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Two novel static and dynamic voltage stability based indexes for power system contingency ranking

2017; Institution of Engineering and Technology; Volume: 12; Issue: 8 Linguagem: Inglês

10.1049/iet-gtd.2017.0991

1751-8695

Aref Doroudi, Ali Motie Nasrabadi, Reza Razani,

Power Systems Fault Detection

IET Generation, Transmission & DistributionVolume 12, Issue 8 p. 1831-1837 Research ArticleFree Access Two novel static and dynamic voltage stability based indexes for power system contingency ranking Aref Doroudi, Corresponding Author Aref Doroudi doroudi@shahed.ac.ir Electrical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this authorAli Motie Nasrabadi, Ali Motie Nasrabadi Biomedical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this authorReza Razani, Reza Razani Electrical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this author Aref Doroudi, Corresponding Author Aref Doroudi doroudi@shahed.ac.ir Electrical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this authorAli Motie Nasrabadi, Ali Motie Nasrabadi Biomedical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this authorReza Razani, Reza Razani Electrical Engineering Department, Shahed University, Tehran, IranSearch for more papers by this author First published: 19 February 2018 https://doi.org/10.1049/iet-gtd.2017.0991Citations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The goal of contingency ranking is to accurately choose a list of critical contingencies and rank them due to their severity. In this study, two novel and effective approaches are presented for contingency ranking. The first method proposes a static voltage stability based index which is calculated from the modal analysis and the branch participation matrix. Based on average participation factor for each line, the average branch influence factor is calculated and then root sum square index is applied to rank the contingencies. The second approach is based on dynamic voltage stability. The method employs the largest Lyapunov exponent, a stability tool adapted from the theory of chaos and non-linear dynamical system, for time series. Time-series data of the weakest bus voltage are employed for the largest Lyapunov exponent computing and based on this quantity, contingency ranking will be carried out. Test results on the IEEE 14 bus system are presented and compared with three previous indices to show the effectiveness of both methods. 1 Introduction 1.1 Problem description Power system security analysis is an important issue in the planning and operation of power networks [1]. In the planning stage, security analysis is used to check out the performance of a power system and need for new transmission expansion due to generation and load growth. In operation stage, valuable assessment of system security conditions will lead to full and reliable utilisation of network capability and deliver power to loads with acceptable quality. Contingencies are considered as system risk states and threat for system security. For highly loaded power networks, contingency analysis is essential for network security. In this analysis, the voltage stability status of the power system is determined for the occurrence of different urgent outage events (contingencies). Contingencies are then ranked based on a proper system performance index (PI). Contingency ranking shows bottlenecks of the power system in the priority order. Suitable preventive and corrective actions can then be made to control the system [2]. 1.2 Previous works For more than three decades many papers have been presented on contingency selection and ranking. Traditionally, the contingency ranking-based indexes measure the violation of various limits related to power system security such as bus voltage limits and transmission lines overload. PI-based methods use a scalar PI value to describe the effect of a contingency on the system and to penalise severely any violations that exist. A PI index based contingency ranking technique can be mainly focused on verifying voltage limits violations during power system operation [3–5]. A well-known index in this category is the index which is defined as follows [6]: (1) where is the voltage magnitude at busbar i at contingency, is the voltage magnitude at busbar i at normal conditions, is a weighting factor, N is the number of busbars, is the voltage deviation limit and n is an exponent. Two parameters, weight and exponent, can be tuned in order to get a better performance. Analytical hierarchy process was used for adjusting the appropriate and unequal values for weighting factors [7]. The authors in [8–10] focused on the contingency ranking with respect to overhead lines overload severity. The contingency ranking based on the simultaneously real power flow and the voltage deviations for better ranking was done in [11, 12]. A serious drawback of these methods which are almost calculated from dc power flow is that there is no information about voltage stability and relationship between voltages and reactive powers. Some of the other papers in contingency analysis have focused on the closeness of system to voltage collapse (bifurcation point) [13–16]. In [13], a methodology for contingency ranking based on mixing the P –V and Q –V curves has been introduced. P –V curves are employed for load margin determination, and Q –V curves are calculated for a particular bus of interest, keeping the remainder of the system as invariant. The outcome is the reactive load margin associated with that bus. The results showed that the buses with the larger reactive load margin are the ones connected to the most severe transmission line outages. Ajjarapu and Christy [14] present an approach to find a continuous power flow solution starting at a base load and leading to the critical point of the system. It employs a tangent predictor to find a solution path of a set of power flow equations and a voltage stability index as an indicator of weak buses. The basic idea in [15, 16] is to use three (or more) known points in the V –Q curve and to fit an appropriate curve to those points. The maximum reactive power and its corresponding voltage magnitude are then determined based on the estimated Q –V curve. Distance from the maximum point represents the load margin. These methods suffer from the convergence problems. Jasmon and Lee [17] present a technique for ranking power system contingencies taking into account power system voltage collapse. The contingency ranking is done using the single-line method for reducing a distribution network and a method to outage simulation. The result of the outage is then further analysed to determine the level of voltage instability (L index) of the network as a result of the outage (2) where and are the total active and reactive loads in the distribution network, respectively. and are also the equivalent resistance and reactance of the reduced two buses network. The higher the L is, the closer the system is to instability. Fast voltage stability index (FVSI) was introduced in [18] to predict the occurrence of voltage collapse and contingency analysis caused by line outage in a power system. Taking the symbols 'i' as the sending bus and 'j' as the receiving bus, FVSl can be defined by (3) where is the reactive power at the receiving end, is the sending end voltage and Z and X are the line impedance and reactance, respectively. The value of FVSl that is evaluated close to 1.00 indicates that the particular line is closed to its instability point which may lead to voltage collapse in the entire system. Introducing the artificial intelligence, methods for contingency ranking is done based on neural networks [19] and fuzzy logic [20]. In [19], an integrated algorithm has been proposed for ranking contingencies in the deregulated network. Voltage violation, line flow violation, locational marginal price and congestion cost indices have been simultaneously considered to rank the contingencies. The algorithm uses neural networks method to estimate the power system parameters. Fuzzy system based voltage ranking methods are promising solutions to overcome the shortcomings of low exponent PI methods [20]. Based on their fuzzy rules, the fuzzy systems transform voltage and generated reactive power calculated values into a severity index. These methods can almost achieve the same performance as PI methods with high exponents. Another approach for contingency analysis is to use power flow based static and dynamic voltage stability analysis [21]. The static voltage stability analysis allows checking up of a broad range of system conditions and, if properly used, can provide much insight into the nature of the network. These methods can also provide a measure of degree of stability, like a proximity indicator, dominant eigenvalues, real and reactive power margin and so on. Dynamic voltage stability analysis, on the other hand, is useful for detailed study of specific voltage problems and examines how the system can reach the steady-state equilibrium point. Therefore, it is able to shed light on the mechanism of voltage stability and also provide corrective strategies to improve voltage stabilities [21, 22]. Static methods are time saving compared with those according to dynamic ones, but they give more optimistic results in evaluating loadability limit than dynamic methods [21]. 1.3 Proposed methods In this paper, two novel indices based on the static and dynamic voltage stability analysis are proposed for contingency ranking. In the static-based analysis, modal approach and branch participation matrix [23] will be utilised and a new index called root sum square index (RSSI) is proposed which can be ranked power system contingencies in an excellent and time-saving manner. In the dynamic approach, time-series data of weakest bus voltage after a fault occurrence are employed for the computation of finite-time Lyapunov exponents [24] and then contingency ranking will be done based on the largest Lyapunov exponent (LLE). The differences between the results of two approaches have been also analysed. The paper is organised as follows. In Section 2, the modal analysis is described and a new static voltage stability based index is introduced. Section 3 presents the contingency ranking algorithm based on the dynamic voltage stability index. In Section 4, the simulation results are shown and finally in Section 5 conclusions are drawn. 