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Electric Power Systems Research 121 (2015) 109–114

Contents lists available at ScienceDirect

Electric Power Systems Research

j o ur na l ho mepage: www.elsev ier .com/ locate /epsr

ynamic state estimation in power systems: Modeling, and challenges

amed Tebianian, Benjamin Jeyasurya ∗

aculty of Engineering and Applied Science, Memorial University, St. John’s, Newfoundland, A1B 3X5, Canada

r t i c l e i n f o

rticle history:eceived 10 June 2014eceived in revised form9 September 2014ccepted 3 December 2014vailable online 24 December 2014

eywords:

a b s t r a c t

This paper proposes Extended Kalman Filter (EKF) based dynamic state estimator for power systems usingphasor measurement unit (PMU) data. Dynamic state estimation in power systems provides synchronizedwide area system history of the dynamic events which is key in the analysis and understanding of thesystem performance, behavior, and the types of control decisions to be made for large scale power systemcontingencies. In this paper, 2-axis-fourth-order state space modeling and validation of the synchronousmachine is explained in detail. The model is then used for dynamic state estimation using EKF in IEEE3-Generator-9-Bus Test System. The simulation results show that the model and estimation approach are

ower system dynamic state estimationower system transient stabilityynchronous machine 2-axis-fourth-orderodel

xtended Kalman Filterhasor Measurement Unit

capable to provide accurate information about the states of the machine and eliminate the noise effectson the measurement signal. The main challenges of dynamic estimation in large power systems are alsoaddressed in this paper.

© 2014 Elsevier B.V. All rights reserved.

onlinear control

. Introduction

Extended Kalman Filter (EKF) has been among the most referredstimation approaches for dynamic state estimation in powerystems [1,2]. The advent of Phasor Measurement Units (PMUs)3] has facilitated online state estimation in large scale powerystems which was previously impossible using low rate andon-synchronous data provided by Supervisory Control and Datacquisition (SCADA) systems. As the number of installed PMUsre gradually increasing worldwide, real time estimation in largenterconnected power grids is becoming more realistic [2]. PMUs a recently developed power system measurement device thatamples input three phase voltage and current waveforms, using aommon synchronizing signal received by Global Positioning Sys-em (GPS), and calculates the phasors (magnitudes and angles) ofhe bus by deploying Discrete Fourier Transform [3]. Different esti-

ation approaches and case studies have been used to investigateynamic state estimation in power systems. Feasibility studies ofpplying Extended Kalman Filter (EKF) to IEEE 3-Generator-9-Bus

est System using classical model of the synchronous generatorre investigated in [1]. EKF with unknown input is the estimationpproach for Single-Machine-Infinite-Bus (SMIB) in [2]. Another

∗ Corresponding author. Tel.: +1 709 864 8902; fax: +1 709 864 4042.E-mail addresses: hamed.tebianian@mun.ca (H. Tebianian), jeyas@mun.ca

B. Jeyasurya).

ttp://dx.doi.org/10.1016/j.epsr.2014.12.005378-7796/© 2014 Elsevier B.V. All rights reserved.

form of nonlinear Kalman Filter, Unscented Kalman Filter (UKF),is used to design an observer for different power system casestudies using the PMU installed on the main bus of the genera-tor [4–6]. In [7], a divide-by-difference-filter based algorithm isproposed for dynamic estimation of the generator rotor angle ina large power system. The results of state estimation in a SMIBusing extended particle filter are also presented in [8]. Simula-tions are performed on 2-axis-fourth-order state space model ofthe synchronous machine in [2,6] using either EKF or UKF to designdynamic state estimator for various power systems.

The paper is organized as follows. In Section 2, mathemati-cal description of the synchronous generator is explained and the2-axis-fourth-order state space model of the machine is derived.EKF principles and equations are presented in Section 3. Section 4presents simulation results for IEEE 3-Generator-9-Bus Test Sys-tem. Section 5 presents an application and the major challenges ofdynamic state estimation in power systems. Section 6 concludesthe paper.

2. Single-machine-infinite-bus state space model

Fig. 1 shows a simplified equivalent model of a general powersystem which is a single generator connected through a trans-

former and parallel transmission lines to infinite bus. The classicaldynamic model of the synchronous machine is as follows [9]:

dı

dt= ω0�ω (1)

110 H. Tebianian, B. Jeyasurya / Electric Power S

�Pesgotcd

e

e

xdTttxtm[{

{

(

i

i

tT

T

P−k+1 = FkP+

kFT

k+ LkQkLT

k(17)

Fig. 1. Single-Machine-Infinite-Bus (SMIB) diagram [2].

d�ω

dt= 1

2H(Pm − Pe − D�ω) (2)

In this model, D and H are damping factor and inertia constant,ω is the per unit rotor speed deviation, ı is the rotor angle, and

m and Pe are the power provided by the prime mover and thelectrical output power of the generator both in per unit. The nexttep is to develop the 2-axis-fourth-order model of the synchronousenerator which includes e′

q and e′d, the q and d axis components

f the generator internal voltage. Based on the phasor diagram ofhe synchronous machine, equations describing the q and d axisomponents of the generator internal voltage and their first orderifferential equations are given in Eqs. (3)–(6) [9].

′q = eq + Raiq + x′

did (3)

′d = ed + Raid + x′

qiq (4)

de′q

dt= 1

T ′do

(Efd − e′q − (xd − x′

d)id) (5)

de′d

dt= 1

T ′qo

(−e′d + (xq − x′

q)iq) (6)

d and xq are direct and quadratic axis reactances, and x′d

and x′q are

irect and quadratic axis transient reactances, all in per unit. Also,′do and T ′

qo are direct and quadratic axis transient open circuitime constants in second. ı is defined as the angle such that e′

q,he q axis component of the voltage behind the transient reactance′d, leads the terminal bus Et or Vt, and Efd is the field voltage of

he machine. Considering the phasor diagram of the synchronousachine, the d-axis and q-axis voltages (ed, eq) can be expressed as

2,10]

ed = Vt sin(ı)

eq = Vt cos(ı)→ Et = Vt =

√e2

d+ e2

q (7)

In addition, the d-axis and q-axis currents (id, iq) are [2,10]

id = It sin(ı + �)

iq = It cos(ı + �)→ It =

√i2d

+ i2q (8)

Using Eqs. (3), (4) and (7) and by neglecting the stator resistanceRa = 0), id and iq can be written as

d = e′q − Vt cos(ı)

x′d

(9)

q = Vt sin(ı) − e′d

x′q

(10)

The air gap torque Te of the generator in per unit is equal to the

erminal power Pe or Pt (generator terminal electrical power) [2].herefore, it is obtained

e = Pt + RaI2t

Ra=0−→Te ∼= Pt = edid + eqiq (11)

ystems Research 121 (2015) 109–114

Eqs. (7), (9) and (10) are inserted into Eq. (11) to obtain

Te ∼= Pt = Vt

x′d

e′q sin(ı) − Vt

x′q

e′d cos(ı) + V2

t

2

(1x′

q− 1

x′d

)sin(2ı) (12)

Using Eqs. (1), (2), (5), (6), (9) and (10), the fourth order modelof a synchronous generator is derived as follows:

dı

dt= ω0�ω

d�ω

dt= 1

2H

(Pm − Vt

x′d

e′q sin(ı) + Vt

x′q

e′d

cos(ı) − V2t

2

(1x′

q− 1

x′d

)sin(2ı) − D�ω

)de′

q

dt= 1

T ′do

(Efd − e′

q + (xd − x′d)

(e′

q − Vt sin(ı)

x′d

))de′

d

dt= 1

T ′qo

(−e′

d+ (xq − x′

q)

(Vt sin(ı) − e′

d

x′q

))(13)

Eqs. (12) and (13) are used in a recursive EKF estimation programafter being discretized with the first term of the Taylor Series.

3. Extended Kalman Filter algorithm

EKF is a powerful recursive algorithm for dynamic state esti-mation in nonlinear systems. This optimal estimation approachminimizes the covariance of squared error between real states andestimated ones. A nonlinear discrete stochastic difference equa-tion and measurement equation can be generally presented in thefollowing form [11]:

xk+1 = fk(xk, uk, wk)

yk+1 = hk+1(xk+1, vk+1)

wk∼(0, Qk)

vk∼(0, Rk)

(14)

f is the nonlinear function of the states and inputs, xk+1 representsstate vector, uk is the control input vector, yk+1 is the output vector,wk and vk are the process and measurement noise, Qk and Rk arethe process and measurement noise covariance, and k is the timestep for each iteration. EKF recursive algorithm is performed in twostages: time update and measurement update. As a result, the fol-lowing steps can be applied to nonlinear system for dynamic stateestimation [11,12].

1. The filter is initialized as follows:

x+0 = E(x0)

P+0 = E[(x0 − x+

0 )(x0 − x+0 )T ]

(15)

For k = 1, 2, 3, . . ., n the following stages are performed.2. Partial derivative matrices of the system equation are obtained

by Eq. (16).

Fk = ∂fk∂X

∣∣∣∣x+

k

Lk = ∂fk∂w

∣∣∣∣x+

k

(16)

• Time update equations of EKF are as follows:

x−k+1 = fk(x+

k, uk, 0)

• Partial derivative matrices of output equation are derived byEq. (18).

H. Tebianian, B. Jeyasurya / Electric Power Systems Research 121 (2015) 109–114 111

F

ig. 2. Complete diagram of an EKF based estimator for a synchronous machine.

Hk+1 = ∂hk+1

∂X

∣∣∣∣x−

k+1

Mk+1 = ∂hk+1

∂v

∣∣∣∣x−

k+1

(18)

• The measurement update is carried out using Eq. (19).

Kk+1 = P−k+1HT

k+1(Hk+1P−k+1HT

k+1 + Mk+1Rk+1MTk+1)

−1

x+k+1 = x+

k+1 + Kk+1[yk+1 − hk+1(x−k+1, 0)]

P+k+1 = (I − Kk+1Hk+1)P−

k+1

(19)

Fig. 4. States and output estimatio

Fig. 3. IEEE 3-Generator-9-Bus Test System in PowerWorld Simulator.

In Eq. (17), x−k+1 is a priori state estimate at step k+1 given

knowledge of the process prior to this step, x+k+1 is a posteriori

state estimate at step k+1 given measurement yk+1, P−k+1 and

P+k+1 are the a priori and a posteriori estimate error covariance,

Fk is the jacobian matrix of f with respect to X, and Kk+1 is theKalman gain that minimizes the error covariance [12].

Fig. 2 shows the complete idea of a Kalman filter based estima-tor designed in this paper for a synchronous machine. This diagramshows that the basic model simulated in PowerWorld simulator

[13] is being controlled by two separate control feedback loopsfor exciter and governor. Instead of PowerWorld Simulator, anytransient stability analysis software can also be used. Some of the

n of Generator 1 using EKF.

112 H. Tebianian, B. Jeyasurya / Electric Power Systems Research 121 (2015) 109–114

imatio

oott(m(cpe

c

′d

)s

x1))

))

Fig. 5. States and output est

utputs of the simulator (Pm, Efd, Vt, and Pt) are used as inputs for theptimal estimator block, and the others (ı, �ω, e′

q, e′d) are obtained

o be compared with the estimated states. The observer is designedo accurately estimate the main states of the synchronous machineı, �ω, e′

q, e′d), and eliminate the effect of noise on the measure-

ent signal, which in this case is externally added to this signalPt) before injecting to the observer block. This is done to make thease study much more similar to a practical case. The estimatedower is also used to evaluate the ability of the estimator block forliminating noise of input signals.

Considering Eqs. (12)–(14), the state space model of the machinean be obtained as follows:

X = [ ı �ω e′q e′

d ]T = [ x1 x2 x3 x4 ]T

U = [ Pm Efd Vt ]T = [ u1 u2 u3 ]T

⎡⎢⎢⎢⎣

x1

x2

x3

x4

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ω0x2

12H

(Pm −

(Vt

x′d

x3 sin(x1) − Vt

x′q

x4 cos(x1) + V2t

2

(1x′

q− 1

x

1T ′

do

(Efd − x3 − (xd − x′

d)

(x3 − Vt cos(

x′d

1(

−x + (x − x′ )

(Vt sin(x1) − x4

T ′qo

4 q q x′q

[y1] = [Pt] =[

Vt

x′d

x3 sin(x1) − Vt

x′q

x4 cos(x1) + V2t

2

(1x′

q− 1

x′d

)sin(2x1)

]

n of Generator 2 using EKF.

in(2x1)

)− Dx2

))

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

4. Simulation results

In this part, EKF is used for dynamic state estimation in IEEE3-Generator-9-Bus Test System [13]. To have reliable simulationresults, original data obtained from contingency analysis of thesystem in PowerWorld Simulator [13] is injected to the designedestimator in MATLAB [14]. Model of the system in PowerWorldSimulator is presented in Fig. 3.

It is assumed that three PMUs are installed on bus 1, bus 2,and bus 3, and the data provided by PMUs is received at the sametime by an assumed Phasor Data Concentrator (PDC). All threesynchronous generators are considered with 2-axis-fourth-order

H. Tebianian, B. Jeyasurya / Electric Power Systems Research 121 (2015) 109–114 113

imatio

mToesspamipeefc

oags

TP

Fig. 6. States and output est

odel and have the same characteristics as presented in Table 1.he simulation scenario is a symmetrical three phase fault at t = 0.5 sn bus 8 which is cleared after 0.1 s. Results of the dynamic statestimation for each generator in this stable case study are pre-ented in Figs. 4–6. It should be noted that Eq. (20) is the statepace model considered for each synchronous generator in thisart; consequently, 12 state variables and 3 measurements (4 statesnd 1 measurement for each generator) are simultaneously esti-ated in each iteration. In addition, each model needs to have 3

nput variables as indicated in Eq. (20). Main parameters of theower system and other simulation characteristics, for examplerror covariance, and process and measurement noise matrices arexpressed in Table 1. The initial states of the system can be foundrom the steady state power flow solution; nevertheless, EKF canonverge to the real values even with zero initial states.

Simulation results in this part reveal the ability of EKF based

bserver for accurately tracking all of the states of the machinesnd eliminating noise effect on the output power signals of theenerators during transient and steady state response. The steadytate error of some estimated states especially of Generator 3 can

able 1ower system parameters and simulation characteristics [2,10].

Symbol Details

D, H Damping factor and inertia conxd , xq Direct and quadratic axis reactax′

d, x′

q Direct and quadratic axis transiT ′

qo, T ′qo Direct and Qdo , Tqo Direct and quadratic axis open

P0 Error covariance matrix

Qk Process noise covariance

Rk Measurement noise covarianceTS Sampling time, s

n of Generator 3 using EKF.

be reduced by adjustment of the measurement and process noisecovariance matrices as it is done for Generator 1 and 2.

5. Application and challenges of dynamic state estimationin power systems

A practical application of the dynamic state estimation for apower system is to put the estimator block in the feedback loop ofthe governor of the machine. This can be considered as a sensorlesscontrol of the machine where the input signal of the governor (� ω)is not provided by a physical sensor and is actually the estimatedspeed provided by the EKF based estimator. The complete diagramof this application is presented in Fig. 7. The main advantage ofthis control approach is its sensorless property which eliminatesthe speed sensor and the related physical wiring. In addition, it is

capable of input signal noise rejection which enhances the totalreliability of the decision made by the control block. Also, the otherestimated states of the synchronous machine (ı, e′

q, e′d) can be effec-

tively used in more complicated control schemes.

Values

stant, per unit 0.05, 5nce, per unit 2.06, 1.21ent reactance, per unit 0.37, 0.37circuit time constant, s 7, 0.75

50 × I4×4

0.072 × I4×4

0.012 × I0.001

114 H. Tebianian, B. Jeyasurya / Electric Power S

Fb

sPtcl

aEmcrtpnwimtpmpht

[

[

ig. 7. Block diagram of the sensorless control of a synchronous machine using EKFased estimator.

The main current challenges of dynamic state estimation in largecale power systems are the inadequate number of the installedMUs and the quite low rate data provided by current PMUsechnology. Although the number of installed PMUs in large inter-onnected power systems is gradually increasing, there is still aong way to equip all buses of the system to an advanced PMU.

In addition, data rate of the PMUs is still low which decreases theccuracy of the estimation to some extent. State space model andKF estimation approach deployed in this study are sensitive to loweasurement data rate; as a result, the optimum sampling rate is

onsidered as 1000 packages per second to get accurate estimationesults. However, this sampling rate is higher than the capability ofhe existing PMUs, which may be resolved in near future. A com-lete advanced dynamic control system for a large scale power grideeds a movement from conventional SCADA to PMU based system,hich needs huge investment in power systems, communication

nfrastructure, and more advanced PMU technology. However, itight be possible to implement a complete local dynamic con-

rol system based on PMU measurement in a small area of a largeower system. There are numerous other models for synchronous

achine with higher degrees of accuracy which may have better

erformance than the model used in this study. Nevertheless, theseigh order models need powerful processors for real time opera-ion, making the implementation more expensive. As the authors

[

[

[

ystems Research 121 (2015) 109–114

foresee, the accurate models of the exciter and governor are alsoneeded for state feedback control of the synchronous machine;therefore, these models should be derived and validated by realdata, which might be a challenging task.

6. Conclusion

In this paper, after a brief introduction about the dynamicstate estimation in power systems, 2-axis-fourth-order state spacemodel of Single-Machine-Infinite-Bus (SMIB) is derived using thebasic mathematical description and phasor diagram of the syn-chronous machine. Then, Extended Kalman Filter as an optimalestimation approach for nonlinear systems is discussed. The 2-axis-fourth-order state space model of the synchronous machineis then used for dynamic state estimation in IEEE 3-Generator-9-Bus Test System, and the simulation results are presented. Theobtained results reveal the capability of the proposed estimatorfor dynamic state estimation and measurement noise rejection inpower systems using PMU high rate data. A possible applicationof the dynamic state estimation in power system is proposed, andsome major current challenges of the dynamic state estimation inlarge power grids are also addressed.

Acknowledgment

This work is supported by the Natural Sciences and EngineeringResearch Council (NSERC) of Canada.

References

[1] Z. Huang, K. Schneider, J. Nieplocha, Feasibility Studies of Applying KalmanFilter Techniques to Power System Dynamic State Estimation, in: Power Engi-neering Conference, IPEC, Singapore, December, 2007, pp. 376–382.

[2] E. Ghahremani, I. Kamwa, Dynamic state estimation in power system by apply-ing the extended Kalman filter with unknown inputs to phasor measurements,IEEE Trans. