Numerical Algorithms for Calculating Temperature, Layered Stress, and Critical Current of Overhead Conductors

Accurate calculation of temperature, stress, sag, and critical current (corresponding to critical temperature) of operational overhead conductors is important for ensuring the strength and sag safety of overhead lines. Based on 2D steady-state heat transfer equations, this article studies the temperature fields of the cross section of typical electrified conductors and establishes numerical simulation methods for calculating the layered stress, sag, and critical temperature. Using the algorithm, the relationship between the critical temperature and characteristics of conductors (e.g., the sag and tensile force) is studied. -e results are verified by a comparison with the test results for heat-resistant aluminum alloy conductors JNRLH1/G1A-400/65 and JNRLH1/G1A-630/55. Finally, the paper studies the relationship between the critical temperature of the conductor and its most sensitive factors.


Introduction
During peak times, it is often necessary to dynamically increase the current of overhead lines to meet the power supply demands. In these cases, the stress and sag of conductors must be accurately calculated to avoid breakage due to excessive stress, multicircuit wire mixture [1], or nonenough safety distance to the ground caused by large sag. In addition, the stress at different layers of strands (called layered stress) and the sag are generally affected by the temperature fields on the conductor cross sections. Only when the temperature fields are simulated accurately can the layered stress and sag be calculated actually. Moreover, due to the uneven thermal expansion in different layers during operation, the mechanical properties of the conductor's cross section will greatly change at definite temperature which is defined as the critical temperature (CT) [2,3]. However, to the author's knowledge, there is limited research on the safety assessment of overhead conductors at the CT. erefore, calculation of the temperature fields, layered stress, sag and tensile force, and the analysis of the safety of overhead lines at CT are of great significance.
Many experimental studies [4][5][6][7][8][9][10][11][12] have shown that the radial temperature of conductors is not evenly distributed. IEEE specifications [13] proposed that there is radial temperature gradient among each layer. In recent years, some methods [14][15][16] based on the finite element method were proposed to simulate the conductor radial temperature field. With these methods, models better conforming to the actual sectional structure of conductors can be established to avoid calculation errors due to improper values for effective thermal conductivity. However, these studies failed to fully consider the impact of cross-section gap distribution, interstrand contact, and convection conditions on the radial temperature field. Furthermore, there is little research that systematically studied the temperature field, layered stress, sag, and critical temperature of conductors.
is paper proposes a complete set of numerical methods to calculate the temperature field, layered stress, and critical current (or temperature) of overhead conductors. e temperature field with voids is calculated by using the finite element method, and the layered stress and sag are calculated by using the temperature results. e CT and the corresponding critical current (CC) are calculated through the temperature and stress calculative method iteratively. By comparing the results of calculation with those of the experiments for a typical ACSR (aluminum-conductor steel reinforced), the algorithms are verified. In order to facilitate the dynamic assessment of safety of the conductors during the operation, this paper studies the relationship between the critical temperature of the conductor and its sensitive factors with the heat-resistant aluminum alloy conductor JNRLH1/G1A-400/65 as an example.

