Petri nets applications Part 14 pdf

PetriNets:Applications512 Sequence-detectability implies the knowledge of all firing sequences of an IPN. In others words, there is a function ),(),(),(),(: 0000 MQLMQMQMQ B k s  where .)),(,( 0 ws MQw  The problem of determining whether or not a system is sequence- detectable has a high computational complexity. However, the following definition provides conditions that reduce the computational complexity. Definition 12. An IPN given by ),( 0 MQ is event detectable if the firing of any transition Tt k  at a marking  k M ,(QR ) 0 M can be uniquely determined through the information provided by the input symbol )( k t and the output signals )( k M , where ),( kC  is the column of C corresponding to transition k t . Note that this definition implies that, all events can be detected and distinguisable, i.e. their firing can be detected and distinguisable from each other after its occurrence and before another event occurs. Thus, an event-detectable IPN system is also sequence-detectable. Event detectable has a structural characterization captured in the next lemma that can be tested in a polynomial time. Lemma 1: An IPN given by ),( 0 MQ is event detectable if only if ],,3,2,1[ mi  , 0),(   iC and kj  ,,3,2,1[  ]m such ),(),( kCjC  , then )()( kj tt  . Proof. You can find the proof in (Ramírez-Treviño et al., 2003).  Observe that an IPN given by ),( 0 MQ where it has transitions Ttt jk , with )()( jk tt  and 0),(),(  jCkC the firing of jk tt , cannot be distinguisable, in this case jk tt , are called indistinguishable. Definition 13. An IPN given by ),( 0 MQ is marking-detectable if there is an integer k such that ),( 0 MQ k w  it holds that the information provided by w and ),( 0 MQ suffices to uniquely determine the marking i M reached by firing w  . In other works, there is a function ),(),(),(),(: 0000 MQRMQMQMQ B k M  where iM MMQw  )),(,( 0 , ws MQw  )),(,( 0 and i MM w   0 . Example 4: Consider the IPN shown in figure 11.a where },,{ 651 pppP m  and },{ 51 ttT c  . Its incidence matrix shown in the figure 11.b and its ouput function is the matrix of the equation (26). In this case, C is the matrix:               10100 01010 10001 C (27) Note that the IPN system in the figure 11.a is event detectable by the lemma 1. Then the fire of all events in this IPN can be detected and distinguisable after its occurrence and before another event occurs. To illustrate as the fire of any event in IPN system is detected consider that the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]000[ ]010[ 1 0 (28) In this case, T yy ]010[ 01  and )4,(   C is the column of C  corresponding to transition 4 t . Then the fire of the transition 4 t is detected trough the information of the input, outputs and structure. Also )( 4 t w    where )]000[,)(]010[,( TT w  , and the IPN system in the figure 11.a is sequence-detectable. Note that also it IPN system is marking-detectable in 2-steps. Finally, consider that set of fault marking is },{ 72 MMF  (see the figure 12) in this case the information provided by w and ),( 0 MQ suffices to uniquely determine the marking FM i  reached by firing w  , then the IPN system shown in the figure 11.a is input-output diagnosable in 2-steps.  4.2 Characterization Sequence-detectability and marking-detectability are both necessary and sufficient conditions for input-output diagnosability, for provide a sufficient condition of input-output diagnosable with a reduced computational complexity is considered a class of IPN defined as follows. Definition 14. A subnet of an IPN given by ),( 0 MQ with ),,,,,(  NQ and ),,,( OITPN  is a net ) ~ , ~ ( 0 MQ such that PP  ~ , TT  ~ ,  ~ ,  ~ and functions OI ~ , ~ are restrictions of I and O over T P ~~  respectively, }{ ~ ~ : ~  T , }{ ~ ~ : ~  P and q ZMQR )() ~ , ~ (: ~ 0   . Definition 15. The set of fault transitions of an IPN given by ),( 0 MQ with ),,,,,(  NQ and ),,,( OITPN  is the set 0),(|{   kkf tOtT and })(  k t . Definition 16. Let ),( 0 MQ with ),,,,,(       NQ , ),,,( OITPN  an IPN where {} f T . The subnet induced by f T is ),( 0 Tf Tf MQ with ),,,( ][ TfTfTfTf Tf OITPN  where fTf TT  , f k i iTf TtP    1 and Tf I , Tf O are restrictions of I and O over TfTf TP  respectively, {} Tf ,  Tf and functions  TfTf T: , }{:  TfTfTf P and q Tf Tf ZMQR )(),(: ~ 0   . This subnet is named fault model. Definition 17. Let ),( 0 MQ with ),,,,,(       NQ , ),,,( OITPN  an IPN where {} f T . The subnet of normal behavior is ),( 0 NN MQ with QQ N  where fN TTT  , }{ 1 f k i iN TtPP    ,    N ,    N and function N I , N O are restrictions of I and O over NN TP  respectively. In this case }{:      NN T , }{:     NNN P and qNN N ZMQR )(),(: 0   . This subnet is named diagnoser model. Example 5: Consider the IPN shown in Figure 13a. The input and output alphabets are },{ ba and },,{ 321      respectively. Functions  ,  , C and  are given by: (29) FaultDiagnosisonElectricPowerSystemsbasedonPetriNetApproach 