Linear State Estimation Model Using Phasor ... - Semantic Scholar
Linear State Estimation Model Using Phasor Measurement Unit (PMU) Technology Sandeep Soni1 , Sudhir Bhil2 , Dhirendra Mehta3 , Sushama Wagh4 1,2,3,4 Department
of Electrical Engineering, V.J.T.I, Mumbai, India e-mail: 1 [email protected], 2 [email protected] 3 [email protected],4 [email protected]
Abstract—Power system state estimation is the important basis of the advanced applications in the power system monitoring, operation and control. The unique ability to calculate synchronized phasors at high precision makes the phasor measurement unit (PMU) an important measuring device to improve the accuracy of state estimator. In this paper, under given placement of PMUs in the power system, the linear model of state estimation is proposed to show the important contribution of PMU measurements to the power system state estimation along with the bad data detection and identification. The effectiveness of the proposed state estimation algorithm is validated using standard IEEE 14 bus test system. Keywords: Bad data, Power system state estimation, PMU
I. I NTRODUCTION The aim of power system state estimation (SE) is to obtain the best possible values of the bus voltage magnitudes and relative phase angles at the system nodes by processing the available network data. It is a critical component in the functioning of a power system. SE provides the platform for advanced security monitoring applications in control centers, for example, contingency analysis and optimal power flow rely on the state of the power system derived from the SE [1]. Traditional methods of estimation use measurements of bus voltage magnitudes, power flows and injections which are taken through the network provided by Supervisory Control and Data Acquisition (SCADA) system at certain times. These parameters are related to state variables through measurement functions and the measurement noise, estimation is then proceeds to obtain the estimated state variables of the power system. The magnitude and phase angle of voltages (|V |∠δ ) at each bus are considered as state variables. This method is a non-linear problem which is solved through an iterative weighted least square (WLS) method which converts nonlinear equations into approximations by using a first-order Taylor series expansion. The addition of synchronized phasor measurements to the estimator will enhance the accuracy of the estimation process and will become faster [2], [3]. With the use of synchronized measurements, modifications are introduced in order to achieve a higher degree of accuracy of the solution at the cost of some additional computations. The role of SE is to provide an estimation for all measured electrical quantities of power network and to filter out all the errors which corrupt analog measurements [1]. One of the basic functions of SE is to detect and identify bad data in
measurement set. A bad data processing is carried out after the SE in order to identify those bad data and thus to eliminate them from the set of measurements. These eliminations will result in improved SE parameters. In this paper, the effect of asynchronicity in the measurement set and addition of synchronized measurements for the accuracy of power system SE is studied. II. P HASOR M EASUREMENT U NIT (PMU) A Phasor measurement unit (PMU) is a device which measures the electrical waves on an electricity grid, using a common time source for synchronization. This provides real time information in a synchronized way and the valuable potential to improve the monitoring, protection and control of power system. These qualities enable state estimation equations to be modelled in a linear manner and thus significantly simplifies the traditional SE.
Fig. 1.
Configuration of modern PMU.
Time synchronization allows synchronized real-time measurements of multiple remote measurement points on the grid. In power engineering, these are also commonly referred to as synchrophasors and are considered as one of the most important measuring devices in the future of power systems. The functions of PMU can be incorporated into a protective relay or other devices. Phasor measurements that observed at the same time are called “synchrophasors”. In typical applications, PMUs are sampled from widely dispersed locations in the power system network and synchronized from the common time source of a global positioning system (GPS) radio clock. Synchrophasor technology provides a tool for system operators and planners to measure the state of the electrical system and
manage power quality. Due to the truly synchronized phasors, comparison of two quantities is possible in real time which can be used to assess system conditions. PMUs were first introduced by A. G. Phadke [4] at Virginia Tech. in early 1980s. Fig. 1 is based on the configurations of the first PMUs build at Virginia Tech [5]. The analog signals of voltage and currents are converted into digital form. The input signal may have harmonic components so the task of the PMU is to separate the fundamental frequency component and find its phasor representation. Data samples are taken from the waveform, and apply the discrete fourier transform (DFT) to compute the phasor representation of input signal. To limit the bandwidth of the pass band (less than half of the data sampling frequency) an anti-aliasing filters are applied to the signal before data samples are taken. The nonharmonic signals and any other random noise present in the input signal leads to an error in estimation of the phasor. III. L ITERATURE R EVIEW Utilization of PMU measurements results in replacement of non-linear SE into linear SE, which in turn directly manipulates the jacobian matrix. The simple power system SE model and algorithm to show the reduced model and the reliability of state variables from PMU has been explained in [6]. Synchronized measurements are preferable to traditional state estimation, it is found that placement of PMUs must meet optimal number of PMUs so that the system is completely observable [7]. As proposed in [8], Phasor measurements integration was non-invasive allowing flexibility for implementation in wide area measurement platforms. The resulting model was linear and the solution is direct and non-iterative. The author of [9], shows that by increasing the number of phasor measurements on a power system, the quality of the estimated state is progressively improved. a double SE model whereby the nonlinear SE was done on the condition of taking the state variable measured or calculated by PMU as the state variables of the nodes. The accuracy of the estimator depends considerably on the accuracy of the synchronization and the accuracy of the applied technique for calculating the difference between estimator and measurements [10].
