Hybrid Petri Nets: Stochastic and Deterministic Modeling for Power Systems

1. Hybrid Petri Nets:Stochastic and Deterministic Modeling for Power Systems Jared Luffman MSIM 752 11/26/2007 • Primary Document: • Dependability Analysis of Power System Protections Using Stochastic Hybrid Simulation in Modelica (2007) • Written By: • Luca Ferrarini, Juliano S.A. Carneiro, Simone Radaelli, and Emanuele Ciapessoni

2. Topics • Petri Nets • Deterministic and Timed • Stochastic • Hybrid Models • Application to Power Systems • Limitations • Literature Comparison • Deterministic and Stochastic Petri Net Models of Protection Schemes (1992) • Probabilistic Assessment of Transmission System Reliability Performance (2006) • Critique • Conclusion

3. Petri Nets • A bipartite graph G(V,E) where • V = PT • P is the set of places (represented with circles) • T is the set of transitions (represented with vertical bars) • E is the set of edges between P and T • Marking function M. Given μЄM, each μ is a function which assigns a positive integer value to each element of P • Μ is the marking of the graph • Μ is a function from P to the non-negative numbers giving the marking of the net • The marking is a vector μ = (μ1, μ2,… μn), where μi is the marking for the place pi • f(p) is the marking of the place p • Marking is represented on the graph with tokens (i.e. dots)

4. Petri Nets (cont’d) • Alternatively, a Petri Net is a 5-tuple • PN = (P,T,F,W,M0) • P = {p1,p2,…pm} is a finite set of places • T = {t1,t2,…tn} is a finite set of transitions • F  (P  T)  (T  P) is a set of arcs • W: F  {1,2,3,…} is a weighting function • M0: P  {1,2,3,…} is the initial marking • P  T =  and T  P = 

5. Deterministic and TimedPetri Nets • Basic Petri Nets are Deterministic in Nature • Each transition is defined precisely based on connectivity and tokens needed for transition • Given an initial condition, the exact system state at an arbitrary future time T can be determined • Timed Petri Nets becomes a 6-tuple system • PN = (P,T,F,W,M0,) •  = {1, 2,… n} is a finite set of deterministic time delays to corresponding ti • A transition tican fire at time T if and only if • For any input place p of this transition, there have been the number of tokens equal to the weight of the directed arc connecting p to tiin the input place continuously for the time interval [T − i, T], where iis the associated firing time of transition ti • After the transition fires, each of its output places, p, will receive the number of tokens equal to the weight of the directed arc connecting tito p at time T

6. Stochastic Petri Nets • Transition times can stochastic in nature • Deterministic transitions are still applicable • Given an initial condition, the exact system state at an arbitrary future time Tcannot be determined • Stochastic Timed Petri Nets become a 6-tuple system • PN = (P,T,F,W,M0,) •  = {1, 2,… n} is a finite set of stochastic distributions representing the time delays to corresponding ti • Each i can be a different distribution (i.e. uniform, normal, exponential) defining the necessary attributes for that distribution (i.e. (,), (,), (,)) • Transitioning follows the same rules as a Deterministic Petri Net, but i is defined by its i random distribution

7. Hybrid Models • Systems with continuous-time behavior that also experience event-driven behavior for discontinuous phenomena • Continuous-time systems are made up of elements that are dynamic in nature and must be recomputed at each time increment • Markings are real numbers • Transition firing is continuous • Discontinuous phenomena are discrete model elements that are introduced at random intervals into the system and typically have a short lifespan

8. Power System Modeling • Power systems are made up of many components that react differently to system events based on their design using reactive impedances (inductance and capacitance) • Power flow based on non-linear equations • System events (i.e. lightning strikes, equipment failures, grounded lines) can introduce voltage and current and/or change the network topology • Sample System • G: Generators • B: Buses • L: Lines • T: Transformers • I: Circuit Breakers • C: Consumers/Load • Protection Schemes are designed to open circuit breakers to limit equipment exposure to undesirable conditions

