Reliability and Vulnerability Analysis of Electrical Substations and

Transmission Towers for Definition of Wind and Seismic Damage Maps for

Mexico

Alberto López López1, Luis E. Pérez Rocha

2, David de León Escobedo

3 and Jorge Sánchez

Sesma4

1Researcher, Civil Engineering Department, Instituto de Investigaciones Eléctricas, Cuernavaca,

Morelos, México, alopezl@iie.org.mx 2Researcher, Civil Engineering Department, Instituto de Investigaciones Eléctricas, Cuernavaca,

Morelos, México, lepr@iie.org.mx 3Engineering School Director, Universidad Autónoma del Estado de México, México,

daviddelen@gmail.com 4Researcher, Instituto Mexicano de Tecnología del Agua, Cuernavaca, Morelos, México,

jsanchez@tlaloc.imta.mx

ABSTRACT

Reliability and vulnerability analyses are proposed as a step forward to develop optimal

recommendations for design of electrical facilities under seismic and wind hazard in Mexico.

Annual failure probability maps are built according to the dominant hazard for the different

regions in Mexico. These maps are based on the calculated annual failure probabilities for a

typical electrical substation and a typical tower for power transmission. The failure probability

considers the vulnerability features of these structures and the hazard occurrence for specific

sites in Mexico. Two design levels are considered for these structures in order to make cost-

effective recommendations. In future stages of the work, structure and failure consequences

costs will be introduced as well as combination of wind and seismic hazards, to generate optimal

design strategies for these structures in Mexico.

INTRODUCTION

Recent damages on electrical facilities in Mexico, after the occurrence of strong hurricanes, has

produced a growing interest on updating the design and assessment procedures to control the

safety of these structures, by incorporating risk and reliability tools [1,2] that have given proof of

their capabilities on the seismic and offshore engineering fields.

CFE, the Federal Electrical Commission in Mexico, has requested to the IIE (Institute of

Electrical Research) to update his current guidelines to design [3,4] substations and power

transmission towers located at ones with high wind and seismic hazards.

The structural reliability is the probability that the structure does not reach a limit state

(e.g. failure state) during a given period. One advantage of measuring the structural safety

through this reliability is that it may be a representation of the overall level of safety of complex

structural systems through a single number. Another advantage is that the uncertainties inherent

in the design process are taken into account explicitly, objectively, and systematically. Such

uncertainties are related mainly to the randomness of the intensities of loads which, in the case of

our study, are referenced to wind speeds or earthquake accelerations, and to a lesser extent to the

variability in resistance of the material elements of the structure.

Since 1969, the use of reliability concepts to determine structural safety [5,6], which are

now implicit in the structural design specifications in Mexico, have been proposed for the first

time. Recently, for the petroleum industry in Mexico [7], design specifications and assessment of

offshore platforms and oil pipelines with reliability bases are being proposed. To handle the

reliability of structures in a systematic way [8] and its application in major infrastructure works

[1,2,9] proposals have also been presented for use in design standards.

The purpose of this paper is to present the reliability assessment that has been applied to

lattice frame and truss structures used in electrical transmission systems in order to evaluate their

wind and seismic vulnerability as a function of failure probability of the structure, as well as the

corresponding wind and seismic damage maps for Mexico. This methodology for two different

typical structure types used in Mexico, one for a lattice self supported transmission tower and the

other for a lattice frame of an electrical substation, has been applied. In these cases, the structural

systems (electrical subsations or transmission line towers) have been considered as independent

of the whole electrical system.

RELIABILITY ANALYSIS OF LATTICE STRUCTURES

The failure probability of a structural component is defined as the probability of occurrence of a

failure event when the load C exceeds the resistance R. Taking C and R as statistically

independent and normally distributed random variables, the failure probability is expressed as:

R)P(Cf

P >= (1)

The evaluation of this probability is simplified when a structural system is modelled as a

finite multi-component structure, as the lattice framed and truss structures, and their structural

failure is governed by several dominant failure modes. For the structures of interest, these failure

modes are in terms of the axial and/or bending forces distributed on the structural members or

components. The failure modes due to joint failures and foundation failures must be also

considered but, at present, they are out of scope for this study.

