# Strategy to minimise the load shedding amount for voltage collapse

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Strategy to minimise the load shedding amount for voltage collapse

www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 12th May 2010 Revised on 23rd September 2010 doi: 10.1049/iet-gtd.2010.0341 ISSN 1751-8687 Strategy to minimise the load shedding amount for voltage collapse prevention Y. Wang1 I.R. Pordanjani1 W. Li1 W. Xu1 E. Vaahedi2 1 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 British Columbia Transmission Corporation, Vancouver, BC, Canada E-mail: [email protected] 2 Abstract: This study presents a practical approach for determining the best location and the minimum amount of the load to be shed for the event-driven-based load shedding schemes. In order to ﬁnd the above key parameters, a non-linear optimisation problem needs to be solved. A multistage method is proposed to solve this non-linear problem. The main idea of this method is to solve the optimisation problem stage by stage and to limit the load shedding to a small amount at each stage. Using this approach, the non-linear optimisation problem will be converted into a series of linear optimisation problems. By solving these linear optimisation problems stage by stage, the optimal solution to the original non-linear problem is obtained. Furthermore, in order to quickly identify the candidate load shedding locations in the multistage method, a novel multiport network model is proposed. Based on the multiport network model, fast ranking of the load locations and of the generators’ participation factors can be done with little calculation efforts. Details of the problem formulation and the solution strategy are presented here. The proposed strategy is illustrated and veriﬁed by using the IEEE 14-bus system, IEEE 118-bus system and a real 2038-bus power system. 1 Introduction With the increasing demand for electrical power and due to economic and environmental constraints, power systems are currently being operated closer to their limits than they were previously. This has led to an ever-increasing risk of voltage instability, which is the most important limiting factor for power transmissions. Among different countermeasures for the prevention of the voltage instability, load shedding is the last line of defence when there is no other alternative to stop an approaching voltage collapse [1]. The growing concern on voltage instability incidents has attracted a great deal of attentions. Signiﬁcant progress has been made in the research on the implementation of the load shedding schemes over the past few decades [2 –8]. Undervoltage load shedding scheme is normally the primary choice for most of the utilities due to its simplicity [9 – 14]. However, it has been proven that the bus voltage level alone is not a good indicator to assess the security of the operating conditions, especially when the shunt compensation and/or some other voltage control devices are heavily used in the modern power systems [10]. Moreover, the load shedding amount is difﬁcult to be minimised only based on the voltage levels at some particular buses. In order to optimise the load shedding schemes, several methods which are aimed at minimising the load shedding amount have been proposed in recent years [9 – 11]. The sensitivities of the load-ability margin and the sensitivities of voltage with respect to the load parameters are often IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307– 313 doi: 10.1049/iet-gtd.2010.0341 used to determine the optimum load locations. By using the sensitivities information, an optimisation problem is usually formed and the minimum load shedding amount is determined by solving it. However, since these sensitivities vary signiﬁcantly when the operating condition changes, the load shedding amount which is computed based on the sensitivities obtained before the load shedding is applied might not be optimal. At the same time, associated with the load shedding, the system also needs to shed the same amount of generation to maintain the power balance. However, the problem of how to distribute the amount of generation shedding has rarely been discussed in the literature. Although the modal analysis method [8] is normally used to calculate the generators’ participation factors, it is difﬁcult to directly use these participation factors to shed the generations. In this paper, the variation in these sensitivities with respect to the load shedding amount is ﬁrst investigated. This