9. Power System Modeling:Stochastic Hybrid Petri Nets • Each component is designed as a Stochastic Petri Net • Petri Net states and transitions designed to incorporate protection schemes • Inputs from circuit analysis incorporated into Petri Net states and transitions • Stochastic Events pre-calculated and turned into a deterministic event array • Continuous-time model reduced by deterministic event array • Limits steady-state solutions being analyzed between events • Initiates continuous-time solving at t prior to the event • Stops continuous-time solving at t after system has stabilized after the event

10. Application • Run simulations to determine probabilistic indices • Expected Power Loss (EPL) • Total load detached from the power system in MW • Ci is the load lost (MW) in the ith simulation cycle • N is the number of cycles • Indicates the impact of hidden failures and cascading effect on system reliability • Expected Un-served Energy (EUE) • Expresses the total un-served energy to the utility in MWh • Ei is the un-served energy (MWh) in the ith simulation cycle • N is the number of cycles • Indicative of system damage by unavailability of service • Bus Isolation Probability (BIL) • Probability that one or more buses have been disconnected • Ii = 1 if one or more bars are disconnected in the ith simulation cycle Ii = 0 otherwise • N is the number of cycles • Identifies critical components/scenarios that isolate buses

11. Limitations • Power Flow Analysis • Non-linear solver needed to determine system power flows • Transient Fault Analysis • PDEs that require very precise data for each system component • Protection Schemes • Layers of protection • Zone 1, 2, and 3 coverage • Transient analysis needed with each protective measure taken • Cascading Failures • Need transient fault analysis and protection schemes to be precise • Reconstitution Schemes • How to return the system to an optimal steady state after equipment failures

12. Literature Comparison • Focused on Petri Nets applied to modeling Power System reliability • Document 1 • Deterministic and Stochastic Petri Net Models of Protection Schemes (1992) • L. Jenkins & H.P. Khincha • Document 2 • Probabilistic Assessment of Transmission System Reliability Performance (2006) • A.A. Chowdhury & D.O. Koval

13. Document 1 • Deterministic and Stochastic Petri Nets to model power system protection schemes • Zones of protection as timed stochastic processes • Interaction between different system elements

14. Document 2 • Probabilistic Reliability Modeling • Maintenance Outaging • Contingency Modeling • Specific Transmission System Outage Data • Probabilistic Indices • Reliability Specifications and Requirements

15. Document 2 (cont’d) • Multi-step Load Model • Annualized Probabilistic Indices • Total Expected Energy Not Supplied • EENS is computed for each load level, based on all contingencies, which caused a load loss at that load level

16. Critique • Major Premise • Stochastic Hybrid Modeling for Power System Contingency Analysis • Weak due to lack of concrete conclusions and too many simplifications to the system • No mention of system solution methods, protection scheme reactions, or reconstitution processes after the event occurs • Potential for Applications • Limited for power system analysis by software package (Modelica) shortfalls • Power System analysis requires transient/non-linear solver along with protection schema • Needs to analyze reconstitution of the system post-event • Use of a Hybrid Petri Net for deterministic events on a continuous model is promising

17. Critique (cont’d) • State of the Art • No impact on power system analysis • Already have tools to run contingencies for every element in the system • Not stochastic, but required by federal regulations • N-1 and N-2 analysis with projected stochastic events and load growth used for planning purposes • Writing • Average for an academic paper • Major premise and limited technical content could be followed • Broken English made some concepts hard to follow • Probably translated directly from Italian without major review

18. Conclusion • Overall • Left a lot to be desired • No definitive conclusion or discussion of how a stochastic hybrid model might improve power system dependability analysis techniques • Authors had limited electric power background • Alternate papers • Better job explaining how to apply Petri Nets to power system failures • Detailed discussion on the dependability of a power system based on simulated events

19. Questions