2 Static voltage stability based index for contingency ranking The static stability analysis captures snapshots of system conditions at different time frames along the time-domain trajectory. At each time frame, time derivatives of the system state variables assumed to be zero. Stability characteristics can be determined by computing the P –V and Q –V curves at selected load buses [13]. The method is time consuming and needs executing a large number of power flows. The modal analysis approach described in [23] has also been applied to voltage stability analysis of practical systems. The modal analysis mainly depends on the power-flow Jacobian matrix. The method can predict voltage collapse in complex power networks. It involves mainly the computing of the smallest eigenvalues and associated eigenvectors of the reduced Jacobian matrix (RJM) obtained from the load flow solution. The eigenvalues are associated with a mode of voltage and reactive power variation which can provide a relative measure of proximity to voltage instability, whereas the eigenvectors provide information related to the mechanism of voltage collapse. In general, system is voltage stable if the eigenvalues of RJM are all positive and the system is considered voltage unstable if at least one of the eigenvalues is negative. The smallest eigenvalue of RJM () determines the degree of system proximity to voltage instability [23]. 2.1 Modal analysis The Jacobian matrix can be written as [23] (4) By letting in (4), we have (5) where (6) is called RJM of the system. Equation (5) can be rewritten as (7) and can be given by (8) where is the right eigenvector, is the left eigenvector and is the diagonal eigenvalue matrix of . Equation (8) results in (9) By combining (5) and (9), the following expression can be obtained: (10) where is the i th eigenvalue, is the i th column right eigenvector and is the i th row left eigenvector of RJM. Each eigenvalue and corresponding right and left eigenvectors and , defined the i th mode of the system. If (), the i th modal voltage and i th modal reactive power variations are along the same (opposite) direction, indicating that the voltage is stable (unstable). The magnitude of each modal voltage variations equals to the inverse of times the magnitude of modal reactive power variations. The smaller the magnitude of positive , the closer the i th modal voltage is to being unstable. means the collapse of the i th modal voltage [23]. The smallest eigenvalues () show that how much the system is on the verge of instability. 2.2 Branch participation factors (BPFs) BPF indicates that for each mode, which branches consume the most reactive power in response to an incremental change in the reactive load [23]. The relative participation of branch k in mode i is given by (11) BPF associated with mode i is computed by assuming that the vector of modal reactive power variations has all elements equal to zero except for the i th element, which equals one. Branches with high participations are either weak links or are heavily loaded. Branch participations are useful for identifying remedial measures to alleviate voltage stability problems [23]. 2.3 Proposed static-based approach Clearly, alone cannot be employed for contingency ranking. The reason is that this parameter belongs only to the one mode and the mode corresponds to can be different from one contingency to the other. Furthermore, applying one contingency may omit one or more modes of the system. In this paper, the average weighted of eigenvalues for each mode and average BPF (ABPF) are used for contingency ranking. The proposed index can provide a good description of the overall stability nature of the system considering the contribution of each mode and branch. The algorithm is as follows. First, the eigenvalues of the system under normal conditions are calculated and for each mode, (average weighted of eigenvalues) is computed as follows: (12) where n is the number of modes. Second, the ABPF is calculated for each line (13) Third, the average branch influence factor (ABIF) is defined as (14) where is the number of system branches. Now, the first contingency is applied and is calculated again for this contingency. The RSSI is then computed by the following equation: (15) where and are related to normal and contingency cases, respectively. The value of RSSI will fall between 0 and 1. Finally, the contingencies are applied one by one and RSSI is calculated for each of them. RSSI can be used for ranking of contingencies. The higher the RSSI is, the closer the system is to instability. 