Power System State Estimation - A Review
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Power System State Estimation - A Review

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The aim of this article is to provide a comprehensive ...

The aim of this article is to provide a comprehensive
survey on power system state estimation techniques. The
algorithms used for finding the system states under both static
and dynamic state estimations are discussed in brief. The
authors are opinion that the scope of pursuing research in the
area of state estimation with PMU and SCADA measurements
is the state of the art and timely.

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Power System State Estimation - A Review Power System State Estimation - A Review Document Transcript

  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 Power System State Estimation - A Review Dudekula Sai Babu1, K Jamuna1, and B.Aryanandiny2 1 2 VIT University, Chennai campus/SELECT, Chennai, India College of Engineering, Trivandrum/EEE Department, Thiruvananthapuram, India Email: jamunakamaraj@gmail.com Abstract— The aim of this article is to provide a comprehensive survey on power system state estimation techniques. The algorithms used for finding the system states under both static and dynamic state estimations are discussed in brief. The authors are opinion that the scope of pursuing research in the area of state estimation with PMU and SCADA measurements is the state of the art and timely. tion of robust estimation, hierarchical estimation, with and without the inclusion of current measurements, etc. The SE uses only volt- age magnitude, real and reactive power injections and flows of SCADA measurements. The inclusion of branch currents measurements in SE deteriotes the performance of estimators. It also leads to non uniquely observable which produces more than one state for the given one set of measurements [6], [7]. Distributed SE for very large power system has been taken for study since the very beginning. The computational procedure involved in SE is an optimization function. The optimization function can be first order or second order of the derivative. The first order methods are the classical weighted least squares [8], the iteratively reweighted least squares [9] and the linear programming based on least absolute value estimator. The second order method involves the evaluation of the Lagrangian Hessian matrix. The primal-dual interior-point method and Huber M estimator are the solution metholodigies are available in literature to solve the second order method. The system states are evaluated either by statically or dynamically. The above methods mentioned are static state estimators. At the given point of time, the set of measurements are used to estimate the system state at that instant of time. The common method used to solve the static SE is weighted least square and weighted least absolute value methods. The heuristic methods are also applied to find the states. In the dynamic state estimation, the system states are continuously monitored at the regular intervals. The DSE uses the Kalman filter, Leapfrog algorithm, non-linear observer technique and invariant imbedding method to estimate the system states dynamically. Index Terms— State Estimation, LAV, WLS, Kalman, Synchronized phasor measurements, PMU. I. INTRODUCTION State Estimation is the process of assigning a value to an unknown system variables based on the measurements obtained from the system. State estimator has been widely used as an indispensable tool for online monitoring, analysis and control of power systems. It is also exploited to filter redundant data, to eliminate incorrect measurements and to produce reliable state estimates. Entire power system measurements are obtained through Remote Terminal Unit (RTU) of Supervisory Control and Data Acquisition (SCADA) systems which have both analog and logic measurements. Logic measurements are used in topology processor to determine the system configuration. State estimator uses a set of analog measurements such as bus voltage magnitude, real power injections, reactive power injections, active power flow, reactive power flow and line current flows along with the system configuration. These measurements are obtained from SCADA systems through remote terminal unit. The phasor concept was introduced by Throp [1]. Based on this the new measuring device was developed, namely Phasor Measurement Unit (PMU). The PMUs use a navigational satellite system to synchronize digital sampling at different substations. The Global Positioning Sys- tem is used to synchronize the measurement units. The PMU records fifteen different channels of the real and imaginary parts of the secondary voltages