IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 4, OCTOBER 2007
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A Method for Voltage Sag State Estimation in Power Systems Elisa Espinosa-Juárez, Member, IEEE, and Araceli Hernández, Member, IEEE
Abstract—This paper presents a new technique for voltage sag state estimation. The proposed method is based on estimating the number of voltage sags occurring at nonmonitored buses from the number of sags recorded at a limited number of monitored buses. This problem, contrary to the traditional state estimation where redundant measurements are available, is formulated as an underdetermined system of equations. In this paper, linear integer programming techniques are employed to solve this estimation problem. The performance of the proposed method is tested by means of several case studies applied in the IEEE 24-bus Reliability Test System (RTS) and in the IEEE 118-bus Test System. Index Terms—Power quality, power system, state estimation, stochastic techniques, voltage sags (dips).
I. INTRODUCTION OLTAGE sags are one of the power quality disturbances which causes the most concern to customers and utilities. Even a short duration voltage sag could cause malfunctioning or failure of a continuous process leading to a significant economic impact [1]. In recent years, different stochastic approaches, such as the well known method of fault positions, have been proposed for estimating the voltage sag performance at a certain bus of an electrical system [2]–[7]. Stochastic methods make use of historical fault statistic data which are usually known for an existing system. However, the system fault rates can vary widely from one year to another depending on different factors, such as weather conditions, maintenance, etc. [1], [6]. Therefore, the probabilistic nature of stochastic methods makes them suitable for long term estimations, but in a specific year, the predicted number of voltage sags can differ substantially from the number measured. In addition, in some cases, historic data are not available when analyzing a part of the system recently introduced or modified. An alternative approach to overcome the aforementioned shortcomings of the stochastic procedures, is the monitoring of the power supply. This method can provide a direct assessment of the voltage sags performance of the monitored site.
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Manuscript received July 26, 2006. This work was supported in part by PROMEP and in part by the Universidad Michoacana de San Nicolás de Hidalgo on behalf of the first author. Paper no. TPWRD-00400-2006. E. Espinosa-Juárez is with the Electrical Engineering Faculty, Universidad Michoacana de San Nicolás de Hidalgo, Morelia 58060, México (e-mail:
[email protected]). A. Hernández is with the Department of Electrical Engineering, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, Madrid 28006, Spain (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRD.2007.905587
However, due to cost constraints, there is always a limitation to the number of instruments a utility can afford to place in a power system. Therefore, it is not possible to monitor the whole network. Voltage sag state estimation (VSSE) approaches are intended to fill this gap since they are based on estimating the voltage sags frequency at unmetered buses by using the data collected at a limited number of metering points. Due to the limitation on the number of available monitors, VSSE differs from the traditional state estimation where redundant metering data are available. As a result, the problem of VSSE must be approached differently. The VSSE concept has been recently introduced by some authors although further development is required to enhance its applicability. In [8], a VSSE method is proposed that estimates the voltage profiles of a feeder based on a limited number of metering points. This is achieved by making use of the radial connection characteristic of a distribution feeder. In [9], the expected number of sags at a low voltage grid connection point can be estimated by using the information about sag characteristics at other low voltage grid points at which sag meters are installed. This method is based on multiple regression algorithms which considers variables such as altitude, lighting density, etc. The objective of this paper is to propose a new VSSE method which estimates the sags frequency within given bins of residual voltage at nonmonitored buses. The proposed method is valid for any network geometry and for any type of fault. First, some aspects about the available metering technology and the monitoring placement are discussed in Section II. The basis of the proposed method is presented in Section III and the details of the formulation are described in Sections IV and V. Section VI describes the extension of the method for unsymmetrical types of faults. The details of the implementation are provided in Section VII. Finally, some case studies are shown in Sections VIII and IX.
