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Method of local characteristics for calculating electric field and ion current of HVDC transmission lines with transverse wind

2016; Institution of Engineering and Technology; Volume: 11; Issue: 4 Linguagem: Inglês

10.1049/iet-gtd.2016.1541

1751-8695

Ji Qiao, Jun Zou, Jiansheng Yuan, Jaebok Lee, Mun-No Ju,

Thermal Analysis in Power Transmission

IET Generation, Transmission & DistributionVolume 11, Issue 4 p. 1055-1062 Research ArticleFree Access Method of local characteristics for calculating electric field and ion current of HVDC transmission lines with transverse wind Ji Qiao, Corresponding Author Ji Qiao qiaoj15@mails.tsinghua.edu.cn State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJun Zou, Jun Zou State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJiansheng Yuan, Jiansheng Yuan State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJaebok Lee, Jaebok Lee Electrical Environment Group, Korea Electrotechnology Research Institute, Changwon, 641600 KoreaSearch for more papers by this authorMun-no Ju, Mun-no Ju Electrical Environment Group, Korea Electrotechnology Research Institute, Changwon, 641600 KoreaSearch for more papers by this author Ji Qiao, Corresponding Author Ji Qiao qiaoj15@mails.tsinghua.edu.cn State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJun Zou, Jun Zou State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJiansheng Yuan, Jiansheng Yuan State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments, Department of Electrical Engineering, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorJaebok Lee, Jaebok Lee Electrical Environment Group, Korea Electrotechnology Research Institute, Changwon, 641600 KoreaSearch for more papers by this authorMun-no Ju, Mun-no Ju Electrical Environment Group, Korea Electrotechnology Research Institute, Changwon, 641600 KoreaSearch for more papers by this author First published: 01 March 2017 https://doi.org/10.1049/iet-gtd.2016.1541Citations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The electric field and ion flow are significantly influenced by the transverse wind around the high-voltage direct current (HVDC) transmission lines. In this study, a new method using local characteristics is proposed to analyse the effect of wind. The present method, called method of local characteristic, updates the ion density based on the local characteristic curve in each second-order finite element method (FEM) element without Deutsch assumption. The defining equation of the characteristics is able to take into account the directional character of the information propagation in convective transport and has a better performance to satisfy the conservation law. The present approach is stable and efficient even in the presence of high-speed transverse wind. Calculations show good agreement with the measurement values of the reduced-scale unipolar model and the full-scale bipolar test line. Finally, the influence of wind on the electric field and ion current is discussed. 1 Introduction The corona phenomena and the resulting electromagnetic environment are important design factors of high-voltage direct current (HVDC) transmission lines. When the potential gradient at the surface of the conductor exceeds the corona-onset value, corona discharge may occur in the vicinity of the wires and generates the ions. In absence of wind, the ions drift outwards mainly under the electric field force while other ones (for instance, gravity force) are neglected. It has been well known that the distribution of ionic space charge will be strongly altered by the external velocity field because the ions drift due to the wind might be dominating far away from the conductors. The effect of wind on the corona losses was first studied on a reduced model [1]. The results have shown that the total corona losses increased with wind speed for the bipolar model while the change could be neglected for the conductor-to-plane arrangement at low wind speed. However, the increase of the corona loss was also observed in [2, 3] for the unipolar model with a longer gap between the conductor and plane. It was explained by the fact that a larger number of ions were blown away from the inter-electrode space when the electric field was weak. For the full-scale test line [4], the measurements indicated that corona current increased with the wind speed for the downwind pole while that of upwind pole had the reverse tendency. However, the reason has not been explained. Another important issue is the influence of wind on the electric field and ion current on the ground level, profiles of which are markedly shifted by wind. Experimental studies have shown that wind made the peak value of the ion current considerably increase on the leeward for both unipolar [2, 5] and bipolar lines [4–6] and its profile-tail was elongated downwind [2, 7]. The effect on the electric field