Question

Determine the bus admittance matrix ${\mathit{Y}}_{\mathbf{bus}}$for the three-phase power system shown in Figure 6.23 with input data given in Table 6.11 and partial results in Table 6.12. Assume a three-phase per unit base.

Short answer

The bus admittance matrix is

$Y_{\mathrm{bus}}=\left[\begin{array}{ccccc}6.25-18.69& -5+\mathrm{j}15& -1.25+\mathrm{j}3.75& 0& 0\\ -5+j15& 12.92-j38.66& -1.667+\mathrm{j}5& -1.25+\mathrm{j}3.75& -5+\mathrm{j}15\\ -1.25+j3.75& -1.667+j5& 8.79-j\mathit{32}\mathit{.}\mathit{23}& -5.88-\mathrm{j}23.53& 0\\ 0& -1.25+\mathrm{j}3.75& -5.88-j23.53& 9.88-j36.4& -2.75-\mathrm{j}9.17\\ 0& -5+\mathrm{j}15& 0& -2.75-j9.17& 7.75-j24.14\end{array}\right]$

Step-by-step solution

Write the given data.

The base MVA, $S_{\mathrm{base}}=100\mathrm{MVA}$

Determine the formula to calculate the admittance matrix for three phase power system.

The equation to calculate the diagonal element of the $Y_{bus}$matrix of the system is

given as follows.

${\mathbf{Y}}_{\mathbf{kk}}\mathbf{=}$Sum of the admittance connected to bus . ......(1)

The equation to calculate the non-diagonal element of the $Y_{bus}$matrix of the system

is given as follows.

${\mathbf{Y}}_{\mathbf{kn}}\mathbf{=}$(-sum of admittance connected between bus and )

Calculate the value of the elements of admittance matrix.

Calculate the diagonal element$Y_{11}$.

$$\begin{gathered} Y_{11}=\frac{1}{0.02+j0.06}+\frac{1}{0.08+j0.24}+j\frac{0.06}{2}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{05}}{\mathit{2}} \\ {Y}_{\mathit{11}}\mathit{=}\mathit{5}\mathit{-}j15+1.25-j3.75+j0.03+j0.025 \\ {\mathrm{Y}}_{\mathit{11}}=6.35-j18.69\mathrm{pu} \end{gathered}$$

Calculate the diagonal element$Y_{22}$.

$$\begin{gathered} Y_{22}=\left(\begin{array}{c}\frac{1}{0.02+j0.06}+\frac{1}{0.06+j0.18}+\frac{1}{0.08+j0.24}+\frac{1}{0.02+j0.06}\\ j\frac{0.06}{2}+j\frac{0.05}{2}+j\frac{0.04}{2}+j\frac{0.02}{2}\end{array}\right) \\ {Y}_{22}=\left\{\begin{array}{c}5-j15+1.667-j5+1.25-j3.75+5-j15+j0.03+j0.02\\ +j0.025+j0.01\end{array}\right\} \\ {Y}_{22}=12.92-j38.66\mathrm{pu} \end{gathered}$$

Calculate the diagonal element$Y_{33}$.

localid="1655358417124" $$\begin{gathered} Y_{33}=\frac{1}{0.08+\mathrm{j}0.24}+\frac{1}{0.06+\mathrm{j}0.18}+\frac{1}{0.01+\mathrm{j}0.04}+j\frac{\mathit{0}\mathit{.}\mathit{05}}{\mathit{2}}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{04}}{\mathit{2}}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{01}}{\mathit{2}} \\ {Y}_{\mathit{33}}=1.25-j3.75+1.667-j5+5.8823-j23.5294+j0.025+j0.02+j0.05 \\ {\mathrm{Y}}_{33}=8.79-j32.23\mathrm{pu} \end{gathered}$$

Calculate the diagonal element$Y_{44}$.

$$\begin{gathered} Y_{44}=\frac{1}{0.08+\mathrm{j}0.24}+\frac{1}{0.01+j0.04}+\frac{1}{0.03+j0.10}+j\frac{\mathit{0}\mathit{.}\mathit{05}}{\mathit{2}}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{04}}{\mathit{2}}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{01}}{\mathit{2}} \\ {Y}_{44}=1.25-j3.75+5.8823-j23.5294+2.7522+j0.025+j0.02+j0.005 \\ {\mathrm{Y}}_{44}=9.88-j36.4\mathrm{pu} \end{gathered}$$

Calculate the diagonal element$Y_{55}$.

$$\begin{gathered} Y_{55}=\frac{1}{0.02+j0.06}+\frac{1}{0.03+j0.10}+j\frac{\mathit{0}\mathit{.}\mathit{02}}{\mathit{2}}\mathit{+}j\frac{\mathit{0}\mathit{.}\mathit{04}}{\mathit{2}} \\ {Y}_{55}=5-j15+2.7522-j9.1743+j0.01+j0.02 \\ {Y}_{55}=7.75-j24.14\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{12}\&Y_{21}$,

$$\begin{gathered} Y_{12}=Y_{21} \\ {Y}_{12}=\frac{-1}{0.02+j0.06} \\ {Y}_{12}=-5+j15\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{13}\&Y_{31}$,

$$\begin{gathered} Y_{13}=Y_{31} \\ {Y}_{13}=\frac{-1}{0.08+j0.24} \\ {Y}_{13}=-1.25+j3.75\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{23}\&Y_{32}$,

$$\begin{gathered} Y_{23}=Y_{32} \\ {Y}_{23}=\frac{-1}{0.06+j0.18} \\ {Y}_{23}=-1.667+j5\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{24}\&Y_{42}$,

$$\begin{gathered} Y_{24}=Y_{42} \\ {Y}_{24}=\frac{-1}{0.08+j0.24} \\ {Y}_{24}=-1.25+j3.75\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{25}\&Y_{52}$,

$$\begin{gathered} Y_{25}=Y_{52} \\ {Y}_{25}=\frac{-1}{0.02+j0.06} \\ {Y}_{25}=-5.88+j23.53\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{34}\&Y_{43}$,

$$\begin{gathered} Y_{34}=Y_{43} \\ {Y}_{34}=\frac{-1}{0.01+j0.04} \\ {Y}_{34}=-5.88+j23.53\mathrm{pu} \end{gathered}$$

Calculate the element$Y_{45}\&Y_{54}$,

$$\begin{gathered} Y_{45}=Y_{54} \\ {Y}_{45}=\frac{-1}{0.03+j0.1} \\ {Y}_{45}=-2.75+j9.17\mathrm{pu} \end{gathered}$$

There are no admittance is connected between (1,4),(1,5)and (3,5). Therefore, the admittance between them is zero.

Therefore the admittance matrix is

$Y_{\mathrm{bus}}=\left[\begin{array}{ccccc}6.25-18.69& -5+\mathrm{j}15& -1.25+\mathrm{j}3.75& 0& 0\\ -5+j15& 12.92-j38.66& -1.667+\mathrm{j}5& -1.25+\mathrm{j}3.75& -5+\mathrm{j}15\\ -1.25+j3.75& -1.667+j5& 8.79-j\mathit{32}\mathit{.}\mathit{23}& -5.88-\mathrm{j}23.53& 0\\ 0& -1.25+\mathrm{j}3.75& -5.88-j23.53& 9.88-j36.4& -2.75-\mathrm{j}9.17\\ 0& -5+\mathrm{j}15& 0& -2.75-j9.17& 7.75-j24.14\end{array}\right]$