  • Figures 6 and 7 are identical except for the last end-state in path 7.
  • the paths are numbered according to the index of the end-state in the event- tree - from top to bottom, using the notation, P pP or P pg , where i is the path counter and pP or pQ indicates that the path is followed by a primary process or quality variable.
  • the primary process/quality variables follow the uppermost path, P pP /P pQ , when the BPCS takes successful controlling actions and keeps them within their green-belt zones (that is, no abnormal events occur).
  • the path is represented by S pP /S pQ or simply S 1 , where S pP /S pQ /S* denotes the success of safety, quality, and operability system k.
  • the combined path and its end-state is denoted as P pP /P pQ -CO.
  • F 1 ⁇ S 2 where F* denotes the failure of SQOS k. Together with its end-state, this least-critical abnormal event is denoted as P p 2 p -/P p 2 Q -CO.
  • the p s/pgs follow the third path, P pP / P pQ , when they enter their orange-belt zones, marking the failures of the BPCS and the first level of corrective actions by operators. However, the second level (more rigorous) corrective actions by the operators successfully return them to their green-belt zones. This path is represented by F 1 ⁇ F 2 ⁇ S 3 . Together with its end-state, this moderately-critical abnormal event is denoted as P p 3 p / P p 3 Q -CO.
  • the primary variables follow the remaining paths 4-7 when they enter their red-belt zones - because of the failures of BPCS and operator corrective actions (both levels), often due to insufficient response times resulting from rapid transients.
  • the p s/pgs follow the fourth path, P p 4 p / P p 4 Q , when the override controller successfully removes them from their red-belt zones and returns them to their green-belt zones.
  • paths 5-7 are traced when the override controller fails to remove the pPs/pQs from their red-belt zones and automatic/manual emergency shutdown sequences are triggered, resulting in safe shutdowns or accidents (RAs or QMs), depending upon their successes or failures.
  • RAs or QMs safe shutdowns or accidents
  • Figure 8 shows the event-tree for secondary process/quality variables (s s or sQs) that enter their yellow- or orange-belt zones.
  • the tree represents the actions of the three SQOSs associated with the secondary variables, shown across the top.
  • the combined paths and their end-states are denoted as -CO, P S 2 P / 3 ⁇ 4 -CO, and
  • the override controllers and ESD system are associated only with the primary variables.
  • the effects of their corrective actions are channeled to the latter, causing them to return to their green-belt zones as well.
  • all of the sPs/sQs return to their green-belt zones.
  • their recovery times often vary significantly.
  • MPCs multivariable, nonlinear model-predictive controllers
  • SISO single-input, single-output
  • override controllers exist only for select primary variables to reduce costly shutdowns. Also, some variables have no first- or second-level alarms to reduce alarm flooding. For those variables having fewer SQOSs, the notation is modified to show the missing systems. For example, when the override controller (SQOS 4 ) is not included in the SUS event-trees ( Figures 6 and 7), only six paths are possible, with their end-states, denoted as: P pP -CO, P pp -
  • event-trees are applicable only for continuous processes, wherein process/quality variables return to their green-belt zones eventually (except when shut downs or accidents occur).
  • Event-trees for batch processes are developed similarly. They have more end-states because variables may remain out of their green-belt zones after the batches are terminated, leading to end-states having safety problems or quality defects.
  • Step 1 abnormal events in the raw data are tracked to extract abnormal-event histories for each variable, pP flesh sP foul p ⁇ 2 cuisine sQi - each involves a, most-, b moderately-, and c, least-critical abnormal events.
  • the set-theoretic formulation is described with reference to Table 2, a process report for a typical continuous process over a brief period (consisting of minute-by- minute status updates in which a disturbance drives a few process and quality variables out of their green-belt zones). Between 1 :00 and 1 :01 PM, the process enters an OOUS. In the next minute, it moves from an OOUS to a SUS as two of its primary process variables move out of their green-belt zones.
  • this abnormal events history which shows several variables experiencing abnormal events over a period of time, is represented by a collection of paths, traced by the variables, leading to the same end-state, and therefore, is referred to as a process record.
  • An abnormal events history may include more than one process record, corresponding to different end-states attained by the variables; e.g., CO, ESD, etc.
  • Step 2a propagation paths through the safety and quality systems are extracted in Step 2a using the event-tree formulations discussed earlier.