investigation reveals that a non-linear optimisation problem needs to be solved in order to obtain the best locations and the minimum amount for the load shedding. Several difﬁculties are involved in solving this non-linear optimisation problem. In order to overcome these difﬁculties, a multistage method is proposed. The main idea of the proposed method is to solve the optimisation problem stage by stage, and limit the load shedding to a small amount at each stage. Using this approach, the nonlinear optimisation problem will be converted into a series of linear optimisation problems. By solving them stage by stage, the optimal solution to the original non-linear 307 & The Institution of Engineering and Technology 2011 www.ietdl.org problem can be obtained. A new multiport network model is also proposed in this paper in order to be used for quick identiﬁcation of the most effective locations for load shedding. Using this model, not only can the loads be ranked with little calculation efforts, but also the required generation shedding can be easily distributed among the generators. The rest of the paper is organised as follows. Section 2 investigates the sensitivities of load-ability margin with respect to load parameters under different load shedding amount. Based on the ﬁndings in Section 2, a multistage method is proposed in Section 3. In Section 4, the multiport network model is introduced to rank the loads and generators. In Section 5, the proposed method is applied on several power systems and its performance is veriﬁed. Finally, Section 6 consists of the conclusion. 2 Sensitivites of load-ability margin with respect to load shedding amount The load-ability sensitivity with respect to load parameters can be deﬁned by (1), which is modiﬁed from the sensitivity formula in [12]. Seni = Dl DSi i = 1, 2, . . . , n (1) where DSi is the load shedding amount at load bus i, Dl is the load-ability margin increment after the load shedding and n is the total number of loads. In order to achieve the required load-ability margin increment Dlreq, the minimum load shedding amount is determined by shedding loads from the most sensitive loads until the achieved margin increase Dl∗ exceeds Dlreq. The margin increment Dl∗ is calculated by (2) [9]. Dl∗ = m Seni × DSi m≤n (2) i=1 where DSi ¼ f × Si , f is the shedding fraction of the selected load, Si and DSi are the load demand and the shedding amount at bus i, respectively. As indicated in [9], the minimum amount of load shedding obtained from (2) relies on the following two assumptions: 1. Linearity of (2): the load-ability margin increments from any single load shedding can be added up. Fig. 1 Actual and the expected load-ability margins 2. Constant sensitivities: the load-ability sensitivities remain constant no matter how much load is shed at the selected locations. To investigate the validity and accuracy of the above two assumptions, the load-ability sensitivities are studied by using the IEEE 14-bus system. Equations (1) and (2) are used in this study. The investigation results on the linearity of (2) are shown in Fig. 1. In this ﬁgure, Dl∗ is calculated by using (2) and Dlact is obtained using the continuation power ﬂow method in the commercial software PSS/E. Fig. 1 clearly reveals that the validity of the linearity assumption is doubtful. To study the second concern, the load-ability sensitivities of the six loads in the IEEE 14-bus system are studied. For this purpose, two different operation conditions – with and without considering the non-linear effects – are studied. The non-linear effects are the reactive power limit, actions of the switched shunts and the movements of the tap changers. The margin increment results with different load shedding amount are shown in Fig. 2. According to Fig. 2a, when the non-linear effects are not considered, the relationship between the margin increments and the load shedding amount is almost linear. In other words, without considering the nonlinear effects in the system, the sensitivities remain relatively constant. On the other hand, when the non-linear effects are considered, these sensitivities vary signiﬁcantly, as shown in Fig. 2b. This ﬁgure also indicates that shedding more loads does not necessarily lead to a higher margin increment. According to what was explained above, none of the assumptions considered in (2) can be conﬁrmed in power systems. Therefore the solution obtained by (2) may not be even close to the optimal load shedding results. Many strategies have been proposed