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Dynamicpropertiesofsubstationsupportstructures

ARTICLEinJOURNALOFCONSTRUCTIONALSTEELRESEARCH·NOVEMBER2012

ImpactFactor:1.32·DOI:10.1016/j.jcsr.2012.06.016

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RezaKaramiMohammadi

KhajeNasirToosiUniversityofTechnology

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VahidAkrami

AmirkabirUniversityofTechnology

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FarzadNikfar

McMasterUniversity

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Journal of Constructional Steel Research 78 (2012) 173–182

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Journal of Constructional Steel Research

Dynamic properties of substation support structures

R. Karami Mohammadi a,⁎, V. Akrami b, F. Nikfar c

a Civil Eng. Dep., K.N. Toosi University of Technology, Tehran, Iranb Amirkabir University of Technology, Tehran, Iranc McMaster University, Hamilton, Canada

⁎ Corresponding author. Tel.: +98 21 88770006.E-mail address: rkarami@kntu.ac.ir (R.K. Mohamma

0143-974X/$ – see front matter © 2012 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2012.06.016

a b s t r a c t

a r t i c l e i n f o

Article history:Received 20 July 2011Accepted 29 June 2012Available online xxxx

Keywords:Dynamic amplification factor (DAF)Substation structuresFinite element method (FEM)Fundamental natural frequencySpectral shape

Substation electrical equipment is usually mounted on different kinds of structures, which can have a verysignificant effect on the motion that the supported equipment will experience during an earthquake. Inthis paper, a parametric study is implemented in order to assess dynamic properties of substation supportstructures.In the first phase of this study, a simplified Four Degrees-of-Freedom (4-DOF) system is proposed in order tomodel different support-equipment systems and is verified through finite element method (FEM). Based onthe proposed model, a practical equation is proposed to calculate the fundamental natural frequency ofsupport-equipment systems and compared to the one proposed by ASCE (substation structure designguide, 2008). Furthermore, dynamic amplification factor (DAF) of substation support structures is calculated,and effect of different parameters (e.g. support mass, height and stiffness) is discussed. Finally, a new crite-rion is developed to restrict maximum acceleration at support-equipment intersection.Results of this paper can be utilized in order to design more proper support structures to preserve substationelectrical equipment from seismic induced damages.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Experiences obtained from previous earthquakes have demon-strated that the substation equipment is seismically vulnerable andis not in the safe state [1]. In the USA, during the 1989 Loma PrietaEarthquake, some 230 kV and 550 kV high voltage substation equip-ment items were destroyed severely. In 1994 Northridge Earthquakedestruction on 230 kV and 550 kV transformer bushings was serious[2]. Direct cost for repair and replacement of earthquake damage topower system facilities of the USA from some well-known earth-quakes is displayed in Table 1.

In 1995 Kobe Earthquake in Japan, 275 kV and 770 kV substationequipment suffered severe damages [3]. In 1990 Manjil Earthquakein Iran, 159 substations were destroyed completely [4]. During thisearthquake and only in Loshan substation (230 kV), 23 disconnectswitches, 11 circuit breakers, 11 current transformers and 5 post in-sulators were destroyed severely [5]. Some damages to substationequipment induced by the 2003 Bam Earthquake in Iran (Mw=6.5),are presented in Fig. 1 [6].

Many different parameters can affect seismic behavior of substa-tion equipment. However, performance is strongly influenced by spe-cific equipment design and installation practice. One of the importantparameters affecting seismic performance of any substation equip-ment is the dynamic properties of supporting structure [7,8]. The

di).

l rights reserved.

necessity of considering dynamic effects of supporting structure onseismic behavior of such interconnected systems is confirmed insome studies [9–11].

Dynamic properties of support-equipment sets have been studiedby different researchers. Among the rest, Gilani et al. [12] studied theeffects of various supports with distinctive heights and stiffness onthe seismic behavior of 230 kV disconnect switches (DS). Amplifica-tion factors of 2–3 were reported for structures studied in this report.However, the amplification factor for one of the studied structureswas in excess of 3. In a similar research Takhirov et al. [13], studiedthe seismic behavior of 550 kV disconnect switches through fragilitytesting. Another work has been done by Matt and Filiatrault, [14],which studied the spectral amplification of different transformertanks and their effects on seismic behavior of bushings.

In this research, a parametric study is implemented in order toassess dynamic properties of substation support structures. In thefirst part of the study, the support-equipment system is modeledas a Four Degrees-of-Freedom (4-DOF) system and its accuracy isverified by the finite element method (FEM). Using the simplified4-DOF model, fundamental natural frequency of different support-equipment systems is calculated through modal analysis, and a newequation is proposed to calculate fundamental natural frequency ofdifferent support-equipment sets with various top and bottommasses. The proposed equation is compared then to the approximateequation presented in the ASCE design guide [15].

