A Novel Waveform Tracking Monitor for Power ... - Semantic Scholar
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
A Novel Waveform Tracking Monitor for Power Systems Bei Gou, Member, IEEE, Cheng Luo, and Ferdinanda Ponci, Member, IEEE
Abstract—This paper presents a novel real-time monitor for power systems. This monitor is designed to track the waveforms of signals of interest during power system operations, including the transients caused by load changes or faults. State estimation techniques are used to detect sampling errors in order to best track the trajectory of the original signal waveforms. Our previously proposed algorithms are then used to perform an observability analysis for a given set of measurements. A new measurement equation is introduced to detect bad data processing. Numerical results illustrate the effectiveness of the proposed tracking monitor. Index Terms—Fault, observability analysis, virtual test bed, waveform tracking monitor.
I. INTRODUCTION HE electronmechanical and electromagnetic dynamics of power systems are complex, and the propagation of major system disturbances is difficult to analyze. However, it is essential to understand these characteristics in order to prevent catastrophic failures and blackouts. Furthermore, it is necessary to monitor the parameters of operation to interpret the system behaviors. The most important parameter for interpreting the system behaviors is the system frequency [2]. The real-time system-wide frequency can be monitored using the estimated voltage waveforms. In certain small systems, such as one onboard a ship, the ratio of load demand to the total amount of power generation is relatively large [3]. So the system frequency may vary considerably, depending on changes in load demand. Monitoring the system frequency in real time allows the power system to be controlled and operated both more quickly and more precisely. Power quality is a growing concern in power systems. With the increased number and size of installed nonlinear loads [4], the issue of distorted voltages and currents is much more severe than ever before. It may be possible to maintain satisfactory power quality by monitoring harmonic components in real time. At present, the existing real-time monitors are formulated in the steady-state domain where the variables are the voltage magnitudes and angles at the fundamental frequency; as a consequence, none of these monitors can identify short-term voltage
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Manuscript received August 8, 2005; revised April 6, 2006. Paper no. TPWRS-00500-2005. B. Gou and C. Luo are with the Energy Systems Research Center, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76010 USA (e-mail: [email protected]; [email protected]). F. Ponci is with the Departmental of Electrical Engineering, University of South Carolina, Columbia, SC 290208 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2006.882459
dynamics, which may last only a short time. The waveform tracking monitor proposed in this paper provides a way to detect the transient processes during large disturbances like transmission line faults. Traditionally, all the meters installed in power systems have provided the rms values of the sampled signals. The rms values are normally calculated by discrete Fourier transform (DFT) or fast Fourier transform (FFT), which cause a delay due to the calculations. Another drawback to DFT and FFT is that they have difficulty calculating the damping dc component in the transient signals. Therefore, it is difficult for these two algorithms to obtain the complete and exact transient behaviors of short-term disturbances, as well as those that pertain to the beginning of general disturbances. Progress has been made in recent years using the wavelet transform on the sampled waveforms to detect, classify, and characterize disturbances [5]. In [5], the main focus is on how to use these samples, but this literature does not fully address how to obtain the samples, especially in the midst of a disturbance. In this paper, we propose a novel real-time monitor that is able to detect the exact dynamic and transient behaviors of power systems in a fast manner based directly on the voltage and current waveforms. The concept for this proposed monitor originated with the authors’ previous work on the Tracking State Estimator presented in [6]. The proposed monitoring system provides not only a platform for postcontingency control but also a tool for the power system operators to acquire system-wide information such as frequency, harmonic components, voltage dynamics, angle dynamics, and fault types and locations. The results of numerical tests prove this proposed monitor effective for both steady-state and transient-state conditions.
II. MEASUREMENT EQUATIONS Considering that power system measurements tend to be redundant and may exhibit gross errors, we utilize the state estimation technique to best estimate the signal waveforms. To speed the calculation and to guarantee its accuracy, meters can be installed appropriately to avoid nonlinear elements in the problem formulation. Specifically, nonlinear elements can be replaced in the problem formulation by measuring their terminal injecting currents and/or their terminal voltages. This is one of the main advantages of the state estimation technique. The main elements we need to include in the formulation are the transmission lines. We first introduce the modeling of transmission lines combined with current and voltage measurements. Our purpose is to show the relationship between the voltage and current measurements of the system under analysis.
0885-8950/$20.00 © 2006 IEEE
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
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and the capacitance as ; then, (3) becomes
Fig. 1. One transmission line model.