II. VOLTAGE SAGS MONITORING INSTRUMENTS Prior to establishing the method for voltage sag estimation, it is important to bear in mind the characteristics and constraints imposed by the existing sags monitoring technology. On this regard, one of the main limitations is caused by the number of instruments an utility can afford to place on an electrical system. The more voltage sags monitoring instruments used, the more accurate the estimates can be, but the higher the cost. As a result, the problems faced by VSSE are not the same as those of traditional power system estimation where redundant metering data are available [8]. On the contrary, in VSSE the problem is
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characterized by an underdetermined system of equations. Consequently, the problem must be approached differently. In addition, the sag monitors are unlikely to be synchronized, which further complicates the problem. Another important issue arises from the limits imposed on the number of available monitors: how to select their optimal location and number. Clearly, these two factors (location and number of monitors) will significantly influence the results of the VSSE. In the context of VSSE, monitors placement can be considered as optimal if it provides the best possible estimates. In this paper, the method presented in [10] is adopted for the selection of the number and position of the meters. This method is based on guaranteeing that the monitoring locations cover the entire network. That is, the placement obtained by this approach makes sure that every fault in the system triggers at least one voltage sag monitor. Apart from the number and location of monitors, their measuring capabilities must also be taken into account. In this work, it is assumed that the voltage sags monitors report uniquely the number of voltage sags classified in several discrete bins of remaining voltage magnitude. This requirement follows the recommendations given for voltage sags measurements in IEC 61000-2-8 [11] and can be easily met by most of the metering equipment. III. BASIS OF THE PROPOSED VSSE METHOD VSSE aims to estimate the number of voltage sags at nonmonitored buses from the number of voltage sags recorded at monitored buses. Traditionally, the general mathematical formulation used in state estimation is based on the following relation [12] (1) where is a measurement vector, is the state vector to be estimated, is a measurement matrix, and is the vector of measurement noise. The general expression (1) will be applied here in the proposed VSSE method. In this approach, the measurement noise in (1) will be initially ignored [13]. The construction of the terms of (1) is analyzed in Sections III-A–C. A. Measurement Vector The elements of the measurement vector of (1) will indicate the number of voltage sags registered by the monitors with residual voltage equal or lower than a voltage threshold . Therefore, the vector is formed directly from the collected measurements data. B. State Variables Vector Elements of vector of (1) are formed by the state variables to be estimated. In this approach, the state variables will indicate the number of faults occurring at certain segments of the system lines during the period under study. An important characteristic of this method is that these segments of lines are established by determining the positions in which if a fault occurs, the residual
voltage magnitude caused at the buses of interest reaches exactly the value of the voltage threshold. Therefore, all the faults occurring at any location within a certain segment limited by these positions produce sags either over or below the considered threshold. Another important aspect is that, since state variables indicate the number of faults and not the number of voltage sags, the latter must be calculated next by establishing and adequate correlation between faults and sags. Additionally, this approach implies that only sags caused by faults are considered in the proposed method. C. Measurement Matrix that relates the state variables to The measurement matrix refers the measurements is a binary matrix. Each element of to a bus of the system and to a segment of a transmission line. An element takes the value one if a fault occurring in the associated line segment causes residual voltage below at the considered bus. The element is zero if the fault does not cause a sag. The first step of the proposed method consists on defining the aforementioned line segments, that is, on determining their limit positions. This task is achieved by applying an analytical method that provides an expression for the residual voltage at a bus of the system when faults take place at any location along a line [4], [7]. Although this method does not constitute the object of this paper, it is briefly described in Section IV for the sake of convenience. Next, the calculation of segments is explained and, after that, the solution process of the VSSE is presented. IV. VSSE FORMULATION In this section, the steps of the proposed formulation for VSSE are described in detail. A generic electrical system with buses and lines will be assumed from here on. denotes the number of metered buses. . Throughout this paper, we will consider that A. Analytical Method for Voltage Sags Calculation In [4] and [7], a method is presented that provides an analytical expression for the residual voltage resulting at a bus when a fault occurs at any position of the system. This method is used here for calculating the segments of interest of the lines. When faults take place at a bus of the electrical system, the sag magnitude (remaining voltage) at a bus of interest, bus , can be easily calculated according to the classical short circuit calculations, based on the -bus matrix. If the faults take place at any location along the lines in the system, for example, at a generic transmission line between two generic buses and (Fig. 1), the location at which the fault occurs can be identified by means of the parameter . This parameter varies continuously from 0 to 1, as the fault position moves from bus to bus , therefore is defined as (2) where (as shown in Fig. 1) is the distance between bus and the location where the fault occurs and is the total length of transmission line .
