Ampabl - Methodology of Ampacity Calculation for Overhead Line Considering the Effect of Atmospheric Boundary Layer

IEEE PES Transactions on Power Systems

AmpABL - Methodology of Ampacity Calculation for Overhead Line Considering the Effect of Atmospheric Boundary Layer

Journal: IEEE Transactions on Power Systems

Manuscript ID: draft

Manuscript Type: Transactions Date Submitted by the Author:

Complete List of Authors: DO NASCIMENTO, CARLOS; 1) UFMG - FEDERAL UNIVERSITY OF MINAS GERAIS 2) CEMIG, 1) ELETRICAL ENGINEEING 2)OVERHEAD LINE MANAGEMENT

Computational techniques and analytical methods for planning, operations, and control < Power System Analysis, Computing and Technical Topic Area : Economics, Computing applications < Power System Analysis, Computing and Economics

Transmission lines, Terrestrial atmosphere, Power transmission Key Words: meteorological factors, Power transmission reliability, Power system availability, Terrain mapping, Thermal factors, Wind

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1 2 3 AmpABL - Methodology of Ampacity 4 5 Calculation for Overhead Line Considering the 6 7 8 Effect of Atmospheric Boundary Layer 9 10 C.A.M. do Nascimento and J.A. Vasconcelos 11 12 13 Abstract — This paper presents a new methodology for speed. It is important to mention that there is still no 14 calculating the steady-state thermal rating of a given overhead methodology proposed in the literature for finding critical 15 line, which uses the wind speed data obtained from numerical spans in Overhead Lines [5]–[7]. 16 analysis of the atmospheric boundary layer - ABL. The In this context, the main goal of this work is to present a 17 motivation behind this work is based on the difficulty and the new methodology for calculating the steady-state rating - 18 high cost for monitoring the main weather parameters along the AmpABL, which is more realistic than the traditional methods 19 overhead line, especially the variation of wind speed. In the new of calculation [7], [9]. In the presentation that follows, are 20 methodology proposed in this paper, the spans with lower wind speed and lower clearance are defined as critical spans and, briefly presented the method for calculating the steady-state 21 naturally, the thermal capacity of the overhead line is limited by 22 thermal rating, the proposed criterion for determining the them. The results obtained in applying this methodology in a 138 critical span, an introduction of the ABL model, a detailed 23 kV overhead line, with 133 spans, demonstrate the consistency of 24 the proposed methodology and its applicability. Therefore it will description of the methodology AmpABL and finally its 25 be discussed how this new methodology can be applied both at application that was originally proposed in this work. 26 design and at operation of overhead lines. In this case, the 27 operation team will improve security with possible maximization II. STEADY -STATE THERMAL RATING CALCULATION 28 of electric energy transmission. The IEEE Std 738 ™ -2006 defines the term “steady-state 29 Index Terms — Overhead Line, Weather Parameter, Steady- thermal rating” as "the constant electrical current that would 30 State Thermal Rating, Ampacity, Atmospheric Boundary Layer . yield the maximum allowable conductor temperature for 31 specified weather conditions and conductor characteristics 32 I. INTRODUCTION under the assumption that the conductor is in thermal 33 n important point in improving the process of calculating equilibrium (steady state)". Considering only the most 34 significant terms in the thermal equilibrium as in (1), 35 A steady-state rating is the need to establish the main weather parameters such as wind speed, temperature and solar P + P = P + P 36 J S R C (1) 37 radiation [1] throughout all spans of the overhead line. The 38 monitoring of weather conditions throughout the overhead line it is possible to determine what is the maximum current that 39 by weather remote stations has a high cost [2], especially for can flow in a conductor when considering the rate of loss by 40 long lines. Thus, a plausible starting point to work in Joule effect. That is, 41 determining the wind speed, which is the parameter of major T(R I) 2 + P = P + P 42 weather influence in the calculation of thermal rating, is the C S R C (2) 43 numerical study of atmospheric boundary layer - ABL [3], [4] Then, 44 around the overhead line. 45 In order to validate the model of the ABL, it is necessary to P + P − P 46 I = R C S have the digital maps of topography and measurements of (R T ) 47 wind speed at some points on the boundary of the region and C (3) 48 inside the ABL. Once the model is validated, the next step is Through (1)-(3), P , P , P and P are respectively the rate 49 J S R C to generate a database by simulations with different boundary of heat gain by Joule effect, the rate of heat gain by solar 50 conditions and then to apply it in the determination of the radiation, the rate of heat loss by radiation and the rate of 51 worst weather spans, i.e. spans with the lowest values of wind 52 convective heat loss (W/m). The parameters R and T C are the 53 resistance per unit length ( /m) and the conductor Manuscript received May 25, 2009. This work was supported in part by temperature (°C). 54 Cemig and Evolutionary Computation Laboratory from the Electrical 55 Engineering Department of UFMG. The steady-state thermal rating calculation of overhead line 56 C. A. M. do Nascimento is with the Cemig in the Engineering is usually done for a maximum allowable conductor Management of Overhead Lines ([email protected]). 57 temperature specified in the design stage (T C < 100°C) and J. A. Vasconcelos is with the Electrical Engineering Department from 58 UFMG and is the head of the Evolutionary Computation Laboratory conservative weather conditions, that is, bounds of both wind 59 ([email protected]). speed of [0.6,1.2] (m/s) and summer temperature [30,45] (°C). 60 IEEE PES Transactions on Power Systems Page 2 of 8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2