Calculation of Conductor Temperature Field
e thermal balance equation of the entire cross section Ω of the conductor can be established [14]: k T ,xx + T ,yy + s � 0, (x, y) ∈ Ω. (1) And the boundary condition on the surface of the conductor Γ can also be given [14]: q n � −k T ,x n x + T ,y n y � α T − T a , l(x, y) ∈ Γ, (2) where T ,xx , T ,yy , T ,x , and T ,y indicate the second-order derivatives and first-order derivatives of the 2D temperature field T in the conductor on the x and y coordinates, respectively; q n indicates the heat loss rate along the normal direction n on the outer surface of the conductor; n x and n y are the components of n in the direction of x and y; α indicates the composite heat loss coefficient; T a indicates the ambient temperature; k is the thermal conductivity, represented by k a in the area of the aluminum strand, k s in the area of the steel core, and k air in the area of the air; and s indicates the rate of heat per unit volume, represented by s a in the area of the aluminum strand, s s in the area of the steel core, and 0 in the area of the air. Herein, k a and k s are irrelevant to temperature field, while k air is relevant to it, and the relationship is as follows: where T av indicates the average temperature of the cross section. e voltage of steel core per unit length is the same as that of aluminum strand. e current is inversely proportional to the resistance which is inversely proportional to the conductor cross section and proportional to the resistivity [14]. e ratio between the current I s of steel core and that I a of aluminum strand is where A s and ρ s are the cross-sectional area and electrical resistivity of steel core, respectively, and A a and ρ a are the area and electrical resistivity of aluminum strand respectively. e rate of heat of steel core per unit volume s s is External aluminum strands should consider solar energy [17], so the rate of heat per unit volume s a is In equation (6), the first term on the right end of the equation is caused by the current, and the second term is caused by the solar radiation, P indicates its total Joule heat gain [17] which is the function of the current, average temperature, and integrated resistance rate of the conductor, S indicates the solar intensity, c s indicates the solar absorption of the conductor with a value of 0.23-0.9 and is usually valued at 0.5, and D indicates the conductor diameter.
An aluminum-conductor steel-reinforced cable (ACSR) mainly loses heat by means of convective cooling P c and radiative cooling P r . e calculation of heat loss is provided in the literature [17]. According to the principle of constant heat loss per unit volume, the composite heat transfer coefficient of conductor surface in equation (2) can be determined: where A indicates the lateral surface area of conductor per unit length (that is, the interface of the conductor and the external environment).

Calculation of Conductor Stress Field
is article assumes that conductors can only be resistant to tension, ignoring their resistance to shear, bend and torsion [18], and the tensile force on each strand of the same layer is equal [19]. ree basic conditions are required when performing calculations for overhead conductors: (1) deformation compatibility; (2) material constitutive relationship; and (3) balance of internal and external forces.

Deformation Compatibility.
Using the condition of deformation compatibility, the strain of each strand can be represented by the longitudinal strain. For a strand with a length of S i , if unfolding it, as shown in Figure 1, the deformation compatibility equation can be obtained as follows [20]: where ε 0 indicates the axial strain of the overhead conductor; L s indicates the length of the overhead conductor corresponding to a strand with a length of S i ; α i indicates the angle between the tangent of layer i of the strand and the conductor cross section; ΔR i φ i indicates the lateral deformation of the unfolded strand; and φ i indicates the ratio between projected length of the unfolded strand on the cross section and the outer radius R i of layer i of the strand as follows: From the geometrical relationship, equation (10) can be obtained as follows: 2 Mathematical Problems in Engineering Dividing the left and right sides of equation (8) by S i and simplifying it with equations (9) and (10), the relationship between the strain of layer i of the strand and the conductor ε 0 is

Lateral Deformation.
From the inside to outside, the lateral deformation ΔR i of conductors of layer i includes three parts: (1) the deformation Δr i1 ′ arising from crosssection shrinkage due to the Poisson effect when the strands of layer i are stretched along their own axis [20]; (2) the deformation Δr i2 ′ due to squeezing between layers; and (3) the deformation Δr iT ′ due to thermal expansion (or contraction). Studies have shown that the lateral deformation ΔR i has a nonlinear functional relation with axial strain ε 0 . e calculation of ε 0 with ΔR i requires iteration, which leads to complicated procedures. Furthermore, study on the lateral deformation of ACSR (JL/LB1A-300/50) found that the lateral deformation of the first two parts is very small (the lateral deformation of the first two parts is less than 1%) [21] and can be neglected compared with that of the third part. e equation of lateral deformation can be simplified as follows: e term Δr iT ′ in equation (12) indicates the total deformation of layer i due to changes in temperature.
e equation of Δr jT (strand deformation due to thermal expansion or contraction) in equation (13) is as follows: where α jT and T j indicate the coefficient of thermal expansion and the average temperature of j th layer, respectively. e strain ε iT due to changes in temperature is as follows:

