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:
Remember Y_{pp} so that it can be backed out directly, or
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.
Determine the transformer’s original primary admittance Y_{pp}, subtract it from the corresponding diagonal term of the nodal admittance matrix y_{pp}.
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}.
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.
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.
Storage | |||
---|---|---|---|
Operation | FLOPS | Required | Assumed |
Update Y_{bus} Diagonals | |||
Recompute | 19n_{t}+4n_{t} | n_{t}n_{float} | 4n_{t} |
Store | 19n_{t}+2n_{t} | 2n_{t}n_{float} | 8n_{t} |
Update Y_{ps} and Y_{sp} | |||
Recompute | 19n_{t} | n_{t}n_{float} | 4n_{t} |
Store | 9n_{t} | 4n_{t}n_{float} | 16n_{t} |
n_{t} is the number of TCUL devices.
n_{float} is the size of a floating point number in bytes.
19 FLOPS computes Y_{pp}, Y_{ps}, and Y_{sp}. Updates y_{pp}.
9 FLOPS computes Y_{pp}, Y_{ps}, Y_{sp}. Updates y_{pp} w/constant angle.
2 FLOPS moves Y_{pp} into or out of y_{pp}.
2 FLOPS recomputes Y_{pp} from ${t}_{old}^{2}$.