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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 4, DECEMBER 2011 1099

Online State Estimation of a Synchronous GeneratorUsing Unscented Kalman Filter From Phasor

Measurements UnitsEsmaeil Ghahremani and Innocent Kamwa, Fellow, IEEE

AbstractThe most important reference quantities for monitor-ing and controlling transient stability in real time are the rotorangle and speed of the synchronous generators. If these quanti-ties can be estimated with sufficient accuracy, they can be used inglobal and local control methods. In the classic state estimationmethods, such as the extended Kalman filter (EKF) technique, thelinear approximations of the system at a given moment in time mayintroduce errors in the states. In order to overcome the drawbacksof the EKF, the authors of this paper have applied the unscentedKalman filter (UKF) to estimating and predicting the states of asynchronous machine, including rotor angle and rotor speed, usingphasor measurement unit (PMU) quantities. The UKF algorithmpropagates the pdf of a random variable in a simple and effectiveway and is accurate up to the second order in estimating the meanand covariance. The overall impression is that the performance ofthe UKF is better than the EKF in terms of robustness, speed ofconvergence, and also different levels of noise. Simulation resultsincluding saturation effects and grid faults show the accuracy andefficiency of the UKF method in state estimation of the system,especially at higher noise ratios.

Index TermsPhasor measurements units, power system mon-itoring, power system operation, state estimation, synchronousmachine, unscented Kalman filter.

I. INTRODUCTION

INCREASED demand for sustainable and reliable energy insupport of a massively digital economy stimulates the in-vestigation of new control techniques to enhance power systemstability and security. This aim can certainly be achieved byadding bulk equipment such as FACTS [1], [2] but the leastexpensive strategies are generally based on more intensive andinnovative use of global and local controllers which tend toadd more intelligence to the grid, instead of iron and cop-per. Indeed, Wide Area Measurement and Control (WAMAC)system are contemplated worldwide to advance system securityand stability, while modernizing the power grid technologicalbackbone at the same time. Enhancing system stability through

Manuscript received December 23, 2010; revised May 8, 2011; acceptedSeptember 7, 2011. Date of publication October 13, 2011; date of current ver-sion November 23, 2011. Paper no. TEC-00503-2010.

E. Ghahremani is with the Department of Electrical and Computer Engi-neering, Laval University, Quebec, Canada (e-mail: [email protected]).

I. Kamwa is with Hydro-Quebec/IREQ, Power Systems and Mathematics,Varennes, Quebec, Canada (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/TEC.2011.2168225

a more sophisticated control requires, however, accurate infor-mation about synchronous generator rotor speeds and angles inorder to carry out online high-sampling rate dynamic securityassessments.

Unfortunately, the existing Supervisory Control and DataAcquisition (SCADA) system can only provide steady, low-sampling density, and non-synchronous information about thenetwork, and its measurements are too infrequent and non-synchronous to capture information about the system dynam-ics [3][5]. The WAMAC system using phasor measurementunits (PMUs), on the other hand, enables control systems to havean accurate picture of the power system both synchronously andin a precise sample time. Using wide-area measurements fromPMUs installed on the generator buses, new dynamic state es-timators could generate the dynamic states of a power system,e.g., generator speed and rotor angles, which could next be usedin various advanced control methods.

Recent literature reveals that there are two different ap-proaches in this area of research: 1) model-free rotor angle orrotor speed estimator [6], [7], and 2) model-based state estimatorof machine states [8]. In the first category, artificial intelligencemethods, such as neural networks, have usually been adoptedfor the state estimation. These AI-based model-free speed esti-mators generate the rotor angle estimation without requiring amathematical model or any machine parameters [6]. However,it is sometimes preferable to assume a physical model of thesynchronous machine and then estimate its states as part of alarge power network.

Different variants of the latter approach with different lev-els of complexity can be obtained by selecting a synchronousgenerator model of various orders. For example, in [4] a gainscheduling scheme is used for the state observer in a third-ordersingle-machine infinite-bus system. In [5], a dynamic state es-timation method for the second-order synchronous machine isproposed. Also, in [9], a dynamic state estimation process for asixth-order power system assuming a third-order model for thesynchronous machine was presented. Additionally, [9] assumedthat the exciter output voltage Efd and rotor angle were mea-surable. In [10], a parameter estimation procedure based on theunscented Kalman filter (UKF) was presented for the third-ordermodel of a synchronous generator. In contrast with the presentresearch, the goal of [10] was not to estimate the states but theparameters. Furthermore, the output power, Pe , was assumedin [10] to be a dynamic state and the only measured output,while Efd was the measured input, and Tm , a constant inputsignal.