  • n-tuples are ordered lists of finite length n. Like sets and multisets (discussed later), tuples contain objects. However, the latter appear in a certain order (which differentiate them from multisets) and an object can appear more than once (which differentiates them from sets).
  • n denotes the number of SQOSs
  • the objects are Boolean variables with permissible values, 0 (FALSE) and 1 (TRUE), for the failure and success of the SQOSs, respectively.
  • ⁇ When any system is not activated, a null value, ⁇ , is used.
  • P pP (-IV) is (0, 0, 0, ⁇ , 1, ⁇ ).
  • Table 2 denoted as a m , followed by the process and quality variables is referred as an underlying set of paths.
  • a m a multiset
  • the elements are repeated with a multiplicity equal to the number of repetitions; and the cardinality of the multiset is the sum of the multiplicities of its elements. Note that a set is a multiset with unique elements.
  • the abnormal events history in the process report in Table 2 is represented as a process record, A m , represented using a multiset of cardinality 6, [ P pp , P pp , P s 2 p , P s 2 p , P s 2 p , P s 2 Q ] , or in the standard format for multisets, [P pp , P pp , P s 2 , P 2 Q ] j j 3 1 , where the multiplicities of P pp , P p 3 p , P s 2 , and P p 2 Q are
  • any abnormal events history comprised of abnormal events involving different process/quality variables
  • process records that is, multisets of paths (modeled as 6-tuples herein) traced by the process and quality variables as the SQOSs take actions.
  • process records for any process record, a unique and non-empty underlying set of paths is defined; whose elements are the various paths of the event- trees.
  • Step 2b the overall abnormal events history is summarized as a multiset of paths.
  • the second block from the bottom shows a multiset, for the entire alarm database, showing typical paths, P p 2 p and P p 3 p , and their multiplicities, i and M 2 .
  • Step 2c the likelihood data for the safety, quality, and operability systems is obtained from the overall abnormal events history using a tuple formulation, as illustrated for a fluidized catalytic cracking unit in the next section. These contain the failure and/or success counts to be used in Bayesian analysis.
  • Basis set The sets of all of the paths of event-trees, traced by process/quality variables are defined as basis sets, analogous to the basis in linear algebra.
  • B PP denotes the set of possible paths, traced by the primary process variables, as the SQOSs respond to their abnormal events. Consequently, the set of possible paths traced by all of the process variables is:
  • Bp Bpp u
  • B s p ⁇ P sp , P s 2 , 3 ⁇ 4 , p , P p 2 p , ... , P p 7 p ⁇
  • the basis set is:
  • B Q B P Q U
  • B S Q ⁇ P SQ , 3 ⁇ 4 , 3 ⁇ 4 , P ⁇ , P P 2 Q , .... P P 7 Q ⁇
  • the underlying sets of paths are subsets of BP + Q.
  • Consequence set The sets of end-states, attained by the paths of the event- trees, are known as consequence sets. It follows that the consequence sets for the primary and secondary process and quality variables are: c pP ⁇ CO, ESD, RA ⁇ ; C pQ ⁇ CO, ESD, QM ⁇
  • Universal set A universal set is an infinite set that consists of all possible process records, including the basis set. From the perspective of probability theory, universal sets are the sample space for all possible process records.
  • Two universal sets associated with the process and quality variables are defined and denoted as Up and UQ.
  • Up includes the basis set, B P , plus all possible process records for processes in their SUS or OOUS.
  • B P the basis set
  • B Q the basis set
  • B p c [/ P , B Q c [ Q .
  • the universal sets for the primary and secondary, process and quality variables are defined, denoted as U p p, U P Q, 1 ⁇ 4>, and U S Q, respectively.
  • the universal set for the process and quality variables, t/p+g includes all possible process records for processes in their upset states (S+QUS, QUS, SUS and OOUS). Note that,
  • the surjective, non- injective, end-state function, ES: U ⁇ C maps the elements of the universal set U
  • ES(u) denotes the end-state, when: (i) a is a path followed by any process/quality variable, or (ii) u is a process record, represented as a multiset of paths followed by the process/quality variables.
  • the event-trees in Figures 5-8 for the primary and secondary process and quality variables, respectively are represented using the following functional dependences: ES: U p p ⁇ C p p, ES: U P Q ⁇ C P Q, ES: U s p ⁇ C s p, and ES: U S Q ⁇ C S Q.
  • these functional dependences permit a condensed representation of the event-trees.