was similar, though the change was not pronounced such as the current [2]. The external wind could change the trajectory and the distribution of the ions. The electric field will also be altered indirectly. For the wind with high speed, the stability problem of numerical methods usually exists [8, 9]. The numerical oscillations always occur for the full finite element method (FEM) methods [9]. Takuma first developed the upstream FEM to solve the ionised field including wind [10]. The upwind scheme was derived to overcome the non-physical instability. However, the space-charge density on the conductor surface was fixed artificially and the influence of wind on the corona losses cannot be considered. This method was later developed for analysing the ion flow of ultra high-voltage direct current (UHVDC) overhead lines [11]. Li [8] proposed an FEM-based optimisation approach for the unipolar model but calculations could not converge when the wind speed is high. Then, the upwind finite volume method was applied for unipolar overhead lines with high-velocity wind [8, 12]. However, the bipolar model with bundle conductors was not studied, which is more important for the operating lines. Qin et al. [13] developed the charge simulation method for ion flow field analysis. Only the calculation for the unipolar case was compared with measurement under windy condition. Recently, the ionised field of bipolar lines has been further studied by the upwind FEM-finite difference method (FDM) combined method [14] and the upstream FEM [5]. The calculations were well compared with experimental values. The method of characteristics (or called flux tracing method) provides another methodology to solve the ion flow problem. Maruvada and Janischewskyj first proposed this method based on the Deutsch assumption which states that the space charges do not affect the direction of the electric field [15, 16]. The modified approach, called flux tracing method (FTM)-wind method, has been recently applied for the case with the wind [17]. The finite difference method was later used to solve the differential equations of FTM to improve the efficiency [18]. To reduce the error caused by the Deutsch assumption, an iterative process is introduced to modify the characteristic lines including wind [9]. However, procedure for tracing flux lines in the whole domain has to be carried out in every iterative step. In this paper, a new method, called method of local characteristics (local-MOCs), is present for analysing the two-dimensional (2D) ion flow in the presence of wind. This approach is derived from the property of the convective transport and has a clear physical picture. The local characteristic line in every triangular element, instead of the global lines, is used to update the ion density. The second-order element is used for a quadratic approximation to potential distribution and thus a smoother characteristic trajectory can be traced. The local-MOC could overcome the stability problems posed by the strong transverse wind and has a good performance on the conservation law of ion current. Its process is similar to the upstream FEM, except the computation of the ion density and thus the calculation is efficient. The calculations show good agreement with the measurements of both unipolar and bipolar models including the wind. Finally, the influence of wind on the ion current and electric field is further analysed. 2 Electric field-ion flow coupled problem The ion flow of HVDC transmission lines is a fully coupled problem, which means that the electric field and the ion current will strongly interact with each other. When the wind is blowing, the pattern of the flux lines of ions will be significantly modified and the electric field will also be changed, which will be more complicated. Some common assumptions are used here: The calculation domain is simplified to a 2D region. Thus, the effects of wires sag and towers are neglected. The diffusion of ions is neglected. The turbulence of the wind around the wires or the ground is not considered. Thus, the motion of space charges can be described as pure transport equations, which are hyperbolic. The ionisation layer is negligible because the thickness of ionisation layer is very small with respect to the whole calculation domain. So, the radius of the corona-onset conductor is same as that of the corona-free one in the calculations [8–21]. For the proposed method, the Deutsch assumption, which is necessary for the flux tracing method [15–17], is not needed. 