In the second part, dynamic amplification of different support-equipment sets is calculated at support-equipment intersection and

Table 1Financial damage from earthquakes to power system facilities of the USA [1].

Earthquake Date Momentmagnitude (Mw)

Million dollar

San Fernando 1971 6.6 45North Palm Springs 1987 6.1 9Loma Prieta 1989 6.9 100Northridge 1994 6.7 183

174 R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

compared to the target DAF proposed by IEEE Std. 693 [16]. Based onthis comparison, a minimum stiffness ratio is proposed for differentsupport-equipment combinations, which is required in order to havea DAF smaller than the target DAF. Effect of various top and bottommasses on DAF of supporting structure is also studied in this section.

Results of this study can be utilized to improve seismic design ofsubstation support structures and decrease earthquake damages toelectrical equipment.

Fig. 2. Finite element model of two 230 kV and 400 kV capacitive voltage transformers.

2. Modeling

2.1. Finite element models

The most precise numerical approximation for an electrical equip-ment item and its supporting structure is to simulate the entire sys-tem as a finite element model. The number, location, and propertiesof elements and nodes shall be such that an adequate representationof the real support-equipment system is obtained in the analysis. Inthis part, the numerical finite element modeling and verification oftwo capacitive voltage transformers (CVT) are explained. Fig. 2 dis-plays 3 dimensional (3D) finite element models for two 230 kV and400 kV capacitive voltage transformers.

In thesemodels, the porcelain insulator, cementmaterial and alumi-num flanges are considered as 8-node solid elements with a maximumdimension of 20 mm. The oil tank is modeled as 4-node shell elementswith a maximum dimension of 50 mm. The geometry and thicknessof each part are obtained through the manufacturer's catalog. Themechanical properties of the materials assumed in the modeling ofthe specimens are presented in Table 2. These properties are based onthe IEEE Std. 693 recommendations. The damping of these equipmentitems is very low and rarely increases 2%, so according to IEEE recom-mendations the damping ratio is assumed to be 2%.

To account for the mass of the oil within the tank and porcelaininsulator, additional masses are symmetrically added to the innerperimeter of the insulator and tank walls. The dimensions, mass andstiffness of the specimens are summarized in Table 3. In order to

Fig. 1. Failure of electrical equipment in Bam Earthquake; a)

verify the analytical models for further complementary assessments,the fundamental natural frequency of these models is calculatedusing modal analysis and compared to the ones stated in equipmentcatalog. As it can be seen in Table 3, results imply the close accord-ance between calculated and actual values of equipment vibrationfrequency.

Since, it is intended in this paper to evaluate the effect of variousparameters on dynamic properties of supporting structures, it seemsnecessary to analyze a huge number of support-equipment combina-tions. This makes it difficult to use complicated finite element models.

230 kV porcelain parts; b) 230 kV current transformer.

Table 2Mechanical properties of the materials.

Material Young's modulus (MPa) Poisson's ratio

Porcelain 70,000 0.24Aluminum 69,000 0.33Cement 25,000 0.2Steel 210,000 0.3

175R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

In the next section, a simplified parametric model is presented whichcan ease the way to study various support-equipment with differentgeometric and structural properties. The finite element models shownin Fig. 2 will be used as reference models to verify proposed simplifiedmodel and assess its accuracy. For this purpose, aforementionedtransformers will be mounted on two kinds of lattice steel structureas the support (Fig. 3) which gives a total number of 4 differentsupport-equipment sets.

The vertical members of support structures are made of L60×60×6 angles, while the diagonal members are made of L40×40×4angles. Other parameters of these structures are given in Table 4. Inthis table stiffness of supporting structure, EIs, is calculated throughapplying a unit force at the support-equipment intersection in the fi-nite element model (EIs=Ls

3/(3Δ), where Δ is the displacement dueto unit force at the top of supporting structure).

Fig. 3. Dimensions of support structures.

2.2. Four Degrees-of-Freedom model

For candlestick support-equipment systems, the electrical equip-ment and its supporting structure can be modeled as a 4-DOF system(Fig. 4). The illustrated model is composed of two beam elementswith distributedmass and constantmaterial properties (bending stiff-ness). In this figure, EIs, Ls and ms are bending stiffness, length and dis-tributed mass of supporting structure, respectively. EIe, Le and me arethe same quantities corresponding to the mounted equipment.

Two concentrated masses, namelyMb andMt are representative ofdead and live tankmasses installed at top or bottom of equipment, re-spectively. It should be noted that for most equipment types such ascurrent transformers (CT), post insulators (PI), CVT and other equip-ment with negligible tank mass, the value of Mt and Mb can be set aszero.