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1100 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 4, DECEMBER 2011

Fig. 1. Synchronous machine connected to infinite bus via the transmissionlines.

In this paper, the proposed online state estimator in the model-based category according to above classification, uses the UKFalgorithm to generate the estimated states from the availablesignals obtained from a PMU, which is assumed to beinstalledin the substation of a power plant. The benefits of the UKF overthe extended Kalman filter (EKF) result from the fact that inthe UKF method, there is no linearization step in the algorithm[16].

The paper is organized as follows. The fourth-order nonlinearmodel structure considered for the synchronous machine is pre-sented in Section II. Section III describes the UKF mathematicalalgorithm while Section IV presents the online state estimationof the synchronous generator using UKF. Section V comparesthe EKF and UKF methods. Section VI is the discussion sectionand Section VII concludes the paper.

II. SINGLE-MACHINE INFINITE-BUS POWER SYSTEMCompared with higher-order nonlinear structures, the effect

of damper windings and stator dynamics are neglected in thefourth-order nonlinear model of a synchronous machine. Thisis possible when very fast dynamic (subtransient) are not ofinterest. However, the effect of damper windings is consid-ered approximately in the rotor-damping factor, D [1] (see Ap-pendix for symbol nomenclature). The single-machine infinite-bus (SMIB) power system, shown in Fig. 1, is considered hereas the benchmark system. Giving a classical model for the syn-chronous generator and neglecting the transmission line resis-tance (Re = 0), all the active power produced by generator Pt ,is delivered to the infinite bus (Pt = PB ). Also, is the angle bywhich eq , the q-axis component of the voltage behind transientreactance xd , leads the terminal bus of machine Et (or Vt). WithVt as the reference phasor, the SMIB power system in Fig. 1,can be represented in per unit (pu) by the fourth-order nonlinearequation in the dqo domain as (1):

= o

=1J

(Tm Te D)

eq =1

T do(Efd eq (xd xd) id)

ed =1

T qo(ed + (xq xq ) iq ). (1)

Equation (1) can be rewritten by determining the state vari-ables and the inputs in the following form:

x = [ eq ed ]

T = [x1 x2 x3 x4 ]T

u = [Tm Efd ]T = [u1 u2 ]T

x1 = o x2

x2 =1J

(u1 Te D x2)

x3 =1

T do(u2 x3 (xd xd) id)

x4 =1

T qo(x4 + (xq xq ) iq ) (2)

where o = 2fo is the nominal synchronous speed (elec. rad/s), the rotor speed (pu), Tm the mechanical input torque (pu), Tethe air-gap torque or electrical output power (pu), Efd the exciteroutput voltage or the field voltage, as seen from the armature(pu) and the rotor angle in (elec.rad). Other variables andconstants are defined in Appendix.

Based on the phasor diagram of Fig. 1 in the dqo domain, theair-gap torque Te will be equal to the terminal electrical powerPt (or Pe = rTe ) by neglecting the stator resistance (Ra = 0)and assuming r = 1.0 p.u.:

Te = Pt + RaI2tRa =0Te = Pt = edid + eq iq (3)

where the voltages (ed , eq ) can be expressed as (4):ed = Vt sin

eq = Vt cos

Et = Vt =

e2d + e2q . (4)

Also, the d-axis and q-axis currents (id , iq ) are

id = It sin( + ) =eq Vt cos

xd

iq = It cos( + ) =Vt sin

xq

It =

i2d + i2q . (5)

Replacing the variables and eq by the state variables x1 andx3 , we will have

id =x3 Vt cosx1

xd

iq =Vt sinx1

xq. (6)

Replacing (4) and (5) in (3) and after mathematical simplifi-cation, the electrical output power Pe at the terminal bus (Pe =Pt) can be presented as

Pt = Vtxd

eq sin +V 2t2

(1xq 1

xd

)sin 2. (7)

GHAHREMANI AND KAMWA: ONLINE STATE ESTIMATION OF A SYNCHRONOUS GENERATOR USING UNSCENTED KALMAN FILTER 1101

In terms of states x1 and x3 we will have

Pt = Vtxd

x3 sinx1 +V 2t2

(1xq 1

xd

)sin 2x1 . (8)

