Uncontrollable decay of system voltage at one or more load buses or even over a significant portion of the network as a response to load variations, generation or structure disturbances etc has been observed in Power systems world wide. This has been termed as voltage Instability and the process of voltage decrease has been termed as a voltage collapse process.
With the rest of the system conditions remaining in charged, if the load at a particular bus is varied , the voltage at the bus will also vary. [Voltage at other nodes will also vary as a response to this load change]. In other words voltage at a load bus is elastic with respect to the active power and reactive power delivered at that node. This elasticity may be quantitatively represented by & where P, Q, V are active power, reactive power and voltage of the bus. Under stable conditions these factors are generally negative.
Power system loads are generally dependent on voltage and frequency. The constant active and reactive power loading (i.e. constant, independent of bus voltage) quite often employed in load flow and other similar studies, is , at best a mathematical idealization of the power system load.
In general, the loads take active power and reactive power as functions of load voltage. For example, a constant admittance load draws active & reactive power, which are proportional to square of voltage.
A load is called static if the power taken by the load is dependent only on voltage (we ignore frequently changes in voltage collapse analysis) and not on time. For example a constant impedance load, lighting load, a constant current load etc are static loads. If the load power drawn by these loads vary in time it is because the voltage varies with time and the power varies in step with the voltage in such loads.
A load is called dynamic if the power drawn by the load is a function of voltage as well as time. For example, consider an induction motor driving a constant toque load. If the voltage changes suddenly, the motor decelerates and the power (both the P & Q) drawn by motor become functions of speed (slip) and hence of time. P & Q will stabilize at new values corresponding to new value of voltage after sufficient time decided by mechanical time constant of the motor. [Quite possibily, the stabilised values may correspond to a stalled condition]. Another dynamic load is a static load stabilised by using a continuous tap changer [i.e. automatic tap changing voltage stabiliser, automatic electronic voltage stabiliser etc]. These tap changers change taps only with a certain delay. Thus the load presented to the power system after a voltage dip will appear to be time varying due to time varying turns ratio in the tap changer.
The interplay of the two factors viz. bus voltage elasticity with load power and load power elasticity with bus voltage results in abnormal sustained voltage drops (when load increases slightly or one of the two parallel lines trips, or local generation is lost etc.) in the case of static loads. In the case of dynamic loads, this interplay causes uncontrollable decay of voltage even in the absence of further disturbances. This is true voltage instability. A qualitative analysis of both types of voltage collapse is attempted in the following sections using a long radial power link fed from an infinite bus as an example.
Receiving and characteristics of a radial power link
The system under consideration is shown in fig.
The load current following in the line impedance causes a voltage drop in the line and hence load bus voltage changes with the load. Line regulation depends strongly on the reactive component of the load. The behavior of the line with u.p.f load at receiving end (i.e. Q=0) and infinite bus maintained at 1.0pu is shown in above figure. The variation of voltage with active power for various power factors given in fig.
These curve are obtained by assuming that a conductance load is connected at load bus and calculating the time flow for various power factors. Fig 2& 3 reveal that at a given power factor and constant sending end voltage, there is a maximum power that the line can transmit to load. Also, there are two operating points for the line for any power below this limit. At the upper operating point the voltage is high, efficiency is high and line current is small. At the lower point voltage & efficiency are low and current is high. What happens when load demands a power more than the maximum that can be delivered by the line ?
Which of the two operating points will the line work at ?
The answer to these questions will need an analysis of load power elasticity with voltage.
An examination of curves in fig reveals that load voltage at any power is very sensitive to reactive demand of the load. The maximum power limit comes down drastically with lagging power factor at load. It is also evident that both the voltage drop and maximum power limit are very sensitive to power factor around u.p.f.
Voltage Collapse with a Static Load
- A static load of constant admittance is assumed in this section. In Section-2 we noticed that with a constant power load there can be two operating points. Which one does the line choose? The answer is that we can not know. Similarly we can not know what will happen if the constant power load is above the maximum power that can be supplied by the line. But the answers to both these questions are found in the fact that there can not be a constant power load at all. All physical loads exhibit a nonzero elasticity with voltage. In this section we consider an elastic load which is static and of constant admittance nature. In fact the conclusions arrived at will be applicable for any static load where and where n0 and m0.
n=m=2 represent constant admittance, n=m=1 represent constant current load and so on.
- The load power characteristics of a constant admittance load is shown in fig by the dotted curve A. In this case there will be in general, only one operating point. Also it is clear that there is no question of the load demanding more power that the maximum power. As the load increases the operating point shifts along the line characteristics and gets in to positive slope region as shown for load B in fig . Also the effect of power factor on the operating point can be clearly seen in fig.