2.1 Governing equations For the bipolar ion flow field problem, the electric field is determined by Poisson equation (1) where Φ is the potential (V) in the presence of space charge, ρ+ and ρ− are the ion densities of positive and negative polarities (C/m3), respectively and ɛ0 is the permittivity of free space (F/m). The conservation laws of the current densities are [10] (2a) (2b) where j +, j − and j are the positive, negative and total current densities, respectively, R is the recombination coefficient (m3/s) and e is the electron charge (C). The terms G± denote the rates at which the ions are recombined [(C/m3)/s]. When the ion flow reaches a steady state, the term ∂ρ/∂t is equal to zero. The current density is defined as (3) where k+ and k− are positive and negative ion mobilities [m2/(Vs)], respectively and W is the wind velocity (m/s). The ion current is defined as the electric current. 2.2 Boundary conditions The calculation domain for the bipolar model is shown in Fig. 1a. A ±250 kV bipolar DC line configuration, which is same as the one in [17], is selected for analysing the ion flow field. The conductor's diameter is 2 cm, the pole spacing S is 10 m and the conductors' height H is 10 m above the ground. The corona-onset potential gradient Ec is 27.3 kV/cm (the roughness coefficient m is 0.7, which is also same as the value in [17]). Fig. 1Open in figure viewerPowerPoint Local-MOCs a Configuration of a ±250 kV DC line and the mesh b Tracing approach of the characteristic line in one element c Configuration of a ±900 kV DC line and the observed paths For the electric field, the boundary conditions on the conductor and on the ground are the corresponding potentials. The potential on the artificial boundary is set as the space-charge-free one. This can be done because there are few ions in the region far away from the conductors and the electric field due to the space charges can be neglected. For the ion transport equation, the inflow part Γin and the outflow part Γout of the boundary Γ are defined as (4a) (4b) where V (x, y) is the velocity vector of the ion at position (x, y) and n (x, y) is the unit outward normal vector to Γ. On the inflow boundary, the Dirichlet boundary condition should be set. If the inflow part is located on the conductor surface, the value ρc is imposed; otherwise, the BC is set as zero because there is no new space charge generated outside the bounding box. For the ion flow, which is the hyperbolic case, it should be noted that the boundary conditions on the outflow portion cannot be specified [22]. In the case with wind, the inflow and outflow parts on the artificial boundary might vary with different wind speeds. For instance, without wind, the right-hand side of the artificial boundary in Fig. 1a is the outflow part for positive ions. However, if the wind blows from right to left and the speed is high enough, the magnitude of W might be larger than that of k+ ·E and thus this side will turn into the inflow part. This case is unlikely to happen on the conductor surface because the k+ ·E is about two orders of magnitude greater than the nature wind speed there. On the conductor surface, ρc should be determined carefully. The measurement of corona current may provide an accurate way but it cannot be used for predicting and designing. A widely used and well established approach is based on the Kaptzov's assumption [8, 9, 11–21] which means that the electric field strength on the conductors' surface will remain at the onset gradient Ec after the corona discharge occurs. It reflects the physical fact that the generated ions will weaken the electric field in the ionisation layer and suppress the corona activity. An initial guess for the charge density ρc is chosen by [19, 20] (5) where Eg is the ground-level electric field strength under the conductor, Uc is the onset voltage of the conductor, Ucon is the applied voltage and Hcon is the height of the conductor. Then an iterative process is applied to modify ρc using [11] (6) where Emax(n −1) is the maximal electric field intensity on conductor surface at (n − 1)th iterative step and μ is the relaxation coefficient. 3 Method of local characteristics The characteristics play an important role in the solution of convection problems. It has been derived in different ways [9, 16, 21]. The physical meaning, linking the space and time through the characteristic line, however, has not been interpreted clearly. In this section, the basis of the characteristic line is present. Afterwards, the local-MOC is proposed, which combines the methodology of the MOC and the upstream FEM. 3.1 Characteristics of transport equation Substituting (3) into (2a), we get (7) where V + = −k+ ·gradΦ+W and V − = k− ·gradΦ+W are the mobility velocities of positive and negative ions, respectively. Here we define the material (or total) derivative [22] as (8) which is just the left-hand side of the convection (7). The operator ∂/∂t +V ± ·∇ is replaced by a simple time derivative D/Dt along a certain characteristic line, which is defined as (9) where r + and r − are the characteristic lines (or streamlines) of positive and negative ions, respectively. The direction dr ±/dt = V ± are termed the characteristic directions. In the case without wind, positive and negative characteristic directions are same. However, they