# Problems With Power System Instabilities Engineering Essay

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Power system instabilities occur in a number of different forms and depend on a variety of factors such as the size of the disturbance, the time span required to assess stability, and the physical nature of the instability [3]. The classification of stability problems into appropriate categories greatly facilitates identification and analysis. A widely used classification system developed by Kundur [4] distinguishes between voltage stability, angle stability and frequency stability phenomena, depending on the main system variable in which the instability can be observed. These three categories are further classified into long term and short term stability. A time scale of several minutes is regarded as long term; study on this time scale typically focuses slower acting devices such as thermostatically controlled loads and tap-changing transformers. Short term covers time scales of a few seconds or less and focuses on faster acting components such as induction motors, generator excitation systems and electronically controlled loads[2].

With reference to the scale of the disturbance, it is usual to distinguish between small disturbance and large disturbance stability. Small disturbances refer to the constant small perturbations that occur during normal system operations. For these small disturbances, it is considered acceptable to linearise the system equations for the purposes of analysis. Large disturbance stability is concerned with the system's ability to return to equilibrium following a large upset, such as a fault on a transmission line or the loss of a generator. The system response to such disturbances often includes large deviations, thus it is necessary to use non linear techniques for analysis [5].

Figure 1: Stability classification system [4]

1.1.1 Rotor angle stability

Rotor angle stability refers to system's ability to maintain synchronism following a disturbance. In steady state conditions, there is equilibrium between input mechanical torque and output electrical torque for all machines, and the system runs at synchronous speed. Disturbance to the system can cause machines to accelerate or decelerate, resulting in rotor angle deviations and in severe cases, loss of synchronism. Loss of synchronism can occur between one machine and the rest of the system or between groups of machines.

Large disturbance angle stability is commonly referred to as transient stability [5]. Significant disturbances to the system are most commonly in the form of a short circuit in one or more elements. The most severe disturbance to a system would occur with the loss of a synchronous generator or high voltage induction motor following a three phase fault.

Small disturbance rotor angle stability can involve either rotor oscillations due to insufficient damping torque or a non-oscillatory increase in rotor angle due to lack of synchronizing torque. In practical systems, small signal rotor instability is generally associated with insufficiently damped oscillations.

1.1.2 Frequency stability

Frequency stability is concerned with the system's ability to maintain a steady frequency within the nominal range. Frequency stability is dependant on maintaining the proper balance between active power generation and active power load demand. A severe upset to the system can cause an imbalance between generation and load; if the system is frequency stable with regards to the given disturbance, it will be able to maintain steady frequency with minimum unintentional loss of load[6]. Frequency instability can result in large deviations from nominal frequency and may possible in tripping of generators or loads.

1.1.3 Voltage stability

Voltage stability refers to the ability of the system to maintain acceptable voltage levels at all buses when subjected to disturbance. The instability commonly occurs in the form of a progressive and uncontrollable drop in voltage, although cases of instability due to overvoltage are possible [1].

Reactive power demand tends to increase following disturbance as power consumption by loads is restored by the action of motor slip adjustment, tap changing transformer and voltage regulators [5]. Voltage instability can occur when the increased reactive power demands exceeds reactive power supply capacity. Possible outcomes of large voltage deviations are loss of load in an area or tripping of transmission lines.

1.2 Transient (Rotor angle) Stability

In order to outline the principles of transient rotor angle stability, it useful to consider the simple case of a two machine system, as shown below. Each machine is represented by an internal voltage behind an effective reactance.

(a)

(b)

Figure 2 : Two synchronous machines connected by a transmission line: (a) single line diagram and (b) idealized model

1.2.1 The power angle relationship

In figure 2, the power transferred from the generator to the motor is a function of the rotor angle separation δ between the two machines. The real power output from the generator on a steady state basis can be given by:

( 1 )

where XT= XG+ XL+ XM

It can be seen that the relationship is sinusoidal, with zero power transfer at δ=0° and power transfer increasing to a maximum at δ=90°. Beyond 90°, the power transfer decreases until Pe=0 at δ=180°. This is the so called power-angle relationship which applies to a single generator system. In a multi-machine system, there is an analogous relationship between rotor and power transfer, however the limiting values on Pe and δ are a more complicated function of generation and load[7].

1.2.2 The swing equation

Figure 3: Mechanical and Electrical torque at generator shaft

The rotation of a synchronous generator is driven by its prime mover which exerts a mechanical torque Tm on the shaft. As the rotation occurs, the reaction of the stator and rotor fields in the generator produce an opposing electromagnetic torque, Te. In steady state, the electrical torque or power produced by the generator is equal to the mechanical torque or power exerted by the prime mover, neglecting losses. Thus and Te and Tm are equal in magnitude and the generator rotates at synchronous speed ωsync.

In the following discussion, the terms torque and power will be used interchangeably, as is common in power systems literature {Kundur}. Power is simply the product of torque and angular velocity. Since the average rotational velocity of a synchronous machine is largely constant, the per unit values of torque and power are very nearly equal.