- With a static load, all operating points are generally stable i.e. line operates stability (but maybe with a very low load voltage, high current and low efficiency) everywhere. The voltage drop is in step with load change. The voltage variation with time is decided by load variation with time. For every load there is a fixed voltage and voltage does not run away with load conductance remaining at a constant level. Thus, in this case collapse can take place only if the load varies. See fig 4. The load is assumed to be at upf and is assumed to vary from a low value to high value in 30 minutes linearly. The corresponding voltage variation is shown.
- If the line is operating near the critical point but in the upper portion of the curve, a small change in load conductance can cause a disproportionate drop in the voltage. However increasing the conductance will result in an increase in power delivered, as it should.
- But if load conductance is taken to such a level where operating point is in the bottom portion of the curve, not only that there is abnormal voltage drop and excessive heating of the line (due to high current) but also the power delivered comes down. In this region, when power is sought to be increased by increasing the load , what we get is a decreased power delivery. In this sense we will lose the controllability of power in this operating region.
- The same kind of voltage collapse and inefficient operation can result in the cases of (i) loss of a line in a parallel line link (ii) loss of capacitor in a series compensated link even without a real load power change. The effect of increase in the link impedance is to shift maximum power of the curve to the left.
- To sum up, with a static load, abnormal reduction of voltage and stable operation at such low voltages can result as a consequence of over-loading or increase in line impedance. Line operation under such conditions will be quite lossy due to high line currents. However there is no voltage run-away phenomenon associated with static load in general. Of course, where line operation is near the critical point, small changes in load will result in large changes in voltage and large reduction in line efficiency. The collapse is more severe and faster if the load is at a low lagging power factor.
- Improving the power factor of the static load by using shunt capacitor can avoid collapse with static load variation. Also, the system can be pulled out of low-voltage condition by switching on the capacitors after the collapse has occured. The static nature of collapse ensures that the voltage will recover and go to acceptable level on switching-in shunt capacitors. Since collapse is very sensitive to p.f in the region of upf, full shunt compensation may be the only solution to this kind of collapse problem due to static load growth.
Voltage Collapse with Static Load - Effect of open loop tap changing at receiving station
The system is shown in fig5 and upf load is assumed for convenience. 'A' is the operating point before tapping up the secondary voltage. The conductance of the load gets multiplied by turns ratio (secondary divided by primary turns) squared as viewed from primary side. Hence the effect of tapping up is to shift the load power curve from (1) to (2). The operating point gets shifted to 'B'
When secondary voltage is stepped up, secondary current increases. With an increase in turns ratio this increased secondary current gets reflected to primary line with higher multiplier. This increased primary line current causes larger drop in link impedance and thereby results a still lower primary voltage. But with this reduction in primary voltage do we get any tapping up effect in the secondary side at all ?
In the case shown, power delivered after tapping up - P2 - is greater than P1. Since transformer does not absorb active power and load conductance is assumed to be constant, it follows that secondary side voltage has indeed increased by a factor . But is the percentage increase equal to expected value? No, because of drop in primary voltage. In particular, as the initial operating point moves closer to critical point 'C', the change in power due to tapping up is marginal and hence secondary voltage remains nearly constant even after tapping up. But the line current always goes up on tapping up and hence using a tap change on an overloaded line (i.e. operating point close to critical point) results in a lossier and still more over loaded line with almost no increase in secondary voltage even with tap up.
What if the initial operating point is in the lower portion of line operating curve
Power after tap change - P2 is less than P1. Thus secondary voltage after tapping up is actually less than the value before tap changing. But the line current will go up and line efficiency will come down drastically.
Now let us approach the case of continuous tap changing on closed tap control of secondary voltage. See fig7. Here the load curve (1) represents the load at the 6’o clock in the evening. Within half an hour the load grows to load curve 2 and operating point shift to 'B'.
Now the operator wants to improve the secondary voltage to the level that was there at 6 ‘o clock. By examining fig7 it is evident that, if he taps up by suitable percentage such that curve 2 gets shifted to curve 3 and operating point goes to 'C', he will succeed in bringing secondary voltage to same old value. So, let us assume that he does that and secondary voltage comes back to normal at 6.30 PM. Now, the load increases to curve 4 and operating point to 'D' by 7:00 PM. He decides to tap-up again and brings the secondary voltage to the same old level. Will he succeed? No, he can not any longer increase the secondary voltage by tapping up since on tapping up the operating point will move on to 'E' on curve 5 and power delivered decreases and hence secondary voltage must have come down on tap changing. And if the operator continues to tap-up, voltages everywhere will come down and current in the line will go up.
Voltage Collapse with static load effect of closed loop continuous tap changing at receiving station.
5.1 Now, let us replace the operator in Section-4 by a continuous regulating transformer with closed loop control at substation. The secondary voltage is always kept constant by continuously changing the tap by a closed loop control system. What happens as the load grows?