Balance Equation of Internal and External Force.
Linear elastic material is supposed in this article. e resultant axial internal force of the conductor section can be obtained by summing up all axial components of internal forces of strands on the cross section. e internal force of each strand of layer i is as follows: where E i indicates the modulus of layer i of the conductor; A i indicates the cross-sectional area of one strand in layer i. What needs to be mentioned is that the temperature strain ε iT (see equation (15)) of each layer of strands may differ due to the difference of the thermal expansion coefficient. As the temperature arises, aluminum strands with a higher thermal expansion coefficient may get zero stress, and even some of them get negative stress. As a result, all the external force is resisted by the steel strands. is may cause changes to the composite property of the overhead conductors and will be studied in detail in Section 4.2 later. e equation of the axial internal forces on the cross section can be obtained as follows: where n i indicates the number of layer i strands. Figure 2 is the diagrammatic sketch of single span overhead conductor, in which the letters "A", "B," and "O" represent the suspension points on the left and right and the lowest point, respectively. e span of the overhead conductor is l and the height difference is h (when suspension point B on the right is higher than A on the left, h is positive, otherwise it is negative). e ratio between h and l is denoted as β. e axial tensile external force N(x) at the distance x from the left suspension point A (as shown in Figure 2) can be calculated according to the axial tensile stress at the lowest point O σ 0 [18]. In equation (18), c denotes the gravity of conductor per unit length: us, the force balance equation of the conductor section at distance of x from the origin can be obtained as follows:  Figure 2). Studies showed that the sag of overhead conductors can be approximately represented by a parabola [18]. If suspension point A on the left is regarded as the origin, the sag f x at the location x is as follows: e relation between the sag f M in the midpoint of the conductor and its arc length L is as follows [18]: If the deformation (due to the gravity and other external loads) is neglected (that is, the axial rigidity of the conductor is assumed to be infinite), its initial arc length L 0 is as follows: If the sag f M at the midpoint is known, the horizontal tensile stress can be calculated with the following equation: If the sag f M at the midpoint is unknown but the height difference f AO between the left suspension point and the lowest point is known, the horizontal tensile stress can be obtained by solving the following equation: In the operation, the arc length of the overhead line will change with the effect of current and load. If the average strain of the whole overhead conductor is ε, the arc length L is

Critical Temperature and Critical Current.
Overhead conductors are generally twisted by multilayers of strands. e deformation of each strand is caused by stress or due to changes in temperature. e deformation due to changes in temperature will not cause internal force in the strands. As the coefficients of thermal expansion and temperature of each layer of strands are different, their temperature deformations are also different. With the increase of temperature, the distribution of stresses in the strands of each layer will vary. If the resultant external force on the conductor is constant, when the temperature rises, the tensile stresses of strands with greater thermal deformation (aluminum strands with higher thermal expansion coefficients) will decrease while the stress of other strands (generally steel core with lower coefficients) will increase. When the temperature reaches a certain critical value, the resultant axial forces on all aluminum strands are zero. At this point, the average temperature of the cross section and the corresponding current are defined as critical temperature (CT) and critical current (CC), respectively. Note that the CT is the same concept as the knee-point temperature [22][23][24].
e CC I c and CT T c of the conductor meet the following conditions: where P a (I) indicates the resultant force on the aluminum strands when the current is I; m i indicates the number of aluminum strands of layer I; α i indicates the angle between the layer i aluminum strands and the cross section; A i indicates the cross-sectional area of a single strand of the layer i aluminum strand; σ i indicates the stress of layer i aluminum strands; and n a indicates the total layers of aluminum strands. CT and CC are related to span, height difference, external load (including its own weight), ambient temperature, wind speed, and solar heating.

