One of the recent instances of practical implementation of a CCS has clearly demonstrated the system’s feasibility and effectiveness in supplying electrical power to dedicated loads [13] . Through rigorous testing and operation, practical studies have shown that CCS can reliably meet the energy needs of specific consumers or applications, thereby, highlighting its potential as a viable solution for electrification in various contexts such as in sparsely populated areas or low dedicated loads [14] . A CCS can be integrated into an electrical transmission network using either a nominal-π or nominal-T configuration. The nominal-T configuration is typically employed for shorter transmission lines, extending up to 80 km, while the nominal-π configuration is preferred for medium-length transmission lines exceeding 80 km [15] . In essence, current studies do not provide any limitations on the type of transmission line where a CCS can be adopted.

Figure 1 below illustrates the simplified depiction of a typical CCS. This approach is similar to that of a Capacitive Voltage Transformer (CVT), which is a transformer commonly used in power systems to reduce extra-high voltage signals to low voltage signals for tasks such as metering or operating protective relays [16] [17] .

The simplified CCS, as shown in Figure 1 above, employs a capacitor-divider configuration where capacitors (C₁ and C₂) are linked across the incoming voltage (V_in) to generate the desired tap-voltage (V_T), which is measured from the tapping node located between the two capacitors. The voltage output (V_out) is calculated by subtracting the voltage drop across the inductor (L) from V_T. It is important to note that C₁ and C₂ refer to capacitor banks rather than individual capacitors where C₁ represents Capacitor Bank 1, while C₂ represents Capacitor Bank 2.

The tap-voltage (V_T) is calculated as follows:

$V_T=V_{in}\times \frac{C_1}{C_1+C_2}$ (1)

The output voltage is calculated as follows:

$V_{out}=V_T-V_{L_1}$ (2)

The output voltage can also be represented as follows:

$V_{out}=\frac{C_1}{C_1+C_2}\times V_L$ (3)

The equations provided, as (1), (2), and (3), serve as the basis for determining the elements of a CCS. The objective is to ensure the stability of V_out as the CCS delivers power to a downstream transformer within the distribution network.

In Figure 2 below, the equivalent CCS connected to a standard transmission line is depicted.

The aim of this study is to develop and simulate an 80 kW loaded Capacitor Coupled Substation (CCS) connected to a standard transmission network. The selected transmission network is operated at 132 kVrms. To define the parameters of the CCS, predetermined values were used, and background calculations were conducted and the results were used for the model. The selected parameters are detailed in Table 1 .

The parameters in Table 1 are used to calculate the variables required in order to achieve the desired V_TAP, and thus, the CCS load voltage of 400 V rms. The voltage selected is based on readily available and commonly used transmission lines and distribution transmission voltages in South Africa.

Using (1), (2) and (3), with known supply voltage and the desired tap voltage, the respective capacitors C₁ and C₂ are calculated.

The process of modeling the system involves the utilization of MATLAB/Simulink software, a powerful computational tool widely employed in engineering research. Furthermore, to ensure a comprehensive analysis, different voltage levels were used to determine the impact on the capacitors to keep tapped voltage within acceptable tolerances. This approach provided a deeper understanding of the system’s behaviour under varying conditions. The parameters used in the simulated model, crucial for accurately capturing the system behaviour, are outlined in Table 1 .

The model utilized for simulating the system is depicted in Figure 4 in the preceding section. This model was streamlined by employing internal MATLAB/Simulink parameters within the subsystem. The primary parameter values were extracted mainly from six model scopes, as outlined in Table 2 .

The simulation was conducted with all circuit breakers in the system remaining closed for the entire duration of the 1.5-second simulation run. The findings from this simulation are elaborated upon in Section 4.

Using the known parameters as per Table 1 , to calculate the values of the capacitor banks $C_1$ and $C_2$ with the given parameters, the following results were achieved.

