Probabilistic simulation of power transmission systems affected …s-space.snu.ac.kr/bitstream/10371/153291/1/73.pdf · 2019-05-30 · Probabilistic simulation of power transmission

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13Seoul, South Korea, May 26-30, 2019

Probabilistic simulation of power transmission systems affected byhurricane events based on fragility and AC power flow analyses

Liyang MaPh.D. Student, Civil & Environmental Eng., Lehigh University, Bethlehem, USA

Vasileios ChristouPh.D. Student, Civil & Environmental Eng., Lehigh University, Bethlehem, USA

Paolo BocchiniAssociate Professor, Civil & Environmental Eng., Lehigh University, Bethlehem, USA

ABSTRACT: This paper presents a technique for probabilistic simulation of power transmissionsystems under hurricane events. The study models the power transmission system as a network ofconnected individual components, which are subjected to wind-induced mechanical failure and powerflow constraints. The mechanical performance of the transmission conductors are evaluated using anefficient modal superpoistion method and extreme value analysis. The fragilty model is then deveopedusing first order reliability theory. The asssumptions of the method are discussed, and its accuracy isthoroughly investigated. The component fragilities are used to map the damage of hurricane events tothe failure probabilities. The electircal performance of the components is modeled through an AC-basedpower flow cascading failure model, to capture the unique phenomena affecting power systems, such asline overflow and load shedding. The methodology is demonstrated by a case study involving ahurricane moving across the IEEE 30-bus transmission network. This technique aims at helpingdecision makers gain fundamental insights on the modeling and quantification of power systemperformance during hurricane events.

1. INTRODUCTION

Power transmission is the bulk transportation ofelectrical energy from power plants to electricalsubstations. The power transmission systems areamong the most critical infrastructures in mod-ern society, since most of the other infrastructuresrely on the electrical power supply to be func-tional. However, power transmission systems areextremely vulnerable to hurricane events. DuringHurricane Harvey, hundreds of high-voltage trans-mission lines experienced storm-related outagesand left thousands of customers without power.Power loss (i.e., unsatisfied power demand) is of-ten initiated by failure of individual components,such as support structures and conductors, trigger-ing cascading failures of the network.

At the component level, research has been con-ducted to study the fragility of poles (Shafieezadehet al., 2014) and towers (Fu et al., 2016). However,fragility curves for transmission conductors are notavailable. At the system level, failures of structuralcomponents are usually considered using fragilitymodels, and cascading failures were captured usinga DC power flow model by Ouyang and Duenas-Osorio (2014). Recently, AC power flow models,which render a better estimation of the system per-formance, have been adopted by Li et al. (2017) tosimulate power network dynamics. However, theaccuracy of these system analyses is undermined bythe lack of detailed structural analyses.

This paper first presents a framework for effi-cient fragility analysis of transmission conductors

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against hurricanes. The wind induced demands onconductors are characterized probabilistically usinga random wind field to capture the uncertainties ofthe wind fluctuation process and correlations. Ran-dom vibration theory is used to derive the closed-form distribution of extreme loading on conductors.Fragility curves are computed using first order reli-ability theory. With the obtained fragility models, atechnique is introduced for the probabilistic simula-tion and objective quantification of power transmis-sion systems performance. The proposed techniquetakes into account both the structural and electri-cal properties of the components and their networktopology. The relationship between reliability of in-dividual components and power network resilienceis addressed. Finally, a case study is conducted us-ing Monte Carlo simulation, studying the effect ofHurricane Katrina on an IEEE benchmark network.

2. FRAGILITY MODELS OF TRANSMIS-SION CONDUCTORS

The failure mode considered in this study is rup-ture of the conductor due to excessive wind load.A power network is formed by multiple transmis-sion lines, and each transmission line consists ofthousands of conductors connected in series. Al-though the failure of a single conductor can be con-sidered as a rare event, the probability of failure fora transmission line is not negligible, and such fail-ure can cause severe damage to the system. Thereare two difficulties in deriving the fragility modelfor transmission conductors: (1) predicting the loadresponse of a conductor with accuracy and effi-ciency, (2) and computing the probability of failurefor a conductor, a rare event, with enough accuracy.Figure 1 presents the conceptual flowchart of themethodology that has been developed to derive thefragility models of transmission conductors.