and currents [2]. The phasor reporting rate is 60 phasors/sec for a voltage, 5 currents, 5 watt measurements, 5 var measurements, frequency and the rate of change of frequency. The state estimators are used both measurements together or separately for estimating system states. The detailed explanations are given in this review paper. A. Weighted Least Square(WLS) Method In the WLS method, the objective is to minimize the sum of the squares of the weighted deviations of the estimated measurements from the actual measurements. The system states are estimated from the available measurements. The objective function is expressed as follows, (1) where fi (x) and σ2 is the system equation and the variance of the ith measurement respectively. J(x) is the weighted residuals. m is the number of measurements and zi is the ith measured quantity. If fi (x) is the linear function then the solution of eqn. (1) is a closed form. Usually, the power flow and power injection equations are described by nonlinear function. Hence the solution leads iterative procedure to determine the state of the system. The steps pro II. STATIC STATE ESTIMATION SE is a very useful tool for the economic and secure operation of transmission networks. From early days of Scheweppe [3]–[5], developments of SE are done as a no © 2014 ACEEE DOI: 01.IJEPE.5.1.12 10
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 cessed in WLS are (1) to find the gradient of J(x) and (2) force it into zero and solved by Newton’s method. The optimal state estimate is found using eqn. (2). In the normal equation method, the numerical instability leads the ill-condition of the linear equation. This can be solved effectively by Peters-Wilkinson method [16]. The condition number of a symmetric, positive definite matrix (G) is defined as whose eigen values are real and positive can be computed using eqn. (4). It can also be defined as the eqn. (5). Ideally the value of κ should be 1 for a well conditioned system. A very high value of κ indicates that the system is ill-conditioned. The ill-conditioning occurs due to the following reasons[17].  The position of the reference slack bus.  Existence of the negative line reactance. Improper choice of line flow measurements. The network strategy create the first and second problem. The metering strategy affects the line flow measurements. The WLS algorithm anticipated to converge very slowly when the network strategy is wrong and the metering strategy is correct. The diagonal elements of the information matrix is increased and solved by LevenbergMarquardt (LM) algorithm, leads to rapid convergence. Both the metering and network strategy is wrong, the WLS algorithm will not converge. Even LM algo- rithm provides a solution but not the right solution. In this case, the household or orthogonal transformation in conjunction with LM escorts to the right solution. The Newton’s method is one of the solutions for the ill-conditioning problem. The actual WLS technique, the first order gradient of measurement function only considered and higher order terms are neglected. The Taylor series expansion of a function g(x) is represented in eqn (6). The second order has been included in the measurement function, and it has solved by Newton’s method, the efficient gradient convergent technique [18] and the states are calculated by the eqn.(8). i represents the iteration. The singular value decomposition is one among the solution for this case [19]. (2) (3) where R Measurement error covariance matrix x System state vector H Jacobian matrix z Measurement vector The covariance matrix R reflects the relative relation between the measurements. If there is no interaction between the various measurements then R will be a diagonal matrix. The diagonal elements are the variances of the individual measurements (ri = σ2 ). Recently the SE with measurement dependencies is solved with WLS technique [10]. Due to the sparsity of the information matrix (G), the WLS solution is obtained quickly by triangular factorization method. This technique is called as normal equation method. It can also solved by orthogonal transformation method and combination of these two methods (hybrid method) also presented in [11]. The steps involved in these methods are explained in Table. I. The normal equation method fails when (1) H is in ill- condition and (2) The high measurement redundancy affects the sparsity. The numerical behavior has improved in orthogonal transformation method. It has to store Q1 matrix, hence memory size increases. The hybrid method solves iteratively the normal equation, where the triangular factorization is carried out using orthogonal transformations. It maintains the good sparsity and numerical robustness. (4) (5) TABLE I. WLS