3 Dynamic voltage stability based index for contingency ranking Power network is a large dynamic and non-linear system and its dynamic behaviour has enormous effect on the voltage stability. Consequently, in order to get more realistic results it is necessary to take into account the full dynamic network model. Dynamic voltage stability includes the dynamic nature of the system and how it affects bus voltage magnitudes from a given load and contingency configuration or a change in such a configuration. Static voltage stability is more optimistic than the dynamic one in evaluating loadability limit [22]. Although static methods are very suitable for screening, and determine the weakest bus by calculating bus participation factor in the system [23], final decisions should however be confirmed by more accurate time-domain simulations in which different characteristics of multiple controller, tap changer and so on are taken into account. Lyapunov exponents as a generalisation of eigenvalues may be employed as a stability criterion for dynamic voltage stability analysis [25]. Lyapunov exponents provide information about the divergence or convergence of nearby system trajectories and they can be used as a measure of average growth rate of instability. In this paper, a novel index based on the LLE is proposed which can be used for ranking of system contingencies. 3.1 Lyapunov exponents The general power system equations for voltage stability analysis, comprising a set of first-order differential equations, can be expressed in the following dynamical general form: (16) and a set of algebraic equations (17) with a set of known initial conditions , where is the state vector of the system, is the bus voltage vector, is the current injection vector and is the network node admittance matrix. The state of system is perturbed away from equilibrium point after a disturbance event. A Lyapunov exponent is a measure of divergence of two nearby trajectories at any point in the phase space. The attractor is a limit cycle or is a fixed point whenever all points in a neighbourhood of a trajectory converge into the same orbit. However, if the attractor is strange (not stable), any two trajectories that start out very close to each other separate exponentially with time. This path from initial conditions can be mathematically written as (18) where , the rate of separation of system's trajectories, is termed LLE. According to (18), the LLE can be obtained as (19) The nearby system trajectories will converge (diverge) to each other whenever the LLE of the system is negative (positive) [25]. In general, the LLE can be computed using the time-series data (model-free) or the system's state equations (model-based) [26, 27]. In this paper, time-series voltage data are employed for computation of LLE because it does not need the mathematical model of the system. Voltage data can be gathered by making use of advanced system measuring devices such as phasor measurement units (PMUs). 3.2 Calculation of LLE from a time-series data In this section, calculation of LLE from a one-dimensional time series of data is described. Most of the existing approaches for calculating LLE using time-series data are suited for offline computation. In this paper, the algorithm proposed in [26] has been adopted for offline computation of LLE. The time-series voltage v (t 0), v (t 1), v (t 2),… are labelled as v 0, v 1, v 2,… for the sake of simplicity. Let the time intervals between samples are all equal. Hence (20) where m and T are the number of samples and the time interval between samples, respectively. The algorithm is as follows: select some value from the voltage time series, say, ; search another value, say, that is close to and then create the sequence of differences as in the following: (21) It is assumed that the sequence of differences rises exponentially, at least on the average and as n increases (22) Taking logarithm of both sides of (22), we get the LLE as (23) In this method of LLE calculations, two nearby trajectory points are located in space and the differences between the two trajectories that start from each of these initial points are then followed. In a dynamical system, if the LLE of the system is negative, the steady state and destination of the system dynamics is a stable equilibrium point. Positive value of the LLE indicates a chaotic behaviour and unstable dynamics [26]. The higher the absolute value of LLE is, the closer the system is to instability. 