  • FIGS 3A-3B shows a schematic of the steps to create the compact representation of data, beginning with the raw data at the top and, after the steps described herein, resulting in likelihood data required for Bayesian analysis to estimate the failure probabilities of the safety and quality systems.
  • Bayesion theory decomposes a system or plant into individual components - for example, power generators, pumps, and heat exchangers - and their failure probabilities are estimated using statistical analysis and expert knowledge. The latter is especially helpful when the data are limited
  • copulas which are multivariate functions used to model the joint probability distribution of the random variables have been utilized to model the dependencies among different components of the system.
  • the copula functions have many attractive features, as will be discussed in the subsections on SQOS Interactions and Copulas, Copulas, and Multivariate Normal and Cuadras-Auge Copulas - especially in permitting the combination of univariate marginal distributions from different distribution families through their correlations.
  • Bayesian analysis using copulas may be employed for dynamic risk analysis to estimate the failure probabilities of various critical incident scenarios for chemical plants using hypothetical accident precursor data in response to abnormal events. Prior distributions can be updated dynamically with incident consequence data to yield mean failure and incident probabilities (using Monte-Carlo integrations) for fixed correlation matrices.
  • the present invention employs a novel Bayesian analysis method using copulas and that utilizes the near-miss information in set-theoretic formulations, to yield risk levels in chemical plants and estimates for failure probabilities of the safety, quality, and operability systems as well as their correlation matrix.
  • the Bayesian model accounts for the interdependences among the safety, quality, and operability systems (SQOSs) using copulas, which occur due to the nonlinear relationships between the variables and behavior-based factors involving human operators.
  • the DCS and ESD databases for a large-scale fluidized-catalytic-cracking unit (FCCU) are utilized over a study period divided into a predetermined number of equal-time periods, such as, for example, 13 equal time periods.
  • Bayesian analysis is a statistical approach to reasoning under uncertainty.
  • the uncertainty associated with the failure probability of a SQOS, ⁇ is modeled initially using a probability density function, f ⁇ ⁇ ), called the prior distribution.
  • Improved failure probability distribution, /( ⁇ ⁇ Data), called posterior distribution (expressed in Eq. (1)), are inferred using the near-miss data obtained from the DCS and ESD databases.
  • 0 ⁇ data is proportional to the distribution of data conditional upon ⁇ , g(Data ⁇ ⁇ ), the likelihood, multiplied by the prior distribution of ⁇ ,J[0 ):
  • ⁇ prior probability distributions for unobservable parameters
  • the priors are updated to give posterior distributions.
  • the influence of the prior distribution on the resulting posterior distribution decreases.
  • the failure probabilities of the SQOSs are estimated with the help of prior knowledge and shared information from other SQOSs (using copulas).
  • the near-miss data, in the set-theoretic formulation are used to update the prior failure probabilities to obtain posterior failure probabilities.
  • the latter probabilities permit a projection of the frequency of incidents.
  • conjugate priors is used to model the unobservable parameters.
  • Their posterior distributions belong to the same family of distributions as their prior distributions.
  • the prior distributions are called conjugate priors to their likelihood distributions.
  • the Gamma distribution is a family of conjugate priors to the Poisson distribution.
  • the posterior estimates of the correlation matrix of the SQOSs are computed herein.
  • the Bayesian simulation to obtain the marginal posterior distributions ford 's and W is carried out using the Metropolis-Hastings algorithm.
  • the algorithm to implement these steps is shown in Figure 10.
  • the results of the statistical analysis can be obtained using the R and MATLAB packages.
  • the number of abnormal events in a time period is the failure count of the first SQOS (i.e., the basic process control system denoted as SQOS 1 ) in that period.
  • Likelihood data on n t can be the number of abnormal events for all of the primary variables, ti (see Table 5, Part I), or the number of abnormal events for p i, « pP (see
  • the Gamma family of distributions is the conjugate family for Poisson observations (that is, a Gamma prior when combined with Poisson likelihood leads to a Gamma posterior). Therefore, to simplify calculations, is assumed to have a Gamma prior distribution with a ⁇ and as shape parameters:
  • the interactions between the performances of the SQOSs are due to one or more of the following:
  • Nonlinear relationships between the variables Due to the interdependent relationship between the variables of a process, the effects of the controlling actions of the SQOSs on variables/group of variables get channeled to other variables/groups of variables, thereby, influencing the controlling actions of the other SQOSs.
