Newton-Raphson Load Flow Analysis

Important Formulae

Powers

\[\begin{aligned} P_i & = \sum_{k=1}^n |V_i||V_k||Y_{ik}|\cos(\theta_{ik} - \delta_i + \delta_k) \\ Q_i & = -\sum_{k=1}^n |V_i||V_k||Y_{ik}|\sin(\theta_{ik} - \delta_i + \delta_k) \end{aligned}\]

Equation

Bus-1 is assumed to be slack bus.

\[\begin{bmatrix} \Delta P_2^{(p)} \\ \vdots \\ \Delta P_n^{(p)} \\ \hline \Delta Q_2^{(p)} \\ \vdots \\ \Delta Q_n^{(p)} \end{bmatrix} = \left[ \begin{array}{ccc|ccc} \left(\frac{\partial P_2}{\partial \delta_2}\right)^{(p)} & \cdots & \left(\frac{\partial P_2}{\partial \delta_n}\right)^{(p)} & \left(\frac{\partial P_2}{\partial |V_2|}\right)^{(p)} & \cdots & \left(\frac{\partial P_2}{\partial |V_n|}\right)^{(p)} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \left(\frac{\partial P_n}{\partial \delta_2}\right)^{(p)} & \cdots & \left(\frac{\partial P_n}{\partial \delta_n}\right)^{(p)} & \left(\frac{\partial P_n}{\partial |V_2|}\right)^{(p)} & \cdots & \left(\frac{\partial P_n}{\partial |V_n|}\right)^{(p)} \\ \hline \left(\frac{\partial Q_2}{\partial \delta_2}\right)^{(p)} & \cdots & \left(\frac{\partial Q_2}{\partial \delta_n}\right)^{(p)} & \left(\frac{\partial Q_2}{\partial |V_2|}\right)^{(p)} & \cdots & \left(\frac{\partial Q_2}{\partial |V_n|}\right)^{(p)} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \left(\frac{\partial Q_n}{\partial \delta_2}\right)^{(p)} & \cdots & \left(\frac{\partial Q_n}{\partial \delta_n}\right)^{(p)} & \left(\frac{\partial Q_n}{\partial |V_2|}\right)^{(p)} & \cdots & \left(\frac{\partial Q_n}{\partial |V_n|}\right)^{(p)} \end{array} \right] \begin{bmatrix} \Delta \delta_2^{(p)} \\ \vdots \\ \Delta \delta_n^{(p)} \\ \hline \Delta |V_2|^{(p)} \\ \vdots \\ \Delta |V_n|^{(p)} \end{bmatrix}\]

Short Jacobian matrix form

\[\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}\]

System Specifications

Total buses: \(n\)
Voltage-controlled buses (PV buses): \(m\)
PQ buses: \(n - m\)

Submatrix Dimensions

\(\Delta P\) size: \((n - 1) \times 1\) (P specified for \(n-1\) buses)
\(\Delta Q\) size: \((n - m - 1) \times 1\) (Q for PQ buses only)
\(J_1\) size: \((n - 1) \times (n - 1)\)
\(J_2\) size: \((n - 1) \times (n - m - 1)\)
\(J_3\) size: \((n - m - 1) \times (n - 1)\)
\(J_4\) size: \((n - m - 1) \times (n - m - 1)\)

Total Jacobian Size \(\boxed{(2n - m - 2) \times (2n - m - 2)}\)

For \(\mathbf{J}_1\) (P-\(\delta\) terms)

Off-diagonal: \(\dfrac{\partial P_i}{\partial \delta_n} = -|Y_{in}||V_i||V_n|\sin(\theta_{in} + \delta_n - \delta_i) \quad (i \neq n)\)
Diagonal: \(\dfrac{\partial P_i}{\partial \delta_i} = \sum\limits_{\substack{n=1 \\ n \neq i}}^N |Y_{in}||V_i||V_n|\sin(\theta_{in} + \delta_n - \delta_i)\)

For \(\mathbf{J}_2\) (\(P-|V|\) terms)

Off-diagonal: \(\dfrac{\partial P_i}{\partial |V_n|} = |Y_{in}||V_i|\cos(\theta_{in} + \delta_n - \delta_i) \quad (i \neq n)\)
Diagonal: \(\dfrac{\partial P_i}{\partial |V_i|} = 2|Y_{ii}||V_i|\cos\theta_{ii} + \sum\limits_{\substack{n=1 \\ n \neq i}}^N |Y_{in}||V_n|\cos(\theta_{in} + \delta_n - \delta_i)\)