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Fig. 1. Electrical system.
When a fault takes place at a generic position , between buses and , the voltage at bus [7] can be calculated by
Fig. 2. Remaining voltage V ( ) at buses m and i, for faults in line l .
(3) equation will be used next for the calculation of the line segments used in the VSSE approach.
where pre-fault voltage at bus ; transfer bus impedance between busbar of the system and the position of the fault ; fault current phasor at . According to [4] and [7], impedance of (3) can be obtained from the transfer impedances of the existing buses of the system as (4) where and are the -bus matrix elements corresponding to buses and . has to be calculated according to the type of fault analyzed. For three phase faults, is calculated as (5) is the driving bus impedance of the location situated where along the line. Expressions similar to (5) can be formulated for other types of faults [7]. can be calculated [4], [7] from the impedances of existing buses of the system as
(6) where is the impedance of the transmission line . Finally, assuming unitary pre-fault voltage and considering (4), (5) and (6), (3) can be expressed as shown in (7) at the bottom of the page. Equation (7) provides an analytical expression for the voltage during the fault at bus in terms of parameter , which varies . This continuously defining the fault position along line
B. Calculation of Line Segments: State Variables Assignation According to (1), the proposed VSSE method must relate the number of faults occurring at certain line segments (vector ) with the number of sags recorded by the measuring instruments (vector ). Initially, a voltage magnitude threshold will be assumed. Therefore, the elements of the measurement vector will be formed by the number of sags registered with a remaining voltage below the threshold . The measurement vector built from here onwards. for will be named The total length of the lines of the system must be divided into segments in which a fault produces remaining voltage either metered completely below or completely above at the ) and at the analyzed unmetered buses (buses bus (bus ). This division is made as follows. • Appling (7), it is possible to obtain the remaining voltage and at the monitored buses at the bus of interest , when a fault takes place along a generic line . • The intersection of these voltage curves with the threshold provides the ends of the segments of the line . This can be easily understood by means of the example shown in Fig. 2. For the sake of simplicity, in this example only one , and the bus of interest, bus , are repremonitored bus, bus sented. Fig. 2 shows the remaining voltage magnitude at buses and when a fault takes place along a line of the system. For a voltage threshold of value , line is divided into four , and . segments , This process should be carried out in a similar way for the lines of the power system. The total number of segments obtained for the whole system will be named . segments have been established, a state variable Once the is associated to each segment. Therefore, a vector with variables is defined. Each state variable represents the number
(7)
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of faults that takes place in the corresponding segment during the period under study. C. Calculation of Measurement Matrix In the previous step, the state variables have been assigned to the line segments. Next, according to (1), it is neccorresponding essary to construct the measurement matrix to a voltage threshold , which relates these variables with the measurement vector . is a binary matrix of order referred to Matrix segments. The elements of the monitored buses and to the are formed according to the expression shown at the bottom of the page. If the measurement matrix is multiplied by the number of buses is obtained, faults, the number of sags observed at the that is (8) For the example shown in Fig. 2, elements of matrix would result as follows:
(9)
takes the value 1 which indicates that as it Element can be observed in Fig. 2, faults at segment of line cause at bus . Analogously, element sags with voltage below is 0 because faults at segment do not cause sags for the considered threshold. at bus This process must take into account all the lines of the system by adding new columns corresponding each to a line segment. Each row is associated to a monitored bus. It is important to note that, due to the method selected for contains at least one monitors placement, all the columns of element different from 0. This means that the monitors cover the whole network, as it was demanded by the placement algorithm [10]. In other words, any fault leading to a sag triggers at least one monitor. D. Extension of the Method to the Application of Several Voltage Thresholds In the previous section, terms of (1) have been defined for magnitudes calculated with a voltage threshold . However, as discussed earlier in Section II, usually sags monitors report the number of voltage sags occurred in some discrete
Fig. 3. Voltage magnitude at bus m and bus i for faults in line l .