2.1. Wind field simulationIn a hurricane event, the largest component of theconductor’s displacement is due to the transversalwind flow in the direction perpendicular to the con-ductor’s plane. In this study, the wind direction isassumed to be perpendicular to the longitudinal di-rection of the conductor, which is the worst case

Time History Analysis Based on Nonlinear FEM

Catenary cable element Nonlinear time history analysis Aerodynamic damping

Time History Analysis Based on Modal Superposition

Modal frequencies and mode shapes Linear time history analysis Coupled aerodynamic damping

Extreme Value Analysis SDF of the response First passage problem

Wind Field Simulation Kaimal Spectrum Davenport coherence Multivariate simulation

Distribution of Demand Conductor type Span length Wind speed

Distribution of Capacity Rated strength Truncated gaussian Steel to aluminum ratio

Fragility curves MC simulation FORM

Figure 1: Flowchart of the methodology for fragilitymodels of transmission conductors

scenario. The mean wind profile follows the fa-mous power law (Hellman, 1916):

V (z) = V0(z/10)α (1)

where V0 is mean wind speed at 10 meters high, z isthe height of the conductor above the ground, andα is the an exponent that depends on the rough-ness of the terrain. The spectral density functionof the wind turbulence is modeled by Kaimal spec-trum (Kaimal et al., 1972):

f Sv( f )u2∗

=200 f z/V

(1+50 f z/V )5/3 (2)

where f is the frequency in Hertz, and u∗ is theshear velocity of the flow. The coherence func-tion, which is a measure of the degree to which tworecords of wind fluctuations at different locationsare correlated is defined by Davenport’s exponen-tial function model (Davenport, 1968). The windfield is modeled by a stationary, Gaussian, one-dimension and multivariate random process. Thesimulation algorithm used in this study is the onedeveloped by Deodatis (1996) based on the spec-tral representation method and fast Fourier trans-form technique. Figure 2 presents one realization

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Figure 2: Realization of wind field simulation

of the wind field simulation correlated in time andspace. The simulated wind field covers 500 meterdistance and a time span of 10 minutes.

2.2. Time history analysis with nonlinear FEMThe FEM model is built in the OpenSees platform(McKenna et al., 2010), and catenary element isused in this study to model the conductor (Abadet al., 2013).To account for the pretension force ofthe conductor, the unstrained length of the cable isdetermined. The conductor is then divided into anumber of catenary cable elements equally spacedalong the x direction, as shown in Figure 3. They coordinates for each node can be determined fol-lowing the procedure described by Irvine (1981).After specifying the nodal coordinates, the gravityload are applied to the conductor and the preten-sion force is automatically implemented as reactionforce at the supports. The nonlinear FEM time his-tory analysis is performed by solving the equationof motion. The external force per unit length iscomputed as :

Ft+∆t =ρ

2CDD

[(V +Vt+∆t − zt+∆t)

2 +(yt+∆t)2](3)

y

x

z

Figure 3: Finite element model of the conductor sub-jected to dynamic wind force

where ρ = 1.25kg/m3 is the air density; CD = 0.9is the static drag coefficient; D is diameter of theconductor; Vt+∆t is the turbulence at time instantt + ∆t; zt+∆t and yt+∆t are the nodal velocities attime instance t +∆t in z and y directions which canbe extracted during the time history analysis. Thenodal velocities at time t +∆t are approximated bythose at the previous time step, given that the timestep ∆t is small enough. Since the external forcedepends on the nodal velocities of the cable, theaerodynamic damping effect has been accountedfor. Even though the time history analysis usingnonlinear FEM is time consuming, it is consideredthe most accurate, as the geometrical nonlinearityof the cable is taken into consideration.