To summarize, using (6) and (8) in (2), the fourth-order non-linear synchronous machine state-space model in (2), can bepresented as (9)x = [ eq e

d ]

T = [x1 x2 x3 x4 ]T

u = [Tm Efd Vt ]T = [u1 u2 u3 ]T

x1 = o x2

x2 =1J

[u1

(Vtxd

x3 sin(x1) +V 2t2

(1xq 1

xd

)sin(2x1)

)

D x2]

x3 =1

T do

[u2 x3 (xd xd)

(x3 Vt cosx1

xd

)]

x4 =1

T qo

[x4 + (xq xq )

(Vt sinx1

xq

)]

y1 =Vtxd

(x3) sin(x1) +V 2t2

(1xq 1

xd

)sin(2x1). (9)

In (9), the electrical output power Pt selected as the measur-able output and all the parameters and input-output quantitiesare (or assumed to be) known and measurable. Only the statesare unknown. This structure is suitable for nonlinear state esti-mation with the UKF approach. We can, therefore, represent (9)in the following global structure:

x1 = f1(x,u)

x2 = f2(x,u)

x3 = f3(x,u)

x4 = f4(x,u)

y1 = h1(x,u)

{

x = f(x,u,w)

y = h(x,u,v)(10)

where x is the state variable vector defined in (9), u is the inputvariable vector defined in (9), w is the process (state) noise,v is the measurement noise, f is the system function, h is theoutput function, and y is the measurable output. For online stateestimation of the system presented in (9) or in (10), the availableterminal bus signals like Vt , Pt , and Qt are used. These signalsare accessible using a PMU device, which is assumed to beinstalled at the terminal bus of the synchronous generator.

The PMU is a power system device that provides measure-ments of the real-time phasors of the bus voltage and line cur-rents. Basically, it samples input voltage and current waveformsusing a common synchronizing signal from the GPS, and cal-culates the phasors (magnitude and angle) using the discreteFourier transform (DFT) [11][13].

The measurement set is composed of the bus voltage magni-tude and angle, as well as the line and injection currents magni-tudes and angles. By measuring the quantities of the three-phasevoltages and currents Vabc , Iabc , the signals Vt , Pt, and Qt can

Fig. 2. Overall structure of the online state estimator for synchronous machineusing phasor measurement unit signals.

be extracted using known formulas for state estimation pur-poses [7]. The overall plan of the estimation process is illustratedin Fig. 2. After receiving the signals Vt , Pt, and Qt , obtainedfrom a PMU, the UKF state estimator generates the dynamicstates estimates. The principles of the UKF algorithm [14] arepresented in the next section.

III. UNSCENTED KALMAN FILTER ALGORITHM

The extended Kalman filter is probably the most widely usedestimation algorithm for nonlinear systems. However, previousexperience in the estimation community has shown that theEKF is often difficult to implement, difficult to tune, and reli-able only for systems that are almost linear on the time scale ofthe updates. Many of these difficulties arise from its use of lin-earization. To overcome this limitation, unscented transforma-tion (UT) was applied in [15] to propagate mean and covarianceinformation by nonlinear transformation. It is more accurate,easier to implement, and uses the same order of calculations aslinearization [15].

In this section, the state estimation of the system will bepresented using the UKF algorithm. Considering a system in theform of y = g(x), the question is, given the pdf of x, how theUKF computes the mean (yUKF ) and covariance (PUKFy ) of arandom variable (y). The unscented transformation is foundedon the intuition that it is easier to approximate a probabilitydistribution than it is to approximate an arbitrary nonlinearfunction or transformation [16]. The basic idea of the UKFapproach is illustrated in Fig. 3.

In Fig. 3, a set of points x(i) (i= 1, . . . , 2n+ 1), termed sigmapoints, is chosen so that its mean and covariance are yUKFand PUKFy . The nonlinear function is applied to each point,in turn, to yield a cloud of transformed points. The statisticsof the transformed points can then be calculated to form anestimate of the nonlinearly transformed mean and covariance[16]. In unscented transformation, each sigma point is associatedwith a weight W(i) . The following steps are then involved inapproximating the mean and covariance [17]:

1) Propagate each sigma point through the nonlinearfunction:

y(i) = g(x(i)). (11)


Fig. 3. Principle of the unscented transformation (UT).

