Load Flow Analysis Using Newton-Raphson Method

Assume initial estimates of the \(n\) variables as \(x_{1}^{(0)}, x_{2}^{(0)}, \cdots, x_{n}^{(0)}\) and add corrections \(\varDelta x_{1}^{(0)},\varDelta x_{2}^{(0)}, \cdots,\varDelta x_{n}^{(0)}\) to get correct solution of these variables \[\begin{aligned} x_{1}^{*} & =x_{1}^{(0)}+\varDelta x_{1}^{(0)}\\ x_{2}^{*} & =x_{2}^{(0)}+\varDelta x_{2}^{(0)}\\ & \vdots\\ x_{n}^{*} & =x_{n}^{(0)}+\varDelta x_{n}^{(0)} \end{aligned}\]

• The functions \(g\) can be written in terms of the variables as \[g_{k}\left(x_{1}^{*},\cdots,x_{n}^{*}\right)=g_{k}\left(x_{1}^{(0)}+\varDelta x_{1}^{(0)},\cdots x_{n}^{(0)}+\varDelta x_{n}^{(0)}\right)=0,\] \(~\mbox{where}~k=1,\cdots,n\)

• Expanding the equation in Taylor’s series around the nominal values \(x_{1}^{(0)}, x_{2}^{(0)}, \cdots, x_{n}^{(0)}\) and neglecting the second and higher order terms of the series, we get \[ \begin{aligned} g_{k}\left(x_{1}^{*},\cdots,x_{n}^{*}\right)=g_{k}\left(x_{1}^{(0)},\cdots,x_{n}^{(0)}\right)+\varDelta x_{1}^{(0)}\left.\dfrac{\partial g_{k}}{\partial x_{1}}\right|^{(0)}+\varDelta x_{2}^{(0)}\left.\dfrac{\partial g_{k}}{\partial x_{2}}\right|^{(0)}+\cdots+\varDelta x_{n}^{(0)}\left.\dfrac{\partial g_{k}}{\partial x_{n}}\right|^{(0)} \end{aligned} \] where \(\left.\dfrac{\partial g_{k}}{\partial x_{i}}\right|^{(0)}\) are the partial derivatives of \(g_k\) evaluated at \(x_{1}^{(1)}, x_{2}^{(1)}, \cdots, x_{n}^{(1)}\).

Equation can be written in vector-matrix form as \[\left[\begin{array}{cccc} \dfrac{\partial g_{1}}{\partial x_{1}} & \dfrac{\partial g_{1}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{1}}{\partial x_{n}}\\ \dfrac{\partial g_{2}}{\partial x_{1}} & \dfrac{\partial g_{2}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{2}}{\partial x_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \dfrac{\partial g_{n}}{\partial x_{1}} & \dfrac{\partial g_{n}}{\partial x_{2}} & & \dfrac{\partial g_{n}}{\partial x_{n}} \end{array}\right]^{\left(0\right)}\left[\begin{array}{c} \varDelta x_{1}^{(0)}\\ \varDelta x_{2}^{(0)}\\ \vdots\\ \varDelta x_{n}^{(0)} \end{array}\right]=\left[\begin{array}{c} 0-g_{1}\left(x_{1}^{(0)},\cdots,x_{n}^{(0)}\right)\\ 0-g_{2}\left(x_{1}^{(0)},\cdots,x_{n}^{(0)}\right)\\ \vdots\\ 0-g_{n}\left(x_{1}^{(0)},\cdots,x_{n}^{(0)}\right) \end{array}\right]\] Jacobian matrix, \(J\) : the square matrix of partial derivatives

\[\left[\begin{array}{c} \varDelta x_{1}^{(0)}\\ \varDelta x_{2}^{(0)}\\ \vdots\\ \varDelta x_{n}^{(0)} \end{array}\right]=\left[J^{\left(0\right)}\right]^{-1}\left[\begin{array}{c} \varDelta g_{1}^{(0)}\\ \varDelta g_{2}^{(0)}\\ \vdots\\ \varDelta g_{n}^{(0)} \end{array}\right]\]

• Taylor ’s series is truncated by neglecting the 2nd and higher order terms, we cannot expect to find the correct solution at the end of first iteration. We shall then have

  \[\begin{aligned} x_{1}^{(1)} & =x_{1}^{(0)}+\varDelta x_{1}^{(0)}\\ x_{2}^{(1)} & =x_{2}^{(0)}+\varDelta x_{2}^{(0)}\\ & \vdots\\ x_{n}^{(1)} & =x_{n}^{(0)}+\varDelta x_{n}^{(0)} \end{aligned}\]

• These are then used to find \(J^{(1)}\) and \(\varDelta g_k^{(1)},~ k = 1,\cdots , n\) .

