# 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 Ybus. The following discussion examines techniques for updating Ybus when a transformer changes tap settings.

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

• Ybus itself,

• YL the transformer’s leakage admittance, and

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

The transfer admittances (off-diagonal elements of Ybus) 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 Ybus.

• 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 Ypp out of ypp (the self admittance of vertex p). There are two ways to proceed:

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

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

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

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

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

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

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

Examining Equation 30, you should observe that the transformer’s secondary admittance Yss 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 gL sin $\delta$, gL cos $\delta$, bL sin $\delta$, bL 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 Ybus and computing the transfer admittances Yps and Ysp in the absence of phase angle regulators.

Table 4: Ybus Maintenance Costs
Storage
Operation FLOPS Required Assumed
Update Ybus Diagonals
Recompute 19nt+4nt ntnfloat 4nt
Store 19nt+2nt 2ntnfloat 8nt
Update Yps and Ysp
Recompute 19nt ntnfloat 4nt
Store 9nt 4ntnfloat 16nt
• nt is the number of TCUL devices.
nfloat is the size of a floating point number in bytes.
19 FLOPS computes Ypp, Yps, and Ysp. Updates ypp.
9 FLOPS computes Ypp, Yps, Ysp. Updates ypp w/constant angle.
2 FLOPS moves Ypp into or out of ypp.
2 FLOPS recomputes Ypp from ${t}_{old}^{2}$.