Assessment of branch outage contingencies using the continuation method

Assessment of branch outage contingencies using the continuation method

Electrical Power and Energy Systems 55 (2014) 74–81 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: ...

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Electrical Power and Energy Systems 55 (2014) 74–81

Contents lists available at ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Assessment of branch outage contingencies using the continuation method R.R. Matarucco a, A. Bonini Neto b,⇑, D.A. Alves b a b

Votuporanga University Center (UNIFEV), Votuporanga, SP, Brazil Department of Electrical Engineering, Univ Estadual Paulista (UNESP), Ilha Solteira, SP, Brazil

a r t i c l e

i n f o

Article history: Received 19 February 2013 Received in revised form 25 August 2013 Accepted 28 August 2013

Keywords: Continuation methods Voltage collapse Load flow Voltage stability margin Contingency analysis Maximum loading point

a b s t r a c t This paper provides a contribution to the contingency analysis of electric power systems under steady state conditions. An alternative methodology is presented for static contingency analyses that only use continuation methods and thus provides an accurate determination of the loading margin. Rather than starting from the base case operating point, the proposed continuation power flow obtains the post-contingency loading margins starting from the maximum loading and using a bus voltage magnitude as a parameter. The branch selected for the contingency evaluation is parameterised using a scaling factor, which allows its gradual removal and assures the continuation power flow convergence for the cases where the method would diverge for the complete transmission line or transformer removal. The applicability and effectiveness of the proposed methodology have been investigated on IEEE test systems (14, 57 and 118 buses) and compared with the continuation power flow, which obtains the post-contingency loading margin starting from the base case solution. In general, for most of the analysed contingencies, few iterations are necessary to determine the post-contingency maximum loading point. Thus, a significant reduction in the global number of iterations is achieved. Therefore, the proposed methodology can be used as an alternative technique to verify and even to obtain the list of critical contingencies supplied by the electric power systems security analysis function. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Modern electric power systems are becoming subject to an increase in the number of insecure contingency cases in which the power flow equations have no feasible solution. Both the static analysis and dynamic analyses have gained acceptance by the electric utilities and are now considered the two most common methods for analysing power system stability [1,2]. Long-term dynamic simulations are used for benchmarking contingencies, the validation of results for steady-state analysis and load shedding strategies [1–3]. However, the static analysis is used to reveal the loss of an equilibrium point of a system [3], to rapidly provide valuable information and to find critical areas for establishing preventive measures and the amount of control actions. The static and dynamic tools complement rather than compete with each other. For a better use of generation resources and the transmission capacity, the voltage stability margins and control actions must be determined in the planning and in the real-time operation ⇑ Corresponding author. E-mail addresses: [email protected], alfredoboninineto@hotmail. com (A. Bonini Neto). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.08.029

phases, not only for normal operating conditions (base case) but also for different operating points and contingency conditions. Therefore, it is common to run hundreds of contingency cases. Thus, for each contingency and several operating conditions, the LM must be obtained through P–V curve tracing [1–4]. The WSCC requires its member utilities to possess at least a 5% P–V margin under the worst single element contingency [1]. In [5], an asymptotic numerical method was used to solve branch outage continuation power flow (CPF) problems, and the method can be considered as a higher-order predictor without any corrections. In [6], three new schemes using Fuzzy Logic were developed to determine the maximum load margin. The iterative process can be started with random initialisation using the proposed Fuzzy Logic schemes, which reflects the superiority of the proposed schemes over the traditional Newton–Raphson technique. The computation of the load margin (LM) using power flow (PF), or the Continuation Power Flow (CPF), is a very time-consuming process when a considerable number of contingencies need to be analysed. Over the past several years, many approaches have been proposed in this subject [9–24]. A large number of research studies have attempted to develop faster and more accurate algorithms for the computation of the post-contingency margin [7,8]. Many other methods have been proposed for voltage stability contingency