ALGORITHM SOLUTION BY TRIANGULAR SOULTION Normal Eqn. Orthogonal Transformation 1. Information matrix G=HTWH Form Jacobian Matrix (H) Form Jacobian Matrix (H) 2. Factorize G=UTU Factorize H Factorize H and Q is not stored 3.? X=U-1 UT-1(B) Where, B=HTW? Z ? X=R-1C C=Q1W0.5? Z (6) Hybrid model ? X=R-1R T-1B B=HTW? Z (7) The WLS algorithm effectively provides the system states under normal operation and identifies the bad data measurements. The numerical instabilities are also handled easily. The on line system state estimation is done with WLS algorithm and implemented in all energy management systems. The least square solution is only optimal when the measurement noise is Gaussian and has well known statistics. In practical applications, this is not the case. Similarly, WLS estimator does not account for the random event like the failure in instrument and the communication channels [12]. Data obtained under these circumstances are named as bad data. The bad data are eliminated by normalized residual method [13]–[15]. © 2014 ACEEE DOI: 01.IJEPE.5.1.12 B. Decoupled State Estimation The main drawback in the WLS solution algorithm is the computational burden that is the calculation and the triangularization of the gain matrix or the information matrix. The approximation of the constant gain matrix reduces the 11
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 computational burden. In WLS solution steps, the elements of the gain matrix do not significantly change between the flat start initialization and the converged solution. The sensitivity of the real (reactive) power equations to changes in the magnitude (phase angle) of bus voltages is very low, especially for high voltage transmission systems [20]. These two observations lead to the fast decoupled formulation of the state estimation problem. The actual measurement, Jacobian and covariance matrices are partitioned based on the real power and reactive power separately. The measurement equations are represented in eqn. (8). bad data and outlier are discussed in robust state estimation but they are closely related. Bad data occurs due to failure of communication link, meter intermittent fault and incorrect recoding. An outlier does not contain any error due to structure of its corresponding equation. The identification of such kind of outliers is very difficult. Robust estimators are expected to remain unbiased despite the existence of different types of outliers. The robustness of an estimator is evaluated based on the breakdown point. It is the largest ratio of (s=m/mb ) mb is the number of bad data present in the measurements. (8) (11) The off diagonal blocks Hqδ and HpV are neglected. The gain matrix is calculated with these approximations and the solution vectors are shown in eqs. (9) and (10). The constant decoupled gain matrixes are estimated at the flat start. (12) (13) The maximum bad data is limited by the maximum norm of the estimated states with and without bad data and it is expressed in eqn. (11). The largest possible breakdown point will be limited by the measurement redundancy. The outlier is represented as leverage point in regression which is located away from the rest of the measurements. By the elimination of leverage point, the system observability will not be affected. It has been identified by finding the hat matrix (K). The least square state estimate is evaluated using the eqn. (12). The measurement estimates are represented in eqn. (13). If the Kii 2( m ) then the particular measurements are suspected as leverage. The following condition creates the leverage measurements. • An injection measurement placed at a bus which is incident to a large number of branches. • An injection measurement placed at a bus which is incident to branches having very different impedance values. • Flow measurements of a branch having different impedances from other branches in the system. • Assigning large weights for specific measurements. The projection statistics is the robust measure of the effect of the leverage measurements and it is applied to power system state estimation. 1) M estimator: The robust state estimation is solved by M estimator which is introduced by Huber. M estimator is a maximum likelihood estimator as a function of the measurement residual f(r) subject to the measurement equation is a constraint. The state estimation problem is defined as follows. (9) (10) The decoupled SE has been widely accepted in the industry and implemented in control canters all over the world. The computation is fast and requires less memory. The decoupled SE drawbacks are described below.  The network parameters or operating conditions violate the stated decoupling assumptions in certain cases. The decoupled solution produces inaccurate solutions or may not converge.  The decoupling properties are not valid for the branch current magnitude measurements. Hence it cannot be included in decoupled SE. C. Robust State Estimation The major function of the state estimator is to detect, identify and eliminate errors in the measurements, network model and parameters. The state estimator is robust when the estimated states are insensitive to the deviations in a limited number of redundant measurements. The concept of © 2014 ACEEE DOI: 01.IJEPE.5.1.12 (16) (17) 12