As long as the power demanded by the load at the voltage maintained constant by the regulator does not exceed the maximum power limit of the line, the regulator will succeed in settling down at a tap position stably. But if load conductance is such that the power taken by it will exceed maximum power limit of line if the secondary voltage is maintained constant, the regulator will fail to find a stable tap position. It will get into the lower portion of line operating curve. But since it is a closed loop control system, it will continue to tap-up and will finally take the line to a short circuit theoretically. But practically it will go to the maximum tap position & will remain there. And voltage would have collapsed to a low value by then. Assuming the tap changer control is instantaneous, all this will happen instantaneously i.e. a fast voltage collapse will take place when load grows to a particular critical value. But tap changers have delays incorporated in them. Then, the time evolution of collapse is decided by tap changer dynamics.
What we have described now is a true voltage instability situation. The static load has been converted into a dynamic load by continuous control of secondary voltage when the load reaches a critical value, with no further disturbance or increase of load, an uncontrollable decay of voltage sets in and takes the system to a low voltage condition.
Can we pull the system out of an ongoing voltage collapse due to closed loop tap changer action
The initial load curve is (1) & initial operating point is (A). VA is the primary voltage corresponding to desired secondary voltage at current tap position. Now the load increased and tap change adjusted tap such that new load curve is (2) and new operating point is (B). But (B) is on the verge of instability and due to small load-transient collapse sets in and operating point is moving along 0.8 pf lag operating line. Assume that when operating point is at (C) a capacitor, which changed, the load admittance power factor to unity was suddenly switched on. Now the operating point shift to (E) as shown. But E is in the stable tap changing region and operating moves to (G) and settles down. System recovers from collapse and tap changer settles down at a tap-down position, secondary voltage is maintained at same old level.
But, if the collapse process is allowed to reach (D) before capacitor is switched-in, the operating point goes to (F) on capacitor switching. (F) is in the unstable region for tap changing and collapse continues. System can not be pulled out by this capacitor switching.
In the case of a dynamic voltage collapse situation
A recovery is possible only if sufficient quantities of shunt capacitor compensation is switched-in fast even before tap changers can go on a spree. Tap Changer control system should be made sluggish and capacitor-switching control should be fast. Preferably, compensation should be in the form of SVC or synchronous condensers.
In a system which suffers from a low voltage problem
The customers show a tendency to regulate their service voltage by using automatic voltage stabilisers. They also use low voltage rated equipments thus taking the power they want inspite of low voltage conditions. Customer automatic regulation of voltage has more severe effect than continuous tap changing. When a customer regulates the voltage he destroys the elasticity of his load and in addition adds to the reactive flow in distribution lines. The higher current levels in distribution feeder due to customer voltage boosting and additional reactive demand produces higher active and reactive loses in these lines. Thus the aggregate effect with voltage closer to zero or even negative. This leads to voltage collapse just as tap changing at substation can lead to collapse.
The effect of the so called ‘voltage improvement transformers’ and additional distribution substations are no different. They also boost up the load admittance seen at receiving end station and make it more prone to voltage collapse.
The solution for the ills of an over loaded radial power link (or for that matter for any overloaded power system) is not ‘voltage improvement transformers’ or additional distribution substations. They can make the problem still worse and can result in mounting power losses.
Effect of generator excitation control on voltage collapse
Till now an infinite bus was assumed at the sending end. But no bus is an infinite bus. In this section it is assumed that a synchronous generator is driving the sending end bus.
With increasing load, the line current increases and machine excitation increase to maintain terminal voltage constant. But soon the machine field current reaches the ceiling level and with further increase in line current the terminal voltage decreases. Decreasing terminal voltage causes further decrease in receiving end voltage and results in further voltage collapse. Thus effect of upper limit on field current of generator is to aggravate the voltage collapse at receiving end.
In fact the problem is more complicated than that. When the collapse process is fast , line current increases fast. If the excitation system of generator is slow, the terminal voltage won’t be able to keep up with the increasing line current and collapse process will be further intensified due to low sending voltage conditions - even though field current has not reached ceiling levels. A fast excitation system simulates infinite bus better.
Voltage Collapse with Induction motor loads.
7.1 The load at the receiving end is assumed to be an induction motors driving a constant torque load. The inertia of the motor is neglected at present. As the bus voltage varies the motor speeds adjust to as new value quickly and power flows change suitably. Since the speed variation before pull out occurs is small in a normal induction motor driving a constant torque, the active power remains a constant with voltage variation. But the reactive power varies differently. Fig 9 gives the active and reactive power variation with voltage for a motor driving constant torque load.
Notice that at lower voltages motor takes more reactive power than at normal voltage.
Consider the following system in fig.
The motor was being fed by the parallel lines and motor was operating at 0.8pf at operating point B. Now one line goes open. Will there be voltage collapse?
