a review of power system dynamic state estimation...

A Review of Power System Dynamic State Estimation Techniques

by

Shivakumar N. R., Amit Jain

in

Power System Technology and IEEE Power India Conference, 2008. POWERCON 2008. Joint InternationalConference on 12-15 Oct. 2008 Page(s):1 to 6, Digital Object Identifier 10.1109/ICPST.2008.4745312

Report No: IIIT/TR/2009/43

Centre for Power SystemsInternational Institute of Information Technology

Hyderabad - 500 032, INDIAMarch 2009

A Review of Power System Dynamic StateEstimation Techniques

Shivakumar N. R., and Amit Jain, Member, IEEE

Abstract-- State estimation is a key Energy ManagementSystem (EMS) function, responsible for estimating the state ofthe power system. Power system is a quasi-static system andhence changes slowly with time. Since state estimation iscomputationally expensive, it is not easy to execute it repetitivelyat short intervals to achieve real time monitoring of such a

changing system. Dynamic State Estimation (DSE) techniquesmodel the time varying nature of the system, which allows it topredict the state vector in advance. This proves to be a majoradvantage for the operator in performing security analysis andother control center functions. Various techniques for dynamicstate estimation are available in the literature. This paper

presents a bird's eye view on different methodologies anddevelopments in DSE, based on our comprehensive survey of theavailable literature.

Index Terms-- dynamic state estimation, kalman filter, realtime monitoring, square root filter, static state estimation

I. INTRODUCTION

AS the power system grows larger and more complex, realtime and monitoring and control becomes very

significant in order to achieve a reliable operation of thepower system. The Energy Management System (EMS)functions are responsible for this task of monitoring andcontrol. State estimation forms the backbone of the energy

management system by providing a database of the real timestate of the system for using in other EMS functions [1].Hence, an efficient and accurate state estimation is a pre

requisite for an efficient and reliable operation of the power

system.The vector consisting of bus voltage magnitudes and phase

angels is called the state of an electric power system. Eversince the concept of state estimation was introduced in to thefield of power systems, by Schweppe et al [2]-[4], in the early1970s, many methods have been proposed to calculate thestate vector of the power system. The state estimation for a

power network involves collecting the real time measurementdata, which includes line flows, injection measurements and

Shivakumar. N. R is a research student in the Power Systems ResearchCenter at International Institute of Information Technology (IIIT),Gachibowli, Hyderabad, India. (e-mail: shivakumarnrgresearch.iiit.ac.in)

Amit Jain is an Assistant Professor with the Power System ResearchCenter at the International Institute of Information Technology (IIIT),Gachibowli, Hyderabad, India (e-mail: amitX iiit.ac. in).

978-1-4244-1762-9/08/$25.00 C2008 IEEE

voltage measurements, through the SCADA and calculatingthe state vector using a predefined state estimation algorithm.If the state vector is obtained for an instant of time 'k' fromthe measurement set for the same instant of time, then such anestimation technique is called the Static State Estimation(SSE). In order to know the state of the power systemregularly, this process of calculating the state vector isrepeated at suitable intervals of time. Static state estimatorsare widely used in power systems and play a very importantrole for the reliable operation of the transmission anddistribution systems.Under normal conditions, power system is said to be a

quasi-static system and hence changes slowly but steadily.These changes in the power system are driven by the loads.As the loads in the system change, the generations also haveto be adjusted accordingly, which in turn changes the flowsand injections across the system and hence making entiresystem dynamic. Therefore, in order to have a continuousmonitoring of the power system, state estimation must beperformed at short intervals of time. But as the power systemexpands, with addition of generations and loads, the systembecomes extremely large for the static state estimation to becarried out at short intervals of time as it consumes heavycomputing resources. Hence, static state estimators may notefficiently capture this dynamic behavior of the power system.This lead to the development of another set of algorithmscalled the "Dynamic State Estimation" (DSE) techniques,where the actual physical modeling of the time varying natureof the power system is used. These algorithms have dualadvantages of being more accurate and possessing the abilityto predict the state of the system one step ahead. That is, fromthe knowledge of the state vector at an instant of time "t", andthe mathematical model of the system, the DSE techniquespredict the state vector of the power system at the next instantof time "t+1". This forecasting ability has tremendousadvantages, as security analysis can now be performed onetime stamp ahead and hence allows more time for the operatorto take control actions, especially in cases of any emergency[5]. Hence, DSE algorithms for power systems form animportant branch of power system state estimation techniques,with a potential to impact the very nature of operation of thereal time monitoring and control of power systems.