bins of remaining voltage. Therefore, the process carried out before for a threshold , can be conducted again in an analogous way for a number of different voltage thresholds. For each threshold, a new division of the line into segments is obtained. Fig. 3 illustrates graphically this idea. Line is divided into , , , and when considering remaining segments is considered, line voltage threshold . If the threshold is divided into segments , , and . Similarly, for , , the threshold , this line is divided into segments and . In an analogous way, it could be made for additional voltage threshold values. Again, a state variable is associated to each segment. A matrix of measurement can be built for each of the considered thresholds. Consequently, systems of equations similar to (8) can be obtained by constructing the measurement vector and the measurement matrix for the considered voltage thresholds. In the example of Fig. 3 these systems are (10)
where , , , are the measurement vectors (number of sags) recorded at monitored buses, for , , and , respectively. , , , are the binary measurement submatrices built for segments corresponding to thresholds , and , respectively. , , , are the state variables vectors corresponding to the number of faults occurred at each line segment. These line segments are obtained for thresholds , and .
ESPINOSA-JUÁREZ AND HERNÁNDEZ: A METHOD FOR VOLTAGE SAG STATE ESTIMATION IN POWER SYSTEMS
In general, for any analyzed system, considering threshold values for sags measurement and classification, (10) can be expressed in a compact form as
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According to the previous analysis, the value of the state must be estimated under the following variables vector constraints:
(11) equations. The The system of (11) is constituted by number of unknowns is equal to the total number of segments . obtained for all the considered thresholds, that is Taking this into account, it can be deduced that generally (11) is an underdetermined system of equations. On one hand, the is limited. On the other, innumber of available monitors creasing the considered number of voltage sags classification ranges implies introducing new unknowns associated to the new segmentation of lines. However, it is important to notice that the state variables associated to new thresholds are not completely independent from the previous ones. These new state variables are subjected to some restrictions that are analyzed in Section IV-E. E. Solution Constraints As it is well known, an underdetermined system has multiple solutions. The feasible region of this underdetermined problem can be reduced to a large extent by taking into account certain relations that must be fulfilled by the defined state variables. That is, a solution of (11), in order to be also a valid solution of the VSSE, must be subjected to the following constraints. C1) The addition of the number of faults occurred in the segments of a line provides the total number of faults occurred in the whole line. This value must be the same regardless of the division into segments performed. Hence, the sum of the values of the state variables assigned to the segments of a line for a certain voltage threshold must be equal to the sum of the state variables assigned to segments corresponding to a different threshold for the same line. For instance, for the example shown in Fig. 3, this restriction would lead to the following relations between the state variables associated to segments obtained for thresholds , and
C2) Faults at system lines can only take zero or positive integer values. That is, the elements of the state variables vector with . must fulfill that: C3) Finally, some inequalities can be established between state variables. Bearing in mind that variables represent the number of faults in segments and that some segments are completely contained inside others, necessarily certain state variables must have a value lower or equal than others. For instance, for the example shown in Fig. 3, as segment is completely included into , the variable must fulfill . In general, if is the variable associated to a segment with and is associated to segment , it can be limits said that
(12) where and
and
and
values of defining the initial and final positions of the segment associated with the state variable ; values of defining the initial and final positions of the segment associated with ; the state variable subscript indicating the considered line and it takes values from 1 to ; subscript indicating the considered voltage threshold value and it takes values from 1 to ; segments in which a line is divided. They take values between 1 and ; segments of all lines of the system and it takes values between 1 and .
