Question

Express the complex power ${\mathit{S}}_{\mathbf{S}}$delivered to the sequence networks in terms of sequence voltages and sequence currents, where ${\mathit{S}}_{\mathbf{S}}\mathbf{=}$___________.

Short answer

Answer

The complex power in terms of sequence voltage and current is $S_S=3\left(V_0{I}_0^{*}+V_1{I}_1^{*}+V_2{I}_2^{*}\right)$.

Step-by-step solution

Determine the equation to calculate the complex power delivered in terms of sequence voltages and current.

The expression to deliver the total power to three phase network is given as follows.

${\mathit{S}}_{\mathbf{P}}\mathbf{=}{\mathit{V}}_{\mathbf{a}\mathbf{g}}{\mathit{I}}_{\mathbf{a}}^{\mathbf{*}}\mathbf{+}{\mathit{V}}_{\mathbf{b}\mathbf{g}}{\mathit{I}}_{\mathbf{b}}^{\mathbf{*}}\mathbf{+}{\mathit{V}}_{\mathbf{c}\mathbf{g}}{\mathit{I}}_{\mathbf{c}}^{\mathbf{*}}$ …… (1)

Here ${\mathit{V}}_{\mathbf{a}\mathbf{g}}\mathbf{,}{\mathit{V}}_{\mathbf{b}\mathbf{g}}\mathbf{,}{\mathit{V}}_{\mathbf{c}\mathbf{g}}$are the phase voltages and ${\mathit{I}}_{\mathbf{a}}\mathbf{,}{\mathit{I}}_{\mathbf{b}}\mathbf{,}{\mathit{I}}_{\mathbf{c}}$are the phase current.

The relationship between the sequence and phase voltage is given as follows.

${\mathit{V}}_{\mathbf{P}}\mathbf{=}\mathit{A}{\mathit{V}}_{\mathbf{s}}$ …… (2)

The relationship between the sequence and phase current is given as follows.

${\mathit{I}}_{\mathbf{P}}\mathbf{=}\mathit{A}{\mathit{I}}_{\mathbf{s}}$ ….. (3)

The identities of the operator,

$$\begin{gathered} \mathbf{1}\mathbf{+}\mathit{a}\mathbf{+}{\mathit{a}}^{\mathbf{2}}\mathbf{=}\mathbf{0} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{a}}^{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{\angle }{\mathbf{0}}^{\mathbf{°}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{a}}^{\mathbf{4}}\mathbf{=}\mathit{a} \end{gathered}$$

Derive the expression for complex power in terms of sequence voltage and current.

The phase power can be written as,

$$\begin{gathered} S_P=\left[V_{ag}{V}_{bg}{V}_{cg}\right]\left[\begin{array}{c}{I}_a^{*}\\ I_b^{*}\\ I_c^{*}\end{array}\right] \\ {S}_P={\left(V_P\right)}^T{\left(I_P\right)}^{*} \end{gathered}$$

Here T represent the transposed and * represents the conjugate of the current.

Substitute $A{V}_s$ for $V_P$ and $A{I}_S$for $I_P$into above equation.

$$\begin{gathered} S_P={\left(A{V}_s\right)}^T{\left(A{I}_s\right)}^{*} \\ {S}_P=V_S^T{A}^T{A}^{*}{I}_S^{*} \\ {S}_P=\left(V_S^T{I}_S^{*}\right)\left(A^{T}A*\right) \end{gathered}$$ …… (4)

Calculate the term $A^{T}A*$,

$$\begin{gathered} A^{T}A*=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right] \\ {A}^{T}A*=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right] \\ {A}^{T}A*=\left[\begin{array}{ccc}1+1+1& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 1+a^3+a^3& 1+a^4+a^2\\ 1+a+a^2& 1+a^2+a^4& 1+a^3+a^3\end{array}\right] \end{gathered}$$

Substitute 1 for $a^3$, a for $a^4$into above equation. 0

$$\begin{gathered} A^{T}A*=\left[\begin{array}{ccc}1+1+1& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 1+1+1& 1+a+a^2\\ 1+a+a^2& 1+a^2+a& 1+1+1\end{array}\right] \\ {A}^{T}A*=\left[\begin{array}{ccc}3& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 3& 1+a+a^2\\ 1+a+a^2& 1+a^2+a& 3\end{array}\right] \end{gathered}$$

Substitute 0 for $1+a+a^2$into above equation.

$$\begin{gathered} A^{T}A*=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right] \\ {A}^{T}A*=3I \end{gathered}$$

Substitute $3I$for $A^{T}A*$into equation (4).

$$\begin{gathered} S_P=\left(V_S^T{I}_S^{*}\right)\left(3I\right) \\ {S}_P=3\left(V_S^T{I}_S^{*}\right) \\ {S}_P=3\left[V_0{V}_1{V}_2\right]\left[\begin{array}{c}{I}_0^{*}\\ I_1^{*}\\ I_2^{*}\end{array}\right] \\ {S}_P=3S_S \end{gathered}$$

Therefore, the three-phase complex power is 3 times the sequence power.

The complex power in terms of sequence voltage and current is $S_S=3\left(V_0{I}_0^{*}+V_1{V}_1^{*}+V_2{I}_2^{*}\right)$