Therefore, the injection current
(6) at bus can be written as
A. Equations for Current Measurement Fig. 1 depicts a generic single transmission line. As this figure shows, the ammeter is located close to bus . The shunt admittance of the transmission line is omitted in this case without loss of generality. In fact, it can be included in the formulation; in particular, EMTP models of transmission lines (such as the distributed-parameter model) can be used to formulate the problem. For the inductance , we have
(1)
(7) where is a set containing all the associated bus numbers for bus . Because the bus voltages are the variables, we can rewrite the above equation as follows:
The current is evaluated through the integral defined in (8) (2) III. STATE ESTIMATION FORMULATION The integration is performed by the trapezoidal rule approach as follows:
(3) So the resulting relationship between bus voltages and branch current is
For an -bus power system, it is assumed that the system has a total of measurements, of which are current measurements , and are voltage measurements. The voltages at buses are defined to be the state at time . If the measurement variables equations as stated in Section II are applied to the -bus power system, then the generalized state estimation formulation can be written as follows:
(9)
(4)
where is an
is an
matrix, vector, is an ,
matrix, is an
B. Equations for Voltage Measurement If a voltage measurement is taken at bus of the system under consideration, and the measured voltage is , then voltage at bus is
(5)
C. Equations for Current Injection Measurement Current injection measurements in a bus equal the sum of all the current measurements of the connected branches. The impedance of the branch between bus and is defined as
matrix, is an
, matrix, and . Eq. (9) contains two sets of terms: those corresponding to the current measurements and those corresponding to the voltage measurements. For each current measurement, (4) has a direct correspondence with its matrix form. The first rows in matrices and the first columns in are corresponding to the current measurements; matrix is corresponding to the current measurements. For each voltage measurement, the matrix form rows in and the last of (5) is directly used. The last columns in are corresponding to voltage measurements; macorresponds to voltage measurements. Because all the trix matrices in (9) are sparse, the solution to (9) is computationally very efficient.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
A. Current Meter Placement Whenever a line is a fault, it is impossible to identify the fault location at the present sampling time. The faulty line thus has to be removed from the formulation because it cannot be exactly modeled without knowing the fault location. Therefore, a proper current meter placement has to be developed to detect the faulty line. A possible way to identify the faulty line is to install the branch current meters in pairs. Current Meter Placement Requirement: Branch current meters are required to be installed in pairs, i.e., at both ends of lines. If the positive direction for each current measurement is defined from the close end to the remote end, then the sum of these two measurements at both ends of the branch must be equal to zero. If, at a sampling time step, the sum of these two measurements is not zero, then there are two possibilities: 1) a fault has occurred on the line; or 2) one of these two measurements is bad data. In either case, these two specific measurements cannot be used for the purpose of state estimation and have to be discarded from the set of measurements. Then matrix must be modified to reflect the measurement configuration change. The algorithm to update the factors of matrix is provided in [14]. B. Determination of Matrices Coefficients First let us consider current measurements. If the th current measurement connects bus and bus (assume that the meter is located close to bus ), the resistance of line is , and the inductance of line is . From (4), we have
The next step is to estimate the initial values for the formulation. C. Initial Values Estimation When the system is in steady state, it is assumed that , where is the integration step, i.e., , and is the total number of samples in period . Because matrix is of full rank, we can use the weighted least square (WLS) method on (9) and obtain the following:
Eq. (4) implies that the diagonal elements of , which correspond to buses without voltage measurements, are equal to 1. Details are given in the following theorem. Theorem 3.1: For a measured power system, the columns of corresponding to buses without voltage measurements are vectors whose diagonal elements equal to 1, and whose other elements are zero. The proof of the above theorem is provided in the Appendix. , Regarding the other nonzero elements in their columns always correspond to the buses with voltage can be substimeasurements. At this point, . In other words, matrix is defined tuted for , then we get
(12) therefore (10) therefore
(13) steps and sum up all the If we extend (13) to the next equations, then a new equation is obtained. The LHS of the new equation is
For voltage measurements, we assume that voltage measurement is located at bus . Then (5) becomes (11) therefore
At this point, the coefficient matrices , , and have been determined. From what was stated above, we know that if current injection measurements are excluded, the elements of can take only values 1, 1, or zero. Just as the phase angle measurement can increase the numerical stability for the steadystate estimator, the existence of voltage measurements implies that the condition number of matrix is small, and the solution of (9) is numerically stable.
LHS (14) The following theorem estimates the initial value of the voltage variable vector. Theorem 3.2: The initial value of the voltage variable vector can be estimated as follows with a sufficient small error:
(15) The proof of the above theorem is provided in the Appendix.
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D. Observability Analysis Sometimes the communication problems can cause the monitor to miss measurements. For this reason, a preprocedure called observability analysis should be executed before solving (9). The algorithm developed in [7] and [8] can determine the observable islands for a given set of measurements. The algorithm must be run whenever a measurement is reported to be missing. The faster algorithm developed in [9] can also be applied to matrix in (9). If more than two observable islands are found in the above algorithms, there are two possible ways to merge those observable islands. One way is to choose additional measurements and add them to the existing set of measurements. The algorithm developed in [10] can be used to select the additional measurements. Another way is to add the missed measurement to the existing set of measurements by using the samples from the previous period.
Fig. 2. Schematic diagram of power system network.