1 III. STEADY -STATE CONDUCTOR TEMPERATURE the spans that belong to S have less cooling capacity of the 2 ww CALCULATION conductor and therefore greater thermal risk. 3 4 The rates of heat loss by convection and radiation are not Now, let Sw∩e be the set of spans obtained by the linearly dependent on the temperature of the conductor, thus 5 intersection of S ww and S we as in (4). 6 the equation of thermal balance (1) can be solved to determine 7 the temperature of the conductor in terms of the current and Sw∩e = Sww ∩ Swe (4) 8 the weather variables through an iterative process [10], i.e. according to the following steps: 9 Eliminating from the Sw∩e the dominated spans by a) Take an initial conductor temperature T , the current I for 10 R C applying a non-dominance criterion, the result is a set of non- 11 which the conductor temperature is expected and a small 1 dominated spans with small clearance and wind speed. Let 12 tolerance number ε. this set of non-dominated spans be S . 13 b) Calculate the corresponding heat losses P and P , the heat c R C Note that the spans belonging to S form a non-dominated 14 gain by solar radiation P and the conductor resistance c S frontier in the space “wind speed versus clearance”. They are 15 R(T ). R those with simultaneously smallest clearances and lowest wind 16 c) Calculate the conductor current (I R) for the reference speeds, among all spans of the overhead line. That is, they are 17 temperature (T R) using (3). strong candidates to be the critical critical spans in terms of 18 d) Compare the values of I C and I R. risk of failure and thermal risk. Once determined the set S c, it 19 • If |I C-IR| ≤ ε, stop the iteration and consider the T R value is necessary to determine the critical spans, by calculating the 20 as the solution, i.e., T C = T R. conductor temperature of each span and their new 21 • If |I C-IR| > ε, increase T R, otherwise T R is decreased. 22 e) Return to step “b”. corresponding clearance. The critical span is the one that 23 would always be hottest and/or nearest to clearance limits for 24 IV. CRITERIA FOR DETERMINING CRITICAL SPANS a given current loading. 25 The determination of critical spans is an important task V. ATMOSPHERIC BOUNDARY LAYER - ABL 26 because if an electrical failure occurs, the major probability is 27 that it will take place in one of these spans. The definitions of The region of ABL is defined as a thin layer of the 28 critical spans, with the factors that influence them, are in a atmosphere, adjacent to the surface (up to 2 km in height), 29 refined manner discussed in [2]. The critical spans are where wind speed show a high Reynolds number. This flow of 30 basically those with slowest values of wind speed and wind occurs at different scales, each scale is described in 31 concurrently lowest clearance between conductor and nearest terms of computational domain with a different model. These 32 models use the same governing equations [3], showing 33 grounded object. differences only in terms of simplifications of source, which 34 In the proposed AmpABL methodology, the determination depends mainly on the computational domain and the thermal 35 of critical spans is carried out in an original and low cost way stratification. The equations governing these flows are the 36 if compared to the cost of instrumenting every span, which is 37 necessary for monitoring the whole transmission line, as geophysical equations of continuity, conservation of 38 mentioned in [2]. Basically, the difference is that the values of momentum, conservation of energy and conservation of 39 wind speed considered in the proposed methodology is chemical species. The wind speed simulation models are 40 obtained from simulations of atmospheric boundary layer, classified into four categories: global circulation, weather 41 with prescribed boundary conditions (wind speed = 1 m/s at forecast, meso-scale and micro-scale. The global circulation 42 different directions of incidence). The spans that have model is used in domains between 200 and 500 km along the 43 numerical results of wind speed less than 1 m/s are considered earth’s surface and is used to analyse the vector of wind speed. 44 worst weather spans. The reference value of 1m/s for the wind The models for weather forecast are used for domains between 45 speed at the boundary surfaces is a technical criterion based on 50 and 100 km to solve problems of weather fronts. The meso- 46 the region of the world considered. Of course, this value scale models are used typically for domains from 2 to 50 km 47 should be different for other region in the world. and employed in the wind speed studies influenced by the 48 Mathematically, let S we be the set of critical spans with topography, and finally the micro-scale models are used for 49 respect to electrical clearance [11], i.e., the set composed by small specific domains and employed in studies where the 50 spans with clearance less than a certain clearance value of wind is influenced by the regionalization of the earth's surface. 51 reference, which is dependent of the transmission line design. The numerical study of the ABL in complex topography for 52 Let S ww be the set of worst weather spans, i.e., the set use in climatological studies was introduced in [4]. The details 53 composed by spans with numerical results of wind speed of modelling the boundary layer are presented in Appendix A. 54 normal to the axis of the conductor lower than the reference 55 value of 1 m/s. 1 One vector u = (u 1, . . . , u k) dominate another vector v = (v 1, . . . , v k) 56 Note that a span belonging to S has a higher probability 57 we (represented by u v) if and only if u is partially less than v. i.e., of violating the minimum safety clearance compared to 58 ∀i ∈{1,..., k }, ui ≤ vi and ∃i ∈{1,..., k u|} i < vi } . If the vector u is another not belonging to S . Therefore, one span of S will 59 we we not-dominated by any other vector in the space, the vector u is a non- have a greater probability of electrical failure. Furthermore, 60 dominated vector. Page 3 of 8 IEEE PES Transactions on Power Systems > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3

