Hybrid Petri Nets Stochastic and Deterministic Modeling for

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

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

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 = (μ 1, μ 2, … 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)

Petri Nets (cont’d) • Alternatively, a Petri Net is a 5 -tuple – PN = (P, T, F, W, M 0) • P = {p 1, p 2, …pm} is a finite set of places • T = {t 1, t 2, …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 • M 0: P {1, 2, 3, …} is the initial marking • P T = and T P =

Deterministic and Timed Petri 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, M 0, ) – = { 1, 2, … n} is a finite set of deterministic time delays to corresponding ti • A transition ti can 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 ti in the input place continuously for the time interval [T − i, T], where i is 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 ti to p at time T

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 T cannot be determined • Stochastic Timed Petri Nets become a 6 -tuple system – PN = (P, T, F, W, M 0, ) – = { 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

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

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

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

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

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

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

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

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

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

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

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

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

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