Study on Heat Transfer Agent Models of Transmission Line and Transformer

When using heat transfer simulation to study the dynamic overload of transmission line and transformer, it needs to establish the mathematical expression of heat transfer. However, the formula is a nonlinear differential equation or equation set and it is not easy to get general solutions. Aiming at this problem, some different temperature change processes caused by different initial conditions are calculated by differential equation and equation set. New agent models are developed according to the characteristics of different temperature change processes. The results show that the agent models have high precision and can solve the problem that the original equation cannot be directly applied in some practical engineers.


Introduction
With economic development, the demand for electricity is continuous increasing. The rated capacities of existing transmission line and transformer do not meet the increase of demand for electricity. Meanwhile, the actual capacity of the existing power transmission and transformation equipment is often not fully utilized when it is in normal operation, and there are still hidden capacity available, so it has become the best way to meet the power demand that the equipment is scientifically and safely dynamic overload [1][2][3]. Temperature of transmission line and insulation temperature inside the transformer are the key factors that directly affect their load capacity. At present, direct temperature measurement and indirect temperature measurement can get the transmission line temperature and transformer hot spot temperature. In the direct temperature measurement, sensors were directly installed on the transmission line and the transformer and monitored their temperature in real time [4,5]. Besides, using the relationship between the wire temperature and its impedance, the temperature was calculated by directly measuring the line current and voltage [6,7]. Indirect temperature measurement mainly includes heat transfer simulation [4,8], numerical calculation [9,10] and intelligent calculation [11,12] methods. However, this mathematical expression of heat transfer simulation is a nonlinear differential equation or equation set, and it is not easy to obtain its general solution directly. Based on the original 2 1234567890 ''""

Agent model of heat transfer in transmission line
Equation (1) is a first-order nonlinear differential equation, not easy to find its general solution, but a particular transmission line can be used numerical solution given approximate solution. For ACSR LGJ400/35 (steel wire wrapped in aluminum wire), the aluminum cross sectional area is 400mm² , and the steel cross sectional area is 35mm² . The M is 1.349kg/m; The C is 880J/(kg·℃); The R T is 0.00007389Ω; The T 0 is 40℃; The V is 0.5m/s; the S I is 1000W/m 2 ; The k e is 0.9; The γ is 0.9; The outer diameter is 0.02682m; The line voltage is 138kV. Several different initial conditions are given, which are different initial temperature and load power. Their temperature change process is shown in figure 1.  In figure 1, the first "*" from the left of each curve is the node at which it changes from the initial temperature to a quarter of the difference from the initial temperature to steady-state temperature. The abscissas of first "*" of each curve are almost the same. The second "*" from the left of each curve is the node at which it changes from the initial temperature to a half of the difference from the initial temperature to steady-state temperature. The abscissas of second "*" of each curve are also almost the same. The n-th "*" from the left of each curve is the node at which it changes from the initial temperature to 1-1/(2n-1) of the difference from the initial temperature to steady-state temperature. Meanwhile, the abscissas of n-th "*" of each curve are almost the same (figure 1). According to this feature, a linear piecewise function can be constructed as its agent model. The coordinates of the segment nodes are shown in table 1.  Where, ΔT = T s -T p , T s is the steady-state temperature, T p is the initial temperature. The equation (1) is assumed to equal to 0, and the steady-state temperature can be obtained. The steady-state temperature is calculated per 1MW at load power from 1MW to 200MW. According to the obtained scattered values, the function have fitted with the binomial function, which is as shown in equation (2).
Where, a is 0.0006, b is 0.0043, c is 325.4. In this way, it is possible to construct an agent model as shown in equation (3), and at a given initial temperature and the load power, an approximate function of the temperature at any time can be obtained. , The internal heat transfer process of the transformer is not same with transmission line, and the temperature of the iron cores and windings, the transformer oil and the housing are quite different, as shown in figure 2.  First, the iron cores and windings generate thermal energy due to power loss. Some of the heat, Φ 1 , is transferred to the oil due to the convection heat transfer. Some of the heat in the oil, Φ 2 , is dissipated by the convection heat transfer to the transformer housing. The heat in the housing, Φ 3 , is delivered to the surrounding air due to convection and thermal radiation. From the above heat transfer process, the heat transfer differential equations of transformer can be written as showing in equation set (4) -(6).

Agent model of heat transfer in the transformer
Equation set (4) -(6) are first-order nonlinear differential equation set. It is more difficult to solve than equation (1), but it can also be solved by numerical solution under the given initial conditions. For S11-M-type transformer, its iron core is silicon steel sheet; the windings are oxygen-free copper; the windings interlayer insulating material is rhombus adhesive tape; the rated capacity is 400kVA; The maximum iron cores and windings temperature is 15℃ higher than the average temperature of the transformer; The maximum temperature should not exceed 95℃. To ensure long-term operation, the average temperature should not exceed 80℃. The M I is 295kg; The C I is 450J/(kg·℃); The P is 7500(0.   The equation set (4) -(6) are assumed to equal to 0, the steady-state temperature of the transformer can be obtained. The steady-state temperature is calculated per 1kW at load power from 1kW to 600kW. The steady-state temperature function (T s ) is fitted with a linear function according to the obtained scattered values, which is as shown in equation (7).
In this way, the linear piecewise function can be constructed as an agent model of the temperature change process of the iron cores and windings. Given the initial temperature of iron cores and windings and the load power of the transformer, the temperature at any time can be calculated by equation (3) , Where t 1 , t 2 , ..., t 7 nodes are shown in table 2, T s is as shown in equation (7). For the S11-M transformer in this paper, in order to verify the accuracy of the agent model (3), two different initial conditions are selected to draw the theoretical temperature changing curve and the approximate temperature changing curve as shown in figure 4. For the two kind of initial conditions (the 1 st one and the 2 nd one in figure 4)

Conclusions
In this paper, explicit agent function models of heat transfer of transmission line and transformer have been established respectively. After an initial temperature and load power are given, the temperature at any time can be calculated. The results show that the error between the agent model and the theoretical model is very small (in table 3 and figure 4), which can be applied to dynamically confirm capacity.