## 2 Surface Charge Accumulation  
In this section, the mechanisms of surface charge accumulation are briefly recalled. The electro-quasistatic (EQS) formulation is suited for modeling the capacitive-to-resistive transition of the electric field in HVDC systems [13; 14].  
It is known from the literature that the charge accumulating along the interface between dielectric materials with different properties, i.e., the solid insulators and the dielectric gas, is responsible for the degradation of insulation performances of HVDC-GIS components and represents one of their major fault causes [15; 16]. The surface charge not only distorts the electric field distribution within the apparatus but also may trigger the surface flashover of the insulator [17; 18]. The charge deposited on the solid insulator surface is subdivided into hetero-charge and homo-charge [19], where the former has the opposite polarity of the applied DC voltage, while the latter has the same polarity of the closest electrode. Experimental results show that hetero-charge is mainly due to conduction within the dielectric gas. Homo-charge, instead, is mainly driven by the nonlinear conductivity of the solid insulator and interface layer.  
The surface charge density \(\varrho_{S}\) is due to the jump of the normal components of the electric displacement fields in the solid insulator, labeled with suffix \(I\), and the dielectric gas, with suffix \(G\):  
\[\varrho_{S}=\mathbf{n}\cdot(\mathbf{D}_{I}-\mathbf{D}_{G}), \tag{1}\]  
where \(\mathbf{n}\) is the unit normal vector pointing in the outward direction of the solid insulator domain and \(\mathbf{D}=\epsilon\mathbf{E}\), where \(\epsilon=\varepsilon_{0}\varepsilon_{r}\) is the electric permittivity, with \(\varepsilon_{0}\) being the vacuum permittivity and \(\varepsilon_{r}\) being the relative permittivity. The dynamics of surface charge accumulation are recognized to be driven by three conduction mechanisms [20]:  
* Electric conduction within the gas.
* Electric conduction in solid insulators.
* Electric conduction of the interface layer.  
Considering a portion of the interface layer of thickness \(\delta\) as illustrated in Figure 1, the dynamic of surface charge accumulation is governed by the following ordinary differential equation (ODE):  
\[\frac{d\varrho_{S}}{dt}=\mathbf{n}\cdot(\mathbf{J}_{I}-\mathbf{J}_{G})-\
abla_{S }\cdot(\sigma_{S}E_{t}), \tag{2}\]  
where \(\mathbf{J}_{I}\) is the current density in the solid domain, \(\mathbf{J}_{G}\) is the current density in the gas, \(\sigma_{S}\) [S] is the interface layer electric conductivity, \(E_{t}\) the tangential electric field strength, and \(\
abla_{S}\) represents the surface divergence. It is worth mentioning that, generally, \(\sigma_{S}\) is a nonlinear function of electric field itself in the form [21]:  
\[\sigma_{S}(E_{t})=\sigma_{S0}\exp(\alpha_{S}E_{t}), \tag{3}\]  
where \(1.2<\alpha_{S}<1.4\) [mm/kV] is the usual range of the nonlinearity exponent.  
The charge accumulation can be reduced by an accurate design of the solid insulator shape [22] or by using innovative insulating materials, for example, field grading materials [23]. From the point of view of the authors, this last point requires intense industrial participation to reduce construction costs and make this technology available for future HVDC-GISs.