F. Solution In the case of other disturbances such as harmonics, different methods have been proposed to deal with underdetermined systems in state estimation problems [14], [15]. Here, taking into a binary matrix, this account that is an integer vector and problem is solved by applying linear integer programming techniques. The optimization algorithm can be applied to solve the problem subjected to the constraints (12) without specifying and objective function. That is, the algorithm is applied just to find a feasible solution. A different strategy would be to consider an objective function in order to find an optimal solution according to some optimization criterion. A possible optimization criterion would be to select the solution which fits the best to the faults probability distribution along the lines. For instance, if uniform probability distribution along the lines is assumed, the optimal solution would be the one which, being subjected to the constraints (12), distributes the number of faults in proportion to the length of the associated segments. For this case, the objective function can be formulated as (13)
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where is a parameter defined as the quotient between the and length of the segment associated to the state variable the total length of the line in which this segment is included. In cases where nonuniform fault distribution along the line must be considered, (13) can be conveniently adapted. V. ESTIMATION OF NUMBER OF VOLTAGE SAGS The procedure previously described allows estimating the state variables vector . The last step consists in calculating the expected number of voltage sags at the bus of interest. This number can be easily calculated by means of a matrix that relates the number of faults estimated to the number of sags. This matrix is formed which was built for the metered buses. exactly as the matrix would be For the example shown in Fig. 2, matrix
(14) Therefore, the number of sags bellow at the bus of interest is
with a remaining voltage (15)
Analogously, the number of sags at bus can be calculated by means for other voltage threshold by means of the appropriate and corresponding to the analyzed voltage submatrix of range. VI. EXTENSION TO DIFFERENT TYPES OF FAULTS The mathematical formulation presented in this paper can be extended in an analogous way to consider any type of fault. Therefore, a system of equations similar to (11) should be formed for each type of fault, that is (16) (17) (18) (19) ,
,
measurement vectors formed by the number of sags produced, respectively, by phase-to-ground faults, two-phase-to-ground faults, phase-to phase faults and three-phase faults;
and
,
,
, measurement matrices formed for phase-to-ground faults, two-phase-to-ground faults, phase-to-phase faults, and three-phase faults, respectively.
,
,
,
and
and
state variables vector formed for the number of phase-to-ground faults, two-phase-to-ground faults, phase-to-phase faults, and three-phase faults, respectively.
Fig. 4. IEEE RTS 24- bus test system.
In order to consider asymmetrical types of faults, the monitors reach area determined by the measurement matrix should be redefined. In this paper, the lowest phase voltage has been used to delimit the sags occurrence. In addition, this approach implies that an appropriate method for faults identification at the metered buses should be used in order to classify the different types of faults [1], [16]. VII. ALGORITHM IMPLEMENTATION The proposed VSSE method has been implemented using MATLAB. The optimization package used for solving the linear integer problem is MOSEK [17]. In Section VIII, some case studies are presented in order to show the performance of the proposed method. VIII. STUDIES IN THE IEEE 24-BUS RELIABILITY TEST SYSTEM The proposed VSSE method is applied to the IEEE 24-bus reliability test system. This network (Fig. 4), consists of 11 generating stations, 24 buses interconnected by 33 lines and five transformers. The data system is provided in [18]. The balanced three phase fault distribution detailed in the Appendix has been considered. The number of faults assumed at each line corresponds to the faults/year rates of the lines proposed in [18] rounded down to an integer number. The positions of the faults have been obtained by applying a random algorithm and assuming an uniform fault probability distribution along the line. The faults/year rate at buses has been assumed negligible [19]. As it was explained in Section II, the monitors location and number has been obtained by applying the methodology proposed in [10], for a threshold of 0.7 p.u. According to this, any fault causing a voltage sag with magnitude below 0.7 p.u.
