Algorithm to Determine Minimum Contingency Set in Voltage Collapse Scenarios Jeremiah D. Smith, Member, IEEE, Peter W. Sauer, Fellow, IEEE Abstract—This investigation explores how many system elements must be outaged in order to cause system-wide voltage collapse. The IEEE 118-bus case was used to conduct simulations which implemented continuous steady-state load flows and power-voltage curves as indicators. The base case maintained constant power factor load throughout, and applied given generator capability curve data and automatic generation control data to each unit. Using the n – 1 contingency list, sensitivities were found for the system with respect to increased system load given a unique n – 1 topology. The minimum number of system elements can be found by systematically repeating this process while outaging the most sensitive element at each step. Index Terms—continuous load flow, p-v curve, p-v study, static bifurcation, voltage stability.
I. INTRODUCTION
V
OLTAGE stability is a growing concern in today’s transmission systems. Recent worldwide blackouts have raised many industry-wide concerns. A voltage collapse is characterized by the loss of voltage control throughout a power system [1]. Oftentimes, a slight decrease in voltage is normal as system load increases. These gradual increases cause slow changes in the system operating point. However, there comes a point when the load increases enough to where the voltage drops off drastically, and a sharp change occurs. At this point, the voltage is highly sensitive to even the smallest perturbations in load. Projections indicate that the load in the United States will increase between 1.3% and 1.9% every year until 2010 [1]. Even though this is a relatively modest increase, it still has the potential to cause problems with the current transmission grid. These problems include added stress on the system stemming from losses in generation capacity, the shift in power flows associated with changing fuel costs [3], and the present trend of transmitting larger amounts of power over longer distances. An ongoing debate currently exists on whether voltage stability is of a static or dynamic nature. Power systems are subject to many parameters that change over time. As these This work was supported by the Power Affiliates Program and Grainger Foundation Endowments to the University of Illinois. J.D. Smith is with the Electrical and Computer Engineering Department at the University of Illinois, Urbana, IL 61801 USA (e-mail:
[email protected]). P.W. Sauer is with the Electrical and Computer Engineering Department at the University of Illinois, Urbana, IL 61801 USA (e-mail:
[email protected]).
parameters change, the system undergoes fluctuations and usually settles into a new stable equilibrium point in a normal operating state. However, there can be times when the system will not settle, and hence, will become unstable. It is accepted that this can occur in one of two ways: the sudden appearance of self-sustained oscillations or the disappearance of the equilibrium point [3], [4]. The disappearance of the equilibrium point is known as a static bifurcation. Bifurcation theory looks at how solutions of the power system branch as the system parameters vary [5]. This investigation stems from the relationship between real power load and voltage at transmission voltage levels, and the occurrence of steady-state bifurcations within a system. It deals with only static analysis, neglecting the dynamics that can contribute to voltage instability. The assumption is that changes in system parameters, specifically load growth, occur in a quasi-static fashion, allowing for the use of continuous load flow calculations to collect data. The power-voltage (p-v) relationship at a given bus is used to illustrate p-v curves, which give insight to system robustness related to individual bus voltage. These data, which provide voltage information with respect to a system-wide parameter, are then used to form an elemental sensitivity for each contingency. This sensitivity is then used to form an algorithm that can predict the minimum number of elements which can be outaged to cause system-wide voltage collapse. Examples are provided on a small-scale system in order to show the flexibility and accuracy of the algorithm. II. P-V STUDIES USING P-V CURVE INDICATORS A p-v curve is an excellent indicator of the voltage stability in a given system. It displays the voltage at a given bus or node, and plots it with respect to the overall system real power load. It is used extensively in p-v studies in order to indicate how system voltage is behaving with respect to system load, and at what value of load system voltage and stability will be compromised. Basically, a p-v study is the process used to construct a p-v curve. It uses a continuous load flow to grow overall system load at user defined increments throughout the system. This load increment can be distributed evenly throughout the system (known as distributed load) or it can be distributed according to user settings in order to stress certain areas of a system more than others. As the system load increases at each step, sources must also increase, and the system must find a
new operating point. This increase in source real and reactive generation can also be distributed evenly throughout the source set (known as distributed generation) or can be distributed according to particular user settings. This process continues until the load grows too high, and system voltage can no longer be sustained. This point of collapse is referred to as the “nose” of the p-v curve, or the point of maximum power transfer for the system. A p-v curve is most easily constructed from the continuous load flow algorithm used in a p-v study. Fig. 1 illustrates the behavior of an example 2-bus, 1-machine system.
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distributed throughout the system, 14 switched shunts at various load buses, and 54 generating units, some of which were synchronous condensers. The base case load was summed to be 3668 MW and 1438 MVAR, whereas in the base case, generation was measured to be 3725 MW and 333 MVAR. A one-line diagram of the IEEE 118-bus test system can be found in [9] for reference. The 118-bus test system provides a great example for analyzing a p-v study. Other test systems such as the IEEE 14bus, 30-bus, and 57-bus provide less meaningful results because of their lack of system complexity, and their small overall size. The 118-bus test system is large enough to mimic the behavior of realistic systems. At the same time, it is small enough that exhaustive search methods can be performed in order to verify results. A p-v study of the 118-bus system was done to the base case in order to find normal operating voltage limitations and transfer limitations of the system. First, the simulation was set-up by defining all the userdefined parameters in the study. Fig. 2 shows the initial set-up stage for a p-v study using PowerWorld Simulator 11.0 software.
Nominal Shift (MW)
Fig. 1. Example of a complete p-v curve. It shows the trend that voltage exhibits with the increase of system load at a given bus.
In this system, the generator was attached to bus 1, load was drawn from bus 2, and a single transmission line with impedance 0.02828 + j0.2074 per unit joined the two buses. This p-v curve shows the behavior of the voltage at the load bus, given the growth in load. The x-axis values indicate the nominal shift of the system. The nominal shift is the growth greater than the base case load. In this model, the base case load was 52.2 MW. Therefore, from this plot the system collapsed at approximately 160 MW nominal shift, or at a total load of 212 MW. The lower part of the curve is only included for completeness. Operation on the bottom side of the curve seldom, if ever, occurs due to the lower associated voltages. This is usually undesirable, and reflects the unstable operating points of the system. Typically, once the system reaches the nose, or maximum power transfer point, it collapses. However, corrective actions of the system, such as under-voltage relays, usually act long before the system reaches its maximum power transfer point. III. EXAMPLE IEEE 118-BUS SYSTEM P-V STUDY The test system used to conduct specific analysis of p-v curves was the IEEE 118-bus test system. In constructing this case, data were taken from various sources [6]-[8]. It was found that the data between sources differed slightly, so data were used from [6] whenever a discrepancy was encountered. The 118-bus system was an actual power system within AEP (American Electric Power) in late 1962, early 1963. The final base case system that was used in this study had 118 buses of nominally 345 kV and 138 kV, along with 186 lines, nine of which were transformers. In addition, it had 91 loads
Fig. 2. Snapshot of p-v study set-up screen in PowerWorld 11.0. It shows all the user-defined parameter settings for the example study.
The source area group was composed of generators that compensated for the increase in load. This means that machines such as synchronous condensers, which typically only provide voltage support, were omitted from the set. The sink group was defined by any or all loads within the system. The user specified which loads within the system were grown in order to stress different areas of the system. Both the source and sink elements were set to contribute to the growth at a given percentage by using participation factors that were set within the simulation. The “AGC Gen” source set included all generation units (less synchronous condensers) that contributed to the growth in load. The participation factor for each machine was set to the percentage of base case generation it contributed. For instance, if a unit were to generate 50 MW in the base case where total generation was 2000 MW, the unit would have a factor of 0.25. This means that for a 1 MW increase in load, its output would increase by 0.25 MW until other units hit physical
limits. When a given unit no longer increased its output, the needed generation came from units not yet at their limits. This effect propagated until either all units reached their limits, or the system collapsed.
Fig. 3. Results menu for p-v study. It shows system collapse point, total megawatt transfer, and worst voltage bus.
The results menu yielded the reason for collapse, the point of collapse or nose of the system, the number of buses with unacceptable voltages, and the bus with the lowest voltage. Collapse occurred at 3235 MW nominal shift. At base case collapse, the system had fifteen buses with inadequate voltage below 0.9 p.u. This minimum voltage limit was user-defined. The worst voltage occurred at Bus 86, where the voltage was 0.7 p.u. Detailed p-v curves were constructed to show the behavior of such bus voltages. For clarity, only seven curves are displayed. These curves are chosen because out of all 118, their voltages are the most sensitive to growth in system load. Fig. 4 illustrates the p-v curves of these buses. 1.05
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Generator capability curves were also defined in order to make the system more realistic. This curve, one per each generator, assumes that real power output is the highest priority, and estimates how much reactive power can be produced in either actual output or system reserve if it is needed. Also, the “Distributed Load” sink set included all 91 loads within the system. The load participation factors were set in a similar fashion. In this analysis, a distributed load scenario was used. The participation factor of each load was set to be its percentage of base case load. Again, if a load made up 20% of the base case load, its factor would be 0.2. The limiting effect shown by the generation was not an issue here, so these participation factors held constant throughout the study. The various step sizes were also set before the study was started. The initial step size was set at an amount that maximized efficiency and accuracy for a given system. A high step size will increase the load in greater increments, causing faster simulation times. However, fewer points will be taken, which may result in less accuracy. Conversely, a smaller step size will provide high accuracy but will increase overall computation time. The minimum step size was needed in order to provide some stopping criteria for the study. This parameter, combined with the reduction in step factor, provided the means to find the nose of the system. The study continued to iterate around the nose of the system until the current step size was less than or equal to the minimum step size parameter. In this case, the initial step size was an exact multiple of the reduction in step factor and the minimum step size, so decimals or imprecise stopping points were avoided. Finally, the power factor was held constant throughout the study at 0.93. This value was the overall power factor in the base case given the base case real and reactive loads. Once all these parameters were set, the user had to decide which variables to track throughout the study. Many different variables exist and can be monitored from step to step as the study progresses. Obviously, the user needed to track the system megawatt load as well as the voltage at certain buses in order to construct a detailed p-v curve, but many other variables can be observed during the study. The next step in the process was to actually run the study. Fig. 3 shows the results menu after running the p-v study for the base case.
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Fig. 4. The p-v curves for the chosen buses. This plot depicts some of the buses which exhibit sharp voltage decreases near the system collapse point.
Fig. 4 shows that the voltage at Bus 86 fell off sooner than the other bus voltages in the system. The load buses, Buses 8386, seem to be most affected as suspected. Their voltages remained somewhat constant until sharp changes occurred closer to the nose point of the system. These results show that p-v curves can adequately show voltage stability limits of a system, and can predict the system collapse point as a function of system load. Ultimately in the 118-bus case, the line losses associated with the line connecting Bus 86 and Bus 87 became too large, and the voltage at Bus 86 began to degrade. Eventually, this led to the collapse of the entire system when the equilibrium operating point disappeared, and no alternative stable solution could be found.
IV. CONTINGENCY EFFECTS ON SYSTEM COLLAPSE POINT In realistic instances, voltage stability is normally compromised under heavy loading conditions in a contingency state of operation. Many blackout cases document a cascading effect of contingencies as the underlying cause of system voltage collapse. The definition of a contingency is “an event that may occur but that is not intended.” Contingency situations in a power system can occur for a multitude of reasons, but most are planned for and are often manually instigated for maintenance or upgrading. Unplanned outages are more of a concern to system operators. The outaging of an element causes perturbations in the system, and causes it to regain equilibrium at a new operating point. Problems arise when the system cannot find a new stable operating point. The importance of contingency analysis in steady-state realtime operation and in long-term transmission planning is important in designing a robust system [10]. Each contingency puts added stress on other system elements. It can cause other elements to violate thermal limits and stability margins. Fig. 5 shows how different contingencies affected the maximum power transfer point of the system. The final difference is hard to see in the plot, but by disconnecting Line 88-89, the system nose decreased to 3215 MW nominal shift. The same can be said when Generator 82 was disconnected. The maximum power transfer point decreased 40 MW resulting in a collapse at 3195 MW nominal shift. 1.05
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( j k l)! (( j k l ) 2) ! 2 !