Using:

From Equation (1), $C_1$ and $C_2$ are calculated as:

$\frac{C_1}{C_1+C_2}=\frac{V_{tap}}{V_s}$

With the known supply voltage and the desired $V_{tap}$, the ratio is:

$\frac{C_1}{C_1+C_2}=\frac{11\text{kV}}{132\text{kV}}=\frac{11}{132}=\frac{1}{12}$

Therefore, the relationship between $C_1$ and $C_2$ is:

$C_2=11C_1$

The individual capacitor banks capacitance is calculated using the system frequency of $f=50\text{Hz}$ and the relationship between the reactance $X_C$ and capacitance $C$ as:

$X_C=\frac{1}{2\pi fC}$

Ignoring the load impedance gives the ratio of the capacitors as 1:11 (from $C_2=11C_1$). Therefore:

$C_1=\frac{C_{total}}{12}$ and $C_2=\frac{11C_{total}}{12}$

The actual values of $C_1$ and $C_2$, with the CCS power of 80 kW at 0.8PF is calculated from:

$\text{Apparent}\text{Power}S=\frac{P}{PF}=\frac{80\text{kW}}{0.8}=100\text{kV}\cdot \text{A}$

Assuming nominal total reactive power is balanced:

$C_{total}\approx \frac{S}{2\pi f{V}_{tap}^2}\approx \frac{100\times {10}^3}{2\times 3.14146\times {\left(11\times {10}^3\right)}^2}\approx 2.63\mu \text{F}$

Therefore:

$C_1\approx \frac{2.63\mu F}{12}\approx 0.22\mu \text{F}$ and $C_2\approx \frac{11\times 2.63\mu F}{12}\approx 2.41\mu \text{F}$.

From the calculated value of $C_1$ and $C_2$, the simulation values used are presented in Table 3 .

The calculated values of $C_1$ and $C_2$ are then used to simulate the system and observe its behaviour. The results obtained from the simulation are shown in Figures 4-8 and Figure 9 .

Figure 5 presents the supply voltage parameters with 132 kV rms as the supply and the resulting current, active power and reactive power based on the given CCS and downstream load. The downstream load parameters are also presented in Figure 6 .

Figure 7 presents the CCS tap node with the $V_{tap}$ being approximately 15.6 kV peak ≈ 11 kV rms, with Figure 8 representing the resulting CCS transformer secondary voltage of approximately 565 V peak ≈ 400 V rms, while Figure 9 represents the CCS load parameters with the selected 80 kW load.

The study primarily focused on three key voltage levels: the supply voltage (V_in), the tap voltage (V_T), and the load voltage (V_L). These voltage levels play a critical role in the operation and performance of the system under investigation.

From the simulation results, the summarised results based on the main variable of focus, $V_{tap}$, are expressed in Figure 10 , Figure 11 and Figure 12 , where Figure 10 and Figure 11 present the supply, $V_{tap}$ and transformer secondary voltages, while Figure 12 presents the numerical representation of the peak values for the supple and $V_{tap}$, respectively.

The measured voltages, extrapolated from Figure 11 and Figure 12 are:

$V_s=V_{in}=1.866e^{+05}=1.866\times {10}^5=186.6\text{kV}$

$V_t=V_{trf,primary}=1.540e^{+05}=1.540\times {10}^4=15.4\text{kV}$

Since the MATLAB results waveform represents the peak value, the root mean square (RMS) is calculated as:

$V_{s,rms}=\frac{V_{peak}}{\sqrt{2}}=\frac{186.6\text{kV}}{\sqrt{2}}=131.9\text{kV}$

$V_{t,rms}=\frac{V_{peak}}{\sqrt{2}}=\frac{15.4\text{kV}}{\sqrt{2}}=10.89\text{kV}$

These results show that the selected C₁ and C₂ gives the desired $V_{tap}$.

In Figure 12 , V_line represents the incoming voltage from the voltage source supply, serving as the primary source of electrical energy. V_t, or tap voltage, denotes the voltage level obtained after tapping from the transmission line, which is essential for regulating and delivering electrical power to the subsequent components of the CCS system. Figure 12 is simplified as:

$V_{Line}=1.866\times 10e^5=186\text{kVpeak}=132\text{kVrms}$

$V_{tap}=1.540\times 10e^4=15.4\text{kVpeak}=11\text{kVrms}$

The calculated results can also be used as the basis for any CCS design. The $V_{tap}$ can be achieved by adjusting $C_1$ and $C_2$ from any HV supply. A basic MATLAB Code that can be used is presented in Table 4 .