Calculation Process.
e CT, temperature field, sag, and layered stress proposed above can be calculated according to the flowcharts shown in Figure 3. e main calculative module indicates the process for calculating CT and CC. e submodules involve the calculation process of temperature field (submodule 1 in Figure 3) and the calculation process of sag and stress (submodule 2 in Figure 3). e two submodules need to be repeatedly called in the calculation of critical temperature. e calculation steps of the main module are as follows: Step 1: set the initial values of current step size d I , number of iterations i � 0, and allowable error tol. Calculate the upper and lower limits of current. Perform steps 1.1-1.3: Figure 2: Diagrammatic sketch of the overhead conductors.
Step 1.1: call the submodule 1 to calculate the temperature field of the conductor cross section with a given current I � i · d I .
Step 1.2: call the submodule 2 to calculate the layered stress of the conductor and the resultant force P a on aluminum strands (see equation (27)).
Step 1.3: if P a ≥ 0, i � i + 1, go to Step 1.1. Otherwise, the upper and lower limits of CCI (2) � i · d I and I (1) � (i − 1) · d I and the corresponding resultant force P (2) a and P (1) a on the aluminum strands are obtained.
Step 2.2: if |I − I (2) | ≥ tol, perform step 2.2.1∼step 2.2.3: Step 2.2.1: call the submodules 1 and 2 to calculate the resultant force P a on the aluminum strands when the current is I.
Step 2.2.2: if P a · P (2) a < 0, update the lower limit, i.e., I (1) Given values for paremeters k s , k a , I, T a and v w ; Set initial values It max , tol, i = 0,T (i) = T a ; Import the mesh of the cross section Solve T (i) using finite element method   Output: ε (j) = ε, P a (j) , layered stress and strain Given ε (1) and ε (2) . Read in parameters of i th layer: Else exit and promt an error If f (ε)f (ε) (2) < 0 then: ε (1) = ε (2) , f (ε (1) ) = f (ε (2) ) Yes No Yes No Submodule 2: the calculation process of sag and stress Solve the stress and strain for each cross section Calculate ε -= n j=1 ε (j) /n, L (i) and f m (i) Figure 3: e calculation methods for critical temperature, temperature field, sag, and layered stress.

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Step 2.2.3: update the upper limit, i.e., I (2) � I, P (2) a � P a , and go to step 2.1. Step 2.3: otherwise, output the critical current I c , critical temperature T c , and corresponding layered stress. e calculation of temperature field is performed with fixed point iteration, and the calculative process is as follows: Step 1: set the initial values of allowed maximum number of iterations It max , allowed error tol, coefficient of thermal conductivity k s for steel core and k a for aluminum strands, current I, ambient temperature T c , and wind speed v w , respectively. Import the crosssectional finite element mesh.
Step 2: update the number of iterations i � i + 1. If i ≤ It max , continue; otherwise, the calculation fails; exit and prompt an error.
Step 3: calculate the average temperature T av of the conductor cross section and the average temperature T sur of the conductor surface. Calculate k air , s s , s a , and α with equations (3)-(7).
Step 4: solve the temperature field of the conductor cross section by using the finite element program derived from equations (1) and (2).
Step 5: if the temperature field T (i) of the iteration is very different from the result T (i− 1) of the previous iteration (i.e., ‖T (i) − T (i− 1) ‖ ≥ tol), perform step 2-5.
Step 6: otherwise, converge the calculation and save the results. e iterative process for calculating the sag and stress is as follows: Step 1: import the geometric information of weight per unit length c, span l, height difference h, and initial sag f AO . Set the initial value of the iteration number i � 0 and the maximum number of iterations It max .
Step 2: solve equation (24) to obtain the initial value of horizontal tensile stress σ 0 . Calculate the initial iterative values f (0) m of sag at the midpoint according to equations (22) and (21), respectively.
Step 3: equally divide the whole span of conductors into n − 1 sections (totally n nodes). For each node j ∈ [1, n], calculate the strain and layered stress and the resultant force P (j) a of aluminum strands. Perform the following steps: Step 3.1: calculate the resultant external force N(x) at the node j of x with equation (18).
Step 4: calculate the average ε � n j�1 ε (j) /n of strain at n nodes. Calculate the arc length L (i) and sag f (i) m at iteration i with equations (25) and (21) Step 5: if the sag f (i) m at the midpoint in the iteration is very different from the result f (i−1) m of the previous iteration (i.e., ‖f (i) m − f (i−1) m ‖ ≥ tol), execute the steps 5.1-5.3: Step 5.1: update the iteration number i � i + 1.
Step 5.2: if the iteration number i ≤ It max go to step 3.
Step 5.3: else the calculation fails, exit and prompt an error.
Step 6: otherwise, (i.e., ‖f (i) m − f (i−1) m ‖ < tol), output the sag f (i) m at the midpoint, the resultant force min(P (j) a , j � 1 ∼ n) on the aluminum strands at the lowest point in the span, etc.

