9.6 Power System Analysis

9.6 Power System Analysis

Power system analysis involves mathematical modeling and computational techniques to ensure the power system operates safely, reliably, and economically under normal and abnormal (fault) conditions.

1. Fault Analysis

Fault analysis is the study of abnormal conditions (short-circuits) to determine the magnitudes of fault currents and voltages, which is essential for designing protection systems and selecting equipment ratings.

1.1 Symmetrical (Balanced) Faults

• Definition: A three-phase fault where all three phases are short-circuited to each other (and possibly to ground) symmetrically. The system remains balanced.

• Characteristics:

  • Most severe in terms of fault current magnitude (highest).

  • Simplest to analyze because the system is balanced. Can be analyzed using a single-phase equivalent circuit (per-phase analysis).

• Analysis Steps:

  1. Draw the single-line diagram of the system.

  2. Represent all components by their per-unit (p.u.) impedance on a common MVA and kV base.

  3. Reduce the network to a single Thevenin equivalent impedance (Z_th) as seen from the fault point.

  4. The symmetrical (steady-state) fault current is given by: $I_f = \frac{V_{th}}{Z_{th}}$ Where $V_{th}$ is the pre-fault voltage at the fault point (usually 1.0 p.u.).

  5. Account for DC offset (transient component) to find the asymmetrical fault current (first cycle current), which is higher due to the unidirectional DC component.

• Application: Used for sizing circuit breaker interrupting capacity, busbars, and other equipment that must withstand the highest possible thermal and mechanical stresses.

1.2 Unsymmetrical (Unbalanced) Faults

• Definition: Faults where the symmetry of the three-phase system is broken (e.g., single line-to-ground, line-to-line, double line-to-ground). These are more common than three-phase faults.

• Analysis Tool: Symmetrical Components Method (developed by C.L. Fortescue).

• Symmetrical Components Theory:

  • Any set of unbalanced three-phase phasors can be resolved into three balanced sets:

    1. Positive Sequence Components (1): Three phasors equal in magnitude, 120° apart, with the same phase sequence (ABC) as the original system.

    2. Negative Sequence Components (2): Three phasors equal in magnitude, 120° apart, but with the opposite phase sequence (ACB).

    3. Zero Sequence Components (0): Three phasors equal in magnitude and in phase with each other.

  • Mathematical transformation: $\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{bmatrix} \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix}$ Where $a = 1 \angle 120^\circ$ is the complex operator.

• Sequence Networks:

  • Each power system component (generator, line, transformer) has distinct impedances for positive ($Z_1$), negative ($Z_2$), and zero ($Z_0$) sequence currents.

  • For fault analysis, we construct three separate networks: Positive, Negative, and Zero Sequence Networks.

• Fault Type & Connection of Sequence Networks:

Fault Type	Interconnection of Sequence Networks at Fault Point (F)	Fault Current (p.u.)
Single Line-to-Ground (L-G)	All three networks connected in series.	$I_f = 3I_0 = \frac{3V_f}{Z_1+Z_2+Z_0+3Z_f}$ ($Z_f$=fault impedance)
Line-to-Line (L-L)	Positive and Negative networks connected in parallel (no Zero network).	$I_f = I_b = -I_c = \frac{\sqrt{3} V_f}{Z_1+Z_2+Z_f}$
Double Line-to-Ground (L-L-G)	Positive, Negative, and Zero networks connected in parallel.	$I_f = I_b + I_c = 3I_0 = \frac{3V_f (Z_2 // Z_0)}{Z_1(Z_2+Z_0)+Z_2Z_0}$ (approx.)

• Application: Used for setting protective relays (especially ground fault relays), and analyzing system behavior during common unbalanced conditions.

2. Load Flow (Power Flow) Analysis

• Definition: The numerical analysis of a power system in steady-state normal operation to determine:

  • Voltage magnitudes and angles at all buses (nodes).

  • Real ($P$) and Reactive ($Q$) power flows in all lines and transformers.

  • System losses.

• Purpose: For planning, operation, and optimization. Ensures voltages are within limits, equipment is not overloaded, and the system operates economically.

• Types of Buses (Nodes):

  1. Load Bus (PQ Bus): $P$ and $Q$ are specified (known). Voltage magnitude ($V$) and angle ($\delta$) are unknown.

  2. Generator Bus (PV Bus): $P$ and $V$ are specified. $Q$ and $\delta$ are unknown.

  3. Slack (Swing) Bus: $V$ and $\delta$ (usually 0°) are specified. $P$ and $Q$ are unknown. It balances the real and reactive power in the system (makes up the losses).