3.3 Minimum time separation There are restrictions on the choosing of that should be selected for a given . If time series is from a closely spaced sampling of a smoothly varying variable, should not select too close to , because the two values occur close in time. In this case, a closing performance is expected and the process ends up with an anomalously small value of . This problem can be avoided by insisting on selection such that it does not too closely follow in the sequence. Several criteria have been proposed for choosing a minimum time separation. The method based on the 'correlation times' concept is used in this paper [26]. The auto-correlation function measures the correlation of a signal with itself shifted by some time delay (time lag). Normally, there is a similarity between observations as a function of the time lag between them, which may be described in terms of the autocorrelation function (24) The autocorrelation function is a mathematical tool for signal processing, e.g. picking out the hidden fundamental frequency in a harmonic signal or specifying a periodic signal in a signal obscured by noise. The compares one observation in the series with another observation placed away n unit of time. When, on the average, they are uncorrelated, equals 0 and instead if they are nearly the same, then . One of the most useful descriptive tools in time-series analysis is to generate the correlogram plot which is simply a plot of the serial correlations versus the time lags n for n = 0, 1,…, M, where M is usually much less than the sample size. For time-series data obtained from stochastic or chaotic systems, the autocorrelation function is expected to fall off exponentially with time, i.e. , where is called the autocorrelation time. Given , one can access that have minimum correlation with selected . 3.4 Proposed dynamic-based approach The proposed dynamic stability based index is the LLE. For dynamic voltage stability analysis, an event capable of capturing the dynamic behaviour of the power networks should be first defined. Short circuit event in power grids has the greatest effect on all buses within the system, reveals the system dynamic nature and stimulates all modes. Therefore, in this approach, three-phase fault at one important bus is chosen for producing the severe conditions. Second, the first contingency is applied immediately following fault clearance and time-series voltage data of the weakest bus (at selected contingency case) are employed for the LLE computing. Computation of the LLE using time-series data was mentioned in Section 3.2. After finding the most uncorrelated observations, is computed for a large number of n using (21). The natural logarithm of is then plotted as a function of n. The slope of the fitted straight-line gives the LLE. Finally, the contingencies are applied one by one and LLE is calculated for each of them. The LLE can be used for ranking of contingencies. The higher the absolute value of the LLE is, the closer the system is to instability. 4 Numerical results The IEEE standard 14-bus system (Fig. 1) is selected to show the efficiency and effectiveness of the proposed methods. The network has 15 transmission lines which are indicated in the figure. Only transmission line outages have been considered as contingency. All generators are equipped with an identical automatic voltage regulator, over excitation limiters and turbine governor. Fig. 1Open in figure viewerPowerPoint Single-line diagram of the IEEE 14-bus system 4.1 Static voltage stability based method The eigenvalues of the 14-bus network under normal conditions and participation factors of each line for all modes should be first calculated. Table 1 shows the BPF for one of the modes (mode 1). The table depicts that for mode 1, lines 2-3 and 12-13 have the greater and lower influence on power network stability, respectively. Table 1. BPF for mode 1 in normal conditions Line BPF Line BPF 1-2 10.61843 6-12 1.506485 1-5 26.35031 6-13 3.95055 2-3 33.77175 9-10 1.981075 2-4 19.47501 9-14 4.762383 2-5 15.95718 10-11 0.19186 3-4 4.644839 12-13 0.034518 4-5 3.815618 13-14 0.219902 6-11 0.382025 — — Subsequently, the ABIFs are calculated by (14). ABIFs are listed in Table 2. The higher the ABIF is, the more effective the line is for that mode. Table 2. ABIF for mode 1 in normal conditions Line ABIF Line ABIF 1-2 0.0477 6-12 0.0086 1-5 0.1652 6-13 0.0195 2-3 0.3 9-10 0.0245 2-4 0.1231 9-14 0.0626 2-5 0.0892 10-11 0.0153 3-4 0.1009 12-13 0.0043 4-5 0.0282 13-14 0.0045 6-11 0.0066 — — Table 3. RSSI for each contingency Line outage RSSI Rank Line outage RSSI Rank 1-2 NAN 1 6-12 0.0253 11 1-5 0.2365 3 6-13 0.0571 10 2-3 0.4042 2 9-10 0.091 9 2-4 0.1424 6 9-14 0.0941 8 2-5 0.1798 5 10-11 0.0209 12 3-4 0.2232 4 12-13 0.015 15 4-5 0.129 7 13-14 0.0178 14 6-11 0.0179 13 — — — Now, the contingencies are applied one by one and RSSI is calculated for each of them. The results are indicated in Table 3. For contingency no. 1, the load-flow solution is unobtainable and the program is diverged. The RSSI can be used for ranking of contingencies. The higher the RSSI is, the closer the system is to instability. The table depicts that outage of the lines 1-2, 2-3, 1-5, 3-4, 2-5 and 2-4 are the severest contingencies cases in consequence. 