Layered Temperature.
With reference to the experimental study [26] of conductor JL/G1A-630/55 in the different current and wind speed conditions, this article conducts numerical simulation of the conductor and sets up a finite element model, as shown in Figure 4. e model is performed with meshing by using triangular elements with a maximum size of 2.4 mm and with 8080 elements and 4184 finite element nodes. e results of numerical simulation and experiments are compared in Table 1. In the measurement of the conductor's temperature, the surface temperature of the conductors is recorded. However, in the numerical simulation, the temperature field on the whole cross section of the conductor can be calculated and then the average temperature of strands of each layer can be obtained. Table 1 and Figure 5 show that (1) the surface temperature simulation results of the conductor at each wind speed fit well with the experimental values apart from individual data when the wind speed is 1 m/s and the current is 922 A. e simulation results are different from the experimental ones by 9.91% and the experimental data at this point are obviously unreasonable. Under the currents of 1106 A and 1290 A, when the wind speed increases from 1 m/s to 2 m/s, the surface temperature decreases by 9°C and 10°C, respectively, compared to a 1°C reduction when the current is 922 A. erefore, the experimental value when the wind speed is 1 m/s and the current is 922 A is lower than expected. (2) e difference of the cross-sectional temperature is small (about 1°C) and the temperature of the internal strands is slightly higher than that of external strands. erefore, the temperature gradients can be ignored in the calculation of layered stress. (3) With the increase of current, the cross-section temperature of the conductor rises, while with the increase of wind speed it drops.

Layered Stress.
e layered stress experiment on JL/ G1A-400/35 ACSRs is performed through stripping, that is, stretching the conductor to a certain strain and successively stripping the conductors by layer to calculate the difference of axial force, so the layered stress of the conductor can be calculated under the strain. Only the layered stresses of the aluminum strands are studied, as the axial force is mainly resisted by the aluminum strands (more than 3 times than steel core). e experiment is performed at temperature 20°C without applying current. e calculation and test results are compared in Figure 6, where the legend layer 1, 2, and 3 represent the external, middle, and internal layers of aluminum strands respectively. As shown in the figure: (1) the numerical simulation results in the elastic phase (with strain of less than 0.2%) and   the experimental and numerical results agree each other with an acceptable accuracy; (2) the tensile stresses of the internal aluminum strands are slightly greater than external ones. ere are still some differences between the two results (with maximum error of 17.6%) which may be because the displacement deformation of strands under tension is random as strands of each layer cannot be closely arranged without any gap in the ACSR construction. It should be noted that although layered stress does not perform well, the resultant tensile stresses fit well with the experimental results, which will be discussed later.

Temperature-Sag Characteristics.
e sag-temperature curves of heat-resistant aluminum alloy conductor JNRLH1/ G1A-400/65 (with initial tension of 38.8 kN) and JNRLH1/ G1A-630/55 (with initial tension of 48.5 kN) are compared in Figures 7 and 8. e curves reflect the change law of sag (at the midpoint) with the cross-sectional temperature. e experiment is performed as follows: fix the overhead lines (with a height difference of 0) on the equipment with a span of 60 m. en, gradually increase the temperature of the conductors by charging with electricity and measure the sag at the midpoint. Figures 7 and 8 show that the numerical and experimental results of the two examples fit well (the maximum difference is 7.37% and 4.49%, respectively). With the increase of temperature, the sag increases with a rate that initially increase and then decrease. When the temperature increases to a certain value (i.e., Point A in Figures 7 and 8), the resultant tensile force on the aluminum strands reduces to zero and the sag gradually increases. According to equation (26)   Mathematical Problems in Engineering and 142.06 A, respectively. erefore, the slope of the sagtemperature curves of the conductors near the critical temperature is in the transition phase, i.e., the slope of the sag-temperature curves increases below and decreases above the critical temperature, but does not change greatly near the critical temperature. When the conductor temperature is close to the critical temperature, the steel core in the conductor starts to bear all the loads. However, around CT, the sag reaches a relatively large value, in which the axial force along the conductor becomes relatively lower and the rate of sag increase becomes smaller.

