Methodology of Ampacity Calculation for Overhead Line - Technical ...

IEEE PES Transactions on Power SystemsAmpABL - Methodology of Ampacity Calculation forOverhead Line Considering the Effect of AtmosphericBoundary LayerJournal: IEEE Transactions on Power SystemsManuscript ID: draftManuscript Type: TransactionsDate Submitted by theAuthor:Complete List of Authors: DO NASCIMENTO, CARLOS; 1) UFMG - FEDERAL UNIVERSITY OFMINAS GERAIS 2) CEMIG, 1) ELETRICAL ENGINEEING 2)OVERHEADLINE MANAGEMENTTechnical Topic Area :Key Words:Computational techniques and analytical methods for planning,operations, and control < Power System Analysis, Computing andEconomics, Computing applications < Power System Analysis,Computing and EconomicsTransmission lines, Terrestrial atmosphere, Power transmissionmeteorological factors, Power transmission reliability, Power systemavailability, Terrain mapping, Thermal factors, Wind

Page 1 of 8IEEE PES Transactions on Power Systems> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960Abstract— This paper presents a new methodology forcalculating the steady-state thermal rating of a given overheadline, which uses the wind speed data obtained from numericalanalysis of the atmospheric boundary layer - ABL. Themotivation behind this work is based on the difficulty and thehigh cost for monitoring the main weather parameters along theoverhead line, especially the variation of wind speed. In the newmethodology proposed in this paper, the spans with lower windspeed and lower clearance are defined as critical spans and,naturally, the thermal capacity of the overhead line is limited bythem. The results obtained in applying this methodology in a 138kV overhead line, with 133 spans, demonstrate the consistency ofthe proposed methodology and its applicability. Therefore it willbe discussed how this new methodology can be applied both atdesign and at operation of overhead lines. In this case, theoperation team will improve security with possible maximizationof electric energy transmission.Index Terms — Overhead Line, Weather Parameter, Steady-State Thermal Rating, Ampacity, Atmospheric Boundary Layer.AAmpABL - Methodology of AmpacityCalculation for Overhead Line Considering theEffect of Atmospheric Boundary LayerI. INTRODUCTIONn important point in improving the process of calculatingsteady-state rating is the need to establish the mainweather parameters such as wind speed, temperature and solarradiation [1] throughout all spans of the overhead line. Themonitoring of weather conditions throughout the overhead lineby weather remote stations has a high cost [2], especially forlong lines. Thus, a plausible starting point to work indetermining the wind speed, which is the parameter of majorweather influence in the calculation of thermal rating, is thenumerical study of atmospheric boundary layer - ABL [3], [4]around the overhead line.In order to validate the model of the ABL, it is necessary tohave the digital maps of topography and measurements ofwind speed at some points on the boundary of the region andinside the ABL. Once the model is validated, the next step isto generate a database by simulations with different boundaryconditions and then to apply it in the determination of theworst weather spans, i.e. spans with the lowest values of windManuscript received May 25, 2009. This work was supported in part byCemig and Evolutionary Computation Laboratory from the ElectricalEngineering Department of UFMG.C. A. M. do Nascimento is with the Cemig in the EngineeringManagement of Overhead Lines (caxandre@cemig.com.br).J. A. Vasconcelos is with the Electrical Engineering Department fromUFMG and is the head of the Evolutionary Computation Laboratory(jvasconcelos@ufmg.br).C.A.M. do Nascimento and J.A. Vasconcelosspeed. It is important to mention that there is still nomethodology proposed in the literature for finding criticalspans in Overhead Lines [5]–[7].In this context, the main goal of this work is to present anew methodology for calculating the steady-state rating -AmpABL, which is more realistic than the traditional methodsof calculation [7], [9]. In the presentation that follows, arebriefly presented the method for calculating the steady-statethermal rating, the proposed criterion for determining thecritical span, an introduction of the ABL model, a detaileddescription of the methodology AmpABL and finally itsapplication that was originally proposed in this work.II. STEADY-STATE THERMAL RATING CALCULATIONThe IEEE Std 738 -2006 defines the term “steady-statethermal rating” as "the constant