The following points highlight the two main stability conditions for power system. The conditions are:- 1. Steady State Stability 2. Transient Stability.
Condition # 1. Steady State Stability:
Steady state stability is the capability of the power system to restore to its initial condition after a small disturbance or to reach a condition very close to the initial one when the disturbance is still present and steady state stability limit refers to the maximum flow of power possible through a particular point without causing the loss of stability when the power is increased very gradually.
The terms steady state stability and ‘steady state stability limit’ are frequently used interchangeably. Since the electrical system is always subject to small disturbances, the steady state stability requirement is essential for the system to operate properly.
The assessment of steady state stability is very important in planning and designing of electrical power systems, in developing special automatic control devices, putting into operation new elements of the system, or modifying its operating conditions (such as interconnection of the power systems anti commissioning of new power plants, transformer substations, transmission lines etc.).
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Furthermore, the assessment of the steady state stability may also be required in the power system analysis (including checking of stability of an electrical power system in a specified steady state, determination of its stability limit, qualitative estimation of transients) and synthesis (involving the choice of type of the excitation system and its control, the mode of control, the parameters of the excitation and automatic control systems).
This choice is made on the basis of requirements to the stability limit or quality of electrical energy (i.e. accuracy of maintaining the voltage) under steady-state conditions or during transients.
When all the machines in one part of a power system act together, they are treated as one large machine connected at that point, for the purpose of study of power system stability. Even if the machines are not connected on the same bus bars but are separated by large percentage reactances, they are considered as one machine.
A large system in the power system stability problems is always supposed to have constant voltage and constant frequency and is treated like infinite bus. In this way, it is always possible to work out a complicated system on the basis of two machines or one machine connected to an infinite bus.
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Let us consider a system consisting of a generator G, transmission line and a synchronous motor M in the form of load (Fig. 7.3).
Power developed by the generator G and synchronous motor M are given by the expressions –
and the maximum power developed by the generator G and synchronous motor M are given by the expressions –
where A, B, and D are the generalized constants of a two terminal pair network. The above expressions will give power in watts per phase in case voltages are taken as phase voltages in volts. However, if the voltages are taken as line voltages in kV the above expressions will give total power (power for all 3-phases) in MW.
Now consider a synchronous motor connected to an infinite bus and running at a constant speed. The power input is equal to power output plus losses. If a small increment of the shaft load is added to the motor, the output power of the motor increases whereas the input to the motor remains unchanged. Hence, there is a net torque on the motor tending to retard it and its speed falls temporarily.
As a result of momentarily reduction in motor speed, the phase angle between the internal voltage of the motor and the system voltage increases until the electrical power input again equals power output plus losses.
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During the transient interval electrical power input to the motor being less than mechanical load, the excess power required is supplied by the stored energy in the rotating system. The motor thus oscillates around the new point of equilibrium and may finally either come to rest or may lose synchronism. The latter will be the case when the load applied will be too large or will be applied too suddenly.
PMmax given by Eq. (7.24) is the maximum power that motor can develop. This value of power can be obtained only when load angle δ = β. The load on the motor may, therefore, be increased till this condition is reached.
After this condition if the load still keeps on increasing, load angle δ will increase further and the motor may lose synchronism and the excess of power required over power developed will come from the stored energy of the rotating system and the speed goes down. Larger and larger angle δ becomes, the smaller and smaller will be the power developed, till the motor comes to rest.
It is to be noted that δ = β is not the condition for maximum power developed by the generator but it needs not to be considered that condition since the motor will lose synchronism earlier (i.e. at an angle δ = β). The difference between the generator and motor power developed for any value of δ is equal to the line losses.
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If the line resistance and shunt admittance are negligible, we get the following expression for the power transferred between the alternator and the motor:
The power transferred will be maximum for δ = 90°
Omission of resistance gives a somewhat optimistic results (higher than actual value) for maximum steady state stability limit while in transient stability calculations the results obtained by neglecting resistance are pessimistic (lower than actual value) since resistance is an important damping element.
Methods of Improving Steady-State Stability Limit:
From Eq. (7.26) it is obvious that the maximum power that can be transferred between an alternator and a motor is directly proportional to product of internal emfs, of the two machines and inversely proportional to the line reactance.
Hence the steady state stability limit can be increased by:
i. Increasing the excitation of a generator or motor or both so that the internal emfs are increased and consequently the maximum power transferred between the two machines increases. Further for a given power transfer and with the increased values of internal emfs, the load angle δ decreases.
ii. Reducing the transfer reactance—increasing the number of parallel lines between the transmission points is a common method of reducing the reactance. In general more power is transferred during a fault on one of the lines if there are two lines in parallel than would be transferred over a single faulted line.