4.2 Dynamic voltage stability based method In this algorithm, the LLE should be calculated for each contingency. A three-phase fault is inserted in the bus 5 to motivate dynamic of the system. The fault time is chosen 100 ms to fully reveal the dynamic behaviour of the system. Now, the first contingency is applied and the weakest bus is determined. Line outage is taken place immediately following the fault clearance. Fig. 2 shows buses 3, 4, 5 and 13 of instantaneous voltages which are obtained during outage of line 1–5 after a fault on bus 5. The figure shows that the weakest bus is the bus 3. The weakest bus can also be obtained by a static analysis similar to weakest line except that instead of branch participation matrix, the bus participation matrix should be employed [23]. Subsequently, time-series voltage data of this bus are employed for the LLE computing. After finding the most uncorrelated observations, is computed for a large number of n using (22). The natural logarithm of is then plotted as a function of n (Fig. 3). The slope of the fitted straight-line gives the LLE. Fig. 2Open in figure viewerPowerPoint Instantaneous voltages of bus nos. 3, 4, 5 and 13 during outage of line 1-5 Fig. 3Open in figure viewerPowerPoint versus n for voltage of bus no. 3 Finally, the contingencies are applied one by one and the LLE is calculated for each of them in a similar manner. Table 4 shows the results of the proposed dynamic method. The obtained LLEs can be used for ranking of contingencies. The higher the LLE is, the closer the system is to instability. Table 4. LLE for each contingency Line outage LLE Rank Line outage LLE Rank 1-2 NAN 1 6-12 — — 1-5 0.2329 2 6-13 0.1748 8 2-3 0.2085 3 9-10 — — 2-4 0.2003 6 9-14 0.1665 11 2-5 0.2013 5 10-11 — — 3-4 0.2074 4 12-13 — — 4-5 0.1995 7 13-14 0.1714 10 6-11 0.1723 9 — — — The results are similar to the static-based approach for the first seven contingencies except number two and three. For a dynamic index, line 1-5 is the severest contingency after the line1-2. To judge which one is correct, loadability margins criteria is selected. At first, the weakest buses at lines 2-3 and 1-5 outage contingencies are determined and P –V curve of this bus is obtained. Figs. 4 and 5 are the P –V curves at these contingencies. As it can be seen in the figures, maximum loadability in outage of line 1-5 is less than the maximum loadability in outage of line 2-3. So, it can be concluded that the line 1-5 outage is severe than line 2-3 outage and the dynamic approach estimates better the severe contingency than the static approach. Fig. 4Open in figure viewerPowerPoint P–V curve of weakest bus – outage of line 1–5 Fig. 5Open in figure viewerPowerPoint P–V curve of weakest bus – outage of line 2–3 It is noteworthy that, as it can be seen in Table 4, for some cases the LLE cannot be calculated. These are relevant to the cases that the voltage signal is so smooth and changes slowly. In these cases, however, contingencies are not important. As a result, this method can also be used for screening of contingencies in networks with several existing contingencies. 4.3 Comparing with other works In order to validate the proposed static and dynamic indexes, three certified indexes including [6], L [17] and [18] were chosen and simulated on the IEEE 14-bus systems. Table 5 shows the results of contingency ranking of the three methods. The results agree with the proposed methods with minor differences indicating that the proposed indexes are reliable. Table 5. Comparison with three cited indexes Item Line outage Ranking PI Ranking L Ranking FVSI 1 1-2 1 1 1 2 1-5 4 3 3 3 2-3 2 2 2 4 2-4 7 4 4 5 2-5 6 6 5 6 3-4 3 7 10 7 4-5 5 5 6 8 6-11 9 12 12 9 6-12 10 11 11 10 6-13 8 8 8 11 9-10 15 10 9 12 9-14 12 9 7 13 10-11 14 14 15 14 12-13 13 15 14 15 13-14 11 13 13 5 Conclusion Two novel indexes are introduced for power system contingency ranking. The first index is based on static voltage stability and modal analysis. The average weighted of eigenvalues for each mode and ABPF are used for contingency ranking. The proposed method can estimate the whole stability nature of the system considering the contribution of each mode and branch. It is not time consuming and very simple. The second method uses dynamic simulation and the LLE. Time-series voltage data were employed for computation of LLE, because it does not need a mathematical model of the system. In modern power systems voltage data can be simply gathered by making use of PMUs. LLE can fully capture the dynamic nature of power systems. Simulations on the IEEE 14-bus system show the results of both methods. The static based method is a fast and reliable approach. The dynamic algorithm is more time consuming. However, it can cover whole dynamic and non-linearity nature of the power system. Static voltage stability is more optimistic than the dynamic one in evaluating loadability limit and final decisions should be confirmed by more accurate time-domain simulations or dynamic analysis. 6 References 1Moa, A., Iravani, M.R.: 'A trend-oriented power system security analysis method based on load profile', IEEE Trans. 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