  • the impact of corrective actions of the override controllers on the primary variables get channeled to the secondary variables, which return the majority of the secondary variables to their green-belt zones (or normal operating ranges) - this clearly, improves the failure probabilities of the corrective actions of the human operators.
  • Shutdown system also involves human interventions, its failure probability, ft, is also likely to be influenced by changes in ft, ft, ft and ft.
  • rank-correlation measures e.g., Spearman's rho and Kendall's tau
  • rank-correlation measures e.g., Spearman's rho and Kendall's tau
  • Pearson's or linear correlations which assume normality
  • the important feature of copula modeling is that it permits the use of rank-correlation measures.
  • dependence matrix Another advantage of the dependence matrix is that it enables the random variables to interact among each other and share information. This is particularly important when data on random variables are limited; for example, associated with rare events. In such situations, information associated with other random variables is channeled through the elements of dependence matrix (i.e., interaction parameters). The choices of copulas determine the dependences among random variables, as discussed below.
  • a copula is a function that links a multivariate cumulative distribution to its univariate cumulative marginals. Stated formally, according to the Sklar Theorem, given a joint cumulative distribution function, ⁇ ( ⁇ ⁇ , ⁇ 2 ,..., ⁇ ⁇ ) , for the random variables, ⁇ ⁇ , ⁇ 2 ,..., ⁇ ⁇ , with marginal cumulative distribution functions (CDFs),
  • F l (e i ), F 2 (e 2 ),..., F N (0 N ) can be written as a function of its marginals:
  • F(e i ,e 2 ,...,e N ) C[F l (e i ), F 2 (e 2 ),..., F N (e N )] (4) where C is the copula (function).
  • F t and C are differentiable, the joint density distribution, /( ⁇ ⁇ , ⁇ 2 ,..., ⁇ ⁇ ) , can be written as:
  • the "independence copula" is known as the "independence copula”. Note that no restrictions are placed on the marginal distributions; for example, a bivariate distribution can be constructed using beta and gamma distributions. The copulas do not impact the marginal distributions of the associated random variables and only model the dependences between them.
  • Elliptical Copulas are derived from common multivariate (elliptical) distributions, which can be extended to arbitrary dimensions. They have many parameters, which facilitate data regression.
  • the Gaussian copula derived from the multivariate normal distribution
  • the i-copula derived from multivariate Student i-distribution
  • the Gaussian copula has a nearly full range (-1, 1) of pairwise correlation coefficients - yielding a general and robust copula used in most applications, including dependences between the SQOSs.
  • the Gaussian copula however, lacks the tail dependence; that is, the probability of observing extreme observations in all random variables at once. This limitation can be addressed by using t-copula or Archimedean copula.
  • Archimedean Copulas are suitable for low-dimensional systems (n ⁇ 2) because of their simple closed functional forms. For n-dimensional distributions, serial iterates of Archimedean copulas are constructed, but these do not provide arbitrary pairwise correlations. Restrictions on Kendall's tau limits reduce the usefulness of these copulas. Examples include the Gumbel, Frank, and Clayton copulas. Marshall-Olkin Copulas may be derived from a simple stochastic process model called a Poisson shock model, where components are subject to fatal shocks, following Poisson processes. One special copula in this family is the Cuadras-Auge copula having an n-dimensional form capable of modeling arbitrary pairwise correlations. It is used herein to model the dependences between the SQOSs.
  • ⁇ ) is the product of likelihood distributions of the SQOSs: where Ns is the number of SQOSs (i.e., five for the FCCU), and K t J and L denote the failure and success counts for SQOS J in time-period, t.
  • Ns is the number of SQOSs (i.e., five for the FCCU)
  • K t J and L denote the failure and success counts for SQOS J in time-period, t.
  • the K T J and L j T are the failure and success counts for all of the primary variables, as given in Table 5, Part I, and I/ pP are for p i (Table 6, Part I), and K J pV and I/ pP are for all of the primary process variables (Table 7, Part I).
  • the joint likelihood distribution is a function of the 0's.
  • the joint posterior distribution (or target distribution), which is a function of the failure rate and probabilities of the SQOSs and their correlation matrix, is proportional to their joint prior distribution multiplied by their joint likelihood distribution: f ⁇ e i ,e 2 ,...,e Ns , W ( 7 )
  • W is the Spearman rank correlation matrix between the failure rate and probabilities of the SQOSs
  • Data is the set of success and failure counts for SQOS 1"5 , as given in Tables 5-7.