Temperature-Tension Force
Characteristics. e experimental and numerical temperature-tensile force characteristics of heat-resistant aluminum alloy conductor JNRLH1/ G1A-400/65 (with initial tension of 34 kN) and JNRLH1/ G1A-630/55 (with initial tension of 42 kN) are compared in Figures 9 and 10. e experiment is performed using the same process in Section 5.3. Figures 9 and 10 show that the numerical and experimental results fit well (the maximum difference is 5.19% and 5.63%, respectively). e tensile force of the overhead conductors decreases gradually with the increase of temperature.
e CT and CC are, respectively, (83.56°C, 827.64 A) and (100.96°C, 1391.86 A) for the two examples. Below the CT, the tensile force rapidly decreases as the temperature increases and gradually decreases until the critical temperature. It should be noted that the critical point cannot be directly observed according to the curve of temperature vs. tensile force but needs be calculated.

Layered Stress and Sag at the Critical Point.
In this section, the experiment in Section 5.4 is simulated, and the relationship between the current and the layered stress (and sag) at the midpoint of overhead conductors is studied. At CT (or CC), the characteristics of the layered stress and sag are discussed. Figures 11 and 12 show the law of the current with the layered stress on the cross section at the midpoint of JNRLH1/G1A-400/65 and JNRLH1/G1A-630/55, respectively. e legend layer 1∼2 (in Figures 11 and 12) represent the internal and external layer of steel core, respectively, the legend layer 3∼4 (in Figure 11) and layer 3∼5 (in Figure 12) indicate the internal and external layers of aluminum strands, respectively. Figures 13 and 14 show relationships between their sags at the midpoint and the current, respectively. e "A" point in Figures 11-14 indicates the critical current. Figures 11 and 12 show that the stresses of the steel core of the internal layers are basically the same and the stresses of the aluminum strands of the external layers are basically the same too. With the increase of current, the stresses of the external aluminum strands gradually decrease and those of the external steel core decrease first and then increase. When the current is greater than the critical current, the stresses of the steel core increase rapidly. In Figure 11, the stress of the internal steel core at critical current is less than the initial stress (no current). However, its stress is far greater than the initial stress in Figure 12. It is shown (in Figures 13 and 14) that the sag (at the midpoint) gradually increases with the increase of current and that, under the critical current, they are 3.8 and 4.7 times more than the initial sag, respectively.
is is because when the current is far below the CC, with the increase of current (the temperature rises), the length and sag gradually increase and the tensile force on it decreases leading to gradual reduction in the stress of the internal steel core. When the current (or temperature) is higher than the CC (or CT), the strands (such as aluminum strands) with a larger thermal expansion coefficient will be under pressure rather than tension. e strands (steel core) with a smaller thermal expansion coefficient will resist the tensile force caused by the aluminum strands under pressure but not that of external load, which leads to a rapid increase in the stress  of the steel core and linear increase in the sag. It is obvious that this stress state is not conducive to the synergy of each layer of strands. In actual operation, the current (or temperature) of the conductor should be controlled to be lower than the critical current (temperature) as much as possible.

