Question

Using a 100 MVA base, each of the three transmission lines has an impedance of 0.05 + j0.1 pu. There is a single 180 MW load at bus 3, while bus 2 is a PV bus with generation of 80 MW and a voltage setpoint of 1.0 pu. Bus 1 is the system slack with a voltage setpoint of 1.0 pu. Manually solve this case using the Newton-Raphson approach with a convergence criteria of 0.1 MVA. Show all your work. Then verify your solution by solving the case with PowerWorld Simulator.

Bus 1
Bus 3
180 MW
Bus 2
80 MW

Answer 1

The Newton-Raphson method is used for power flow analysis in electrical power systems by iteratively correcting voltage guesses at PQ buses until the power mismatch is sufficiently small. This iterative process involves updating voltages based on the inverse of the Jacobian matrix. The manual solution can be verified using software like PowerWorld Simulator.

Step-by-step explanation:

The Newton-Raphson method is a powerful algorithm used to find roots of real-valued functions and is particularly useful in power flow analysis for electrical power systems. In the context of solving for bus voltages in a power system, the method iteratively updates the voltage guess until the power mismatch, which is the difference between the actual power flow and the desired power flow, is below the specified tolerance. The initial step is to set up the power mismatch equations for each bus without a specified voltage. These buses are typically referred to as PQ buses. The changes in voltage are related to the changes in power by the Jacobian matrix. This matrix is comprised of derivatives of the active and reactive power equations with respect to voltage magnitude and angle.

For an example calculation, let's assume we have one PQ bus (Bus 3) with a known load and one PV bus (Bus 2) with specified voltage and active power. The slack bus (Bus 1) has a specified voltage but an unknown power. The given data indicates we are working with a 100 MVA base and each line's impedance is 0.05 + j0.1 pu. Using the Newton-Raphson method, we would set up our power mismatch equations based on the active and reactive power injections at the buses that do not have both specified and do our first iteration using a flat start (e.g., initial voltage estimates of 1.0 pu for all buses). We continually update the voltages using the inverse of the Jacobian matrix until the power mismatch at Bus 3 is below the 0.1 MVA threshold.

To verify with PowerWorld Simulator, the model would be set up with the given generation and load data, and the simulator would solve using its version of the power flow algorithm, which may also utilize the Newton-Raphson method among others. The convergence criteria would be set to 0.1 MVA, and the solution would be compared to the manually calculated result to ensure consistency.