Using the consistent mass and stiffness matrices of the Euler'sbeam element [17], the mass and stiffness matrices of the entire sys-tem (Fig. 4) can be written as:

M ¼ 156meLe420

αmαL þ 1þ 420γb

156926

11 1−αmαL2

� �Le

78− Le

12926

1þ 420γt

156Le12

−11Le78

11 1−αmαL2

� �Le

78Le12

1þ αmαL3

� �L2e

39− L2e

52

− Le12

−11Le78

− L2e52

L2e39

26666666666664

37777777777775

ð1Þ

Table 3Parameters of the modeled CVT.

Description of parts Height (m) Total mas

CPA 245 Porcelain insulator and bottom flange 2.4 330Oil tank 0.6 200

CPA 420 Porcelain insulator and bottom flange 3.9 550Oil tank 0.6 200

a Calculated from Eq. (3).

and:

K ¼ 2EIeL3e

6αEI þ αL

3

αL3 −6 3

αL2−αEI

αL2 Le 3Le

−6 6 −3Le −3Le

3αL

2−αEI

αL2 Le −3Le 2

αEI þ αL

αLL2e L2e

3Le −3Le L2e 2L2e

266666664

377777775

ð2Þ

where:

αm ¼ msme

αEI ¼EIsEIe

αL ¼LsLe

γb ¼ Mb

meLeγt ¼

Mt

meLe :

Parameters of this model (i.e. height, mass and stiffness) should berationally adjusted to the ones corresponding to the actual support-equipment system. For support structures with the standard cross sec-tion, the value of EIs can be calculated easily. However, for other types ofsupporting structures, e.g. latticed structures, this value should be cal-culated using analysis (as in Table 4). Although for some substationequipment the value of EIe can be found in equipment technical specifi-cation, it is not a commonly used quantity and hence it is not usuallyaddressed in equipment technical specifications. If it is not given, the

s (kg) Natural frequency (Hz) Stiffnessa (N.m2)

Equipment catalog Finite element model

8.9 8.95 2.81×106

3.8 3.73 2.66×106

Table 5Value of λe for different top masses.

γt 0.0 0.5 1.0λe 12.36 4.066 2.426

Table 4Parameters of the support structures.

Model Height (m) Total mass (kg) Stiffness (N.m2)

St 170 1.7 100 1.93×107

St 250 2.5 150 2.48×107

176 R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

value of this parameter can be calculated using the equation presentedfor natural frequency of a uniform cantilever beam with distributedmass, constant material properties and a lumped mass at the free end[17,18]:

EIe ¼�meL

4eω

2e

λe: ð3Þ

where, the parameters �me, Le, and ωe (natural frequency of equipment)are well-known quantities, which can be found easily in any equipmenttechnical specifications. The parameter λe is a coefficient obtained fromsolving governing equation of beam vibration. The value of this param-eter is given for different top masses in Table 5.

Since the proposed simplified model will be used to evaluate fun-damental natural frequency and DAF of different support-equipmentcombinations, its degree of accuracy should be verified prior to anyanalysis. For this purpose, the parameters of the 4-DOF systemcorresponding to each reference finite element model are calculatedusing the values listed in Tables 3 and 4, and shown in Table 6. Inpreparation of this table it is assumed that �me is equal to the massper unit length of porcelain part, e.g. for CPA 245 �me=330/2.4=137.5 kg/m. Accordingly, the bottom mass of each equipment itemwill be equal to the total mass menus �meLe, where Le is the totalheight of equipment.

Table 7 represents the fundamental natural frequency and DAF ofreference structures for finite element model and corresponding4-DOF system. These values are obtained for simplified 4-DOF systemimplementing mass and stiffness matrices of Eqs. (1) and (2) in aneigen value problem, and for finite element model using conventionalmodal analysis (mode shapes of reference model CPA 420-St 250 arerepresented in Fig. 5). DAFs are calculated using spectral shape givenin IEEE Std. 693 (Eq. (14)). As it is apparent from this table, the resultsfor two models are very close to each other and it can be said that the

Fig. 4. Example of electrical equipment and its 4-DOF model.

simplified 4-DOF system is well estimating the fundamental naturalfrequency and DAF of the modeled reference support-equipment.

3. Estimation of system fundamental natural frequency

It is essential to determine natural frequencies of a support-equipment combination in order to calculate seismic loads acting onthe entire system. On-site testing, laboratory testing and finite ele-ment analysis are some common methods to evaluate these frequen-cies. As these methods are expensive and complicated, usually someapproximate methods are being used by design consultants. Accord-ing to recommendations of the ASCE design guide, for simple candle-stick structures, the fundamental natural frequency of the system canbe approximated from the following equation:

1=ω2sys ¼ 1=ω2

e þ 1=ω2s : ð4Þ

where,ωe,ωs andωsys are the fundamental frequencies correspondingto the equipment, structure and system, respectively. Eq. (4) is a sug-gested simplification for combining the individual fundamental fre-quencies of the structure and equipment to obtain an approximateestimation of system fundamental frequency.