This paper presents a review of the developments indynamic stat estimation techniques based on our survey ofresearch papers available in literature. Such a survey ofdynamic state estimation techniques will be of greatimportance to the power engineers working on these topics.

Authorized licensed use limited to: INTERNATIONAL INSTITUTE OF INFORMATION TECHNOLOGY. Downloaded on February 3, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

We have done a comprehensive survey of the literatureavailable on the dynamic estimation techniques and in thispaper an attempt has been made to give an overview of thevarious algorithms, their applications and future trends, onthese topics.

II. DYNAMIC STATE ESTIMATION

An easy way of following the changes in a power systemon a real time basis is by using the state estimation techniqueson a continuous basis with a certain time step. But thesetechniques, though computationally very efficient, do not useany physical modeling of the time varying nature of the powersystem, and hence may not be as accurate as desirable.Dynamic state estimation uses the present (and sometimesprevious) state of the power system along with the knowledgeof the system's physical model, to predict the state vector forthe next time instant. This prediction feature of the DSEprovides vital advantages in system operation, control, anddecision making. It allows the operator more time to act incases of emergency, helps in detection of anomalies, bad dataetc [5].

The main steps involved in DSE algorithms to achieve anoptimized estimate of the state vector are presented in thefollowing subsections. Through out the discussion, 'k' is usedto suggest the present instant of time and 'k+1' to suggest thenext instant of time.

A. Mathematical ModelingThis is the first step in DSE and involves the

identification of correct mathematical model for the timebehavior of the power system. The general mathematicalmodel used for a dynamic system is given by [1], [6]:

Xk+l ff(Xk, Uk,Wk, k) ()

Where, 'k' is the time sample, 'x' is the state vector, 'u' is thecontrol actions, 'w' represents the uncertainties in the modeland 'f represents the nonlinear function. But such a model isextremely complex, costly and impractical. Hence, certainassumptions are made to ease the implementation (some havealready been mentioned earlier). They are:* The system is quasi static and hence changes extremely

slowly* Time frames considered are small enough, for the usage of

linear models to describe the transition of states betweenconsecutive instants of time

* The uncertainties are described using white Gaussian noisewith zero mean and constant covariance QConsidering these assumptions, we can obtain a generic

linear model for the DSE as:xk+ =Fx +Gk+wk (2)

Where Fk is the function representing the state transitionbetween two instants of time, Gk is associated with the trendbehavior of the state trajectory and wk is the white Gaussiannoise with zero mean and covariance Q [1], [6].

B. Parameter IdentificationThis step involves the calculation of various parameters

mentioned in the mathematical model of the DSE. Fk, Gk and

dynamic model shown in (Eq. 2). Debs and Larson [6],credited with the seminal paper on DSE, and Nishiya et al [7]have assumed a simple linear model with Fk assumed to be an

identity matrix and Gk assumed to be zero. But this makes theestimator very simplistic and hampers the forecasting abilityof the estimator [1]. In the model proposed by Debs andLarson [6], the change in state vector is considered to be so

small that it is replaced by a zero mean, white Gaussian noise.Hence, the equation (11) reduces to:

Xk+l = Xk + Wk (3)

Other authors in [8], [9], [10] etc have also used similarmathematical models for describing time update of the statevector.

Linear Exponential Smoothing (LES) technique has alsobeen found to be a widely used technique for this purpose.