513 Sequence-detectability implies the knowledge of all firing sequences of an IPN. In others words, there is a function ),(),(),(),(: 0000 MQLMQMQMQ B k s  where .)),(,( 0 ws MQw   The problem of determining whether or not a system is sequence- detectable has a high computational complexity. However, the following definition provides conditions that reduce the computational complexity. Definition 12. An IPN given by ),( 0 MQ is event detectable if the firing of any transition Tt k  at a marking  k M ,(QR ) 0 M can be uniquely determined through the information provided by the input symbol )( k t and the output signals )( k M , where ),( kC  is the column of C corresponding to transition k t . Note that this definition implies that, all events can be detected and distinguisable, i.e. their firing can be detected and distinguisable from each other after its occurrence and before another event occurs. Thus, an event-detectable IPN system is also sequence-detectable. Event detectable has a structural characterization captured in the next lemma that can be tested in a polynomial time. Lemma 1: An IPN given by ),( 0 MQ is event detectable if only if ],,3,2,1[ mi    , 0),(   iC and kj  ,,3,2,1[  ]m such ),(),( kCjC   , then )()( kj tt    . Proof. You can find the proof in (Ramírez-Treviño et al., 2003).  Observe that an IPN given by ),( 0 MQ where it has transitions Ttt jk  , with )()( jk tt  and 0),(),(    jCkC the firing of jk tt , cannot be distinguisable, in this case jk tt , are called indistinguishable. Definition 13. An IPN given by ),( 0 MQ is marking-detectable if there is an integer k such that ),( 0 MQ k w  it holds that the information provided by w and ),( 0 MQ suffices to uniquely determine the marking i M reached by firing w  . In other works, there is a function ),(),(),(),(: 0000 MQRMQMQMQ B k M  where iM MMQw  )),(,( 0 , ws MQw    )),(,( 0 and i MM w   0 . Example 4: Consider the IPN shown in figure 11.a where },,{ 651 pppP m  and },{ 51 ttT c  . Its incidence matrix shown in the figure 11.b and its ouput function is the matrix of the equation (26). In this case, C  is the matrix:               10100 01010 10001 C (27) Note that the IPN system in the figure 11.a is event detectable by the lemma 1. Then the fire of all events in this IPN can be detected and distinguisable after its occurrence and before another event occurs. To illustrate as the fire of any event in IPN system is detected consider that the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]000[ ]010[ 1 0 (28) In this case, T yy ]010[ 01  and )4,(C is the column of C corresponding to transition 4 t . Then the fire of the transition 4 t is detected trough the information of the input, outputs and structure. Also )( 4 t w  where )]000[,)(]010[,( TT w  , and the IPN system in the figure 11.a is sequence-detectable. Note that also it IPN system is marking-detectable in 2-steps. Finally, consider that set of fault marking is },{ 72 MMF  (see the figure 12) in this case the information provided by w and ),( 0 MQ suffices to uniquely determine the marking FM i  reached by firing w  , then the IPN system shown in the figure 11.a is input-output diagnosable in 2-steps.  4.2 Characterization Sequence-detectability and marking-detectability are both necessary and sufficient conditions for input-output diagnosability, for provide a sufficient condition of input-output diagnosable with a reduced computational complexity is considered a class of IPN defined as follows. Definition 14. A subnet of an IPN given by ),( 0 MQ with ),,,,,(  NQ and ),,,( OITPN  is a net ) ~ , ~ ( 0 MQ such that PP  ~ , TT  ~ ,  ~ ,  ~ and functions OI ~ , ~ are restrictions of I and O over T P ~~  respectively, }{ ~ ~ : ~  T , }{ ~ ~ : ~  P and q ZMQR )() ~ , ~ (: ~ 0   . Definition 15. The set of fault transitions of an IPN given by ),( 0 MQ with ),,,,,(  NQ and ),,,( OITPN  is the set 0),(|{   kkf tOtT and })(  k t . Definition 16. Let ),( 0 MQ with ),,,,,(  NQ , ),,,( OITPN  an IPN where {} f T . The subnet induced by f T is ),( 0 Tf Tf MQ with ),,,( ][ TfTfTfTf Tf OITPN  where fTf TT  , f k i iTf TtP    1 and Tf I , Tf O are restrictions of I and O over TfTf TP  respectively, {} Tf ,  Tf and functions  TfTf T: , }{:  TfTfTf P and q Tf Tf ZMQR )(),(: ~ 0   . This subnet is named fault model. Definition 17. Let ),( 0 MQ with ),,,,,(  NQ , ),,,( OITPN  an IPN where {} f T . The subnet of normal behavior is ),( 0 NN MQ with QQ N  where fN TTT  , }{ 1 f k i iN TtPP    ,  N ,  N and function N I , N O are restrictions of I and O over NN TP  respectively. In this case }{:  NN T , }{:  NNN P and qNN N ZMQR )(),(: 0   . This subnet is named diagnoser model. Example 5: Consider the IPN shown in Figure 13a. The input and output alphabets are },{ ba and },,{ 321  respectively. Functions  ,  , C and  are given by: (29) PetriNets:Applications514 (a) (b) (c) Fig. 13. a) Interpreted Petri Net System C, b) A subnet of Interpreted Petri Net