Fig. 2.
Proposed State estimation process.
The linear SE is carried out using traditional SE results and PMU measurements, All these values are merged together
to form a measurement vector. Estimation is then done with proper weight given to the errors. The solutions are found with minimum errors. This method improves processing speed as there is no iterations required. In this paper, phasor-based measurements are used so as to make the estimation process linear after the bad data excluded from the non-linear estimation as shown in Fig. 2. IV. E STIMATION P ROCEDURES A. Traditional State Estimation The traditional non-linear estimation model is formulated as follows: z = h(x) + e
z1 z2 . . . zm
=
h1 (x1 , x2 ....xn ) h2 (x1 , x2 ....xn ) . . . hm (x1 , x2 ....xn )
+
e1 e2 . . . em
(1)
where z is a set of measurements consisting of voltage magnitudes (|Vi |), real and reactive power injections (Pi , Qi ) and line power flows (Pi j , Qi j ) which are related to state variables x thorough non-linear measurement function h and the measurement error e which is assumed to have zero mean distribution and constant variance σ . The number of measurements is greater than the number of state variables in order to have a unique solution. The relevant non-linear functions can be given by these equations: N
Pi = |Vi | ∑ |V j | (Gi j cos(δi − δ j ) + Bi j sin(δi − δ j ))
(2)
j=1 N
Qi = |Vi | ∑ |V j | (Gi j sin(δi − δ j ) − Bi j cos(δi − δ j ))
(3)
j=1
Real and reactive power flow from bus i to bus j are, Pi j = |Vi |2 (gsi + gi j )−|Vi ||V j | (gi j cos(δi − δ j ) + bi j sin(δi − δ j )) (4) Qi j = −|Vi |2 (bsi + bi j )−|Vi ||V j | (gi j cos(δi − δ j ) − bi j sin(δi − δ j )) (5) were Yi j = Gi j + jBi j is admittance of line which is connected from bus i to bus j. The estimation is then manipulated through weighted least square method (WLS), which is computed by minimizing an objective function J given by sum of the squares of the weighted errors e as m 2 ei J(x) = ∑ (6) σ i i=1 J(x) = [z − h(x)]T R−1 [z − h(x)]
(7)
where m is the number of measurements, i is the corresponding numbers of measurement, R is the covariance matrix of measurements used as weighting matrix in order to get the best estimation of the state variables and is given by R−1 = diag 1/σi2 (8) The minimization of this objective function is achieved by setting the derivative of J with respect to state variables x to zero ∂ J(x) = H T (x)R−1 [z − h(x)] = 0 (9) ∂x where H(x) = ∂ h(x) ∂ x is the Jacobian matrix. The solution of (9) is found using Gauss-Seidel iterative method. Iteration process is terminated when the specified error criterion is reached. The measurement function h calculates estimated values of voltage magnitude and phase angle at each bus in the system. If xk is the state variable at kth iteration, the next estimation xk+1 is computed as G(xk )(xk+1 − xk ) = H T (x)R−1 [z − h(x)]
(10)
where G(xk ) = H T (x)R−1 H(x) is the gain matrix, the dimension of G matrix is (2N −1)X(2N −1), where N is the number of buses at power system B. Linear State Estimation Model
Vireal Viimag Vireal Viimag Iireal Iiimag
II 0
est M11 = M21 Gi j Bi j pmu Z
0 II M12 M22 −Bi j Gi j
Vireal +[ε] Viimag X H
(14) where n is the number of state variables and II is the identity matrix of dimension nxn because measurements are directly related to state variables. Starting with the sparse matrix, depending on the connectivity of networks and placement of PMUs, the zero elements of sparse matrix are replaced by 1 to form M11 and M22 , while M12 and M21 represent the null matrix. ε is the vector of errors. Unlike traditional method where iterations are needed, the solution of linear model is computed directly as: −1 X = [H]T [R]−1 [H] [H]T [R]−1 [Z]
(15)
here R is a diagonal co-variance matrix with σi as standard deviation of errors, i stands for the corresponding measurement number of Z vector. V. BAD DATA R ECOGNITION
For the given optimal placement of PMUs [11], Linear model of state estimation is formed as [Z] = [H]X + [ε]
(11)
were [Z] is a set of measurements, which consists of magnitude and phase angle of voltages (|V |,δ ) pmu and line current flows (|I|,θ ) pmu measured by PMUs located at different places of the system and the traditionally estimated states (|V |,δ )est as measured vector. [H] is a matrix which relates state variables (|V |,δ )new to these measurement values. All the quantities in this approach are expressed in rectangular co-ordinates so the standard deviations of variables must also be numerically converted to rectangular format. Voltage measurements are directly related to the states, while current at any bus is calculated by sum of the line currents connected to that bus. Currents in transmission lines are related to voltages by series admittance Yi j = Gi j + jBi j of the line which connects bus i to j. Transmission lines are represented by an equivalent π-model. Mathematically, real and imaginary parts of injected currents can be expressed as:
When the system model is correct and the measurements are accurate, there is good reason to accept the estimated states calculated by the WLS estimator, but if a measurement is bad, it should be detected and then identified so that it can be removed from the estimator calculations. It is essential to detect and identify the bad data for traditional measurements as well as for phasor measurements since these errors will have significant impact on the estimated state [12]. Detection and identification is done in two sections as: A. Chi-Squared test As the errors due to SE are normally distributed, it can be shown that the performance index J has a χk2 -distribution (Chi-squared distribution) where k is degree of freedom and is obtained as m − n. Measurements containing bad data will result in unexpectedly large value of objective function J. For a certain confidence level α the objective function is required to be below the corresponding value of chi-squared distribution as shown in Fig.3. Thus if objective function
n
Iireal = Vireal Gii −Viimag Bii +
∑
V jreal Gi j −V jimag Bi j
(12)
j=1, j6=i n
Iiimag = Vireal Bii +Viimag Gii +
∑
V jreal Bi j −V jimag Gi j
(13)
j=1, j6=i
As a result the state model in (11) will take the form as follows:
2 J(x) ≥ χk,α
(16)
then during the estimation process, it is detected as bad data in measurements which may come out as wrong power system SE solutions. Hence it becomes necessary to eliminate these bad-data and continue estimation process. The chi-square values corresponding to the various probabilities and degrees of freedom are detailed in Appendix-I.