Authors in [5], [10], [11] and [12], [13] [14] have all used theHolt's Exponential Smoothing technique [15] to obtain thevalues of Fk and Gk, based on the previous values of the statevariables. In this case the mathematical model used is of theform:

Xk+l FkXk +Gk+Wk (4)

The next parameter to be identified is the error covarianceQ. Under normal operating conditions of a quasi static powersystem, the Fk and Gk are adjusted such that the value of Qalmost remains constant or varies within a very small range ofvalues or at a constant noise level. This value of Q can beobtained by offline simulation studies. In general the vale ofQis taken to be extremely low, in the order of 10-6 [1] [5].

The other parameter to be modeled is the measurementfunction, which helps to observe the system. The SCADAsystem measures the network data and continuously transmitsthe data over a communication medium to the centralmonitoring station. The measurement data includes the activeand reactive power flows, power injections and voltagemagnitude at the buses. Let the measurements so obtained berepresented by 'Z. The line flows, injections and loads have a

nonlinear relationship with the state of the system. Hence,given the state vector of a system, the measurement vector can

always be calculated back. Such a calculated vector is givenby h(x) and is a nonlinear function of the state vector. Hence,the actual measurement vector (Z) can be represented as, thesum of the calculated measurement vector (h(x)), and an error

vector v, as shown below. This is given as:

Z = h(x) + v (5)Equation (5) is a nonlinear equation. But as we know,

solutions to only linear equations can be found explicitly.Hence, most of the techniques linearise the abovemeasurement model and use the equation after linearization as

shown in (Eq.6).Z = Hx+v (6)

C. State Prediction or State ForecastingState forecasting is the next step in DSE where, the nodal

voltages or the measurements at the next instant of time are

predicted. State vector corresponding to these predicted valuesis called the predicted state at the instant 'k+1'. The forecasted

Q are the parameters to be calculated online to evaluate the


state vector is obtained by performing a conditional operationon equation (4) [5], [13].

The forecasted state vector, along with its errorcovariance is given by:

xk+l = Fkxk +Gk (7)

Where, x is the predicted value at instant 'k+1'. If Ek is the

covariance of the estimate at k, then the covariance ofpredicted value is given by:

Mk+l = FkikFk + Qk (8)Many techniques can be used to predict the state vector.

Some authors have also used Artificial Neural Networks(ANN) [16] and Fuzzy Logic techniques [17] to predict thefuture values of parameters. Other techniques like autoregression and Box & Jenkins method have also been used asmentioned in [1].

D. State FilteringOnce the predicted state vector for the next time instant

'k+1' is obtained, it can be purified or filtered as and when themeasurements arrive at the instant 'k+1' to obtain anoptimized and a higher quality estimate for the instant 'k+1'.The most commonly used filtering technique is the KalmanFilter (KF) technique [18]. As the measurement vector isnonlinear in nature, Extended Kalman Filter (EKF) is used inmost cases. Since the predicted state vector is used forobtaining the final estimate in DSE, the objective function istaken as a sum of measurement residuals and the predictedstate vector residuals. Hence, the objective function becomes:

J(x) = [z - h(x)]T R -[z -h(x)] + [X-XjTM'[x-(9)

Where, M is the covariance of the predicted state vector.Minimizing this objective function will yield an expression forupdate of the state vector as:

Xk Xk+l + Kk+l[Zk+l h(Xk+l)1 (10)

Where K = XHTR1 = [HTR-1H + M-l ]-l HTR-l(1 1)

Where H is the Jacobean of h evaluated at xk, L is the

covariance of the estimated state vector and M is thecovariance of the predicted state vector. The term K is calledthe Kalman gain, which acts as the gain matrix for themeasurement residuals at instant 'k+1'. The measurementupdate is carried out either for the entire measurement set orfor each such sequential measurement. Once all themeasurements at that instant have been processed, the entireprocedure (time and measurement update) is repeated again,by replacing time instant k with k+1. Majority of thetechniques on DSE, available in literature, have used theKalman filter technique or techniques based on Kalman filter.Other techniques have also been proposed and implemented inthe literature, still Kalman filter techniques seem to dominatein most of the DSE algorithms.The advantages of using a predicted state vector in filtering

are, that it provides additional measurement redundancy,reduces the effects bad data and reduces uncertainty levels ofthe final estimated values [1].

