State Estimation

942 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 2, MAY 2012

An Alternative Method for Power System DynamicState Estimation Based on Unscented Transform

Shaobu Wang, Wenzhong Gao, Senior Member, IEEE, and A. P. Sakis Meliopoulos, Fellow, IEEE

Abstract—An efficient, timely, and accurate state estimation isa prerequisite for most energy management system (EMS) appli-cations in power system control centers. The emerging wide-areameasurement systems (WAMSs) offer new opportunities for devel-oping more effective methods to monitor power system dynamicsonline. Recently, alternative methods for power system state es-timation have caught much attention. Due to the nonlinearity ofstate transition and observation equation, linearization and Jaco-bian matrix calculation are indispensible in the existing methodsof power system state estimation. This makes WAMS’ high per-formance compromised by burdensome calculation. In order toovercome the drawbacks, this study tries to develop an effectivestate estimation method without the linearization and Jacobianmatrix calculation. Firstly, unscented transformation is introducedas an effective method to calculate the means and covariances of arandom vector undergoing a nonlinear transformation. Secondly,by embedding the unscented transformation into the Kalman filterprocess, a method is developed for power system dynamic state es-timation. Finally, some simulation results are presented showingaccuracy and easier implementation of the new method.

Index Terms—Nonlinear filter, power systems dynamics, state es-timation, unscented transform, wide-area measurement systems.

I. INTRODUCTION

I N electric power systems, an efficient, timely, and accuratestate estimation is a prerequisite for most energy manage-

ment system (EMS) applications in control centers. However,due to the slow updating rate of SCADA systems (on the orderof several seconds), traditional state estimators based on steadystate system model cannot capture the dynamics of powersystem very well. Since the 1990s, phase measurement units(PMUs)-based wide-area measurement systems (WAMSs) hasemerged, featured with synchronous sampling and high dataupdating rates. The emerging WAMSs evoke many researchers’interests on developing more effective state estimation methodsbased on WAMS to monitor power system dynamics [1]–[17].This is also important for future smart grid to ensure efficientand reliable operations and controls of the grid. Recently,alternative methods for power system state estimation have

Manuscript received May 02, 2011; revised August 29, 2011 and October 04,2011; accepted November 02, 2011. Date of publication December 26, 2011;date of current version April 18, 2012. Paper no. TPWRS-00404-2011.

S. Wang is with the Center for Energy Systems Research, Tennessee Techno-logical University, Cookeville, TN 38501 USA.

W. Gao is with the Department of Electrical and Computer Engineering, Uni-versity of Denver, Denver, CO 80208 USA (e-mail: [email protected]).

A. P. S. Meliopoulos is with the School of Electrical and Computer En-gineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2011.2175255

been studied; for example, the feasibility of applying extendedKalman filter (EKF) to dynamic state estimation is discussedin [6]; supercalibrator approaches are proposed in [8]–[11]enabling the state estimation to be performed at the substationlevel and then simply synthesize the system state at the controlcenter. However, because power system state equations are non-linear and some measurement equations are nonlinear functionsof state variables, linearization and Jacobian matrix calculationare indispensable in the above methods. Sometimes virtual- andpseudo-measurement technology must be introduced to accesscertain variables of a generator such as its rotor angle andinternal voltages— . For example, when we usethe measured output voltage and injected current of a generatorto solve some observation equations to get those variables,the observation equations are nonlinear with respect to theabove variables. Although the model quadratization process isintroduced to eliminate the third order and higher truncationerrors in [8] and [12], the computation of Jacobian matrix ofobservation equations is still necessary and the second ordertruncation error still exists.

It is well known that the linearization and Jacobian matrixcalculation can lead to the following drawbacks:

• Linearization is only reliable if the higher order terms in theTaylor series expansion can be ignored. If this conditiondoes not hold, the linearized approximation can be poor.As a result, linearized transformations may produce highlyunstable filtering performance if time-step intervals are notsufficiently small.

• Derivation of Jacobian matrices may be nontrivial. Lin-earization can be applied only if the Jacobian matrix exists.However, this is not always the case.

• Computation demanded for the generation of the Jacobianmatrices and the predictions of state estimation and covari-ance is large.