The value of ωe to be used in Eq. (4) can be found in equipmenttechnical specification (or seismic qualification report) which shouldbe presented by the equipment manufacturer. The correspondingvalue for supporting structure, i.e. ωs, should be calculated using thestiffness of structure and mass of both structure and mounted equip-ment. As it is recommended in the ASCE design guide, for cantileversubstation structures the value of ωs can be calculated from the fol-lowing equation:

ω2s ¼ 3EIs

F þ 0:243 �msLsð ÞL3s: ð5Þ

where, the parameter F is the total mass of mounted equipment,which is consisted of distributed mass of equipment plus top and bot-tom masses. Substituting the value of F in Eq. (5) and simplifying theobtained expression will result in:

ω2s ¼ EIe

meL4e� 3αEI

1þ γt þ γb þ 0:243αmαLð ÞαL3 : ð6Þ

Substituting Eq. (3) in Eq. (6) gives an expression for calculatingfundamental natural frequency of supporting structure:

ω2s ¼ 3αEI

1þ γt þ γb þ 0:243αmαLð ÞαL3λe

ω2e : ð7Þ

Since Eq. (4) is an approximate equation, its application for differ-ent support-equipment combinations should be assessed. In the next

Table 6Parameters of the 4-DOF system.

Model Massratio (αm)

Stiffnessratio (αEI)

Heightratio (αL)

Bottom massratio (γb)

CPA 245-St 170 0.427 6.87 0.56 0.285CPA 245-St 250 0.436 8.825 0.83 0.285CPA 420-St 170 0.417 7.25 0.377 0.182CPA 420-St 250 0.425 9.32 0.555 0.182

Table 7Comparison of the natural frequencies and dynamic amplification factors.

Model Natural frequencies (Hz) Dynamicamplificationfactor

FEM 4-DOF Eqs. (9) and (10) FEM 4-DOF

CPA 245-St 170 6.98 6.88 6.85 1.39 1.33CPA 245-St 250 6.18 6.15 6.14 1.84 1.73CPA 420-St 170 3.27 3.29 3.28 1.32 1.35CPA 420-St 250 3.09 3.12 3.08 1.79 1.71

177R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

section, a more precise equation is proposed for evaluation of the sys-tem fundamental natural frequency considering the effect of top andbottom masses. Results of this equation are compared to the onesobtained from the equation presented by the ASCE design guide(Eq. (4)).

3.1. Equipment without top and bottom masses

For the support-equipment system shown in Fig. 4, the natural fre-quencies can be computed using the mass and stiffness matricesgiven in Eqs. (1) and (2). Solving the eigen value problem for thissimplified parametric model, the resultant fundamental frequencyof the system will be in the form:

ω2sys ¼ λsys �

EIemeL

4e: ð8Þ

where, λsys is a coefficient obtained from solving governing equation ofsystem vibration. In general, this parameter is a function of systemproperties (i.e. αm, αEI, αL, γt and γb). Since the fundamental natural fre-quency of equipment is usually known, the term EIe= �meL

4e in Eq. (8) can

Fig. 5. Vibration modes of refere

be substituted from Eq. (3). Accordingly, fundamental natural frequen-cy of the system can be indicated as:

ω2sys ¼

λsys

λe�ω2

e or ωsys ¼ ωe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλsys=λe

q: ð9Þ

Above expression is an alternative for Eq. (4) which can be used toevaluate the system fundamental frequency. For a support-equipmentcombination without any top or bottommasses, the coefficient λsys/λe

is calculated for different system properties and plotted in Fig. 6.As illustrated in the magnified window at the right bottom of Fig. 6,

when both parameters αm and αL are less than 1.0, the effect of param-eter, αm, on the coefficient λsys is negligible and the curves presented inthis figure are valid for different values of αm (0bαmb1.0).

As it can be seen from Fig. 6, as the stiffness ratio of the system(namely the parameter αEI) increases, the value of coefficient λsys

tends to λe (λsys/λe→1.0). This general trend indicates that forsupport-equipment systems with very rigid supporting structure orflexible equipment, the natural frequency of the system will beclose to the equipment natural frequency. Fig. 7 displays the relationbetween system and structure fundamental frequencies each as aratio of equipment fundamental frequency. In this figure, the ratios(ωs/ωe)2 and (ωsys/ωe)2 are calculated from Eqs. (7) and (9), for thesame support-equipment properties (for 1bαEIb100). The curvecorresponding to Eq. (4) (proposed by ASCE design guide) is alsoplotted in this figure for comparison. As it is apparent from this figure,ASCE equation overestimates the fundamental frequency of the sys-tem for all support-equipment combinations.

In general, it can be said that for a special equipment type with agiven fundamental frequency, as the supporting structure becomesstiffer (higher ωs/ωe ratios) the results of ASCE equation get closerto the exact system fundamental frequency. It should be noted thatthe application of this equation for support-equipment combinations

nce model CPA 420-St. 250.