where matrix
is defined as the residual sensitivity matrix [12]
E. Bad Data Processing When a state estimation model fails to yield estimates with an appropriate degree of accuracy, we must conclude either that the measured quantities contain spurious data, or that the model is unfit to explain that measured quantities or that both are true [11]. Many algorithms have been developed to detect these two kinds of errors. One widely used technique is the largest normalized residual test [12], which is utilized in this paper to detect the gross errors in measurements. However, because the measurement equations of the proposed state estimator are different from those of the traditional static state estimator, the following adjustment have to be made. as follows: First, we define a new measurement vector (16) Then the measurement equation (9) becomes (17) where is the error vector. Eq. (17) implies that the solution of the voltage vector is (18) Then the estimated values for
are (19)
The residual at sampling time can be calculated as follows:
(20)
(21) Therefore, the largest normalized residual test [12] can be applied to the proposed measurement equations. IV. SIMULATION RESULTS The single phase diagram of a 13-bus power system is shown in Fig. 2. This system is modeled in the virtual test bed (VTB) [15]. Two generators are installed at buses 1 and 2. The system has 12 transmission lines (C0, C1, , C11). There are totally 12 current measurements, which are represented by circles, and five voltage measurements, which are shown as triangles. A. Monitoring of Continuous Faults In order to illustrate the monitoring of continuous faults, we present a case in which the system is operating in steady-state conditions and a single phase (Phase A) to ground fault oc. After 0.01 s have passed, curs on bus 3 at time another fault (the same type as fault ) occurs at time . The duration of both faults is 0.02 s. In other words, at time , the fault is cleared, and at time , the fault is cleared. The real-time state estimation algorithm proposed in this paper is used to track the voltage waveforms of all buses. Fig. 3 in comparison to the shows the estimated voltage of bus real value (Phase A) as obtained in simulation. Fig. 4 shows the estimated and simulated/real voltage waveforms while the faults are in progress. Since all measurements are normally contaminated by noise, we investigate the robustness of the proposed monitoring algorithm for signals contaminated by noise by considering a number of different cases. In particular, two typical cases are examined in detail here. Fig. 5 depicts the estimated voltage waveform at bus 1 when the measurements are contaminated by Gaussian white noise. The signal-to-noise ratio (SNR) is 10 dB, and the power of the noise is 10 W. As Fig. 5 shows, the proposed monitor exhibits satisfactory behaviors, even under noisy conditions.
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Fig. 3. Comparison of state estimation and the real voltage value of bus 2.
Fig. 4. Comparison of the state estimation and the real voltage value of bus 2 during faults.
Fig. 6 gives the estimated voltage waveform at bus 1 when the SNR of the noise is 1 dB and its power reaches 40 W. This figure illustrate that even in the case of a small SNR and high power, the proposed monitor is still able to capture the transient behaviors during the fault. B. Bad Data Analysis In Section IV, we define a new measurement (17) that applies the largest normalized residual test [12] to execute bad data processing. In order to increase the measurement redundancy, voltage measurements are assigned to all the buses; meanwhile, the current measurements are left unchanged as in the above cases. In this test, we introduce a gross error at the voltage mea. The largest norsurement of bus 6 at the time malized residual test shows that the voltage measurement at bus 6 detects the greatest value at 7.472, which is greater than the threshold 3.0. Thus, the new measurement equation successfully detects the gross error added to the voltage measurement at bus 6.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
Fig. 5. Comparison of state estimation and the real voltage value of bus 1.
Fig. 6. Comparison of state estimation and the real voltage value of bus 1 during faults.
V. ERROR ANALYSIS The proposed waveform tracking monitor is subject to inaccuracy when the initial values are not appropriately selected or when there are errors in the system parameters. In this section, we investigate the extent of such malfunctions in the proposed monitor, by examining its performance under abnormal conditions. A. Oscillation Error Inappropriately selecting initial values can lead to oscillation errors. For those buses without voltage measurements, their initial voltage values must be entered individually into the monitor. Any errors in these initial values will generate oscillation errors. Fig. 7 shows the comparison of state estimation results and the real values, which are obtained in the VTB simulation when the oscillation error exists. Fig. 8 shows the difference between the state estimation and the real value. It can be seen that the difference is a triangle waveform, and its period is , where is the integration step during the simulation. It
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
Fig. 7. Comparison of state estimation result and the real value due to initial value error.
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Fig. 9. Comparison of state estimation result and the real value due to parameter errors.
Fig. 8. Error between the state estimation result and the real value. Fig. 10. Difference between the estimated waveform and the original waveform due to parameter error.
should be noted that the amplitude of the triangle wave is a constant value (310 V). This error stems directly from the inappropriate selection of the initial values. We assume the error of the , initial value at time is . When . It is a triangle from (4), we find that the error is waveform, and the period is . The amplitude of the triangle waveform is , as shown in Fig. 8. The results of the oscillation error analysis show that exact initial values are necessary to obtain precise estimation of signal waveforms. Section III-C presented a method to exactly estimate initial values for the tracking estimator. B. Parameter Error Errors in parameters and in (9) will also generate large estimation error. In this test, large errors are introduced into meter and the original parameters H/meter, resulting in meter H/meter. and Fig. 9 shows both the state estimation result and the real value. The discrepancy between them is due to parameter er-
rors. Fig. 10 displays the difference itself, between the estimated waveform and the original waveform. This comparison illustrates that parameter errors have significant impact on the monitoring accuracy during the fault period. Fig. 9 also shows that the proposed real-time monitor is robust enough to exactly track the trajectory of the original signal waveforms after the fault is cleared, even when the parameters bear large errors. VI. CONCLUSIONS In this paper, we proposed a real-time monitor that can track the waveforms of bus voltage signals in power systems. Transient processes caused by significant disturbances, like load changes or faults, can be detected by the proposed monitor. Tracking voltage waveforms, especially under transient conditions, is critical for coordinating protective relays, analyzing fault and power quality, and performing system-wide monitoring. The waveform tracking monitor provides a platform that allows for the development of post-contingency controls, such as fast protection and control strategies. Numerical tests
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show that the proposed waveform tracking monitor is exact and robust and is ready for practical applications.
where
APPENDIX PROOF OF THE THEOREMS
(24)
A) Proof of Theorem 3.1: Proof: Let us first re-order the rows and columns of such that the buses without voltage measurements and the current branch and injection measurements are ordered first.
and
(25) The inverse of the lower triangular matrix
is
where
(26) matrix; From (22), we obtain
total number of measurements; number of buses; matrix; number of current branch and injection measurements; number of buses without voltage measurements;
(27)
matrix. Similarly, we can partition matrix
The partitioning of can get
where
in the same way
is obtained based on (10). Therefore, we
Now we need to conduct matrix product
(28)
(22)
It is easy to check that the left upper submatrix with the dimension in is a negative unit matrix as follows:
and (29)
Since matrix is of full rank, then must have full column is of full rank and can be factorized rank . Therefore into , where is a lower triangular matrix is a diagonal matrix. Then matrix can be factorand ized into the following:
are matrices decided by where , and This ends our proof. B) Proof of Theorem 3.2: Proof: Assume that is given by
and
.