to solve this problem in recent years. In this paper, a practical strategy called the multistage method is proposed. Fig. 2 Variation of the sensitivities under different load shedding amount a Margin increments with different load shedding amount (without considering the non-linear effects) b Margin increments with different load shedding amount (with considering the non-linear effects) 308 & The Institution of Engineering and Technology 2011 IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307 –313 doi: 10.1049/iet-gtd.2010.0341 www.ietdl.org 3 Proposed multistage method The above analysis indicates that a non-linear optimisation problem, as described by (3), needs to be solved to obtain the best load shedding location and the minimum load shedding amount. min Sshed = m DSi i=1 Dl∗ = f (DS1 , DS2 , . . . , DSm ) Dl∗ ≥ Dlreq s.t. power flowf(z, s) = 0 power system components limits limits proposed by the load shedding providers (3) where Sshed is the total load shedding amount, m is the number of available load shedding providers, z is the system state vector and s is the vector of active and reactive powers consumed by the loads. As seen in (3), several power system operation constraints including power ﬂow equations, power system components limits and the limits of load shedding providers are considered in the optimisation problem. More factors, such as generators’ cost functions and load characteristics, can also be considered as long as they are properly represented in (3). However, the principle of solving this non-linear optimisation problem remains the same. In this paper, we mainly focus on introducing the principles of the multistage method using the constraints described in (3). The analysis in Section 2 reveals that the relationship between the margin increment and the load shedding amounts is an unknown non-linear function, which is represented as the function f. Therefore there is a big challenge in solving this non-linear optimisation problem. To overcome this difﬁculty, a practical multistage method is proposed. It is called multistage because it is going to solve (3) stage by stage. For each stage, two circumstances are considered: 1. The load shedding is applied at only one location. 2. The load shedding amount is limited to a small value (say 10%) so that the sensitivities can be considered constant. Considering the above conditions, (3) can be converted to a series of linear optimisation problems by using the piecewise linear method. At each stage, the linear optimisation problem can be described by (4). By solving these linear optimisation problems one by one, the load-ability margin is improved stage by stage. Until the last stage, the desired load-ability margin is obtained. The solution to the original problem is the combination of the solutions to all these linear optimisation problems. Max{Dli } s.t. Dli = Seni × DSi , Fig. 3 Flowchart of the proposed multistage method Fig. 3. As seen in this ﬁgure, the sensitivities are calculated at each stage and the load with the highest sensitivity is selected as the most effective location. The load shedding is then applied at the selected location. After this load shedding, a new operation case is constructed and the next stage starts. This process will be repeated until the required margin is obtained. The ﬁnal load shedding rule is the combination of the results from all stages. It is worthwhile to mention that the term of ‘multistage’ is only to describe the design procedure, not to reﬂect the load shedding stages in implementation. The load shedding sensitivities can be calculated by using any existing method such as the method proposed in [12]. The main problem in this kind of methods is that they are very time-consuming. This problem becomes more important in the proposed multistage method because the sensitivities need to be calculated at each stage, and a high number of stages might be necessary for a large system. Moreover, after a load shedding is applied at each stage, an active power shedding should be applied to the generators in order to construct a new base case. In other words, a proper generation shedding should be assigned for the selected load shedding at each stage. In order to make the proposed multistage method more practical, a new algorithm is required which can not only ﬁnd the most effective location for the load shedding in a fast manner, but also obtain a proper generation shedding associated with the selected load shedding. For this purpose, a new procedure based on a multiport network model is proposed in the next section. 