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Elfadil.Z. Yahia^{1}, Mustafa A. Elsherif ^{2}, Mahmoud N. Zaggout ^{2}

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Keywords 
Lindex method, voltage collapse, National Grid of Sudan, Voltage stability. 
INTRODUCTION 
Extensive work has achieved on static voltage stability, and several voltage stability indices were derived from static power flow analysis. Power systems operate most of the time under quasisteady state conditions. Disturbances also occur in power systems. Examples of such disturbances are sudden changes in load demand, generator failures or change in transmission system configuration due to faults and line switching. Voltage stability is concerned with the ability of a power system to maintain acceptable voltages in the system under normal conditions and after being subjected to a disturbance [1]. Voltage stability analysis through simulation often requires examination of several system states and many contingency scenarios, for this reason, numerous works have been conducted recently for the prediction of voltage stability and collapse based on the steady state analysis by conventional ways [26]. Some of these focus on static voltage stability analysis with the use of instability measuring indicators. Among them the suitable and simple way of finding the stability margin and the limit by means of the popular Lindex is as described in [7]. This index gives a sufficiently accurate and more practical means of the assessment and can express the stability analysis in a simple way. Huang & Nair [8] discuss a voltage stability indicator whose value changes between zero (no load) and one (voltage collapse). The indicator incorporates the effect of all other loads in the system on the evaluation of index at individual load buses. The overall voltage stability of the system could be identified by the largest value of the index evaluated amongst all the load buses. This indicator can also be used as a normalized quantitative measure, for estimation of the voltage stability margin from the operating point. The indicator uses information of a normal load flow. Static voltage instability is actually related to the reactive power imbalance. The reactive power support which the bus receives from the system, can limit loadability of that bus and hence the entire system. If the reactive power support reaches below the limit, the system will approach to maximum loading point or voltage collapse point due to high real and reactive power losses [911]. Hence, the reactive power supports should be local and adequate in order to avoid problem associated with its transmission [12, 13]. The advantage of the method lies in the simplicity of the numerical calculation and the expressiveness of the result. This paper discusses the formulation of the Lindex in detail. The Lindex simulation results are presented for IEEE 14bus, IEEE 30bus as benchmark and Sudan National Grid. In this paper after detection of proximity to voltage collapse compensation methods of mitigating the voltage collapse by supplying the reactive power at load buses is discussed and compare the results with and without compensation. . On comparison of the results at the load buses it is seen that with capacitor bank at the load buses voltage collapse is mitigated in spite of heavily loading the system. This paper is organized as follows. In section II, describes the problem formulation of the Lindex. Section III presents the case study for IEEE 14bus, IEEE 30bus as benchmark and Sudan National Grid. In section IV, presents results and discussions, and finally section V a conclusion is drawn by discussing the results. 
II. PROBLEM FORMULATION 
In order to become of practical value the indicator L has to be extended to the multinode system, there are two categories of nodes which have to be distinguished. One is characterized by the behaviour of the PQnode which stands for a type of consumer node, the other comprise the generator nodes which may be given by a PVnode or by the slack node. The transmission system itself is linear and allows a matrix representation of the power systems. The power system can be expressed as [14]: 
Where ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ = − ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ are the required values which are obtained by repeated solutions of sparse linear equations. An Lindex value away from 1 and close to 0 indicates improved system security. For an unloaded system with generator/ load buses voltage 1.0 ï00 , the L indices for load buses are close to zero, indicating that the system has maximum stability margin. For a given network, as the load/ generation increases, the voltage magnitude and angles change near maximumpowertransfer condition and the voltage stability index j L (at bus j) values for load buses approaches unity, indicating that system is close to voltage collapse. While different methods give a general picture of the proximity of the system to voltage collapse, the Lindex gives a scalar number to each load bus. Among the various indices for voltage stability and voltage collapse predication, the Lindex gives fairly consistent results. The Lindex for a given load condition is computed for all the load buses and the maximum of the Lindices gives the proximity of the system to voltage collapse. 