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 Programming (LP). (20) (21) Fig.1 Huber Function The estimators are mainly designed for automatically detecting measurements with rapidly growing residuals and sup- pressing their influence on the state estimate. The measurement location, type and network parameters have a influence on the creation of the leverage points which have circuitous influence on the state estimate [26]. The objective function of the M estimate is chosen according to overcome the influence of all the above. Hence, the M estimator objective function should satisfy the following condition and it is represented in Fig. 1. The objective functions can be quadratic constant, quadratic linear, square root, schweppeHuber generalized-M, and least absolute valve. The solution methodology can be done by Newton’s method and iteratively re weighted least square technique.  f(r)=0 for r=0.  f(r) 0 for any r.  f(r) is monotonically increasing in both +r and -r directions.  It should be symmetric around r=0. Newton’s Method the Newton’s solution methodology is explained in steps. 1) Compute the gradient f’(r). 2) Compute xk = H -1 f 2 (r). 3) Compute αk using a line search. 4) Update the solution xk+1 = xk +αk xk . 5) Update the Hessian matrix H. 6) Go to step 1. Iteratively re-weighted least square estimation: Weighted Least Absolute Value The weighted least absolute value estimator minimizes the sum of the absolute values of the residuals and its cost function is given in eqn.(19). The LP problem can be solved by any well developed LP solution methods. These formulations have the advantages of the noise filtering and bad data elimination. Linear approximation of the power system equations are accomplished by replacing the non linear equations with their first order Taylor series approximation expanded at the operating point. By linearizing the nonlinear functions at each intermediate solution, a sequence of system states is generated which under suitable circumstances converges to an optimal solution of the original non linear programming problem. (22) (23) (24) The objective is to minimize the effect of measured noise in the evaluation of the system states. This method is quite fast and efficient. The residual should be below the prescribed value (ëi ). The in equality eqn. (22) is made into equality by adding the non negative slack variables and the eqs. (23) and (24) are obtained. The |r| in the objective function in eqn (49) can be replaced by ëi . The LAV estimate is given by the solution to the following linear programming problem, (25) (26) (18) Interior Point Method The basic steps nvolved in the interior point method is explained as follows. 1. Formulate the problem (19) (27) 2. Introduce the slack variables to make all the inequality where z is the set of the observed measurements, A is the constraints into nonnegative. vector that are linearly related with the measurement vector, e is the random measurement error, r= z - h(x)”x is the residual vector, Wi is the reciprocal of the measurement (28) error variance. The state estimate of WLAV can be described as optimization problem and formulated as Linear 13 © 2014 ACEEE DOI: 01.IJEPE.5.1.12
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 3. Replace the non-negative constraints with logarithmic barrier terms in the objective. (29) 4. Incorporate the equality constraints into the objective using (30) the next sampling time. Prediction capacities of DSE automatically ensure system observability even in the presence of measurement observable islands. Also predicted values help to identify topology errors, gross bad data and sudden change of states. The kalman filter is the best tool for dynamic state estimation. A. Kalman Filter based DSE The power system state and observation equations are represented in eqs. (42) and (40) respectively. xk+1 = Axk + ωk (39) (40) The steps involved in Kalman filter is follows. _ x ˆ˙ = f (xˆ, u, 0) + L(y - yˆ) zˆ = h(xˆ, u, 0) 5. Set all derivatives to zero. zk = H xk + υ k (43) (44) (31) (32) (33) 6. Rewrite the system (34) (35) (36) 7. Apply Newton’s method to compute the search directions x, w, y. (37) (38) 8. For solving w, use (38). Reduced Karush Khun Tucker (KKT) system. 