Fig. 8. Error of using ASCE equation in estimation of the system fundamental frequency.

Fig. 6. The ratio of λsys/λe for different system properties (0≤αm≤1.0).

178 R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

with a low stiffness ratio (i.e. parameter αEI) may result in completelywrong values for system natural frequency (see Fig. 8).

Another result which can be deduced from Fig. 7 is that the differ-ence between the results of Eqs. (4) and (9) decreases as the value ofheight ratio (namely the parameter, αL) increases. This is also trueabout the mass ratio (parameter, αm), however, the changes in thesystem fundamental frequency due to the changes in the height ratioare more evident.

Regarding the design spectrum presented in the ASCE designguide, it can be said that although the results of approximate equationpresented by ASCE are far from the actual system frequency in somecases, this equation usually results in conservative loads except forvery rigid support-equipment systems. In other words, for most typ-ical support-equipment systems application of Eq. (4) will overesti-mate seismic loads acting on the entire system.

3.2. Effect of top and bottom masses

This section focuses on the effect of different top and bottommasses on fundamental natural frequency of support-equipment sys-tems. The examples of such equipment items are live and dead typetank circuit breakers.

The effect of top and bottom masses on the system fundamentalfrequency is displayed in Fig. 9 (for 1bαEIb100). As it can be inferredfrom this figure, for a specific ωs/ωe ratio, as the bottom mass ofequipment increases the fundamental frequency of the entire system

Fig. 7. Comparison between Eqs. (9) and (4) (γt=γb=0).

gets closer to equipment fundamental frequency. This is vice-versafor live tank type equipment where the system natural frequency de-creases as the value of top mass increases. Comparing the illustratedresults with the ones obtained from Eq. (4), it can be said that forequipment with large amount of top mass (e.g. live tank circuit break-ers), application of ASCE equation may result in a poor prediction ofthe system fundamental frequency.

Using different systemproperties for themodel shown in Fig. 4, thevalues of coefficient λsys/λe are calculated and a mathematical expres-sion is fitted to the resulting data. For a typical candlestick support-equipment system, the coefficient λsys/λe can be estimated with agood accuracy using:

λsys

λe¼ r2−r1αEI

r3αEI þ r4αEI 1≤ αEI ≤ 100 : ð10Þ

where, the parameters r1, r2, r3 and r4 are summarized for differentsystem properties in Table 8. For comparison purpose the curvescorresponding to eachαL value are obtained using Eq. (10) and plottedwith gray lines in Fig. 5. This reveals the close accordance betweenproposed equation and the curves obtained from modal analysis. Itshould be noted that Eq. (10) is adjusted for the range 1≤αEI≤100(which is typical of most substation support-equipment systems)and its application beyond this range may decrease its accuracy, tosome extent.

As an example, consider a support-equipment system with givenproperties: γb=γt=0, αm=αL=1.0 and αEI=6.0. For such a support-equipment set, the system fundamental frequency will be equal toωsys=0.3×ωe (see Fig. 4). This value can be compared to the oneobtained from Eq. (10) which is ωsys=0.302×ωe (error=0.7%).

Fig. 9. Effect of different top and bottom masses on the system fundamental frequency.

Fig. 10. Normalized response spectrum of IEEE Std. 693 [16].

Table 8Parameters of Eq. (10) for different system properties.

γb γt αL r1 r2 r3 r4

1.0 0.0 0.2 0.33 8818 8701 95540.4 1.09 5119 4969 15,2650.6 1.65 2742 2546 16,8960.8 2.04 1761 1508 19,5541.0 1.38 822 630 14,906

0.5 0.0 0.2 0.04 1106 1088 11870.4 0.12 9817 9690 27,3280.6 2.39 5510 5204 32,0000.8 1.05 1282 1149 13,0151.0 2.96 2448 2039 39,725

0.0 0.0 0.2 0.19 7250 7169 77500.4 0.54 4791 4697 13,4970.6 0.54 1969 1891 10,7660.8 1.39 2865 2682 26,4871.0 3.61 5097 4586 73,208

0.0 0.5 0.2 0.04 3513 3505 29480.4 0.09 1999 1988 41840.6 0.84 7368 7274 28,4820.8 1.12 5264 5135 32,9961.0 0.19 581 557 5461

0.0 1.0 0.2 0.01 2127 2125 16880.4 0.04 1326 1322 25890.6 0.45 6299 6254 22,3820.8 0.26 1945 1917 11,0581.0 0.41 1937 1890 16,329

179R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

For the above example, application of approximate equation pro-posed by the ASCE design guide will result in ωsys=0.54×ωe whichis apart from exact system fundamental frequency (error=80%).Again, it should be emphasized that the application of ASCE approx-imate equation for support-equipment combinations with a lowstiffness ratio may result in completely wrong values for system fun-damental frequency. Therefore, it is recommended to use Eqs. (9) and(10) to calculate fundamental frequency of support-equipment com-bination in similar engineering aspects.