2) The mean is approximated by the weighted average of thetransformed points (Fig. 3):

yUKF =p

i=0

W(i)y(i) (12)

where the weighting coefficients W(i) are defined asfollows:

pi=0

W(i) = 1. (13)

3) The covariance is computed from the weighted outer prod-uct of the transformed points:

PUKFy =p

i=0

W(i)(y(i) y)(y(i) y)T . (14)

Both the sigma points and the weights are computed deter-ministically using a set of conditions given in [16]. The UKFalgorithm is presented briefly below. For more details and thebackground theory, readers are referred to [16].

Let the n-state discrete-time nonlinear system be representedby (15):

xk+1 = f(xk ,uk , tk ) + wk

yk = h(xk ,uk , tk ) + vk

wk (0,Qk )vk (0,Rk ). (15)

The UKF algorithm for this system can be expressed as [15]Step I: Initialization of the filter at k = 0 as follows:

{x+0 = E(x0)

P+0 = E[(x0 x+0 )(x0 x+0 )T ](16)

where E indicates the expected value and the + subscriptdenotes that the estimate is an a posteriori estimate.

Step II: The following time update equations are used topropagate the state estimate and covariance from one measure-ment time to the next.

(a) To propagate from time step (k 1) to k, first choosesigma points x(i)k1 as specified in (17), with appropriate changessince the current best guesses for the mean and covariance ofxk are x+k1 and P

+k1 :

x(i)k1 = x+k1 + x

(i) i = 1, . . . , 2n

x(i) =(

nP+k1)Ti

i = 1, . . . , n

x(n+i) = (

nP+k1)Ti

i = 1, . . . , n. (17)(b) Use the known nonlinear system equation f(.) to trans-

form the sigma points into x(i)k1 vectors as shown in (11), withappropriate changes since the nonlinear transformation is f(.):

x(i)k = f(x(i)k1 ,uk , tk ). (18)

(c) Combine the x(i)k1 vectors to obtain the a priori stateestimate at time k. This is based on (12):

xk =12n

2ni=1

x(i)k . (19)

(d) Estimate the a priori error covariance as shown in (14).However, we should add Qk1 to the end of the equation to takethe process noise into account:

Pk =12n

2ni=1

(x(i)k xk )(x(i)k xk )T + Qk1 . (20)

Step III: Now that the time update equations are done, weimplement the measurement-update equations.

(a) Choose sigma points x(i)k1 as specified in (17), with ap-propriate changes since the current best guess for the mean andcovariance of xk is xk andP

k :

x(i)k = xk + x

(i) i = 1, . . . , 2n

x(i) =(

nPk)Ti

i = 1, . . . , n

x(n+i) = (

nPk)Ti

i = 1, . . . , n. (21)This step can be omitted if desired. Instead of generating new

sigma points, we can reuse the sigma points that were obtainedfrom the time update. This will save computational effort if weare willing to sacrifice performance.

(b) Use the known nonlinear measurement equation h(.) totransform the sigma points into y(i)k vectors (predicted measure-ments) as shown in (11):

y(i)k = h(x(i)k , tk ). (22)

(c) Combine the y(i)k vectors to obtain the predicted measure-ment at time k. This is based on (12):

yk =12n

2ni=1

y(i)k . (23)

(d) Estimate the covariance of the predicted measurement asshown in (14). However, we should add Rk to the end of theequation to take the measurement noise into account:

Py =12n

2ni=1

(y(i)k yk )(y(i)k yk )T + Rk . (24)


(e) Estimate the cross covariance between xk and yk :

Pxy =12n

2ni=1

(x(i)k xk )(y(i)k yk )T . (25)

(f) The measurement update of the state estimate can be per-formed using the normal Kalman filter equations:

Kk = PxyP1y

x+k = xk + Kk [yk yk ]

P+k = Pk KkPyKTk . (26)

The algorithm above assumes that the process and measure-ment equations are linear with respect to the noise, as shown in(15). In general, the process and measurement equations mayhave noise that enters the process and measurement equationsnonlinearly. That is,

xk+1 = f(xk ,uk ,wk , tk )

yk = h(xk ,uk ,vk , tk ). (27)

In this case, the UKF algorithm presented above is not rigor-ous because it treats the noise as additive, as seen in (20) and(24). To handle this situation, we augment the noise on the statevector as shown in (28) [16]:

xa+k = [xk wk vk ]T . (28)

We then use the UKF to estimate the augmented state xa+k .The UKF is initialized as:

xa+0 = [E(x0) 0 0]T (29)

Pa+0 = diag(E[(x0 x0)(x0 x0)T

],Q0 ,R0). (30)

Then we use the UKF algorithm presented above but, since weare estimating the augmented state with mean and covariance,we can remove Qk1 and Rk from (20) and (24).