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V. MINIMUM CONTINGENCY SET ALGORITHM The p-v curves and studies, along with the effects of contingency analysis can be used in monitoring system voltage stability. A minimum contingency set algorithm (MCSA) can be used to determine which system elements are most vulnerable to compromising system voltage stability. MCSA gauges the sensitivity of a system to an increase in system load for the loss of an element. By using a p-v study that incorporates contingencies, an operator can effectively identify the number of elements which will cause voltage collapse in the system. Each time the study is run different contingencies will cause the system to reach the nose at different values of load. The contingency that causes the greatest difference between the collapse point of a system and the base case collapse point will be deemed the most sensitive element in the system. Once that element is outaged and removed from the study, the process is rerun, and a new element is found. The system will eventually collapse by systematically removing the most sensitive elements. The flowchart in Fig. 6 shows the process of determining the minimum number of elements needed to produce collapse. This algorithm is an iterative process that continues to remove elements until no more load growth can be supported.
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Fig. 5. System collapse under different contingencies. These curves show the effect different contingencies have on the voltage at Bus 86.
The biggest difference can be seen in the trajectory of the curves leading up to the nose point. All three curves exhibit a fairly constant rate of voltage decay until they are in close proximity to the point of collapse. At this point, the voltage begins to drop off very quickly for all cases. In most control systems, contingencies are considered when designing and planning a power system. Many utilities use n – 1 and n – 2 security criteria to ensure proper operation and reliability. The n – 1 contingency set refers to all single element outages in a system. Therefore, a system with j generators, k transformers, and l transmission lines will have a n – 1 contingency set of size n where n j k l . The n – 2 contingency set can be thought of in this same manner, except that it takes into account all double element outages in the system. It incorporates every combination of two elements which can be outaged simultaneously. So for the
Determine parameters (step size, power factor, AGC units)
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Fig. 6. Flowchart for minimum contingency set algorithm.
One benefit to this algorithm is that it only involves the n – 1 contingency set. Normal contingency analysis is done using exhaustive methods, where all combinations of elements are analyzed. This greatly increases complexity when dealing with large-scale systems. The complexity of MCSA only increases in the amount of iterations the algorithm must progress through in order to find the minimum number of elements. Since it only depends on the n – 1 contingency set, the complexity will increase proportionally to system size instead of increasing exponentially. The 118-bus test case was analyzed using MCSA. The results are shown below:
This scenario was analyzed using MCSA. The results are shown below: TABLE II 118-BUS TEST CASE BRANCH ELEMENT ANALYSIS Element 1
Element and Nose (MW) T_8-5
1955
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TABLE I MINIMUM CONTINGENCY SET Element
According to MCSA, the 118-bus case collapsed when two elements were outaged simultaneously. The value of 0 MW nominal shift indicates that no more load can be added to the base case load. Therefore, the system could not sustain even base case load at this point, which implies collapse. This does not mean that collapse is imminent for all combinations of double element contingencies; it only means that the algorithm found that possibly two elements could cause collapse. This is also the same minimum contingency set that was obtained when the system was examined exhaustively. Observing the 118-bus test case one-line diagram, it can be seen that when Transformer 8-5 and Generator 12 are simultaneously outaged, the reactive support in that region becomes inadequate due to the increased load and losses. This causes the system voltage to degrade uncontrollably at high system loading. Assume that a given utility is concerned about the physical security of their system. Since generators are typically monitored frequently, have a high level of security, and exhibit many safety features, they are not very vulnerable to physical sabotage. Transmission lines, however, typically span hundreds of miles, and traverse remote locations. A utility might wish to only determine how many lines or transformers it would take in order to cause a voltage collapse. A benefit of MCSA is that by manipulating the contingency set, the operator can choose which elements get included in the algorithm. For example, if only lines and transformers are to be analyzed, the operator can initially load the n – 1 contingency set and then remove all generators from the set. This would leave only lines and transformers to be analyzed. Now, each eliminated element will be a branch device, and eventually the system will collapse due to only the loss of branch devices.