For \(\mathbf{J}_3\) (Q-\(\delta\) terms)

Off-diagonal: \(\dfrac{\partial Q_i}{\partial \delta_n} = -|Y_{in}||V_i||V_n|\cos(\theta_{in} + \delta_n - \delta_i) \quad (i \neq n)\)
Diagonal: \(\dfrac{\partial Q_i}{\partial \delta_i} = \sum\limits_{\substack{n=1 \\ n \neq i}}^N |Y_{in}||V_i||V_n|\cos(\theta_{in} + \delta_n - \delta_i)\)

For \(\mathbf{J}_4\) (\(Q-|V|\) terms)

Off-diagonal: \(\dfrac{\partial Q_i}{\partial |V_n|} = -|Y_{in}||V_i|\sin(\theta_{in} + \delta_n - \delta_i) \quad (i \neq n)\)
Diagonal: \(\dfrac{\partial Q_i}{\partial |V_i|} = -2|V_i||Y_{ii}|\sin\theta_{ii} - \sum\limits_{\substack{n=1 \\ n \neq i}}^N |Y_{in}||V_n|\sin(\theta_{in} + \delta_n - \delta_i)\)

Newton-Raphson Method Procedure

Initialization
- PQ buses: \(|V_i^{(0)}|=1.0\), \(\delta_i^{(0)}=0.0\)
- PV buses: \(\delta_i^{(0)}=0.0\)
Power Mismatch
- PQ: Compute \(\Delta P_i^{(k)}\), \(\Delta Q_i^{(k)}\)
- PV: Compute \(\Delta P_i^{(k)}\)
Jacobian Matrix
- Calculate \(J_1\)-\(J_4\) elements
Solve Linear System
- \(J\cdot\Delta x = \Delta S\) via factorization
Update Variables
- New \(|V_i^{(k+1)}|\), \(\delta_i^{(k+1)}\)
Convergence Check
- Repeat until \(\max(|\Delta P|,|\Delta Q|) \leq \epsilon\)

Problem

Obtain the power flow solution by the Newton-Raphson method for the system shown by the SLD (impedances in pu on 100-MVA base).

Solution

Admitance Matrix - Rectangular Form

\[Y_{bus} = \begin{bmatrix} 20 - j50 & -10 + j20 & -10 + j30 \\ -10 + j20 & 26 - j52 & -16 + j32 \\ -10 + j30 & -16 + j32 & 26 - j62 \end{bmatrix}\]

Admitance Matrix - Polar Form

\[= \begin{bmatrix} 53.85165 \angle -1.9029 & 22.36068 \angle 2.0344 & 31.62278 \angle 1.8925 \\ 22.36068 \angle 2.0344 & 58.13777 \angle -1.1071 & 35.77709 \angle 2.0344 \\ 31.62278 \angle 1.8925 & 35.77709 \angle 2.0344 & 67.23095 \angle -1.1737 \end{bmatrix}\]

The power flow equations for buses 2 and 3:

\[\begin{aligned} P_2 &= |V_2||V_1||Y_{21}| \cos(\theta_{21} - \delta_2 + \delta_1) + |V_2|^2|Y_{22}| \cos\theta_{22} \\ &\quad + |V_2||V_3||Y_{23}| \cos(\theta_{23} - \delta_2 + \delta_3) \\ P_3 &= |V_3||V_1||Y_{31}| \cos(\theta_{31} - \delta_3 + \delta_1) + |V_3||V_2||Y_{32}| \cos(\theta_{32} - \delta_3 + \delta_2) \\ &\quad + |V_3|^2|Y_{33}| \cos\theta_{33} \\ Q_2 &= -|V_2||V_1||Y_{21}| \sin(\theta_{21} - \delta_2 + \delta_1) - |V_2|^2|Y_{22}| \sin\theta_{22} \\ &\quad - |V_2||V_3||Y_{23}| \sin(\theta_{23} - \delta_2 + \delta_3) \end{aligned}\]