4 i = 1, 2, . . . , n (4) Since (4) is a linear optimisation problem, it is quite easy to solve it. It sufﬁces to calculate the sensitivities and select the load with the highest sensitivity. According to what was explained above, the procedure of the proposed multistage method will be as depicted in IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307– 313 doi: 10.1049/iet-gtd.2010.0341 Multiport network model Associated with the multistage load shedding method proposed in the previous section, a new multiport network model is presented in this section to solve the following two problems with little calculation effort: 1. ﬁnd the most effective location for the load shedding at each stage; 309 & The Institution of Engineering and Technology 2011 www.ietdl.org The impedance ratio Zratio, j is calculated by (9), where ZLj is the impedance of load j. Zratio,j Z j = , ZLj ZLj = VLj , ILj j = 1, 2, . . . , n (9) Fig. 4 Multiport network model 2. obtain the generators’ participations associated with the selected load shedding in order to calculate the proper generation shedding for each generator. The proposed multiport network model is to equivalent the power system by a model shown in Fig. 4. All the generator and load buses are separated from the transmission network, which is converted to an equivalent impedance matrix Z. The equation to describe the above multiport network model can be written as V L = KE − ZI L n ZLLji ILi = Eeqj − Zj ILj (6) i=1,i=j Zj = ZLLjj ILj + Sni=1,i=j ZLLji ILi ILj Eeqj = [KE]j (7) (8) where Zj is the Thevenin impedance of the network at bus j and Eeq j is the Thevenin equivalent voltage seen at bus j. The Thevenin equivalent parameter Eeq j represents the contributions from all the generators. As well, the Thevenin equivalent parameter Zj represents the impacts from all the other loads. Using these parameters, the power system can be converted into an equivalent circuit shown in Fig. 5. For each load bus, the impedance matching theory can be applied in order to ﬁnd the weakest load bus. In other words, the load with the highest impedance ratio value is the weakest load [13]. The weakest load is obviously the most effective location for the load shedding. Fig. 5 Multiport network equivalent circuit 310 & The Institution of Engineering and Technology 2011 Eeqj = Kj1 E1 + Kj2 E2 + · · · + Kjm Em (10) Equation (10) shows the composition of the Thevenin equivalent voltage. Based on (10), the contribution of each generator on the selected load j can be deﬁned by (11). Cji = (5) where T E = E1 E2 . . . Em , V L = VL1 VL2 . . . VLn ]T , T I L = IL1 IL2 . . . ILn , K is an n × m matrix obtained from system admittance matrix, and Z is an n × n system impedance matrix. For the jth load bus, we can obtain VLj = [KE]j − ZLLjj ILj − Furthermore, the generators’ participations for the selected load shedding can also be calculated based on the contribution of each generator on this selected load. From (6), we obtain an expanded expression of the Thevenin equivalent voltage by (10). |Kji Ei | cos(uji ) |Eeq j | (11) where Cji is the contribution of generator i on the load j and uji is the angle difference between the Thevenin equivalent voltage Eeqj and the generator voltage KjiEi . Thus, the participation of each generator on the selected load shedding amount DSj will be calculated by (11). The generation shedding of each generator associated with this load shedding can then be obtained by (12). Meanwhile, the capacity of the automatic frequency control of each generator has to be taken into account in order not to introduce any other problem, such as angular stability problem. For this purpose, the amount of generation shedding for each generator needs to be limited by its corresponding control capacity DGi,max . If the calculated generation shedding (DGji) is over DGi,max , the shedding is ﬁxed at DGi,max . The leftover (DGji 2 DGi,max) will be distributed to the other generators based on the same ratio provided by (12). DGji = Cji k Si=1 Cji DPj i = 1, 2, . . . , k (12) where DGji is the active power shedding of generator i, DPj is the real part of the load shedding amount DSj and k is the total number of generation plants in the systems. In order to verify the performance of the proposed method, the method is applied to different power systems. IEEE 14-bus system: This case study is the stressed version of IEEE 14-bus system. The studied case has 1.85 times more load demand than the base case which can be found in [15]. After an N 2 2 contingency (the loss of the line between bus 2 and bus 4, and the line between bus 2 and bus 5), the system power ﬂow diverges. Both the modal analysis method [8] and the proposed method are used to ﬁnd the top ﬁve critical (weakest) load buses. The results are listed in Table 1, which reveals that both the methods lead to bus 14 as the weakest bus in the system. As well, the generators’ contributions Cji associated with the load shedding at load bus 14 are shown in Table 2. IEEE 118-bus system: The studied power system is the stressed IEEE 118-bus system. The studied case has two times more load demand than the base case which can also be found in [15]. After an N 2 1 contingency (the loss of the line between bus 74 and bus 75), the system margin IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307 –313 doi: 10.1049/iet-gtd.2010.0341 www.ietdl.org Table 1 Results of the bus ranking for IEEE 14-bus system No. Load bus ranking 1 2 3 4 5 Table 2 Modal analysis based on the critical mode Proposed method 14 10 9 4 5 14 9 10 4 5 0.01237 0.0 Results of the bus ranking of IEEE 118-bus system No. Load bus ranking 1 2 3 4 5 Table 4 Modal analysis based on the critical mode Proposed method 44 45 43 22 21 44 45 43 95 21 Generators’ contribution on the load bus 44 No. generator 49 generator 54 Modal analysis based on the critical mode Proposed method 4220 4219 19 314 4361 19 388 4220 4219 4361 19 314 18 393 Generators’ contribution on the load bus 4220 No. shrinks to 4%. Table 3 shows the top ﬁve critical load buses for this case. According to this table, both the proposed method and modal analysis method result in the load 44 as the weakest load in the system. As well, the non-zero generators’ contributions associated with the load shedding at the load bus 44 are shown in Table 4. Real 2038-bus power system: A real large system (the Alberta Interconnected Electric System) is considered as the last case study. It consists of 208 PV buses, 688 load buses and 2366 branches. The total load demand is 10222.4 MW and 3349.3 MVar. After an N 2 1 contingency or loss of the line between bus 74 and bus 814, the system loses its power ﬂow solvability. The bus ranking results obtained from the modal analysis method and the proposed method are listed in Table 5. It can be seen that both of the methods result in the same weakest load. Also, the top ﬁve contributed generators associated with the load shedding at load bus 4220 are shown in Table 6. The above case studies clearly verify the performance of the proposed method in terms of the identiﬁcation of the most sensitive load. The results from modal analysis and those from the proposed method are perfectly consistent with each other. Therefore the proposed multiport network model can be used in the procedure of the multistage method. For this purpose, the procedure which was previously shown in Table 3 Load bus ranking 1 2 3 4 5 Table 6 Generators’ contribution generator 1 generator 2 Results of the bus ranking of the real power system No. Generators’ contribution on the load bus 14 No. Table 5 Generators’ contribution 0.4518 0.3172 IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307– 313 doi: 10.1049/iet-gtd.2010.0341 Generators’ contribution generator 1495 generator 1497 generator 1496 generator 3248 generator 2248 0.1058 0.1044 0.0965 0.0430 0.0429 Fig. 6 Flowchart of the proposed strategy Fig. 3 can be modiﬁed to the one depicted in Fig. 6. In the new procedure, multiport network model is used at each stage of the multistage method to ﬁnd the most effective location for the load shedding and the associated generation shedding. In the next section, the procedure described in Fig. 6 will be demonstrated by the IEEE 14-bus system, the IEEE 118-bus system and the real 2038-bus power system. 5 Illustration studies on the selected power systems The proposed strategy is applied to several test power systems and the results are investigated in this section. Both the proposed method and the conventional method described by (2) are tested. The results are compared with 311 & The Institution of Engineering and Technology 2011 www.ietdl.org As seen in this table, in order to restore the system, the proposed method shed 0.43 MW less active power and 2.28 MVar less reactive power. IEEE 118-bus system: The test system is the stressed IEEE 118-bus system. The studied case has two times more load demand than the base IEEE 118-bus system. The stability margin of the studied case is 6%. After the N 2 1 