2.3. Time history analysis with modal superposi-tion

In this section, the time history analysis is carriedout using the modal superposition method (Wanget al., 2017). The first step of the method is to de-termine analytically the state of the conductor un-der static mean wind speed and calculate the naturalfrequencies and mode shapes based on the theoret-ical solutions (Irvine, 1981). The coupled modaldamping ratios are derived to take into account theaerodynamic effect. Then, the conductor is mod-eled as a linear system characterized by those dy-namic properties, and the time history analysis canbe performed. To validate the linear assumption ofthe modal superposition, the time history responseof the modal superposition method is compared tothe response of nonlinear FEM, where geometricnonlinearity is considered. Multiple tests with dif-ferent cables and winds have shown that the modalsuperposition method is always able to accuratelycapture the peak force, but deviates from the FEMresults when the response is relatively small, asshown in Figure 4. This is probably due to the factthat the conductor dynamic properties at the meanwind situation are similar to the dynamic proper-ties under extreme loadings, but different from thedynamic properties under relatively small loadings.At the mean wind state the conductor is alreadystretched tight, therefore in the high wind statethe dynamic properties do not change significantly.However, under low wind loading the conductor

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Time(sec)

Tension(kN)

Figure 4: Comparison of conductor tension responsebetween FEM and modal superposition method

becomes slack and its dynamic properties deviatefrom the linear estimation. Since the assessmentof fragility curves requires to determine only thepeak response, the modal superposition method isconsidered sufficiently accurate to be used to eval-uate the maximum response of conductors during ahurricane event. The modal superposition methodnot only reduces the computational time for timehistory analysis, but more importantly, facilitatesthe computations in the frequency domain, whichis important for the next step.

2.4. Extreme value analysisThe spectral density matrix of the modal displace-ment vector is calculated as:

Sq( f ) = H( f )SQ( f )H( f )∗ (4)

where H( f ) is the transfer matrix and SQ( f ) is thespectral density matrix of the generalized force vec-tors. The spectrum of the total force response canbe determined as:

SN( f ) = rTNSq( f )rN (5)

where rN is the modal participation coefficient forthe conductor tension response N. The variance ofN is then computed by integration of SN( f ) and thevariance of N can be computed by integration ofSN( f ). Then the problem falls under the categoryof "first-passage problems" in the theory of randomvibrations. The expected number of up-crossings oflevel a per unit time is:

v+a =1

2π

σNσN

exp(− a2

2σ2N) (6)

It is well known that the probability of up-crossinglevel a in the interval 0 < t < T0 is

P(T0) = 1− exp(−v+a T0) (7)

For a fixed T0, Eqs. (6)–(7) provide the distributionof v+a , a, and, in turn, of the peaks N +a:

Si = N +

[−2σ

2N ln

(2π(v+a )iσN

σN

)]0.5

(8)

where N is the mean tension response under meanwind speed, σN and σN are functions of the prop-erties and location of the conductor as well as theintensity measure of the hurricane. (v+a )i is the ithrealization of the random variable v+a and Si is theith realization of the conductor demand. Maximumsustained wind speed, which is the highest aver-age wind over a one-minute time span, is chosenas the intensity measure in this study. The maxi-mum sustained wind speed is then convert to theaverage wind speed over 1 hour. The first rea-son for this conversion is that the Kaimal spec-trum requires the wind speed averaging period tobe at least 10 minutes. The second reason lies inthe concept of spectral gap. According to Van derHoven (1957) there exists a spectral gap at around1 hour. The presence of this spectral gap indicatesthat there is much less variability in the mean windspeed of 1 hour than the mean wind speed averag-ing over other periods. Figure 5(a) shows the de-mand PDFs of conductor ‘Drake’ subjected to dif-ferent maximum sustained wind speeds with samespan length. The probability distributions of the de-mand are sharp when the wind speed is relativelylow, but under large wind speed the probability dis-tribution becomes wide and the variability increasessignificantly. It is noted that most codes are de-veloped based on the work by Davenport Daven-port (1964), assuming extremely narrow probabil-ity distributions of the demand. The assumption isvalid for relatively low wind intensity, but may notbe valid when the intensity measure is large. Fig-ure 5(b) presents the demand PDFs of five typesof conductors with the same span length and sub-jected to the same maximum sustained wind speedof 50m/s.

