A new contingency analysis approach for voltage collapse assessment

Electrical Power and Energy Systems 25 (2003) 781–785 www.elsevier.com/locate/ijepes

A new contingency analysis approach for voltage collapse assessment A.C. Zambroni de Souzaa,*, A.P. Alves da Silvaa, Jorge L.A. Jardimb, C.A. Silva Netob, G.L. Torresa, Claudio Ferreiraa, L.C. Araujo Ferreirac a

Escola Federal de English de Itajuba´—EFEI, Grupo de Engenharia de Sistemas—GESis, CP 50-CEP 37500-000, Itajuba´, MG, Brazil b Control Centre Technologies, Grid Operation, BC Hydro, 6911 Southpoint Dr., Burnaby, BC, Canada V3N 4X8 c Real Grandeza, 219/307, CEP 22283-900 Rio de Janeiro, RJ, Brazil Received 3 May 1999; revised 27 February 2001; accepted 14 February 2002

Abstract Voltage collapse analysis in power systems has been the subject of concern of many researchers. Recently, a new technique based on tangent vector behavior has shown to be attractive, because of the accuracy and the low computational effort required. In this paper, tangent vector is employed for contingency analysis. The idea consists of monitoring tangent vector norm associated with each contingency, identifying the most critical ones. The method is tested with the help of the IEEE-118 bus and the Southeastern Brazilian power systems, considering the generators reactive power limits. q 2003 Elsevier Ltd. All rights reserved. Keywords: Voltage collapse; Tangent vector; Contingency analysis

1. Introduction Voltage collapse is a phenomenon that may result in serious consequences for power systems, as observed in many reported occurrences around the world. The literature has shown that algebraic (‘static’) power system model may suffice for analysis of long-term voltage collapse [1]. Such a model is used in this work. Of course, for a rigorous analysis of large and fast disturbances, the dynamic characteristics of the system should be considered [12,13]. However, as the methodology proposed in this paper concerns only with the post disturbance equilibrium point, it does not take the dynamic aspects into account. It is also considered that angular and short-term voltage stability can be analyzed using time domain simulation. Several indices based on algebraic models have been proposed in the literature. Those indices identify a voltage collapse equilibrium point in which the Jacobian matrix of the algebraic equations becomes singular. Such an equilibrium point has been identified as being of the saddle-node bifurcation type. To locate this point, Lof et al. [2] and Barquı´n [3] propose to monitor the smallest Jacobian singular value, whereas Gao et al. [4], Marannino et al. [5] * Corresponding author. 0142-0615/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0142-0615(03)00030-9

assess voltage collapse through the calculation of the smallest eigenvalue. Other methodologies, like family of test functions, reduced Jacobian determinant [6], direct and continuation methods [7], energy functions [8] and optimization techniques [9] have also been proposed. A technique based on tangent vector behavior as a function of load increase has also shown to be effective for the determination of voltage collapse points [10,11]. That method has the advantage of calculating these points with short computational effort and identifying the system critical bus for other regular operating points, being therefore suitable for on-line environments. In this work, the norm of the tangent vector is used as an index for contingency screening in a voltage security assessment. It is well known that this norm tends to infinity as the operating condition approaches a bifurcation point, i.e. incremental load variations produce large voltage changes. This characteristic enables one to monitor the tangent vector norm as a voltage security index when topological changes occur. For each fault, a new equilibrium point and the associated tangent vector are calculated. The contingencies associated with the largest tangent vector norms are considered as the critical ones. This paper is organized as follows: Section 2 reviews the tangent vector method and proposes the methodology for

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contingency analysis. Section 3 shows results obtained with an IEEE-118 bus test system and the Brazilian South/ Southeast power system. The conclusions are in Section 4.

2. Contingency analysis methodology 2.1. Tangent vector calculation The power flow model used in this paper is represented by the set of algebraic equations (1): f ðx; lÞ ¼ 0

ð1Þ

where l is the parameter that drives the system from one equilibrium point to another and x represents the state variables, i.e. angle ðuÞ of all buses and voltage ðVÞ of load buses). The tangent vector [10,11] is the variation of the state variables variation with respect to the parameter l: For a known operating point, the inverse of the load flow Jacobian provides: 2 3 2 3 Dug DPg 6 7 6 7 6 Dul 7 ¼ ½J21 6 DPl 7 ð2Þ 4 5 4 5 DVl

DQl

where g and l represent the generators and load buses, respectively. The load is increased as follows: Pi ¼ Pi0 ð1 þ Kpi DlÞ

ð3Þ

Qi ¼ Qi0 ð1 þ Kqi DlÞ where Pi and Qi are the active and reactive loads as a function of the parameter l; and Pi0 and Qi0 are the initial active and reactive loads at bus i: The active power generation varies in the same way. Constants Kpi and Kqi represent the direction of increasing active and reactive power on bus i: Hence, Eq. (2) represents a generic load increase direction. Therefore: DPi ¼ Pi0 Kpi Dl