contingency, that is, the loss of the line between bus 74 and bus 75, the system load-ability margin shrinks to 4%. The results for the load shedding rules are listed in Table 8. In this case, the results of the proposed strategy are the same as the conventional method. The reason is that this contingency is not severe and very small amount of load shedding is enough to restore the system. As a result, as seen in Table 8, the proposed strategy is completed in only one stage. In this situation, the performance of the proposed strategy might be improved by reducing the size of the load shedding amount at each stage. Real 2038-bus power system: The real power system used in Section 4 is studied here. After the N 2 1 contingency, that is, loss of the line between bus 74 and bus 814, the system each other. The detailed information of the conventional method can be found in [9]. The load shedding step size is considered as 10% of the selected load for both of the two methods. IEEE 14-bus system: This test power system is a stressed system from the IEEE 14-bus system. The studied case has 1.85 times more load demand than the base IEEE 14-bus system. The voltage stability margin of the studied case is 42%. After the selected N 2 2 contingency, that is, the two transmission lines outages (the branch between bus 2 and bus 4, and the branch between bus 2 and bus 5), the system loses its power ﬂow solvability. The load shedding strategy is then applied to ensure that the load-ability margin is no less than the required 5% (WSCC standard [14]). Based on the results from Table 1, load bus 14 will be chosen at the ﬁrst stage and the load shedding amount is 10% for this stage. This procedure is then repeated until the required voltage stability margin is obtained. Table 7 shows the load shedding results obtained from the proposed strategy and those obtained by the conventional strategy. Table 7 Results of the load shedding rules for IEEE 14-bus system Methods Load shedding rules Stage the proposed strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 the conventional method Table 8 Margin after shedding, % Location Amt, % 14 14 9 9 9 14 14 9 14 10 14 9 4 10 10 10 10 10 10 10 10 10 10 10 10 10 29.36 MW, 11.84 MVar 5 14 10 9 100 100 20 29.79 MW, 14.12 MVar 6 Results of the load shedding rules for IEEE 118-bus system Methods Load shedding rules the proposed strategy the conventional method Table 9 Total shedding amount Location Amt, % Total shedding amount Margin after shedding, % bus 44 bus 44 10 10 1.6 MW, 1.2 MVar 1.6 MW, 1.2 MVar 5 5 Results of the load shedding rules for the real large system (outage of the line 74 –814) Methods the proposed strategy the conventional method Load shedding rules Location Amt, % Total shedding amount Margin after shedding, % 4220 4219 99 393 100 30 10 34.89 MW, 15.49 MVar 5 4220 4219 100 40 36.56 MW, 16.54 MVar 6 312 & The Institution of Engineering and Technology 2011 IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307 –313 doi: 10.1049/iet-gtd.2010.0341 www.ietdl.org Table 10 Results of the load shedding rules for the real large system (outage of the line 1164–1165) Methods Load shedding rules Location the proposed strategy the conventional method Amt, % Total shedding amount Margin after shedding, % 1169 19 185 60 10 6.85 MW, 2.4 MVar 8 1169 100 9.5 MW, 3.7 MVar 8 loses its power ﬂow solvability. To save space, only the ﬁnal results of the load shedding rules are listed in Table 9, which shows that the proposed method shed 1.67 MW less active power and 1.05 MVar less reactive power. In order to further verify the advantage of the proposed multistage method, another N 2 1 contingency which is the loss of the line between bus 1164 and bus 1165 is studied. After this contingency, the system again loses its power ﬂow solvability. The top ﬁve weakest load buses for this case are the buses 1169, 19 156, 19 185, 4590 and 19 371. Both the conventional method and the proposed multistage method are studied to determine the optimal load shedding rules. The results are listed in Table 10. As seen in this table, the proposed method shed 38.6% less active power and 54.1% less reactive power. The above case study results clearly conﬁrm the advantages of the proposed multistage method. According to the above results, the important features of the proposed strategy can be summarised as follows: 1. The proposed multiport network model accurately identiﬁes the best load shedding locations with little calculation effort. This model also enables us to ﬁnd a proper generation shedding associated with the selected load shedding. 