• We can then find \(\varDelta x_2^{(1)} , \cdots , \varDelta x_n^{(1)}\) and subsequently calculate \(x_2^{(1)}, \cdots , x_n^{(1)}\).

• The process continues till \(\varDelta g_k,~ k = 1, \cdots , n\) becomes less than a small quantity.

The Jacobian matrix is divided into sub-matrices as \[J=\left[\begin{array}{cc} J_{11} & J_{12}\\ J_{21} & J_{22} \end{array}\right]\] Size of the Jacobian matrix is

\[\left(n + n_p -1\right) \times \left(n + n_p -1\right)\]

Dimensions of the sub-matrices are as follows \[\begin{array}{cc} J_{11}: & \left(n-1\right)\times\left(n-1\right)\\ J_{12}: & \left(n-1\right)\times n_{p}\\ J_{21}: & n_{p}\times\left(n-1\right)\\ J_{22}: & n_{p}\times n_{p} \end{array}\]

\[ \begin{aligned} J_{11} & =\left[\begin{array}{ccc} \dfrac{\partial P_{2}}{\partial\delta_{2}} & \cdots & \dfrac{\partial P_{2}}{\partial\delta_{n}}\\ \vdots & \ddots\\ \dfrac{\partial P_{n}}{\partial\delta_{2}} & & \dfrac{\partial P_{n}}{\partial\delta_{n}} \end{array}\right]~~~~~~~J_{12}=\left[\begin{array}{ccc} \left|V_{2}\right|\dfrac{\partial P_{2}}{\partial\left|V_{2}\right|} & \cdots & \left|V_{1+n_{p}}\right|\dfrac{\partial P_{2}}{\partial\left|V_{1+n_{p}}\right|}\\ \vdots & \ddots\\ \left|V_{2}\right|\dfrac{\partial P_{n}}{\partial\left|V_{2}\right|} & \cdots & \left|V_{1+n_{p}}\right|\dfrac{\partial P_{n}}{\partial\left|V_{1+n_{p}}\right|} \end{array}\right]\\ J_{21} & =\left[\begin{array}{ccc} \dfrac{\partial Q_{2}}{\partial\delta_{2}} & \cdots & \dfrac{\partial Q_{2}}{\partial\delta_{n}}\\ \vdots & \ddots\\ \dfrac{\partial Q_{1+n_{p}}}{\partial\delta_{2}} & \cdots & \dfrac{\partial Q_{1+n_{p}}}{\partial\delta_{n}} \end{array}\right]\:J_{22}=\left[\begin{array}{ccc} \left|V_{2}\right|\dfrac{\partial Q_{2}}{\partial\left|V_{2}\right|} & \cdots & \left|V_{1+n_{p}}\right|\dfrac{\partial Q_{2}}{\partial\left|V_{1+n_{p}}\right|}\\ \vdots & \ddots\\ \left|V_{2}\right|\dfrac{\partial Q_{1+n_{p}}}{\partial\left|V_{2}\right|} & \cdots & \left|V_{1+n_{p}}\right|\dfrac{\partial Q_{1+n_{p}}}{\partial\left|V_{1+n_{p}}\right|} \end{array}\right] \end{aligned} \]

Formation of \(J_{11}\)

\[J_{11}=\left[\begin{array}{ccc} L_{22} & \cdots & L_{2n}\\ \vdots & \ddots & \vdots\\ L_{n2} & \cdots & L_{nn} \end{array}\right]\]

\[\begin{aligned} L_{ik} & =\dfrac{\partial P_{i}}{\partial\delta_{k}}=-\left|Y_{ik}V_{i}V_{k}\right|sin\left(\theta_{ik}+\delta_{k}-\delta_{i}\right),~i\neq k\\ L_{ii} & =\dfrac{\partial P_{i}}{\partial\delta_{i}}=\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{i}V_{k}\right|sin\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\\ & =-Q_{i}-\left|V_{i}\right|^{2}B_{ii} \end{aligned}\]

Formation of \(J_{21}\)

\[J_{21}=\left[\begin{array}{ccc} M_{22} & \cdots & M_{2n}\\ \vdots & \ddots & \vdots\\ M_{n2} & \cdots & M_{nn} \end{array}\right]\] \[\begin{aligned} M_{ik} & =\dfrac{\partial Q_{i}}{\partial\delta_{k}}=-\left|Y_{ik}V_{i}V_{k}\right|cos\left(\theta_{ik}+\delta_{k}-\delta_{i}\right),i\neq k\\ M_{ii} & =\dfrac{\partial Q_{i}}{\partial\delta_{i}}=\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{i}V_{k}\right|cos\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\\ & = P_{i}-\left|V_{i}\right|^{2}G_{ii} \end{aligned}\]