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screening and ranking [11,12]. The purpose of the algorithm is to accelerate the process of contingency analysis by identifying a relatively short list of critical contingencies from a large list of credible contingencies and rank them according to the degree of severity. In [13–15], methodologies to quickly calculate or estimate P–V margins by using a curve fitting technique that requires one to three PF solutions for each contingency were proposed. The main disadvantage of this method is that it relies heavily on the shape of the P–V curve, and thus it may fail when the tap limits of the OLTCs and the reactive power limits on the generators are considered. Moreover, a PF solution close to the MLP and a bus that presents a P–V curve with an appropriate geometry for curve fitting are required. The multiple PF solution method computes a first-order approximation of the systems margin using voltage gradients determined at a pair of PF solutions [16]. However, the test results obtained in [13] showed that the accuracy of the method is not satisfactory. In [17], a second-order approximation of the Q–V curve was proposed that requires three PF solutions to estimate the approximate margin to collapse. The method proposed in [10] uses the linear and quadratic sensitivities for a faster post-contingency P–V margin calculation. Although it is a very rapid method and could be reasonably good for contingency ranking, the results showed that the obtained margins are practically unacceptable for many analysed contingencies. The convergence of the CPF is associated with the chosen parameter and the solution path. Therefore, in contingency analysis, depending on the variable chosen as the continuation parameter, the CPF cannot converge. In this case, using k as a parameter is the cause of the PF divergence. In [25], a new, robust and efficient CPF was presented that uses branch admittance as a continuation parameter to evaluate the effects of the branch parameter variations rather than estimating or predicting the effects of their removal. The technique provides a robust second-stage verification tool. In the technique presented in [9] for LM determination, the pre-contingency maximum loading point (i.e., the MLP of the base case) is first computed using a CPF. Next, the post-contingency voltage magnitude of a chosen bus (the reference bus voltage) is estimated, and its value is fixed (adopted as parameter) while the loading factor is considered as a dependent variable in the CPF. According to these considerations, the contingency is applied, and the post-contingency maximum loading point (MLPpost) and the respective margin are computed. In the cases where the procedure fails to obtain a solution, the authors propose to use a damped Newton method to identify the systems post-contingency reference bus and to estimate its voltage magnitude, which is used as an estimate of the actual voltage at the MLPpost for the reference bus. However, if the post-contingency critical bus is not known a priori, the voltage stability margin determination can become a difficult and computationally heavy process [9]. In this paper, the features of the proposed alternative methodology for the static contingency evaluation are presented. The post-contingency loading margins are obtained starting from the pre-contingency (base case) maximum loading point (MLP) and by using the voltage magnitude of an appropriated bus as the continuation parameter during the transition from one P–V curve to another. First, the numerical difficulties that can appear when a PF or a CPF is used for the post-contingency loading margin determination are presented. Next, the proposed methodology used to assure the CPF convergence for the analysis of any transmission line (TL) or transformer contingency is presented. Finally, the results obtained with the new methodology for the IEEE test systems are presented and discussed. Even though a few cases required a few more iterations, the main advantages of the proposed methodology are the characteristic of guaranteeing the computation of the post-contingency solution and the significant

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reduction in the global number of iterations. Thus, the method can be used as an alternative technique to verify and even to obtain the list of critical contingencies supplied by the electric power systems security analysis function. 2. Formulation of the proposed continuation power flow The objective of this section is to highlight the difficulties that can appear when using the PF and CPF methods for the static contingency analysis of electric power systems. 2.1. Characterisation of the problem Fig. 1 presents the base-case P–V curve (curve 1) and the postcontingency P–V curves for the outage of a transmission line of a system. Consider the system operating at point ‘‘P’’ in the pre-contingency curve (case base). From the loading margin (LM) definition, the system presents a positive LM (LM > 0). Three contingencies will be analysed: the first one is related to the positive LM (curve 2) of the operating system and the two others are under negative LM conditions (curves 3 and 4). If the system loading (k) of the base case is maintained fixed, i.e., k is considered as a parameter, in the case of curve 2, the system will remain stable and will operate at point ‘‘A’’. However, for both curves 3 and 4, it will collapse because there is no post-contingency feasible solution for this k and either the PF or CPF will diverge. Starting from the solved base case, the conventional PF or the CPF using k do not converge to a solution because the post-contingency MLP is smaller than the MLP of the base case operating point (k = 1), i.e., in the base case, there will be no local solution to the PF equations when the network faces the transmission line outage. Therefore, for the cases where the LM is negative, it will be necessary to establish a load shedding strategy to maintain voltage stability, i.e., to move the system to a secure voltage operating point. Thus, using the CPF parameterised by k, the LM determination is possible only for curve 2, and the determination of the other two LM is not possible. Although both the conventional PF and the CPF using k as the parameter do not converge to a solution, there are no guaranties that this situation is due to either a bad initial voltage setting, a singularity (MLP), a deficiency of the numerical method, the existence of multiple solutions, or unsolvability for the desired operating point. The user has to resort to either a trial and error process or to using some heuristic techniques to determine which parameters

