Newton-Raphson Power Flow Analysis Quiz

Power System Dynamics and Stability

• The power system dynamics and stability analysis is performed on a two-bus system with a slack bus (Bus 1) and a load bus (Bus 2).

Two Bus Newton-Raphson Example

• The two-bus power system has a line impedance of Z = 0.1j and a base power of SBase = 100 MVA.
• Bus 1 has a voltage magnitude and angle of 1.000 pu, and a real power and reactive power of 200 MW and 100 MVR, respectively.
• The unknown variables are the voltage magnitude and angle at Bus 2, which are represented by x = [θ2, V2].

Power Balance Equations

• The general power balance equations for the two-bus system are:
• Real power balance: ∑n Vk (Gik cosθik + Bik sinθik) - PGi + PDi = 0
• Reactive power balance: ∑n Vk (Gik sinθik - Bik cosθik) - QGi + QDi = 0
• For Bus 2, the power balance equations are:
• Real power balance: V2 V1 (10sinθ2) + 2.0 = 0
• Reactive power balance: V2 V1 (-10cosθ2) + V2 (10) + 1.0 = 0

Power Flow Analysis

• The power flow equations are rewritten in terms of the unknown variables x = [θ2, V2]:
• P2(x) + 2.0 = V2 (10sinθ2) + 2.0
• Q2(x) + 1.0 = V2 (-10cosθ2) + V2 (10) + 1.0
• The power flow Jacobian is calculated as:
• J(x) = [[∂P2/∂θ2, ∂P2/∂V2], [∂Q2/∂θ2, ∂Q2/∂V2]]

Jacobian Matrix

• The Jacobian matrix is:
• J(x) = [[10 V2 cosθ2, 10 V2 sinθ2], [10 V2 sinθ2, -10 V2 cosθ2 + 20 V2]]

First Iteration

• The initial guess for x(0) is [θ2(0), V2(0)] = [0, 1].
• The power flow equations are calculated using the initial guess:
• V2(0) (10sinθ2(0)) + 2.0 = ?
• V2(0) (-10cosθ2(0)) + V2(0) (10) + 1.0 = ?