1 A. ABL Simulation determined in step 4. 2 8) Recalculate the clearances for all spans of S . 3 The ABL simulation was performed under the use of c 9) Identify the critical spans, i.e., least clearance and/or 4 ANSYS CFX computational system [12]. Only four direction of the wind were considered at angles of 45°, 135°, 225° and hottest conductor as described in section III. 5 6 315° (see Fig. 1). All these simultations were performed using 10) Calculate the steady-state thermal rating for the set of 7 on the inlet surface (see Fig. 6), a logarithmic boundary critical spans identified in step 9. 8 condition with wind speed equal to 1 m/s at 10 m height from 9 the ground and neutral atmosphere, i.e. the Froude number The advantage of AmpABL methodology is to allow the 10 exceeds 1000 (as in (11)). Analysing the numerical results, the identification of critical spans without monitoring any spans of the whole overhead line. The monitoring and calculation of 11 set of spans belonging to S ww was identified, i.e., spans with 12 wind speed less than 1 m/s. Of course, many other angles of the dynamic rating in real time by conductor temperature 13 incidence could have been considered to precisely identify the sensors are now possible to be made simply monitoring the 14 spans with lowest wind speed. critical spans. The financial cost to supervise them is expected 15 to be lower if compared to other methodologies. 16 17 VII. AP PLICATION OF AMP ABL 18 The proposed methodology AmpABL was applied in 19 determining the steady-state thermal rating [10] of a real 20 overhead line designed for the following parameters: 21 1) Conductor Linnet or 336 MCM 22 2) Line Voltage = 138 kV 23 3) Solar radiation = 1000 W/m 2 24 4) Design wind speed of 1 m/s at 10 m height from the 25 ground 26 5) Angle between the wind speed direction and the 27 conductor axis equal to 90 degrees 28 29 6) 1 PU = 510 A

30 7) Ambient temperature = 30 °C

31 8) TN: Normal Design Temperature of Condutor = 70 °C

32 9) TE: Emergency Design Temperature = 90 °C 33 Fig. 1. ABL simulation for different directions of wind speed.