Fig. 1. P-V curves for the base-case (pre-contingency) and for the outage of transmission lines.

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or controls have to change to return the case to solvability. Versions of the damped Newton method have been applied to PF calculations where there is a bad initial guess. In [26], the use of scaling factors was proposed, such as the optimal multiplier to prevent PF divergence by guaranteeing that the sum of the squares of the power flow mismatches decrease in each iteration. In [27], an algorithm was further developed to compute a measure for quantifying the system unsolvability based on the special convergence property of the method proposed in [26]. The use of the CPF for tracing the correspondent P–V curve of each contingency, starting from a loading factor smaller than that of the base case (k = 1), is a process that requires a large computational time. It is well known that the convergence of CPF does not depend only on the contingency but also on the parameter choice in the parameterisation step. In addition, a power system under normal operating conditions is generally able to take the highest amount of load increase, i.e., it presents the highest MLP. Thus, in [9], the determination of the post-contingency loading margin (MLPpost) from the pre-contingency maximum loading point (i.e., the MLPpre of the base case, point B on curve 1) is proposed. For a given load pattern and generation dispatch scenario, the MLP of the base case is first computed using a CPF [23,24,28]. Next, the post-contingency voltage magnitude of a chosen bus (the reference bus voltage) is estimated, and its value is fixed (adopted as a parameter) while the loading factor is considered as a dependent variable in the CPF. Finally, the contingency is applied, and the MLPpost and the respective loading margin are computed. In other words, the post-contingency maximum loading point is a backwards computation process. In the case of curve 2 in Fig. 1, the MLPpost, point ‘‘C’’ on curve 2, is easily obtained with this procedure. Only one power flow calculation is necessary to compute the MLPpost because this system presents an ideal situation where the MLPpost voltage magnitudes of the reference bus practically maintain the same base case values. Nevertheless, as shown in Fig. 1, when this technique is applied to obtain the MLPpost on curve 4, the new post-contingency operating point will move to point ‘‘D’’ in the lower part of the P–V curve. Therefore, additional refinement is needed to accurately compute the MLPpost (point E). This refinement is accomplished using a CPF starting at the estimated solution point ‘‘D’’. However, in some cases, the procedure fails to obtain a solution. Then, as in the case of curve 3 in Fig. 1, a damped-Newton method can be used to identify the post-contingency reference bus of the system and to estimate its voltage magnitude, as proposed in [9]. The optimal multiplier selection scheme has exhibited very good overall performance with unsolvable cases. The damped-Newton method is used to determine the minimum mismatch solution for each contingency. Next, the bus with the lowest voltage magnitude is selected as the reference bus, and its value is used as an estimate for the actual voltage magnitude at the MLPpost for the reference bus. It is observed that the magnitude of reference bus voltage obtained using this process is just an approximation of the actual value; moreover, the implementation of reactive power limits of the generators makes it more difficult to estimate the voltage magnitude of the reference bus accurately. The branch kl selected for contingency evaluation is gradually removed by using a scaling factor (l) [10,29]. For the outage of a transmission line, the series admittance (ykl) and the line-charging susceptance ðysh kl Þ are scaled down by (1  l). The difference in the proposed methodology is the use of this technique associated with the CPF parameterised by the voltage magnitude to obtain the post-contingency loading margin from the pre-contingency maximum loading point. The goal is to guarantee the computation of a solution without the use of a damped-Newton method to identify the system’s post-contingency reference bus and to estimate its voltage magnitude, as proposed in [9]. Thus, in the proposed methodology, the new set of CPF equations is represented by