electrical current that wouldyield the maximum allowable conductor temperature forspecified weather conditions and conductor characteristicsunder the assumption that the conductor is in thermalequilibrium (steady state)". Considering only the mostsignificant terms in the thermal equilibrium as in (1),P + P = P + PJSRCit is possible to determine what is the maximum current thatcan flow in a conductor when considering the rate of loss byJoule effect. That is,2R (T )I P = P +CS RThen,I =C(1)+ P(2)PR+ PC− PSR(T )CThrough (1)-(3), P J , P S , P R and P C are respectively the rateof heat gain by Joule effect, the rate of heat gain by solarradiation, the rate of heat loss by radiation and the rate ofconvective heat loss (W/m). The parameters R and T C are theresistance per unit length (Ω/m) and the conductortemperature (°C).The steady-state thermal rating calculation of overhead lineis usually done for a maximum allowable conductortemperature specified in the design stage (T C < 100°C) andconservative weather conditions, that is, bounds of both windspeed of [0.6,1.2] (m/s) and summer temperature [30,45] (°C).(3)

IEEE PES Transactions on Power SystemsPage 2 of 8> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960III. STEADY-STATE CONDUCTOR TEMPERATURECALCULATIONThe rates of heat loss by convection and radiation are notlinearly dependent on the temperature of the conductor, thusthe equation of thermal balance (1) can be solved to determinethe temperature of the conductor in terms of the current andthe weather variables through an iterative process [10], i.e.according to the following steps:a) Take an initial conductor temperature T R , the current I C forwhich the conductor temperature is expected and a smalltolerance number ε.b) Calculate the corresponding heat losses P R and P C , the heatgain by solar radiation P S and the conductor resistanceR(T R ).c) Calculate the conductor current (I R ) for the referencetemperature (T R ) using (3).d) Compare the values of I C and I R .• If |I C -I R | ≤ ε, stop the iteration and consider the T R valueas the solution, i.e., T C = T R .• If |I C -I R | > ε, increase T R , otherwise T R is decreased.e) Return to step “b”.IV. CRITERIA FOR DETERMINING CRITICAL SPANSThe determination of critical spans is an important taskbecause if an electrical failure occurs, the major probability isthat it will take place in one of these spans. The definitions ofcritical spans, with the factors that influence them, are in arefined manner discussed in [2]. The critical spans arebasically those with slowest values of wind speed andconcurrently lowest clearance between conductor and nearestgrounded object.In the proposed AmpABL methodology, the determinationof critical spans is carried out in an original and low cost wayif compared to the cost of instrumenting every span, which isnecessary for monitoring the whole transmission line, asmentioned in [2]. Basically, the difference is that the values ofwind speed considered in the proposed methodology isobtained from simulations of atmospheric boundary layer,with prescribed boundary conditions (wind speed = 1 m/s atdifferent directions of incidence). The spans that havenumerical results of wind speed less than 1 m/s are consideredworst weather spans. The reference value of 1m/s for the windspeed at the boundary surfaces is a technical criterion based onthe region of the world considered. Of course, this valueshould be different for other region in the world.Mathematically, let S we be the set of critical spans withrespect to electrical clearance [11], i.e., the set composed byspans with clearance less than a certain clearance value ofreference, which is dependent of the transmission line design.Let S ww be the set of worst weather spans, i.e., the setcomposed by spans with numerical results of wind speednormal to the axis of the conductor lower than the referencevalue of 1 m/s.Note that a span belonging to S we has a higher probabilityof violating the minimum safety clearance compared toanother not belonging to S we . Therefore, one span of S we willhave a greater probability of electrical failure. Furthermore,the spans that belong to S ww have less cooling capacity of theconductor and therefore greater thermal risk.Now, let S ∩be the set of spans obtained by thewintersection of S ww and S we as in (4).S= S∩ Sew∩ e ww we(4)Eliminating from theS ∩the dominated spans bywapplying a non-dominance criterion, the result is a set of nondominated1 spans with small clearance and wind speed. Letthis set of non-dominated spans be S c .Note that