In order to overcome the above drawbacks introduced by lin-earization and Jacobian matrix calculation, we try to developa more effective dynamic state estimation method without lin-earization or calculation of Jacobian matrices. In this paper,power systems are still modeled as systems with nonlinear statetransition and observation models; and a deterministic samplingtechnique known as the unscented transform (UT) [19]–[22]is used to calculate the mean and covariance of the nonlinearfunctions of state transition and observation models. The per-formance benefits are demonstrated in an example applicationshowing the ease implementation and more accurate estimationfeature of the new method.

This paper is organized as follows. The theory of unscentedtransformations is introduced in Section II. A new filter algo-rithm based on the unscented transformation is described in

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WANG et al.: ALTERNATIVE METHOD FOR POWER SYSTEM DYNAMIC STATE ESTIMATION 943

Section III. A new effective state estimation method withoutlinearization and Jacobian matrix calculation is developed inSection IV. Then a detailed study-case is presented to demon-strate the effectiveness of our method in Section V. Specially,the method performance is tested under different disturbancesand data update rates. Finally, some conclusions are drawn inSection VI.

II. UNSCENTED TRANSFORMATION

UT is an effective method for calculating the statistics of arandom variable undergoing a nonlinear transformation. Sup-pose that is an -dimensional random variable with mean

and covariance another random variable, isrelated to through a nonlinear function

(1)

where is a vector of nonlinear functions. With the unscentedtransformation method, the mean and covariance of arecalculated through the following steps.

1) Use (2) to yield points (or sigma points)

(2)

(3)

where is the th row or column of thematrix square root of is the weight which isassociated with . Here

is a scaling factor; is another parameter of the method,and can be chosen as if is Gaussian [20],[22]; the matrix square root of positive definite matrixmeans that a matrix exists such that

.

2) Instantiate each point through the nonlinear function toyield the set of transformed sigma points as shown in

(4)

3) Calculate the mean of by (5), where is defined by(3):

(5)

4) Calculate the covariance of using

(6)

As can be seen from the above steps, the mean and covari-ance are calculated using standard vector and matrix operations;it is not necessary to evaluate the Jacobians which are usuallyneeded in existing state estimation of nonlinear systems. Thismeans that the algorithm can be implemented rapidly and suit-able for online application. Moreover, the algorithm can guar-antee a higher order of accuracy than traditional linearizationalgorithm [20]–[22]. The more detailed derivation and proper-ties of the unscented transformation algorithm can also be foundin [20]–[22].

III. FILTERING ALGORITHM BASED

ON THE UNSCENTED TRANSFORMATION

In the previous section, we introduced a method for deter-mining the mean and covariance of a random vector that un-dergoes a nonlinear transformation. The method is superior tolinearization methods in many important respects [20]–[22]. Inthis section, we describe how UT is embedded into the recursiveprediction and update structure of Kalman filter.

A system with nonlinear functions of state transition and ob-servation models is shown in (7):

(7)

where is the state variable at the time stepis the measurement at the time step and are vectors con-sisting of nonlinear functions; is the Gaussianprocess noise at the time step , and is theGaussian measurement noise at the time step and arecovariance of and at the time step . Using UT describedin the previous section, the prediction and update steps of thenew filter algorithm can be formulated in standard matrix oper-ations [23] as follows:

1) Prediction: according to (2)–(6), use as the statemean to yield sigma points and calculate the predicted statemean and the predicted covariance shown in (8)at the bottom of the page, where isdefined in (3), is an matrix, is an

-dimensional vector.2) Update: according to (2)–(6), use as the state mean

to yield sigma points and calculate the predicted meanand covariance of the measurement, and the cross-co-variance of the state and measurement shown in (9),

(8)


where is defined in (10), is an ma-trix, the operation of is similar toin (8). Here is a parameter of the method, and if

is Gaussian [19]. Then calculate the filter gain andthe updated state mean and covariance as shownin (11):

(9)

(10)

(11)

IV. ONLINE DYNAMIC STATE ESTIMATION

METHOD FOR POWER SYSTEMS

In this section, we describe how the new filtering method asintroduced in the previous section is used for power system dy-namic state estimation. Power systems can be modeled as a setof differential-algebraic equations given in (12):

(12)

where is a set of nonlinear functions of state transition;is a set of nonlinear observation equations; is a set of alge-braic equations, representing the passive network of the powersystem. In this paper, loads are modeled as constant impedancesand is formulated as a nodal equation with a nodal admittancematrix. Given can be solved from function .