34 A. Step 1: Parameters of the Overhead Line 35 VI. AMP ABL METHODOLOGY The overhead line has 133 spans distributed along 52 km of 36 37 The methodology proposed in this paper was developed length in the region of Acuruí-MG, southeast of Brazil. The based on information of wind speed (obtained from ABL minimum clearances, in all spans, were extracted directly from 38 ™ 39 simulation) and span clearance of the whole transmission line the design or calculated by using PLS-CADD [11]. For 40 (obtained from the transmission line design). AmpABL purposes of illustration one span is shown in Fig. 2. 41 methodology can be summarized in the following set of steps: 42 1) Select the appropriate parameters of the overhead 43 transmission line to be analysed. 44 2) Select the region of study (by digitalizing the topographic 45 region containing all spans of the overhead line) and 46 discretize it using an appropriate mesh generator. 47 3) Carry out simulations of the ABL as described in Section 48 V. 49 4) Identify the spans to constitute the set S ww , as described in 50 section IV. 51 5) Identify the spans to compose the set S we , as described in 52 the Section IV. 53 6) Take the intersection between S and S to have the set 54 ww we S . Eliminate the dominated spans to have the non- 55 w∩e dominated ones. Store the identification number of non- 56 dominated spans in a vector S . 57 c 7) Calculate the conductor temperature for all spans Fig. 2. Range of original profile of the 138 kV overhead line. 58 59 belonging to S c, using the iterative method described in 60 section III, considering the corresponding wind speed IEEE PES Transactions on Power Systems Page 4 of 8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4

1 B. Step 2: Mesh Generation 2 3 The region of interest was properly discretized using an 4 appropriate mesh generator [12]. The Fig. 3 shows the mesh of 5 part of the region containing part of the overhead line. 6 7 8 9 10 11 12 13 14 15 16 17 18 Fig. 4. Numerical results of wind speed for the whole transmission line. 19 20 The spans of S ww are shown in Table II. These spans are 21 Fig. 3. Discretized domain with part of the overhead line. (20-21), (40-41 and (51-52), which correspond to wind speeds 22 less than 1.0 m/s. 23 C. Step 3: ABL Simulation

24 The simulation was performed for each of the 4 wind TABLE II 25 directions (45°, 135°, 225° and 315°) with the wind speed on WORST WEATHER SPAN - SWW 26 the inlet surface (see Fig. 6) equal to 1 (m/s) at 10 m height Numerical results of Clearance (m) extracted from the 27 using a logarithmic profile (see (11)). The numerical wind Span 28 Wind Speed (m/s) design with wind speed equals to 1 m/s speed on each span was than extracted from the numerical 20-21 0.9 10 29 results. The lowest numerical wind speed result obtained 40-41 0.9 10 30 51-52 0.9 14 among the four simulations was considered as the minimum 31 wind speed at the corresponding span. Table I shows the 32 results for a few spans. 33 E. Step 5: Span Identification to Constitute the Set Swe

34 The detection of which spans have small electrical TABLE I clearance to constitute the set S is obtained from the 35 RESULTS OF ABL SIMULATIONS . we 36 electromechanical design of the overhead line. This is made Incidence Angle of Wind 37 Lowest Numerical Wind taking into account the clearance between conductor and Span Speed at the Conductor 38 Speed Result (m/s) ground (or grounded object if nearest to the conductor), and (Fig. 1) (°) 39 19-20 45 3.8 then checking whether this distance is less than a certain value 40 19-20 135 1.4 of reference (h ref ), which should be higher than the minimum 41 19-20 225 1.1 19-20 315 4.8 42 TABLE III 20-21 45 3.5 43 WORST ELECTRICAL SPAN - SWE (SEE FIG . 2) 20-21 135 1.2 44 20-21 225 0.9 Simulated Wind Speed by Clearance (m) for Design 20-21 315 4.1 Span 45 ABL (m/s) Wind Speed equal 1 m/s 21-22 45 3.7 46 3-4 1.1 10 21-22 135 1.2 19-20 1.1 10 47 21-22 225 1.0 20-21 0.9 10 48 21-22 315 3.8 39-40 1.0 10 49 40-41 0.9 10 50 D. Step 4: Span Identification to Constitute the Set Sww 51 safety clearance specified in the project (h ). For example, a The Fig. 4 shows the numerical results of wind speed at min 52 clearance of h = 10 (m) is the minimum safety distance for each span. These values are the lowest wind speeds obtained ref 53 lines of 138 kV. Table III shows the results of this analysis 54 from the simulations as described early. It is observed that based on safety standard requirements for design. It is 55 only 3 spans present wind speed lower than the reference observed that among the 133 spans, only 5 of them have 56 value of 1 m/s. On these spans with lower wind speed it is heights equal to 10 (m). None of the spans of this overhead 57 expected the highest conductor temperatures, reduction of line can have a height less than this reference value. 58 clearance and augmentation of both thermal and electrical