sp kðCpg Psp gen  Cpl Pload Þ  Pðh; VÞ ¼ 0

ðQ gen  kCql Q sp load Þ  Q ðh; VÞ ¼ 0

ð1Þ

where V and h are the vectors of the voltage magnitudes and the sp sp phase angles, respectively. Psp load , Q load and Pgen are specified values for the base case (k = 1) for the real and reactive powers of PQ buses and the real power for PV buses, respectively. The pre-specified parameters Cpg, Cpl and Cql are used to adjust a specific loading scenario, describing the rate of changing of real power (Pgen) for generation buses (PV buses) as well as the real (Pload) and reactive (Qload) power for load buses (PQ buses). The loading factor (k) is used to scale up the loading and generation level. When l = 0, the original set of PF equations (1) is obtained, and when l = 1, a new set of PF equations is obtained, which represent the network with the outage of branch kl. In the procedure, when l is preset, one chooses the P–V curve that will be traced while, with the pre-setting of one of the other parameters (hk, Vk, or k), the desired curve is traced. This approach avoids retracing the P–V curve to compute the LM with the outage TL. The P–V curves are obtained by computing the PF solution for successive k increments according to a predefined direction. After obtaining the base case, a predictor step is performed to find an estimate for the next solution point. Among the several different predictors found in the literature, the tangent [23] and the secant [24] methods are the most popular ones. Finally, after the prediction has been made, it is necessary to correct the approximate solution to avoid error accumulation. Because the point obtained by a good predictor is very close to the correct solution, few iterations will be performed to obtain the exact solution. Newton’s method is the most widely used in the corrector step, although any other numeric method could be used [7]. A parameterisation procedure is used to overcome the singularity of J and thus to allow an accurate determination of the MLP without numerical difficulties [1– 8,23,24]. 2.2. Computational procedure for determining the post-contingency loading margin Fig. 2 presents the P-V curves of two buses (31 and 35) corresponding to the outage of the TL between the buses 31 and 35 (branch 48). It corresponds to one of the most severe contingencies of the IEEE-57 bus system. It will be used to clarify the steps of the computational procedure used to determine the post-contingency

Fig. 2. Performance of the proposed CPF considering the parameter changing for obtaining the LM after the contingency of the TL between buses 35 and 36 of the IEEE-57.

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Fig. 3. Performance of the proposed CPF for single branch outages of the IEEE-14: (a) P–V curves and (b) number of iterations.

loading margin by the proposed methodology. Curves 1 and 5 correspond to the pre-contingency (the base case, l = 0) and post-contingency (the branch is completely removed, l = 1) of the TL, respectively. Curves 2, 3 and 4 correspond to the partial contingencies (for l = 0.5, l = 0.875 and l = 0.9687). Buses 31 and 35 are the pre- and post-contingency critical buses, respectively [2,23]. Each of these curves was obtained with a CPF starting from a very low value of the loading factor k. According to these curves, it is clear that, as the TL is removed, there will always be a post-contingency operating point. Thus, even though it is necessary to perform a load shedding to obtain a physically feasible operating point in some cases, finding a solution will depend on the adopted parameterisation strategy. The curvatures of the P–V curves shown in Fig. 2 indicate that, as the branch is gradually removed by increasing the scaling factor (l), the voltage magnitude of the post-contingency critical bus (V35) begins to decrease while the voltage magnitude of the precontingency critical bus (V31) begins to increase. Thus, in the criterion adopted to change the parameter, the voltage magnitude of the bus that presents the highest rate of change in voltage magnitude (one of the characteristics of a critical bus [2]) during the change from one P–V curve to another is the chosen continuation parameter. Its corresponding voltage magnitude at the MLP of the current P–V curve must be the value that will be fixed during the change from one P–V curve to another. The post-contingency loading margin is obtained by following the main computational steps of the proposed continuation power flow (PCPF) method: i. Obtain the base case operating point P and the pre-contingency maximum loading point (MLPpre) by using the CPF presented in [28]. The critical bus (k) of the base case and its corresponding voltage magnitude value (Vk) of the MLPpre are identified in this step;