System C, c) Subnet induced by f T . Thus, the controlled transitions are },{ 21 ttT c  and the uncontrolled ones are }{ 3 tT u  . The measured places are },,{ 321 pppP m  and the non-measured are {} nm P . In this case, the figure 13.b is a subnet of normal behavior Interpreted Petri Net system of the figure 13.a with functions N  , N  , N C and N  are given by: i 1 2 )( iN p 1  2  k 1 2 )( kN t a b          11 11 N C        10 01 N (30) The subnet induced by }{ 3 tT f  is showed in the figure 13.c where functions Tf  , Tf  , Tf C and Tf  are given by: i 1 3 )( iTf p 1  3  k 3 )( kTf t         10 01 Sf (31) In this chapter to emphasize the fact that the IPN system captures the normal and fault behaviour (illustrated as the IPN systems of the figures 13.b and 13.c respectively) the next definition is proposed. Definition 18. An IPN system to diagnose given by ),( 0 DD MQ is a net where {} f T and the funtions I , O and C are:        Tf N I I I 0  ;        Tf N O O O 0  and        Tf N C C C 0  (32) where N I , N O , N C are restrictions of I and O over NN TP  , i.e. over the subnet indiced by normal behavior ),( 0 NN MQ ; and Tf I , Tf O , Tf C are restrictions of I and O over TfTf TP  , i.e. over the subnet induced by fault behavior f T ( ),( 0 Tf Tf MQ ). Note that the incidence matrix PN system 13.a is:               1 0 1 00 11 11 C with          11 11 N C and             1 0 1 Tf C (33) The characterization of input-output diagnosable is based on the idea from the IPN system captures the normal and fault behaviour (illustrated as the IPN systems of the figures 13.b and 13.c respectively. Note that in the figure 13.c non-manipulated transition marking a faulty place. Then it is possible to known when a faulty place is marked and to determinate which is the marked place. This idea is formalized in the next theorem. Theorem 1: Let ),( 0 DD MQ be an IPN, live, strongly connected and event detectable with ),,,,,(  NQ and ),,,( OITPN  . Let }, ,{ 1 r XX be the set of all T-invariants of ),( 0 DD MQ . Let ),( 0 NN MQ be subnet induced by f T . If Tfi Pp   , where ji tp  and Tfj Tt  (i.e. i p are predecessors of any fault transition) the following conditions hold: 1. r  , j  1)( jX r , where Tfij Tpt   )( ; 2. Tfik Tpt  )( , )}{()( ik pt  and  )( k t . Then the IPN ),( 0 DD MQ is input–output diagnosable. Proof. You can find the proof in (Ramírez-Treviño et al., 2007).  4.3 Diagnoser Design The issue with detection and localization of faults consist in identify the abnormal behavior in the systems and locate the root cause or resources that are working in a wrong way. Diagnoser design proposed in (Santoyo-Sanchez et al., 2008) is shows in the figure 14, this scheme consist of six components 1. Diagnoser model, which is an IPN, denoted by ),( 0 NN MQ that represents the normal behavior of the system. 2. System model, it is an IPN denoted by ),( 0 DD MQ which contains the normal and abnormal behavior from the system, where the diagnoser IPN is embedded into the IPN system. 3. Firing events detector block, which detect and determine which transitions was fired into the system. 4. Error block, it is an Error IPN defined between ),( 0 DD MQ and ),( 0 NN MQ , which compares the behavior between both IPN systems. 5. Detecting Fault Marking algorithm, it detects and locates fault through the Error IPN, also indicating faulty state. 6. Diagnoser Fault algorithm, it indicate the component fault and specify the kind of fault that occur. FaultDiagnosisonElectricPowerSystemsbasedonPetriNetApproach 515 (a) (b) (c) Fig. 13. a) Interpreted Petri Net System C, b) A subnet of Interpreted Petri Net System C, c) Subnet induced by f T . Thus, the controlled transitions are },{ 21 ttT c  and the uncontrolled ones are }{ 3 tT u  . The measured places are },,{ 321 pppP m  and the non-measured are {}  nm P . In this case, the figure 13.b is a subnet of normal behavior Interpreted Petri Net system of the figure 13.a with functions N  , N  , N C and N  are given by: i 1 2 )( iN p 1  2  k 1 2 )( kN t a b          11 11 N C        10 01 N (30) The subnet induced by }{ 3 tT f  is showed in the figure 13.c where functions Tf  , Tf  , Tf C and Tf  are given by: i 1 3 )( iTf p 1  3  k 3 )( kTf t          10 01 Sf (31) In this chapter to emphasize the fact that the IPN system captures the normal and fault behaviour (illustrated as the IPN systems of the figures 13.b and 13.c respectively) the next definition is proposed. Definition 18. An IPN system to diagnose given by ),( 0 DD MQ is a net where {} f T and the funtions I , O and C are:        Tf N I I I 0  ;        Tf N O O O 0  and        Tf N C C C 0  (32) where N I , N O , N C are restrictions of I and O over NN TP  , i.e. over the subnet indiced by normal behavior ),( 0 NN MQ ; and Tf I , Tf O , Tf C are restrictions of I and O over TfTf TP  , i.e. over the subnet induced by fault behavior f T ( ),( 0 Tf Tf MQ ). Note that the incidence matrix PN system 13.a