Newton-Raphson power flow Jacobian matrix:

\[\begin{aligned} \frac{\partial P_2}{\partial \delta_2} &= |V_2||V_1||Y_{21}| \sin(\theta_{21} - \delta_2 + \delta_1) + |V_2||V_3||Y_{23}| \sin(\theta_{23} - \delta_2 + \delta_3) \\ \frac{\partial P_2}{\partial \delta_3} &= -|V_2||V_3||Y_{23}| \sin(\theta_{23} - \delta_2 + \delta_3) \\ \frac{\partial P_2}{\partial |V_2|} &= |V_1||Y_{21}| \cos(\theta_{21} - \delta_2 + \delta_1) + 2|V_2||Y_{22}| \cos\theta_{22} + |V_3||Y_{23}| \cos(\theta_{23} - \delta_2 + \delta_3) \\ \frac{\partial P_3}{\partial \delta_2} &= -|V_3||V_2||Y_{32}| \sin(\theta_{32} - \delta_3 + \delta_2) \\ \frac{\partial P_3}{\partial \delta_3} &= |V_3||V_1||Y_{31}| \sin(\theta_{31} - \delta_3 + \delta_1) + |V_3||V_2||Y_{32}| \sin(\theta_{32} - \delta_3 + \delta_2) \\ \frac{\partial P_3}{\partial |V_2|} &= |V_3||Y_{32}| \cos(\theta_{32} - \delta_3 + \delta_2) \\ \frac{\partial Q_2}{\partial \delta_2} &= |V_2||V_1||Y_{21}| \cos(\theta_{21} - \delta_2 + \delta_1) + |V_2||V_3||Y_{23}| \cos(\theta_{23} - \delta_2 + \delta_3) \\ \frac{\partial Q_2}{\partial \delta_3} &= -|V_2||V_3||Y_{23}| \cos(\theta_{23} - \delta_2 + \delta_3) \\ \frac{\partial Q_2}{\partial |V_2|} &= -|V_1||Y_{21}| \sin(\theta_{21} - \delta_2 + \delta_1) - 2|V_2||Y_{22}| \sin\theta_{22} - |V_3||Y_{23}| \sin(\theta_{23} - \delta_2 + \delta_3) \end{aligned}\]

Initial System Conditions

Scheduled power in per unit:

\[S_{2}^{sch} = -\frac{400 + j250}{100} = -4.0 - j2.5 \text{ pu}\]

\[P_{3}^{sch} = \frac{200}{100} = 2.0 \text{ pu}\]
Voltage specifications:

\[\begin{aligned} V_1 &= 1.05\angle 0 \text{ pu (Slack bus)} \\ |V_3| &= 1.04 \text{ pu (PV bus)} \end{aligned}\]
Initial estimates:

\[|V_2^{(0)}| = 1.0,\ \delta_2^{(0)} = 0.0,\ \delta_3^{(0)} = 0.0\]

First Iteration Calculations

\[\begin{aligned} \Delta P^{(0)}_{2} &= P^{\text{sch}}_{2} - P^{(0)}_{2} = -4.0 - (-1.14) = -2.8600 \\ \Delta P^{(0)}_{3} &= P^{\text{sch}}_{3} - P^{(0)}_{3} = 2.0 - (0.5616) = 1.4384 \\ \Delta Q^{(0)}_{2} &= Q^{\text{sch}}_{2} - Q^{(0)}_{2} = -2.5 - (-2.28) = -0.2200 \end{aligned}\]

Power Residuals

\[\begin{bmatrix} -2.8600 \\ 1.4384 \\ -0.2200 \end{bmatrix} = \begin{bmatrix} 54.28000 & -33.28000 & 24.86000 \\ -33.28000 & 66.04000 & -16.64000 \\ -27.14000 & 16.64000 & 49.72000 \end{bmatrix} \begin{bmatrix} \Delta \delta_2^{(0)} \\ \Delta \delta_3^{(0)} \\ \Delta |V_2^{(0)}| \end{bmatrix}\]

Jacobian Matrix Equation

\[\begin{aligned} \Delta\delta^{(0)}_{2} &= -0.045263 & \delta^{(1)}_{2} &= 0 + (-0.045263) = -0.045263 \\ \Delta\delta^{(0)}_{3} &= -0.007718 & \delta^{(1)}_{3} &= 0 + (-0.007718) = -0.007718 \\ \Delta|V^{(0)}_{2}| &= -0.026548 & |V^{(1)}_{2}| &= 1 + (-0.026548) = 0.97345 \end{aligned}\]