; then
(30) where is the total number of samples in a period. 1) When is an even number
or
(23)
(31)
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
The similar result can be achieved in the following steps. , there is In other words, for
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So
(40) (32) In conclusion, for any
then
, the following is always true:
(41)
(33) 2) When
Similarly
is an odd number, from (30), there are
(42) If we approximately suppose
, then
(43)
If we define equations becomes
It should be noted that the analysis shows that, compared with the real value, the result obtained from the above assumption is satisfactory.
, the sum of the above
REFERENCES (34) Considering number, we can conclude that
and
is an odd
(35) and when
, there is
(36) It can be also known that
(37) Substituting (36) and (37) into (35)
(38) Therefore
(39)
[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, U.K.: Addison-Wesley, 1999. [2] B. Liu, L. Chen, V. Centeno, X. Dong, and Y. Liu, “Internet based Frequency Monitoring Network (FNET),” in Proc. IEEE Power Eng. Soc. General Meeting, 2001, pp. 1166–1171. [3] K. L. Butler, N. D. R. Sarma, C. Whitcomb, H. Do Carmo, and H. Zhang, “Shipboard systems deploy automated protection,” IEEE Comput. Appl. Power, vol. 11, no. 2, pp. 31–36, Apr. 1998. [4] I. Jonasson and L. Soder, “Power quality on ships-a questionnaire evaluation concerning island power system,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Jul. 2001, pp. 216–221. [5] P. Daponte, M. Di Penta, and G. Mercurio, “TransientMeter: a distributed measurement system for power quality monitoring,” IEEE Trans. Power Del., vol. 19, no. 2, pp. 456–463, Apr. 2004. [6] B. Gou and A. Abur, “A tracking state estimator for nonsinusoidal periodic steady-state operation,” IEEE Trans. Power Del., vol. 3, no. 4, pp. 1509–1514, Oct. 1998. [7] ——, “A direct numerical method for observability analysis,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 625–631, May 2000. [8] ——, “A simple method to determine observable islands for state estimation,” in Proc. IEEE Int. Symp. Circuits Systems, Orlando, FL, 1999. [9] B. Gou, “Observability analysis by measurement Jacobian matrix for state estimation,” in Proc. IEEE Int. Symp. Circuits Systems, Kobe, Japan, 2005, accepted for publication. [10] B. Gou and A. Abur, “A non-iterative numerical method of measurement placement for observability analysis,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 819–824, Nov. 2001. [11] A. Monticelli, State Estimation in Electric Power Systems: A Generalized Approach. Norwell, MA: Kluwer, 1999. [12] A. Abur and A. G. Exposito, Power System State Estimation: Theory and Implementation. New York: Marcel Dekker, 2004. [13] A. Monticelli and F. F. Wu, “Network observability: identification of observable islands and measurement placement,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 5, pp. 1035–1041, May 1985. [14] B. Gou, “Monitoring and optimization of power transmission and distribution systems,” Ph.D. dissertation, Texas A&M Univ., College Station, May 2000. [15] T. Lovett, A. Monti, E. Santi, and R. A. Dougal, “A multilanguage environment for interactive simulation and development controls for power electronics,” in Proc. IEEE 32nd Annu. Power Electronics Specialists Conf., Jun. 17–21, 2001, vol. 3, pp. 1725–1729.
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Bei Gou (M’00) received the B.S. degree in electrical engineering from North China University of Electric Power, Hebei, China, in 1990, the M.S. degree from Shanghai JiaoTong University, Shanghai, China, in 1993, and the Ph.D. degree from Texas A&M University, College Station, in 2000. From 1993 to 1996, he taught at the Department of Electric Power Engineering in Shanghai JiaoTong University. He worked as a Research Assistant at Texas A&M University beginning in 1997. He worked at ABB Energy Information Systems, Santa Clara, CA, for two years and at ISO New England for one year as a Senior Analyst. He is currently an Assistant Professor at the Energy Systems Research Center, University of Texas at Arlington. His main interests are power system state estimation, power market operations, power quality, power system reliability, and distributed generators.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
Cheng Luo received the B.S. and M.S. degrees from Huazhong University of Science and Technology, Wuhan, China, in 2000 and 2002, respectively, both in electrical engineering. He is currently pursuing the Ph.D. degree with the Energy Systems Research Center, Department of Electrical Engineering, University of Texas at Arlington. His area of interests are state estimation and subsynchronous resonance.
