Power Systems
Comprehensive guide to electrical power systems, generation, transmission, and distribution.
Three-Phase Systems
Three-phase systems are the most common method for electric power transmission and distribution. In a balanced three-phase system:
V_L = √3 × V_P
I_L = I_P (for Y-connection)
I_L = I_P / √3 (for Δ-connection)
Where:
- V_L = line voltage (V)
- V_P = phase voltage (V)
- I_L = line current (A)
- I_P = phase current (A)
The total power in a balanced three-phase system is:
P = √3 × V_L × I_L × cos(φ)
Q = √3 × V_L × I_L × sin(φ)
S = √3 × V_L × I_L
Where:
- P = active power (W)
- Q = reactive power (VAR)
- S = apparent power (VA)
- cos(φ) = power factor
Power Flow Analysis
Power flow analysis is used to determine the steady-state operating condition of a power system. The power flow equations for a bus i are:
P_i = |V_i| × ∑|V_j| × |Y_ij| × cos(θ_ij + δ_j - δ_i)
Q_i = |V_i| × ∑|V_j| × |Y_ij| × sin(θ_ij + δ_j - δ_i)
Where:
- P_i, Q_i = active and reactive power at bus i
- |V_i|, |V_j| = voltage magnitudes at buses i and j
- |Y_ij| = magnitude of the (i,j) element of the bus admittance matrix
- θ_ij = angle of the (i,j) element of the bus admittance matrix
- δ_i, δ_j = voltage angles at buses i and j
Power flow studies typically use iterative methods such as Newton-Raphson or Gauss-Seidel to solve these nonlinear equations.
Fault Analysis
Fault analysis is used to determine the currents and voltages in a power system during fault conditions. The symmetrical components method is commonly used for analyzing unbalanced faults:
[V_0, V_1, V_2]ᵀ = [A] × [V_a, V_b, V_c]ᵀ
[I_0, I_1, I_2]ᵀ = [A] × [I_a, I_b, I_c]ᵀ
Where:
- V_0, V_1, V_2 = zero, positive, and negative sequence voltages
- I_0, I_1, I_2 = zero, positive, and negative sequence currents
- V_a, V_b, V_c = phase voltages
- I_a, I_b, I_c = phase currents
- [A] = transformation matrix
For a three-phase balanced system, the transformation matrix is:
[A] = [1, 1, 1; 1, a, a²; 1, a², a]
Where a = e^(j2π/3) is a complex operator representing a 120° phase shift.