ii. Select a branch for contingency evaluation and set the initial step size of the scaling factor (Dl) equal to 0.5; iii. Set l = 1  Dl and determine the next system operating point on the new P–V curve by using the PCPF parameterised by the voltage magnitude of the critical bus (a = 0 and b = 1), the value of which is set to be equal to its corresponding value at the maximum loading point of the previous P–V curve (at the pre-contingency maximum loading point if l = 1 or at the partial-contingency maximum loading point if l = 1  Dl); iv. Identify the critical bus during the execution of step (iii), i.e., the one that presents the highest rate of change in its voltage magnitude during the change from one P–V curve to another. The voltage magnitude of this bus will be used as the continuation parameter to determine the new system operating point at the next P–V curve for the new l value; v. Next, apply the refinement step to obtain the partial-contingency maximum loading point of the current P–V curve. The step size (DVk) of 0.01 p.u. is used to change the continuation parameter (Vk) in the refinement step; vi. Reduce the step size of the scaling factor by dividing it by 4 (Dl/4); vii. Repeat steps (iii) to (vi) until Dl is smaller than a predefined value (Dl < 0.03125); otherwise, set l equal to 1 and obtain the post-contingency maximum loading point (MLPpost) and the respective loading margin for the full contingency of the branch; viii. If there is another branch for contingency evaluation, go back to step (ii); otherwise, end the procedure. 3. Test results To demonstrate the effectiveness of the proposed methodology, numerical tests were conducted on the IEEE test systems (14, 30,

Table 1 Post-contingency loading margin and loading margin reduction for the IEEE-14. Branch

1 2 3 4 5 6 7 9

Bus

LM (p.u.)

From

To

1 1 2 2 2 3 4 4

2 5 3 4 5 4 5 9

MCpre = 0.768 0.019 0.397 0.299 0.587 0.660 0.717 0.593 0.679

LM reduction (%)

0 102.5 48.3 61.1 23.5 14.0 6.7 22.9 11.6

Branch

10 11 12 13 16 17 18 19 20

Bus From

To

5 6 6 6 9 9 10 12 13

6 11 12 13 10 14 11 13 14

LM (p.u.)

LM reduction (%)

0.342 0.751 0.743 0.661 0.721 0.618 0.766 0.767 0.748

55.5 2.2 3.2 13.9 6.1 19.5 0.3 0.2 2.6

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Fig. 4. Performance of the proposed CPF for single branch outages of the IEEE-57: (a) P–V curves for pre (curve 1, l = 0) and post-contingencies branch outages (l = 1), (b) post-contingency P-V curves for outages of branch numbers 41, 67, and 80, (c) number of iterations needed to compute each LM starting from MLPpre and from the base case solution.

57 and 118 buses). For each branch, the post-contingency loading margin and the number of iterations obtained using the proposed methodology were compared to those obtained from the base case solution. The overall performance for each system is also presented. The analyses are performed for the cases of TL and transformers whose removals did not result in system islanding or the simultaneous removal of generators or lines. The mismatch convergence threshold was 104 p.u. Upper and lower tap limits of respectively 1.05 and 0.95 were adopted. The load and generation variations occur according to the loading factor, i.e., proportional to their corresponding base case values. The adjustment of tap in the On Load-Tap Changing (OLTC) transformers consisted of including the tap position as a dependent variable, whereas the controlled bus voltage magnitude was considered as an independent variable. The method of accounting for Q-limit at PV buses and tap limits is the same as in conventional load flow methods. At every iteration the reactive generation at all PV buses

are compared to their respective limits. In case of violation, a PV bus is switched to type PQ. This bus can be switched back to PV in future iterations. Tap limit violations are also checked. The branch is removed starting from the MLP of the base case while setting l equal to 0.5. By making two successive reductions in the step size of l, which is divided by 4 (Dl/4), the remaining post-contingency maximum loading points MLPpost for the Dl = 0.125 and Dl = 0.03215 step sizes are obtained. Finally, using l = 1, the full contingency of the branch is obtained. 3.1. IEEE 14-bus system Fig. 3 shows the impact of each branch outage for the IEEE-14. Fig. 3(a) presents the P–V curves for all contingencies analysed, where curve 1 is the P–V curve of the base case and curve 2 is the post-contingency P–V curve of the most critical contingency, i.e., the one that produces the smallest load margin is related to