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(a)

(b)

Figure 5: (a) Conductor ‘Drake’ demand distributionwith different intensity measure (b) Demand distribu-tion of five types of conductor with the same intensitymeasure

2.5. Capacity of transmission conductorsMost of the conductor producers specify conduc-tor strength as a single value: rated strength. Therated strength represents the lower exclusion limitof the conductor strength and is calculated in ac-cordance with specification requirements. There-fore, rated strength cannot be directly used to rep-resent the breaking force (capacity) of the conduc-tor. The breaking force of the conductor in thisstudy is modeled by Monte Carlo simulation withthe ASTM rule as:

Ri = nawS(aw)

πd2(aw)

4+nswS(sw.1%)

πd2(sw)

4(9)

where Ri is the ith realization of the conductorbreaking force; daw and dsw represent the aluminumwire diameter and steel wire diameter respectively;S(aw) is the breaking stress of individual aluminumstrands; S(sw.1%) is the stress in steel strands at1% extension. daw, dsw, S(aw), S(sw.1%) are allrandom variables with truncated normal distribu-tion. The parameters of these distributions are takenfrom Farzaneh’s research (Farzaneh and Savadjiev,2007). Figure 6 shows the realizations of the MonteCarlo simulation of the conductor breaking force.

Figure 6: Distribution of the conductor capacity

Each vertical line in the figure represents the ratedstrength of one conductor.

Once the distribution of demand and capacity ofthe conductor are obtained, the probability of fail-ure can be computed by Monte Carlo simulation.However, the failure probability of the conductor isvery small (often in the order of 10−7), thereforeconducting Monte Carlo simulation can be verycostly and the first order reliability method (FORM)is used to replace Monte Carlo simulation. Theimplementation of FORM is facilitated by UQlab(Marelli and Sudret, 2014). To test the validity ofFORM, a large scale Monte Carlo simulation hasbeen conducted and its results are compared againstthe results from FORM. FORM has very good ac-curacy for the computation of the failure probabil-ity. In this study, the fragility curves of conductorsare computed and the uncertainties of wind loadand conductor capacity are considered. The nonlin-ear FEM dynamic model is replaced by the modalsuperposition method. Figure 7 shows the fragilitycurves of five types of transmission conductor withspan length of 300 meters. The wind direction isoften unknown and therefore assumed to have uni-form distribution. The fragility curves indicate thatthe failure of the conductors is very sensitive tothe intensity measure. Once the hurricane intensityreaches the critical range, large amount of conduc-tor damage can be expected. Span length also hassignificant impact on the fragilities.

3. PROBABILISTIC SIMULATION OFPOWER TRANSMISSION SYSTEMS

This section presents the methodology to conductprobabilistic simulation of power transmission sys-

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Figure 7: Fragility curves of five types of conductorwith span length of 300 meters

tems incorporating the structural fragility modelsdeveloped in Section 2 with a power system modeland power flow models. Figure 8 presents theflowchart of the methodology. The first step is toidentify the critical components in the system andextract the hurricane intensity measure from histor-ical data. The network topology (i.e., connectivity)is also modeled. The second step is to conduct dam-age assessment of the structural components withthe fragility model. Finally, the failure of the powersupply due to disconnection of the electrical com-ponents and load balancing is simulated by an AC-based power flow model. Monte Carlo simulationis conducted to quantify the power loss and assessthe probabilistic characteristics of the system per-formance.

Hazard and power system model➢ HAZUS-MH hurricane record➢ Power system components

Component fragility model➢ Transmission tower fragility function➢ Conductor fragility function

Power network response model➢ AC based power flow model ➢ Cascading failures and load shedding

Methodology

➢ Monte Carlo simulation ➢ Probability assessment of power system

Numerical Application

Figure 8: Flowchart of the methodology for probabilis-tic simulation of power transmission systems

3.1. Hazard and power system modelThis methodology requires to model the regionalhazard with a specific scenario, which can be a

historical event, or a simulated intensity measuremap. In particular, the intensity measure used forthis analysis are the peak gust speed and maximumsustained wind speed. The power transmission sys-tem is modeled by power plants and transmissionsubstations connected by high-voltage transmissionlines, which include the transmission support struc-tures and the conductors between the support struc-tures. In the graph model, power plants are nodesgenerating real and reactive power with specificpower generation capacity; transmission substa-tions are nodes receiving power from power plantsand then supplying the low-voltage power to dis-tribution systems with certain load demand; trans-mission lines are links between power plants andtransmission substations, with certain load capac-ity. All the electrical properties of the componentsin power transmission systems are modeled usingMATPOWER (Zimmerman et al., 2011). Failure ofa link is determined by the damage state of trans-mission support structures (i.e. transmission towersetc. ) or conductors along the link. The conductorsand support structures are considered in series, thuscollapse of a single support or conductor trips thewhole line.