Fig. 3.
Chi-square distribution.
B. Largest Normalized Residual The Normalized residual matrix computed for each measurement is as |ri | (17) riN = √ Rii Sii The measurement residual is expressed as r = Se, where S = 1 − HG−1 H T R−1 is sensitivity matrix and e is measurement error vector as in (1), The wrong measurement is responsible for the largest normalized residual. The largest normalized residual has no relation with the scale of power system, and it only depend on measurement redundancy. The higher the redundancy level, the diagonal element has more advantages in the matrix, and the bad data is easier to be figured out. VI. C ASE S TUDY The simulations are carried out from MATLAB software package for the IEEE 14 bus test network shown in Fig. 4 to validate the results. Bus 01 is considered as slack bus so the phase angle in this bus is referred as reference angle and all other phase angles are obtained with respect to this. The measured values are obtained from the power flow solution computed by PSAT [13]. Power flows, power injections and voltages are considered as measurement data. The errors are assumed to have normally distributed and zero mean and are of statistical nature. The proper values of weight matrix is calculated corresponding to its accuracy classes. The whole process is carried out in two scenarios. In scenario I, Traditional SE method is carried out where three types of measurements are observed which might have been corrupted with some noises. The process is carried out until it converged with a tolerance of 10−5 . Linear SE model is studied in scenario II to deal with the inclusion of synchronized phasor measurements observed from PMUs. The real-time measurement sets are generated with the steady state power flow condition. The proposed algorithm for the linear SE using PMU data is implemented as shown by the flowchart in Fig. 5. Measurements are composed in two major blocks, PMU readings are found in one block while another block
Fig. 4.
IEEE 14 bus test network.
contains estimated states obtained from the traditional SE. These measurements are combined together and proceeded to form z vector and their covariance matrix R. The process is continued to form jacobian matrix H and then final solution is found at the end.
Fig. 5.
Flow chart of the proposed algorithm.
For the given optimal placement of PMU as in [7],[11] four PMUs were placed at bus numbers 2, 6, 7 and 9 so that the entire system is observable. Here measurement vector consists of all 14 bus voltages with added noises in it along with 8 measurements of real and reactive power injection and 12 measurements of real and reactive power flows as summarized in Appendix-II.
time taken by the direct solution obtained from linear model algorithms. Voltage magnitudes by scenario II as in Fig. 6 approaching much closer to the true values than the scenario I which results in more accurate states. dimension of the model is also reduced in scenario II which results in faster process. The resulting estimated voltage angles are shown in Fig. 7.
VII. R ESULTS A. Bad-data processing results Various measurements were undergone testing for bad-data identifications. If in case there is no bad data in the measurements the objective function is subjected to be minimum. The normalized residuals are found to be larger value if bad-data exist in their corresponding measurements. Two Tests were performed to identify bad-data. Test-I, contains measurements without any bad-data while bad-data are considered to be in the Test-II with probability of 95% (α= 0.05). The objective function J(x) in Test-II seems to be larger than the corresponding chi-square χ 2 value which is an indication of bad-data existence in the measurements. It is clear from the Test result summarized in TABLE I, voltage measurements having largest normalized residual are identified as bad data. Bad data is eliminated from the measurements and objective function is checked again for further bad data if exist in the measurements. T EST RESULTS FOR BAD - DATA Test-I rN 0.1814 0.0499 0.3097 0.0066 0.0303
J(x)
0.8003
Test-II Measurements rN Vi 1.5762 Pi 0.0502 Qi 0.3118 Pi j 0.0068 Qi j 0.0303
Simulation result of voltage angles for all IEEE 14bus.
The average squared voltage error along with the number of iterations required for both the scenarios are mentioned in TABLE II. TABLE II E STIMATION RESULTS Scenario I II
TABLE I
Measurements Vi Pi Qi Pi j Qi j
Fig. 7.
Iterations 5 nil
Squared voltage errors (p.u.) 2.5840e-004 1.7663e-004
J(x)
VIII. C ONCLUSION 327.6
B. Estimation results Estimation results were obtained for both the scenarios showing the voltage magnitude and phase angles for all 14 buses.
This paper shows the importance of including PMUs in traditional estimator. A linear state estimator model with direct state measurements from PMUs were presented. The model was tested and validated in IEEE 14 bus test system. PMUs output brings the significant improvement in power system estimation. Simulation results reveal PMU as an important device for the power system applications. Linear approach of estimation has the advantage of faster response due to the smaller dimension formed by incorporating PMUs readings in estimation process. The unique feature and reliance of PMUs were pointed out in this paper. Further research work in this paper is ongoing for larger power system. ACKNOWLEDGEMENTS The authors wish to acknowledge the authority of Electrical Engineering department of V.J.T.I., Mumbai, India for providing infrastructure necessary for this research activity.
Fig. 6.
Simulation result of voltage magnitudes for all IEEE 14bus.
After the 5 iterations, results were obtained for traditional estimation process which takes significantly more time than
A PPENDIX -I
R EFERENCES
Chi-square distribution values for various probabilities.
[1] F.C.Schweppe, at all, “Power System Static State Estimation, Part I, 11, and 111, IEEE Transactions on PAS, Vol. PAS-89, No.1, January 1970, pp. 120-135 [2] A.G.Phadke, at all, “State Estimation with Phasor Measurements, IEEE Transactions on Power Systems, Vol. PWRS-1, So.1, February 1986, pp.233-241 [3] J.S.Thorp, at all,“Real-time Voltage Phasor Measurements for Static State Estimation, IEEE Transactions on PAS, Vol. PAS-104, No.11, November 1985, pp.3098-3106 [4] A. G. Phadke, J. D. L. Ree, S. Member, V. Centeno, J. S. Thorp and L. Fellow, “Synchronized Phasor Measurement Applications in Power Systems”,IEEE Transactions on smart grid, vol. 1, no. 1, pp. 20-27, 2010. [5] A.G. phadke and J.S. thorp, “Synchronized Phasor measurements and their applications”, ISBN 978-0. Springer, 2008, p. 249. [6] F. Chen, X. Han, Z. Pan, and L. Han, “State Estimation Model and Algorithm Including PMU”, DRPT 2008 no. April 2008, pp. 1097-1102. [7] B. Xu and A. Abur, “Observability Analysis and Measurement Placement for Systems with PMUs”,Power Systems Conference and Exposition, 2004. IEEE PES, no. 1, pp. 1-4, [8] R. F. Nuqui, A. G. Phadke, and L. Fellow, “Hybrid Linear State Estimation Utilizing Synchronized Phasor Measurements,” PowerTech 2007, pp. 1665-1669, 2007. [9] M. Zhou et al., “An Alternative for Including Phasor Measurements in State Estimators”,IEEE Transactions on power systems, vol. 21, no. 4, pp. 1930-1937, 2006. [10] T. Pretoria, “Implementation of PMU technology in state estimation: An overview”, IEEE AFRICON 4th, pp. 1006-1011, 1996. [11] K. Jamuna and K. S. Swarup, “Critical Measurement Set with PMU for Hybrid State Estimation”,16th National power systems conference, pp. 254-259, 2010. [12] J. Zhu and A. Abur, “Bad data Identification when using Phasor Measurements”,Power Tech, 2007 IEEE Lausanne, pp. 1676-1681, 2007. [13] Federico Milano, in “Power System Analysis Toolbox [PSAT]”, 2005, p. 439.
Probabilities Degree of freedom 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100
0.10
0.05
0.025
0.01
0.005
2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 51.805 63.167 74.397 85.527 96.578 107.565 118.498
3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 55.758 67.505 79.082 90.531 101.879 113.145 124.342
5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 59.342 71.420 83.298 95.023 106.629 118.136 129.561
6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 63.691 76.154 88.379 100.425 112.329 124.116 135.807
7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169
A PPENDIX -II Measured values for estimation of IEEE 14 bus system.
Voltage
Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Value (p.u.) 1.062 1.0432 1.01 1.014 1.016 1.07 1.032 1.08 1.010 1.015 1.076 1.055 1.056 1.019
Power injections
Power flows
Bus
Value (p.u.)
Bus
Value (p.u.)