In this way, the failure probability might be obtained through different ways, this can be

done through a safety margin, denoted as M = R-C, which is considered as normally distributed

variable. However, if the variables C and R are considered lognormal (which frequently occurs

given that the lognormal variable is only defined for positive values and many engineering

quantities as well), the failure probability is defined by considering a safety factor,θ , which is

usually expressed as R/Cθ = .

As the system failure probability requires the combination of component failure

probabilities, 1st order bounds on

fP [8] are often used in practice, and here the Cornell’s

reliability index, β, and its relationship with the failure probability, f

P , are introduced:

( ) 2CCV2

RCV/CM~

/RM~

lnβ += (2)

Φ(β)1Φ(-β)f

p −== (3)

where R

M~

and C

M~

are the medians of the resistance and the load, while R

CV and C

CV are

the coefficients of variation of the same variables, respectively. The medians are calculated with:

2X

CV1/Xµ

XM~

+= (4)

where Xµ is the mean value and

XCV is the coefficient of variation of the variable X, which

can be C or R. This approach is convenient in practice and can be applied for lattice structures

exposed to seismic and wind hazard. Thus, an event called failure occurs when R < C and thus

the failure probability is related to Equation 3.

However, when the failure event depends on the independent failure of several structural

members, usually seen in statically indeterminate structures, or when the system failure may

have several realistic independent ways to reach failure, the failure probability requires the

consideration of combination of probabilities of events that make possible the global failure

event. For example, the following expression:

)4

F3

F2

F1

P(Ff

P ∩∩∩= (5)

indicates that the failure of a structure depends on four local failures occurring simultaneously

which are independent, given by the events 1F to 4F and, therefore, the system failure is

governed by the failure of these four critical structural components. For this study, the

component failure event is defined as the one in which the load exceeds the resistance of the

critical structural components, namely those with the largest working stress ratios. Also,

according to conventional practice, it has been assumed that both the load and the resistance are

lognormal random variables. In the next subparagraph a practical procedure to evaluate global

failure probability for framed or truss lattice structures is presented.

PROCEDURE TO EVALUATE THE PROBABILITY OF FAILURE

The typical procedure for evaluating the probability of failure is as follows: 1) The type of probability distributions of variables of load and resistance are

established as lognormal.

2) The averages, standard deviations (and other statistical parameters, such as

medium, if necessary) of variables of load and resistance are calculated. Data

from literature or experience is compiled to estimate the statistics of the

resistance. For this work the relations of standard deviations that were considered

are: for loads V

2σFσ = and for resistances:

yfσ

Rσ = , where

Vσ and

yfσ

are the standard deviations of the maximum wind velocity and the steel yielding

stress, respectively. The corresponding coefficients of variation considered, for

force and resistance, are: V

2CVF

CV = and

yf

CVR

CV = . Therefore, the mean

value is ( ) σ/CVIµ = , for which one value is evaluated for each structural response

analysis for the variable of intensity "I".

3) Several response analyses of the structure of interest are carried out to obtain a

series of loads that take the structure to an instability condition or out of limit

service condition. For this, for each structural analysis and intensity level of

hazard, I, the critical component is taken out from the model (this correspond to

the element that have the highest working relationship) and when global failure

condition is reached, the reliability index, β, and corresponding probability failure

is evaluated by Equation 3 for each critical component from the initial model.

4) Then, Equation 5 is applied to evaluate the global failure probability.

5) Finally, to build a vulnerability function for multiple values of the hazard

intensities, the procedure is repeated.

This may be applied to the case of seismic and wind hazard, both for substations as towers and

other structures. The application of this procedure is shown in the next section to build

vulnerability functions in terms of the failure probability of structures.

LATTICE STRUCTURES VULNERABILITY ANALYSIS

The terms vulnerability and fragility are often used interchangeably. Both are intended to

describe the susceptibility to structural damage. Recently, a probabilistic character was added to

both terms. The difference is slight and the vulnerability may be interpreted, in a general sense,

when a damage limit state is exceeded, while the fragility tends to apply to collapse limit states.

Although there are several concepts on structural vulnerability, the ones that are often

used involve descriptions of the variation of an index of damage or the failure probability

regarding the intensity of the hazard. In this case, the present study considers the function of the

failure probability as function of the maximum wind speed or the maximum pseudoacceleration

associated to a structural period of interest. The characteristics of this function represent the

peculiarities of how the tower or substation structures are vulnerable to increasing levels of

maximum wind speed or pseudoaccelerations. The usefulness of knowing these peculiarities is

that it is possible to identify the ranges of wind speed for which the failure probability increase

faster. The ordinate, the slope, and the curvature of the vulnerability function allow for the

calculation of cost / benefit rates for which it is possible to establish measures of the reliability

cost and make a decision about choosing the most efficient structural alternative for the

geographical zone where the facility will be installed.