Ferdinanda Ponci (M’99) was born in Milano, Italy. She received her M.S. and Ph.D. degrees in electrical engineering from Politecnico di Milano in 1998 and 2002, respectively. Since 2003, she has been an Assistant Professor with the Department of Electrical Engineering, University of South Carolina, Columbia. Her current research interests are in the fields of virtual instruments applied to diagnosis of motor drives, integrated environment for the monitoring of complex systems, and multiresolution modeling.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
A Novel Waveform Tracking Monitor for Power Systems Bei Gou, Member, IEEE, Cheng Luo, and Ferdinanda Ponci, Member, IEEE
Abstract—This paper presents a novel real-time monitor for power systems. This monitor is designed to track the waveforms of signals of interest during power system operations, including the transients caused by load changes or faults. State estimation techniques are used to detect sampling errors in order to best track the trajectory of the original signal waveforms. Our previously proposed algorithms are then used to perform an observability analysis for a given set of measurements. A new measurement equation is introduced to detect bad data processing. Numerical results illustrate the effectiveness of the proposed tracking monitor. Index Terms—Fault, observability analysis, virtual test bed, waveform tracking monitor.
I. INTRODUCTION HE electronmechanical and electromagnetic dynamics of power systems are complex, and the propagation of major system disturbances is difficult to analyze. However, it is essential to understand these characteristics in order to prevent catastrophic failures and blackouts. Furthermore, it is necessary to monitor the parameters of operation to interpret the system behaviors. The most important parameter for interpreting the system behaviors is the system frequency [2]. The real-time system-wide frequency can be monitored using the estimated voltage waveforms. In certain small systems, such as one onboard a ship, the ratio of load demand to the total amount of power generation is relatively large [3]. So the system frequency may vary considerably, depending on changes in load demand. Monitoring the system frequency in real time allows the power system to be controlled and operated both more quickly and more precisely. Power quality is a growing concern in power systems. With the increased number and size of installed nonlinear loads [4], the issue of distorted voltages and currents is much more severe than ever before. It may be possible to maintain satisfactory power quality by monitoring harmonic components in real time. At present, the existing real-time monitors are formulated in the steady-state domain where the variables are the voltage magnitudes and angles at the fundamental frequency; as a consequence, none of these monitors can identify short-term voltage
T
Manuscript received August 8, 2005; revised April 6, 2006. Paper no. TPWRS-00500-2005. B. Gou and C. Luo are with the Energy Systems Research Center, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76010 USA (e-mail: [email protected]; [email protected]). F. Ponci is with the Departmental of Electrical Engineering, University of South Carolina, Columbia, SC 290208 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2006.882459
dynamics, which may last only a short time. The waveform tracking monitor proposed in this paper provides a way to detect the transient processes during large disturbances like transmission line faults. Traditionally, all the meters installed in power systems have provided the rms values of the sampled signals. The rms values are normally calculated by discrete Fourier transform (DFT) or fast Fourier transform (FFT), which cause a delay due to the calculations. Another drawback to DFT and FFT is that they have difficulty calculating the damping dc component in the transient signals. Therefore, it is difficult for these two algorithms to obtain the complete and exact transient behaviors of short-term disturbances, as well as those that pertain to the beginning of general disturbances. Progress has been made in recent years using the wavelet transform on the sampled waveforms to detect, classify, and characterize disturbances [5]. In [5], the main focus is on how to use these samples, but this literature does not fully address how to obtain the samples, especially in the midst of a disturbance. In this paper, we propose a novel real-time monitor that is able to detect the exact dynamic and transient behaviors of power systems in a fast manner based directly on the voltage and current waveforms. The concept for this proposed monitor originated with the authors’ previous work on the Tracking State Estimator presented in [6]. The proposed monitoring system provides not only a platform for postcontingency control but also a tool for the power system operators to acquire system-wide information such as frequency, harmonic components, voltage dynamics, angle dynamics, and fault types and locations. The results of numerical tests prove this proposed monitor effective for both steady-state and transient-state conditions.
II. MEASUREMENT EQUATIONS Considering that power system measurements tend to be redundant and may exhibit gross errors, we utilize the state estimation technique to best estimate the signal waveforms. To speed the calculation and to guarantee its accuracy, meters can be installed appropriately to avoid nonlinear elements in the problem formulation. Specifically, nonlinear elements can be replaced in the problem formulation by measuring their terminal injecting currents and/or their terminal voltages. This is one of the main advantages of the state estimation technique. The main elements we need to include in the formulation are the transmission lines. We first introduce the modeling of transmission lines combined with current and voltage measurements. Our purpose is to show the relationship between the voltage and current measurements of the system under analysis.
0885-8950/$20.00 © 2006 IEEE
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
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and the capacitance as ; then, (3) becomes
Fig. 1. One transmission line model.
Therefore, the injection current
(6) at bus can be written as
A. Equations for Current Measurement Fig. 1 depicts a generic single transmission line. As this figure shows, the ammeter is located close to bus . The shunt admittance of the transmission line is omitted in this case without loss of generality. In fact, it can be included in the formulation; in particular, EMTP models of transmission lines (such as the distributed-parameter model) can be used to formulate the problem. For the inductance , we have
(1)
(7) where is a set containing all the associated bus numbers for bus . Because the bus voltages are the variables, we can rewrite the above equation as follows:
The current is evaluated through the integral defined in (8) (2) III. STATE ESTIMATION FORMULATION The integration is performed by the trapezoidal rule approach as follows:
(3) So the resulting relationship between bus voltages and branch current is
For an -bus power system, it is assumed that the system has a total of measurements, of which are current measurements , and are voltage measurements. The voltages at buses are defined to be the state at time . If the measurement variables equations as stated in Section II are applied to the -bus power system, then the generalized state estimation formulation can be written as follows:
(9)
(4)
where is an
is an
matrix, vector, is an ,
matrix, is an
B. Equations for Voltage Measurement If a voltage measurement is taken at bus of the system under consideration, and the measured voltage is , then voltage at bus is
(5)
C. Equations for Current Injection Measurement Current injection measurements in a bus equal the sum of all the current measurements of the connected branches. The impedance of the branch between bus and is defined as
matrix, is an
, matrix, and . Eq. (9) contains two sets of terms: those corresponding to the current measurements and those corresponding to the voltage measurements. For each current measurement, (4) has a direct correspondence with its matrix form. The first rows in matrices and the first columns in are corresponding to the current measurements; matrix is corresponding to the current measurements. For each voltage measurement, the matrix form rows in and the last of (5) is directly used. The last columns in are corresponding to voltage measurements; macorresponds to voltage measurements. Because all the trix matrices in (9) are sparse, the solution to (9) is computationally very efficient.
