Among steady-state techniques, one of the most common methods for evaluating voltage stability is through the venerable P-V curve, also known as the “knee” curve. This method is used to identify the real power, or megawatt, margin to the point when the transmission grid is no longer able to support voltages, a state the industry has referred to as “voltage collapse.” Many analysts have long suspected that the P-V curve at best provided an approximate indicator, and that there was a lot more it was not capturing. Recent indications from near voltage collapse events seem to confirm that there is indeed a lot more, which may perhaps require a re-thinking of the whole process of using steady-state methods for voltage analysis.
Flexing the Knee
In the traditional P-V curve analysis, the power across a set of transmission paths is gradually increased by either increasing load or reducing generation at the receiving end. This would simulate, for example, the gradual rise in demand over time, such as the weekday load rise, or the adjustments to 
increase imports into a region. Several nodes or substations are selected where voltage is monitored as a series of power flow solutions is performed. The result is the P-V curve with its characteristic “knee” shape.
The power flow solution assumes at least one critical factor that gives analysts concern — generators are modeled by static reactive limits. For periods of a second to several minutes, generators are capable of delivering reactive power of up to 2 to 3 times the static limits, so-called transient vars [1]. What really happens in the short timeframes? Is the effect sufficient to alter the P-V curve?
The answer to the second question is apparently “yes.” But surprisingly, the basis goes beyond just the transient reactive capability of generators.
Generators deliver extra vars at the expense of thermal heating. Oversized generators such as steam turbines in coal and nuclear plants are able to deliver more transient vars since they are able to tolerate more heating than smaller units such as gas turbines. However, in a timeframe of minutes, rotor temperature control requires the reduction of transient vars through either overexcitation limiter protection or operator action. When generators reduce var output, voltages drop. At this point, the action shifts away from the observation of transmission network operators over to the distribution system!
Tap-changing transformers serving distribution circuits mask dropping voltages in the transmission network by maintaining setpoint voltages at customer loads. The tap-changers will do this until they exhaust the available tap steps, at which point customer loads are exposed to a rapidly deteriorating voltage. Capacitors do not help during this time as they themselves are losing reactive output at the rate of square of the drop in voltage. In addition, the following effects seek to push voltages further downwards:
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Loads that have low voltage tolerance that allow them to recover to their normal demand level at lower terminal voltages, such as variable speed motors, or thermostat-controlled loads
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Motor loads that stall, causing an increase in reactive demand
However, there are other effects that help curb the deterioration of voltage:
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The natural response of loads to decrease power demand as terminal voltages decrease
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The dropout of contactors due to low voltage, most notably in motors, such as air conditioners and pumps
The combined effects of the above, as seen from the transmission system, produce a quasi-equilibrium characterized by voltages holding steady at the 
transmission level, somewhere in the .9 to 1.0 per unit range, while distribution voltages experience widely fluctuating and damaging voltages. This condition may remain in place over an extended range of transfers and load levels, and thus prevail over several hours. The net effect is a sort of voltage “ledge” in the P-V curve where transmission voltages appear to be “normal.” One could liken this to the Brownian movement that occurs in the molecules of a liquid that from all outward appearances is sitting still.
Case Study
In July 1999, a situation characterized as a “near voltage collapse event” provides a good, practical example of the voltage ledge.
On this particular summer day, extremely high temperatures were projected along the Eastern US seaboard. By noon, emergency dispatch conditions and a 5% voltage reduction (where the distribution voltage setpoints are reduced by 5%) had been called for. At about 1 PM, voltages at the 500 kV network had dropped to 1.0 per unit, an unusual event, perhaps the first time ever, and yet not outside acceptable operating limits. This condition remained for about 3 hours until load outages and cooling outside temperatures allowed 
voltages to recover. During 3 hours on the voltage ledge, over a thousand pole-top distribution transformers failed, numerous motors stalled and re-started, some repeatedly, primarily air conditioners, and reports of damaged computers and TV sets and other sensitive equipment clogged call centers. Distribution connected gas turbines failed to startup because of low terminal voltage conditions.
In the aftermath of this event, root cause analysis indicated the need for certain critical measures including more accurate modeling of generator reactive capability, the need for static var devices, must-run generators and possibly undervoltage load shedding and a broader study of the conditions that led to the disturbance.
Additional Comments
There is no doubt that the transient effects have an impact on the P-V curve. Most of the time, the steady-state results are more conservative in terms of providing margin to voltage instability. However, it is significant to note that the voltage ledge starts around 1.0 per unit voltage, above where most P-V curves would consider voltage collapse.
Clearly, dynamic simulation techniques provide a clearer view of grid response during the transient period. Models for self-restoring loads, overexcitation limiters, motor stall and re-start and tap-changing transformers extend the reach of simulations beyond a few seconds to the minutes timeframe that is needed to see how systems behave in the voltage ledge.
References
The following references provide additional information on the concepts and events described in this article.
- A. Murdoch, G.E. Boukarim, B.E. Gott, M.J. D’Antonio and R.A. Lawson, “Generator Overexcitation Capability,” Panel Session Summary for the IEEE/PES 2001 WPM, Columbus, OH, jointly sponsored by the Excitation System Subcommittee and the
Power System Stability Controls Subcommittee. - Ricardo Austria, et al, “Voltage Stability Assessment of the National Grid System Using Modern Analytical Tools,” presented at the 2001 Transmission and Distribution Conference and Exposition, October 28 – November 2, 2001, Atlanta, Georgia, USA.
- Robert T. Eynon, Thomas J. Leckey, and Douglas R. Hale, “The Electric Transmission Network: A Multi-Region Analysis,” EIA, DOE.
- “Report of the U.S. Department of Energy’s Power Outage Study Team, Findings and Recommendations to Enhance Reliability from the Summer of 1999,” Washington, DC, March 2000.
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