Formation of \(J_{12}\) \[J_{12}=\left[\begin{array}{ccc} N_{22} & \cdots & N_{2n}\\ \vdots & \ddots & \vdots\\ N_{n2} & \cdots & N_{nn} \end{array}\right]\] \[ \begin{aligned} N_{ik} & =\left|V_{k}\right|\dfrac{\partial P_{i}}{\partial\left|V_{k}\right|}=\left|Y_{ik}V_{i}V_{k}\right|cos\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)=-M_{ik},i\neq k\\ N_{ii} & =\left|V_{i}\right|\dfrac{\partial P_{i}}{\partial\left|V_{i}\right|}=\left|V_{i}\right|\left[2\left|V_{i}\right|G_{ii}+\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{k}\right|cos\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\right]\\ & =2\left|V_{i}\right|^{2}G_{ii}+\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{i}V_{k}\right|cos\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\\ & =2\left|V_{i}\right|^{2}G_{ii}+M_{ii} \end{aligned} \]

Formation of \(J_{22}\) \[J_{22}=\left[\begin{array}{ccc} O_{22} & \cdots & O_{2n}\\ \vdots & \ddots & \vdots\\ O_{n2} & \cdots & O_{nn} \end{array}\right]\] \[ \begin{aligned} O_{ik} & =\left|V_{i}\right|\dfrac{\partial Q_{i}}{\partial\left|V_{k}\right|}=-\left|V_{i}\right|\left|Y_{ik}V_{i}V_{k}\right|sin\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)=L_{ik},~i\neq k\\ O_{ii} & =\left|V_{i}\right|\dfrac{\partial Q_{i}}{\partial\left|V_{k}\right|}=\left|V_{i}\right|\left[-2\left|V_{i}\right|B_{ii}-\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{k}\right|sin\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\right]\\ & =-2\left|V_{i}\right|^{2}B_{ii}-\sum_{\begin{array}{c} k=1\\ k\neq i \end{array}}^{n}\left|Y_{ik}V_{i}V_{k}\right|sin\left(\theta_{ik}+\delta_{k}-\delta_{i}\right)\\ & =-2\left|V_{i}\right|^{2}B_{ii}-L_{ii} \end{aligned} \]

\[J_{11}^{(0)}=\left[\begin{array}{cccc} 18.8269 & -4.8077 & 0 & -3.9231\\ -4.8077 & 11.1058 & -3.8462 & -2.4519\\ 0 & -3.8462 & 5.8077 & -1.9615\\ -3.9231 & -2.4519 & -1.9615 & 12.4558 \end{array}\right]\] \[\begin{aligned} M_{23}^{(0)} & =-\left|Y_{23}V_{2}^{(0)}V_{3}^{(0)}\right|cos\left(\theta_{23}+\delta_{2}-\delta_{3}\right)=-G_{23}=0.9615\\ P_{2}^{(0)} & =\left|V_{2}^{(0)}\right|^{2}G_{22}+{\displaystyle \sum_{\begin{array}{c} k=1\\ k\neq2 \end{array}}^{n}}\left|Y_{2k}V_{2}^{(0)}V_{k}^{(0)}\right|cos\left(\theta_{2k}+\delta_{k}-\delta_{2}\right)\\ & =G_{22}+1.05G_{21}+G_{23}+G_{24}+1.02G_{25}=-0.1115\\ M_{22} & =P_{2}^{(0)}-\left|V_{2}^{(0)}\right|^{2}G_{22}=-3.7654 \end{aligned}\]

\[ J_{21}=M\left(1:3,1:4\right)~and~J_{12}=-M\left(1:4,1:3\right)\Longleftarrow M=\left[\begin{array}{cccc} M_{11} & M_{12} & M_{13} & M_{14}\\ M_{21} & M_{22} & M_{23} & M_{24}\\ M_{31} & M_{32} & M_{33} & M_{34}\\ M_{41} & M_{42} & M_{43} & M_{44} \end{array}\right] \]

\[J_{21}^{(0)}=\left[\begin{array}{cccc} -3.7654 & 0.9615 & 0 & 0.7846\\ 0.9615 & -2.2212 & 0.7692 & 0.4904\\ 0 & 0.7692 & -1.1615 & 0.3923 \end{array}\right]\]

\[J_{12}^{(0)}=\left[\begin{array}{ccc} 3.5423 & -0.9615 & 0\\ -0.9615 & 2.2019 & -0.7692\\ 0 & -0.7692 & 1.1462\\ 0.7846 & -0.4904 & -0.3923 \end{array}\right]\]

\[J_{22}^{(0)}=\left[\begin{array}{ccc} 17.5615 & -4.8077 & 0\\ -4.8077 & 10.8996 & -3.8462\\ 0 & -3.8462 & 5.5408 \end{array}\right]\]