ð4Þ

and DQi ¼ Qi0 Kqi Dl Replacing Eq. (4) in Eq. (2) yields: 2 3 2 3 Kpi Pg0 Dug 6 7 1 7 21 6 7 6 7 TV ¼ 6 4 Dul 5 Dl ¼ ½J 4 Kpi Pl0 5 Kqi Ql0 DVl

ð5Þ

Notice that calculating tangent vector is computationally cheaper than a power flow iteration. Tangent vector TV is used as a predictor step in continuation methods, and shows how state variables change as a function of parameter l: Note that Dl is related to the system parameter variation, whereas Kpi and Kqi stand for a particular load increase direction associated with bus. It, therefore, may be a scalar,

if a single load direction is adopted, or a matrix, when each load bus is assigned to a different load increase. Note that the load increase direction also affects the tangent vector calculation. This feature drives one to conclude that tangent vector information is dependent on the parameter variation. The behavior of this vector as a function of load increase has been the concern in Refs. [10,11] where a comparison between this vector and the right-eigenvector has been carried out for the identification of the critical bus. It is shown that tangent vector identifies this bus for operating points others than the bifurcation one, a problem not addressed before in the literature. Because tangent vector may provide some discontinuities during the voltage collapse process, it cannot be used to estimate the voltage collapse point. This drawback has been overcome in Refs. [10,11], where an algorithm to calculate the voltage collapse point in a short computational time is proposed. It is shown that the computational time involved in this process is about one third of the time required by continuation methods. It is important to stress that the results obtained using this technique are the same as those obtained with the use of continuation methods. Therefore, the method handles accurately discontinuity problems due to reactive power limits. Ref. [14] proposes a method to detect the voltage collapse point, tracking the state variables (voltage level magnitude) as a function of system parameter. In order to obtain accurate results, the method proposed in Ref. [14] should follow the same iterative steps presented in Refs. [10,11]. Another use of tangent vector is proposed in Ref. [15], where a loss sensitivity analysis is carried out. In that paper tangent vector identifies the buses whose reactive power compensation reduces at most the system losses. It is also shown that even though voltage collapse may be associated with high values of system losses, control actions suggested to reduce those losses do not necessarily alleviate voltage collapse problems. 2.2. Contingency analysis tool Fig. 1 shows the PV curve associated with Bus 14 from the IEEE-14 Bus System, when the generators reactive power limits are considered. Three cases are considered: (1) system with no topological changes, (2) dotted line, when the transmission line connecting Buses 6 and 13 is switched off, and (3) dashed-dotted line, when the transmission line connecting Buses 5 and 6 is removed. Fig. 1 shows that third case is the most critical contingency, since the voltage collapse margin is the smallest. The figure also shows the tangent vector norm ðk k2 Þ calculated for the base case and for the postcontingency equilibrium point. Notice that the most critical contingency (smallest load margin) is associated with the largest tangent vector norm. It is expected, as a consequence of this result, that larger the tangent vector norm, the smaller the associated load margin.

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Fig. 1. Result IEEE-14 bus system (V_Bus 14).

Based on the fact that the most critical cases are associated with the largest norm, the following method is proposed:

rendering this methodology as inadequate as a tool for voltage collapse analysis.

(I) for each contingency, calculate the equilibrium point and its tangent vector norm; (II) sort the norms, in order to identify the most critical contingencies.

3. Test results

The discussion about the contingency analysis problem may follow two approaches: (a) Screening process, where the most critical contingencies are identified through a specified index (in this paper, voltage collapse proximity). For this purpose, load margin calculation is not required. In general, approximations on load flow computation are considered at this stage. In this paper, however, all the system static limits are taken into account. (b) Evaluation process, where the most critical contingencies are thoroughly analyzed. For voltage collapse analysis, it means the load margin calculation. This process is computationally more expensive than the previous one. However, because load margin is computed for few contingencies, the computational time involved is reduced dramatically. It is important to remind that the voltage collapse point is calculated with the help of the technique proposed in Refs. [10,11]. As stressed in foregoing sections, such a technique requires a smaller computational time than continuation methods. Notice that successive load flows do not provide good results, since step size determination is not trivial,

The contingency analysis methodology proposed in Section 2 has been tested with IEE-118 bus system and Brazilian Southeastern system, with 1768 buses. For both systems, the generators reactive power limits have been considered. 3.1. IEEE-118 Bus system Initially, a base case is considered. The tangent vector norm (TVN) is: TVN ¼ 2:8824 Employing the extrapolation technique proposed in Refs. [10,11], yields the following load margin: Dl ¼ 0:9 For this test system, the outage of all transmission lines is considered, one by one. Table 1 shows the outages associated with tangent vector norms larger than 3.25. In order to check the consistency of the results, the voltage collapse point is then calculated through the extrapolation technique proposed in Refs. [10,11]. This result is shown in the second column. It is important to stress that none of the contingencies associated with a tangent norm smaller than 3.25 is critical. This is proven when the extrapolation technique is employed and a large load margin is obtained.