Applying numerical integration scheme (e.g., the trapezoidalmethod), we discretize (12) and convert it into algebraic formsat each time step. The filtering process shown in Section III canbe incorporated into power system state estimation as follows:

1) According to (8), use as the state mean to yieldsigma points and calculate the predicted state meanshown in (13):

(13)

2) Calculate by solving

(14)

3) According to (9), use as the state mean to yield sigmapoints and calculate the predicted mean and covariance

Fig. 1. Single machine infinite bus system.

of the measurement, and the cross-covariance of thestate and measurement given as follows:

(15)

4) Calculate the filter gain and the updated state meanand covariance given as follows:

(16)

Remark: The update rate of traditional SCADA systems isabout 4 to 5 s. This means that the in (13) is about 4 s. Asa result, dynamic state estimation based on SCADA is prone todivergence. However, PMUs can send their data at a rate up to240 frames per second, which lays the basis for the implemen-tation of dynamic state estimation in power systems.

V. CASE STUDIES

A. Case 1

In this subsection, we compare our method with other lin-earization-based methods. In fact, the new method’s advantagescompared with other linearization methods have been studiedin [20]. Here, we show the advantages through a power systemcase. For the sake of clarity, we use a single machine infinite bussystem to compare our method with the method in [6] which isbased on the linearization. A multi-machine case will be studiedin the next subsections.

A single machine infinite bus system is shown in Fig. 1 andthe system dynamic equations are given in (17) wherep.u., kWs/kVA, p.u., p.u.,

p.u., rad/s.The system measurement equations are given in (18) whereand represent measurement noise. It can be seen from (17)and (18) that both the state transition and observation equationscontain nonlinear term :

(17)

(18)

Letting the right side of (17) be equal to zero, we getrad, as an equilibrium of the system.

Givenand s in (13), (15), and


Fig. 2. Estimated results from the method in [6].

Fig. 3. Estimated results using the new method.

Fig. 4. Comparison of estimation errors of the two methods.

(16), we test the two methods’ performance by tracking systemdynamics around the equilibrium, respectively.

By introducing a small increment to the equilib-rium, we get as initial valuesaround the equilibrium. The system oscillations derived fromthese initial values are shown in Figs. 2 and 3. The solid line inFig. 2 represents actual values; and the dashed line is the esti-mated result from the method in [6]. In Fig. 3, the solid line rep-resents actual values; and the dashed line is the estimated resultfrom the method in this paper. Fig. 4 gives the comparison of es-timation errors of the two methods. Both methods use the samemeasured values with noises that are shown in Fig. 5 in which

Fig. 5. Two measured variables with noises.

Fig. 6. Estimated result from the method in [6].

Fig. 7. Estimated result from the new method.

a Gaussian noise with a zero mean and variance of 0.00001 p.u.to the measured electric power, and another Gaussian noise witha zero mean and variance of 0.0001 rad/s to the measured rotorvelocity . It can be seen from Figs. 2–4 that both methods canwork well.

Now we introduce a larger increment to the equi-librium. As a result, we getas initial values around the equilibrium. The system oscillationsderived from these initial values are shown in Figs. 6 and 7. Thesolid line in Fig. 6 represents actual values; and the dashed lineis the estimated result from the method in [6]. In Fig. 7, the solidline represents actual values; and the dashed line is the estimated


Fig. 8. Comparison of estimation errors of the two methods.

Fig. 9. Two measured variables with noises.

result from the method in this paper. Fig. 8 gives the compar-ison of the estimation errors of two methods. Both methods usethe same measured variables with noises are shown in Fig. 9 inwhich a Gaussian noise with a zero mean and variance of 0.001p.u. added to the measured electric power, and another Gaussiannoise with a zero mean and variance of 0.01 rad/s added to themeasured rotor velocity.

It can be seen from Fig. 6 that a phase shift (lag) emergesbetween the actual values and the estimated values from themethod in [6], while Fig. 7 shows that the new method canstill work well. This lag, when the estimated values are usedas feedback signals, can deteriorate controller performance, oreven threat stability of closed-loop systems [24]–[26].