ESPINOSA-JUÁREZ AND HERNÁNDEZ: A METHOD FOR VOLTAGE SAG STATE ESTIMATION IN POWER SYSTEMS
TABLE I VSSE PERFORMANCE (MONITORS PLACED AT BUSES 2, 3, 8, AND 17)
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TABLE II AVERAGE NUMBER OF VOLTAGE S SAGS IN THE WHOLE SYSTEM
Fig. 5. Voltage sags with magnitude below 0.9 p.u. obtained with different number of monitors.
is recorded by, at least, one monitor. The application of this method leads to placing monitors at buses 3, 6, 8, and 17 of the system shown in Fig. 4. The measurement vector of the VSSE algorithm is formed by the number of sags experienced at the metered buses. This number of sags is classified in six voltage ranges (corresponding to sags with magnitudes below 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 p.u.). These values are equivalent to the measurements that would be recorded by monitors located at the indicated buses. The performance of the method is analyzed in Table I. In this table the number of voltage sags estimated by using the proposed approach is compared to the actual number of voltage sags caused by the assumed faults distribution and calculated by classical short-circuits theory. In this case, the VSSE algorithm has been applied by considering the constraints (12) and the objective function (13). As expected, the real number of sags and the number predicted by VSSE matches exactly at buses 3, 6, 8 and 17 where the monitors are placed (Fig. 4). In the rest of buses, some discrepancies can appear between the real and the estimated values but, in general, there is a good agreement. On average, the error between the actual and the estimated values for all the system buses is 2.47% when sags with magnitude below 0.9 p.u are estimated. If sags below 0.7 p.u. are considered, the average estimation error is 8.88%. The reason why the error increases when considering lower thresholds, is that the number and location of monitors guarantees that any sag with magnitude below 0.7 p.u is recorded by, at least, one monitor. Therefore, sags with magnitude over 0.7 p.u. can be recorded by a larger number of monitors, and consequently, the estimation is better determined and more precise. On the contrary, deeper sags require more monitors in order to achieve a similar precision.
Table II shows the comparison between the average number of voltage sags experienced in all the buses of the system and the average number obtained by means of the VSEE. A very good agreement can be observed, which indicates that VSEE can be very useful to estimate the average system performance regarding voltage sags. The preceding study was performed considering 4 monitors. It is interesting to analyze the influence of the number of monitors on the accuracy of the results. To this aim, two cases are compared with the previous one. • Only two monitors are placed at buses 9 and 24. This location is obtained for a voltage threshold of 0.8 p.u. (that is, guaranteeing that any fault leading to a sag with magnitude below 0.8 p.u. triggers at least one monitor). • Eight monitors are placed at buses 3, 4, 6, 8, 12, 14, 20 and 22. this location is obtained for a voltage threshold of 0.6 p.u. Figs. 5–7 show the performance of the proposed VSSE method in the above cases. The average error obtained in each case is indicated in Table III. Obviously, the more monitors used, the more accurate the results are. IX. STUDIES IN THE IEEE 118-BUS TEST SYSTEM The IEEE 118-bus test system consists of 36 generating stations, 118 buses interconnected by 177 lines and 9 transformers. The data system is provided in [20]. To perform a voltage sags analysis, a random number between 0 and 5 of three-phase faults has been assumed at each line. The positions of faults occurrence have also been obtained by applying a random algorithm. The monitors location and number have been calculated by applying the methodology proposed in [10], with a threshold of 0.7 p.u. According to this, any fault causing a voltage sag with magnitude below 0.7 p.u. is recorded by, at least, one monitor.
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Fig. 6. Voltage sags with magnitude below 0.8 p.u. obtained with different number of monitors.
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 4, OCTOBER 2007
Fig. 9. Actual and estimated number of voltage sags at all the buses of the system for a voltage threshold of 0.8 p.u.