III. ALTERNATIVE FORMULATIONS OF DSE

So far the basic steps of the dynamic state estimationtechnique available in the literature have been presented. Nowwe can look various other techniques, which try to improveupon the existing technique or have provided a new directionto it.

A. Extensions to Kalman Filter based TechniquesAs discussed earlier, the nonlinear measurement function

and the nonlinear system model is approximated to a linearmodel in the conventional dynamic state estimation. Butwhenever there are large changes in the load or generations,the nonlinearity in the system becomes large, which results inreducing the performance of the Kalman filter basedtechniques. Hence, to take care of the error arising due to this,J. K. Mandal et al in [19] have proposed two schemes toincorporate nonlinearities in to the Kalman filter based DSE.In the first method, local iterations are carried out during thecalculation of measurement residuals, at each time sample.This increases the reference trajectory and gives a betterestimate in presence of nonlinearities. In the second schemethe Tailor' series expansion of the measurement function isretained until the second order in rectangular co ordinates.This helps in retention of full nonlinearity as the measurementfunction is related to state vector through nonlinear quadraticfunctions [19].

Another method of tackling the problem of nonlinearitieshas been proposed by Sakr. M. M.F et al in [20], [21]. Thistechnique is based on the nonlinear transformation of themeasurement vector, which makes the estimation process alinear estimation problem. The mathematical modeling of thesystem state vector and the measurement vector are bothsimilar to the conventional Kalman filter algorithm. Butinstead of linearizing the measurement vector, a nonlineartransformation of the measurements is carried out. Thistransformation is carried out at each node, in three steps. Thefirst step involves transforming the power flows on linesconnected to a bus (say j) to the angular difference A0Lj.Next, the adjacent voltage of bus bar j is calculated by usingthe voltages of neighbors of bus bar 'j' and the angulardifference A OLj. In the last step this adjacent voltagecalculated in the previous step is used along with the businjection at bus 'j' to obtain a transformed voltage V1. This

procedure is repeated at all the buses of the network to obtaina transformed measurement vector Z . The resultantmeasurement model will now be used to estimate the statevector using the Kalman filter technique. The advantages ofthis technique are that the matrix Z is related to the statevector by a constant matrix and not a Jacobean. This constantmatrix is very sparse, the errors due to linearization areremoved and also the technique of calculating transformedmeasurements can be used to detect anomalies in the system.The above technique has one difficulty that the

measurement covariance matrix (R) is non sparse and hencecomputationally intensive. Though, the matrix R can beapproximated to a partially sparse matrix (by decoupling), the


approximation may take its toll on the accuracy of theestimate. Hence, the authors proposed another method called atwo level estimation technique [8]. Here, in the first level,called as the Low Level (LL) estimation, the overall networkis divided in to several small sub systems, which have smallnumber of buses, and hence there will be no computationproblems related to calculation of R. The next level ofestimation, called the Upper Level (UL) estimation, is only forcoordination among the various sub systems to obtain a singlestate vector for the entire system. The performance of the two-level estimation technique is comparable to the one levelestimator in terms of the accuracy but the computation time isfar lesser. The other important features are that there are noiterations involved in the processes and that the majorcomputational burden is handled at the LL where the systemsize is lesser.The Kalman filter method has the disadvantage that it

cannot handle large changes in the load and generation. K. R.Shih and S. J. Huang in [10], [12] have proposed an algorithmwhere the weight vector Wk is replaced with the

termWke lz-h(k)'. Their technique embeds a weighted

function as an error suppressing aid, i.e. instead of using theconventional Wk weight vector; the technique uses an

exponential function, W§ e( lzk-h(¾k)1) as the new weight

function. Hence, whenever there is a large difference in theestimated and actual values, due to sudden load/generationchanges, the weight associated with that measurementautomatically gets reduced as the term 'lz-h(x)l' appears as anegative exponential term. When normal conditions areprevalent, the residual will be a small value, making theweight function very close to Wk. In this way the errors arereduced and robustness of the algorithm is increased.