9. III. DYNAMIC STATE ESTIMATION State estimators are performed by static approaches which have a single set of measurements to estimate the system states. The estimator has to be reinitialized for every new set of measurements without any state prediction from previous estimations. By the increase in power demand and deregulation market, the system state has to be monitored continuously for the better power system operation. The dynamic state estimators (DSE) have the capability of tracking the current system states and also predicting the state vector at 14 © 2014 ACEEE 12 DOI: 01.IJEPE.5.1. where x is the state vector, u is the input variable, z is the measured variable vector, ω, ë are the system noise and measurement noise respectively. xˆ is the estimated state vector, zˆ is the estimated measured vector, L is the observer gain. The non linear observer gain is depends on xˆ. The observer gain is shown in eqn. (45). The value od P is found using Ricctaci equation is represented in (46). (45) L = P C T R -1 AP + P AT + GQGT -P C T R -1 C P = 0 (46) The Q and R are the spectral densities of ω and respectively. In the optimal state estimate, the observer gain is also denoted as Kalman gain. The non linear observer is applied for finding the states of single machine- infinite busbar (SMIB) power system. In which, the system states are assumed as machine angle and the frequency deviation. The system states are estimated with and without observer gain and it has been compared with the result produced by machine angle synthesis method. It has been proved that the non linear observer provides better estimate than those methods. The discrete non linear observer is also designed for SE. The iterative sequential observer is also used for estimating the system states of SMIB power system. In which the observer gain is evaluated like Kalman gain in Kalman filter. B. Invariant Imbedding Method: The non-linear continuous time varying functions are defined as in eqs. (47) and (48). x(˙t) = f (x(t)) + G(t)ω(t) (47) z(t) = h(x(t)) + υ(t) (48) The system state estimation is determined by minimizing the objective functions J as in eqn. (49). The state estimate of this optimization problem is obtained by applying Pontryagin Maximum principle. (49) (50) (51)
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 (52) The Hamiltonian of this optimization is represented in eqn.(52). By minimizing the eqn. (52), the Euler-Lagrange function is applied. The eqns. (51) and (52) are obtained is estimated and shown in eqn. (55). The initial and final states are unknown. The transvarsatlity condition associated with the minimization of ë leads to the boundary value condition. Hence both Λ(t0 ) and Λ(tf ) are made equal to zero. The eqns. (53) and (54) defines a two point boundary value problem, the solution (59) • Block diagonal constant gain: (60) (61) (62) (53) • Full diagonal gain (decoupled δ and V): (54) (63) (55) In this paper, the single machine infinite busbar power system model is utilized. The machine state estimates are obtained in the hunting and out-of step region. (64) C. Tracking estimator Tracking state estimator is the one of the dynamic state estimation, which identifies the on line states with the changes occured in the measurements. The LAV based static state estimator is discussed which have the ability to incorporate the changes in the measurement set, operating states and the network configuration. The estimator handles the number of measurements at a time. Masiello and Scheweppe is proposed the tracking static estimator [36]. If the mathematical model of time structure of the state (x(tn )) and the observation error (υ(tn )) is known then the estimator could be designed well. The tracking estimator is started with the flat start (unity bus voltage magnitude and zero phase angle), which is a recursive estimator, not an iterative one and the related equation is shown in eqn. (56). In digital for, F(x(ˆtn))R-1[z(tn+1) - f(x(ˆtn))] is named as the feedback correction signal and P(tn) is feedback gain. The tracker has computed f(x(ˆtn), F(x(ˆtn) and P(tn) at every instant of time. The state estimate x(ˆtn) is found using eqn. (56). The major computation time of the recursive estimator is the finding out the in eqn. (56) and it is a full matrix. By the properly designing this as sparse matrix, the computation time has been reduced. The five categories has been proposed and explained in eqs. (58) - (64). • Full diagonal constant gain: Eqn. (64) is diagonalized with S(x0 ) The discretized state estimation is used for tracking the system states [37]. Both the bus and branch current measurements are sampled and the rectangularized formation was maintained. In addition to the