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Load margin reduction (%)

150 150 100 50

100 Branch: 48 42 46 47 41 50 49 43 40 67 8 39 80 36 35 15 44 33 37 38 From: 35 25 34 34 7 37 36 30 28 29 8 27 9 24 24 1 31 22 24 26 To 36 30 32 35 29 38 37 31 29 52 9 28 55 25 25 15 32 23 26 27

50

0

0

10

20

30

40

50

60

70

80

Branch number Fig. 5. Simulation results of single branch outage for IEEE-57: branch number vs. loading margin reduction and the 20 most critical contingencies according to the LM.

(a)

1.9

Maximum loading point

Maximum loading point

(a) 1.8 1.7 1.6 1.5 1.4 1.3 0

20

40

60

80

100

120

140

160

2.1 2

1.9 1.8 1.7 1.6 1.5 1.4 1.3

180

0

Branch number

Number of iterations

50 40 30 20 10 0

20

40

60

starting from base case starting from MLPpre

80

100

120

140

160

Load margin reduction (%)

40

100

120

140

160

180

50 40 30 20 10 0

Branch: 8 185 97 96 51 163 94 93 104 71 38 118 102 70 99 98 174 105 67 66 From: 8 75 64 38 38 100 63 63 65 49 26 76 65 49 49 49 103 47 42 42 To 5 118 65 65 37 103 64 59 68 51 30 77 66 50 66 66 110 69 49 49

30 20 10 60

80

100

120

40

60

140

160

180

Branch number Fig. 7. Simulation results of a single branch outage for IEEE-118(1): branch number vs. loading margin reduction and the 20 most critical contingencies according to the loading margin reduction.

branch number 1. The LM of the base case is equal to 0.768 p.u. For each branch outage, Fig. 3(b) compares the numbers of iterations for the proposed methodology with those obtained starting from the base case solution. In the case of curve 2, the CPF fails to obtain

80

100

120

140

160

180

Branch number

Fig. 8. Performance of the proposed CPF for single branch outages of the IEEE118(2): (a) branch number vs. maximum loading point (MLP), (b) number of iterations needed to compute each MLP starting from MLPpre and from the base case solution.

80

40

20

starting from base case starting from MLPpre

60 50 40 30 20 10

20

60

Branch number

Fig. 6. Performance of the proposed CPF for single branch outages of the IEEE-118 (1): (a) branch number vs. maximum loading point (MLP), (b) number of iterations needed to compute each MLP starting from MLPpre and from the base case solution.

50

80

Branch number

70

0

180

Load margin reduction (%)

Number of iterations

60

60

60

80

70

70

40

(b)

(b)

0

20

70 60 50 40 30 20 10 Branch: 8 185 51 163 118 174 38 116 3 From: 8 75 38 100 76 103 26 69 4 To 5 118 37 103 77 110 30 75 5

70 60 50 40

36 167 108 94 173 33 178 141 29 32 31 30 100 69 63 108 25 17 89 22 26 23 17 106 70 64 109 27 113 92 23 25 25

30 20 10 0

0

20

40

60

80

100

120

140

160

180

Branch number Fig. 9. Simulation results of a single branch outage for IEEE-118(2): branch number vs. loading margin reduction and the 20 most critical contingencies according to the loading margin reduction.

the maximum loading point starting from the base case, and thus its number of iterations is not shown in Fig. 3(b). This figure shows that the proposed methodology requires a lower global number of iterations. Table 1 shows the post-contingency loading margins

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Fig. 10. (a) Comparison for each system of the overall number of iterations needed to compute the loading margin by the proposed methodology and by a CPF starting from the base case solution, (b) reduction in the global number of iterations obtained by using the proposed methodology.