the spans belonging to S c form a non-dominatedfrontier in the space “wind speed versus clearance”. They arethose with simultaneously smallest clearances and lowest windspeeds, among all spans of the overhead line. That is, they arestrong candidates to be the critical critical spans in terms ofrisk of failure and thermal risk. Once determined the set S c , itis necessary to determine the critical spans, by calculating theconductor temperature of each span and their newcorresponding clearance. The critical span is the one thatwould always be hottest and/or nearest to clearance limits fora given current loading.V. ATMOSPHERIC BOUNDARY LAYER - ABLThe region of ABL is defined as a thin layer of theatmosphere, adjacent to the surface (up to 2 km in height),where wind speed show a high Reynolds number. This flow ofwind occurs at different scales, each scale is described interms of computational domain with a different model. Thesemodels use the same governing equations [3], showingdifferences only in terms of simplifications of source, whichdepends mainly on the computational domain and the thermalstratification. The equations governing these flows are thegeophysical equations of continuity, conservation ofmomentum, conservation of energy and conservation ofchemical species. The wind speed simulation models areclassified into four categories: global circulation, weatherforecast, meso-scale and micro-scale. The global circulationmodel is used in domains between 200 and 500 km along theearth’s surface and is used to analyse the vector of wind speed.The models for weather forecast are used for domains between50 and 100 km to solve problems of weather fronts. The mesoscalemodels are used typically for domains from 2 to 50 kmand employed in the wind speed studies influenced by thetopography, and finally the micro-scale models are used forsmall specific domains and employed in studies where thewind is influenced by the regionalization of the earth's surface.The numerical study of the ABL in complex topography foruse in climatological studies was introduced in [4]. The detailsof modelling the boundary layer are presented in Appendix A.e1 One vector u = (u 1, . . . , u k) dominate another vector v = (v 1, . . . , v k)(represented by u v) if and only if u is partially less than v. i.e.,∀ i ∈{1,...,k },ui ≤ vi and ∃ i ∈{1,...,k }| ui < vi} . If the vector u isnot-dominated by any other vector in the space, the vector u is a nondominatedvector.

Page 3 of 8IEEE PES Transactions on Power Systems> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960A. ABL SimulationThe ABL simulation was performed under the use ofANSYS CFX computational system [12]. Only four directionof the wind were considered at angles of 45°, 135°, 225° and315° (see Fig. 1). All these simultations were performed usingon the inlet surface (see Fig. 6), a logarithmic boundarycondition with wind speed equal to 1 m/s at 10 m height fromthe ground and neutral atmosphere, i.e. the Froude numberexceeds 1000 (as in (11)). Analysing the numerical results, theset of spans belonging to S ww was identified, i.e., spans withwind speed less than 1 m/s. Of course, many other angles ofincidence could have been considered to precisely identify thespans with lowest wind speed.Fig. 1. ABL simulation for different directions of wind speed.VI. AMPABL METHODOLOGYThe methodology proposed in this paper was developedbased on information of wind speed (obtained from ABLsimulation) and span clearance of the whole transmission line(obtained from the transmission line design). AmpABLmethodology can be summarized in the following set of steps:1) Select the appropriate parameters of the overheadtransmission line to be analysed.2) Select the region of study (by digitalizing the topographicregion containing all spans of the overhead line) anddiscretize it using an appropriate mesh generator.3) Carry out simulations of the ABL as described in SectionV.4) Identify the spans to constitute the set S ww , as described insection IV.5) Identify the spans to compose the set S we , as described inthe Section IV.6) Take the intersection between S ww and S we to have the setS w∩e . Eliminate the dominated spans to have the nondominatedones. Store the identification number of nondominatedspans in a vector S c .7) Calculate the conductor temperature for all spansbelonging to S c , using the iterative method described insection III, considering the corresponding wind speeddetermined in step 4.8) Recalculate the clearances for all spans of S c .9) Identify the critical spans, i.e., least clearance and/orhottest conductor as described in section III.10) Calculate the steady-state thermal rating for the set ofcritical spans identified in step 9.The advantage of AmpABL methodology is to allow theidentification of critical spans without monitoring any spans ofthe whole overhead line. The monitoring and calculation ofthe dynamic rating in real time by conductor temperaturesensors are now possible to be made simply monitoring thecritical spans. The financial cost to supervise them is expectedto be lower if compared to other methodologies.VII. APPLICATION OF AMPABLThe proposed methodology AmpABL was applied indetermining the steady-state thermal rating [10] of a realoverhead line designed for the following parameters:1) Conductor Linnet or 336 MCM2) Line Voltage = 138 kV3) Solar radiation = 1000 W/m 24) Design wind speed of 1 m/s at 10 m height from theground5) Angle between the wind speed direction and theconductor axis equal to 90 degrees6) 1 PU = 510 A7) Ambient temperature = 30 °C8) TN: Normal Design Temperature of Condutor = 70 °C9) TE: Emergency Design Temperature = 90 °CA. Step 1: Parameters of the Overhead LineThe overhead line has 133 spans distributed along 52 km oflength in the region of Acuruí-MG, southeast of Brazil. Theminimum clearances, in all spans, were extracted directly fromthe design or calculated by using PLS-CADD [11]. Forpurposes of illustration one span is shown in Fig. 2.Fig. 2. Range of original profile of the 138 kV overhead line.

IEEE PES Transactions on Power SystemsPage 4 of 8> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960B. Step 2: Mesh GenerationThe region of interest was properly discretized using anappropriate mesh generator [12]. The Fig. 3 shows the mesh ofpart of the region containing part of the overhead line.Fig. 3. Discretized domain with part of the overhead line.C. Step 3: ABL SimulationThe simulation was performed for each of the 4 winddirections (45°, 135°, 225° and 315°) with the wind speed onthe inlet surface (see Fig. 6) equal to 1 (m/s) at 10 m heightusing a logarithmic profile (see (11)). The numerical windspeed on each span was than extracted from the numericalresults. The lowest numerical wind speed result obtainedamong the four simulations was considered as the minimumwind speed at the corresponding span. Table I shows theresults for a few spans.SpanTABLE IRESULTS OF ABL SIMULATIONS.Incidence Angle of WindSpeed at the Conductor(Fig. 1) (°)Lowest Numerical WindSpeed Result (m/s)19-20 45 3.819-20 135 1.419-20 225 1.119-20 315 4.820-21 45 3.520-21 135 1.220-21 225 0.920-21 315 4.121-22 45 3.721-22 135 1.221-22 225 1.021-22 315 3.8D. Step 4: Span Identification to Constitute the Set S wwThe Fig. 4 shows the numerical results of wind speed ateach span. These values are the lowest wind speeds obtainedfrom the simulations as described early. It is observed thatonly 3 spans present wind speed lower than the referencevalue of 1 m/s. On these spans with lower wind speed it isexpected the highest conductor temperatures, reduction ofclearance and augmentation of both thermal and electricalfailure risks.Fig. 4. Numerical results of wind speed for the whole transmission line.The spans of S ww are shown in Table II. These spans are(20-21), (40-41 and (51-52), which correspond to wind speedsless than 1.0 m/s.TABLE IIWORST WEATHER SPAN - SWWSpanNumerical results of Clearance (m) extracted from theWind Speed (m/s) design with wind speed equals to 1 m/s20-21 0.9 1040-41 0.9 1051-52 0.9 14E. Step 5: Span Identification to Constitute the Set S weThe detection of which spans have small electricalclearance to constitute the set S we is obtained from theelectromechanical design of the overhead line. This is madetaking into account the clearance between conductor andground (or grounded object if nearest to the conductor), andthen checking whether this distance is less than a certain valueof reference (h ref ), which should be higher than the minimumTABLE IIIWORST ELECTRICAL SPAN - SWE (SEE FIG. 2)SpanSimulated Wind Speed by Clearance (m) for DesignABL (m/s) Wind Speed equal 1 m/s3-4 1.1 1019-20 1.1 1020-21 0.9 1039-40 1.0 1040-41 0.9 10safety clearance specified in the project (h min ). For example, aclearance of h ref = 10 (m) is the minimum safety distance forlines of 138 kV. Table III shows the results of this analysisbased on safety standard requirements for design. It isobserved that among the 133 spans, only 5 of them haveheights equal to 10 (m). None of the spans of this overheadline can have a height less than this reference value.

Page 5 of 8IEEE PES Transactions on Power Systems> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960F. Steps 6, 7, 8 and 9The intersection between the sets S ww and S we , i.e.,S w∩e = S ww ∩ S we , results in the following two spans {(20-21),40-41}. When applying the criterion of non-dominance on thisset by considering the pairs of (wind speed, clearance), it ispossible to realize that these spans have same values of windspeed and clearance. As the wind speed is less than the oneconsidered during the overhead line design, new calculationsof the conductor temperature and consequently for theclearance are necessary. After perfoming this calculus theresults obtained are respectively 72.6 (°C) and 9.9 (m). Boththese values violate the limits required by theelectromechanical design of this overhead line [70 (°C) and10 (m)].Another alternative analysis to determine the nondominatedset is to plot the data from all spans S ww and S we ina graph of wind speed versus clearance, after correcting thetemperature and clearance and taking the non-dominatedpoints. Fig. 5 shows this procedure considering the number ofspans belonging to S ww and S we . Clearly the same final resultsare obtained, that is, clearance of 9.9 (m).Fig. 5. Identification of non-dominated spans.G. Step 10: Deterministic Thermal Rating in Critical Spans-ScThe steady-state thermal rating for the critical spansdefines the maximum capacity of the overhead line. In thismanner, for the wind speed of 0.9 (m/s) and conductortemperature equal to 70 (°C) from the overhead line design,the maximum capacity results in 0.965 (PU). This result is3.5% smaller than the original ampacity which was calculatedfor the design with wind speed equal to 1 (m/s) and conductortemperature equal to 70 (ºC).This result shows that the overhead line monitoring couldbe done in real time just on the two critical spans. Thismonitoring should be done via sensors for measuring theconductor temperature or clearance. This information iscertainly of great value for the company, because the wholeoverhead line monitoring would be much expensive.Of course, more ABL simulations with many winddirection in the boundary condition is worthwhile. Forexample, if more data from ABL simulations are considered,with small angles between adjacent main wind directions,results much more precise are expected. However, the fewsimulations presented in this paper are intended to show theapplication of the proposed AmpABL in a real overhead line.This methodology can be applied both during the designand the operation of the overhead line. At the design, theproposed methodology allows the designer to get a realisticview of the line when the critical spans are found. At theoperation, the AmpABL allows the monitoring of these criticalspans. Moreover, applying this methodology, are expected thatboth the reliability and transmission capacity are improved.VIII. CONCLUSIONThe AmpABL methodology was presented in details in thiswork. The core of this methodology is the procedure to findthe critical spans which define the overall capacity ofelectrical energy transmission. The methodology was appliedin one real overhead line of 138 kV, with 133 spans in theregion of Acuruí (southeast-Brazil). This application wasdiscussed step by step to contrast the main characteristics ofthe proposed methodology. The analysis of the results firstpointed out that it is possible, after applying the AmpABL, tosupervise the overhead line in real time with small investment(2 critical spans found among 133 ones) and finally that themaximum capacity or thermal rating of this overhead linecould be better explored. One expects that this methodology isglobal because the usual way of designing overheadtransmission lines is by using the steady-state procedure. Inthis procedure, the allowable maximum conductortemperature, wind speed, weather conditions, for the wholeoverhead line, are defined in a conservative way.APPENDIX A – MATHEMATICAL MODEL OF ABLThe mathematical model of the ABL is determined byequations (5) to (8). They represent, respectively, conservationof mass or continuity equation (5), momentum (6), stateequation (7) and thermal energy (8) based on Reynoldsaveraging and Boussinesq approximation [15], [17], [18].∂u∂xi=i0∂ui+ uj∂t0T 0∂ui∂xj∂ ⎛ δp2 ⎞= −kx⎜ +i 03⎟∂ ⎝ ρ ⎠∂+∂xj⎡ ⎛ ui⎢ ⎜∂νt⎢⎣ ⎝ ∂xj∂u⎞⎤j+ ⎟⎥− S∂xi ⎠⎥⎦δρ δT= −(7)ρ⎛ T T ⎞ ⎛ v ∂T⎞c ⎜∂ ∂ ∂ t+ u ⎟⎟j= ⎜t xj∂xjPrTx⎝∂ ∂⎠ ⎝∂j ⎠p+where u i is i-th mean wind speed component (m/s), δρ thedeviation of fluid density from its reference value ρ 0 , ρ 0 thefluid density (kg/m 3 ), δp is the deviation of pressure densityQw(5)(6)(8)

IEEE PES Transactions on Power SystemsPage 6 of 8> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960from its reference value p (N/m 2 ), k turbulent kinetic energy(TKE) (m 2 /s 2 ), t the time (s), ν t the effective kinematicviscosity (m 2 /s), S w the source term (m/s 2 ), C p the specific heator air at constant pressure (dimensionless), T the temperature(°C), Pr T the Prandt number of turbulence (dimensionless) andQ the heat generation rate.The term S w represents a flotation term, which considersturbulence efect. This fluctuation characterizes many modelsavailable in literature. Usually, the models presented in[13]-[15] use the Boussinesq approximation and the models ofturbulence are derived from k-ε model.A. Domain of AnalysisThe simulation of Atmospheric Boundary Layer - ABL ismade on a domain that has its base on the ground. Thevegetation, water, or buildings impose difficulties for themovement of the air layer. The top of the layer must be highenough, in order to be considered boundary conditions ofNeumann natural type, without loss of accuracy in results. Atthe inlet, are imposed both the turbulent k-ε model and thelogarithmic distribution of wind speed profile. On both sides,are also imposed the condition of symmetry and in the outletthe outlet condition. Fig. 6 shows all sides of the domainwhere the boundary conditions should be defined.x 3inletx 2x 1upperwallFig. 6 Illustration of boundary conditions.B. Boundary Conditionslateralsoutlet1) Boundary Condition: k-ε ModelThe boundary conditions used in the ABL simulation by themodel of turbulence k-ε are associated with the turbulentkinetic energy k and its dissipation rate ε. In this model, theinlet conditions are given by (9) and (10).k =2u *2C µ3u*ε = (10)κxwhere u* is the wind speed of friction (m/s), κ Von Karmanconstant (κ=0.41) (dimensionless), Cµ empirical constant(dimensionless) and x 2 is the height (m).(9)2) Boundary Condition at InletThe logarithimic wind speed at inlet of ABL is given by(11) based on height x2.u( )⎛ x ⎞ ⎛ 2ref⎞x = u .ln2ln⎜⎟⎠⎜⎝⎟⎠2 in refxx2o2o⎝x(11)where u( x2) in is the wind speed (m/s), u ref wind speed ofreference at 10 m in height (m/s), x2ref reference height equalto 10 (m) and x2the aerodynamic roughness length (m).0Typical values of roughness are presented in Table IV.TABLE IVGENERAL VALUES OF ROUGHNESSZ 0 (m)Parameters1 City0.3 Forest0.03 Low grass0.0001 Water3) Outlet and Lateral Boundary ConditionThis boundary condition is given by natural Neumanncondition, that is,∂ui∂xj∂k∂xj= 0∂ε= 0∂xj= 0 ∀i= 1,2,3(12)where j is the direction normal to the surface, that is, j = 1 forthe oulet surface and j = 3 for the lateral ones.4) Wall Boundary ConditionThe wind speed in the ground is considered without friction,that is, all speed components are null, that is,u1 2 3== u = u 0(13)5) Upper Boundary ConditionAll the boundary conditions are Neumann type except forthe main speed component that is Dirichlet boundarycondition,u =U ∞(14)Table V shows the boundary conditions that should beimposed on the domain boundaries.