  • the joint prior distribution of ⁇ conditional on their correlation matrix, combines the marginal prior distributions of 6s and their dependences using the copula density distribution, c, which is a function of the elements of the dependence matrix; that is:
  • the number of degrees of freedom is preferably based on the number of pair-wise correlation coefficients for the SQOSs. In the example of the present application, the number degrees of freedom is 10 because there are 10 different pair- wise correlation coefficients for five SQOSs.
  • the multivariate normal copula has been widely used in risk analysis. Like multivariate normal distributions, it models dependencies (using pair-wise correlations), but does so for random variables with arbitrary marginals.
  • R, ⁇ "1 , and / are the product-moment correlation matrix, the normal inverse transformation, and the 5x5 identity matrix.
  • the Spearman correlation matrix, W, with elements 1 ⁇ 43 ⁇ 4 can be constructed from R, with elements r kq , using the following relationship:
  • Cuadras-Auge Copula The Cuadras and Auge copula density for positive correlations is:
  • the powerful M-H algorithm produces a correlated sequence of draws from the target density that may be difficult to sample using independence sampling methods.
  • a suitable starting point eP for which/ ⁇ l Data) > 0; e.g., maximum likelihood estimates (MLEs) of the failure rate and probabilities.
  • MLEs maximum likelihood estimates
  • a random number is obtained between 0 and 1.
  • ri ⁇ 1
  • the proposal value is accepted when that random number is less than r ⁇ . Otherwise, the proposal value is rejected and the current ⁇ ⁇ is retained.
  • the ratio, ri is proportional to the relative change in the posterior distribution using the proposal value and indicates the goodness of the match of the proposal value to the target distribution compared to the previous value. This relative change is multiplied by J' (d° I ⁇ * )
  • a single block M-H algorithm that converges rapidly to the target density can be difficult to construct.
  • the variate space can be discretized into smaller blocks, with Markov chains constructed in the smaller blocks.
  • This modified algorithm is referred to as the multiple-block, Metropolis-Hastings algorithm.
  • the proposal distributions are chosen to be random-walk distributions with the mean at iteration t equal to the previous iteration value.
  • the variances of the proposal distributions are displayed as a function of the failure rate and probabilities in Figures 1 1A-11B.
  • the variances of the proposal distribution are very small (using high A k ), the acceptance rates of samples (i.e., the fraction of proposed samples accepted) are very high.
  • the proposal distribution for the correlation matrix is chosen to be a random-walk, Inverse- Wishart distribution with its expected value at iteration t equal to its previous iteration value. Therefore,
  • ⁇ * ⁇ Inv-Wishart v GT 1 ) (20a)
  • ⁇ * is the unconstrained correlation matrix (proposed)
  • S is the scale matrix
  • v is the number of degrees of freedom (assumed to be 10).
  • the elements of ⁇ * are scaled to obtain the scaled correlation matrix, W*
  • This transformation sets the diagonal elements of ⁇ * to unity and scales the other elements to the range, [-1 , 1 ]. Furthermore,
  • elements of the correlation matrix can be sampled individually from different univariate proposal distributions (e.g., beta distributions) - without maintaining a positive-definite correlation matrix. Then, the sample elements that destroy the positive-definiteness of the matrix are rejected and re-sampled until the proposal matrix becomes positive-definite. The entire proposal matrix is accepted or rejected using the M-H algorithm. This individual sampling algorithm yielded a much lower acceptance rate for the correlation matrix than that obtained using the Inverse-Wishart distribution. In addition, the posterior variances of the elements were much higher, introducing more uncertainty and convergence problems. Hence, the random-walk, Inverse-Wishart distribution was used as the proposal distribution for the correlation matrix.
  • beta distributions e.g., beta distributions
  • the Bayesian analysis method presented herein is capable of handling large numbers of abnormal events over extended periods of time. This use of dynamic alarm databases for risk analysis permits a more rigorous and reliable assessment of the performance of the various regulatory and protection systems, and importantly, calculation of the probabilities of shutdowns and accidents.
  • SQOSs safety, quality, and operability systems
  • copulas are needed. Although they add an additional complexity in calculations, they provide a mechanism to utilize the shared information between the SQOSs and to estimate the posterior values of elements of dependence matrix.