The typical shape of the curve is continuously growing with positive curvature for low

wind speeds and, after an inflection point for certain hazard intensity, the curvature becomes

negative and the ordinate values become asymptotic to the value 1. The vulnerability of

structures is important in proposing improvements in current design specifications, such as those

for electrical substations structures, self supported towers and metallic posts [4].

WIND VULNERABILITY FUNCTIONS

TYPICAL SUBSTATION

The substation type known as the double-switch, capable of 400kV, was analysed and two

designs were considered: for wind speeds of 200 to 300 km/h. The response analysis was

performed using a structural model in a commercial code [10] for several maximum wind

speeds. The complex consists of five frames made of columns and beams, where frame 1 and

frame 3 are repeated, as shown in Figure 1.

Figure 1: Substation of 400kV

Following the procedure described above, the resulting vulnerability curves, for the

designs of 200 to 300 km/h, considering the probability of failure of the critical elements are

shown in Figure 2.

Figure 2: Substation failure probability for the designs of 200 to 300 km/h, considering wind hazard.

This figure shows the typical behavior in "S" where the curve grows with a positive

curvature for small values of wind speeds, then undergoes a change of curvature to finish in an

asymptotic trend towards the unitary probability of failure, for high values of wind speed. Also,

it appears that the vulnerability curve for the design 300 km/h lies below that of 200 km/h, as

expected, and the difference becomes more significant for wind speeds between 180 and 380

Km/h. It can be remarked that tension in cables were taken into account regarding the wind

intensity considered. By obtaining several vulnerability curves for different designs of the same

type of substation, the appropriateness of the safety standards for the structure or of a particular

site may be checked to find out if it is under-or over-designed.

FRAME 2

FRAME 3

FRAME 1

FRAME 3

FRAME 1

TYPICAL TOWER

The types of tower called: 4YR23 N20C1 (taller-size) and 4YR23 N0C1 (lower-size) were

considered. The tower 4YR23 is for 400 kV, with 2 circuits, 3 drivers per phase. The

vulnerability curves, for two designs, 120 km/h and 160 km/h (tall and low tower), are shown in

Figures 3 and 4. In the latter case, the vulnerabilities are almost equal for wind speeds less than

or equal to 180 km/h. For speeds between 180 and 280 km/h there are significant differences

between vulnerabilities of designs, being higher for 240 km/h. For speeds greater than 300 km/h,

the vulnerabilities reach the failure probability. Also it can be noted that tension in cables were

taken into account regarding the wind intensity considered.

Figure 3: Vulnerabilities for low (left) and tall (right) towers under wind hazard, designs of 120 km/h

Figure 4: Vulnerabilities for towers under wind hazard, design of 160 km/h

SEISMIC VULNERABILITY FUNCTIONS

TYPICAL SUBSTATION

In a similar way to what has been described for wind, vulnerability analysis of the substation of

400 KV double switch, was performed assuming that the seismic intensity is the spectral

pseudoaccelerations to the corresponding structural period, which in this case is 0.5 s. These

pseudoaccelerations were: 0.15g, 0.25g, 0.35g, 0.45gy 0.55g. The vulnerability was measured as

the probability of failure of the most critical element (see Figure 5).

The pseudoacceleration is the product of the frequency to the square and the relative

displacement. This amount is commonly used because it is proportional to displacement and in

magnitude, which is close to the absolute acceleration.

Here, the typical "S" curve behaviour is again observed, where the curve grows with

positive curvature for small values of soil pseudoacceleration; later on it experiences a change of

curvature to finish in the asymptotic form, for high values of soil pseudoacceleration. Since the

earthquake does not govern the design of the towers, seismic vulnerability of the towers was not

calculated.

Figure 5: Seismic vulnerability of the substation for the designs of 200 and 300 km/h (Dis200 and Dis300)

APPLICATIONS OF THE FUNCTIONS OF VULNERABILITY FOR RISK EVALUATION IN

STRUCTURE SUBSTATION

This section describes how to obtain the probability of failure from the annualized rate of failure,

once the resistance design is described as an uncertain variable. It will be taken as a starting

point, the exceeding rate of expected value of the losses.