59 failure risks. 60 Page 5 of 8 IEEE PES Transactions on Power Systems > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5

1 F. Steps 6, 7, 8 and 9 simulations presented in this paper are intended to show the 2 application of the proposed AmpABL in a real overhead line. 3 The intersection between the sets S ww and S we , i.e., This methodology can be applied both during the design 4 Sw∩e = Sww ∩ S we , results in the following two spans {(20-21), and the operation of the overhead line. At the design, the 5 40-41}. When applying the criterion of non-dominance on this 6 set by considering the pairs of (wind speed, clearance), it is proposed methodology allows the designer to get a realistic 7 possible to realize that these spans have same values of wind view of the line when the critical spans are found. At the 8 speed and clearance. As the wind speed is less than the one operation, the AmpABL allows the monitoring of these critical 9 considered during the overhead line design, new calculations spans. Moreover, applying this methodology, are expected that 10 of the conductor temperature and consequently for the both the reliability and transmission capacity are improved. 11 clearance are necessary. After perfoming this calculus the

12 results obtained are respectively 72.6 (°C) and 9.9 (m). Both VIII. CONCLUSION 13 these values violate the limits required by the The AmpABL methodology was presented in details in this 14 electromechanical design of this overhead line [70 (°C) and work. The core of this methodology is the procedure to find 15 10 (m)]. the critical spans which define the overall capacity of 16 Another alternative analysis to determine the non- electrical energy transmission. The methodology was applied 17 dominated set is to plot the data from all spans S ww and S we in in one real overhead line of 138 kV, with 133 spans in the 18 a graph of wind speed versus clearance, after correcting the region of Acuruí (southeast-Brazil). This application was 19 temperature and clearance and taking the non-dominated discussed step by step to contrast the main characteristics of 20 points. Fig. 5 shows this procedure considering the number of the proposed methodology. The analysis of the results first 21 spans belonging to S and S . Clearly the same final results pointed out that it is possible, after applying the AmpABL, to 22 ww we are obtained, that is, clearance of 9.9 (m). supervise the overhead line in real time with small investment 23 (2 critical spans found among 133 ones) and finally that the 24 maximum capacity or thermal rating of this overhead line 25 could be better explored. One expects that this methodology is 26 27 global because the usual way of designing overhead 28 transmission lines is by using the steady-state procedure. In 29 this procedure, the allowable maximum conductor 30 temperature, wind speed, weather conditions, for the whole 31 overhead line, are defined in a conservative way. 32 33 APPENDIX A – MATHEMATICAL MODEL OF ABL 34 The mathematical model of the ABL is determined by 35 equations (5) to (8). They represent, respectively, conservation 36 Fig. 5. Identification of non-dominated spans. of mass or continuity equation (5), momentum (6), state 37 equation (7) and thermal energy (8) based on Reynolds 38 averaging and Boussinesq approximation [15], [17], [18]. 39 G. Step 10: Deterministic Thermal Rating in Critical Spans- 40 Sc ∂u i = 0 (5) 41 The steady-state thermal rating for the critical spans ∂xi 42 defines the maximum capacity of the overhead line. In this 43 manner, for the wind speed of 0.9 (m/s) and conductor ∂u ∂u ∂  δp 2  i i   44 temperature equal to 70 (°C) from the overhead line design, + u j = −  + k  ∂t ∂x j ∂xi  ρ0 3  45 the maximum capacity results in 0.965 (PU). This result is (6) 46 3.5% smaller than the original ampacity which was calculated ∂   ∂u ∂u  + ν  i + j  − S 47 for the design with wind speed equal to 1 (m/s) and conductor t   w ∂x j  ∂x j ∂xi  48 temperature equal to 70 (ºC).    49 This result shows that the overhead line monitoring could δρ δT 50 be done in real time just on the two critical spans. This = − (7) ρ T 51 monitoring should be done via sensors for measuring the 0 0 52 conductor temperature or clearance. This information is 53  ∂T ∂T  ∂  v ∂T  certainly of great value for the company, because the whole c  + u  =  t  + Q (8) 54 p  j    overhead line monitoring would be much expensive.  ∂t ∂x j  ∂x j  Pr T ∂x j  55 Of course, more ABL simulations with many wind 56 direction in the boundary condition is worthwhile. For 57 example, if more data from ABL simulations are considered, where u i is i-th mean wind speed component (m/s), δρ the 58 with small angles between adjacent main wind directions, deviation of fluid density from its reference value ρ , ρ the 59 0 0 results much more precise are expected. However, the few fluid density (kg/m 3), δp is the deviation of pressure density 60 IEEE PES Transactions on Power Systems Page 6 of 8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6

