Understanding Voltage Stability in Power Systems: Large vs. Small Disturbance and Stability Limits

Definition of Voltage Stability

Voltage stability in a power system is defined as the ability to maintain acceptable voltages at all buses under both normal operating conditions and after being subjected to a disturbance. In normal operation, the system’s voltages remain stable; however, when a fault or disturbance occurs, voltage instability may arise, leading to a progressive and uncontrollable voltage decline. Voltage stability is sometimes referred to as "load stability."

Voltage instability can trigger voltage collapse if the post-disturbance equilibrium voltage near loads falls below acceptable limits. Voltage collapse is a process in which voltage instability results in an extremely low voltage profile across critical parts of the system, potentially causing a total or partial blackout. Notably, the terms "voltage instability" and "voltage collapse" are often used interchangeably.

Classification of Voltage Stability

Voltage stability is categorized into two main types:

• Large-Disturbance Voltage Stability：This refers to the system’s ability to maintain voltage control following significant disturbances, such as system faults, sudden load loss, or generation loss. Assessing this form of stability requires analyzing the system’s dynamic performance over a timeframe long enough to account for the behavior of devices like on-load tap-changing transformers, generator field controls, and current limiters. Large-disturbance voltage stability is typically studied using nonlinear time-domain simulations with accurate system modeling.

• Small-Disturbance Voltage Stability：A power system operating state exhibits small-disturbance voltage stability if, after minor disturbances, voltages near loads either remain unchanged or stay close to their pre-disturbance values. This concept is closely linked to steady-state conditions and can be analyzed using small-signal system models.

Voltage Stability Limit

The voltage stability limit is the critical threshold in a power system beyond which no amount of reactive power injection can restore voltages to their nominal levels. Up to this limit, system voltages can be adjusted through reactive power injections while maintaining stability.The power transfer over a lossless line is given by:

• where P = power transferred per phase

• Vs = sending-end phase voltage

• Vr = receiving-end phase voltage

• X = transfer reactance per phase

• δ = phase angle between Vs and Vr.

Since the Line is lossless

Assuming the power generation to be constant,

For maximum power transfer:δ = 90º, so that as δ→∞

The above equation determines the position of the critical point on the curve of δ versus Vs, with the assumption that the receiving - end voltage remains constant.A similar result can be derived by assuming the sending - end voltage to be constant and treating Vr as a variable parameter when analyzing the system. In this scenario, the resulting equation is

The reactive power expression at the receiving-end bus may be written as

The above equation represents the steady - state voltage stability limit. It indicates that, at the steady - state stability limit, the reactive power approaches infinity. This implies that the derivative dQ/dVr becomes zero. Thus, the rotor angle stability limit under steady - state conditions coincides with the steady - state voltage stability limit. Additionally, the steady - state voltage stability is also influenced by the load.