Hybrid Petri Nets:Stochastic and Deterministic Modeling for Power Systems

Jared LuffmanMSIM 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 = PT

• 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)

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 =

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 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 numberof tokens equal to the weight of thedirected 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,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

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 cycleIi = 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 eachload level, based on allcontingencies, which causeda 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

Questions