Due to rotor inertia, changes in mechanical torque can only occur relatively slowly, while electrical torque is able to change almost instantaneously following a disturbance. Due to this difference in adjustment times, a disturbance is likely to cause a mismatch between mechanical and electrical torques, resulting in an accelerating torque Ta:

( 2 )

Ta causes the motor to accelerate or decelerate and thus deviate from synchronous speed.

The rotor motion can be described by Newton's second law, given by

( 3 )

Where the J (kg m2) denotes the total moment of inertia of the generator and prime mover and ωm (rad/s) is the angular velocity of the rotor.

It is common to work with a normalized inertia constant H, which is defined as the ratio of the stored kinetic energy at synchronous speed to the rated apparent power. After some manipulations, ( 3 ) can be re-written as the swing equation:

( 4 )

where ωsync is the synchronous electrical angular velocity of the rotor. (Full workings in appendix)

The swing equation is so called because it describes the swings of the rotor angle δ during transient disturbances. For a system to be transiently stable as described by the swing equation, it must oscillate about an equilibrium point. On the other hand if the rotor angle magnitude increases indefinitely, this indicates that the generator has lost synchronism and the system is said to be transiently unstable.

Frequently, an additional term -DΔ ω is added to the right hand side of ( 4 ) to represent a damping torque, where D is a small positive number, typically between 0 and 2 [8]. The presence of the damping term allows the oscillations of a stable system to attenuate and gradually die down, whereas in the undamped version, a stable system will simple oscillate indefinitely about a central point. In practical generator-turbine units, damping is contributed by friction and windage, the governor action on the prime mover, and eddy currents in the rotor electrical circuits [9].

1.2.3 Equal area criterion

For a multimachine system, the swing equation must be solved using numerical integration techniques. In the simpler case of a two machine system, it is possible to determine first swing stability using a graphical method called the equal area criterion. Figure 4 illustrates the method. A generator in steady state operates on the power angle curve as discussed in 1.2.1. Thus initial operation is at point 1, where rotor angle is δ0 and mechanical and electrical powers are in balance (Pmo=Peo). A three phase transmission line fault interrupts current flow and causes the electrical power to become zero (point 2). The mechanical power is now greater than the electrical power, causing the rotor to accelerate. As changes in mechanical power occur much more slowly than changes in electrical power, Pm can be assumed to remain constant over the duration of the study. The fault is cleared at time t, and electrical power is restored. By this time the rotor angle has advanced to δ1, and the corresponding electrical power is at Pm1 (point 5). The electrical power output now exceeds the mechanical input power, creating a retarding torque; however rotor inertia causes the rotor angle to continue advancing.

Figure 4: The acceleration and deceleration areas: (a) short clearing time and (b) long clearing time.[2]

If the retarding torque is sufficient, δ reaches a maximum value and then swings back towards the prefault position. If the system is stable, an oscillatory period follows, with the outcome dependant on system damping. In the absence of damping, the rotor will swing back and forth about point a. For positive damping, the oscillations will attenuate and die down to zero. In both cases the system does not lose synchronism and stability is maintained.

If the retarding torque is insufficient, δ advances to δcr=(π- δ0) where Pm is again greater that Pe, so the rotor has entered another accelerating region. The rotor will accelerate and stability is lost. Thus δcr is the critical rotor angle, beyond which loss of synchronism occurs.

Area 1-2-3-4 corresponds to the accelerating region (Pm>Pe) , and is proportional to the kinetic energy gained by the rotor during acceleration. Area 4-5-6-7 is proportional to the kinetic energy lost during deceleration. Thus the rotor again reaches synchronous speed when

( 5)

As the generator loses stability if the rotor travels past the crititical angle δcr, area 4-5-8 is therefore the available deceleration area with which to stop the swinging rotor. Figure 4a the generator did not use the the whole available deceleration area, whereas in Figure 4b, the available deceleration area is smaller that the acceleration area, thus synchronism is lost. We can use the available deceleration area to define a transient stability margin[2].

( 6)

The transient stability of the system is dependent on the nature and severity of the disturbance. Prefault loading of the generator will affect both initial rotor angle and the mechanical power input to the generator during a fault. Location of the fault with respect to the generator terminals is also important, as the transmission circuits provide fault severity attenuation[7].

Finally fault clearing time determines the duration of the accelerating torque, and thus how far the rotor angle will be displaced. In fact, the swing equation dictates that the accelerating area A1 is proportional to the fault clearing time squared[4]. The longest fault clearing time for which the generator will remain in synchronism is termed the critical clearing time. The relative difference between the critical clearing time and actual clearing time can be used to give another measure of the stability margin.

( 7)

where tf and tcr are the actual and critical clearing times

Stability also depends in the excitation speed and forcing capacity [7]. A fast acting AVR and exciter increase the excitation voltage to its ceiling level before the fault is cleared. This means that the system will follow a higher power angle charateristic when the fault is cleaed, leading to a larger decelerating area. It can be seen that the critical rotor angle is greater for the altered power angle curve, thus the system will be able to maintain stability for longer duration faults.