The tapped voltage ( $V_{tap}$) can be influenced by the load [21] . $V_{tap}$ on a steady-state circuit is derived by Equation (1). However, when the load connected changes, it affects the impedance seen by the capacitors, thus altering the $V_{tap}$. The load introduces an additional impedance parallel with $C_2$ and can be represented by $Z_{C2}$. If $Z_L$ is the impedance of the load, then the effective impedance across $C_2$ and the load is given by:

$Z_{eff}={\left(\frac{1}{Z_{C2}}+\frac{1}{Z_L}\right)}^{-1}$

where: $Z_{C2}=\frac{1}{j\omega C_2}=\frac{1}{j2\pi f{C}_2}$

The $V_{tap}$ under varying load is thus derived from:

$V_{tap}=V_S\times \frac{Z_{eff}}{Z_{eff}+Z_{C1}}\to V_{tap}=V_S\times \frac{\frac{1}{j\omega C_2}\times Z_L}{\frac{1}{j\omega C_2}+Z_L+\frac{1}{j\omega C_1}}$

In order to maintain the desired value of $V_{tap}$:

Using the data from Table 4 , a state-space representation for maintaining the desired tapped voltage ( $V_{tap}$) on a CCS under dynamic loads can be modelled as follows, using state variables:

The state-space equations is therefore defined using Kirchhoff’s Current Law (KCL) at the $C_1$, with $I_s\left(t\right)$, being the current from the source, as:

$C_1\frac{\text{d}{V}_{C1}\left(t\right)}{\text{d}t}=I_s\left(t\right)-I_L\left(t\right)$

Using Kirchhoff’s Voltage Law (KVL) across L, gives:

$L\frac{\text{d}{I}_L\left(t\right)}{\text{d}t}=V_{C1}\left(t\right)-V_{C2}\left(t\right)$

While using KCL at $C_2$, where $I_{load}\left(t\right)$ is the current drawn by the dynamic load connected to the system, gives:

$C_2\frac{\text{d}{V}_{C2}\left(t\right)}{\text{d}t}=I_L\left(t\right)-I_{load}\left(t\right)$

The state-space equations are thus given as:

${\stackrel{\dot{}}{x}}_1\left(t\right)=\frac{1}{C_1}\left(I_s\left(t\right)-x_2\left(t\right)\right)$

${\stackrel{\dot{}}{x}}_2\left(t\right)=\frac{1}{L}\left(x_1\left(t\right)-x_3\left(t\right)\right)$

${\stackrel{\dot{}}{x}}_3\left(t\right)=\frac{1}{C_2}\left(x_2\left(t\right)-I_{load}\left(t\right)\right)$

The matrix representation is thus:

$\left(\begin{array}{c}{\stackrel{\dot{}}{x}}_1\left(t\right)\\ {\stackrel{\dot{}}{x}}_2\left(t\right)\\ {\stackrel{\dot{}}{x}}_3\left(t\right)\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{1}{C_1}& 0\\ \frac{1}{L}& 0& -\frac{1}{L}\\ 0& \frac{1}{C_2}& 0\end{array}\right)\left(\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right)+\left(\begin{array}{c}\frac{1}{C_1}\\ 0\\ 0\end{array}\right)I_s\left(t\right)+\left(\begin{array}{c}0\\ 0\\ -\frac{1}{C_2}\end{array}\right)I_{load}\left(t\right)$

The output equation, which relates the state variables to $V_{tap}=x_3\left(t\right)$, is:

$y\left(t\right)=\left(\begin{array}{ccc}0& 0& 1\end{array}\right)\left(\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right)$

In order to maintain the desired $V_{tap}$, control input $u\left(t\right)$ is introduced as part of $I_s\left(t\right)$, making $I_s\left(t\right)$ a function of $u\left(t\right)$. The input vector B and control law can be modified accordingly to achieve the desired $V_{tap}$. The state-space model to use when designing control strategies to maintain the desired $V_{tap}$ under varying load conditions is given by:

${\stackrel{\dot{}}{x}}_1\left(t\right)=A_x\left(t\right)+B_u\left(t\right)+B_{load}{I}_{load}\left(t\right)$

$y\left(t\right)=C_x\left(t\right)$

where:

$A=\left(\begin{array}{ccc}0& -\frac{1}{C_1}& 0\\ \frac{1}{L}& 0& -\frac{1}{L}\\ 0& \frac{1}{C_2}& 0\end{array}\right)$, $B=\left(\begin{array}{c}\frac{1}{C_2}\\ 0\\ 0\end{array}\right)$, $B_{load}=\left(\begin{array}{c}0\\ 0\\ -\frac{1}{C_2}\end{array}\right)$, $C=\left(\begin{array}{ccc}0& 0& 1\end{array}\right)$

The control law may include load compensation to counteract the effects of $B_{load}{I}_{load}\left(t\right)$. Under varying $I_{load}$, in order to achieve and maintain the desired $V_{tap}$, a gain matrix K can be designed to fulfil the control law with feedback controller $u\left(t\right)=Kx\left(t\right)$.