will differ in the presence of wind. If the Deutsch assumption is used, the characteristic lines are in accordance with space-charge-free flux lines (without wind). Thus, they can be traced beforehand and the FTM can be used [15, 16]. However, it may lead to unacceptable errors, especially under windy condition [9, 14]. By definition, the material derivative Dρ±/Dt is in fact the time rate of change felt by an observer moving with the material particles (ions) along the characteristic line. It indicates that the time rate of change consists of two parts according to (7). One is the term ∂ρ±/∂t, local derivative, which is the variation of the ion density at a fixed point. The second one is V ± ·∇ρ±, which is termed convective derivative. It reflects spatial inhomogeneity of the density at two points along one streamline. It is worth noting that even for the stationary state (∂ρ±/∂t = 0), the time rate Dρ±/Dt still exists in the characteristic form (8). This indicates that the spatial variation of the ion density is described using another pattern, i.e. the time rate on one characteristic line. Thus, the method based on the characteristic line substantially reflects inherent connections between time and space for flow problems. The material derivative is equal to the source term s, right-hand side of (7), which also includes two parts. The first one is caused by the divergence of the velocity vector k± E. The other one is the recombination rate for bipolar models. It can be seen that if the source term is zero, the ion density along one characteristic line remains constant (Dρ±/Dt = 0). This case, however, will not exist for the ion flow problem of HVDC transmission line, even for the unipolar model. Substituting (1) into (7), we get (10) The solution of the above ordinary differential equation is (11) where ρ±UpS and ρ±DownS are, respectively, the ion densities at the upstream and downstream points on the characteristic line and t is the ion drift time from upstream point to the downstream one. According to (11), downstream charge densities are completely determined by the upstream densities and the parameters along the characteristic line. This coincides with the direction of the information propagation in the convective transport problem. 3.2 Method of local characteristics The proposed method for updating the ion density and the FEM for calculating the electric field shares the same mesh, as shown in Fig. 1a. The second-order elements are used for the FEM and the computed electric field is linearly varying in each triangle. Thus, a more accurate characteristic line can be traced in each element compared with the straight line. Take calculation of positive ions for example. Triangle ABC is the upstream element of the positive charge at point A. To compute the ion density at point A, the local characteristic line is traced in the upstream direction, according to the distribution of velocity vector V + in this element. This drift path intersects the side BC at point D′ in the second-order element. Thus, in this element the upstream point of point A is point D′, the position of which is more accurate than that of point D. Using (11), the positive ion density at point A is (12a) where Δt is the drift time from point D′ to point A and (12b) (12c) (12d) (12e) where ρA−, ρB−, ρC− are the negative ion densities computed in the last iterative step. The drift time Δt is (13) which can be computed by numerical integration. It can be seen that the accuracy of the streamline is determined by two factors, i.e. the position of point D and the drift time Δt. When the wind blows, the characteristic line or even the upstream element will change. The calculation of negative ion density is similar. For the unipolar case, ρA+ = 1/ (1/ρD′+ + k+/ɛ0 ·Δt). The technique for tracing characteristic line in one element is shown in Fig. 1b. The line is traced from A to D′ in the reverse direction of the drift velocity. The total number of the tracing points is N. For the n th point, the position r n is computed by (14) where V +(n −1) is the velocity vector calculated at (n − 1)th point and ls is the step size, which is determined by the size of the element. It can be seen from (12) that the ion density ρA is calculated by the values of upstream points B and C, i.e. ρB and ρC. The calculation at points B and C has to be earlier in order than point A. Thus, the updating order of mesh points is similar to that of the upstream FEM. From the conductors to outwards points, the ion densities on the nodes are solved using (12a) one by one and the whole domain can be updated. The computational algorithm is described by the following steps: (1) For a certain HVDC overhead line configuration, the calculation domain is specified and the triangular mesh is generated. The updating order of node numbers is shored. (2) The space-charge-free electric field is computed and the initial charge densities are set on the conductors' surface using (5). (3) For a certain point, the characteristic line is traced in the upstream element and charge densities are calculated with local-MOC. All the points are updated. (4) The electric field in the presence of space charge is solved by FEM with the same mesh. (5) Steps (3) and (4) are repeated until the electric field and ion densities are stable between two iterative steps. (6) The ion densities are modified according to the potential gradient on the conductors' surface using (6). Steps (3)–(6) are carried out until it meets the set criterion. 