Comparison of corona degree between the stranded and the shaped wires of HVDC transmission lines

Abstract

Most of high voltage transmission lines use the stranded wire, while the surface of the shaped wire is smoother than the stranded one. It is investigated in this paper that how much the corona can be weakened by the shaped wire, by calculating the ion flow. The steady state corona model is employed, and a local domain surrounding the wire is chosen as the field domain to simulate the corona phenomena near the wire precisely. The equation system of corona model is a multi-physics coupled problem with electric field and particle diffusion and migration. The approach of defining and applying the boundary conditions for negative and positive wires is proposed, and the solution of coupled problem is implemented by using software COMSOL. By the simulation of 500 kV transmission line, the corona current of the shaped wire is around 35% less than that of the stranded wire.

Keywords

Corona loss high voltage transmission line shaped wire stranded wire

1. Introduction

The corona is a complex phenomenon concerning different disciplines and random factors [1, 2], so that it is a big challenge to simulate it precisely. However, the discharge both corona and arc is one of the key issues to investigate in high-voltage power systems. To avoid or control the discharge in the high-voltage transmission lines and substations, it is important or essential to simulate it numerically, on which some work has been conducted [3, 4].

The wires used in the high-voltage transmission line can be classified into two types, the stranded and the shaped wire. The former is composed by winding a few aluminium threads around a few steel threads, whose section is shown in Fig. 1a. The latter is shown in Fig. 1b whose outer aluminium conductors are assembled by Z-shaped structures. Since the former is much easier to be manufactured than the latter, the former is commonly used in engineering. Obviously, the outer surface pattern of the latter is much smoother than that of the former, and the corona degree of the later must be weaker than the former in a certain voltage. That how much the corona can be weakened by the shaped wire relative to the strand one is the task to be investigated in this paper.

The adopted approach to compare the difference of the corona is the numerical simulation of the ion flow or corona current around the wires. In the simulation, the local domain surrounding the wire is taken as shown in Fig. 1, because of the periodic symmetry along the circumferential direction. Since the finite element method is employed for the simulation, a small domain with a radius of $r_{out}$ as shown in Fig. 1 is taken, so that the corona phenomena occurring near the wire can be concentrated and calculated more precisely, for the practical field region bounded by the earth surface far away from the wires.

Figure 1.

Section structure of the stranded and the shaped wires (in left and right respectively).

2. Corona model and boundary condition definition

In order to analyze the difference of corona degree of different surface shapes, it should concentrate on the ionosphere surrounding the wire. Actually, inside the ionosphere there exists randomly varying complex pulse phenomena. In the discharge theory research area, the transient simulation model has been presented [1, 2, 3], but the mathematical model and simulation method are very complex.

2.1 Steady-state mathematical model and solution domain model

Actually, in the study of transmission line corona, it is sufficient to know the average effect of corona, which is called the steady-state model. Woong-Gee demonstrated the consistency of the steady-state model and the transient model for the calculation of corona in device [4]. The steady-state model without time-varying terms is just the fluid model including particle transport equations and Poisson’s equation, i.e.,

$\displaystyle\nabla\cdot[\vec{j}_{e}{-}D_{e}\nabla n_{e}(r)]=-(\alpha-\eta)j_{e}$ (1) $\displaystyle\nabla\cdot\vec{j}_{p}{=}\alpha j_{e}$ (2) $\displaystyle\nabla\cdot\vec{j}_{n}=-\eta j_{e}$ (3) $\displaystyle\nabla^{2}\varphi=-e(n_{p}-n_{e}-n_{n})/\varepsilon_{0}.$ (4)

Equation (1) shows the characteristic of the current density $\vec{j}_{e}$ formed by the electrons, where $\alpha$ is the impact ionization coefficient, $\eta$ is the adsorption coefficient describing the adsorption of electrons by other particles, $D_{e}$ is the diffusion coefficient, and $n_{e}(r)$ is the electron density. Equations (2) and (3) indicate the characteristics of the current density formed by positive and negative ions respectively. Equation (4) is the Poisson’s equation of the potential. For Eqs (1)–(3), the corresponding current density $\vec{j}(r)=\mu n(r)\vec{\bar{E}}(r)$ , where $\mu$ and $n$ are the corresponding mobility ratio and ion density respectively.

The above equation system is the general mathematical model, especially for the ionosphere surrounding the wire. But for the region far from the wire, the variables and their governing equations are much simpler, as there only exits positive ions for the positive polarity wire, and negative ions for the negative polarity wire, and the ionization coefficient as well as the adsorption coefficient are null.

In the simulation, the local domain surrounding the wire with a radius of $r_{out}$ is taken as shown in Fig. 1, where $r_{out}$ is chosen as greater than the ionosphere, only existing positive ions for positive polarity wire, and negative ions for negative polarity wire. Actually, the thickness of the ionosphere is only in millimetre, but $r_{out}$ is chosen in centimetres in the simulation. On the other hand, because of the periodic symmetry along the circumferential direction, only a small sector is taken as the local domain. Two-dimensional models are adopted.