1 from its reference value p (N/m 2), k turbulent kinetic energy 2) Boundary Condition at Inlet 2 (TKE) (m 2/s 2), t the time (s), ν the effective kinematic The logarithimic wind speed at inlet of ABL is given by 3 t 2 2 (11) based on height x . 4 viscosity (m /s), S w the source term (m/s ), C p the specific heat 2 5 or air at constant pressure (dimensionless), T the temperature  x  6 (°C), Pr T the Prandt number of turbulence (dimensionless) and  x2  2ref u()x2 in = uref ln.   ln   Q the heat generation rate. x2 x2 7  o   o  (11) 8 The term S w represents a flotation term, which considers 9 turbulence efect. This fluctuation characterizes many models where u( x2 )in is the wind speed (m/s), u ref wind speed of 10 available in literature. Usually, the models presented in reference at 10 m in height (m/s), x2 ref reference height equal 11 [13]-[15] use the Boussinesq approximation and the models of to 10 (m) and x the aerodynamic roughness length (m). 12 turbulence are derived from k-ε model. 20 13 14 Typical values of roughness are presented in Table IV. A. Domain of Analysis 15 16 The simulation of Atmospheric Boundary Layer - ABL is TABLE IV made on a domain that has its base on the ground. The GENERAL VALUES OF ROUGHNESS 17 18 vegetation, water, or buildings impose difficulties for the Z0 (m) Parameters 19 movement of the air layer. The top of the layer must be high 1 City 20 enough, in order to be considered boundary conditions of 0.3 Forest 21 Neumann natural type, without loss of accuracy in results. At 0.03 Low grass 22 the inlet, are imposed both the turbulent k-ε model and the 0.0001 Water 23 logarithmic distribution of wind speed profile. On both sides,

24 are also imposed the condition of symmetry and in the outlet 3) Outlet and Lateral Boundary Condition 25 the outlet condition. Fig. 6 shows all sides of the domain This boundary condition is given by natural Neumann 26 where the boundary conditions should be defined. condition, that is, 27 ∂ui 28 = 0 ∀i = 3,2,1 ∂x j 29 upper 30 ∂k = 0 (12) 31 inlet outlet ∂x 32 j ∂ε 33 = 0 x2 34 wall ∂x j 35 x3 36 x1 where j is the direction normal to the surface, that is, j = 1 for 37 laterals the oulet surface and j = 3 for the lateral ones. 38 Fig. 6 Illustration of boundary conditions. 39 4) Wall Boundary Condition 40 B. Boundary Conditions The wind speed in the ground is considered without friction, 41 that is, all speed components are null, that is, 42 1) Boundary Condition: k-ε Model 43 The boundary conditions used in the ABL simulation by the u1 = u2 = u3 =0 (13) 44 model of turbulence k-ε are associated with the turbulent 45 kinetic energy k and its dissipation rate ε. In this model, the 5) Upper Boundary Condition 46 inlet conditions are given by (9) and (10). All the boundary conditions are Neumann type except for 47 the main speed component that is Dirichlet boundary u 2 48 k = * condition, 49 C u =U 50 (9) ∞ (14) 51 3 Table V shows the boundary conditions that should be 52 u* ε = (10) imposed on the domain boundaries. 53 κx 2 54 55 where u* is the wind speed of friction (m/s), κ Von Karman 56 constant ( κ=0.41) (dimensionless), C empirical constant 57 (dimensionless) and x2 is the height (m). 58 59 60 Page 7 of 8 IEEE PES Transactions on Power Systems > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7

