This chapter contain the physical model and numerical procedure of calculating the dynamics of space charge and electric field distribution in cable insulation material under DC voltage. Electric field distribution and formation of space charge
The dependency of electric field distribution on time and temperature is obvious in HVDC system. This is due to the resistivity/conductivity dependency on temperature and electric field [1, 44]. It is also clear that the formation of space charge within the insulation material is depended on both temperature and external applied field [15]. Therefore, the dependent conductivity has a great role on space charge and then on field distribution in DC system. Under steady state DC condition the electric field distribution is determined by the electrical resistance and thus by the resistivity/conductivity of insulation material [1]. When a DC cable is energized (voltage on) at the beginning with the zero load and there is no temperature gradient across the cable, at that time the distribution field would be ‘quasi-capacitive or quasi- Laplacian’ [45]. After the cable is loaded, the Joule heating take place at the conductor and then the electric field starts to change from quasi-capacitive to purely resistive and hence become stable after a finite of time, stages of distribution electric field in DC is extensively mentioned in [1, 9]. All above principles can be described using the following equations [1, 32, 45]: Electric field distribution equation E=-∇V (3.1) Ohms law or transport equation J=σE (3.2) Gauss law or Poisson equation ∇.E=ρ/(ε_0 ε_r ) (3.3) Current density continuity equation ∇.J=(-∂ρ)/∂t (3.4) Where, E is electric field in (kV/mm), J is current density in (A/m2), σ is electric conductivity in (S/m), ε_r is relative permittivity of the dielectric and ρ is space charge density in (C/m3). According to equation 3.3, when the voltage is switched on there is no space charge inside the insulation (ρ=0), hence it becomes ∇.(ε_0 ε_r E)=0 (3.5) By this, the initial field distribution is depending on the permittivity variation inside the material [1, 46]. …show more content…
Moreover, after applying the voltage for a long time, if there is no space charge built up, the current density gain the steady state value, and equation 3.2 and 3.4 become:
∇.J=∇.(σE)=0 (3.6)
Based on equation 3.6, the field distribution is depended on conductivity variation inside the material [1, 45, 46]. Furthermore, according to equations 3.5 and 3.6, space charge can be accumulated in insulation material in steady state condition due to the variation of permittivity to conductivity. As a result, by combining equation 3.2, 3.3 and 3.6, the equation of determining space charge in steady state condition can be written
∇.(ε_0 ε_r E)=∇.(ε_0 ε_r J/σ)=J.∇((ε_0 ε_r)/σ)+ (ε_0 ε_r)/σ ∇.J=J.∇ ((ε_0 ε_r)/σ)= ρ (3.7) or ρ=σE.∇( (ε_0 ε_r)/σ) (3.8) it means, space charge can accumulate, when the ratio of ε/σ varies with position and the steady current density is flowing in that position [8, 47]. As mentioned above, a purely resistive (DC) distribution is obtained after an infinite of time [48]. Therefore, finding a suitable time to gain the steady field distribution is vital after changing the condition of DC cable. For instance, loading the cable makes the temperature variation inside the cable [32]. It is clear that the conductivity which is depending on the local temperature and electric field has a great role for estimating the time to reach the steady state operation. The conductivity gradient makes the divergence of field distribution and …show more content…
Based on this, when the time derivation of charge density goes to zero, field distribution and space charge obtained the steady state condition, hence equation 3.9 reduces to 3.8. Furthermore, when the conductivity increases, the dielectric time constant decrease, thus the field distribution can reach the steady state condition rapidly.