From the initial conditions the power and reactive power are computed as \[\begin{aligned} P_{calc}^{(0)} & =\left[\begin{array}{cccc} -0.1115 & -0.0096 & -0.0077 & -0.0098\end{array}\right]^{T}\\ Q_{calc}^{(0)} & =\left[\begin{array}{ccc} -0.6327 & -0.1031 & -0.1335\end{array}\right]^{T} \end{aligned}\] Consequently the mismatches are found to be \[\begin{aligned} \varDelta P^{(0)} & =\left[\begin{array}{cccc} -0.8485 & -0.3404 & -0.1523 & 0.2302\end{array}\right]^{T}\\ \varDelta Q^{(0)} & =\left[\begin{array}{ccc} 0.0127 & -0.0369 & 0.0535\end{array}\right]^{T} \end{aligned}\]

Then the updates at the end of the first iteration are given as \[\left[\begin{array}{c} \delta_{2}^{(0)}\\ \delta_{3}^{(0)}\\ \delta_{4}^{(0)}\\ \delta_{5}^{(0)} \end{array}\right]=\left[\begin{array}{c} -4.91\\ -6.95\\ -7.19\\ -3.09 \end{array}\right]deg~~\left[\begin{array}{c} \left|V_{2}\right|^{(0)}\\ \left|V_{3}\right|^{(0)}\\ \left|V_{4}\right|^{(0)} \end{array}\right]=\left[\begin{array}{c} 0.9864\\ 0.9817\\ 0.9913 \end{array}\right]\] Load flow converges in 7 iterations when all \(\varDelta P\) and \(\varDelta Q\) are below \(10^{-6}\)

• The current flowing between the buses \(i\) and \(k\) can be written as \[I_{ik}=-Y_{ik}\left(V_{i}-V_{k}\right),~i\neq k\]

• The complex power leaving bus-\(i\) is given by \[P_{i}+jQ_{i}=V_{i}I_{i}^{*}\]

• The complex power entering bus-\(k\) is \[P_{k}+jQ_{k}=V_{k}I_{k}^{*}\]

• Therefore the \(I^2R\) loss in the line segment \(i-k\) is \[P_{loss,i-k}=P_{i}-P_{k}\]

• The line \(I^2X\) drop can be calculated by

  \[Q_{drop,i-k}=Q_{i}-Q_{k}\]

\[\begin{aligned} I_{12} & =0.9623-j0.5187=1.0932\angle-28.33^{0}~pu\\ S_{12} & =V_{1}I_{12}^{*}\times100=-101.0395-j54.4645~MW\\ S_{21} & =V_{2}I_{12}^{*}\times100=98.6494+j42.5141~MW \end{aligned}\]

-ve sign in power indicates the power is leaving the bus (leaving the bus-1 and entering the bus-2)

• out of a total amount of 101.0395 MW of real power dispatched from bus-1 over the line segment 1-2, 98.6494 MW reaches bus-2. This indicates that the drop in the line segment is 2.3901 MW.

• Knowing resistance we can calculate \(I^2R\) loss. \[\begin{aligned} \left|I_{12}\right|^{2}\times R_{12}\times100 & =1.0932^{2}\times0.02\times100=2.3901~MW\\ \left|I_{12}\right|^{2}\times X_{12}\times100 & =1.0932^{2}\times0.1\times100=10.9508~MVAr \end{aligned}\]

• Also, \(I^2X\) loss can be calculated by subtracting the reactive power absorbed by bus-2 from that supplied by bus-1

• The above calculation however does not include the line charging

• Since the line is modelled by an equivalent-\(\pi\), the voltage across the shunt capacitor is the bus voltage to which the shunt capacitor is connected

• Therefore the current flowing through line segment is not the current leaving bus-1 or entering bus-2

• It is the current flowing in between the two charging capacitors

• Since the shunt branches are purely reactive, the real power flow does not get affected by the charging capacitors

• Each charging capacitor is assumed to inject a reactive power that is the product of the half line charging admittance and square of the magnitude of the voltage of that at bus

• The half line charging admittance of this line is 0.03. Therefore line charging capacitor will inject

  \[0.03\times100\times\left|V_{1}\right|^{2}=3.3075~MVAr\] at bus-1. Similarly the reactive injected at bus-2 will be \[0.03\times100\times\left|V_{2}\right|^{2}=2.8968~MVAr\]

• The power flow through the line segments 1-2 and 1-5 is shown