7 The Nodal Admittance Matrix during TCUL

Tap-changing-under-load (TCUL) has an impact on power flow studies since it changes the voltage transformation ratio thus changing the impedance of a transformer. These impedance adjustments must be reflected in the nodal admitance matrix Y_bus. The following discussion examines techniques for updating Y_bus when a transformer changes tap settings.

Assume that the following information is required by any procedure that updates Y_bus following a tap change:

• Y_bus itself,

• Y_L the transformer’s leakage admittance, and

• t_new the transformer’s tap ratio after the tap change.

The transfer admittances (off-diagonal elements of Y_bus) can be maintained with this base of information. However, maintaining the self admittances is more complicated. There are two strategies for updating the diagonal of Y_bus.

• Rebuild the entry from scratch, or

• Back out the old transformer admittances, then add in the new values.

The reconstruction strategy is ruled out as computationally inefficient without extensive examination. Computational requirements of rebuilding an entry include:

• Compute the new admittances of the TCUL device.

• Access the incidence list of the primary vertex p.

• Look up shunt elements (e.g. impedance loads) at the primary vertex p.

• Recompute the self admittances of any transformers adjacent to the TCUL device (incident upon its primary vertex).

The direct update strategy is examined more carefully. Its basic computational complexity stems from backing the old value of the transformer’s primary admittance Y_pp out of y_pp (the self admittance of vertex p). There are two ways to proceed:

1. Remember Y_pp so that it can be backed out directly, or

2. Recompute Y_pp, then back it out of y_pp. From Equation 27 or Equation 28, it is seen that the square of t_old (the magnitude of the original tap ratio) is needed to recompute Y_pp.

The Y_bus maintenance algorithm based on a direct update approach follows.

1. Determine the transformer’s original primary admittance Y_pp, subtract it from the corresponding diagonal term of the nodal admittance matrix y_pp.

2. Compute the new primary admittance Y_pp using Equation 27 or Equation 28, add it to the corresponding diagonal term of the nodal admittance matrix y_pp.

3. Compute the new transfer admittance Y_ps from Equation 32, set the corresponding off-diagonal element of the admittance matrix y_ps to this value.

4. Compute the new transfer admittance Y_sp using Equation 36, set the corresponding off-diagonal element of the admittance matrix y_sp to this value.

Examining Equation 30, you should observe that the transformer’s secondary admittance Y_ss is not a function of the tap setting. Therefore, it does not have to be updated when the tap changes.

When $\delta$ does not change during the load flow (i.e. no phase angle regulators), the terms g_L sin $\delta$, g_L cos $\delta$, b_L sin $\delta$, b_L cos $\delta$ in Equation 32 and Equation 36 remain constant throughout the load flow solution. If these terms are precomputed and stored, a considerable reduction in floating point arithmetic results. Obviously, you pay a price in storage. Table 4 summarizes the time and space trade-offs associated with recompututaion vs storage of these products when updating the diagonal of Y_bus and computing the transfer admittances Y_ps and Y_sp in the absence of phase angle regulators.