4. Evaluation of DAF in substation support structures

The acceleration that the equipment experiences on a structurecan be several times the ground acceleration. If the dynamic amplifi-cation of supporting structure is not controlled during the design ofstructure, it may cause the failure of mounted equipment due to theexcessive acceleration which is induced by the supporting structure.

Based on IEEE Std. 693, during seismic qualification, it is generallydesirable to have the equipment mounted or modeled in the identicalmanner as it would be in its in-service configuration. However, fordifferent reasons it is not practical to qualify the equipment in itsin-service configuration. For these equipment types, the qualificationshould be done without supporting structure at 2.5 times the speci-fied requirements by IEEE Std. 693. Accordingly, the users shall designthe structures such that the supports do not amplify the base acceler-ations more than 2.25 times.

Currently, substation support structures are being designedaccording to ASCE design guide, without any consideration about ef-fects of these structures on dynamic behavior of mounted equipment.In this section, it is intended to establish a new criterion in order tocontrol DAF of supporting structures.

Having determined the value of different frequencies, mode shapescorresponding to each frequency can be calculated. The modal partic-ipation factor of DOF j corresponding to nth vibrationmode (Djn), maybe indicated as [18]:

Djn ¼ ϕjnΦ½ �Tn M½ �i

Φ½ �Tn M½ � Φ½ �n: ð11Þ

where, i, is the influence vector as defined in [18] and ϕjn is the modeshape corresponding to jth DOF and nth vibrationmodes. The absoluteacceleration at support-equipment intersection (first DOF of 4-DOFmodel) corresponding to each mode can be calculated as follows:

A1n ¼ D1n � Sa ωn; ξð Þ: ð12Þ

In this equation, Sa(ωn, ξ) is the spectral acceleration which de-fines the response of a single degree of freedom system (S-DOF)with natural frequency of ωn and damping ratio of ξ. The value of dy-namic amplification factor at support-equipment intersection can becomputed using an appropriate combination method of the modal re-sponses. Different combination methods are being used for modalsuperposition of responses. In this paper, modal responses are com-bined using absolute sum method, which is conservative and is thepreferred combination method recommended by IEEE Std. 693.Using this method, dynamic amplification factor of supporting struc-ture can be computed as:

DAF ¼XN

n¼1

���D1n � B ωn; ξð Þ���: ð13Þ

where, N is the number of DOFs, ξ is damping ratio which can be as-sumed to be 2% of critical damping (as recommended by IEEE Std.693) and B(ωn, ξ) is the value of response spectrum normalized byPGA.

As it was mentioned before, according to specifications of IEEE Std.693,when an equipment qualification is being donewithout supportingstructure, the resulting DAF obtained from Eq. (13) should be less than2.25 when the support-equipment system is exposed to required re-sponse spectrum (RRS).

Using either high or moderate RRS of IEEE Std. 693, the value ofnormalized response spectrum to be used in Eq. (13) shall be taken as:

B ωn; ξð Þ ¼ 1:144βωn

π0:0≤ωn ≤ 2:2π

B ωn; ξð Þ ¼ 2:5β 2:2π≤ωn ≤ 16π

B ωn; ξð Þ ¼ 2π 26:4β−10:56ð Þωn

−0:8β þ 1:32 16π≤ωn ≤ 66π

B ωn; ξð Þ ¼ 1 ωn ≥ 66π:ð14Þ

where:

β ¼ 3:21−0:68 ln ξð Þð Þ2:1156

: ð15Þ

It should be noted that both high and moderate response spectrapresented in IEEE Std. 693, have the same spectral shape which is

Fig. 11. DAF of different support-equipment combinations (αm=αL=0.8).

Fig. 13. Effect of different top and bottommasses on the minimum stiffness ratio (αm=αL=0.8).

180 R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

shown in Fig. 10. In the next section, DAF of substation support struc-tures is evaluated using the prescribed 4-DOF system.

4.1. Equipment types without top and bottom masses

In this part, DAF of substation structures is calculated for varioussupport-equipment sets and checked to be less than the target DAFpresented in IEEE Std. 693. Using Eq. (13), DAFs of different support-equipment systems with mass and height ratios equal to 0.8 (αm=αL=0.8) are calculated and shown in Fig. 11.

