In this article we will discuss about:- 1. Introduction to Stability of Power System 2. Maximum Steady State Power.
Introduction to Stability of Power System:
In all power systems, the larger machines are of the synchronous type, these include substantially all the generators and condensers, and a considerable part of the motors. On such systems it is necessary to maintain synchronism, otherwise a standard of service to the consumers will not be achieved. The transient disturbances are caused by the changes in loads, switching operations, and, particularly, faults and loss of excitation.
Thus, maintenance of synchronism during steady state conditions and regaining of synchronism or equilibrium after a disturbance are of prime importance to the electrical utilities. The term ‘stability’ can be interpreted as ‘maintenance of synchronism’. These two terms are quite frequently used interchangeably.
The present trend is towards interconnection of the power systems; resulting into increased lengths and increased reactance of the system, this presents an acute problem of maintenance of stability of the system.
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The term ‘power limit’ is also sometimes interpreted as ‘stability’ because to have the maximum utility of the system it should be capable of supplying maximum power without causing instability. Power system stability, in general terms, may be defined as its ability to respond to a disturbance from its normal operation by returning to a condition where the operation is again normal.
The terms ‘stability’ and ‘stability limit’ are defined by A.I.E.E. as below:
Stability when used with reference to a power system, is that attribute of the system, or part of the system, which enables it to develop restoring forces between the elements thereof, equal to or greater than the disturbing forces so as to restore a state of equilibrium between the elements.
A stability limit is the maximum power flow possible through some particular point in the system or the part of the system to which the stability limit refers is operating with stability.
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For the purpose of analysis there are three stability conditions that must be considered:
1. Steady state stability.
2. Transient stability.
3. Dynamic stability.
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1. Steady State Stability:
‘Steady state stability’ may be defined as the capability of an electric power system to maintain synchronism between machines within the system and external tie lines following a small slow disturbance (normal load fluctuations, the action of automatic voltage regulators and turbine governors).
In case the maximum power transfer exceeds under this condition, individual machines or groups of machines will cease to operate in synchronism, violent fluctuations of the voltage will occur and the steady state limit for the system as a whole would have been reached. The steady state stability limit refers to the maximum power which can be transferred through the system without loss of stability.
2. Transient Stability:
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Loss of synchronism is, however, possible at loads (or disturbances) below the steady state limit, if they are suddenly applied or removed. When dealing with such large sudden disturbances a concept of transient stability is introduced. A sudden large disturbance includes application of faults, clearing of faults, sudden load changes, and inadvertent tripping of lines and generators.
The maximum power which can be transferred through the system without the loss of stability under sudden disturbances is referred as transient stability limit.
Transient stability is the ability of the system to remain in synchronism during the period following a disturbance and prior to the time that the governors can act. Ordinarily the first swing of machine rotors will take place within about one second following the disturbance, but the exact time depends on the characteristics of the machines and the transmission system. Following this period, governors begin to take effect, and dynamic stability conditions are effective.
3. Dynamic Stability:
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It is the ability of a power system to remain in synchronism after the ‘initial swing’ (transient stability period) until the system has settled down to the new steady state equilibrium condition. When sufficient time has elapsed after a disturbance, the governors of the prime movers will react to increase or reduce energy input, as may be required, to re-establish a balance between energy input and the existing electrical load.
This usually occurs in about 1-1.5 seconds after the disturbance. The period between the time the governors begin to react and the time that steady state equilibrium is re-established is the period when dynamic stability characteristics of a system are effective.
It is possible to have transient stable but dynamically unstable conditions. Immediately after a disturbance, the machine rotors will go through the first swing (before governor action) successfully, and then, after governor control is initiated, the oscillations will start increasing until the machine falls out of synchronism.
Such action can occur if the time delays of the governor control are such that, following the sensing of necessity for increasing or reducing energy input, action is delayed sufficiently in time to augment rather than diminish the next swing. If such a condition exists, the oscillations of the machine rotor can continue to build up until the machine falls out of synchronism.
Maximum Steady State Power:
Line diagram of a 3-phase transmission line is shown in Fig. 7.1. The transmission line may supply power from a generating station, grid switching station or grid supply point. The ends of a transmission line are designated as buses. The concept of a bus in a one line diagram is essentially the same as that of a node in a circuit diagram.
Let the receiving-end voltage be VR ∠0° (taking VR as reference phasor) and the sending-end voltage Vs ∠δ where δ is the phase angle between sending- and receiving-end voltages. Let the generalised line constants be expressed as A = A ∠α; = B ∠β; D = D ∠Δ.
The input voltage per phase (voltage at sending end) and the input current (current at sending end) of a transmission line can be expressed as –
The complex power at the receiving end is given by the expression –
where IRC is the conjugate of receiving-end current IR.
Receiving-end current can, however, be expressed in terms of receiving- and sending-end voltages [using Eq. (7.1)].
Substituting the value of IRC from above equation in Eq. (7.3) we have –
Separating real and reactive components we have –
For fixed values of Vs and VR the power received will be maximum when cos (β – δ)= 1 or when δ = β
Substituting the value of IR from above equation in Eq. (7.1) we have –
For fixed values of Vs and VR, the power sent out will be maximum when cos (β + δ) = – 1 or when β + δ = 180°
The noteworthy points about Eqs. (7.5) and (7.11) are:
1. These equations give phase values of SR and Ss in volt-amperes if VR and Vs are phase values in volts (i.e. line-to-neutral voltage).
2. The total 3-phase power is 3 times the power per phase. Since the line voltage is √3 times the phase voltage, these equations yield 3-phase power directly if VR and Vs are expressed in line volts (line- to-line voltage).
3. If the voltages VR and Vs are expressed in kV, the complex power is expressed in mega volt-amperes.
Now let us consider a loss-less transmission line having negligible shunt admittance.
For such a line –
A = D 1 ∠0°, B = X ∠90°; and C = 0
and the expressions for receiving-end power, maximum receiving-end power, sending-end power and maximum sending-end power become –
From above expressions we find that –
(i) PR = Ps;
(ii) PRmax = PSmax; it is because of negligible line losses.
(iii) For fixed values of sending-end voltage Vs, receiving-end voltage VR and line reactance X, the power sent out (or power received) varies directly as the site of the load angle δ (phase shift between sending-end and receiving end voltages).