Another interesting Kalman filter based DSE technique,described and analyzed by Da Silva et al in [14], can be foundin [22] and [23]. The technique is primarily used by Da Silvaet al to compare the results of their proposed technique in [14]with those in [22] and [23]. This technique uses a modelshown in Eq. 12, to represent the time variation of the system.

xk+1 Fkxk +Wk (12)

To reduce the dimension of the problem, Fk is assumeddiagonal and its elements are used to define a parametervectorck . A model for time evolution of ck is defined based

on the measurement model. Then the Kalman filter techniqueis used to find out, the value of Ck . Once Ck is found, we can

calculate the value of Fk based on which the State forecastingstep is carried out. Once the predicted state vector is obtained,the Kalman filter technique is applied in order to filter thepredicted state vector and obtain the estimated state vectors.

B. Square Root Filter basedDSESo far we have looked at Kalman filter based techniques,

But Isabel et al in 1994 proposed a Square Root Filter (SRF)for DSE instead of Kalman filter [11]. The SRF technique isalgebraically equivalent to the Kalman filter technique but isnumerically more stable than the Kalman filter. In their paper[11], the information square root filter has been used. This

technique replaces the covariance vectors with itscorresponding square root vectors. The reason for propagatingthe square root vectors instead of the covariance vectorsthemselves is that the numerical conditioning of square rootsmatrices is much better than the corresponding covariancematrix. More over the product of the square root matrices(which gives the actual covariance vector) can never beindefinite (singular) even in presence of round off errors,which may not necessarily be true for the conventionalcovariance vectors [24]. Hence, this method proves to be moreefficient, especially in ill conditioned systems. The authors in[11] have used a mathematical model similar to the onedescribed in equation (2). Linear exponential smoothingtechnique has been used for calculating the predicted statevector and the SRF for filtering the predicted state vector andobtaining the optimized state vector. The SRF techniquereduces the uncertainty level in measurements, saves time andcomputer memory as compared to the traditional kalman filtertechnique.

C. Robust Dynamic State Estimation

The Kalman filter based techniques assume Gaussiandistribution of noise. But frequently, the noise distributiondeviates from the assumed model resulting in outliers. Theperformance of kalman filter based techniques degrades in thepresence of these outliers. Hence, to counter this, G.Durgaprasad et al [25] and S. S. Thakur et al [26] have comeup with a robust dynamic estimation technique based on M-estimation. The two essential features based on which thetechnique has been developed are:

* Realistic modeling based on nodal analysis* Using M-estimation to have robust filteringThis technique has three main steps, namely the system

modeling, measurement modeling and robust filtering. Themodeling is based on the assumption that the complex busvoltage at a given point is not only dependent on the previousvoltage level but also on the latest available voltage change atthe buses to which it is connected. The method presented usesthe Huber's M-estimates for the estimation process. The M-Estimation technique reduces the uncertainty level under baddata conditions, uses a filtering technique which is moreeffective in presence of outliers and is also easy to implementin comparison to the conventional Kalman filter basedtechniques.

D. Al basedDSE techniquesThe ability of artificial intelligence techniques in

prediction and pattern recognition can be put to good use inDSE.

1) ANN based techniqueAll the previously defined techniques essentially deal

with classical estimation theory. But in the recent years,Artificial Neural Networks (ANNs) have generated a lot ofinterest in the scientific community. ANN is a very goodoptimization tool and since state estimation is essentially anoptimization problem, it can be put to good use. A. K. Sinhaand J. K. Mandal [16] have proposed an Artificial NeuralNetwork (ANN) based DSE algorithm which is based on thepopular Short Term Load forecasting (STLF) technique. This


method is called the Dynamic Load Prediction (DLP) methodand is based on the assumptions that the bus loads andgenerations are the ones which drive the system dynamics andthat the loads follow a pattern and hence can be predicted. Inthis method an ANN is used to predict the active and reactiveloads at all the buses at instant (k+1), given the load at instant'k'. The bus generations cannot be predicted as they aredependant on the loads and adjust themselves depending upon the changes in load. For this reason, the bus generationsare calculated by adapting the load variations to generationsby using generation precipitation factors. Once the injectionsat various buses are known, it is transformed in to complexbus voltages through load flow solution, as they are moresuitable for the filtering step. These predicted state vectors arein turn used in the filtering stage to obtain accurate stateestimates. The ANN based dynamic state estimation techniquedescribed above is found to produce better prediction of statesand better state vector estimates at almost the samecomputational time as the conventional methods.