2 3 6 8 10 11 12 14
0.1830 + 0.3523i -0.9420 + 0.0876i -0.1120 + 0.1501i 0 + 0.2103i -0.0900 - 0.0580i -0.0350 - 0.0180i -0.0610 - 0.0160i -0.1490 - 0.0500i
1-2 2-3 4-2 4-7 4-9 5-2 5-4 5-6 6-13 7-9 11-6 13-14
1.5708 - 0.1748i 0.7340 + 0.0594i -0.5427 + 0.0213i 0.2707 - 0.1540i 0.1546 - 0.0264i -0.4081 - 0.0193i 0.6006 - 0.1006i 0.4589 - 0.2084i 0.1834 + 0.0998i 0.2707 + 0.1480i -0.0816 - 0.0864i 0.0645 + 0.0508i
of Electrical Engineering, V.J.T.I, Mumbai, India e-mail: 1 [email protected], 2 [email protected] 3 [email protected],4 [email protected]
Abstract—Power system state estimation is the important basis of the advanced applications in the power system monitoring, operation and control. The unique ability to calculate synchronized phasors at high precision makes the phasor measurement unit (PMU) an important measuring device to improve the accuracy of state estimator. In this paper, under given placement of PMUs in the power system, the linear model of state estimation is proposed to show the important contribution of PMU measurements to the power system state estimation along with the bad data detection and identification. The effectiveness of the proposed state estimation algorithm is validated using standard IEEE 14 bus test system. Keywords: Bad data, Power system state estimation, PMU
I. I NTRODUCTION The aim of power system state estimation (SE) is to obtain the best possible values of the bus voltage magnitudes and relative phase angles at the system nodes by processing the available network data. It is a critical component in the functioning of a power system. SE provides the platform for advanced security monitoring applications in control centers, for example, contingency analysis and optimal power flow rely on the state of the power system derived from the SE [1]. Traditional methods of estimation use measurements of bus voltage magnitudes, power flows and injections which are taken through the network provided by Supervisory Control and Data Acquisition (SCADA) system at certain times. These parameters are related to state variables through measurement functions and the measurement noise, estimation is then proceeds to obtain the estimated state variables of the power system. The magnitude and phase angle of voltages (|V |∠δ ) at each bus are considered as state variables. This method is a non-linear problem which is solved through an iterative weighted least square (WLS) method which converts nonlinear equations into approximations by using a first-order Taylor series expansion. The addition of synchronized phasor measurements to the estimator will enhance the accuracy of the estimation process and will become faster [2], [3]. With the use of synchronized measurements, modifications are introduced in order to achieve a higher degree of accuracy of the solution at the cost of some additional computations. The role of SE is to provide an estimation for all measured electrical quantities of power network and to filter out all the errors which corrupt analog measurements [1]. One of the basic functions of SE is to detect and identify bad data in
measurement set. A bad data processing is carried out after the SE in order to identify those bad data and thus to eliminate them from the set of measurements. These eliminations will result in improved SE parameters. In this paper, the effect of asynchronicity in the measurement set and addition of synchronized measurements for the accuracy of power system SE is studied. II. P HASOR M EASUREMENT U NIT (PMU) A Phasor measurement unit (PMU) is a device which measures the electrical waves on an electricity grid, using a common time source for synchronization. This provides real time information in a synchronized way and the valuable potential to improve the monitoring, protection and control of power system. These qualities enable state estimation equations to be modelled in a linear manner and thus significantly simplifies the traditional SE.
Fig. 1.
Configuration of modern PMU.
Time synchronization allows synchronized real-time measurements of multiple remote measurement points on the grid. In power engineering, these are also commonly referred to as synchrophasors and are considered as one of the most important measuring devices in the future of power systems. The functions of PMU can be incorporated into a protective relay or other devices. Phasor measurements that observed at the same time are called “synchrophasors”. In typical applications, PMUs are sampled from widely dispersed locations in the power system network and synchronized from the common time source of a global positioning system (GPS) radio clock. Synchrophasor technology provides a tool for system operators and planners to measure the state of the electrical system and
manage power quality. Due to the truly synchronized phasors, comparison of two quantities is possible in real time which can be used to assess system conditions. PMUs were first introduced by A. G. Phadke [4] at Virginia Tech. in early 1980s. Fig. 1 is based on the configurations of the first PMUs build at Virginia Tech [5]. The analog signals of voltage and currents are converted into digital form. The input signal may have harmonic components so the task of the PMU is to separate the fundamental frequency component and find its phasor representation. Data samples are taken from the waveform, and apply the discrete fourier transform (DFT) to compute the phasor representation of input signal. To limit the bandwidth of the pass band (less than half of the data sampling frequency) an anti-aliasing filters are applied to the signal before data samples are taken. The nonharmonic signals and any other random noise present in the input signal leads to an error in estimation of the phasor. III. L ITERATURE R EVIEW Utilization of PMU measurements results in replacement of non-linear SE into linear SE, which in turn directly manipulates the jacobian matrix. The simple power system SE model and algorithm to show the reduced model and the reliability of state variables from PMU has been explained in [6]. Synchronized measurements are preferable to traditional state estimation, it is found that placement of PMUs must meet optimal number of PMUs so that the system is completely observable [7]. As proposed in [8], Phasor measurements integration was non-invasive allowing flexibility for implementation in wide area measurement platforms. The resulting model was linear and the solution is direct and non-iterative. The author of [9], shows that by increasing the number of phasor measurements on a power system, the quality of the estimated state is progressively improved. a double SE model whereby the nonlinear SE was done on the condition of taking the state variable measured or calculated by PMU as the state variables of the nodes. The accuracy of the estimator depends considerably on the accuracy of the synchronization and the accuracy of the applied technique for calculating the difference between estimator and measurements [10].
Fig. 2.
Proposed State estimation process.
The linear SE is carried out using traditional SE results and PMU measurements, All these values are merged together
to form a measurement vector. Estimation is then done with proper weight given to the errors. The solutions are found with minimum errors. This method improves processing speed as there is no iterations required. In this paper, phasor-based measurements are used so as to make the estimation process linear after the bad data excluded from the non-linear estimation as shown in Fig. 2. IV. E STIMATION P ROCEDURES A. Traditional State Estimation The traditional non-linear estimation model is formulated as follows: z = h(x) + e
z1 z2 . . . zm
=
h1 (x1 , x2 ....xn ) h2 (x1 , x2 ....xn ) . . . hm (x1 , x2 ....xn )
+
e1 e2 . . . em
(1)
where z is a set of measurements consisting of voltage magnitudes (|Vi |), real and reactive power injections (Pi , Qi ) and line power flows (Pi j , Qi j ) which are related to state variables x thorough non-linear measurement function h and the measurement error e which is assumed to have zero mean distribution and constant variance σ . The number of measurements is greater than the number of state variables in order to have a unique solution. The relevant non-linear functions can be given by these equations: N
Pi = |Vi | ∑ |V j | (Gi j cos(δi − δ j ) + Bi j sin(δi − δ j ))
(2)
j=1 N
Qi = |Vi | ∑ |V j | (Gi j sin(δi − δ j ) − Bi j cos(δi − δ j ))
(3)
j=1
Real and reactive power flow from bus i to bus j are, Pi j = |Vi |2 (gsi + gi j )−|Vi ||V j | (gi j cos(δi − δ j ) + bi j sin(δi − δ j )) (4) Qi j = −|Vi |2 (bsi + bi j )−|Vi ||V j | (gi j cos(δi − δ j ) − bi j sin(δi − δ j )) (5) were Yi j = Gi j + jBi j is admittance of line which is connected from bus i to bus j. The estimation is then manipulated through weighted least square method (WLS), which is computed by minimizing an objective function J given by sum of the squares of the weighted errors e as m 2 ei J(x) = ∑ (6) σ i i=1 J(x) = [z − h(x)]T R−1 [z − h(x)]
(7)
where m is the number of measurements, i is the corresponding numbers of measurement, R is the covariance matrix of measurements used as weighting matrix in order to get the best estimation of the state variables and is given by R−1 = diag 1/σi2 (8) The minimization of this objective function is achieved by setting the derivative of J with respect to state variables x to zero ∂ J(x) = H T (x)R−1 [z − h(x)] = 0 (9) ∂x where H(x) = ∂ h(x) ∂ x is the Jacobian matrix. The solution of (9) is found using Gauss-Seidel iterative method. Iteration process is terminated when the specified error criterion is reached. The measurement function h calculates estimated values of voltage magnitude and phase angle at each bus in the system. If xk is the state variable at kth iteration, the next estimation xk+1 is computed as G(xk )(xk+1 − xk ) = H T (x)R−1 [z − h(x)]
(10)
where G(xk ) = H T (x)R−1 H(x) is the gain matrix, the dimension of G matrix is (2N −1)X(2N −1), where N is the number of buses at power system B. Linear State Estimation Model
Vireal Viimag Vireal Viimag Iireal Iiimag
II 0
est M11 = M21 Gi j Bi j pmu Z
0 II M12 M22 −Bi j Gi j
Vireal +[ε] Viimag X H
(14) where n is the number of state variables and II is the identity matrix of dimension nxn because measurements are directly related to state variables. Starting with the sparse matrix, depending on the connectivity of networks and placement of PMUs, the zero elements of sparse matrix are replaced by 1 to form M11 and M22 , while M12 and M21 represent the null matrix. ε is the vector of errors. Unlike traditional method where iterations are needed, the solution of linear model is computed directly as: −1 X = [H]T [R]−1 [H] [H]T [R]−1 [Z]
(15)
here R is a diagonal co-variance matrix with σi as standard deviation of errors, i stands for the corresponding measurement number of Z vector. V. BAD DATA R ECOGNITION
For the given optimal placement of PMUs [11], Linear model of state estimation is formed as [Z] = [H]X + [ε]
(11)
were [Z] is a set of measurements, which consists of magnitude and phase angle of voltages (|V |,δ ) pmu and line current flows (|I|,θ ) pmu measured by PMUs located at different places of the system and the traditionally estimated states (|V |,δ )est as measured vector. [H] is a matrix which relates state variables (|V |,δ )new to these measurement values. All the quantities in this approach are expressed in rectangular co-ordinates so the standard deviations of variables must also be numerically converted to rectangular format. Voltage measurements are directly related to the states, while current at any bus is calculated by sum of the line currents connected to that bus. Currents in transmission lines are related to voltages by series admittance Yi j = Gi j + jBi j of the line which connects bus i to j. Transmission lines are represented by an equivalent π-model. Mathematically, real and imaginary parts of injected currents can be expressed as:
When the system model is correct and the measurements are accurate, there is good reason to accept the estimated states calculated by the WLS estimator, but if a measurement is bad, it should be detected and then identified so that it can be removed from the estimator calculations. It is essential to detect and identify the bad data for traditional measurements as well as for phasor measurements since these errors will have significant impact on the estimated state [12]. Detection and identification is done in two sections as: A. Chi-Squared test As the errors due to SE are normally distributed, it can be shown that the performance index J has a χk2 -distribution (Chi-squared distribution) where k is degree of freedom and is obtained as m − n. Measurements containing bad data will result in unexpectedly large value of objective function J. For a certain confidence level α the objective function is required to be below the corresponding value of chi-squared distribution as shown in Fig.3. Thus if objective function
n
Iireal = Vireal Gii −Viimag Bii +
∑
V jreal Gi j −V jimag Bi j
(12)
j=1, j6=i n
Iiimag = Vireal Bii +Viimag Gii +
∑
V jreal Bi j −V jimag Gi j
(13)
j=1, j6=i
As a result the state model in (11) will take the form as follows:
2 J(x) ≥ χk,α
(16)
then during the estimation process, it is detected as bad data in measurements which may come out as wrong power system SE solutions. Hence it becomes necessary to eliminate these bad-data and continue estimation process. The chi-square values corresponding to the various probabilities and degrees of freedom are detailed in Appendix-I.