IV. MATLAB IMPLEMENTATION AND SIMULATION RESULTSThe overall structure of the MATLAB implementation based

on the Simulink embedded function block is shown in Fig. 4.For time-step simulation in MATLAB, we need to convert thecontinuous-time equations into discrete-time equations in orderto be compatible with the Simulink structure. From the basicdefinition of the time derivative of a variable x, we have:

x =x(k) x(k 1)

t x(k) = x.t + x(k 1) (31)

where t is the time step, k and k 1 indicate the time att = kt and t = (k 1)t, respectively. On the other hand,we have the definition of x = f (x, u, w) from (10). Substituting(10) in (31), we obtain

x(k) = x(k 1) + t.f(x,u,w) (32)x(k) = t f(x,u,w) + x(k 1) (33)

Fig. 4. Implementation of UKF algorithm using the embedded MATLABfunction block as the UKF estimator with Tm , Ef d , and Pt as its input signals.

which can be rewritten in the familiar structure presented in (15)or (27) in Section III as

{xk = fk1(xk1 , uk1 , wk1)

yk = hk (xk , uk , vk ).(34)

The embedded MATLAB function was used to develop theUKF approach because this block enables us to predict the dy-namic state (bottom of Fig. 4) while simulating the synchronousmachine (top of Fig. 4). The embedded function block createsan m-file page in the Simulink model, which means we caneasily set and run more or less complicated algorithms in theSimulink file. As shown in Fig. 4, in the embedded MATLABfunction block, the signals Tm and Efd and the system obser-vation signal Pe ( = y1), taken from the machine model, havebeen used as inputs of the UKF block. The latter thus has ac-cess to these input-output signals at each time step in order togenerate the state estimation based on the algorithm describedin the previous section. Further details about MATLAB imple-mentation and initial values of some parameters are describedin the Appendix.

The noise-free results of the state estimation of the UKFmethod are depicted in Fig. 5(a), based on excitation voltagestep responses. The corresponding estimated output signal ykcompared with the actual output signal is shown in Fig. 5(b). De-tails of the Efd step signal are given in the Appendix. Next, theeffectiveness of the UKF method was checked in the presenceof a network fault.

The general configuration of the power system can be reducedto the form of Fig. 6 using Thevenins equivalent theory [1].Based on this equivalent circuit, we will consider that the faultoccurred in the mid-point of the transmission line, as shown in


Fig. 5. UKF state estimation results without noise: (a) Estimated states.(b) Estimated output.

Fig. 6. Mid-point short-circuit fault.

Fig. 6, with the system in the steady state at T = 20 (s). Theanalysis will focus on two scenarios following a disturbance:1) stable and 2) unstable. In the first scenario, the fault needs tobe cleared after 0.1 (s), at T = 20.1 (s), for the system to remainstable.

Fig. 7. UKF state estimation results in stable short-circuit fault: (a) Estimatedstates. (b) Estimated output.

The results of the state estimation process during and fol-lowing the contingency in this case are shown in Fig. 7. Fromthese results, it is clear that broadly speaking, the UKF stateestimator generates very accurate states right after fault clear-ing. However, some variables, notably the angle and speed, aretemporarily wrong during the fault due to the abrupt change inthe external reactance.

For the second scenario, the fault was cleared after 0.3 (s),at T = 20.3 (s) and the system transitioned into an unstablecondition. In this case also, the UKF state estimator generatesthe estimated states with an appropriate accuracy once the faultis cleared (Fig. 8). Generally, it takes several milliseconds forthe state estimator to track the real signal after a sudden changein the network reactance and there is a large error between the


Fig. 8. UKF state estimation results in unstable short-circuit fault: (a) Esti-mated states. (b) Estimated output.

real and the estimated signals within the fault timeframe. Thiserror may be reduced by blocking the slow-changing states totheir pre-fault values during the fault occurrence.

Next, we have the analysis of the fault simulations (Figs. 7and 8) in the presence of noise. The noise sequences were setto have the same magnitudes as in Fig. 10 (see Appendix). Weagain obtained satisfactory results from the UKF state estimationprocess in the presence of noise, similarly to Fig. 10. Based onthe good performance achieved with these two fault scenarios(not shown for the sake of saving space), it could be concludedthat the UKF approach will be able to estimate the state of asynchronous machine whatever the stability condition of thepower system to which it is connected, even when the processand measurement noises are significant.

Fig. 9. Comparison of the exact, linearized, and unscented mean and covari-ance of 300 randomly generated points by (35).

V. COMPARISON OF THE UKF AND EKF METHODSON NOISY DATA AND PROCESSES

The difference in the principles of state estimation using theUKF approach and the EKF method is illustrated in Fig. 9.This figure shows 300 points randomly generated by (35) with runiformly distributed between0.1 and uniformly distributedbetween 0.35 rad:

{y1 = r cos

y2 = r sin .(35)

The points spread in Fig. 9 represent the exact points gener-ated by (35), and the bold point at (0,1) represents the linearizedmean. The true mean and the approximate unscented mean areso close that they are plotted on top of each other and are bothequal to (0, 0.9797) to four significant digits. As a result, thisfigure shows the improved accuracy of the mean and the co-variance estimation when unscented transformations are usedinstead of linear approximations [15].

To demonstrate the superior performance of UKF under noisyconditions, we have implemented separately the EKF method inorder to compare it with the UKF approach. In terms of Simulinkimplementation, we simply replaced the UKF block in Fig. 4 byan equivalent EKF block, implemented according to [15].

As is clear from Fig. 10, when we compare the state estimatesin the presence of noise, the UKF approach definitely outper-forms the EKF in terms of accuracy and smoothness. In fact,at the same level of process and measurement noise, the EKFestimator results are noisier, like the rotor speed , and arealso biased, like the rotor angle, eq and ed voltages while theUKF results are smoother and do not have bias.

VI. DISCUSSION

The results presented in Figs. 5 and 10 assume that the inputsignal Efd is a step function while Tm is a constant input. Toconfirm the UKF effectiveness under more general conditions,we tested it with different kinds of input. Hence, the simulation


Fig. 10. State estimation in the presence of noise: (a) Estimated states with EKF approach. (b) Estimated states with UKF approach, (c) Estimated output withEKF approach. (d) Estimated output with UKF approach.

was repeated for two other configurations: the first for Tm =ramp and Efd = constant, and the second for Tm = constantand Efd = ramp. In both simulation studies, the results ofthe UKF approach showed that the UKF block estimated thestates properly and with enough accuracy. These scenarios wererepeated for the system in the presence of noise and the resultswere accurate again.

Also, the state estimation process was repeated for differentvalues of the external line reactance and for different values ofthe synchronous machine parameters. In all different simulationstudies, we obtained results that showed the robustness andeffectiveness of the proposed method.

Interestingly, it turns out that the UKF method is not so de-pendent on the initial value of the states. For example, if werun the UKF simulation with o = 0.4 (while the nominal valueof rotor angle is = 0.82), we still obtain accurate results. Bycontrast, the standard EKF algorithm is more dependent on theinitial value of the rotor angle. For the same procedure, with a

nominal value of rotor angle at = 0.82, if we run the EKF sim-ulation file with a o less than 0.6, the state estimator loses itsability to track the actual states. It could, therefore, be concludedthat the UKF method is less dependent on the initial values thanthe EKF method.

As is clear from Figs. 2 and 4, in addition to the phasormeasurements of voltages and currents (Vabc , Iabc ), which arerequired to determine Pt , Vt, and It , the UKF approach needsto access Efd and Tm as two other input quantities. In orderto record and measure Efd and Tm near the machine phasorsignals, the PMU should be located in the power plant.

From this point of view, one issue of the proposed UKF ap-proach could reside in the fact that measuring the Efd signalis essential for this method, while it may not be the case inall other methods. On the other hand, for power system engi-neers, it would be interesting if the dynamic states of a syn-chronous machine could be obtained using recorded signals of aPMU installed on the terminal bus or transmission line. We are


TABLE IDEFINITIONS OF VARIABLES AND CONSTANTS

therefore working on this idea as a useful and practical proce-dure to remove these limitations (recording Efd and Tm ) fordynamic state estimation of a synchronous machine. After that,the PMU could be installed on the transmission line but thetransformer reactance would then have to be added to the ma-chine reactance. To develop this idea, the first step would be toestimate the states of the synchronous machine without one ofthese input quantities, for example, the Efd signal. After that,we could improve the idea by developing an estimator withoutthe two input Efd and Tm signals.