Page 7 of 8IEEE PES Transactions on Power Systems> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960TABLE VBOUNDARY CONDICTIONS USED IN SIMPLIFIED CFX MODEL [17]-[18].Boundary u 1 u 2 u 3 K εxu x2 = uref.lnin ⎜ ⎟ ln⎜u 0⎝ x2o⎠ ⎝x2= u3= 02oInlet ( )⎛ x ⎞ ⎛ ⎞22ref⎟⎠∂uOutlet 0∂x1=1∂u∂x2=10∂u∂x31= 0k =2*C µu∂k= 0∂x∂kWall Nonslip conditions couplet with wall functions = 0∂x∂ u 1∂ u2 ∂ u3 Lateral = 0= 0= 0= 0∂x∂x∂xUpper (Symmetry)3u 1= U∞C. Treatment of Vertical Buoyancy StrenghtThe strengh of vertical Buoyancy is considered as a sourceterm S w in the momentum equation (5), which is given by(15).Sg ⋅δρδg ⋅(ρ − ρ )δ0w=2i=2i(15)ρ0ρ0where S w is the source term by unit of fluid density given in(m/s 2 ), g gravity accelaration (m/s 2 ), ρ fluid density (kg/m 3 ),ρ 0 the fluid reference density (kg/m 3 ) and δ is the Kroneckerdelta wich is unity only for i equal to 2 and zero for i = {1,3}.When considering the density variation ∆ρ constant during thesimulation, the Froude number can be calculated by (16).U∞Fr = (16)1( g ⋅ L ⋅δρ ρ ) 20where U ∞ is the wind speed above 500 meters of height (m/s),∆ρ variation between the base of the land to its top (kg/m 3 )and L the height of the simulation domain (m).In this way, the source term on the x 2 direction (height),introduced in the equation of momentum can be written as(17).S2U∞w=2δ2iL⋅Fr2i(17)Finally, the intensity of the strength of buoyancy due tothermal effects of the ground is obtained from values assignedto the Froude number. Table VI shows the relationshipbetween the state of the atmosphere and the range of valuesassigned to the Froude number.TABLE VIGENERAL VALUES OF FROUDE NUMBER - FR.F rType of Atmospheric> 1000 ABL Neutral10 < F r

IEEE PES Transactions on Power SystemsPage 8 of 8> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960a cubical building”, Atmospheri¢ Environment, Vol. 30, No. 8, pp.1327-d 345, 1996.[16] N. G. Wright, and G. J. Easom, “Non-linear k–ε turbulence modelresults for flow over a building at full-scale”, Applied MathematicalModelling, 27, pp. 1013–1033, 2003.[17] D.A. Trifonopoulos and G.C. Bergeles, “Stable stratification effects onthe flow past surface obstructions: A numerical study”, Int. J. Heat andFluid Flow, Vol. 13, No. 2, June 1992.[18] G.A.A. Moreira, R.M. Valle, W.L.S. Carvalho, “Numerical simulationof neutral atmospheric boundary layer flow”, In: ENCIT 2008 - 12thBrazilian Congress of thermal Sciences and Engineering, BeloHorizonte, 2008.BIOGRAPHYNascimento, C.A.M was born in Conselheiro Lafaiete-MG, Brasil, onOctober 26, 1968. He graduated in Mechanical Engineering from the CatholicUniversity of Minas Gerias – PUC-MG in 1994. In 1999, he received a postgraduationdegree from Federal University of Minas Gerais – UFMG as aMaster in Mechanical Engineering. He develops and applies real-timemonitoring technology coupled with the intense use of mathematical,statistical computation tools and optimization algorithms to optimizeinvestments on electrical power delivery engineering.João Antônio de Vasconcelos obtained his Ph.D. in 1994 from EcoleCentrale de Lyon —France. He is Associate Professor at ElectricalEngineering Department of the Federal University of Minas Gerais and CNPqResearcher since 1995. His research interests include linear and non-linearoptimization (evolutionary multi-objective optimization, stochastic anddeterministic methods), computational intelligence, and computationalelectromagnetics (finite element methods, boundary integral equationmethods and others). Currently, he is working in R&D projects aboutapplication and development of new technologies for electrical systems.