Fig. 7. Voltage sags with magnitude below 0.7 p.u. obtained with different number of monitors. TABLE III AVERAGE ERROR. COMPARISON FOR DIFFERENT NUMBER OF MONITORS
Fig. 10. Actual and estimated number of voltage sags at all the buses of the system for a voltage threshold of 0.7 p.u.
TABLE IV AVERAGE ESTIMATION ERROR FOR THE 118-BUS SYSTEM
p.u., 0.8 p.u., and 0.7 p.u., respectively. A good agreement between the actual and the estimated value is achieved as it can be also deduced from the average estimation errors shown in Table IV. X. DIFFERENT TYPES OF FAULTS
Fig. 8. Actual and estimated number of voltage sags at all the buses of the system for a voltage threshold of 0.9 p.u.
The application of this method leads to placing 24 monitors at buses 8, 13, 21, 24, 35, 42, 43, 47, 49, 53, 56, 62, 69, 70, 72, 75, 82, 86, 92, 98, 99, 108, 114, and 117. The measurement vector of the VSSE algorithm is formed, similarly to the previous case study, by the number of sags experienced at the metered buses. In this case study, the VSSE algorithm is applied to find a feasible solution subjected to the constraints (12) but without considering an objective function for the optimization problem. Figs. 8–10 show the real and the estimated number of voltage sags at all the buses of the system for voltage thresholds of 0.9
In this case study, the performance of the method is checked when different types of faults are considered. A random number of faults has been assumed in the IEEE 24-bus system. 70% of them have been assumed as phase-to-ground faults, 15% are phase-to-phase short circuits, 10% are phase-to-phase-to-ground and 5% are three-phase faults. The results obtained are shown in Fig. 11. XI. CONSIDERATIONS ABOUT THE VALIDATION OF THE METHOD The method presented in this paper has shown a good performance in the simulated cases. One of the advantages of the proposed VSSE method is that it is based on simple specifications for the performed measurements. In this method, for instance, recorded voltage sags are classified in discrete bins of residual magnitude, that is, the exact voltage magnitude of the sag is not
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the residual voltage caused by faults occurring along lines. An integer linear optimization method is used to solve the problem. The proposed VSSE method has been implemented and applied to a 24-bus and a 118-bus test systems. By these examples, it is shown that the proposed methodology can be a helpful tool to estimate the voltage sag performance of the system at nonmonitored buses. APPENDIX
Fig. 11. Actual and estimated number of voltage sags with different types of faults for a voltage threshold of 0.9 p.u.
Table V shows the assumed faults distribution for the cases presented in Section VIII. REFERENCES
TABLE V ASSUMED NUMBER AND POSITIONS OF FAULTS ALONG LINES AT THE IEEE 24-BUS SYSTEM
required. This fact contributes to reduce the influence of measuring errors. However, it is important to notice that the results shown on this paper have been validated by means of a pseudo-measurements strategy, considering a random number and a random position of faults. A more precise validation should include the verification of the proposed formulation using field data. The impact of metering errors and the influence of a nonoptimal placement of monitors should also be analyzed. XII. CONCLUSION This paper has described a method for estimating the voltage sags performance at nonmonitored buses using the data collected at a limited number of metering points. The proposed formulation is based on the use of an analytical method to calculate
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Elisa Espinosa Juárez (M’07) graduated in Electrical Engineering from the Universidad Michoacana de San Nicolás de Hidalgo (UMSNH), México, in 1986. She received the M.Sc. degree from the Instituto Politécnico Nacional, México City, México, in 2001, and the Ph.D. degree from the Universidad Politécnica de Madrid (UPM), Madrid, Spain, in 2006. Since 1989, she has been an Associate Professor at UMSNH. Her research interests include monitoring and stochastic assessment of voltage sags.
Araceli Hernández (M’06) graduated in Electrical Engineering from the Universidad Politécnica de Madrid (UPM), Spain, in 1996, where she received the Ph.D. degree in 2000. Currently, she is an Associate Professor with the Department of Electrical Engineering at UPM. Her fields of interest include measurement and analysis of power quality.