3.3 Advantages of the proposed method The procedure of the proposed method is similar to that of the upstream FEM except the calculation of the ion densities, which is derived from the methodology of the MOC. Thus, it has the main features of the upstream FEM and the MOC. The Deutsch assumption, which causes the main error of the flux tracing method, is avoided. Thus, the fully coupled problem is solved considering the strong interaction between electric field and ion flow. Tracing global characteristic lines in the whole space in every iterative step for iterative method of characteristics (IMOC) is not needed and the cases without and with wind do not have to be treated separately [9]. Compared with the upstream FEM, the local-MOC has a more distinct physical picture, which contains the feature of the information propagation of the convective transport. Tracing more accurate streamlines in the second-order elements can provide a simple way for accuracy improvement. The accuracy is compared by testing the performances of these two methods on the conservation law of ion current in the next section. The local-MOC remains stable even in the presence of wind with high speed. Moreover, the calculation has high efficiency, which can be comparable with that of the upstream FEM [10, 11, 14]. 4 Validation case 4.1 Performance on the conservation law of ion current A typical bipolar ±900 kV DC test line is analysed, as shown in Fig. 1c. The configuration is same as the one in [17] and will also be used in Section 4.3. The pole of the line consists of 6 × 4.06 cm bundle conductors with a bundle spacing of 45.72 cm. The minimum conductor height H is 12.2 m and the pole spacing S is 15.2 m. The corona-onset gradient E0 is 18.2 kV/cm (the roughness coefficient m is 0.5), which is also used in [17]. The positive direction of wind is from left to right. Here, the conservation law of the corona current is tested on five typical paths in the left calculation domain in Fig. 1c. Two of them are the boundary Γc on the bundle conductor surface of the negative pole and the boundary of the left domain Γb, respectively. The other three observed paths are the artificial circles ΓO(I) with radius R1 of 1, 3 and 6 m. The performance on the conservation law on these five paths can represent the performance in the whole domain to some extent. The corona currents I on these paths are calculated by (15) Theoretically, the current computed along all the paths are exactly equal. The error δI due to the numerical computation is defined as the maximum error among the corona currents on the observed paths. Comparison between the upstream FEM and the proposed method with different applied voltages and with different wind speeds are, respectively, shown in Figs. 2a and b. The second-order elements for calculating the potential are used for both methods and thus the errors of the electric field are the same. The results without wind in Fig. 2a show that the proposed method has a better consequence of current conservation with various voltages (corresponding to different current densities). In the presence of high-speed wind, the errors of local-MOC slightly increase but still within 2%. For the upstream FEM, the errors grow remarkably, especially with negative wind. Fig. 2Open in figure viewerPowerPoint Comparison of the performances on the conservation law a Errors with different applied voltages b Errors with different wind speeds c Corona current on path ΓO(II) with different wind speeds Another observed path is set between the poles. The height of the circle centre is 12 m and the radius R2 is 0.2 m. The corona current on this path is zero in theory. The computed currents with two methods are compared in Fig. 2c. Only the cases with positive winds are shown because the results are same for negative winds. It shows that the current calculated by upstream FEM is about one order of magnitude greater than that of local-MOC and increases with wind speed. It should be mentioned that all the above errors also include errors of the electric field, the ion densities interpolation on the path and numerical integration. Thus, local-MOC shows a good performance on the conservation law and has a high accuracy. 