Fig. 3.
Chi-square distribution.
B. Largest Normalized Residual The Normalized residual matrix computed for each measurement is as |ri | (17) riN = √ Rii Sii The measurement residual is expressed as r = Se, where S = 1 − HG−1 H T R−1 is sensitivity matrix and e is measurement error vector as in (1), The wrong measurement is responsible for the largest normalized residual. The largest normalized residual has no relation with the scale of power system, and it only depend on measurement redundancy. The higher the redundancy level, the diagonal element has more advantages in the matrix, and the bad data is easier to be figured out. VI. C ASE S TUDY The simulations are carried out from MATLAB software package for the IEEE 14 bus test network shown in Fig. 4 to validate the results. Bus 01 is considered as slack bus so the phase angle in this bus is referred as reference angle and all other phase angles are obtained with respect to this. The measured values are obtained from the power flow solution computed by PSAT [13]. Power flows, power injections and voltages are considered as measurement data. The errors are assumed to have normally distributed and zero mean and are of statistical nature. The proper values of weight matrix is calculated corresponding to its accuracy classes. The whole process is carried out in two scenarios. In scenario I, Traditional SE method is carried out where three types of measurements are observed which might have been corrupted with some noises. The process is carried out until it converged with a tolerance of 10−5 . Linear SE model is studied in scenario II to deal with the inclusion of synchronized phasor measurements observed from PMUs. The real-time measurement sets are generated with the steady state power flow condition. The proposed algorithm for the linear SE using PMU data is implemented as shown by the flowchart in Fig. 5. Measurements are composed in two major blocks, PMU readings are found in one block while another block
Fig. 4.
IEEE 14 bus test network.
contains estimated states obtained from the traditional SE. These measurements are combined together and proceeded to form z vector and their covariance matrix R. The process is continued to form jacobian matrix H and then final solution is found at the end.
Fig. 5.
Flow chart of the proposed algorithm.
For the given optimal placement of PMU as in [7],[11] four PMUs were placed at bus numbers 2, 6, 7 and 9 so that the entire system is observable. Here measurement vector consists of all 14 bus voltages with added noises in it along with 8 measurements of real and reactive power injection and 12 measurements of real and reactive power flows as summarized in Appendix-II.
time taken by the direct solution obtained from linear model algorithms. Voltage magnitudes by scenario II as in Fig. 6 approaching much closer to the true values than the scenario I which results in more accurate states. dimension of the model is also reduced in scenario II which results in faster process. The resulting estimated voltage angles are shown in Fig. 7.
VII. R ESULTS A. Bad-data processing results Various measurements were undergone testing for bad-data identifications. If in case there is no bad data in the measurements the objective function is subjected to be minimum. The normalized residuals are found to be larger value if bad-data exist in their corresponding measurements. Two Tests were performed to identify bad-data. Test-I, contains measurements without any bad-data while bad-data are considered to be in the Test-II with probability of 95% (α= 0.05). The objective function J(x) in Test-II seems to be larger than the corresponding chi-square χ 2 value which is an indication of bad-data existence in the measurements. It is clear from the Test result summarized in TABLE I, voltage measurements having largest normalized residual are identified as bad data. Bad data is eliminated from the measurements and objective function is checked again for further bad data if exist in the measurements. T EST RESULTS FOR BAD - DATA Test-I rN 0.1814 0.0499 0.3097 0.0066 0.0303
J(x)
0.8003
Test-II Measurements rN Vi 1.5762 Pi 0.0502 Qi 0.3118 Pi j 0.0068 Qi j 0.0303
Simulation result of voltage angles for all IEEE 14bus.
The average squared voltage error along with the number of iterations required for both the scenarios are mentioned in TABLE II. TABLE II E STIMATION RESULTS Scenario I II
TABLE I
Measurements Vi Pi Qi Pi j Qi j
Fig. 7.