III. CASE STUDY 
A singleline diagram of the IEEE 14bus standard system extracted from [15]. It consists of five synchronous machines; two of them are providing both active and reactive power at buses 1 and 2, and the generators at buses 3, 4 and 5 are basically synchronous condensers; the base total loading level of the system is 259 MW and 81.3 MVar. The IEEE 30bus test case is one of the widely used test cases available from the power system test case archive [15, 16]. Bus 1 and 2 are providing both active and reactive power, and the generators at buses 3, 4, 5 and 6 are basically synchronous condensers. All the remaining buses are load buses. The base total loading level of the system is 283.4 MW and 126.2 MVar. The simplified one line diagram of Sudan National Grid is given in the Appendix. The system consists of 45bus in two parts: a 500 kV system at bus 17, 18, 26 , 27 and a 220 kV system at the remaining buses ; the 220 kV network forms the backbone of the transmission system in the Sudanese National Grid, the maximum power can be generated in the Sudan is a round 1700 MW which comes from four power plants, Roseires, Khartoum North, Garri, and Merow, it has 41 transmission lines, 35 load buses, 8 tapchanging transformers, the base total loading level of the system is 1550 MW and 807.1 MVar [6]. 
IV. RESULTS AND DISCUSSION 
The Lindex varies in the range between 0 (for no load) to 1 (voltage collapse point). This index was implemented using MATLAB. Results are obtained for the IEEE (14bus, 30bus) systems as benchmark and the Sudan National Grid (SNG). A. IEEE 14bus System The IEEE 14bus system is used as a benchmark test system. Result of a load flow simulation is used to illustrate how Lindex predicts the voltage stability margin, and the results of the Lindex computation are shown in table 1. It can be seen from the table 1 that the order of the worst load buses with the largest Lindices for the base and the critical case without compensation are almost the same. Therefore, bus 14 represents the weakest bus in the system. 
Figures 1, 2 shows the Lindex versus the bus number without compensation; figure 1 represents Lindex method for the base case and figure 2 represents Lindex at voltage stability limit. For both case bus 14 represents the weakest bus in the system. As indicated from Figs. 2 and 3, the maximum Lindex at bus 14 is 1 and 0.588 respectively, therefore fig 3 after compensation the value will reduce from 1 to 0.588. 
Figures 4, 5 shows the Lindex versus the bus number without compensation; figure 4 represents Lindex method for the base case and figure 5 represents Lindex at voltage stability limit. For both cases bus 30, 29 and 26 represents the weak buses; therefore bus 30 is weakest bus in the system. In case of comparing between fig. 5 and 6 the maximum Lindex at bus 30 is reduce from 1 in fig. 5 to 0.475 in fig. 6 it seem very clear the effect of compensation. 
C. Sudan National Grid A single line diagram showing the Sudan National Grid is attached in appendix A. After intensive simulation of base and critical cases, the Lindex computation of the worst buses in the system is listed in the table 3, which provides the results of the base and critical cases. It can be seen from the table 3 that the order of the worst buses with the largest Lindices for the base and the critical cases without compensation are almost the same. Based on these results it can be conclude that the buses of central areas which are representing 29, 28 and 30 are weakest buses for both cases. Figures 7, 8 shows the Lindex versus the bus number without compensation; figure 5 represents Lindex method for the base case and figure 8 represents Lindex at voltage stability limit. For both case the weakest bus is # 29 and further, it can be seen that buses 28 and 30 are standing out as the most critical. Therefore, the central area represents the weakest buses in the grid. Thus indicating an improvement in the voltage stability margin for the system. A comparison is also made between figs. 8 and 9 it is very clear that the maximum value of lindex is reduced from 0.9781 at bus 29 in fig. 8 to 0.489 in fig. 9 a reduction of about 50%, which indicating an improvement in the system voltage stability margin. 
V. CONCLUSION 
This paper has discussed the formulation of Lindex method in detail. The higher values for L indices are indicative of most critical buses and thus maximum of Lindices is an indicator of proximity in the system to represent voltage collapse. Simulation results of the Lindex were carried out on IEEE 14bus system, IEEE 30bus system (as benchmark) and the Sudan National Grid system. The results clearly indicate that Lindex is stable in all cases above. The Lindex based voltage stability index is simple to compute and it provides a quantitative means to review the voltage stability of the system at any given operating point. On comparison of the results at the load buses it is seen that with a capacitor bank at the load buses voltage collapse is mitigated in spite of heavily loading the system in case of comparing the status of compensated and uncompensated it is seen that the power transfer capability of the bus has been increased after the addition of shunt capacitor. The comparison can best be achieved with help of the graph showing the Lindex in figures of compensated and uncompensated and this method of compensation is lowest cost compare with the other methods. 
References 