B. Case 2

In this section, WSCC three-machine nine-bus system [27]shown in Fig. 10 is studied as a case to test the proposed method.In this example system, the dynamics of generators are repre-sented with a 4th order model. The state variables are rotor angle, velocity , and internal voltages and . Each generator

has its own excitation system which is modeled with state vari-ables: . The entire system has 21 state variables, andthe state transition equations are given in (19) wherecorresponding to the three generators, respectively. The relatedparameters can be found in [27]. Now we discuss measurementsfor this case study.

Fig. 10. WSCC three-machine nine-bus system.

Fig. 11. Pseudo-steady state phasor diagram for synchronous generators.

We noticed that all generators are non-salient pole generators.Therefore, we have and generator internal voltagecan be calculated by where and are gen-erator terminal voltage and injection current that can be mea-sured by PMUs directly. Similarly, we can obtain rotor angleby , and then we can get angle between

and , which is shown in Fig. 11. Actually, “the latestPMU released in 2006 can measure excitation voltage, excita-tion current, valve position, output of PSS, etc.” [28]. Thereforestate variables of excitation system can be measured directly orindirectly.

Based on above analysis, we choose electrical power ,rotor velocity , generator internal voltage , angle

(indirectly measured) as measurements for eachgenerator, and the measurement equations are given in (20)where and are algebraic vari-ables depending on predicted state variable etc.;

, represent measurementnoises; . Now we test our method by applying adisturbance as follows.

Now a three-phase metallic short circuit fault is configuredon the line between bus #4 and #5 at the moment ofs, near bus #4. The fault is cleared after 100 ms via opening ofthe circuit breakers at both ends of the line. Then the systemoperation mode is changed with loss of the line between bus #4and #5. Following the disturbance, we get the measured valuesshown in Figs. 12–14 by adding a Gaussian noise to the timedomain simulation results. All the noises have a zero mean and


Fig. 12. Measured values for � .


0.01 p.u. variance for electric power; 0.00001 p.u. variance forrotor velocities; 0.0001 p.u. variance for ; 0.001 rad for ;


0.1 p.u. variance for ; 0.1 p.u. variance for ; 0.1 p.u. vari-ance for :

(19)

Given

in (13), (15), and (16), we use the measuredvalues with noise to estimate the actual values of statevariables through the new method. The results are given inFigs. 15–18 in which the solid lines represent the actual valuesand dashed lines represent the estimated values. Thedynamics of the 2-norm of the Kalman gain matrix isgiven in Fig. 19 from which we can see the Kalman gaindecreases as time increases, which indicates a convergent


Fig. 15. Estimation results for rotor angle difference of the system.

Fig. 16. Estimation results for � .

filter. It can be seen from Figs. 15–18 that the estimatedvalues match well the actual values:

(20)

C. Sensitivity to Different Data Update Rates

In the previous subsections, the simulation results are ob-tained under the condition that the data update rate is 10 ms.Due to the following two reasons, the method performanceunder different data update rates need to be investigated:1) Sometimes missing measurement may happen in powersystem, which means that the filter process cannot obtaintimely update from measurement. Temporary faults in mea-surement are reflected by intermittent data update in the filterprocess, which is represented by different data update rates.2) In real power systems, PMUs send their data at a rate up to240 frames per second with most typical rates being 10, 30,or 60 frames per second (for 60-Hz systems). This means thatthe update rate in control centers is from 4 ms up to 100 ms.



Fig. 19. Convergence of the 2-norm of Kalman gain matrix.

Therefore the method’s sensitivity to different data update ratesis worth further investigation.

Now we continue to study Case 2. Given the noises asshown in Figs. 12–14 and the same disturbance, the methodperformance under different data updating rates are given in


Fig. 20. Performance under the data update rate of 60 ms.



Figs. 20–22 in which the solid lines represent actual values andthe dashed lines represent the estimated values. From thesefigures, we can see that the method is still able to maintain agood performance for update rate up to 80 ms for this casesystem; the method starts to show deteriorated performance forupdate rates beyond 100 ms.

D. Bad Data Detection

Detecting and handling the effects of gross measurementerrors is another important task in power system state estima-tion. The unscented filter identifies the gross errors througha normalized innovation vector. Each element of the vectorcorresponds to a unique measurement. The th measurementcorresponds to the th element shown in (21) where subscript

represents the th time step; subscript represents the thmeasurement; is the th diagonal element of . Theother symbols are the same as defined in previous sections.The results from [29] show that the measurement should bediscarded if is greater than 1.5:

(21)

By letting in (21), now we test if this index can workwell by adding a gross measurement error 0.05 p.u. to the mea-sured rotor velocity of generator No. 1 at s. We get forthis measurement shown in Fig. 23 from which we can see thegross error can be detected.