2) Fuzzy logic based techniqueTo counter the linearization errors in Kalman filter based

techniques, the nonlinearities in the measurement were takenin to account. But this resulted in increased computation time.Hence, to overcome these problems a sliding surfaceenhanced Fuzzy logic based technique has been proposed byJeu-Min Lin et al in [17]. Here the error signals and the rate oferror are integrally used. The input variable is defined by thelinear equation as shown below:

g =Ae+e (13)Where X is a positive constant, 'e' is the state error and e isthe rate of state error. The output variable is expressed as 'uc'.Here 'g' is assumed to be an element of input vector 'G' and'uc' an element of output vector 'U'. Now the input and outputdiscourses are partitioned and tabulated along with the fuzzyrules. Then the crisp input values are fuzzified by assigning anappropriate membership function. Next step in the procedureis the 'fuzzy inference', where the min-max method [27] isused to obtain the fussy output for each input. The next step iscalled the 'Defuzzification' where the fuzzy inference for eachinput variable is converted to a crisp output value 'uc'. Thisvariable 'u*c', for each input, serves as an element of theoutput vector U. From this, the equation for a new state vectorcan be written as:

Xnew =Fkxk +Gk +U (14)

All the terms in the above equation have their usual meaningas applied to a conventional dynamic state estimationproblem. The term U will be close to zero when the system isnormal but assumes significance whenever there are drasticchanges in load and hence accounting for the nonlinearities.The sliding surface enhanced fuzzy control approach to

DSE is found to have a higher computational performancethan other methods.

IV. ADVANTAGES OF DYNAMIC STATE ESTIMATION

The ability of predicting the state vector one step ahead is avery important advantage of DSE. Some of the advantages ofthat includes [1], [5]:* It allows security analysis to be carried out in advance

and hence allows the operator to have more time duringemergencies.

* It helps to identify and reject bad data and henceimproves the estimator performance.

* In cases where pseudo measurements are to be used, DSEreadily provides high quality values and hence avoids illconditioning.

* DSE can be used for data validation as the states arepredicted one time stamp before.

* Similarly, with the help of the predicted state vector wecan identify sudden changes in the system, topologicalerrors and other anomalies.

These advantages coupled with fairly accurate and quickfiltering procedure make the DSE algorithms an importantplayer in the modem day EMS.

V. CONCLUSIONS

Real time monitoring and control of power systems isextremely important for an efficient and reliable operation of apower system. State estimation forms the back bone for thereal time monitoring and control functions. Since powersystem changes continuously, the operator has to be extremelyalert in taking decisions on real time, especially in cases ofemergency. In such a scenario a technique, which can predictthe possible state of a power system in the immediate future isa boon. Hence, researchers have proposed dynamic stateestimation techniques, which provide the predicted statevector at the next time instant to the operator, with which theoperator will be able to take any suitable control actions. Oncethe measurements at the next instant arrive, the predicted statevector is filtered to obtain an optimized estimate. VariousDSE techniques proposed in the literature, their advantages,disadvantages and specialties if any, have been brieflydescribed in this paper.A lot of developments have occurred since the first paper

on DSE was published in the early 1970s. The initialdevelopments in the field started with the Kalman filter basedtechniques. Then, to improve the performance of the filter,especially under sudden load changing conditions, fewauthors proposed extensions to the Kalman filter approach.Attempts were made to improve the mathematical modelingand computational efficiency by following other methods likethe M-estimation technique and the SRF technique. In the lastfew years, artificial intelligence based techniques such as theANN and Fuzzy logic, have also been applied to the dynamicstate estimation problem. Each of the techniques discussedhave their own advantages and disadvantages. But the Kalmanfilter based techniques have proven to be more popular (but


not necessarily more accurate in every case) than the othertechniques and are being used widely in both utility andacademic circles.

