Keywords: Voltage Stability, Voltage Collapse, Reactive Power, Steady-State Voltage Stability, Transient Analysis
Voltage stability is a major concern in planning and operations of power systems. It is well known that voltage instability and collapse have led to major system failures;with the development of power markets, more and more electric utilities are facing voltage stability-imposed limits.The problem of voltage stability may be simply explained as inability of the power system to provide the reactive power or the egregious consumption of the reactive power by the system itself. It is understood as a reactive power problem and is also a dynamic phenomenon. [8,9] The objective of this paper is to develop a fast and simple method, which can be applied in the power system online, to estimate the voltage stability margin of the power system. In general, the analysis of voltage stability problem of a given power system should cover the examination of these aspects:
Voltage stability analysis often requires examination of lots of system states and many contingency scenarios. For this reason, the approach based on steady state analysis is more feasible, and it can also provide insights of the voltage reactive power problems. A number of special algorithms have been proposed in the literature for voltage stability analysis using static approached [2-4], however these approaches are laborious and does not provide sensitivity information useful in a dynamic process. Voltage stability is indeed a dynamic phenomenon. [8,9] Some utilities use Q-V curves at some load buses to determine the proximity to voltage instability [6]. One problem with Q-V curve method is that by focusing on a small number buses, the system-wide voltage stability problem will not be readily unveiled. An approaches, model analysis of the modified load flow Jacobian matrix, has been used as static voltage stability index to determine vulnerable bus’s voltage stability problem [7].
This paper explores the online monitoring index of the voltage stability, which is derived from the basic static power flow and Kirchoff law. A derivation will be given. The index of the voltage stability [3] predicts the voltage problem of the system with sufficient accuracy. This voltage stability index can work well in the static state as well as during dynamic process. It can also be used to find the vulnerable spots of the system, the stability margin based on the collapse point, and the key factors for the voltage stability problem, etc
A simple power system is considered, through which the useful index of the voltage stability is derived. As showed in Fig.1, whereby bus 1 is assumed as a generator bus, and bus 2 is a load bus whose voltage behavior will be our interest.
This simple system can be described by the following equations (where the dot above a letter represents a vector):
Thereby an indicator has been derived which can be used for monitoring the voltage stability problem of the system and for assessing the degree of risk for a potential voltage collapse. When 0 2 S = , the indicator will be zero YL Load V2,I2,S2 YS G,V1 and indicates that there will be no voltage problem. When S2 = 1 , the voltage at load bus will collapse. One example of a single generator and load system was constructed to demonstrate the correctness of the indicator
Continuously change the load at bus 2, and keep the power factor of the load to find the collapse point
As shown in the basic theory of the multi-bus power system, all the buses can be divided into two categories: Generator bus (PV bus and Slack bus) and Load bus (PQ bus). Because the voltage stability problem is reactive power relative problem, and the generator bus can provide the reactive power to support the voltage magnitude of the bus, it is absolutely necessary that the all of buses be distinguished. The power system can be expressed in the form through Kirchoff Law:
Hence, we see that the voltage of the load bus j is affected by an equivalent complex power and by an equivalent generator part
.
To compare the equation (17) and (2), we can observe that they have an identical form, and the voltage stability of the multi-bus system has been equivalent to a simple single generator and load system. The indicator of the voltage stability of the load bus j will be easily obtained:
Thereby it is clear that the indicator of the voltage stability at a load bus mainly influenced by the equivalent load ¢j S , which has two parts: the load at bus j itself, and the ‘contributions’ of the other load buses (showed at equation 20). When the load at a load bus changes, it will influence the indicator. On the other words, the voltage stability problem is a system-wide problem, not a local problem.Through equation (20), the contribution of any other load bus on the load bus j can be numerically updated and computed. It is a very important concept for the deregulated power market, and will help the customers and ISO to evaluate the responsibility of voltage stability problem.
The indicator is an effective quantitative measurement for the system to find how far is the current state of the system to the voltage collapse point. All the derivations are correct as long as that all the generator bus of the system have the enough reactive power supply to maintain the magnitude of voltage as constant. If some of the generators are unable to maintain the voltage magnitude, these generator buses will become load buses, the load bus congregation will expand; the indicator will have a discrete jump, which will be shown at the following demonstration.
The WSCC 9 bus system is taken as a sample system to illustrate the applicability of the indicator L to a multi-bus system. The test system is shown in Figure 2
Fig 2 Case scenarios of the test system:
Buses 1 to 3 are generation buses; there are no generators or loads at buses 4, 6 and 8. Three case scenarios have been simulated to study the steady state voltage collapse at the load buses and their respective L index.
Increasing load at bus 5 and observing the indicator L at bus 5.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus 7 is taken as 100+j35 MVA and the load at bus 9 is taken to be 125+j50 MVA.