3.2. Power network response modelAn AC-based power flow model is used to cap-ture the power system response after componentfailures due to a hurricane. In this study, the per-formance of the power system is measured un-der the following assumptions: (a) the vulnerabil-ity of power plants and substations is negligible;(b) the damage state of the transmission line is bi-nary: functional or failed; (c) network operatorsand automated switches have enough time to inter-rupt power supply to certain areas to prevent fur-ther failures of the network. In this situation, thisstudy leverages MATPOWER to solve the AC op-timal power flow (ACOPF) problem, carrying outan optimization aiming to maximize the total powerdemand satisfaction in the system. This is achievedby changing load demand at the substations andpower generation at the plants, while consideringall the necessary physical constraints. The failedbranches determined by the fragility model are re-moved, and sub-grids are formed automatically by

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Figure 9: Hurricane Katrina wind field and georefer-enced IEEE 30-buses network

appropriate codes. Power redistribution in the sub-grids is computed and load shedding is conductedwhen ACOPF cannot converge. The simulationstops when all the sub-grids achieve a stable powerflow and the power loss for the whole network iscomputed and stored.

4. NUMERICAL APPLICATIONThe IEEE 30-bus system with 6 generators, 20loads and 41 transmission lines which represents aportion of the American Electric Power (AEP) net-work is used as a case study, in its version adaptedby Zimmerman et al. (2011). The wind field dataof Hurricane Katrina, which hit the Southern US in2005 is obtained from the HAZUS database. Theintensity measures are obtained at the resolutionof a census tract. The IEEE 30-bus system withpower demand of 189.2 MW can serve approxi-mately a population of 700,000. The bus system isselected and projected onto costal Mississippi andLouisiana, as shown in Figure 9, where a compa-rable population lives. The IEEE 30-bus system isconsidered a good approximation of the real powernetwork in the studied region. The power networkconsists of 4656 transmission support structures,whose fragility models are governed by peak gusts(Quannta-Techonology, 2009), and 4615 spans ofconductors, with span length of 400 meters andfragility models built following the procedure de-scribed in Section 2. The results of the Monte Carlosimulations are presented in Figure 10. To inves-

Power loss (MW)

PD

F

Figure 10: PDF of system power loss

tigate the influence of each type of structural com-ponent on the system performance, two simulationsare conducted for two different types of conductors.As shown in Figure 10, the probabilistic distribu-tion shifts to the right when the conductors typechanges from ‘Coot’ to ‘Tern’, because conductor‘Tern’ has lower probability of failure. In addition,changing the conductor type also changes the prob-abilistic distribution completely. This indicates thatthis power system is highly sensitive to conductortypes. Therefore, detailed information and rigor-ous fragility models about each component in thepower system are needed in order to evaluate thepower system behavior accurately.

5. CONCLUSIONSThis paper introduced a method for efficient com-putation of transmission conductor fragility and amulti-scale, probabilistic methodology to assess theperformance of a power system subjected to a hurri-cane event. The proposed methodology combines,in a coherent way, the electrical properties of thepower system and the structural behavior of the net-work components. The response of transmissionconductors is computed using the modal superposi-tion method, which enables the extreme value anal-ysis using random vibration theory. The demandand capacity of the transmission conductors are ob-tained and their probabilities of failure are com-puted by FORM. The obtained structural fragilitiesare used to map the hurricane intensity measure tocomponent damage in the power network by dis-

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crete event simulation. Then, an AC-based powerflow analysis is performed to simulate the load re-distribution process within the damaged power sys-tem. A case study using the IEEE 30-bus systemwith wind field data from hurricane Katrina wasconducted to illustrate the methodology. This com-prehensive model, can be used for various types ofanalyses and offers a protocol to quantify powersystem performance.

6. ACKNOWLEDGMENTSThis work is part of the “Probabilistic Re-silience Assessment of Interdependent Systems(PRAISys)” project (www.praisys.org). The sup-port from the National Science Foundation throughgrant CMS-1541177 is gratefully acknowledged.

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Probabilistic simulation of power transmission systems affected …s-space.snu.ac.kr/bitstream/10371/153291/1/73.pdf · 2019-05-30 · Probabilistic simulation of power transmission