  • the Bayesian approach with copulas provides better failure probability estimates, accounting for the prior/expert belief about their failures as appropriate. With sufficient data, copulas permit the reliable estimation of posterior correlation coefficients. Using the maximum entropy principle, the multivariate normal copula, with a given correlation matrix, is less informative and is preferred over the Cuadras- Auge copula.
  • the present invention can be implemented as a method of risk analysis, in which case, one output from the method could be an estimate of one or more probabilities of occurrence of undesirable events.
  • the invention can be implemented for use in any industrial process or facility such as chemical plants, production plants, energy generation facilities, waste treatment facilities, recycling facilities and other similar facilities.
  • the method can also be extended to include a further step of taking some action based on the estimate of probabilities.
  • a change can be made to any of the process variables, quality variables, primary, or key, variables a, secondary variables, and/or any combination thereof. Changes can also be made to any one or more of the Basic Process Control System (BPCS) - SQOS 1 , Operator (machine + human) corrective actions, Level I - SQOS 2 , Operator (machine + human) corrective actions, Level II - SQOS 3 , Override controller - SQOS4, Automatic Emergency Shutdown (ESD) - SQOS 5 or the Manual Emergency Shutdown (ESD) - SQOS 6 , either independently and/or in combination with one or more changes to the variables mentioned above.
  • BPCS Basic Process Control System
  • ESD Automatic Emergency Shutdown
  • SQOS 5 Automatic Emergency Shutdown
  • ESD Manual Emergency Shutdown
  • the present invention can also be implemented as a system using software, hardware, firmware or any combination thereof.
  • the method of the present invention can be implemented on a computer system including one or more general and/or special purpose processors, preferably associated with a suitable memory and/or device for acquiring the data required to carry out the method.
  • Suitable components for such a system are well-known to skilled persons and can be modified to carry out the present invention based on the foregoing disclosure.
  • a suitable system can be designed based on, for example, Figures 3A-3B and/or 10 of the present application.
  • the present invention can also be implemented in the form of an algorithm for carrying out the method of the present invention, preferably stored on a non-transient medium.
  • Said algorithm is preferably in the form of computer readable code stored on a non-transient computer readable medium.
  • the code when executed by a suitable computer system, will cause the computer system to carry out a method in accordance with the present invention.
  • Suitable computer readable code can be developed, for example, based on Figures 3A-3B and/or 10 of the present application.
  • DCS and ESD databases associated with an industrial fluidized-catalytic-cracking unit (FCCU) at a major petroleum refinery that processes over 250,000 barrels of oil per day and over an extended period of time are used.
  • FCCU fluidized-catalytic-cracking unit
  • the unit has on the order of 150-200 alarmed variables and as many as 5,000-10,000 alarm occurrences per day - as a result of 500-1,000 abnormal events daily.
  • These databases are stored on secured servers and updated dynamically with very small time delays (less than 30 seconds). Due to data limitations, the SQOS 5 and SQOS 6 (automatic and manual emergency shutdown systems ), are treated as a single SQOS for the purpose of this example. Also, four of its process and three of its quality variables are associated with emergency shutdown (ESD) systems.
  • ESD emergency shutdown
  • Variables have no high-high a arms and no override controller.
  • a total of 2,545 abnormal events occurred for the primary variables - 2,036 for the primary process and 509 for the primary quality variables.
  • the primary process variable, pPi which involves all five SQOSs, experienced 1 ,857 abnormal events (1 ,720 least-critical, 21 moderately-critical, and 1 16 most-critical abnormal events, with two leading to ESDs).
  • the remaining primary process variables, pP 2 , p 3, and p 4 (which have no high-high/low-low alarms and override controllers), experienced only 179 abnormal events (176 least-critical and three most-critical leading to ESDs).
  • the abnormal events history for individual and groups of variables is represented by the branches (paths) of the event-trees in Figures 5-8.
  • the p s and pQs that experienced least-critical abnormal events that is, entered their yellow-belt zones only), followed the path P p 2 p and P p 2 Q , respectively, leading to CO.
  • the p s and pQs that experienced moderately-critical abnormal events followed P pP and P p 3 Q , respectively, again leading to CO.
  • p i entered its red- belt zone, but was returned to its normal operating range by the override controller, it followed the path, Pi .
  • an ESD was triggered, it followed the path, P pp , leading to an ESD.