1 2 3 TABLE V BOUNDARY CONDICTIONS USED IN SIMPLIFIED CFX MODEL [17]-[18]. 4 5 Boundary u1 u2 u3 K ε 6 2 3  x   x2ref  u* u* 7 Inlet 2 u = 0 k = u()x2 in = uref ln.   ln   u 2 = 0 3 ε =  x2o  x2o C κx 8   2 9 ∂u 1 ∂u 2 ∂u 3 ∂k ∂ε 10 Outlet = 0 = 0 = 0 = 0 = 0 ∂x ∂x ∂x ∂x1 11 1 1 ∂x1 1 12 3 ∂k u* 13 Wall Nonslip conditions couplet with wall functions = 0 ε = ∂x2 κx 14 2 ∂u ∂u ∂u ∂k ∂ε 15 Lateral 1 = 0 2 = 0 3 = 0 = 0 = 0 ∂x ∂x ∂x ∂x ∂x 16 3 3 3 3 3 ∂u ∂u ∂k ∂ε 17 Upper (Symmetry) 2 = 0 3 = 0 = 0 = 0 u1 = U ∞ ∂x ∂x2 ∂x2 ∂x 18 2 2 19 20 C. Treatment of Vertical Buoyancy Strenght ACKNOWLEDGMENT 21 The strengh of vertical Buoyancy is considered as a source The authors thank Dr. Ramon Molina Valle by the 22 term S w in the momentum equation (5), which is given by technical support in the ABL analysis and the electrical 23 (15). engineers Saulo Mariano and Marcello Dantas, and the 24 undergraduate students Moisés Ferber and Alexandre 25 g ⋅δρ g ⋅( ρ − ρ ) S = δ = 0 δ (15) Sifuentes by the ABL simulations. 26 w i2 i2 ρ0 ρ0 27 REFERENCES 28 where S w is the source term by unit of fluid density given in

29 (m/s 2), g gravity accelaration (m/s 2), ρ fluid density (kg/m 3), [1] ELECTRA 174 - WG 22.12, “The thermal behaviour of overhead conductors”, October 1997. 30 3 ρ0 the fluid reference density (kg/m ) and δ i2 is the Kronecker [2] G.J. Ramon, “Dynamic Thermal Line Rating: Summary and Status of 31 delta wich is unity only for i equal to 2 and zero for i = {1,3}. the State-of-the-Art Technology”. IEEE Transactions on Power 32 Delivery, Vol. PWRD-2, No. 3, July 1987. 33 When considering the density variation ρ constant during the [3] T. Uchida and Y. Onya, “Numerical Simulation of Atmospheric Flow simulation, the Froude number can be calculated by (16). Over Complex Terrain, Journal of Wind Engineering and Industrial 34 Aerodynamics”, vol. 81, p. 283-293, 1999. 35 U [4] P.A. Taylor and H.W. Teunissen, “The Askervein Hill Project: ∞ Overview and Background data. Boundary Layer Meteorology”, v.39. 36 Fr = 1 (16) 2 pp. 15-39. 1987. 37 ()g ⋅ L⋅δρ ρ0 [5] S. Jaufer, “Experiences with the Weather Parameter Method for the use 38 where U is the wind speed above 500 meters of height (m/s), in Overhead Line Monitoring systems” – Group B2-105 - Cigre Session 39 ∞ Paris/France/August-2008. ρ variation between the base of the land to its top (kg/m 3) 40 [6] G. Bruno, “Increasing Capacity of Two Italian Lines by the Adoption of and L the height of the simulation domain (m). Devices for Monitoring Environmental Conditions and Conductors 41 Temperature or by Using High-Temperature Conductors” – Group B2- In this way, the source term on the x2 direction (height), 42 102 – Cigre Session Paris/France/August-2008. 43 introduced in the equation of momentum can be written as [7] A. K. Deb, Power Line Ampacity System: Theory, Modeling and 44 (17). Applications , CRC Press LLC – NY-USA, 2000. [8] IEEE Standard for Calculating the Current-Temperature of Bare 45 U 2 Overhead Conductors , IEEE Std 738™-2006. 46 ∞ [9] Guide for selection of weather parameters for bare overhead conductor Sw = 2 δ i2 (17) 47 L⋅ Fr ratings , CIGRE-Task Force B2.12.6, 2005.