e Sensitive Factors of Critical Temperature and Current.
As discussed in Section 5.1, the operational current and temperature should be less than the critical values. As the critical temperature and current are related to layered stress and temperature field of conductors, the following factors    are assumed to be sensitive: (1) solar intensity S with a unit of W/m 2 and range of 0∼1000; (2) solar absorption α s , dimensionless, with a range of 0.23∼0.9, reflecting the age of the conductors; new bright conductors have an absorption rate of 0.23 and old gray conductors have an absorption rate of 0.9, and conductors in service period have an absorption rate between them; (3) ambient temperature T a with a unit of°C and range of −20∼50; (4) wind speed v w with a unit of m/s and range of 0∼7; (5) initial tensile force F with a unit of kN and range of 0∼0.3F max , where F max indicates the ultimate bearing capacity. In other words, critical temperature and critical current have a functional relation with these variables T c (α s , T a , v w , F, S) and I c (α s , T a , v w , F, S). is section takes the heat-resistant aluminum alloy conductor JNRLH1/G1A-400/65 as an example. First, the fitting of the functional relationships (e.g., linear/quadratic/ exponential functions/trigonometric functions) between critical current and the abovementioned five variables is conducted. e relationships between critical current and each of the five factors are, respectively, shown in Figures 15(a)-15(e). en, the relationship between critical current and its five factors can be postulated as equation (29) according to the fittings of the individual factor. e fitting coefficients a i , (i � 1 ∼ 11) with a confidence level of 95% can be obtained with the multifactor method (see Table 2 for specific data). e relationship between critical current and its influencing factors is shown in Figure 16. e two dotted lines in the figure represent the confidence interval of predicted values (with a corresponding confidence level of 95%): I � a 1 F a 2 + a 3 sin a 4 v w + a 5 + a 6 α s + a 7 sin a 8 T a + a 9 + a 10 S + a 11 .

(29)
In the same way, the change law of critical temperature with each influencing factor can be obtained (as shown in Figure 17). e results show that the critical temperature has a great correlation with the initial tensile force and little correlation with other factors (i.e., they can be ignored). For function relationship fitting, the following equation can be obtained: where a � 13.54, b � 0.4206, and c � 24.72. Figures 15 and 16 show that (1) the sensitive factors of critical current are successively wind speed, ambient temperature, horizontal tensile force, solar intensity, and solar absorption according to their significance level. (2) Horizontal tensile force and wind speed are positively correlated with critical current. Solar absorption, solar intensity, and  ambient temperature are negatively correlated with it. Figure 17 shows that the critical temperature is positively correlated with horizontal tensile force and has little correlation with other factors.

Conclusion
is article proposes a set of numerical methods for calculating the temperature field, layered stress, and sag of overhead conductors, based on which the critical temperature and critical current are studied. e calculation methods and procedures are verified through experimental results. e authors also studied the relationship between the five most sensitive factors and the critical temperature (or current). e following conclusions are made: (1) e results of numerical calculation and experiments for conductor temperature agree well. e temperature gradient on the cross section is relatively small; therefore, its impact on the layered stress can be ignored in the calculation. Meanwhile, the average temperature of cross sections rises with the increase of current and drops with the increase of wind speed. (2) e numerical simulation and experimental results for layered stress in the elastic phase (with strain of less than 0.2%) agree each other with acceptable accuracy. e tensile stresses of the internal aluminum strands are slightly greater than those of the external ones. (3) e calculated sages fit well with the experimental results. With the increase of temperature, the sag increases with a rate that first increases and then decreases, that is, when the conductor temperature increases close to the critical temperature, the sag of the conductors gradually increases. e calculated tensile forces agree well with experimental ones. Below the critical temperature, the tensile force decreases rapidly with the increase of temperature, while the changing rate decreases close to the critical temperature.
e critical temperature cannot be directly observed according to the curve of temperature vs. tensile force before calculation. (4) With the increase of current, the stresses of the steel core first decrease and then increase, while those of the aluminum monotonically decrease. When the current (or temperature) of overhead conductors is higher than the critical current (or temperature), the stresses of the steel core rapidly increase and the sag linearly increases with increasing temperature. At the critical current, the sag increases to 3-4 times of its original value. In some cases, the stresses of steel cores in the conductor are far greater than their initial value. In actual operation, the current (or temperature) of conductors should be controlled so that it remains lower than the critical current (or temperature) as much as possible. (5) e critical temperature is positively correlated with horizontal tensile force and has little correlation with other factors. According to their relative sensitivities, the importance ranking of the influencing factors to critical current are (from high to low): wind speed, ambient temperature, horizontal tensile force, solar intensity, and solar absorption. Horizontal tensile force and wind speed are positively correlated with critical current. Solar absorption, solar intensity, and ambient temperature are negatively correlated.

Data Availability
All the data used to support the findings of this study are included within the article.

Disclosure
Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsors.

Conflicts of Interest
e authors declare that there are no conflicts of interest. Mathematical Problems in Engineering 13