  • process record Ai is represented as a multiset of paths, [P p 2 P , P pP , P pP , P p 2 Q , P pQ , ] 1 896> 21 > 1 14> 50 4 > 5 , and process record A 2 as a multiset [P pP , P pP (-III, - IV)] 2 3 .
  • this new set-theoretic framework provides a compact representation in handling 1 ,000s of abnormal events depicting success/failure paths followed by the process and quality variables through the SQOSs.
  • This framework facilitates the Bayesian analysis to compute failure probabilities of the SQOSs and incident probabilities.
  • the success and failure counts for SQOS 3 do not sum to the failure count for SQOS 2 (145), and the success count for SQOS 5 is not equal to the failure count for SQOS 4 - because three of the p s, were not equipped with the high- high/low-low alarms and override controllers. Because the multisets store information for the SQOSs, specific to each variable, in a generic format, the new framework permits effective accounting of the success and failure counts for each SQOS.
  • the abnormal events history for the all primary process variables, pPi, ...,pP 4 is represented as [P p 2 P , P p 3 P , P p 4 P , P p 5 P ] 1896> 21>114>5 .
  • Similar analysis can be done for secondary variables - to calculate the performances and pair-wise interaction coefficients for SQOSs (in terms of failure probabilities) in response to abnormal events of secondary variables.
  • Their values for individual as well as groups of variables can be monitored over a extended period of time to assess and improve the safety and operational performance of the process, for example, whenever their values increase, management and operators be alerted to take actions to address the root-causes; e.g. improved (1) DCS configurations and tuning, (2) operator training, (3) operating regimes, (4) process designs, and (5) alarm system configurations.
  • Alarm data in abnormal event histories were represented efficiently by new event-trees, showing the paths followed by the safety, quality, and operability systems (SQOSs) in handling abnormal events.
  • SQOSs safety, quality, and operability systems
  • the event-trees permit specific SQOSs to be assigned to various process and quality variables.
  • the new multi-set and tuple formulations provide a robust and efficient transformation into a compact
  • the multiset structures have on the order of 10° data entries; that is, a six order-of-magnitude reduction from millions of data entries - sharply increasing the efficiency in storage and handling of data.
  • the FCCU example showed the efficiency of the compaction method in handling large alarm datasets.
  • incident probabilities can be estimated; i.e., probabilities of the occurrence of an ESD or accident.
  • Two types of incident probabilities are computed herein: (a) incident probabilities per period and (b) incident probabilities per abnormal event.
  • the probability of occurrence of an ESD per period, p ⁇ SO is obtained by multiplying the failure rate and the probabilities of SQOS 1"4 and the success probability of SQOS 5 .
  • the probability of occurrence of an accident in each period is obtained by multiplying the failure rate of SQOS 1 and the failure probabilities of SQOS 2"5
  • the probability of the occurrence of an ESD per abnormal event, ⁇ ⁇ is obtained by multiplying the failure probabilities of SQOS 2"4 , and the success probability of SQOS 5
  • the probability of occurrence of an accident per period, p ⁇ cident is obtained by multiplying the failure probabilities of SQOS 2"5 .
  • Figure 14 presents histograms of the probabilities of occurrence of an ESD and an accident, per abnormal event. Their 95% posterior intervals and means are as follows:
  • the probability of the occurrence of an ESD associated with p i is 0.124 per period
  • the probabilities of ESDs and accidents can help to assess the compliance of chemical plants with national and international safety standards.
  • the Instrumentation, Systems, and Automation Society (ISA) 50 presents safety integrity limits (SILs) to measure the level of risk-reduction provided by ESD systems. More specifically, when the probability of failure under demand (PFD) lies between 10 " and 10 " , the SIL is set at 2.
  • PFD probability of failure under demand
  • Figures 15A-15B and 16A-16B present box and whisker plots for the probabilities of occurrence of an ESD and an accident, per period and per abnormal event, respectively, starting from the first period and cumulated through the end of each of the 13 periods.
  • the box and whisker plot for period 1-6 For example, the box and whisker plot for period 1-6
  • the figures show how the risk estimates (cumulative values) change with time periods. For the first four periods, the risk estimates experience a consistent decrease, followed by a sudden increase in the fifth period. This was due to increases in failure probabilities of levels I and II corrective actions, due to special operations in Period 5. Following that, the risk estimates decrease slightly and attain a steady value. These figures provide insights on the relative performances during different time periods and can be used to compare risk levels associated with different plants and work conditions.
  • the mean values of the correlation coefficients are positive, with their distributions skewed to the left.