VII. CONCLUSIONIn this paper, the UKF approach was applied to online state

estimation of a synchronous generator including the rotor an-gle and rotor speed signals, based on two generally availablevariables, which are, the electrical power and field voltage. Thisapproach allows to overcome the limitations of the linearizationprocess required by the traditional EKF method, and also to in-crease the operational range of the system variables around theoperating point by not using the linearization in the state estima-tion algorithm. The implemented UKF-based scheme producedhigh quality results and also showed greater accuracy of the stateestimates in the presence of noise, compared to the traditionalEKF method.

APPENDIX

A. Definitions of Variables and ConstantsThe main variables and constants of the SMIB system de-

scribed in (1) to (9) are presented in Table I.

TABLE IITEST MACHINE AND EXTERNAL SYSTEM DATA

TABLE IIIDEFINITIONS OF VARIABLES OF UKF ALGORITHM

B. Test Machine and External System DataThe test machine parameters and external system data in per

unit (pu) are presented in Table II.

C. Definition of Variables of UKF AlgorithmThe main variables of the UKF algorithm presented in (11)

(30) are presented in Table III.

D. MATLAB ImplementationAs mentioned in Section IV, the embedded MATLAB func-

tion was used for implementation of the UKF method. By run-ning the Simulink file and receiving the signals from UKF blockinputs, the block creates the outputs based on its internal com-mands. At the beginning of the m-file, we can use the commands


persistent, isempty, and if-end for initializing the UKF al-gorithm variables x0 and P0 . These commands in the formbelow, allows the algorithm to be initialized once at the start ofthe simulation:P0 = diag([10,10,10,10]);persistent P.

.

.

if isempty(P)P = P0;

end;The above script loads the initial value of matrix P with P0

in the first iteration of the algorithm. In the next iterations, thecommand isempty(P)will not allow the matrix P to be reloadedagain. After the second iteration, the matrix P will always havea value and it will not be null so the program will pass the loopif-end.

We should mention that the time step t defined in the embed-ded MATLAB function block and the time step of the Simulinkfile must be equal. In our case, the time step for the Simulinkfile was Ts = 0.1 ms and the time step for the EKF block, whichshould be equal to the Simulink file time step, was also t =0.1 ms. To have the same case study signals for an acceptablecomparison in Fig. 10, the time step also was defined t =0.1 ms for the UKF simulation file.

The following initial values were used in implementing theMATLAB function block for the UKF method algorithm: forthe state vector, x0 = [0.4; 0; 0; 0] and for the state covariancematrix, P0 = diag([10, 10, 10, 10]). Also, the values of thenoise characteristics were defined as follows: For the processnoise, we had wk (0, Qk ) = (0, 0.0012 (I44)) and for themeasurement noise, vk (0, Rk ) = (0, 0.012 I). The size ofmatrix I44 is related to the fact that we have four states in themodel.

Also, the initial values, used for the EKF results in Fig. 10were set as follows: x0 = [0.6; 0; 0; 0] for the state variable, andP0 = diag([102 , 102 , 102 , 102 ]) for the state covariance matrix.Finally, the noise covariance matrices were defined as follows:for state noise wk (0, Qk ) = (0, 0.082 (I44)) and foroutput noise vk (0, Rk ) = (0, 0.12 I).

Lastly, for step response simulation studies in Simulink, thevalue of mechanical torque input was fixed at Tm = 0.8, and thestep function settings for Efd signal were defined as follows:initial value = 0, final value = 0.2, rise time = 0.1 (s), and thisvariation was added to a constant value of 2.11.

To mimic near real system conditions for both the UKF andthe EKF methods in Fig. 10, white noise was added to the stateprocess with the mean and covariance (0, 0.0012 (I44)).Likewise, the output measurement noise mean and covariancewere set to (0, 0.012 I).

REFERENCES

[1] P. Kundur, Power System Stability and Control. New York: McGrawHill, 1994.

[2] Y. N. Yu, Electric Power System Dynamic. New York: Academic Press,1983.

[3] M. Anjia, Y. Jiaxi, and G. Zhizhong, PMU placement and data processingin WAMS that complement SCADA, in Proc. 2005 IEEE/PES GeneralMeeting, San Francisco, CA, MA, pp. 780783.