4.2 Reduced-scale DC line in the presence of wind The computed ion current onto the ground are compared with the measured value from Hara [2]. The height of the line is 2 m and the radius of the wire is 2.5 mm. The positive corona-onset voltage measured in [2] is 83 kV. Fig. 3a shows the calculations with different methods when the applied voltage is +200 kV. Without wind, all the methods well predict the current density, even though slight deviations exist. The results of the local-MOC and upstream FEM show reasonable agreement with the measurement in the presence of wind. The current density computed by upstream FEM, however, is higher than that of the local-MOC. The main reason is that upstream FEM does not perform well on the conversation law and thus has a larger prediction of ion current on the ground level. The results of the FTM-wind do not agree well with the experimental results with wind in reduced-scale model, which is also mentioned in [17]. Fig. 3Open in figure viewerPowerPoint Ion current and electric field of the reduced-scale DC line a Errors with different applied voltages b Errors with different wind speeds c Corona current on path ΓO(II) with different wind speeds The ion current density and electric field on the ground level with different applied voltages are shown in Figs. 3b and c, respectively. The measured value with applied voltage of +120 kV is not given in the still air [2]. The agreement between the measurements and calculations of the proposed method is quite good, except for peak value of the current density under windy condition with applied voltage of 300 kV. The profiles of the current density and electric field are shifted downwind and change in shape. Furthermore, the effect of wind is much greater for the former. The peak value of the current density markedly increases with wind for a given voltage. 4.3 Full-scale DC test line in the presence of wind Long-term measurement was made on IREQ ±900 kV bipolar DC line and well published in [17]. Calculations of ion current density on the ground level are made using the local-MOC. Fig. 4a shows the comparison between the measurement with wind and the computed values using different methods. The range of wind speed is from −7.5 to −4.5 m/s. The results of the proposed method show well agreement with the measurement value considering the variable wind speeds and possible change of the onset electric field during the experiment. Compared with the FTM-wind, the local-MOC predicts a more reasonable profile of the current density. The downwind results of upstream FEM are about 100 nA/m2 larger than that of the proposed method. It is because the error of the upstream FEM on conservation law becomes larger with wind, as discussed in the preceding section. Fig. 4Open in figure viewerPowerPoint Ion current density of a ±900 kV bipolar line a Comparison of ion current densities between measurement and calculations on the ground level b Comparison of ion current densities with different artificial boundary position PL c Net space-charge distribution around the overhead line, w = −7.5 m/s In this paper, the distance between artificial boundary and the bundle conductors is about two times as large as the pole spacing. Fig. 4b shows the effect of the left artificial boundary position PL on the results. It indicates that the position of the artificial boundary has negligible effect on the calculations if the boundary is far enough from the conductors. The net space-charge distribution is shown in Fig. 4c. The structure of the ion density drifting from the bundle conductors is like a flower petal, showing the severe change near the conductor. The negative wind strongly pushes the space-charge downwind and the distribution of ions is not symmetric. The results with wind of other speeds are shown in Fig. 5. It indicates that the experimental and computed values agree well with each other. Furthermore, the trend in the variation of the current density profiles observed in the experiment is well reflected in the calculations. Fig. 5Open in figure viewerPowerPoint Current density on the ground level for a ±900 kV bipolar line 5 Effect of wind on electric field and ion flow of HVDC transmission lines The line configuration in Fig. 1a is used for analysing the influence of wind on the electric field and ion flow both in the space and on the ground. The effect on the corona current is also explained according to the change of charge distribution. 5.1 Electric potential and ion density spatial distribution under windy condition The contours of the ion density and electric field in the space with wind speed of −2 m/s are shown in Figs. 6a and b, respectively. It can be seen that the distribution of the space charge are no longer symmetrical and the contour lines are elongated downwind. On the right-hand side of the positive pole, a domain exists where the ion density and the direction of the current change severely. It also indicates that the proposed method still performs well even though the charge distribution is strongly distorted by the wind. The shift of the potential is much slighter because that it is influenced indirectly by the wind. The contour of the potential of 10 kV is closer to the ground downwind, implying that the electric field on the ground increases under the leeward pole. Fig. 6Open in figure viewerPowerPoint Contour of ion density and electric field with wind of −2 m/s a Ion density b Electric field 5.2 Influence of wind on the corona current For the unipolar case, the increase of the corona current has been measured in the experiment [2, 3]. It becomes more complicated for the bipolar lines and no in-depth analysis has been made. Fig. 7a shows the influence of wind on the corona current of the left negative pole. When the wind is blown from right to left, the corona current increases approximately linearly with wind speed. It can be explained by the change of the recombination rate in Figs. 7b and c. Without wind, the recombination occurs in the vicinity of both poles. Under the windy condition, less negative ions move to the right conductor, whereas more positive charges are pushed downwind. The recombination is mainly concentrated around the leeward line and its rate rises. More ions are generated to complement the space charges and thus the current increases. Fig. 7Open in figure viewerPowerPoint Influence of wind on the corona current a Corona current of the negative pole of the ±250 kV bipolar line with different wind speeds b Recombination rate of ion densities when w = 0 m/s c Recombination rate of ion densities when w = −2 m/s With positive wind, the current decreases with low-speed wind. Then, the current will reach a minimum point and increase at the higher wind velocity. In the presence of wind from left to right, less positive current j + from the right pole reaches the upwind pole and the recombination is reduced. Therefore, the ion current of the left wire is reduced at first with low wind speed. After the positive and negative ions are completely 'decoupled' by the wind at the left pole, the total (only negative) current will grow such as the unipolar case [2, 3]. With higher applied voltage or less pole spacing, the decoupling of the positive and negative ions is more difficult and thus the minimum point will be at a higher wind speed, as shown in Fig. 7a. For the operating HVDC lines, the current of upwind pole will decrease in a wide range of wind speed. This tendency is in accordance with the observation of the measurement in [4]. 5.3 Shift of electric field and ion current density on the ground level caused by wind The ground-level ion current density and electric field with wind are computed in Figs. 8a and b. The curves are rather symmetric with zero wind. Under windy condition, the corona current is strongly increased downwind and reduced upwind. The peak magnitude of positive current density rises to about 60 nA/m2 with 4 m/s wind and is approximately three times as large as that without wind. With negative wind, the current under positive pole decreases or even disappears because the positive ions are pushed to the other pole and fewer charges will reach the ground. This tendency coincides with measured values in [5, 17]. The effect of wind on the electric field is much less, which is also claimed in [2, 9–11, 17]. Its peak value is increased by about 39% with wind of 4 m/s. With negative wind, the enhancement of electric field under the positive pole is weaker compared with the case with zero wind, due to the shift of ions distribution. Fig. 8Open in figure viewerPowerPoint Influence of wind on the current density and electric field on the ground level a Ion current density b Electric field 6 Conclusions The proposed local-MOC is used to evaluate the influence of wind on the electric field and ion current density on the ground level and in the space of HVDC transmission lines. The numerical approach can reflect the characteristics of the information propagation of convective transport. The local characteristic line in each element is used to update the ion densities and the accuracy can be improved by tracing more precise drift line in the higher-order element. This method is efficient and stable with or without wind. The accuracy of local-MOC is proved to be high by testing the performance on the conservation law. Moreover, the calculations of the present method show good agreement with the measurements of both unipolar and bipolar models. Analysis has been made over a range of wind speeds. Calculations show that wind strongly shifts the distribution of space charges, leading to the change of the corona current and the distortion of ground-level profiles. The impact of the wind on the current density is larger than the electric field in the space and on the ground. The profiles of electric field and ion current are both enhanced downwind and reduced upwind for the bipolar case. 7 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant 51577103 and in part by the Major Science and Technology Programme of SGCC under grant SGTJ0000KXJS1 400081. 