is:               1 0 1 00 11 11 C with          11 11 N C and             1 0 1 Tf C (33) The characterization of input-output diagnosable is based on the idea from the IPN system captures the normal and fault behaviour (illustrated as the IPN systems of the figures 13.b and 13.c respectively. Note that in the figure 13.c non-manipulated transition marking a faulty place. Then it is possible to known when a faulty place is marked and to determinate which is the marked place. This idea is formalized in the next theorem. Theorem 1: Let ),( 0 DD MQ be an IPN, live, strongly connected and event detectable with ),,,,,(  NQ and ),,,( OITPN  . Let }, ,{ 1 r XX be the set of all T-invariants of ),( 0 DD MQ . Let ),( 0 NN MQ be subnet induced by f T . If Tfi Pp  , where ji tp  and Tfj Tt  (i.e. i p are predecessors of any fault transition) the following conditions hold: 1. r  , j  1)( jX r , where Tfij Tpt  )( ; 2. Tfik Tpt  )( , )}{()( ik pt  and  )( k t . Then the IPN ),( 0 DD MQ is input–output diagnosable. Proof. You can find the proof in (Ramírez-Treviño et al., 2007).  4.3 Diagnoser Design The issue with detection and localization of faults consist in identify the abnormal behavior in the systems and locate the root cause or resources that are working in a wrong way. Diagnoser design proposed in (Santoyo-Sanchez et al., 2008) is shows in the figure 14, this scheme consist of six components 1. Diagnoser model, which is an IPN, denoted by ),( 0 NN MQ that represents the normal behavior of the system. 2. System model, it is an IPN denoted by ),( 0 DD MQ which contains the normal and abnormal behavior from the system, where the diagnoser IPN is embedded into the IPN system. 3. Firing events detector block, which detect and determine which transitions was fired into the system. 4. Error block, it is an Error IPN defined between ),( 0 DD MQ and ),( 0 NN MQ , which compares the behavior between both IPN systems. 5. Detecting Fault Marking algorithm, it detects and locates fault through the Error IPN, also indicating faulty state. 6. Diagnoser Fault algorithm, it indicate the component fault and specify the kind of fault that occur. PetriNets:Applications516 Fig. 14. Scheme for proposed diagnoser. The diagnose process is based on the idea of that the system behavior is modeled as IPN, which contains the normal and fault behavior. When a transition fires, due to system IPN is event-detectable then it is possible to determine its fires (with the firing events detect block). Moreover, when the system does not fire fault transitions the output of both models (system and diagnoser) is equal, i.e. the system behavior only includes the fire of normal transitions. In the oher case, when a fault transition fires in the system, its fire is detected but this transition is not include into the diagnoser model, then the output of both models (system and diagnoser) is not equal. In this case a fault is detected, and the next steps are to locate the fault, indicate the component fault and specify the kind of fault. To illustrate the general diagnose process consider the next example. Example 6: Consider the IPN shown in figure 13a. as the system model, and the IPN shown in figure 13.b as the diagnoser model. Both system the input and output alphabets are },{ ba and },,{ 321  respectively. And its functions  ,  , C and  are given by the equations (29) and (30) respectively. In this case, C and NN C are the matrix:              100 011 111 C ;          11 11 NN C (34) Note that the IPN system in the figure 13.a is event detectable by the lemma 1. Assume that from 0 M and N M 0 the information provide by the IPN system is: bu  1 input ; outpu t        T T y y ]10[ ]01[ 1 0 and bu  1N input ; output N        TN TN y y ]10[ ]01[ 1 0 (35) In this c ase, T yy ]11[ 01  and )2,(   C is the column of C  corresponding to transition 2 t . Also its output error between both systems is TN yyy ]00[ 1 1  . In this case the system does not fire fault transitions. Assume that from 0 M and N M 0 the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]00[ ]01[ 1 0 and   1N input u ; output N        TN TN y y ]01[ ]01[ 1 0 (36) In this case, T yy ]01[ 01  and )3,(  C is the column of C  corresponding to transition 3 t . And its output error between both systems is TN yyy ]01[ 1 1  . In this case a fault transition fires in the system, and a fault is detected, i.e. an error different to zero. In general, the error concept among systems is computed as differences among theirs outputs. In the observer and controller design when the error is zero then the system reach a required behavior (De Jesús & Ramírez-Treviño, 2001). In the context of diagnoser design to localize the fault transition and the place of fault is used an error structure introduced in (De Jesús & Ramírez-Treviño, 2001) and (Santoyo-Sanchez et. al, 2008), which is presented in the next definition. Definition 19 Let ),( 0 DD MQ and IPN where ),( 0 NN MQ and ),( 0 Tf Tf MQ are its subnets IPN diagnoser model and IPN fault model respectively. Structure Error between ),( 0 NN MQ and ),( 0 DD MQ is defined as ),( 0 EE MN where ),,,( EE EEE OITPN  with DE P P  , DE T T  ,   ZTPI EE E : ,   ZTPO EE E : and   ZTPC EE E : defined as:          Tf NN E I II I 00  ;          Tf EE E O OO O 00  and          Tf NN E C CC C 00  (37) The initial marking of IPN ZPM EE : 0 defined as:                  Tf N N E M M M M 0 0 0 0 - 0  (38) The marking at the k-th instant is:                  Tf k N k N k E k M M M M - 0  (39) A transition N j Tt  is enabled at marking E k M if N i Pp  , ),()( ji N i E k tpIpM  ; while that a transition D j Tt  is enabled at marking E k M if D i Pp  , ),()( ji D i E k tpIpM  . When a transition j t is fired, then a new marking 1k M is reached. This new marking is computed as: FaultDiagnosisonElectricPowerSystemsbasedonPetriNetApproach 517 Fig. 14. Scheme for proposed diagnoser. The diagnose process is based on the idea of that the system behavior is modeled as IPN, which contains the normal and fault behavior. When a transition fires, due to system IPN is event-detectable then it is possible to determine its fires (with the firing events detect block). Moreover, when the system does not fire fault transitions the output of both models (system and diagnoser) is equal, i.e. the system behavior only includes the fire of normal transitions. In the oher case, when a fault transition fires in the system, its fire is detected but this transition is not include into the diagnoser model, then the output of both models (system and diagnoser) is not equal. In this case a fault is detected, and the next steps are to locate the fault, indicate the component fault and specify the kind of fault. To illustrate the general diagnose process consider the next example. Example 6: Consider the IPN shown in figure 13a. as the system model, and the IPN shown in figure 13.b as the diagnoser model. Both system the input and output alphabets are },{ ba and },,{ 321      respectively. And its functions  ,  , C and  are given by the equations (29) and (30) respectively. In this case, C  and NN C  are the matrix:              100 011 111 C ;          11 11 NN C (34) Note that the IPN system in the figure 13.a is event detectable by the lemma 1. Assume that from 0 M and N M 0 the information provide by the IPN system is: bu  1 input ; outpu t        T T y y ]10[ ]01[ 1 0 and bu  1N input ; output N        TN TN y y ]10[ ]01[ 1 0 (35) In this c ase, T yy ]11[ 01  and )2,(C is the column of C corresponding to transition 2 t . Also its output error between both systems is TN yyy ]00[ 1 1  . In this case the system does not fire fault transitions. Assume that from 0 M and N M 0 the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]00[ ]01[ 1 0 and  1N input u ; output N        TN TN y y ]01[ ]01[ 1 0 (36) In this case, T yy ]01[ 01  and )3,(C is the column of C corresponding to transition 3 t . And its output error between both systems is TN yyy ]01[ 1 1  . In this case a fault transition fires in the system, and a fault is detected, i.e. an error different to zero. In general, the error concept among systems is computed as differences among theirs outputs. In the observer and controller design when the error is zero then the system reach a required behavior (De Jesús & Ramírez-Treviño, 2001). In the context of diagnoser design to localize the fault transition and the place of fault is used an error structure introduced in (De Jesús & Ramírez-Treviño, 2001) and (Santoyo-Sanchez et. al, 2008), which is presented in the next definition. Definition 19 Let ),( 0 DD MQ and IPN where ),( 0 NN MQ and ),( 0 Tf Tf MQ are its subnets IPN diagnoser model and IPN fault model respectively. Structure Error between ),( 0 NN MQ and ),( 0 DD MQ is defined as ),( 0 EE MN where ),,,( EE EEE OITPN  with DE P P  , DE T T  ,   ZTPI EE E : ,   ZTPO EE E : and   ZTPC EE E : defined as:          Tf NN E I II I 00  ;          Tf EE E O OO O 00  and          Tf NN E C CC C 00  (37) The initial marking of IPN ZPM EE : 0 defined as:                  Tf N N E M M M M 0 0 0 0 - 0  (38) The marking at the k-th instant is:                  Tf k N k N k E k M M M M - 0  (39) A transition N j Tt  is enabled at marking E k M if N i Pp  , ),()( ji N i E k tpIpM  ; while that a transition D j Tt  is enabled at marking E k M if D i Pp  , ),()( ji D i E k tpIpM  . When a transition j t is fired, then a new marking 1k M is reached. This new marking is computed as: PetriNets:Applications518                            F k D k N k F NN E k E k C CC MM 00 1  (40) where N k  is an m-entry firing vector of structure ),( 0 NN MQ , N k  is an m-entry firing vector of structure ),( 0 DD MQ and F k  is an m-entry firing vector of structure ),( 0 Tf Tf MQ . Example 7: Consider the IPNs shown in Figure 13.a, 13.b and 13.c note that this IPN are ),( 0 DD MQ , ),( 0 NN MQ and ),( 0 Tf Tf MQ respectively. Structure Error between ),( 0 NN MQ and ),( 0 DD MQ is:                                                  0 0 0 0 0 1 - 0 0 1 - 0 0 0 0 0 Tf N N E M M M M  ;                 1 0 1 00 11 11 00 11 11 C        10 01 (41) which is showed in the figure 15. In this error marking there not are enabled transitions. Assume that from 0 M and N M 0 the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]00[ ]01[ 1 0 and  1N input u ; output N        TN TN y y ]01[ ]01[ 1 0 (42) In this case, T yy ]01[ 01  and )3,(C is the column of C corresponding to transition 3 t . And its output error between both systems is TN yyy ]01[ 1 1  . In this case an error is detected and the error marking is:                                                  1- 0 1 1 0 0 - 0 0 1 - 0 1 1 1 1 Tf N N E M M M M  (43) Fig. 15. Representation of the structure error model. In this marking, the enabled transitions are N Tt  2 and Tf Tt  3 , if 2 t fires into the structure error the new marking 1k M is reached. This new marking is computed as:                          1 1 0 00010 1- 0 1 2 T E CM (44) In the case of that 3 t fires into the structure error the marking:   T C 10000 1- 0 1 0 0 0                       (45) Then from E M 1 to reach the error zero, it is necessary fires the fault transition 3 t and 33 tp  is the fault place.  The following theorem characterizes the diagnosis based on structure error models. 4.4 Diagnose Fault Theorem 2: Let ),( ND QQ be a pair system-diagnoser with the state equations (46) and (47) respectively. Where the diagnoser is based on the idea that the diagnoser IPN ),( 0 NN MQ is embedded into the IPN system ),( 0 DD MQ and each faulty transition is non-manipulable (see the figure 13.a).  D Q )( 1 D k N D k DD k D k D k M CM y M            (46)  N Q )( 1 N k N N k NN k N k N k M CM y M            (47) If ),( 0 DD MQ is input–output diagnosable (Ramírez-Treviño et al., 2007) and the structure Error between ),( 0 NN MQ and ),( 0 DD MQ is defined as ),( 0 EE MN then it is possible to detect and isolate a fault into the system ),( 0 DD MQ . Proof. You can find the proof in (Santoyo-Sanchez et al., 2008).  Based on the theorem 2 the following algorithm is presented to detect and isolate error marking. Algorithm 1: Detecting and isolate error marking. Inputs: The IPN model of the pair system-diagnoser. Outputs: The error marking E k M , faulty place iF p and faulty transition . Procedure: 1. Define the structure error. 2. When 0   E k M then: FaultDiagnosisonElectricPowerSystemsbasedonPetriNetApproach 519                            F k D k N k F NN E k E k C CC MM 00 1  (40) where N k  is an m-entry firing vector of structure ),( 0 NN MQ , N k  is an m-entry firing vector of structure ),( 0 DD MQ and F k  is an m-entry firing vector of structure ),( 0 Tf Tf MQ . Example 7: Consider the IPNs shown in Figure 13.a, 13.b and 13.c note that this IPN are ),( 0 DD MQ , ),( 0 NN MQ and ),( 0 Tf Tf MQ respectively. Structure Error between ),( 0 NN MQ and ),( 0 DD MQ is:                                                  0 0 0 0 0 1 - 0 0 1 - 0 0 0 0 0 Tf N N E M M M M  ;                 1 0 1 00 11 11 00 11 11 C        10 01 (41) which is showed in the figure 15. In this error marking there not are enabled transitions. Assume that from 0 M and N M 0 the information provide by the IPN system is:  1 input u ; outpu t        T T y y ]00[ ]01[ 1 0 and  1N input u ; output N        TN TN y y ]01[ ]01[ 1 0 (42) In this case, T yy ]01[ 01  and )3,(  C is the column of C  corresponding to transition 3 t . And its output error between both systems is TN yyy ]01[ 1 1  . In this case an error is detected and the error marking is:                                                  1- 0 1 1 0 0 - 0 0 1 - 0 1 1 1 1 Tf N N E M M M M  (43) Fig. 15. Representation of the structure error model. In this marking, the enabled transitions are N Tt  2 and Tf Tt  3 , if 2 t fires into the structure error the new marking 1k M is reached. This new marking is computed as:                          1 1 0 00010 1- 0 1 2 T E CM (44) In the case of that 3 t fires into the structure error the marking:   T C 10000 1- 0 1 0 0 0                       (45) Then from E M 1 to reach the error zero, it is necessary fires the fault transition 3 t and 33 tp  is the fault place.  The following theorem characterizes the diagnosis based on structure error models. 4.4 Diagnose Fault Theorem 2: Let ),( ND QQ be a pair system-diagnoser with the state equations (46) and (47) respectively. Where the diagnoser is based on the idea that the diagnoser IPN ),( 0 NN MQ is embedded into the IPN system ),( 0 DD MQ and each faulty transition is non-manipulable (see the figure 13.a).  D Q )( 1 D k N D k DD k D k D k M CM y M            (46)  N Q )( 1 N k N N k NN k N k N k M CM y M            (47) If ),( 0 DD MQ is input–output diagnosable (Ramírez-Treviño et al., 2007) and the structure Error between ),( 0 NN MQ and ),( 0 DD MQ is defined as ),( 0 EE MN then it is possible to detect and isolate a fault into the system ),( 0 DD MQ . Proof. You can find the proof in (Santoyo-Sanchez et al., 2008).  