Iterations 5 nil
Squared voltage errors (p.u.) 2.5840e-004 1.7663e-004
J(x)
VIII. C ONCLUSION 327.6
B. Estimation results Estimation results were obtained for both the scenarios showing the voltage magnitude and phase angles for all 14 buses.
This paper shows the importance of including PMUs in traditional estimator. A linear state estimator model with direct state measurements from PMUs were presented. The model was tested and validated in IEEE 14 bus test system. PMUs output brings the significant improvement in power system estimation. Simulation results reveal PMU as an important device for the power system applications. Linear approach of estimation has the advantage of faster response due to the smaller dimension formed by incorporating PMUs readings in estimation process. The unique feature and reliance of PMUs were pointed out in this paper. Further research work in this paper is ongoing for larger power system. ACKNOWLEDGEMENTS The authors wish to acknowledge the authority of Electrical Engineering department of V.J.T.I., Mumbai, India for providing infrastructure necessary for this research activity.
Fig. 6.
Simulation result of voltage magnitudes for all IEEE 14bus.
After the 5 iterations, results were obtained for traditional estimation process which takes significantly more time than
A PPENDIX -I
R EFERENCES
Chi-square distribution values for various probabilities.
[1] F.C.Schweppe, at all, “Power System Static State Estimation, Part I, 11, and 111, IEEE Transactions on PAS, Vol. PAS-89, No.1, January 1970, pp. 120-135 [2] A.G.Phadke, at all, “State Estimation with Phasor Measurements, IEEE Transactions on Power Systems, Vol. PWRS-1, So.1, February 1986, pp.233-241 [3] J.S.Thorp, at all,“Real-time Voltage Phasor Measurements for Static State Estimation, IEEE Transactions on PAS, Vol. PAS-104, No.11, November 1985, pp.3098-3106 [4] A. G. Phadke, J. D. L. Ree, S. Member, V. Centeno, J. S. Thorp and L. Fellow, “Synchronized Phasor Measurement Applications in Power Systems”,IEEE Transactions on smart grid, vol. 1, no. 1, pp. 20-27, 2010. [5] A.G. phadke and J.S. thorp, “Synchronized Phasor measurements and their applications”, ISBN 978-0. Springer, 2008, p. 249. [6] F. Chen, X. Han, Z. Pan, and L. Han, “State Estimation Model and Algorithm Including PMU”, DRPT 2008 no. April 2008, pp. 1097-1102. [7] B. Xu and A. Abur, “Observability Analysis and Measurement Placement for Systems with PMUs”,Power Systems Conference and Exposition, 2004. IEEE PES, no. 1, pp. 1-4, [8] R. F. Nuqui, A. G. Phadke, and L. Fellow, “Hybrid Linear State Estimation Utilizing Synchronized Phasor Measurements,” PowerTech 2007, pp. 1665-1669, 2007. [9] M. Zhou et al., “An Alternative for Including Phasor Measurements in State Estimators”,IEEE Transactions on power systems, vol. 21, no. 4, pp. 1930-1937, 2006. [10] T. Pretoria, “Implementation of PMU technology in state estimation: An overview”, IEEE AFRICON 4th, pp. 1006-1011, 1996. [11] K. Jamuna and K. S. Swarup, “Critical Measurement Set with PMU for Hybrid State Estimation”,16th National power systems conference, pp. 254-259, 2010. [12] J. Zhu and A. Abur, “Bad data Identification when using Phasor Measurements”,Power Tech, 2007 IEEE Lausanne, pp. 1676-1681, 2007. [13] Federico Milano, in “Power System Analysis Toolbox [PSAT]”, 2005, p. 439.
Probabilities Degree of freedom 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100
0.10
0.05
0.025
0.01
0.005
2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 51.805 63.167 74.397 85.527 96.578 107.565 118.498
3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 55.758 67.505 79.082 90.531 101.879 113.145 124.342
5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 59.342 71.420 83.298 95.023 106.629 118.136 129.561
6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 63.691 76.154 88.379 100.425 112.329 124.116 135.807
7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169
A PPENDIX -II Measured values for estimation of IEEE 14 bus system.
Voltage
Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Value (p.u.) 1.062 1.0432 1.01 1.014 1.016 1.07 1.032 1.08 1.010 1.015 1.076 1.055 1.056 1.019
Power injections
Power flows
Bus
Value (p.u.)
Bus
Value (p.u.)
2 3 6 8 10 11 12 14
0.1830 + 0.3523i -0.9420 + 0.0876i -0.1120 + 0.1501i 0 + 0.2103i -0.0900 - 0.0580i -0.0350 - 0.0180i -0.0610 - 0.0160i -0.1490 - 0.0500i
1-2 2-3 4-2 4-7 4-9 5-2 5-4 5-6 6-13 7-9 11-6 13-14
1.5708 - 0.1748i 0.7340 + 0.0594i -0.5427 + 0.0213i 0.2707 - 0.1540i 0.1546 - 0.0264i -0.4081 - 0.0193i 0.6006 - 0.1006i 0.4589 - 0.2084i 0.1834 + 0.0998i 0.2707 + 0.1480i -0.0816 - 0.0864i 0.0645 + 0.0508i