2. In order to restore the power system from an emergency operating condition, the proposed multistage strategy provides the optimal solution. Compared to the conventional method, the proposed method needs less amount of load reduction. 6 Conclusions Because of the non-linearity that exists in the relationship between the load-ability margin and the load shedding amount, a non-linear optimisation problem needs to be solved for optimising the load shedding rules, which includes the best load shedding locations, the minimum load shedding amount and the corresponding generation reduction. In order to solve this non-linear programming problem, a practical multistage method was proposed in this paper. By using the piecewise linear method, the multistage method converts the original non-linear problem into a series of linear programming problems and solves these linear problems one by one. At each stage, the load-ability margin is improved and the desired margin is obtained at the last stage. The solution to the original non-linear optimisation problems is the combination of the solutions to all these linear programming problems. In order to reduce the calculation efforts on the location selection, a multiport network model was also proposed in this paper. Using the multiport network model, the most effective locations for the load shedding and proper IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 3, pp. 307– 313 doi: 10.1049/iet-gtd.2010.0341 generation dispatching information can be easily obtained. The results obtained from the multiport network model were veriﬁed using the well-known modal analysis method. Three test power systems including a real power system have been used to examine the effectiveness of the proposed algorithms. The results have veriﬁed the advantages of the proposed method. Compared to the traditional algorithm, the proposed multistage method needs less load reductions to restore the system from emergency conditions. 7 Acknowledgment This work was supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC) and China Scholarship Council under grant 20066035. 8 References 1 Cutsem, T.V.: ‘Voltage instability: phenomena, countermeasures, and analysis methods’, Proc. IEEE, 2000, 88, pp. 208–227 2 Taylor, C.W., Erickson, E.C., Martin, K.E., Wilson, R.E., Venkatasubramanian, V.: ‘WACS—wide-area stability and voltage control system: R&D and online demonstration’, Proc. IEEE, 2005, 93, pp. 892 –906 3 Sinha, A.K., Hazarika, D.: ‘A comparative study of voltage stability indices in a power system’, Int. J. Electr. Power Energy Syst., 2000, 22, pp. 589 –596 4 Zarate, L.A.Ll., Castro, C.A.: ‘A critical evaluation of a maximum loading point estimation method for voltage stability analysis’, Electr. Power Syst. Res., 2004, 70, pp. 195 –202 5 Zambroni de Souza, A.C., Stacchini de Souza, J.C., Leite da Silva, A.M.: ‘On-line voltage stability monitoring’, IEEE Trans. Power Syst., 2000, 15, pp. 1300–1305 6 Haque, M.H.: ‘On-line monitoring of maximum permissible loading of a power system within voltage stability limits’, IEE Proc. Gener. Transm. Distrib., 2000, 150, pp. 107–112 7 Chebbo, A.M., Irving, M.R., Sterling, M.J.H.: ‘Voltage collapse proximity indicator: behavior and implications’, IEE Proc. Gener. Transm. Distrib., 1992, 139, pp. 241–252 8 Gao, B., Morison, G.K., Kundur, P.: ‘Voltage stability evaluation using modal analysis’, IEEE Trans. Power Syst., 1992, 7, pp. 1529–1542 9 Nikolaidis, V.C., Vournas, C.D.: ‘Design strategies for load-shedding schemes against voltage collapse in the Hellenic system’, IEEE Trans. Power Syst., 2008, 23, (2), pp. 582– 591 10 Mozina, C.J.: ‘Undervoltage load shedding’. 60th Annual Conf. on Protective Relay Engineers, March 2007, pp. 16–34 11 Cutsem, T.V.: ‘An approach to corrective control of voltage instability using simulation and sensitivity’, IEEE Trans. Power Syst., 1995, 10, (2), pp. 616– 622 12 Cutsem, T.V., Vournas, C.D.: ‘Voltage stability of electric power systems’ (Springer, New York, 2007) 13 Vu, K., Begovic, M.M., Novosel, D., Mohan Saha, M.: ‘Use of local measurements to estimate voltage-stability margin’, IEEE Trans. Power Syst., 1999, 14, (3), pp. 1029–1035 14 Abed, A.M.: ‘WSCC voltage stability criteria, undervoltage load shedding strategy, and reactive power reserve monitoring methodology’. IEEE PES Summer Meeting, July 1999, vol. 1, pp. 191–197 15 http://www.ee.washington.edu/research/pstca/ 313 & The Institution of Engineering and Technology 2011