Figure 5: Acceleration and decelerating areas when the influence of voltage regulator is (a) neglected; (b) included [2]

1.3 Historical overview of stability problems

Power system stability has been recognised as a problem as far back as the 1920's [4], with different forms of instability gaining importance during different periods. Early systems consisted of remote power plants feeding load centres over long transmission lines. Power systems at the time feature slow exciters and non continuous acting voltage controller, and power transfer limits were often dictated by steady state as well as transient stability considerations. Since calculations were performed with slide rules and mechanical calculators, models were required to be simple. Systems were modelled using 2 machine representations and graphical techniques such as equal area criterion and power circle diagrams were used for analysis.

As systems evolved in complexity, the two machine representation began to be inadequate. The development in the 1930's of the network analyser (a scale model of the system with adjustable resistors, reactors and capacitors to represent transmission networks and loads) allowed the power flow analysis of multimachine systems, however dynamics were still solved using swing equations and manual, step-by-step numerical integration[2].

Developments in fast fault clearing and continuous acting voltage regulators allowed considerable improvement in stability. The continuous acting, high gain excitation systems were effective in both limiting first swing transient instability and increasing steady state power transfer. With these new developments, there was increased dependence on control systems, and a requirement for more detailed representations of machines and excitation systems. The development of digital computers allowed the analysis of larger networks and more realistic modelling of equipment characteristics[2].

Transient stability had been the dominant stability problem in the early decades of stability analysis, and the main focus of industry attention. As high gain excitations systems became more prevalent during the 1950s and 60s, it became clear that these devices could have a destabilising effect of power systems, making them more prone to low frequency, sustained oscillations. Increased use of high gain exciters, together with decreased strengths of transmission systems lead to an increased interest in small signal stability. Analysis of small signal stability problems in turn required the adoption of new techniques, such as modal analysis using eigenvalues . In recent times, supplementary excitation control, static VAR compensation and FACTS (flexible AC transmission Systems) are increasingly used to dampen oscillations and improve small signal performance[6].

2 Modelling

2.1 Synchronous generator

To describe the generator equations it is common to transform all values to the rotating rotor reference frame (dq frame) [4].

The mathematical description of the Digsilent synchronous generator model is as follows[10]:

(8)

where the flux linkages are caluculated as follows:

( 9)

electrical torque te in p.u. can be expessed in terms of generator currents and fluxes :

( 10)

2.1.2 Mechanical equations

The machine mechanics are described by the swing equation as discussed in 1.2.2. The accelerating torque arises from a mismatch between mechanical and eletcrical torque.

( 11)

( 12)

where H is the normalised machine inertia constant.

2.2 Induction Motor

Figure 6: Induction motor equivalent circuit diagram [10]

2.3 Automatic voltage regulator

The AVR( automatic voltage regulator) provides control of the terminal voltage by changing the generator field voltage. The generator terminal voltage is compared with the reference or desired terminal voltage to produce an error signal. This error signal then determines the amount of adjustment to the DC exciting current that is applied to the field. Thus the terminal voltage is continuously adjusted to maintain the correct level.

In the case of a single generator system the AVR regulated terminal voltage just as its name suggests. In the case of two or more generators connected in parallel, the generators are connected to the same busbar and will have the same terminal voltage. It is not possible to regulate the voltage of each generator individually. In this case the AVR of each generator controls the proportional sharing of the reactive power load[11].

2.3.1 AVR block diagram

The IEEE has derived a number of standarized block diagrams to aid in the modelling of AVR systems [12]. The Digsilent model library features built in block diagrams based on the IEEE systems, as well as supporting user defined models.

The model used for this project is the IEEE type 1, which is one of the most common. It represents a continously acting regulator with a rotating exciter system. The automatic voltage regulation is stabilized by the use of derivative feedback as represented in the damping filter block. The exciter saturation function is approximated by a simple exponetial function of the form Se=AeBefd. The constants A and B can be solved by specifying two pairs of data points. It is customary to use the points at the 100% and 75% excitation levels.

Figure 7: IEEE type 1 AVR [10]

2.4 Governor

The governor regulates generator rotation speed by controlling the output power of the turbine. Active power output of the generator is directly related to rotor speed (and hence electrical frequency). The governor error sensing circuit compares shaft speed with a set or reference value and the adjusts fuel input or fuel valve to increase or decrease turbine power. Similar to the case for AVRs, it is not possible to individually control the speed of generators connected to the same busbar. Thus the governors of paralled generators will determine the active power load sharing of the generators[11].

2.4.1 Governor block diagram

This type of governor-turbine system represents the IEEE gas-turbine governing systems.

2.5 Network equations

The relationship between the network voltages and currents can be represented by forming the nodal network equation:

or

( 17)

2.6 Change of reference frame

While network equations are given in the network's complex co-ordinates, for generator equations it is convenient to work in the generator's d,q orthogonal axes. The q axis of a given generator is shifted with respect to the networks real axis by the rotor angle δ. The relationship between the two coordinate systems can be given by:

( 18)

As each generator may operate at a different rotor angle δ, there are as many d,q systems of coordinates as there are generators in the network.