The system stiffness ratio (αEI) is displayed in the abscissa.Corresponding values of DAF are represented in the vertical axis. Thehorizontal line shown in this figure corresponds to the target DAFrecommended by IEEE Std. 693 (DAF=2.25). According to this figure,the value of DAF for some (αEI, Te) combinations is above the targetDAF. The points obtained from intersection of each curve with theline corresponding to target DAF criterion is the minimum stiffnessratio in order to have a DAF smaller than 2.25. Using this value, theminimum bending stiffness of supporting structure, EIs, may be calcu-lated accordingly:

EIs ≥ αEIð ÞminEIe: ð16Þ

Substituting Eq. (3) in Eq. (16), one can write:

EIs ≥ αEIð Þmin�meL

4eω

2e

λe: ð17Þ

Fig. 12. Effect of height and mass ratios on the minimum stiffness ratio;

In above expression, �me is distributedmass of equipment in (kg/m),Le is the height of equipment in (m) and EIs is the minimum allowablebending stiffness of supporting structure in (N.m2). λe can be obtainedfrom Table 5.

As an example, consider a capacitive voltage transformer with givenproperties: Le=4.65 m, �meLe=750 kg and Te=0.18 s (γb=γt=0),and also assume that for supporting structure: αm=αL=0.8. Based oncalculations, the minimum stiffness ratio for this system will be equalto 3.74, which results in EIs≥27,802 kN.m2. Using a steel tubular sec-tion (with modulus of elasticity, E=2.1×106 kg/cm2) with wall thick-ness of 0.5 cm, the minimum outer diameter of supporting structurewill be equal to 51.8 cm (EIs=27,802 kN.m2). Hence, the seismic de-sign of this supporting structure will be acceptable if the diameter ofdesigned section is greater than 51.8 cm.

The above example indicates a special case which may be rarelyfound in actual practice. For other support-equipment systems withvarious mass and height ratios, different border lines can be found be-tween safe and unsafe states as shown in Fig. 12.

According to Fig. 12a, for special equipment as the height of struc-ture decreases the minimum stiffness ratio required to limit DAF tothe IEEE Std. 693 criterion decreases too. The effect of height ratio canbe seen considering the previous example with αm=0.8 and αL=0.4.For this support-equipment combination, the minimum stiffness ratiowill be equal to 0.14, which results in a minimum support diameter of17.5 cm. In a similar manner, as the natural period of equipmentdecreases, the required stiffness ratio of the system decreases as well.

As it can be inferred from Fig. 12b, there is a similar trend for var-ious mass ratios. However the changes in minimum stiffness ratio due

a) effect of height ratio (αm=0.8); b) effect of mass ratio (αL=0.8).

Table 9Parameters of Eq. (18) for different system properties.

αm αL γt=γb=0.0 γt=0.0 γb=0.0

γb=0.5 γb=1.0 γt=0.5 γt=1.0

q1 q2 q3 q1 q2 q3 q1 q2 q3 q1 q2 q3 q1 q2 q3

0.2 0.2 0.00 0.00 0.00 6.00 −2.82 0.39 10.00 −3.99 0.49 0.00 0.00 0.00 0.00 0.00 0.000.4 31.00 −14.15 1.77 51.95 −13.17 1.00 69.12 −11.01 0.50 −12.00 11.40 −2.52 0.00 0.00 0.000.6 82.18 −20.35 1.39 145.40 −17.92 0.64 207.85 −16.62 0.38 33.00 −12.94 1.40 48.11 −32.50 5.580.8 168.26 −24.36 1.01 330.42 −26.59 0.76 480.30 −25.84 0.78 73.50 −20.04 1.61 44.43 −14.36 1.261.0 309.70 −28.76 0.90 613.64 −28.46 0.84 911.52 −34.11 1.91 122.68 −20.12 0.92 81.24 −19.16 1.34

0.4 0.2 0.00 0.00 0.00 8.00 −4.40 0.70 10.57 −4.33 0.54 0.00 0.00 0.00 0.00 0.00 0.000.4 32.19 −13.88 1.64 52.90 −13.22 1.01 70.74 −11.41 0.53 1.60 −0.13 −0.05 0.00 0.00 0.000.6 85.94 −19.35 1.23 146.87 −16.10 0.43 218.85 −19.02 0.56 35.05 −13.22 1.45 38.06 −22.39 3.420.8 179.27 −21.82 0.72 339.52 −23.45 0.56 498.91 −26.97 0.89 73.37 −16.84 1.11 52.57 −18.03 1.861.0 343.76 −26.54 0.86 652.73 −31.24 1.21 957.27 −38.43 2.49 133.14 −19.18 0.83 88.90 −18.73 1.18

0.6 0.2 0.00 0.00 0.00 6.60 −3.13 0.43 11.29 −4.81 0.62 0.00 0.00 0.00 0.00 0.00 0.000.4 32.52 −13.43 1.56 55.84 −14.14 1.09 69.83 −10.28 0.41 7.40 −4.25 0.70 0.00 0.80 −0.250.6 89.49 −18.68 1.14 149.16 −14.59 0.29 214.52 −15.14 0.26 34.88 −11.64 1.11 23.57 −10.25 1.180.8 193.36 −21.19 0.69 360.30 −25.82 0.76 517.58 −30.10 1.24 76.60 −15.45 0.93 52.90 −15.70 1.401.0 372.73 −24.25 0.99 676.97 −27.90 1.35 992.42 −40.89 2.94 149.21 −20.63 1.01 93.99 −17.22 1.03

08 0.2 0.00 0.00 0.00 7.20 −3.67 0.56 11.00 −4.50 0.56 0.00 0.00 0.00 0.00 0.00 0.000.4 34.05 −13.78 1.59 54.10 −12.37 0.86 74.47 −12.41 0.64 12.80 −8.32 1.48 0.00 0.92 −0.290.6 95.25 −18.86 1.10 161.80 −18.33 0.63 224.48 −17.23 0.42 36.86 −11.97 1.14 22.93 −9.00 0.950.8 207.91 −20.31 0.66 357.21 −18.70 0.34 537.03 −31.73 1.38 79.68 −14.20 0.77 51.15 −12.84 1.011.0 408.00 −17.77 0.73 700.30 −19.86 1.16 1018.1 −33.75 2.67 139.93 −8.67 −0.12 92.45 −10.63 0.26

1.0 0.2 0.00 0.00 0.00 6.29 −2.83 0.38 10.64 −4.25 0.52 0.00 0.00 0.00 0.00 0.00 0.000.4 38.29 −15.66 1.82 55.48 −12.70 0.90 73.50 −11.23 0.51 12.57 −7.42 1.20 3.60 −2.26 0.430.6 97.25 −17.29 0.93 158.79 −14.47 0.26 223.58 −15.00 0.28 38.75 −11.98 1.10 21.71 −7.63 0.770.8 214.97 −16.86 0.51 375.33 −20.08 0.52 528.18 −20.83 0.69 82.39 −13.02 0.65 51.64 −10.90 0.701.0 444.42 −7.15 −0.07 762.73 −23.89 1.62 1066.6 −32.15 2.85 164.01 −10.81 0.12 100.62 −8.81 −0.04

181R.K. Mohammadi et al. / Journal of Constructional Steel Research 78 (2012) 173–182

to variation of the parameter αL are more evident than the changesdue to variation of the parameter αm. Again, consider the previousexample, now with αm=0.4 and αL=0.8. In this case, the minimumstiffness ratio and minimum support diameter will be equal to 2.6and 45.9 cm, respectively.

4.2. Effect of top and bottom masses

Effect of different top and bottommasses on theminimum stiffnessratio is illustrated in Fig. 13. As it is clear from this figure, as the bottommass of equipment increases the required stiffness ratio increases ac-cordingly. There is a reverse trend for increasing topmass ratio, i.e. thestiffness ratio decreases as the top mass of equipment increases. To il-lustrate the impact of top and bottom masses, consider previouslymentioned example once with γb=0.5 and another time with γt=0.5. Based on calculations, minimum stiffness ratios corresponding tothese systems will be equal to 8.55 and 0.8, which result in the mini-mum support diameters of 68.2 cm and 31.1 cm, respectively.

As it can be deduced from Figs. 12 and 13, the general relationshipbetween (αEI)min and Te can be well estimated using a second orderpolynomial. Statistical analysis of minimum stiffness ratios requiredfor 125,000 support-equipment sets with different system propertiesled to the following equation:

αEIð Þmin ¼ q1T2e þ q2Te þ q3; αEIð Þmin ≥ 0: ð18Þ

where, the coefficients q1, q2 and q3 are given for different system prop-erties in Table 9. For comparison purpose the curves corresponding toEq. (18) are plotted with gray lines in Fig. 13. As it is apparent fromthis figure, the above expression has an acceptable degree of accuracy.When the resultant (αEI)min obtained from Eq. (18) is more than unitythe error of using this equation will be under 5%. Employing thisequation can help designers to control DAF of any support-equipmentsystem and decrease earthquake damages to electrical equipment.

5. Conclusion

In this paper, a parametric study is implemented in order to improveseismic design of substation structures. In the first phase of this study, a4-DOF system is proposed in order tomodel support-equipment combi-nation and its accuracy is verified using four finite element models. Apractical equation is proposed to calculate the fundamental frequencyof different support-equipment systems and compared to the one pro-posed by ASCE design guide. Based on the results, it is indicated thatapplication of approximate equation presented in ASCE for somesupport-equipment combinations may result in poor prediction of thesystem fundamental frequency.

In the second part, dynamic amplification factor (DAF) of differentsupport-equipment combinations is evaluated and compared to thetarget DAF presented in IEEE Std. 693. A minimum stiffness ratio isobtained for each support-equipment combination, which guaranteesthat the supporting structure will not amplify the ground motionsmore than 2.25 times when subjected to an IEEE Std. 693 compatibleearthquake.

Application of the results presented in this paper in conjunctionwith a seismic design code, can result in a more reliable design ofearthquake resistant support-equipment combinations and decreasethe earthquake induced damages to substations.

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