Based on the theorem 2 the following algorithm is presented to detect and isolate error marking. Algorithm 1: Detecting and isolate error marking. Inputs: The IPN model of the pair system-diagnoser. Outputs: The error marking E k M , faulty place iF p and faulty transition . Procedure: 1. Define the structure error. 2. When 0   E k M then: PetriNets:Applications520 2.1 Faulty places are }1)({  i E k iF pMpp 2.2 Fi pp  faulty transitions are |{ kF tt  })(  F p . 3. Return E k M , F p and F t . According with the scheme of the figure 14 the diagnose algorithm has two parts; in the first one the algorithm 1 detects and locates fault through the structure error, also indicating faulty state. In the second one, it is necessary to define a diagnostic fault algorithm, which indicate the component fault and specify the kind of fault. In this case, it is necessary to consider the characteristics of power electrical network. 4.5 Diagnostic fault in power electrical networks Under the point of view of the analogical-digital conversion, the minimal elements of the power electrical network are: A) lines, B) sources and C) charges; using the methodology proposed in (Santoyo et. al, 2001) each minimal element is represented as Interpreted Petri Net. Additionally in (Santoyo-Sanchez et. al, 2008) the power flow from the generator to charge is considered as an element of the power electrical network. For illustrate the IPN modeling of power electrical systems consider the IPN model of the figure 16, which represent the IPN model of the power electrical network of the figure 8 only of power flow from the generator 1. Note that in the figure 16 for each line with a relay the fault behavior is modeled in two parts. The first one represents the fault window, i.e. the normal rate of relay; the second one represents when the fault condition is reached. To capture the protection zone by each relay (see the figure 6) in the context of IPN, in this chapter is proposed to define a new function Z  given by a matrix, where each row ),(  k Z represents the protection zone, which is defined considering the trajectory of energy distribution (because a relay can detect the fault in from of them) and its fault zone (first, second and third). Definition 20. Let }{:   ZZ a relation that indicates which lines is front other in second and third zone . The protection zone by each relay is defined as:  ),( ik z        case. otherin 0 ).( of place fault the is place if1 relay. of place fault the is place if1 kp thkp i i (48) Example 8. For illustrated the protection zone consider the IPN system of the figure 16.b. . In this case }5,4,3{)2(  , }5,4{)3(  , }5{)4(  and {})5(  . Then the protection zone induced by  is:                10000 11000 11100 11110 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 000000000000000000000000000000 z (49) For the relay 1, note that 32 p is its fault place, 33 p and 34 p are the faulty places of line 3 and 4 respectively. Assume that the marking is ]3,3430,31,32,3,27,28,29,5,20,22,251,3,5,10,1[ k M the compute of potection is   T kz M 0123)(  then relay 1(into the line 2) is in the third protection zone, relay 2 (into the line 3) is in the second protection zone, and finlay the relay 3 (into the line 4) is in the first protection zone.  When a fault occurs, the marking is analized to determine each fault zone (figure 6). Like a resulting of this process, each relay determines its tripping time. The relays acts instantaneously when the fault location at the first zone. Then the protection devices acts as fast as possible to disconnect the faulted element, the element into the first zone. The next algorithm captures this idea. Algorithm 2. Diagnosis Fault. Inputs: The IPN model of system, the structure error, error marking E k M , faulty place F p and faulty transition . Outputs: Faulty component F Comp , the sets Fault, F Comp oteccDPr and F Comp oteccIPr . Procedure: When 0   E k M do: 1. Compute the protection zone using z  . 2. Define  )({ FiiF tpcComp and i p is the place used to represent the system element i c and i c into the protection zone 1}. 3. Define the Protection behavior. 3.1 1)({Pr  iiiComp pMpcoteccD F , where i p is the place used to represent the disconnection of the system element i c by the fault in F Comp }. 3.2 1)({Pr  iiiComp pMpcoteccI F , where i p is the place use to describe how the electrical system distribute power electrical and i c indicates the electrical element that is stressed due to fault into F Comp }. 4. Diagnostic. Return the sets Fault, F Comp oteccDPr and F Comp oteccIPr . The algorithm 2 indicates fault and disconnection of elements by the protections occurrence in the electrical system; thus it is possible to distinguish between fault elements and consequences of the faults. In this case the IPN model and the algorithms (1 and 2) are desingned for a line, due to the line has three phases the model is generated for each phase (sequence positive, negative and zero). Finally using the information of the table 1 is specify the kind of fault. [...]