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A. Current Meter Placement Whenever a line is a fault, it is impossible to identify the fault location at the present sampling time. The faulty line thus has to be removed from the formulation because it cannot be exactly modeled without knowing the fault location. Therefore, a proper current meter placement has to be developed to detect the faulty line. A possible way to identify the faulty line is to install the branch current meters in pairs. Current Meter Placement Requirement: Branch current meters are required to be installed in pairs, i.e., at both ends of lines. If the positive direction for each current measurement is defined from the close end to the remote end, then the sum of these two measurements at both ends of the branch must be equal to zero. If, at a sampling time step, the sum of these two measurements is not zero, then there are two possibilities: 1) a fault has occurred on the line; or 2) one of these two measurements is bad data. In either case, these two specific measurements cannot be used for the purpose of state estimation and have to be discarded from the set of measurements. Then matrix must be modified to reflect the measurement configuration change. The algorithm to update the factors of matrix is provided in [14]. B. Determination of Matrices Coefficients First let us consider current measurements. If the th current measurement connects bus and bus (assume that the meter is located close to bus ), the resistance of line is , and the inductance of line is . From (4), we have
The next step is to estimate the initial values for the formulation. C. Initial Values Estimation When the system is in steady state, it is assumed that , where is the integration step, i.e., , and is the total number of samples in period . Because matrix is of full rank, we can use the weighted least square (WLS) method on (9) and obtain the following:
Eq. (4) implies that the diagonal elements of , which correspond to buses without voltage measurements, are equal to 1. Details are given in the following theorem. Theorem 3.1: For a measured power system, the columns of corresponding to buses without voltage measurements are vectors whose diagonal elements equal to 1, and whose other elements are zero. The proof of the above theorem is provided in the Appendix. , Regarding the other nonzero elements in their columns always correspond to the buses with voltage can be substimeasurements. At this point, . In other words, matrix is defined tuted for , then we get
(12) therefore (10) therefore
(13) steps and sum up all the If we extend (13) to the next equations, then a new equation is obtained. The LHS of the new equation is
For voltage measurements, we assume that voltage measurement is located at bus . Then (5) becomes (11) therefore
At this point, the coefficient matrices , , and have been determined. From what was stated above, we know that if current injection measurements are excluded, the elements of can take only values 1, 1, or zero. Just as the phase angle measurement can increase the numerical stability for the steadystate estimator, the existence of voltage measurements implies that the condition number of matrix is small, and the solution of (9) is numerically stable.
LHS (14) The following theorem estimates the initial value of the voltage variable vector. Theorem 3.2: The initial value of the voltage variable vector can be estimated as follows with a sufficient small error:
(15) The proof of the above theorem is provided in the Appendix.
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D. Observability Analysis Sometimes the communication problems can cause the monitor to miss measurements. For this reason, a preprocedure called observability analysis should be executed before solving (9). The algorithm developed in [7] and [8] can determine the observable islands for a given set of measurements. The algorithm must be run whenever a measurement is reported to be missing. The faster algorithm developed in [9] can also be applied to matrix in (9). If more than two observable islands are found in the above algorithms, there are two possible ways to merge those observable islands. One way is to choose additional measurements and add them to the existing set of measurements. The algorithm developed in [10] can be used to select the additional measurements. Another way is to add the missed measurement to the existing set of measurements by using the samples from the previous period.
Fig. 2. Schematic diagram of power system network.
where matrix
is defined as the residual sensitivity matrix [12]
E. Bad Data Processing When a state estimation model fails to yield estimates with an appropriate degree of accuracy, we must conclude either that the measured quantities contain spurious data, or that the model is unfit to explain that measured quantities or that both are true [11]. Many algorithms have been developed to detect these two kinds of errors. One widely used technique is the largest normalized residual test [12], which is utilized in this paper to detect the gross errors in measurements. However, because the measurement equations of the proposed state estimator are different from those of the traditional static state estimator, the following adjustment have to be made. as follows: First, we define a new measurement vector (16) Then the measurement equation (9) becomes (17) where is the error vector. Eq. (17) implies that the solution of the voltage vector is (18) Then the estimated values for
are (19)
The residual at sampling time can be calculated as follows:
(20)
(21) Therefore, the largest normalized residual test [12] can be applied to the proposed measurement equations. IV. SIMULATION RESULTS The single phase diagram of a 13-bus power system is shown in Fig. 2. This system is modeled in the virtual test bed (VTB) [15]. Two generators are installed at buses 1 and 2. The system has 12 transmission lines (C0, C1, , C11). There are totally 12 current measurements, which are represented by circles, and five voltage measurements, which are shown as triangles. A. Monitoring of Continuous Faults In order to illustrate the monitoring of continuous faults, we present a case in which the system is operating in steady-state conditions and a single phase (Phase A) to ground fault oc. After 0.01 s have passed, curs on bus 3 at time another fault (the same type as fault ) occurs at time . The duration of both faults is 0.02 s. In other words, at time , the fault is cleared, and at time , the fault is cleared. The real-time state estimation algorithm proposed in this paper is used to track the voltage waveforms of all buses. Fig. 3 in comparison to the shows the estimated voltage of bus real value (Phase A) as obtained in simulation. Fig. 4 shows the estimated and simulated/real voltage waveforms while the faults are in progress. Since all measurements are normally contaminated by noise, we investigate the robustness of the proposed monitoring algorithm for signals contaminated by noise by considering a number of different cases. In particular, two typical cases are examined in detail here. Fig. 5 depicts the estimated voltage waveform at bus 1 when the measurements are contaminated by Gaussian white noise. The signal-to-noise ratio (SNR) is 10 dB, and the power of the noise is 10 W. As Fig. 5 shows, the proposed monitor exhibits satisfactory behaviors, even under noisy conditions.