The use of bundle conductors is, of course, another method of reducing reactance. The bundle conductors are used mainly in EHV transmission lines to reduce corona loss and radio-interference. Another method of reducing the line reactance is by using capacitors in series with the lines. Series capacitors are only used in EHV lines to increase the power transfer and are most economical for transmission distances over 350 km.
Condition # 2. Transient Stability:
A synchronous power system has transient stability if, after a large sudden disturbance, it can regain and maintain synchronism. A sudden large disturbance includes application of faults, clearing of faults, switching on and off the system elements (transmission lines, transformers, generators, loads etc.).
Usually, transient stability studies are carried out over a relatively short-time period that will be equal to the time of one swing. Normally, the time period will be one second or less. The analysis is carried out to determine whether the system loses stability during the first swing or not.
In case the power system remains stable, it is assumed that subsequent swings will diminish and that power system will remain stable, as usually happens. However, there is a possibility of power system going unstable in some subsequent swing. For example negative damping is one cause. Control equipment improperly adjusted or applied can produce negative damping.
In practice load is not applied gradually and the machine (motor or generator) is not in a position to meet the demand instantly. The limit of the system to meet such loads, called the transient stability limit, is obviously less than steady state stability.
As said earlier, the power systems have rotating synchronous machines, in order to know whether the system is stable or not, it is necessary to derive swing equation.
Swing Equation:
Let angular displacement = θ radians
Power developed, P = T ω watts (where T is torque in Nm)
Angular momentum, M = 1 ω j-s/radian
where I is moment of inertia in kg-m2 or j-s2/radian2
In case of a generator the input is a mechanical torque (shaft torque), Ts and the output is an electromagnetic torque, TE and both of these torques are reckoned as positive In case of a motor the input is electromagnetic torque. TE and the output is shaft torque, Ts, both of these torques are reckoned as the negative. Torques due to rotational losses is assumed to be negligible in either case during the discussions.
Under normal working, the relative position of the rotor axis and the stator magnetic field axis is fixed. The angle between the two is called load angle (or torque angle) δ, which depends upon the loading of the machine. Larger the loading, larger is the load angle δ. In case load is added or removed from the shaft of the synchronous machine, the rotor will decelerate or accelerate accordingly with respect to the synchronously rotating stator field and a relative motion starts.
Now rotor is said to be swinging with respect to the stator field. The equation giving the relative motion of the rotor (load angle δ) with respect to the stator field as a function of time is called the swing equation.
It is obvious that any difference between the input and output torques (Ts~ TE) will cause acceleration or retardation of the rotor depending whether the input torque is greater than output torque or otherwise. Accordingly for a generator –
TAG = TS – TE … (7.27)
where TAG is the net torque causing acceleration of the rotor and will be positive if TS > TE.
The above expression holds true for a motor also.
A similar relation holds good when expressed in terms of power, i.e.,
PAG = PS– PE where PAG is the accelerating power …(7.28)
Further PAG = TAG ω = 1 α ω = M α ….(7.29)
Since the angular position θ of the rotor is continually varying with time, it is more convenient to measure the angular position and velocity with respect to a synchronously rotating axis.
Angular displacement, θ with respect to time, t can be expressed as –
θ = ω St + δ ….(7.30)
where ωS is the angular velocity of the reference axis rotating synchronously and δ is the angular displacement in electrical degrees from the synchronously rotating reference axis (Fig. 7.6).
Differentiating with respect to time t, we have –
Equation (7.33) is called the swing equation. The angle δ is the difference between the internal angle of the machine and the angle of the synchronously rotating reference axis which in this case corresponds to the infinite bus. For a two-machine system two swing equations are necessary one for each machine, a two machine system can be reduced to a single machine connected to an infinite bus.
Inertia Constant:
The stored kinetic energy of a rotating body is given as –
The above equation gives the unit of M in joule-seconds per radian (Js rad-1) if the angular velocity ω is in radians per second. The energy is more conveniently expressed in MJ and angles are often measured in degrees. So M can be expressed in MJs per electrical degree.
From above equation M, being equal to I ω, is not constant but varies somewhat during the swings because of variation in ω. In practice, the change in co from the normal system angular velocity is not much during swing except of course when the machine falls out of step and, therefore, hardly any significant error is involved if M is assumed to remain constant and is equal to the value Iωn, where ωn is the normal angular velocity of the machine. This value of M is known as the inertia constant of the machine, and is normally used in calculations for stability studies. The inertia constant is truly constant because it is the angular momentum at synchronous speed.