[4] J. Chang, G. N. Taranto, and J. Chow, Dynamic state estimation in powersystem using gain-scheduled nonlinear observer, in Proc. 1995 IEEEControl Appl. Conf., Albany, NY, pp. 221226.

[5] Z. Huang, K. Schneider, and J. Neplocha, Feasibility studies of applyingKalman filter techniques to power system dynamic state estimation, inProc. 2007 Int. Power Eng. Conf. (IPEC), Singapore, pp. 376382.

[6] I. Kamwa, B. Baraboi, and R. Wamkeue, Sensorless ANN-based speedestimation of synchronous generator: improved performance throughphysically motivated pre-filters, in Proc. 2006 IEEE Neural NetworkConf. (IJCNN), Vancouver, BC, pp. 17101718.

[7] A. Del Angel, P. Geurts, D. Ernst, M. Glavic, and L. Wehenkel, Estimationof rotor angle of synchronous machines using artificial neural networksand local PMU-based quantities, Neurocomputing, vol. 70, pp. 26682678, Oct. 2007.

[8] V. Venkatasubramanian and R. G. Kavasseri, Direct computation of gen-erator internal dynamic states from terminal measurements, in Proc. 37thHawaii Int. Conf. Syst. Sci., 2004, Washington, DC, pp. 16.

[9] L. Lin, Linawati, L. Jasa, and E. Ambikairajah, A hybrid state estima-tion scheme for power system, in Proc. 2002 IEEE Circuits Syst. Conf.(APCCAS02), Bali, Indonesia, Oct. 2831, vol. 1, pp. 555558.

[10] M. Huanga, W. Li, and W. Yana, Estimating parameters of synchronousgenerators using square-root unscented Kalman filter, Int. J. ElectricPower Syst. Res., vol. 80, pp. 11371144, Sep. 2010.

[11] I. Kamwa, M. Leclerc, and D. McNabb, Performance of demodulation-based frequency measurement algorithms used in typical PMUs, IEEETrans. Power Del., vol. 19, no. 2, pp. 505514, Apr. 2004.

[12] J. Warichet, T. Sezi, and J. C. Maun, Considerations about synchrophasorsmeasurements in dynamic system conditions, Int. J. Elect. Power EnergySyst., vol. 31, pp. 452464, Oct. 2009.

[13] I. Kamwa, K. Pradhan, and G. Joos, Adaptive phasor and frequency-tracking schemes for wide-area protection and control, IEEE Trans.Power Del., vol. 26, no. 2, pp. 744753, Feb. 2011.

[14] G. Valverde and V. Terzija, Unscented Kalman filter for power systemdynamic state estimation, IET Gener. Transm. Distrib., vol. 5, no. 1,pp. 2937, 2011.

[15] D. Simon, Optimal State estimation; Kalman, H, and Nonlinear Ap-proaches. New Jersey: John Wiley & Sons, 2006.

[16] S. J. Julier and J. K. Uhlmann, Unscented filtering and nonlinear estima-tion, Proc. IEEE, vol. 92, no. 3, pp. 401422, Mar. 2004.

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Esmaeil Ghahremani received the B.Sc. and M.Sc.degrees from Amirkabir University of Technology(Tehran Polytechnic), Tehran, Iran, in 2003 and 2007respectively. He has been pursuing the Ph.D. de-gree under Prof. Kamwas supervision, studying inPower System Stability, at Laval University, Laval,QC, Canada, since 2008.

His research interests include power system stabil-ity studies, dynamic state estimation of power system,FACTS placement and wide area control of powersystem.

Innocent Kamwa (S83M88SM98F05) re-ceived the Ph.D. degree in electrical engineering fromLaval University, Quebec, Canada, in 1988.

Since 1988, he has been with the Hydro-QuebecResearch Institute (IREQ), Power System Analysis,Operation, and Control, Varennes, Quebec, where heis currently a Project Manager for Automation andControl Systems and also Principal Scientist for smartgrids. He has also been an Associate Professor of elec-trical engineering at Laval University, since 1990.

Dr. Kamwa has been active for the last 13 years onthe IEEE Electric Machinery Committee, where he is presently the StandardsCoordinator. A member of CIGR E and a registered Professional Engineer, he isa recipient of the 1998, 2003, and 2009 IEEE Power Engineering Society PrizePaper Awards and is currently serving the IEEE/PES Standards Coordinatingand System Dynamic Performance Committees.

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