In this system, the active power P transferred through the coupling capacitor is influenced by the voltage magnitudes on the primary HV side and the secondary side V_tap, as well as the phase angle difference between these voltages, is given by:

$P=\frac{V_1{V}_2}{X_c}\text{sin}\left(\delta \right)$

where:

The reactive power Q is associated with the maintenance of voltage levels within the system and is influenced by the difference in voltage magnitudes between the two sides of the coupling capacitor and is described by:

$Q=\frac{V_1^2-V_1{V}_2}{X_c}\text{cos}\left(\delta \right)$

This equation shows that reactive power is managed by adjusting the voltage difference and the phase angle $\delta$. These equations demonstrate that by controlling $V_1$, $V_2$ and $\delta$, the CCS-CNT system can provide or absorb reactive power, thereby stabilizing voltage levels in the network and enhancing overall power quality.

In a Capacitor-Coupled Substation (CCS) system integrated with a Controllable Network Transformer (CNT), achieving bidirectional power flow is essential for efficient operation, especially in scenarios where power needs to be both tapped from a high-voltage (HV) line and injected back into the grid from a lower voltage level. The mathematical representation of bidirectional power flow control is developed through different stages. The system parameters are chosen. The following are the chosen parameters:

The active power flow of the system can occur in two primary modes: tapping mode (HV to LV) and injection mode (LV to HV).

The power flow equations are developed as:

Active Power Flow from HV to LV (tapping mode):

$P_{tap}=\frac{V_{tap}^2}{Z_{tap}}\cos \left({\theta }_{tap}\right)-P_{loss}$

where:

$P_{inj}=P_{MG}-P_{loss}$

where:

The reactive power flow of the system also, can occur in two primary modes: tapping mode (HV to LV) and injection mode (LV to HV).

$Q_{tap}=\frac{V_{tap}^2}{Z_{tap}}\sin \left({\theta }_{tap}\right)+Q_{comp}$

where:

$Q_{inj}=Q_{MG}+Q_{comp}-Q_{loss}$

where:

The voltage control is achieved by the CNT adjusting the voltage at the tap node based on the power flow direction and load or MG conditions (whether is it absorbing or injecting power). The voltage at the secondary side of the CNT can be expressed as:

$V_{tap}=\frac{V_{HV}}{a\left(t\right)}\times \frac{1+j{X}_c}{1+j{X}_{tap}}$

where:

To control power flow is both directions, the CNT’s control algorithm adjusts the tap ratio $a\left(t\right)$ and coordinates with the MG and the capacitor bank. The controller maintains the voltages levels withing the desired limits and ensures stable bidirectional power flow.

The control strategy for tapping power involves maintaining the $V_{tap}$ close to the desired value while supplying power to the load by adjusting $a\left(t\right)$ to minimize the difference between $V_{tap}$ and the reference voltage $V_{ref}$ as follows:

$a\left(t\right)=\frac{V_{HV}}{V_{ref}}\times \frac{1+j{X}_{tap}}{1+j{X}_c}$

The control strategy for injecting power involves ensuring that the MG power is injected into the HV line with minimal losses by adjusting $a\left(t\right)$ based in the desired injection power $P_{inj}$ and voltage level at the HV side as follows:

$a\left(t\right)=\frac{V_{ref}}{V_{tap}}\times \frac{1+j{X}_c}{1+j{X}_{tap}}$

where $V_{ref}$ is set based on the desired injection conditions.

The stability of the system is maintained by ensuring that ferroresonance suppression and reactive power compensation is effectively managed. The stability condition is maintained by:

$\frac{V_{tap}}{V_{HV}}\le a_{ref}$

where $a_{ref}$ is the maximum turns ratio for stability.

This mathematical representation and control strategy enables the CCS-CNT system to efficiently manage bidirectional power flow, ensuring reliable operation while adapting to varying load and microgrid conditions.

The control process begins with the initialization phase, where key parameters are set:

To correct any deviations from the desired setpoints:

$e_P=P_{set}-P_{meas}$

$e_Q=Q_{set}-Q_{meas}$

where:

These errors represent the amount by which the actual power flow deviates from the setpoints and are used to determine the necessary adjustments.

A Proportional-Integral-Derivative (PID) controller is used to adjust the control variables based on the calculated errors:

$\Delta V_1=K_{p1}{e}_P+K_{i1}{\displaystyle \int e_P\text{d}t}+K_{d1}\frac{\text{d}{e}_P}{\text{d}t}$

$\Delta V_2=K_{p2}{e}_Q+K_{i2}{\displaystyle \int e_Q\text{d}t}+K_{d2}\frac{\text{d}{e}_Q}{\text{d}t}$

$\Delta \delta =K_{p3}{e}_P+K_{i3}{\displaystyle \int e_P\text{d}t}+K_{d3}\frac{\text{d}{e}_P}{\text{d}t}$

where:

$V_1\leftarrow V_1+\Delta V_1$

$V_2\leftarrow V_2+\Delta V_2$

$\delta \leftarrow \delta +\Delta \delta$

This updating process ensures that the control variables are continuously refined to reduce errors and achieve the desired power flow conditions.

To control power flow within the CCS-CNT:

$\Delta V_{tap}=K_{p1}\times \left(V_{tap,set}-V_{tap,meas}\right)$

$\Delta V_1=K_{p2}\times e_P$

$\Delta V_2=K_{p3}\times e_Q$

$\Delta \delta =K_{p4}\times e_P$

$V_{1,new}=V_1\times \Delta V_1$

$V_{2,new}=V_2\times \Delta V_2$

${\delta }_{new}=\delta \times \Delta \delta$

If $P_{flow}>0$ on the HV side and $P_{flow}<0$ on the tap node, power is flowing from the HV side to the CCS. Conversely, if $P_{flow}<0$ on the HV side and $P_{flow}>0$ on the tap node, power is flowing from the CCS side to the HV network.

This control algorithm ensures that the CCS-CNT system operates efficiently and meets the desired power flow conditions, while also maintaining stability and reliability in the grid.

The power flow direction within the CCS-CNT can be verified through a MATLAB code approach to prove the viability of the claim of power flow direction in Section 5.1.

The gains and setpoints of active and reactive power can be modified as per the system requirements.