Keeping in mind the importance of a review paper forfuture research in the area, a comprehensive survey of theavailable literature on the dynamic state estimation techniqueshas been presented. It is sincerely hoped that this will help thecommunity of power engineers to further research on dynamicstate estimation topics.

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[5] Sunita Chohan, "Static and Tracking State Estimation in Power Systemswith Bad Data Analysis", PhD Dissertation, Centre for Energy Studies,IIT-Delhi, July 1993.

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[17] Jeu-Min Lin, Shyh-Jier Huang, and Kuang-Rong Shih, "Application ofSliding Surface Enhanced Fuzzy Control for Dynamic State Estimationof a Power System", IEEE Transactions On Power Systems, VOL. 18,No. 2, MAY 2003.

[18] Greg Welch and Grey Bishop, "An Introduction to the Kalman Filter",Department of Computer Science, University of North Carolina atChapel Hill, Course 8, SIGGRAPH 2001.

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[20] Sakr. M. M. F., Bahgat. A., and El-Shafei, AR, "Dynamic stateestimation in power systems with abnormalities detection, identificationand correction". IEE 'Control 85' Conference, Cambridge, ConferencePublication 252, 1, pp. 245-251, 9-1 1 July 1985.

[21] Sakr. M. M. F., Bahgat. A., and El-Shafei, AR, "Modified estimatorapplied to electric power systems", Proc. of Int. AMSE Conf Modelingand Simulation, Tunisia, November 1985, pp. 13 1-146.

[22] A. K. Mahalanabis, K. K. Biswas, G. Singh, "An algorithm fordecoupled dynamic state estimators of power systems", IEEE PESsummer meeting, paper A 78 573-8, Los Angeles, CA, July 1978.

[23] M. B. Do Coutto Filho, A. M. Leitte Da Silva, J. F. De Queiroz,"Dynamic state estimation in electric power systems using kalmanfilter", Proceedings of the 4th Brazilian Congress on automatic control(in Portuguese), 1982, pp. 152-157.

[24] Paul G. Kaminski, Arthur E. Bryson and Stanley F. Schmidt, "DiscreteSquare Root Filtering: A Survey of Current Techniques", IEEETransactions on Automatic Control, Vol. AC-16, No. 6, 1971

[25] G. Durgaprasad, S. S. Thakur, "Robost dynamic state estimation ofpower systems based on m-estimation and realistic modeling", IEEETransactions on Power Systems, Vol. 13, No. 4, November 1998.

[26] S. S. Thakur, A. K. Sinha, "A robust dynamic state estimator for electricpower systems ", IE (I) Journal-EL, August 2000.

[27] Shaohua Tan, Li Zhang and Joos Vandewalle, "On The Learning ofMin-Max Fuzzy Systems", Proceedings of the Sixth IEEE InternationalConference on Fuzzy Systems, vol.3 Page: 1581 - 1584, July 1997.

VII. BIOGRAPHIES

Shivakumar N R has obtained his ElectricalEngineering Degree from RVCE Bangalore.Currently he is working towards his masters inPower Systems, in IIIT, Hyderabad. Previously heworked with Schneider Electric, GTCI Bangalore.His areas of interest include, real time monitoringand control of power systems, Al applications topower systems, condition monitoring anddiagnostics of electrical equipment and powersystem economics.

Amit Jain graduated from KNIT, India inElectrical Engineering. He completed his mastersand Ph.D. from Indian Institute of Technology, NewDelhi, India.

He was working in Alstom on the power SCADAsystems. He was working in Korea in 2002 as a Post-doctoral researcher in the Brain Korea 21 proj'ectteam of Chungbuk National University. He was PostDoctoral Fellow of the Japan Society for thePromotion of Science (JSPS) at Tohoku University,

Sendai, Japan. He also worked as a Post Doctoral Research Associate atTohoku University, Sendai, Japan. Currently he is an Assistant Professor inIIIT, Hyderabad, India. His fields of research interest are power system realtime monitoring and control, artificial intelligence applications, power systemeconomics and electricity markets, renewable energy, reliability analysis, GISapplications, parallel processing and nanotechnology.


a review of power system dynamic state estimation...

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