2.2 Boundary conditions

Due to the selection of a small local domain surrounding the conductor to be calculated, the boundary condition is unknown in principle. However, since the outer boundary of the domain is beyond the ionosphere, the particle state is relatively simple, so that the approximated boundary conditions can be set. In the corona mathematical model given above, there are four unknowns, namely the electron density $n_{e}(r)$ , positive ion density $n_{p}(r)$ , negative ion density $n_{n}(r)$ and the potential.

The boundary conditions for the calculation domain are set approximately as Table 1, where $r_{in}$ and $r_{out}$ denote the surface of the conductor and the outer boundary of the domain respectively, which are explained as follows.

Since the secondary electron emission of negative polarity corona comes from the cathode surface, for the negative polarity conductor surface, the number of emitted electrons is taken as known [5, 6]:

$n_{e}(r_{in})=\frac{\gamma_{p1}\int\limits_{r_{in}}^{r_{out}}\alpha(r)n_{e}(r)% \mu_{e}(r)E(r)g(r)e^{-\mu(r-r_{in})}r{d}r}{r_{in}\mu_{e}(r_{in})E(r_{in})}$ (5)

where, ${\gamma}_{{p1}}$ is the efficiency of generating electrons by the photoionization, $\mu$ is the photon absorption coefficient, and $g(r)$ is the area factor representing the photon absorption probability under the influence of the electrode surface geometry, whose values taken for the simulation can be found in [5]. Since Eq. (5) is a nonlinear integral equation and it also includes the unknown $E(r)$ , the solution of the corona model must be an iterative procedure, by setting an initial value of $n_{e}(r_{in})$ .

On the outer boundary of negative polarity conductor, the number of electrons is null, since only negative ions exist over there. Certainly, the positive ions are also null over there.

For the potential, it is known on the conductor surface, but it is unknown on the outer boundary. It can be determined and calculated according to the voltage of the transmission line and the height of the line above the earth simply by the image method. Certainly, this approximate determination will introduce error for the corona solution, since the contribution of space charges are not considered in the solution of potential, but this error can be weakened in the comparison of corona degree for different wire patterns, i.e., the error of the conclusion of the corona degree difference obtained will be much smaller than the initial error.

For the positive conductor, since the secondary electron comes from the photoionization, only the influence to corona by the contribution of photoelectrons entering from the outer boundary is taken into account. The number of electrons on $r_{out}$ has the similar form as Eq. (5), but replacing ${\gamma}_{{p1}}$ by ${\gamma}_{{p2}}$ that is the efficiency of generating electrons by the photoelectric effect [5].

Generally, on the conductor surface, the ions with the same polarity as the conductor is zero, because the electric charges with the same polarity would be of infinite repulsion. On the outer surface, the ions with the opposite polarity as the conductor are zero.

For the particle transport problem, a boundary condition is “outflow” that means the particles flow out the region with the normal direction to the boundary, and without adsorption as well composition phenomena.

Table 1

Boundary conditions for negative and positive polarity wires

Polarity of wire	Particles	$r_{in}$	$r_{out}$
Negative	Electron	$n_{e}=$ const	$n_{e}=0$
	Positive ion	outflow	$n_{p}=$ 0
	Negative ion	$n_{n}=$ 0	outflow
	Potential & field	$\varphi(r_{i})$	$\varphi(r_{0})$
Positive	Electron	$n_{e}=$ 0	$n_{e}=$ const
	Positive ion	$n_{p}=$ 0	outflow
	Negative ion	outflow	$n_{n}=$ 0
	Potential & field	$\varphi(r_{i})$	$\varphi(r_{0})$

3. Implementation of simulation by software COMSOL

In order to solve this coupled problem, the solution of electric field and the solution of particle creating, compounding and transporting are separated into two-step process. Certainly, in the solution of the electric field, in addition to the known potential boundary conditions, the space charge is unknown, which requires the results particle solution. Meanwhile, the solution of the particle process requires the results of electric field. Therefore, an iterative procedure is required to solve this coupled problem.

As mentioned above, since the electron density as the boundary condition on the conductor surface is given by a nonlinear integral equation for the negative conductor by Eq. (5), and it is similar to the outer boundary of the positive conductor, an iterative procedure must be adopted to solve the particle problem. In fact, this boundary condition is a self-consistent process constraint that constrains the relationship of values of the field quantities, i.e., as long as the field quantities satisfy the relationship, the correct solution is obtained. Consequently, the iterative procedure for solving the problem requires two cycles as follows.

Step 1:
Determine the potential on $r_{in}$ and $r_{out}$ , and apply all known boundary conditions.
Step 2:
Solve Poisson’s equation given by Eq. (4).
Step 3:
Set a value for $n_{e}(r_{in})$ for negative polarity conductor, and $n_{e}(r_{out})$ for positive one.
Step 4:
Solve corona ion creating and transporting model governed by Eqs (1)–(3).
Step 5:
Judge whether Eq. (5) is satisfied. If not, adjust $n_{e}(r_{in})$ and $n_{e}(r_{out})$ , and return to Step 4.
Step 6:
Judge the difference of the maximum value of charge between the current step and the previous one. If it is less enough, complete the solution, otherwise return to Step 2.