Ref fig 9. Let the motor pf at the voltage VC (point A) be 0.9 (this is the best pf the motor can have with the specified load.) If V shown in fig 11 is more than VC. The motor will operate stably with some voltage between V & VC . If V is less than VC, the bus fails to meet reactive demand of motor, the voltage drops, motor demand further reactive power and voltage drops still further. These processes continue till the motor goes to stand still and voltage comes to a small value.
The same sequence of events can take place when there is an increase of static load connected to the same bus even without any structure disturbance.
If the motor and load inertia are neglected
The voltage collapse due to motor load has to be instantaneous. But with the inertia coming in, the motor dynamics enters the picture. When voltage decreases, motor speed comes down slowly, its reactive demand goes up and causes further drop in voltage. It is obvious that the time evolution of collapse will depend on time evolution of motor dynamics. The reader is referred to the articles in ref(1) and (4) for an excellent coverage on voltage collapse due to Induction motor loads.
The basic reason for Induction
Motor provoking voltage instability is the inelastictity of active power demand of motor to voltage and negative elasticity to reactive power demand of the motor to voltage when the driven load is of constant torque nature. It is true that active power after motor pulls out would be close to zero. But by that time the motor would have gone through the negative slope region of reactive demand and voltage collapse would have progressed into an irrevocable stage. The tendency to voltage collapse will be less if (i) load on the motor is low (ii) motor has high inertia (iii) load on the motor is elastic with respect to speed (e.g. fan load, blower load etc). Elasticity of motor load with respect to speed imparts the much needed positive elasticity to motor active power and tones down the undesirable negative elasticity of reactive demand of the motor with voltage.
Voltage Collapse with Composite Loads
The load at a load bus varies in a complex fashion with voltage and is neither fully static non-fully dynamic. Modeling the load as a composite of equivalent static load and equivalent induction motor at the bus has been attempted by various researchers.
While induction motor load tends to induce voltage instability and collapse upon system disturbances and load disturbances, static loads tend to tone down the voltage instability at the bus during same conditions. Of course, sustained low voltage conditions will exist even with static loads if line is over-loaded. But dynamic loads can provoke instability even in an otherwise healthy system under contingencies. A good mix of static load and induction motor load at the bus can go a long way towards curbing the tendency to voltage instability.
As already pointed out, it is the interplay between bus voltage elasticity with respect to load active and reactive power elasticity with respect to bus voltage that decides voltage instability & collapse. Any measure that tends to reduce bus voltage sensitivity with respect to P & Q will help to counter voltage instability. These measures will include i) reducing the electrical distance between generator and load buds (use of parallel lines, series compensation etc) ii) reducing the reactive flow in the line by shunt capacitor compensation at load bus. The first method makes the bus tight ( a tight bus approaches infinite bus; its short circuit MVA is high) and thereby increases its maximum power transfer capability whereas the second measure shifts the operating point to regions where bus voltage is less sensitive to power variations. Similarly, anything which tends to make P & Q of load elastic with respect to voltage will aid in maintaining voltage stability. Hence, automatic secondary voltage regulation either by utility or by customer would appear detrimental from voltage stability point of view. Similarly a slow response in the tap changer, fast response in capacitor switching systems or use of SVC, fast responding excitation system for generator etc. are seen to be helpful in curtailing voltage instability. One point worth special mention is that capacitors can save a system from voltage collapse due to instability only if they are switched on right at the inception of instability or preferably before that. It is due to this reason that automatically controlled fast responding static var compensators become preferable to static capacitors for management of voltage collapse.
A typical spontaneous voltage collapse (i.e. not due to overload, but due to instability triggered by system or load disturbances) takes about 15 minutes to 30 minutes for competition. The voltage drop is slow during the beginning but toward the end of the voltage collapse interval it takes a decisive turn and flops down fast. The exact evolution of collapse will be governed by the mix of static and dynamic loads, loading level of equivalent induction motor load on the system, tap changer dynamics, action taken by operators, injection of shunt compensation, protection equipment functioning etc.
Voltage collapse in a general power system
The entire power system behind a particular load bus may be equivaleced to a Thevenin’s equivalent in the form of a voltage source (infinite bus) in series with a reactance (short circuit reactance at that bus). Thus, the voltage stability at a load bus may be studied using the principles covered till now. However, this Thevenin’s equivalent method suffers from a shortcoming. When load at a bus change the voltages everywhere (not only at that particular bus) change and hence voltage stability at a particular node is tied up with voltage stability at other nodes too to a small or large extent depending on electrical distances between the nodes.
The principles enumerated and the quanlitative conclusions arrived at in the case of radial power link are valid for any load bus in a general power system. But the exact quantitative determination of voltage stability margins at the various load buses will require complex modelling and computations due to the coupling which exists between various load nodes.
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