EXCEEDANCE LOSSES RATE

Consider a function of loss that depends on the intensity of the danger, in this case, the seismic

intensity. Such functions are known as functions of vulnerability and they relate the intensity of

the hazard with a measure of expected losses, which can be treated as number of victims, damage

index, the amount of economic losses or any other indicator describing the losses particularly of

interest.

If the rate to exceed the intensities is known, then it is possible to derive the rate to

exceed the losses throughout the following the expression:

∫ >−=∞

0

I)dI|yPr(YdI

(I)dν(y)

λ (5)

where:

ν(y) =exceedance losses rate

λ(I) = exceedance intensity rate

Pr(Y>y|I) = the probability that the losses are greater than y, given that an event intensity “I” has

occurred.

In this expression, the term d (I)

dIdI

λ−

represents the number of events per year who

have intensities in the interval between I and I dI+ . By combining this term with Pr(Y y | I)> , the

product is the number of times that the level of loss is exceeded due to the occurrence of events

with intensities in the mentioned interval. The integral goes from 0 to ∞, to cover the number of

times per year that the loss level “y” is exceeded.

FAILURE RATE AND FAILURE PROBABILITY

Sometimes it is of interest to quantify the number of failures that can be expected, under the

action of the hazard and a restriction in the design as a resistance provided by design. In this

case, the number of times that a failure occurs is equal to the number of times that the intensity is

greater than the resistance. This means, I R> being R the actual resistance of the structure,

which may be an uncertain quantity. Therefore, the failure rate can be represented as:

0

d (I)µ(R) Pr(I R)dI

dI

∞ λ= − >∫ (6)

If the hazard constitutes a Poisson process, then the probability that at least a failure

occurs in a Te period is: µ(R)Te

fP 1 e−

= − (7)

If (R )Teµ is small, then the probability of failure is:

fP µ(R)Te= (8)

This means that, when the annual failure rate is small, then the annual probability of

failure correspond to the annual failure rate. For optimal design maps, the required resistance

varies according to the geographic location. It was assumed that the facility is a structure of the

group B and that design resistance (optimal) is the expected value of a log normal distribution

and the standard deviation of the resistance is Rσ 0.5= .

A failure rate map for hard soil, throughout the country, may be obtained for a given

dominant hazard, by developing all the necessary failure probability functions for the considered

facility. For each case, a failure probability function for given intensity is considered because the

structural resistance is implicit. Therefore, Equation 6 becomes:

0 f0

d (I)µ(R ) P (I)dI

dI

∞ λ= −∫ (9)

PRACTICAL APPLICATIONS TO EVALUATE ANNUAL FAILURE RATES

FAILURE PROBABILITY UNDER EARTHQUAKE IN ELECTRICAL SUBSTATION

Llet´s consider the vulnerability of the substation frames. It is of interest to evaluate the failure

probability under seismic loads for the designs of 200 and 300 km/hr. For these designs the

vulnerability is represented through the curves shown in Figure 5. In this case, the dynamic

response of the framework was evaluated and it was determined that the dominant period is Te =

0.5. Therefore, the seismic hazard was marked with curves threat corresponding to Te = 0.5 s

throughout the country. The map corresponding to 200 km/h is shown in Figure 6. In this case, a

simple examination shows that the failure rates would be very low, on the order of a failure every

5000 years for the areas of greater seismic activity. For the design to winds of 300 Km/h, Figure

7 shows that failure rates would be even lower for these areas, on the order of a failure every

15000 years.

FAILURE PROBABILITY UNDER WIND LOADING IN ELECTRIC SUBSTATIONS

Now, let´s consider the vulnerability of the substation frame against wind action. The failure

probability will now be represented by the curves shown in Figure 2, for designs of 200 and 300

km/h. In this case, the dynamic response was evaluated under the action of wind hazard

considering the hazard curves throughout the country. The map corresponding to 200 km/h is

shown in Figure 8. In this case, the examination shows that failure rates would be higher than

those obtained by earthquake, on the order of a failure every 50 years for the areas of largest

wind hazard. For the design of 300 km/h, which is illustrated in Figure 9, it is shown that failure

probabilities are lower for these areas, on the order of a failure every 500 years.