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Table 1 Results for IEEE 118 Bus

Table 2 Results for the southeastern Brazilian system

Transmission line outage

Load margin ðDlÞ

Tangent vector norm

38– 65 48– 37 30– 17 69– 70 64– 65

0.7120 0.7160 0.7564 0.8490 0.8400

5.2900 3.7496 3.3721 3.3237 3.2609

Based on the results shown in Table 1, it is possible to see that this methodology may identify correctly the most severe contingencies. The methodology will be extended for a larger power system. 3.2. Southeastern Brazilian system The system used here consists of 1767 buses, 118 generators and 2526 transmission lines. This time, not only transmission lines, but also generators outages will be analyzed. The loading considered regards the month of November 1997, evaluated at the peak period. Firstly, transmission lines outages are considered: 3.2.1. Transmission lines analysis Besides tangent vector norm and load margin, the critical buses are also shown at the base case and the bifurcation point, yielding: Dl ¼ 0:097

Transmission line outage

Critical buses

Tangent vector norm

Load margin

106 –107

2611 2642 2627 2611 2642 2678 2611 2642 2678

321.79

0.086

289.37

0.089

288.87

0.096

220 –219

182 –274

analyzed. The second column presents the results obtained when the generator analyzed trips out. For each case, a power flow program is executed and tangent vector norm is stored. At the end of the process, the results are sorted, and the most critical generators outages are determined. When analyzing the contingencies likely to occur, three possibilities must be considered: I. II.

A post-contingency equilibrium point is obtained. No solution is obtained after a contingency is considered. Possibilities above have already been studied in the foregoing examples. III. The system reaches an operating point in the lower part of a PV curve.

TVN ¼ 283:43 Critical Buses ðBase Case and Bifurcation PointÞ ¼ 2611; 2642; 2678 The load margin obtained above indicates that point of voltage collapse is very close to the operating point. Notice that the system critical buses remain the same during the voltage collapse process. Tangent vector norm is then stored for contingency analysis. It is assumed that a list of contingencies is available, i.e. unlike the previous test, the technique is not carried out for all transmission lines. The contingencies on the 750 transmission lines are the most critical, since a feasible post-contingency equilibrium point does not exist for them. Table 2 shows the results obtained for the most critical contingencies among the remaining ones. The results above confirm the efficiency of tangent vector as a contingency severity index. As stressed before, the aim of the method is to identify the most critical contingencies under voltage collapse point of view, even though the system load margin is not calculated. 3.2.2. Generators analysis Table 3 illustrates the results obtained when generators outages are considered. The first column shows the generator

Item (III) above has not been considered yet. That is a situation of a theoretical concern only, since a power system does not operate in the lower part of the PV curve, as shown by eigenvalues analysis. Therefore, identifying such a situation provides important information about voltage collapse mechanism, since this case characterizes a nonoperating region. Hence, this possibility must be considered during a contingency screening with the point of view of voltage collapse. The tangent vector is used, once more, to identify when an outage drives the system to the lower part of a PV curve. For a stable equilibrium point, i.e. in the upper part of a PV curve, it is expected a voltage level decrease as a function of a load increase. Hence, the tangent vector components associated with voltage level are all negative. As Table 3 Generators outage cases Generator

Tangent vector norm

Grajau´ Ibiu´na Embuguacu ˆ ngelo S. A Itaipu

232.39 58.92 330.34 465.64 Diverge

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a consequence of this statement, if positive tangent vector components associated with voltage level are encountered, an undesirable condition is detected, since the system works in the lower part of the curve. The literature has already focused this problem, when continuation methods are used to trace the bifurcation diagram of a system. When some reactive sources are depleted, the system may be driven suddenly to the lower part of another PV curve, and no further load increase is considered. This point is not dependent on the load model, as shown in Ref. [15]. The methodology proposed in this paper enables one to detect immediately if an operating point is feasible for meaning of operation. This situation is illustrated in Table 3 with the cases marked with dark letters. Tangent vector identifies the postcontingency equilibrium points as in the lower part of a PV curve. This result is confirmed through an incremental system load increase. The voltage level obtained is larger than the previous case, rendering the point as unfeasible for real operation. Since the results are obtained with the use of a conventional power flow, the method may be easily incorporated as a real time simulation voltage security assessment. Based on this feature, this tool has been implemented at FURNAS (a Brazilian Hydro Electrical Company) control center for contingency screening analysis, where the Southeastern Brazilian system is monitored. The tests carried out in this paper follow the strategy adopted at FURNAS operation center. The system considered is the same as the one used here. For a list of 25 contingencies previously chosen, the total CPU time (load flow, tangent vector norm, sort the results) was short enough to use the technique for contingency screening purpose.

4. Conclusions A new proposal for contingency analysis from the point of view of voltage stability is shown. Such a proposal is based on tangent vector norm. It is shown that a large tangent vector norm indicates a low load margin. Therefore, for each contingency, the new operating point and tangent vector norm are evaluated. The contingencies associated with the largest tangent vector norms are identified as the critical ones. The results obtained render this technique adequate for this purpose, especially because of the short computational effort required. This technique may also be useful as a tool for short term maintenance schedule.

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Acknowledgements The authors thank the financial support from CNPq, FAPEMIG and FINEP/RECOPE (project 0626/96-SAGE).

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