The solution of the corona ion transport equations can be implemented easily by using the chemical mass transport simulation module of COMSOL Multiphysics, which is named as “chds”. In chds module, cancelling the convective transfer mechanism, and introducing the electric field mobility mechanism, the general form of the equation is as follows:

$\nabla\cdot(-D\nabla c-zu_{m}{F}c\nabla V)=R,$ (6)

where, $D$ is the diffusion coefficient of the particles, $c$ is the particle concentration to solve, $z$ represents the unit charge carried by the particles, $u_{m}$ is particle mobility, $F$ is the Faraday constant (taking 96485.3), $V$ is the potential, and $R$ represents the reaction term.

In the corona model, there exist three types of particles. By adding three chds modules in COMSOL, the three types of particle models can be described.

Figure 2.
Dimensions of the stranded (left) and the shaped (right) wire.

For the electronic model expressed by Eq. (1), since $\vec{j}_{e}(r)={-}\mu_{e}n_{e}(r)\vec{E}(r)$ , replacing $n_{e}$ by $c_{e}$ , setting $\mu_{e}=\mu_{m,e}{F}$ , and the reaction term in Eq. (6) as $R=(\alpha{-}\eta)\cdot c_{e}\cdot\mu{}_{e}\cdot E$ , the COMSOL equation for solving the electron density can be:

$\nabla\cdot(-D_{e}\nabla c_{e}-u_{m,e}{F}c_{e}\vec{E})=(\alpha{-}\eta)\cdot c_% {e}\cdot\mu_{e}\cdot E$ (7)

For the positive ion model expressed by Eq. (2), since $\vec{j}_{p}(r)=\mu_{p}n_{p}(r)\vec{E}(r)$ , and setting $R=\alpha\cdot c_{e}\cdot\mu_{e}\cdot E$ , the COMSOL equation for solving the positive ion density can be obtained, and similar to the negative ion model.

The electric field is calculated using the COMSOL AC/DC electrostatic (es) module for Poisson’s equation. By coupling the above four modules, and setting the boundary conditions and conducting the iterative procedure, we can get the solution.
4. Simulation results

The simulation is verified by the solution of smooth cylindrical wire that is a one-dimensional problem, for which the analytical solution exists [7]. The solutions are in good coincident. Afterwards, a 500 kV HVDC transmission line with two types of wires is simulated, whose height from the earth surface is 20 m. The practical transmission line is composed by four-split wires or conductors. The simulation model only takes one conductor into account. Therefore, the voltage must be decreased to make the maximum electric field intensity on the single conductor surface is approximately the same as that on the split conductors. A potential of 500 kV should be decreased to 91 kV that is used to simulate both types of wires. The dimensions of each thread of wires are illustrated in Fig. 2. Both wires are composed by 24 threads, so that $\alpha$ in the figures is $\pi$ /12, and $R=$ 24.3 mm, $r_{1}=$ 2.8056 mm, and $r_{2}=$ 0.8 mm.

Figure 3.

Distribution of electron density of stranded (left) and shaped wire (right).

In the simulation, the radius of outer boundary is chosen as 3.16 cm. The number of triangular finite elements employed is 119516 and 68346 respectively for the stranded and shaped wire. The distribution of the electron, positive ion and negative ion density of stranded and shaped wires in positive polarity are illustrated in Figs 3, 4, and 5 respectively. Their maximum values and those in negative polarity are given in Table 2. By integrating the ions on the outer boundary and timing the number of threads, the corona current per unit length of wire can be obtained, which indicates the corona degree and the corona loss. For the stranded wire, the corona currents are 1.16 and 2.157 $\mu$ A/cm for negative and positive polarity wires respectively, which are 1.45 and 1.34 times those of the shaped wire.

Figure 4.

Distribution of positive ion density of stranded (left) and shaped wire (right).

5. Conclusion

The steady state corona model is sufficient in the simulation of transmission line corona loss, as the loss is an average value and the line works steadily. A local domain surrounding the wire must be employed to simulate the corona by the finite element method, since small size elements are needed. The definition and implementation of boundary conditions on the wire and domain outer surfaces is the key task. The solution of the corona ion transport equations can be implemented by COMSOL multi-physics modeller. By the simulation of 500 kV transmission line, the corona current of the shaped wire is around 35% less than that of the stranded wire.

Table 2

Maximum values of electron and ion density for two types of wires (unit: electron in G/m ${}^{3}$ , positive and negative ion in T/m ${}^{3}$ )

	Positive polarity wire		Negative polarity wire
	Stranded wire	Shaped wire	Stranded wire	Shaped wire
Electron	98	16	31	8.1
Negative ion	3.2	1.8	28	7.3
Positive ion	33	6.8	33	8.6

Figure 5.

Distribution of negative ion of stranded (left) and shaped wire (right).

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