and the loading margin reduction for the outage of each branch. Only in the case of the outage of branch number 1 is the MLP of post-contingency smaller than the MLP of the base case, i.e., the LM is negative when this transmission line is removed. Therefore, for this case, it will be necessary to establish a load shedding strategy to maintain voltage stability. All loading margins can be directly obtained starting from the MLPpre and applying l = 1, except for branch numbers 3 and 6. 3.2. IEEE 57-bus system Fig. 4(a) shows the base case P-V curve (curve 1) and the postcontingency PV curve of each branch of the IEEE-57. The LM of the base case is equal to 0.7252 p.u. Fig. 4(c) provides a comparison of the number of iterations required to compute each post-contingency loading margin between the proposed methodology and the one that starts from the base case solution. Fig. 4(b) presents P-V curves of branches 41 (transformer connected between buses 7 and 29), 67 (TL connected between buses 29 and 52) and 80 (transformer connected between buses 9 and 55). Only in the case of the removal of one of these branches does the number of iterations of the proposed methodology become slightly larger, as shown in Fig. 4(c). For branches 42, 46, 47, and 48, the CPF fails to obtain the maximum loading point starting from the base case. As shown in Fig. 5, when one of those branches is removed, the LM is negative, i.e., the loading margin reductions are greater than 100%. Therefore, they are classified as the most critical branches, and thus, the removal of one of them leads the system to voltage collapse. 3.3. IEEE 118-bus system Two different operating conditions are considered for the IEEE118 (54 generation buses, 91 load buses and 194 branches). A single branch outage analysis is conducted for all of the transmission lines and transformers. For each contingency, the post-contingency loading margin and number of iterations are computed by considering both starting points: the base case solution and the solution at the maximum loading point of the base case. The exact maximum loading points of all contingencies of both operating conditions are shown in Figs. 6 and 8(a). A comparison between the numbers of iterations spent for both starting points is made for each branch outage. The results presented in Figs. 6 and 8(b) show that the proposed methodology generally requires a lower number of iterations to obtain the post-contingency maximum loading point. Figs. 7 and 9 show the per cent decrease of the loading margin for each branch outage. The details presented in these figures show

the twenty most critical contingencies, which were ranked according to the loading margin reduction. Branch 8, which corresponds to the transformer connected between buses 5 and 8, is the most critical for both operating conditions. Nevertheless, its removal results in a positive loading margin, and thus the system will not be led to voltage collapse. Fig. 10(a) shows the comparisons of the percentage reduction of the total number of iterations required to compute the post-contingency loading margins of all branches for all of the systems. Fig. 10(b) depicts the percentage of number of necessary iterations to perform all of the contingency analyses of all of the systems. The total number of iterations spent by the proposed methodology represents only 17% of the total iterations. When compared to the application of the continuation power flow method starting from the base case loading factor, a reduction in the total number of iterations due to the particular implementation of the proposed methodology was obtained in all the considered cases. Although in some cases the use of continuation power flow method starting from a base case loading factor fails to converge, the proposed methodology succeeds to obtain all of the solutions. 4. Conclusion This paper presented the numerical drawbacks that can appear when either a PF or a CPF is used for the determination of the postcontingency loading margin. The approach is able to provide the post-contingency loading margin with high accuracy by using only the continuation methods. The method overcomes the numerical problem by considering the computation of the post-contingency loading margin starting from the pre-contingency (base case) maximum loading point and choosing the voltage magnitude of the bus that presents the largest variation rate in its magnitude during the change from one P–V curve to another as the continuation parameter. In addition, its corresponding voltage magnitude at the MLP must be the fixed value during the changing of the curves. The method along with a criterion adopted for the parameter changes assures the convergence of the CPF used for the analysis of any TL or transformer outage, which provides a reduction in the global number of iterations. The main advantage of the proposed methodology is that it simplifies the development and implementation of the continuation methods for the static contingency analysis of electric power systems. In other words, the contingencies are efficiently analysed without considerable modifications to the CPF algorithms found in the literature. Furthermore, it was confirmed that the results obtained from the proposed methodology present better performance than those obtained using the method that determines the post-contingency loading margin starting from the base case with a significantly

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