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Fig. 3. Comparison of state estimation and the real voltage value of bus 2.
Fig. 4. Comparison of the state estimation and the real voltage value of bus 2 during faults.
Fig. 6 gives the estimated voltage waveform at bus 1 when the SNR of the noise is 1 dB and its power reaches 40 W. This figure illustrate that even in the case of a small SNR and high power, the proposed monitor is still able to capture the transient behaviors during the fault. B. Bad Data Analysis In Section IV, we define a new measurement (17) that applies the largest normalized residual test [12] to execute bad data processing. In order to increase the measurement redundancy, voltage measurements are assigned to all the buses; meanwhile, the current measurements are left unchanged as in the above cases. In this test, we introduce a gross error at the voltage mea. The largest norsurement of bus 6 at the time malized residual test shows that the voltage measurement at bus 6 detects the greatest value at 7.472, which is greater than the threshold 3.0. Thus, the new measurement equation successfully detects the gross error added to the voltage measurement at bus 6.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
Fig. 5. Comparison of state estimation and the real voltage value of bus 1.
Fig. 6. Comparison of state estimation and the real voltage value of bus 1 during faults.
V. ERROR ANALYSIS The proposed waveform tracking monitor is subject to inaccuracy when the initial values are not appropriately selected or when there are errors in the system parameters. In this section, we investigate the extent of such malfunctions in the proposed monitor, by examining its performance under abnormal conditions. A. Oscillation Error Inappropriately selecting initial values can lead to oscillation errors. For those buses without voltage measurements, their initial voltage values must be entered individually into the monitor. Any errors in these initial values will generate oscillation errors. Fig. 7 shows the comparison of state estimation results and the real values, which are obtained in the VTB simulation when the oscillation error exists. Fig. 8 shows the difference between the state estimation and the real value. It can be seen that the difference is a triangle waveform, and its period is , where is the integration step during the simulation. It
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
Fig. 7. Comparison of state estimation result and the real value due to initial value error.
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Fig. 9. Comparison of state estimation result and the real value due to parameter errors.
Fig. 8. Error between the state estimation result and the real value. Fig. 10. Difference between the estimated waveform and the original waveform due to parameter error.
should be noted that the amplitude of the triangle wave is a constant value (310 V). This error stems directly from the inappropriate selection of the initial values. We assume the error of the , initial value at time is . When . It is a triangle from (4), we find that the error is waveform, and the period is . The amplitude of the triangle waveform is , as shown in Fig. 8. The results of the oscillation error analysis show that exact initial values are necessary to obtain precise estimation of signal waveforms. Section III-C presented a method to exactly estimate initial values for the tracking estimator. B. Parameter Error Errors in parameters and in (9) will also generate large estimation error. In this test, large errors are introduced into meter and the original parameters H/meter, resulting in meter H/meter. and Fig. 9 shows both the state estimation result and the real value. The discrepancy between them is due to parameter er-
rors. Fig. 10 displays the difference itself, between the estimated waveform and the original waveform. This comparison illustrates that parameter errors have significant impact on the monitoring accuracy during the fault period. Fig. 9 also shows that the proposed real-time monitor is robust enough to exactly track the trajectory of the original signal waveforms after the fault is cleared, even when the parameters bear large errors. VI. CONCLUSIONS In this paper, we proposed a real-time monitor that can track the waveforms of bus voltage signals in power systems. Transient processes caused by significant disturbances, like load changes or faults, can be detected by the proposed monitor. Tracking voltage waveforms, especially under transient conditions, is critical for coordinating protective relays, analyzing fault and power quality, and performing system-wide monitoring. The waveform tracking monitor provides a platform that allows for the development of post-contingency controls, such as fast protection and control strategies. Numerical tests
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show that the proposed waveform tracking monitor is exact and robust and is ready for practical applications.
where
APPENDIX PROOF OF THE THEOREMS
(24)
A) Proof of Theorem 3.1: Proof: Let us first re-order the rows and columns of such that the buses without voltage measurements and the current branch and injection measurements are ordered first.
and
(25) The inverse of the lower triangular matrix
is
where
(26) matrix; From (22), we obtain
total number of measurements; number of buses; matrix; number of current branch and injection measurements; number of buses without voltage measurements;
(27)
matrix. Similarly, we can partition matrix
The partitioning of can get
where
in the same way
is obtained based on (10). Therefore, we
Now we need to conduct matrix product
(28)
(22)
It is easy to check that the left upper submatrix with the dimension in is a negative unit matrix as follows:
and (29)
Since matrix is of full rank, then must have full column is of full rank and can be factorized rank . Therefore into , where is a lower triangular matrix is a diagonal matrix. Then matrix can be factorand ized into the following:
are matrices decided by where , and This ends our proof. B) Proof of Theorem 3.2: Proof: Assume that is given by
and
.