8 References 1Khalifa, M.M., Morris, R.M.: 'A laboratory study of the effects of wind in DC corona', IEEE Trans. Power Appl. Syst., 1967, PAS-86, (3), pp. 290– 298 (doi: 10.1109/TPAS.1967.291955) 2Hara, M., Hayashi, N., Shiotsuki, K. et al.: 'Influence of wind and conductor potential on distributions of electric field and ion current density at ground level in DC high voltage line to plane geometry', IEEE Trans. Power Appl. Syst., 1982, PAS-101, (4), pp. 803– 814 (doi: 10.1109/TPAS.1982.317145) 3Abdel-Salam, M., Farghally, M., Abdel-Sattar, S.: 'Monopolar corona on bundle conductors', IEEE Trans. Power Appar. Syst., 1982, 101, pp. 4079– 4089 (doi: 10.1109/TPAS.1982.317086) 4Johnson, G.B.: 'Electric fields and ion currents of a ±400 kV HVDC test line', IEEE Trans. Power Appl. Syst., 1983, PAS-102, (4), pp. 2559– 2568 (doi: 10.1109/TPAS.1983.317775) 5Zhen, Y., Cui, X., Lu, T. et al.: 'A laboratory study on the ion-flow field model of the DC wires in stable wind', IEEE Trans. Power Deliv., 2015, 30, (5), pp. 2346– 2352 (doi: 10.1109/TPWRD.2015.2444372) 6Maruvada, P.S., Dallaire, R.D., Heroux, P. et al.: 'Long-term statistical study of the corona electric field and ion-current performance of a ±900 kV bipolar HVDC transmission line configuration', IEEE Trans. Power Appl. Syst., 1984, PAS-103, (1), pp. 76– 83 (doi: 10.1109/TPAS.1984.318580) 7Chartier, V.L., Stearns, R.D., Burns, A.L.: 'Electrical environment of the uprated pacific NW/SW HVDC intertie', IEEE Trans. Power Deliv., 1989, 4, (2), pp. 1305– 1317 (doi: 10.1109/61.25618) 8Li, X.: ' Numerical analysis of ionized fields associated with HVDC transmission lines including effect of wind'. PhD dissertation, Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, 1998 9Guillod, T., Pfeiffer, M., Franck, C.M.: 'Improved coupled ion flow field calculation method for ac/dc hybrid overhead power lines', IEEE Trans. Power Deliv., 2014, 29, (6), pp. 2493– 2501 (doi: 10.1109/TPWRD.2014.2322052) 10Takuma, T., Ikeda, T., Kawamoto, T.: 'Calculation of ion flow fields of HVDC transmission lines by the finite element method', IEEE Trans. Power Appl. Syst., 1981, PAS-100, (12), pp. 4802– 4810 (doi: 10.1109/TPAS.1981.316432) 11Lu, T., Feng, H., Cui, X. et al.: 'Analysis of the ionized field under HVDC transmission lines in the presence of wind based on upstream finite element method', IEEE Trans. Magn., 2010, 46, (8), pp. 2939– 2942 (doi: 10.1109/TMAG.2010.2044149) 12Li, X., Ciric, I.R., Raghuveer, M.R.: 'Investigation of ionized field due to bundled unipolar DC transmission lines in the presence of wind', IEEE Trans. Power Deliv., 1999, 14, (1), pp. 211– 217 (doi: 10.1109/61.736719) 13Qin, B.L., Sheng, J.N., Yan, Z. et al.: 'Accurate calculation of ion flow field under HVDC bipolar transmission lines', IEEE Trans. Power Deliv., 1988, 3, (1), pp. 368– 376 (doi: 10.1109/61.4266) 14Qiao, J., Zou, J., Yuan, J. et al.: 'Calculation of ion-flow field of HVDC transmission lines in the presence of wind using finite element-finite difference combined method with domain decomposition', IEEE Trans. Magn., 2016, 52, (3), pp. 1– 4 (doi: 10.1109/TMAG.2015.2481725) 15Maruvada, P.S., Janischewskyj, W.: 'Analysis of corona losses on DC transmission lines: I – unipolar lines', IEEE Trans. Power Appl. Syst., 1969, PAS-88, (5), pp. 718– 731 (doi: 10.1109/TPAS.1969.292362) 16Maruvada, P.S.: 'Electric field and ion current environment of HVDC transmission lines: comparison of calculations and measurements', IEEE Trans. Power Deliv., 2012, 27, (1), pp. 401– 410 (doi: 10.1109/TPWRD.2011.2172003) 17Maruvada, P.S.: 'Influence of wind on the electric field and ion current environment of HVDC transmission lines', IEEE Trans. Power Deliv., 2014, 29, (6), pp. 2561– 2569 (doi: 10.1109/TPWRD.2014.2302373) 18Qiao, J., Zou, J., Li, B.: 'Calculation of the ionised field and the corona losses of high-voltage direct current transmission lines using a finite-difference-based flux tracing method', IET Gener. Transm. Distrib., 2015, 9, (4), pp. 348– 357 (doi: 10.1049/iet-gtd.2014.0333) 19Yin, H., He, J., Zhang, B. et al.: 'Finite volume-based approach for the hybrid ion-flow field of UHVAC and UHVDC transmission lines in parallel', IEEE Trans. Power Deliv., 2011, 26, (4), pp. 2809– 2820 (doi: 10.1109/TPWRD.2011.2165344) 20Abdel-Salam, M., Al-Hamouz, Z.: 'A finite-element analysis of bipolar ionized field', IEEE Trans. Ind. Appl., 1995, 31, (3), pp. 477– 483 (doi: 10.1109/28.382106) 21Zhang, B., Mo, J., Yin, H. et al.: 'Calculation of ion flow field around HVdc bipolar transmission lines by method of characteristics', IEEE Trans. Magn., 2015, 51, (3), pp. 1– 4 22Donea, J., Huerta, A.: ' Finite element methods for flow problems' ( Wiley, Chichester, 2005) Citing Literature Volume11, Issue4March 2017Pages 1055-1062 FiguresReferencesRelatedInformation