Voltage Updates

Second Iteration

\[\begin{bmatrix} -0.099218 \\ 0.021715 \\ -0.050914 \end{bmatrix} = \begin{bmatrix} 51.724675 & -31.765618 & 21.302567 \\ -32.981642 & 65.656383 & -15.379086 \\ -28.538577 & 17.402838 & 48.103589 \end{bmatrix} \begin{bmatrix} \Delta \delta_2^{(1)} \\ \Delta \delta_3^{(1)} \\ \Delta |V_2^{(1)}| \end{bmatrix}\]

Updated Residuals and Jacobian

\[\begin{aligned} \Delta\delta^{(1)}_2 &= -0.001795 & \delta^{(2)}_2 &= -0.045263 + (-0.001795) = -0.04706\,\text{rad} \\ \Delta\delta^{(1)}_3 &= -0.000985 & \delta^{(2)}_3 &= -0.007718 + (-0.000985) = -0.00870\,\text{rad} \\ \Delta|V^{(1)}_2| &= -0.001767 & |V^{(2)}_2| &= 0.973451 + (-0.001767) = 0.971684\,\text{pu} \end{aligned}\]

Voltage Updates

Third Iteration Setup

\[\begin{bmatrix} -0.000216 \\ 0.000038 \\ -0.000143 \end{bmatrix} = \begin{bmatrix} 51.596701 & -31.693866 & 21.147447 \\ -32.933865 & 65.597585 & -15.351628 \\ -28.548205 & 17.396932 & 47.954870 \end{bmatrix} \begin{bmatrix} \Delta \delta_2^{(2)} \\ \Delta \delta_3^{(2)} \\ \Delta |V_2^{(2)}| \end{bmatrix}\]

Jacobian Matrix

\[\begin{aligned} \Delta\delta^{(2)}_2 &= -0.000038 & \delta^{(3)}_2 &= -0.047058 + (-0.000038) = -0.047096\,\text{rad} \\ \Delta\delta^{(2)}_3 &= -0.0000024 & \delta^{(3)}_3 &= -0.008703 + (-0.0000024) = -0.008705\,\text{rad} \\ \Delta|V^{(2)}_2| &= -0.0000044 & |V^{(3)}_2| &= 0.971684 + (-0.0000044) = 0.971680\,\text{pu} \end{aligned}\]

Final Voltage Updates

Converged Solution

\[\begin{aligned} V_{2} &= 0.97168\angle -2.696^\circ \text{ pu} \\ V_{3} &= 1.04\angle -0.4988^\circ \text{ pu} \\ \text{Max mismatch} &= 2.5\times 10^{-4} \text{ pu} \end{aligned}\]

Final Bus Voltages

Convergence Achieved

Solution obtained in just 3 iterations
Excellent numerical stability
Meets typical engineering accuracy requirements

Slack Bus and PV Bus Power Calculations

Reactive Power at Bus 3

\[\begin{aligned} Q_{3} & = -|V_{3}||V_{1}||Y_{31}|\sin(\theta_{31}-\delta_{3}+\delta_{1}) -|V_{3}||V_{2}||Y_{32}|\sin(\theta_{32}-\delta_{3}+\delta_{2}) -|V_{3}|^{2}|Y_{33}|\sin\theta_{33} \\ &= 1.4617\,\text{pu} \end{aligned}\]

Slack Bus Power

\[\begin{aligned} P_{1} &= |V_{1}|^{2}|Y_{11}|\cos\theta_{11} + |V_{1}||V_{2}||Y_{12}|\cos(\theta_{12}-\delta_{1}+\delta_{2}) + |V_{1}||V_{3}||Y_{13}|\cos(\theta_{13}-\delta_{1}+\delta_{3}) \\ &= 2.1842\,\text{pu} \\[5pt] Q_{1} &= -|V_{1}|^{2}|Y_{11}|\sin\theta_{11} - |V_{1}||V_{2}||Y_{12}|\sin(\theta_{12}-\delta_{1}+\delta_{2}) - |V_{1}||V_{3}||Y_{13}|\sin(\theta_{13}-\delta_{1}+\delta_{3}) \\ &= 1.4085\,\text{pu} \end{aligned}\]

Solution Summary

Key Results

\(V_2\): 0.97168 pu \(\angle\) -2.696°
\(V_3\): 1.04 pu \(\angle\) -0.4988°
\(Q_3\): 1.4617 pu
\(P_1\): 2.1842 pu
\(Q_1\): 1.4085 pu

Performance

3 iterations to converge
Robust numerical method
Suitable for large systems

Next Steps

Line flow calculations
Loss computation
Voltage stability analysis