The collapse occurs when the load at bus 5 is about 350+j115.5
P5(M.W) | 0 | 100 | 200 | 250 | 300 | 375 |
V5(p.u) | 1.014 | 0.971 | 0.909 | 0.867 | 0.809 | 0.635 |
L55 | 0.1662 | 0.2439 | 0.3177 | 0.3802 | 0.5171 | 0.8959 |
Increasing load at bus 5 and observing the indicator L at bus 7.
As shown in graph the voltage collapse point would not identified with index of L7 for the increasing of load at bus 5. For this simulation, load at Bus 5 is varied, the load at bus 7 is taken as 100+j35 MVA and the load at bus 9 is taken to be 125+j50 MVA
P5(M.W) | 0 | 100 | 200 | 300 | 350 | 375 |
V5(p.u) | 1.014 | 0.971 | 0.909 | 0.809 | 0.707 | 0.635 |
L75 | 0.1124 | 0.1665 | 0.2305 | 0.3171 | 0.3941 | 0.4541 |
Increasing load at bus 5 and observing the indicator L at bus 9.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus 7 is taken as 100+j35 MVA and the load at bus 9 is taken to be 125+j50 MVA.
The collapse occurs when the load at bus 5 is about 375+j123.75
P5(M.W) | 0 | 100 | 200 | 300 | 350 | 375 |
V5(p.u) | 1.014 | 0.971 | 0.909 | 0.809 | 0.707 | 0.635 |
L95 | 0.1532 | 0.2262 | 0.3144 | 0.4392 | 0.5567 | 0.6519 |
Increasing load at bus 7 and observing the indicator L at bus 5.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus 5 is taken as 90+j30 MVA and the load at bus 9 is taken to be 125+j50 MVA.
The collapse occurs when the load at bus 7 is about 425+j148.75
P7(M.W) | 0 | 100 | 200 | 400 | 450 |
V7(p.u) | 1.014 | 0.986 | 0.944 | 0.775 | 0.723 |
L57 | 0.1627 | 0.2357 | 0.3215 | 0.6129 | 0.9976 |
Increasing load at bus 7 and observing the indicator L at bus 7.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus5 is taken as 90+j30 MVA and the load at bus9 is taken to be 125+j50 MVA.
The collapse occurs when the load at bus 7 is about 435+j152.25
P7(M.W) | 0 | 100 | 200 | 300 | 450 |
V7(p.u) | 1.014 | 0.986 | 0.944 | 0.833 | 0.723 |
L77 | 0.1127 | 0.1608 | 0.2216 | 0.3045 | 0.5356 |
Increasing load at bus 7 and observing the indicator L at bus 9.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus 5 is taken as 90+j30 MVA and the load at bus 9 is taken to be 125+j50 MVA.
The collapse occurs when the load at bus 7 is about 425+J148.75
P7(M.W) | 0 | 100 | 200 | 300 | 450 |
V7(p.u) | 1.014 | 0.986 | 0.944 | 0.833 | 0.723 |
L97 | 0.1508 | 0.2185 | 0.2983 | 0.4009 | 0.6572 |
Increasing load at bus 9 and observing the indicator L at bus 9.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus5 is taken as 90+j30 MVA and the load at bus7 is taken to be 100+j35 MVA.
The collapse occurs when the load at bus9 is about 367.8+j147.12
P9(M.W) | 0 | 100 | 200 | 300 | 375 |
V9(p.u) | 1.001 | 0.970 | 0.929 | 0.849 | 0.663 |
L99 | 0.1265 | 0.1982 | 0.2644 | 0.4343 | 0.7305 |
Increasing load at bus 9 and observing the indicator L at bus 5.
As shown in graph index L approaches one at the collapse point. For this simulation, the load at bus 5 is taken as 90+j30 MVA and the load at bus 7 is taken to be 100+j35 MVA.
The collapse occurs when the load at bus 9 is about 379.6+151.84
P9(M.W) | 0 | 100 | 200 | 300 | 377 |
V9(p.u) | 1.001 | 0.970 | 0.929 | 0.849 | 0.663 |
L59 | 0.1352 | 0.2142 | 0.3070 | 0.4338 | 0.7302 |
Increasing load at bus 9 and observing the indicator L at bus 7.
As shown in graph the voltage collapse point would not identified with index of L7 for the increasing of load at bus 9. For this simulation, load at Bus 5 is varied, the load at bus 5 is taken as 90+j30MVA and the load at bus 7 is taken to be 100+j35 MVA
P9(M.W) | 0 | 100 | 200 | 300 | 375 |
V9(p.u) | 1.001 | 0.970 | 0.929 | 0.849 | 0.663 |
L79 | .0926 | 0.1461 | 0.2094 | 0.2929 | 0.4178 |
A real time measurement based voltage stability indicator for monitoring of the power systems is presented. We verify our approach by both static and dynamic simulations. We conclude that
The needed information can be obtained through local measurements and data exchanges of among preset buses.
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