; then
(30) where is the total number of samples in a period. 1) When is an even number
or
(23)
(31)
GOU et al.: NOVEL WAVEFORM TRACKING MONITOR FOR POWER SYSTEMS
The similar result can be achieved in the following steps. , there is In other words, for
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So
(40) (32) In conclusion, for any
then
, the following is always true:
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(33) 2) When
Similarly
is an odd number, from (30), there are
(42) If we approximately suppose
, then
(43)
If we define equations becomes
It should be noted that the analysis shows that, compared with the real value, the result obtained from the above assumption is satisfactory.
, the sum of the above
REFERENCES (34) Considering number, we can conclude that
and
is an odd
(35) and when
, there is
(36) It can be also known that
(37) Substituting (36) and (37) into (35)
(38) Therefore
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[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, U.K.: Addison-Wesley, 1999. [2] B. Liu, L. Chen, V. Centeno, X. Dong, and Y. Liu, “Internet based Frequency Monitoring Network (FNET),” in Proc. IEEE Power Eng. Soc. General Meeting, 2001, pp. 1166–1171. [3] K. L. Butler, N. D. R. Sarma, C. Whitcomb, H. Do Carmo, and H. Zhang, “Shipboard systems deploy automated protection,” IEEE Comput. Appl. Power, vol. 11, no. 2, pp. 31–36, Apr. 1998. [4] I. Jonasson and L. Soder, “Power quality on ships-a questionnaire evaluation concerning island power system,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Jul. 2001, pp. 216–221. [5] P. Daponte, M. Di Penta, and G. Mercurio, “TransientMeter: a distributed measurement system for power quality monitoring,” IEEE Trans. Power Del., vol. 19, no. 2, pp. 456–463, Apr. 2004. [6] B. Gou and A. Abur, “A tracking state estimator for nonsinusoidal periodic steady-state operation,” IEEE Trans. Power Del., vol. 3, no. 4, pp. 1509–1514, Oct. 1998. [7] ——, “A direct numerical method for observability analysis,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 625–631, May 2000. [8] ——, “A simple method to determine observable islands for state estimation,” in Proc. IEEE Int. Symp. Circuits Systems, Orlando, FL, 1999. [9] B. Gou, “Observability analysis by measurement Jacobian matrix for state estimation,” in Proc. IEEE Int. Symp. Circuits Systems, Kobe, Japan, 2005, accepted for publication. [10] B. Gou and A. Abur, “A non-iterative numerical method of measurement placement for observability analysis,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 819–824, Nov. 2001. [11] A. Monticelli, State Estimation in Electric Power Systems: A Generalized Approach. Norwell, MA: Kluwer, 1999. [12] A. Abur and A. G. Exposito, Power System State Estimation: Theory and Implementation. New York: Marcel Dekker, 2004. [13] A. Monticelli and F. F. Wu, “Network observability: identification of observable islands and measurement placement,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 5, pp. 1035–1041, May 1985. [14] B. Gou, “Monitoring and optimization of power transmission and distribution systems,” Ph.D. dissertation, Texas A&M Univ., College Station, May 2000. [15] T. Lovett, A. Monti, E. Santi, and R. A. Dougal, “A multilanguage environment for interactive simulation and development controls for power electronics,” in Proc. IEEE 32nd Annu. Power Electronics Specialists Conf., Jun. 17–21, 2001, vol. 3, pp. 1725–1729.
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Bei Gou (M’00) received the B.S. degree in electrical engineering from North China University of Electric Power, Hebei, China, in 1990, the M.S. degree from Shanghai JiaoTong University, Shanghai, China, in 1993, and the Ph.D. degree from Texas A&M University, College Station, in 2000. From 1993 to 1996, he taught at the Department of Electric Power Engineering in Shanghai JiaoTong University. He worked as a Research Assistant at Texas A&M University beginning in 1997. He worked at ABB Energy Information Systems, Santa Clara, CA, for two years and at ISO New England for one year as a Senior Analyst. He is currently an Assistant Professor at the Energy Systems Research Center, University of Texas at Arlington. His main interests are power system state estimation, power market operations, power quality, power system reliability, and distributed generators.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 21, NO. 4, NOVEMBER 2006
Cheng Luo received the B.S. and M.S. degrees from Huazhong University of Science and Technology, Wuhan, China, in 2000 and 2002, respectively, both in electrical engineering. He is currently pursuing the Ph.D. degree with the Energy Systems Research Center, Department of Electrical Engineering, University of Texas at Arlington. His area of interests are state estimation and subsynchronous resonance.
Ferdinanda Ponci (M’99) was born in Milano, Italy. She received her M.S. and Ph.D. degrees in electrical engineering from Politecnico di Milano in 1998 and 2002, respectively. Since 2003, she has been an Assistant Professor with the Department of Electrical Engineering, University of South Carolina, Columbia. Her current research interests are in the fields of virtual instruments applied to diagnosis of motor drives, integrated environment for the monitoring of complex systems, and multiresolution modeling.