The three equations 3.2, 3.3 and 3.4 demonstrate the electrical behaviour of a weakly conductive of insulation material, while equation 3.9 which is the result of combining these three equations describe the accumulated space charge when the time derivation of current density is not zero and the ratio between permittivity and conductivity ( (ε_"0" ε_"r" )/σ) is not constant across the insulation. The permittivity of insulation material is almost independent of temperature and electric field, hence it can be considered as a constant [29]. However, the electrical conductivity of the dielectrics is strongly dependence on both temperature and electric field
The dependency of electric field distribution on time and temperature is obvious in HVDC system. This is due to the resistivity/conductivity dependency on temperature and electric field [1, 44]. It is also clear that the formation of space charge within the insulation material is depended on both temperature and external applied field [15]. Therefore, the dependent conductivity has a great role on space charge and then on field distribution in DC system. Under steady state DC condition the electric field distribution is determined by the electrical resistance and thus by the resistivity/conductivity of insulation material [1]. When a DC cable is energized (voltage on) at the beginning with the zero load and there is no temperature gradient across the cable, at that time the distribution field would be ‘quasi-capacitive or quasi- Laplacian’ [45]. After the cable is loaded, the Joule heating take place at the conductor and then the electric field starts to change from quasi-capacitive to purely resistive and hence become stable after a finite of time, stages of distribution electric field in DC is extensively mentioned in [1, 9]. All above principles can be described using the following equations [1, 32, 45]: Electric field distribution equation E=-∇V (3.1) Ohms law or transport equation J=σE (3.2) Gauss law or Poisson equation ∇.E=ρ/(ε_0 ε_r ) (3.3) Current density continuity equation ∇.J=(-∂ρ)/∂t (3.4) Where, E is electric field in (kV/mm), J is current density in (A/m2), σ is electric conductivity in (S/m), ε_r is relative permittivity of the dielectric and ρ is space charge density in (C/m3). According to equation 3.3, when the voltage is switched on there is no space charge inside the insulation (ρ=0), hence it becomes ∇.(ε_0 ε_r E)=0 (3.5) By this, the initial field distribution is depending on the permittivity variation inside the material [1, 46]. …show more content…
Moreover, after applying the voltage for a long time, if there is no space charge built up, the current density gain the steady state value, and equation 3.2 and 3.4 become:
∇.J=∇.(σE)=0 (3.6)
Based on equation 3.6, the field distribution is depended on conductivity variation inside the material [1, 45, 46]. Furthermore, according to equations 3.5 and 3.6, space charge can be accumulated in insulation material in steady state condition due to the variation of permittivity to conductivity. As a result, by combining equation 3.2, 3.3 and 3.6, the equation of determining space charge in steady state condition can be written
∇.(ε_0 ε_r E)=∇.(ε_0 ε_r J/σ)=J.∇((ε_0 ε_r)/σ)+ (ε_0 ε_r)/σ ∇.J=J.∇ ((ε_0 ε_r)/σ)= ρ (3.7) or ρ=σE.∇( (ε_0 ε_r)/σ) (3.8) it means, space charge can accumulate, when the ratio of ε/σ varies with position and the steady current density is flowing in that position [8, 47]. As mentioned above, a purely resistive (DC) distribution is obtained after an infinite of time [48]. Therefore, finding a suitable time to gain the steady field distribution is vital after changing the condition of DC cable. For instance, loading the cable makes the temperature variation inside the cable [32]. It is clear that the conductivity which is depending on the local temperature and electric field has a great role for estimating the time to reach the steady state operation. The conductivity gradient makes the divergence of field distribution and …show more content…
Based on this, when the time derivation of charge density goes to zero, field distribution and space charge obtained the steady state condition, hence equation 3.9 reduces to 3.8. Furthermore, when the conductivity increases, the dielectric time constant decrease, thus the field distribution can reach the steady state condition rapidly.
The three equations 3.2, 3.3 and 3.4 demonstrate the electrical behaviour of a weakly conductive of insulation material, while equation 3.9 which is the result of combining these three equations describe the accumulated space charge when the time derivation of current density is not zero and the ratio between permittivity and conductivity ( (ε_"0" ε_"r" )/σ) is not constant across the insulation. The permittivity of insulation material is almost independent of temperature and electric field, hence it can be considered as a constant [29]. However, the electrical conductivity of the dielectrics is strongly dependence on both temperature and electric field