• Power Flow Equations (for a bus k connected to bus m): The net complex power injection at bus k is: $S_k = P_k + jQ_k = V_k I_k^* = V_k \sum_{m=1}^{N} Y_{km}^* V_m^*$ This leads to the fundamental power flow equations: $P_k = \sum_{m=1}^{N} |V_k||V_m||Y_{km}| \cos(\theta_{km} + \delta_m - \delta_k)$ $Q_k = -\sum_{m=1}^{N} |V_k||V_m||Y_{km}| \sin(\theta_{km} + \delta_m - \delta_k)$ Where $Y_{km} = G_{km} + jB_{km}$ is the element of the bus admittance matrix (Y-bus).

• Solution Methods (Iterative):

  1. Gauss-Seidel Method: Simple but slow convergence. Good for small systems.

  2. Newton-Raphson Method: Uses partial derivatives (Jacobian matrix). Fast convergence (quadratic), robust. Most widely used for large systems. $\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}$

  3. Fast Decoupled Load Flow: A simplification of Newton-Raphson that assumes real power flow depends mainly on voltage angles, and reactive power flow depends mainly on voltage magnitudes. Very fast and used for on-line control.

3. Power System Stability

The ability of a power system to maintain synchronism (all generators running at the same electrical speed/frequency) when subjected to a disturbance.

3.1 Steady-State Stability

• Definition: The ability of the system to maintain synchronism under slow, gradual, small changes in load or generation (e.g., daily load variations).

• Criterion: Damping torque > Synchronizing torque for small oscillations.

• Analysis: Uses linearized models and eigenvalue analysis.

• Power-Angle Relationship: The foundation of stability analysis is the power transfer equation for a simple two-machine system: $P = \frac{E V}{X} \sin \delta = P_{max} \sin \delta$ Where $P_{max} = EV/X$ is the steady-state stability limit, and $\delta$ is the power angle.

• Enhancement: Use of Automatic Voltage Regulators (AVR) and Power System Stabilizers (PSS).

3.2 Transient Stability

• Definition: The ability of the system to maintain synchronism when subjected to a large, sudden disturbance (e.g., a fault, loss of a large generator, line switching). The system's response involves large rotor angle swings.

• Analysis Method: Equal Area Criterion (for a simple system) and Numerical Integration of Swing Equations (for multi-machine systems).

• Swing Equation: Describes the motion of a generator rotor. $M \frac{d^2 \delta}{dt^2} = P_m - P_e = P_a$ Where $M$ is inertia constant, $P_m$ is mechanical input power, $P_e$ is electrical output power, $P_a$ is accelerating power.

• Equal Area Criterion: For stability, the decelerating area (A2) must be at least equal to the accelerating area (A1) on the power-angle (P-δ) curve.

  • A1: Area under ($P_m - P_e$) during fault (accelerating).

  • A2: Area under ($P_e - P_m$) after fault clearing (decelerating).

  • Stability Condition: $A_2 \geq A_1$.

• Factors Improving Transient Stability:

  • Fast fault clearing (reduces accelerating area A1).

  • High speed reclosing.

  • Use of series capacitors (reduces line reactance X, increases $P_{max}$).

  • Braking resistors.

  • Generator tripping.

3.3 Dynamic Stability

• Definition: Concerned with small-signal oscillations (0.2-3 Hz) that persist for several seconds to minutes after a disturbance, often due to interaction between generator controls (AVR, governor) and the network.

• Analysis: Involves detailed modeling of control systems and uses linearized models with eigenvalue/participation factor analysis to identify poorly damped oscillation modes.

• Solution: Installation of Power System Stabilizers (PSS) on generator AVRs. The PSS adds a damping torque signal proportional to rotor speed deviation to suppress oscillations.

Stability Type	Disturbance Size	Time Frame	Key Analysis Tool	Main Concern
Steady-State	Very Small, Slow	Seconds to Minutes	Linear Models, Eigenvalues	Damping of small oscillations
Transient	Large, Sudden	First Swing (0-2s)	Equal Area Criterion, Time Simulation	First swing rotor angle stability
Dynamic	Small, Following	Several Seconds	Linear Models, Eigenvalues	Damping of low-frequency oscillations

Conclusion: Power system analysis uses a combination of circuit theory, network mathematics, and dynamic system modeling. Fault analysis ensures protection design, load flow ensures operational feasibility, and stability analysis ensures the system can survive disturbances without collapse. These analyses are interdependent and critical for secure grid operation.