... equilibrium is determined by partial differential equations; equivalent Petri net model for the interaction is given in figure 5 534 Petri Nets: Applications Fig 5 Petri net model of Pre-Predator interaction 5.1 Creating the model Petri net definition file (PDF) that defines the static Petri net graph of figure 5 is given below: % function: Petri Net Definition File (PDF) % filename: predator_prey_def.m... Creating a Petri net model consists of two steps: 1) Defining the static Petri net graph, and 2) Assigning initial dynamics in the main simulation file Step-1) Defining the Petri net graph in one or more Petri net Definition Files (PDF) : this is the static part This step consist of three sub-steps: a Identifying the basic elements of a Petri net graph: the places, b Identifying the basic elements of a Petri. .. Electrical Systems using Interpreted Petri Nets, Proceedings of IEEE International Conference on Emerging Technologies and Factory Automation, pp 538 – 526 Petri Nets: Applications 546, ISBN 978-1-4244-1505-2, Location Hamburg Germany, September 2008, IEEE Press, USA Sheng-Luen C., Chien-Chung W & Muder J (2003) Failure Diagnosis: A case study on Modeling and Analysis by Petri Nets, Proceedings of the IEEE... GPenSIM terminology, whereas in some other literature (e.g Colored Petri Net (CPN)) it is referred to as ‘guard-functions’) There can be a separate transition definition file for each transition in a Petri net model 530 Petri Nets: Applications 3.3 Natural language interface Users need not know Petri net mathematics when creating a Petri net model of a discrete event system GPenSIM offers a natural... http://www.informatik.uni-hamburg.de/TGI/PetriNets/ Pritsker Corporation (1990) SLAM II Quick Reference Manual Pritsker Corporation, West Lafayattee, IN, USA SIMCSRIPTII (2009) Available: http://www.simscript.com/ Wikipedia (2009) Available: http://www.wikipedia.org Wilkinson, D (2006) Stochastic Modelling for Systems Biology Chapman & Hall / CRC, NY 540 Petri Nets: Applications Assessing Risks in Critical Systems using Petri Nets 541... interest using the Assessment Framework 2.2 The Fluid and Stochastic Petri nets and its ability to represent Hybrid Systems 2.2.1 Petri nets and their evolution in the assessment of systems Petri nets (PN) are a graphical and mathematical formalism used in the specification, modeling, verification and analysis of systems properties Particularly, PN are used to model complex systems that have concurrent,... There may be a number of PDFs, if the Petri net model is divided into many modules, and each module is defined in a separate PDF While the Petri net definition file has the static details, the main simulation file contains the dynamic information (such as initial tokens in places, firing times of transitions) of the Petri net In addition to these files (main simulation file and Petri net definition files),... GPenSIM: A New Petri Net Simulator 537 400 Y1 Y2 350 300 Y 250 200 150 100 50 0 0 5 10 15 Time 20 25 30 6a) Composition of specimens Prey-Predator with time 400 350 300 Y2 250 200 150 100 50 0 50 100 150 Y1 200 250 6b) Prey-Predator equilibrium Fig 6 Simulation results 300 538 Petri Nets: Applications 6 Conclusion This chapter presents a new Petri net simulator, called General Purpose Petri Net simulator... voltages and currents measurements and its digital processing with a relay to maintain the operation of the power electric system 524 Petri Nets: Applications 7 References Aguirre-Salas L & Santoyo-Sanchez A (2009) Sequence-detectability analysis of Interpreted Petri nets under partial state observations, To be published in Proceedings of IEEE International Conference on Emerging Technologies and Factory... Stochastic Petri nets (and their evolutions) using reward structures demands the calculation of state probabilities πn(t) = Pr(N(t) = n) Vector π(t) = (1(t), 2(t), , m(t)), S = {1, 2, , m} can be calculated using 2 possible ways: Analytical Techniques and Simulation (Computer Science Department, 2000) Applying analytical techniques to solve stochastic Petri nets (and their evolutions) requires nets with .  , C and  are given by: (29) Petri Nets: Applications5 14 (a) (b) (c) Fig. 13. a) Interpreted Petri Net System C, b) A subnet of Interpreted Petri Net System C, c) Subnet induced. electric system. Petri Nets: Applications5 24 7. References Aguirre-Salas L. & Santoyo-Sanchez A. (2009). Sequence-detectability analysis of Interpreted Petri nets under partial state observations,. one or more Petri net definition files (PDFs); definition of a Petri net graph (static details) is given in the Petri net Definition File. There may be a number of PDFs, if the Petri net model