Fig. 23. Dynamics of � .

VI. CONCLUSION

This paper’s main idea is to develop a method to handlethe nonlinear equations in power system state estimation sothat linearization and Jacobian matrix computation can beavoided, which is necessary in traditional power system stateestimation methods. We propose the use of unscented transformto calculate the mean and covariance of nonlinear functionsof random variables (which represent power system measure-ments as nonlinear functions of the power system state). Inthe new estimation method based on the unscented transform,the linearization and evaluation of Jacobian matrices are notnecessary anymore, and only standard vector or matrix opera-tions are involved. Therefore, the method can be implementedrapidly using WAMS’ higher data update rates. The methodhas been demonstrated with a number of examples. Comparedwith traditional state estimation methods, the results from thenew method demonstrate better performance in computationefficiency, accuracy, and convergence. The results and evalu-ation of the method indicate a promising outlook for onlineapplication.

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Shaobu Wang was born in Shandong Province,China. He received the Ph.D. degree in electricalengineering from Zhejiang University, Hangzhou,Zhejiang province, China, in 2009.

From March 2009 to March 2010, he worked as apostdoctoral fellow in the Department of Electricaland Computer Engineering, University of Alberta,Edmonton, AB, Canada. From April 2010 to April2011, he worked as a postdoctoral research asso-ciate in the Center for Energy Systems Research,Tennessee Technological University, Cookeville,

TN. From May 2011 to July 2011, he worked as a research scholar in theDepartment of Electrical and Computer Engineering, University of Denver,Denver, CO. He is currently with the Pacific Northwest National Laboratory,Richland, WA. His research interests include PMU-based stability analysis andcontrol of power systems; renewable energy integration.

Wenzhong Gao (S’00-M’02-SM’03) received theM.S. degree and Ph.D. degree in electrical andcomputer engineering specializing in electric powerengineering from Georgia Institute of Technology,Atlanta, in 1999 and 2002, respectively.

His current teaching and research interests includerenewable energy and distributed generation, smartgrid, power delivery, power electronics application,power system protection, power system restruc-turing, and hybrid electric propulsion systems.

Dr. Gao is a Member of the Power and Energy Ed-ucation Committee (PEEC) of the IEEE Power and Energy Society (PES). Heis an Editor for the IEEE TRANSACTIONS ON SUSTAINABLE ENERGY.

A. P. Sakis Meliopoulos (M’76–SM’83–F’93) wasborn in Katerini, Greece, in 1949. He received theM.E. and E.E. diploma from the National TechnicalUniversity of Athens, Athens, Greece, in 1972, andthe M.S.E.E. and Ph.D. degrees from the Georgia In-stitute of Technology, Atlanta, in 1974 and 1976, re-spectively.

In 1971, he worked for Western Electric in At-lanta. In 1976, he joined the Faculty of ElectricalEngineering, Georgia Institute of Technology, wherehe is presently Georgia Power Distinguished Pro-

fessor. He is active in teaching and research in the general areas of modeling,analysis, and control and protection of power systems. He has made significantcontributions to power system control and operation, grounding, harmonics,and reliability assessment of power systems. He developed the distributed stateestimator. He applied statistical estimation methods for improved groundingsystem measurements and power quality measurement methods. The work inthis area led to the Smart Ground Multimeter (U.S. patent) and a commerciallyavailable device for grounding system performance assessment. He developed(with Dr. Cokkinides) the first GPS-synchronized harmonic measurement andestimation system, a wide area monitoring system of power quality problemsfor high voltage transmission systems. His present research activities focuson using this technology for advanced and precise monitoring, visualization,and control of power systems. He holds three patents and has published over200 technical papers. He is the author of the books, Power Systems Groundingand Transients (New York: Marcel Dekker, 1988), Lightning and OvervoltageProtection, Section 27, Standard Handbook for Electrical Engineers (NewYork: McGraw Hill, 1993), and the short book (100 pages) Problems andConcise Theory of High Voltage Systems, in Greek.

Dr. Meliopoulos is a member of the Hellenic Society of Professional Engi-neering and the Sigma Xi.

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