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With the exclusion of all generators, Table II shows the minimum contingency set increased to three elements. The first and most sensitive element was still Transformer 8-5, but from the results of the algorithm, Line 16-17 was the next most sensitive. A drawback to this algorithm is that the given minimum contingency set is not unique. Observing Table II, after the initial transformer was outaged, different combinations of double line outages caused the system to collapse. The minimum number remained consistent, but there existed different elements which composed this number. Initially, the region in close proximity to Bus 86 seemed to be the main problem in the test case. However, from this analysis it seemed that if the right combinations of contingencies occurred in the northwestern region of the system, the system could collapse at base case load. This shows the importance of contingency planning and the use of p-v studies in system operation. As expected, when system size increased, the minimum contingency set also increased. In the case of the 118-bus test system, all combinations of three elements caused 3 652 110 contingencies to be introduced. A realistic large-scale case could potentially introduce billions of different contingency combinations which would all have to be analyzed for a complete exhaustive analysis. MCSA allows the use of only the n – 1 contingency set which allows for quicker simulation times. Although a fair amount of time would be needed to complete the additional iterations, other methods can theoretically be applied in order to be more efficient. The use of fast contingency analysis techniques can be implemented in order to speed up the iterative process. By systematically eliminating non-threatening contingency states, the elapsed computation time required by each iteration can be drastically reduced. The implementation of distribution factors could also speed up the iterative process. They could serve as performance
indices that could indicate which line outages will negatively affect the flows on other lines. The contingencies which produce large distribution factors will rank as highly sensitive contingent states, while distribution factors with small values will not create voltage compromising scenarios. VI. CONCLUSION The purpose of this investigation was to effectively show the use of an algorithm used to determine the minimum contingency set of a power system that causes voltage collapse. The p-v studies were utilized in forming sensitivities which gauged the effect of each outaged element on system performance. These p-v studies implemented p-v curves to indicate the nose of each system in the base case, as well as in each contingent state. PowerWorld Simulator 11.0 was used as the primary simulation software. The primary test medium was the IEEE 118-bus test system. MCSA has many benefits in finding the minimum contingency set which induces voltage collapse. It can make major improvements in computation time because it deals with far less contingencies. In order to exhaustively examine the 118-bus test case with triple combinations of elements in the contingency set, over 3.5 million combinations were introduced. MCSA only dealt with a total of less than 1 000 throughout all iterations. Another benefit is that it also has the ability to analyze the system for only user-defined elements. By manipulating the n - 1 contingency set, the user can include any group of elements. These groups can be strictly lines, generators, transformers, or any combination of the elements. Nonetheless, the algorithm is not without a few drawbacks. MCSA does not provide a unique solution set. It is possible for different elements to compose the contingency set. The only thing the method can accurately provide is the size of the contingency set, or the minimum number of elements. There is work that can be looked at in the future on this topic. The ability to parallel p-v studies along with the implementation of fast contingency analysis methods and distribution factors could lead to advancements in computation time. Islanding effects could also cause potential problems. If islanding were to occur, the first thought is that MCSA would have to be run on the two separate subsystems instead of just the one larger system. Finally, a formal proof would be effective in proving the overall accuracy of the algorithm. This work could stem from the work done on automatic contingency selection [10]. A full detailed proof would provide an effective reference for all simulations, and reinforce the results provided by the algorithm. REFERENCES [1] [2]
[3]
M. Crow and B. Lesieutre, “Voltage collapse, an engineering challenge,” IEEE Potentials, vol. 13, pp. 18-21, Apr. 1994. R. Billinton et al., “Reliability issues in today’s electric power utility environment,” IEEE Transactions on Power Systems, vol. 12, pp. 17081714, Nov. 1997. Kwatney et al., “Static bifurcation in electric power networks: Loss of steady-state stability and voltage collapse,” IEEE Transactions on Circuits and Systems, vol. CAS-33, pp. 981-991, Oct. 1986.
V. A. Venikov et al., “Estimation of electrical power system steady-state stability,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, pp. 1034-1040, May/June 1975. [5] H.-D. Chang et al., “On voltage collapse in electric power systems,” IEEE Transactions on Power Systems, vol. 5, pp. 601-611, May 1990. [6] Illinois Institute of Technology, “IEEE118bus_data.xls,” Nov. 2003, http://motor.ece.iit.edu/data/IEEE118bus_inf/IEEE118bus_data.xls. [7] University of Washington, “118 bus power flow test case,” 1993, http://www.ee.washington.edu/research/pstca/formats/cdf.txt. [8] PowerWorld Corporation, “The IEEE 118 bus case,” Sept. 1997, http://www.powerworld.com/cases.asp. [9] Jeremiah Smith, “Algorithm to Determine Minimum Contingency Set in Voltage Collapse Scenario,” M.S. thesis, University of Illinois at Urbana-Champaign, May 2006. [10] T. A. Mikolinnas and B. F. Wollenberg, “An advanced contingency selection algorithm,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, pp. 608-615, Feb. 1981. [4]