M, the inertia constant of the machine, depends on the type and the size of the machine. Another constant H having narrow range of values for each class of machine regardless of its capacity is defined as the ratio of the kinetic energy at rated speed to the rated apparent power of the machine, i.e.,
Thus stored energy in mega-joules equals GH.
A relation between M and H is derived as follows:
From Eqs. (7.35) and (7.36)
where f is the frequency in Hz and co is expressed as 360 f in terms of electrical degrees per second.
Equation (7.38) relates the two inertia constants of the machine. For stability studies it is necessary to determine M which depends upon the size and speed of the machine, but instead H has a characteristic value or a range of values for each class of machines. The average value of H in MJ per MVA for turbo-generators is from 3 to 9; for water-wheel generators between 2 to 4; for synchronous condensers between 1 and 1.25; while for synchronous motors about 2.
It is observed that the value of H is considerably higher for turbo-generators than for water-wheel generators. 30 to 60 percent of the total inertia of a turbo-generator unit is that of the prime mover, while only 4 to 15% of the inertia of a water-wheel generator unit is that of the water wheel, including water.
When a number of synchronous machines are operating in parallel, they can be replaced by a single equivalent machine that has a rating equal to the sum of the ratings of all the machines considered to act together during the transient period. The inertia constant M of the equivalent machine is the sum of the inertia constants of the individual machines, i.e.,
It is evident from the above discussion that the value of H for a given rating varies inversely as the base.
For a multi-machine system solution of several swing equations is required. In fact for each machine a swing equation is required. The solution is tedious and normally the point-by-point or step-by-step method is used for it. Even for a single machine system connected to infinite bus with resistance neglected, the formal solution of the swing equation is possible only when Ps = 0 and by use of elliptic integrals.
The solution of Eq. (7.34) gives a graph of load angle δ versus time t, called the swing curve. This curve is used to determine the stability of the system. In case δ increases indefinitely it indicates instability whereas if it reaches a maximum and starts decreasing it shows that the system will not lose stability and the oscillations of the machine about the equilibrium point will become smaller and smaller and finally it reaches equilibrium.
Many a time, in a multi-machine system one of the machines may stay in step on the first swing and yet go out of step in the subsequent swing. For a two machine system under the assumption of constant input, no damping and constant voltage behind transient reactance, the machines either fall out of step in the first swing or never. Under this condition the two machines are said to be running at standstill with respect to each other.
For solution of swing equation certain assumptions and simplifications are usually made.
These are listed below along with their justification and their effects on accuracy of results:
1. It is assumed that mechanical input for a generator and mechanical load on a motor remains constant during the entire period of analysis. This assumption is quite valid for a synchronous motor because the speed of the motor does not change significantly unless stability is lost.
The validity of this assumption for a generator is based on the fact that the prime-mover governor, steam valve etc. are slow acting and act only after sensing a change in speed which does not change significantly during the transient period. This assumption in case of a generator makes the results somewhat pessimistic.
2. The resistances of the transmission lines and the synchronous machines are neglected. This assumption leads to a pessimistic result as resistance introduces damping term in swing equation and stability is improved.
3. The damping term contributed by synchronous machine damper windings is also ignored. This assumption leads to a pessimistic result for the transient stability limit.
4. All shunt capacitances are neglected. This assumption does not introduce any significant error.
5. Direct axis transient reactance of the synchronous machine is considered in the analysis and the saliency effects of the synchronous machine are neglected. This assumption is quite valid, because during a disturbance the period of mechanical oscillation may be of the order of one second. The sub-transient time constant will be approximately 0.3 second, so it is usually ignored in transient stabilities.
A typical armature time-constant would be about 0.2 s, so in many studies the direct component of armature current is ignored. The transient component of armature current may have a time constant of a few seconds, which is longer than the period of mechanical oscillation, so it is necessary to consider the transient reactance.
6. The voltage behind transient reactance is assumed to remain constant during the entire period of analysis. This means that the action of excitation control system is ignored and this leads to pessimistic stability results.
This is a stronger assumption than assuming Ps constant as the excitation control system is faster than the governor system. However, with the advent of digital computer, the effects of voltage regulators and turbine governors can be incorporated in the transient stability analysis.
The above assumptions simplify the calculations to some extent. Even then the calculations are so lengthy that the use of digital computer is preferred. The solution for a multi-machine system is possible only through a computer simulation.
In case of only one machine connected to an infinite bus-bar, by application of equal area criterion the transient stability of the system can be worked out without detailed solution of swing equation.
Modern power systems are so large that even after lumping of machines, the system remains a multi- machine one. Even then a simple two-machine system greatly aids the understanding of the transient stability problem. The two-machine system can be reduced to one machine connected to infinite bus.