Figure 6: Annual failure probability in substations designed for 200 Km/h, under seismic loading

Figure 7: Annual failure probability in substations designed for 300 km/h, under seismic loading

Figure 8: Annual failure probability in substations designed for 200 km/h, under wind action

Figure 9: Annual failure probability in substations designed for 300 Km/h, under wind action

FAILURE PROBABILITIES OF TALL AND LOW TRANSMISSION TOWERS DESIGNED FOR 160

KM/H UNDER WIND LOADING

Now considering the vulnerability of towers, high and low, designed for 160 Km/h and

submitted to wind action. The failure probability will now be represented by curves of Figure 4.

In this case, the response of these latticed structures submitted to the wind velocities shown in

the abscissas was evaluated. The wind hazard was examined throughout the country and the

resulting map for low tower is shown in Figure 10. A simple examination shows that a failure

would be obtained every 70 years for the areas of largest wind hazard. The corresponding map

for the tall tower, which is more vulnerable, is illustrated in Figure 11. The examination shows

that there are larger failure probabilities for these areas, on the order of a failure every 35 years.

Due to its structural type, these towers are significantly vulnerable to the wind action. In fact, a

calibration with actual failures history is next in order to adjust the reported losses to reality.

Figure 10: Annual failure probability in low towers designed for 160 Km/h, under wind action

Figure 11: Annual failure probability in high towers designed for 160 Km/h, under wind action

DISCUSSION

In Mexico, earthquake loading dominates the design of structures in certain zones whereas in

others it is the wind loading. Given that, in current nationwide practice, some electrical facilities

are overdesigned whereas others are underdesigned; managers from the Mexican electric

industry are trying to adjust their safety regulations in order to put their facilities close to optimal

standards throughout the country. The present study, as others presented simultaneously in this

event, is a part of these efforts. Also, the design Manual for Seismic Design [11] that the sector

subscribes for all types of structures, and which serves as a national reference for many structural

engineering practicers, it is being updated under optimal criteria. From this studies, some lessons

are being learned, for example and as expected, the towers are more vulnerable that the

substations and the cables pretension play an important role in that performance. Annual failure

probability maps, developed for specific structures, may help to select the adequate safety level

required to balance present and future costs according to the expected losses related to the

service provided by the specific facility under consideration. From this viewpoint, the hazard and

reliability assessments serve as a tool to support regional and national planning tasks of the

sector, under optimal criteria in the long term. The annual failure probabilities identified for

electrical substations and power transmission towers provide a judge element to direct design

specifications towards optimal criteria. One of the next steps is the incorporation of costs to

generate practical recommendations to back up the industry needs regarding the expansion

required by the country. In the future, the connections performance needs to be addressed as they

contribute to the overall facilities safety and they have not yet been included in the current

developments.

Maps under failure conditions may be extended to cover damage states for electrical

facilities as an intermediate step towards performance-based design.

CONCLUSIONS

An overview has been presented of developments related to structural reliability applied to the

design of electrical substations and transmission towers. Substations under two design levels,

winds of 200 and 300 Km/h, and tall and low towers designed under 160 Km/h, have been taken

as examples and the vulnerability under seismic and wind hazard throughout their failure

probabilities has been characterized.

The results show the speed ranges where the vulnerability increases faster and where,

eventually, it should be more convenient to “buy” more reliability.

The curves obtained for two typical design levels of a substation open the way towards

the identification and quantification of design improvements that are more efficient in mitigating

the vulnerability of these installations.

Additionally, annual failure probability maps for typical electrical substation and power

transmission towers in Mexico have been developed in terms of the seismic and wind hazards at

different sites. Towers are more vulnerable than substations. It is recommended to extend the

present study to cover other types of structures, connections performance and damage states.

It is recommended also to continue this ongoing research to explain the differences in

vulnerability of various types of substations and towers that the CFE might install in different

areas of our country. This will require the calculation of vulnerability curves for earthquake and

wind hazard for a diversity of substations and towers for several design levels.

ACKNOWLEDGEMENT

The authors would like to thank financial support of Comisión Federal de Electricidad (CFE).

Also, the contributions by and M.Sc. Jorge I. Vilar Rojas, Celso Muñoz and Engr. Armandina

Alanís Velázquez are recognized and thanked. Also, grateful to Eng. Rosa M. Soberanes for

preparing the annual probability failure maps.

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