48 [10] V.T. Morgan, “The Thermal Rating of Overhead-Line Conductors, Part Finally, the intensity of the strength of buoyancy due to I. The Steady-State Thermal Model”, Electric Power Systems Research , 49 thermal effects of the ground is obtained from values assigned 5 (1982) 119 - 139. 50 [11] PLS-CADD ™ (Power Line Systems - Computer Aided Design and to the Froude number. Table VI shows the relationship 51 Drafting - www.powline.com ), USA. between the state of the atmosphere and the range of values 52 [12] CFX-5.5 - AEA Technology plc. Documentation. AEA Technology assigned to the Froude number. Engineering Software Ltd, United Kingdom, 2002. 53 [13] A. Huser, P. J. Nilsen, and H. Skatun, “Aplication of k-ε model to the 54 stable ABL: pollution in complex terrain”, Journal of Wind Engineering TABLE VI and Industrial Aerodynamics , Vol. 67-68, pp. 425-436, 1997. 55 GENERAL VALUES OF FROUDE NUMBER - F . R [14] C. Montavon, “Validation of a non-hydrostatic numerical model to 56 simulate stratified wind fields over complex topography”, Journal of F Type of Atmospheric 57 r Wind Engineering and Industrial Aerodynamics , Vol. 74-76, 1998, pp. > 1000 ABL Neutral 273-282., 1998. 58 10 < F <1000 ABL Stable r [15] Y. Q. Zhang, S. P. Arya and W. H. Snyder, “A comparison of numerical 59 -100 < F r < 10 ABL Unstable 60 and physical modeling of stable atmospheric flow and dispersion around IEEE PES Transactions on Power Systems Page 8 of 8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8

1 a cubical building”, Atmospheri¢ Environment , Vol. 30, No. 8, pp. 2 1327-d 345, 1996. 3 [16] N. G. Wright, and G. J. Easom, “Non-linear k–ε turbulence model 4 results for flow over a building at full-scale”, Applied Mathematical Modelling , 27, pp. 1013–1033, 2003. 5 [17] D.A. Trifonopoulos and G.C. Bergeles, “Stable stratification effects on 6 the flow past surface obstructions: A numerical study”, Int. J. Heat and 7 Fluid Flow , Vol. 13, No. 2, June 1992.

8 [18] G.A.A. Moreira, R.M. Valle, W.L.S. Carvalho, “Numerical simulation of neutral atmospheric boundary layer flow”, In: ENCIT 2008 - 12th 9 Brazilian Congress of thermal Sciences and Engineering , Belo 10 Horizonte, 2008. 11 12 13 BIOGRAPHY 14 Nascimento, C.A.M was born in Conselheiro Lafaiete-MG, Brasil, on 15 October 26, 1968. He graduated in Mechanical Engineering from the Catholic 16 University of Minas Gerias – PUC-MG in 1994. In 1999, he received a post- graduation degree from Federal University of Minas Gerais – UFMG as a 17 Master in Mechanical Engineering. He develops and applies real-time 18 monitoring technology coupled with the intense use of mathematical, 19 statistical computation tools and optimization algorithms to optimize 20 investments on electrical power delivery engineering.

21 João Antônio de Vasconcelos obtained his Ph.D. in 1994 from Ecole 22 Centrale de Lyon —France. He is Associate Professor at Electrical 23 Engineering Department of the Federal University of Minas Gerais and CNPq 24 Researcher since 1995. His research interests include linear and non-linear optimization (evolutionary multi-objective optimization, stochastic and 25 deterministic methods), computational intelligence, and computational 26 electromagnetics (finite element methods, boundary integral equation 27 methods and others). Currently, he is working in R&D projects about 28 application and development of new technologies for electrical systems.

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