Introduction
1.4 Definitions of power system stability
“Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilib- rium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact.”
[24].
This definition is intended to apply to an interconnected system in its entirety, however, it also includes the instability and timely disconnection of an element such as a generator with- out the system itself becoming unstable.
There are three main categories of power system stability: rotor-angle stability, voltage sta- bility and frequency stability. Ensuring stability in all its categories is the primary focus of the design of power system controllers. Such controllers are designed to ensure, through the specifications on the dynamic performance of the system, that adequate margins of stable operation are attained over a range of normal operating conditions and contingencies.
Pe t
8 Introduction Ch. 1 In the following chapters we are concerned with rotor-angle stability which is defined as fol- lows [24]:
“Rotor angle stability refers to the ability of synchronous machines of an interconnected power system to remain in synchronism after being subjected to a disturbance. It depends on the ability to maintain/restore equilibrium between electromagnetic torque and mechanical torque of each synchronous machine in the system. Instability that may result oc- curs in the form of increasing angular swings of some generators leading to their loss of synchronism with other generators.”
Rotor-angle stability, and oscillations of the rotors of synchronous generators, are essentially governed by the equations of motion of the unit; the relevant versions of the equations are derived in Chapter 4. In terms of the per unit rotor speed , synchronous speed and the per unit prime-mover torque and the torque of electromagnetic origin, , the equations of motion are given by (4.58) and (4.59) which are repeated below:
and .
.H is the inertia constant (MWs/MVA) of the generating unit and D (pu torque/pu speed) is the damping torque coefficient. From the above equation it is apparent that a steady-state condition exists when the torques are in balance and thus there is no change in rotor angle or in speed about synchronous speed. However, a disturbance on an element of the electrical system will result in an imbalance in the torques and cause the rotor to accelerate or decel- erate, in turn causing the rotor angle to increase or decrease. The shaft equation is linear so it is applicable to large and small disturbances. For large disturbances the term ‘transient sta- bility’ is defined as follows [24]:
Large-disturbance rotor angle stability or transient stability, as it is commonly referred to, is concerned with the ability of the power sys- tem to maintain synchronism when subjected to a severe disturbance, such as a short circuit on a transmission line.
On the other hand, rotor-angle stability for small disturbances is defined as [24]:
Small-disturbance (or small-signal) rotor-angle stability is con- cerned with the ability of the power system to maintain synchronism under small disturbances. The disturbances are considered to be suffi- ciently small that linearization of system equations is permissible for purposes of analysis.
0
Tm and Tg
p = b – 0
p 1
2H
---Tm–Tg–D – 0
=
Sec. 1.4 Definitions of power system stability 9 Associated with transient stability are severe events or disturbances such as system faults, the opening of a faulted line - or a heavily loaded circuit, the tripping of a large generator, the loss of a large load. As indicated in Section 1.1, for transient stability analysis the dynamic behaviour of certain devices are modelled by their non-linear differential and algebraic equa- tions. The presence of the various types of non-linearities in the equations results in tran- sient stability analysis, in practice, being conducted by simulation studies in the time domain.
The basis of such analyses is a power flow study, an equilibrium or steady-state operating condition to which the relevant disturbance is applied.
Stable, large-disturbance performance of a multi-machine power system depends on ade- quate synchronizing power flows being established between synchronous generators to pre- vent loss of synchronism of any generator on the system. High gain excitation systems are employed to increase synchronizing power flows and torques. The decay of oscillations, not only following the initial transient (usually the first swing) but also following cessation of limiting action by controllers, is dependent on the development of damping torques of an electro-magnetic origin acting on the generator rotors. Damping torques may be degraded significantly by high gain excitation systems such that, if the net damping torque is negative, instability occurs. To counter this type of instability, positive damping torques can be in- duced on generators by installing continuously-acting controllers known as stabilizers.
Small-disturbance or small-signal rotor-angle stability is associated with disturbances such as the more-or-less continuous switching on and off of relatively small loads. The anal- ysis of small-signal rotor-angle stability is conducted for a selected steady-state operating condition about which the non-linear differential and algebraic equations and other non-lin- earities are linearized. This process produces a set of equations in a new set of variables, the perturbed variables. Important features of small-signal analysis are: (i) as shown by Poincaré, information on the stability of the non-linear model at the selected operating condition, based on the stability of the linearized system, is exact; and (ii) all the powerful tools and techniques in linear control system analysis are available for the design and analysis of dy- namic performance. The design of power system stabilizers for inducing damping torques under normal and post-contingency conditions is conducted using such facilities.
Two forms of spontaneous small-signal instability may be: (i) a steady increase in rotor angle due to inadequate synchronizing torque, or (ii) rotor oscillations of increasing amplitude due to insufficient damping torque. Most generally in practice, however, the latter is of concern in small-signal rotor-angle stability analysis.
While we are mainly interested in small-signal rotor-angle stability in the following chapters, the definitions of voltage and frequency stability are quoted from [24] for information and for the sake of completeness.
Voltage stability refers to the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a dis-
10 Introduction Ch. 1 turbance from a given initial operating condition. It depends on the abil-
ity to maintain/restore equilibrium between load demand and load supply from the power system. Instability that may result occurs in the form of a progressive fall or rise of voltages of some buses.
(In system planning studies it is occasionally found that analyses which suggested rotor angle instability are, in fact, associated with voltage instability; at times, it may be difficult to dis- criminate between them.)
Frequency stability refers to the ability of a power system to maintain steady frequency following a severe system upset resulting in a signifi- cant imbalance between generation and load. It depends on the ability to maintain/restore equilibrium between system generation and load, with minimum unintentional loss of load. Instability that may result oc- curs in the form of sustained frequency swings leading to tripping of generating units and/or loads.