  • this multivariate normal copula yields positive correlations between the failure rate and probabilities of the SQOSs.
  • Histograms of the marginal posterior failure rate and probabilities of the SQOSs, ⁇ , for p i were also calculated using the Cuadras-Auge copula. Their 95% posterior intervals and means for ⁇ are:
  • the 95% posterior intervals and means for the elements of correlation matrix are:
  • the distributions are skewed to the left. Although their mean values positive, they are smaller than those using the multivariate normal copula. On average, the correlation strengths between the SQOSs are less than using the multivariate normal copula, as shown in Table 10.
  • the strength of the correlations between the SQOSs is contingent upon the copula used for modeling. It is important to note, however, that the trends obtained from the two copulas are similar - the correlation strengths increase as the SQOSs assume closer proximity. Furthermore, the copula modeling validates the theoretical assumptions that the actions of the SQOSs are related to each other (due to nonlinear relationships between the variables and behavior-based factors). These results can persuade plant personnel to take preventive measures to avoid shutdowns and accidents. For example, when ⁇ or #3 increase, the onset of a special-cause(s) is likely to trigger the ESDs more often, increasing their likelihood of failure
  • Copula selection can be important, especially when data are limited. For multivariate systems, copula properties like tail dependences, closed-form expressions, symmetricity, and similar, can strongly influence the selection. More recently, it is recommended that non-informative copulas be selected. In one approach, an entropy function is maximized. For the multivariate prior
  • ⁇ [ ( ⁇ 1 , ⁇ 2 ,..., ⁇ 5 )] -
  • Ent[ (0 1 ,0 2 ,..., 0 5 )] Ent[ 1 (0 1 )] + ⁇ Ent[ .(0 .)] +Ent[ w (ff)]
  • the entropy of the multivariate prior distribution is the sum of the entropies of the individual marginal PDFs plus the entropy of the copula.
  • the entropy is maximized when the random variables are independent. With no information shared among the univariate marginals, the data are 'most randomly' distributed.
  • the random variables are dependent (given their correlation matrix)
  • the differential entropy is maximized using the multivariate normal copula. It follows that, for a given prior correlation matrix, the multivariate normal copula has a higher entropy than the Cuadras-Auge copula and thus, is less informative. Therefore, herein, the multivariate normal copula is the preferred prior.
  • the safety performance of any process is characterized by its primary variables, which include both primary process and primary quality variables.
  • primary variables include both primary process and primary quality variables.
  • a primary process or quality variable moves outside of its green-belt zone, a safety problem is likely to occur.
  • quality variables are causally related to process variables, abnormal events associated with primary quality variables do not necessarily imply the movement of primary process variables out of their green-belt zones, and vice versa.
  • the abnormal events history of the primary quality variables should also influence the Bayesian risk analysis. In practice, however, quality variable data are often not estimated or recorded.
  • Table 11 compares the mean failure probabilities and the probabilities of ESDs and accidents (calculated using Bayesian analysis with multivariate normal copula) - with and without product- quality data. Table 11. Comparison of the mean failure probabilities and the probabilities of occurrence of ESDs and accidents (calculated using multivariate normal copula)

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Abstract

L'invention concerne une méthodologie d'analyse dynamique de risques, qui utilise des bases de données d'alarme pour améliorer la sûreté des processus et la qualité des produits. La méthodologie comprend trois étapes: i) le suivi d'événements anormaux sur une durée prolongée; ii) l'utilisation d'arbres d'événements et de formulations ensemblistes pour compacter les données relatives aux événements anormaux; et iii) la conduite d'une analyse bayésienne pour calculer la probabilité de survenue d'incidents. Les arbres d'événements et les formulations ensemblistes autorisent un compactage de nombres massifs (des millions) d'événements anormaux. La structure ensembliste regroupe les chemins des événements en un seul enregistrement de données compact, ce qui améliore considérablement l'efficacité des calculs probabilistes et permet une analyse bayésienne en temps réel de grandes bases de données d'alarme. La méthode d'analyse bayésienne utilise des quasi-instances provenant de bases de données de systèmes de contrôle distribués et de systèmes d'arrêt d'urgence pour calculer les probabilités de défaillance des systèmes SQOS en termes de sûreté, de qualité et d'exploitabilité, ainsi que les probabilités de survenue d'incidents; et utilise des copules pour rendre compte des interdépendences des SQOS.
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