system states, both observability and bad data analysis has been performed. D. Leapfrog algorithm The SE problem is formulated mathematically as follows. (65) (66) (67) where r(x)=z-h(x). The residuals are represented as quadratic constraint function in eqn. (65). The weight σi is the measurement error and wi is the down weights assigned for leverage measurements. Hence, t h e objective minimization is done by Gauss-Newton method or leapfrog method. The leapfrog method is also known as Euler forwardEuler back- ward method. The state estimation problem considers analogous the physical dynamical problem of the motion of the unit mass particle in a n-dimensional conservative method. The potential energy J of the particle is to be minimized. The method requires the solution of the particle’s equation of the subject to the initial position and velocity. Initially total • Best Gain: (58) • Full Constant Gain: © 2014 ACEEE DOI: 01.IJEPE.5.1.12 15
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 energy is the sum of kinetic and potential energy. By monitoring the kinetic energy, the inferring strategy is adopted such that potential energy is systemically reduced. The particle forced to move in the minimum potential energy position. The advantages of this method are listed below. • It uses only gradient information. There is no explicit line search performed. • It is very robust and it handles the discontinuous and function steep valley’s and gradients. • The algorithm searches the low local minimum. Hence it used it for global minimum optimization. the linear form of branch current using equation (69), which relates the line current with both ends of voltage phasor. When a PMU is placed at the particular bus, the corresponding element in a first block of matrix H becomes 1, otherwise zero. PMU Branch current measurements (I) and their relation with voltage phasor are shown in eqn. (69). A typical π model of the network is shown in Fig. 2, gij +jbij is the series branch of the transmission line and gsi + jbsi are shunt branch of the transmission line. (70) IV. PMU BASED STATE ESTIMATION The concept of voltage phasor for static state estimation is presented in [1]. Recently the availability of synchro phasors has made it possible to incorporate phase angle also into the measurement set. A phasor based state vector is preferable than SCADA based SE. In many cases, the number of Phasor Measurement Unit (PMU) requirements to meet sufficient measurements poses a problem. Due to the factor of price, technology, communication ability the PMU cannot be implemented at all buses in the system. In SCADA based SE the inclusion of power measurements makes the problem of state estimation highly non-linear, leading to iterative procedure. Hence it is highly time consuming and infeasible for on-line implementation. The Linear State Estimation (LSE) is done with synchronized phasor measurements which overcomes the drawbacks of SCADA based SE. In LSE, the current and voltage phasors are used in the measurement set vector. This makes the problem of SE linear and hence a non-iterative procedure can be used for estimating the states. This also helps in ease of on-line implementation. The measurement vector z is expressed in (68) formed by bus voltage and line current phasors collected by PMU. (71) The measurement equation is presented in rectangular form as z = zA + jzB ; x=E+jF, using which it can be rewritten in matrix notation as, where Hr , Hm are the real and imaginary parts of Hessian matrix (Hnew ). The WLS state estimates can be obtained using eqn. (71). This method is very simple and needs no iterative procedure as the H matrix is linear. The matrix S converting the measurements to the state estimate is constant as long as there is no change in the network topology. This technique is suitable for online implementation except for the time delay in communication. The phasor measurements in SE are implemented in Sevillana de Electricidad, which shows that the more accuracy in state estimates [40]. The power system is clustered into multi area and the PMU measurements are used for SE which is the replication of a real power system monitoring. Multi area SE produces the best accurate estimates and bad data measurements are added to the sufficient numbers of SCADA measurements, the accuracy of the estimate is much improved through the iterative procedure. But measurements are obtained in different time frames. The angle bias correction is done in order to improve the existing SCADA based SE in addition with PMU measurements. (68) where, H is the matrix of linear equations and e is an error vector. The eqn. (68) can be expanded as follows, V. SIMULATION STUDIES The static state estimations are performed with SCADA and PMU measurements which are named as traditional and linear state estimation respectively. The results have been demonstrated on simple four bus system and IEEE 14, 57 bus test systems. For each system, the true values of measurements were generated using load flow. The noise was added to the true value assuming a Gaussian distribution with zero mean. Here, the error of TSE measurements follows the normal distribution with zero mean and standard deviation of 0.004 for voltage magnitudes, 0.01 for power injections and 0.008 for power flows, while SD of real and imaginary value of PMU measurements are taken as 0.0001. 4 bus system: A simple 4 bus test system is comprising 2 generators and loads. The PMU is located at nodes 1 and 4. where, V is the voltage phasors, I is the branch current phasors. r and im denotes the real and imaginary parts of the phasors respectively. N is the number of buses and NL is the number of branches. The notation k1 and k2 obtained from (69) © 2014 ACEEE DOI: 01.IJEPE.5.1.12 16
  • Full Paper ACEEE Int. J. on Electrical and Power Engineering , Vol. 5, No. 1, February 2014 Table 2 gives the comparison of estimated voltage magnitude and angle using both TSE and LSE. It is observed that the estimated voltage magnitudes of the LSE method are almost equal to the actual values. The PMU measurements thus provide an accurate system state. PMUs located at the various nodes. The comparison is made between TSE and LSE in terms of estimated bus voltage magnitude and angle from the true value. Fig. 2 explains the bus voltage magnitude error. LSE produces very little error in bus voltage magnitude. TABLE II. C OMPARISON OF ESTIMATED SYSTEM STATES FOR 4 BUS SYSTEM Bus No TSE Approach V(pu) 1 2 3 4 True Value V (pu) 0.9988 0.9397 0.9420 0.9454 1.0000 0.9414 0.9423 0.9515 Ang (deg) 0.0000 -0.533 -1.631 2.6365 Ang (deg) 0 -0.522 -1.612 -2.631 LSE Approach V(pu) Ang (deg) 0.9982 0 0.9402 -0.532 0.9422 -1.613 0.9520 2.624 IEEE 14 bus system: The single line diagram and bus/branch details are given in Appendix A. In TSE, the measurement set consists of voltage magnitudes, forward/reverse real power flows, forward/reverse reactive power flows and real/reactive power injections. All branch current and few bus voltage phasor measurements are considered as measurements in LSE. The PMU located nodes are 2,6,7,9,11,12,14. Table 3 shows the comparison among the TSE and LSE in terms of estimated system states. The simulation results show the estimated bus voltage magnitude and angle along with the true value. The TSE have some differences in magnitude and angle. But LSE exhibits very minute error in both magnitude and angle. The TSE usually chooses the slack bus as a reference bus for the entire system. Synchro phasor measurements may have a different reference, which is determined by the instant synchronized sampling initiated. The results would be different if the reference problem is not considered. In order to adjust the angle of the slack bus (bus no.1 for IEEE 14), a PMU is installed in the slack bus, which is taken as reference for the TSE. Once the PMU is installed in SB, the particular estimated angle can be directly deducted from all buses. T ABLE III. C OMPARISON OF ESTIMATED SYSTEM STATES FOR IEEE 14 Fig. 2 Errors in the estimated bus voltage magnitude error for IEEE 57 bus systems Fig.3 presents bus spread angle error for IEEE 57 bus system. The angle error for LSE is negligible. Hence more accurate estimate or value of system states is obtained with synchronized phasor measurements. BUS SYSTEM Bus No TSE Approach V(pu) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 True Value V (pu) 1.046 1.032 1.010 1.002 1.003 1.063 1.051 1.082 1.046 1.042 1.046 1.045 1.044 1.029 1.060 1.045 1.010 1.018 1.020 1.070 1.061 1.090 1.054 1.049 1.056 1.055 1.050 1.034 Ang (deg) 0 -4.981 -12.71 -10.32 -8.781 -14.24 -13.35 -13.35 -14.93 -15.09 -14.80 -15.09 -15.17 -16.04 Ang (deg) 0 -4.834 -11.07 -10.21 -8.953 -15.47 -14.77 -15.25 -16.04 -16.48 -16.38 -16.56 -16.52 -17.46 LSE Approach V(pu) Ang (deg) 1.060 0 1.045 -4.979 1.010 -12.71 1.018 -10.32 1.020 -8.780 1.070 -14.24 1.061 -13.35 1.090 -13.35 1.054 -14.93 1.049 -15.09 1.056 -14.80 1.055 -15.09 1.050 -15.17 1.034 -16.03 Fig. 3 CONCLUSIONS IEEE 57 bus system: To demonstrate both traditional and linear state estimation, the results are displayed for the lower order systems. IEEE 57 test system is chosen to ensure effective working of the higher order systems. There are 25 © 2014 ACEEE DOI: 01.IJEPE.5.1.12 Errors in the estimated bus angle error for IEEE 57 bus systems The article presented a classified list of algorithms for state estimation in approximately 40 years of extensive 17
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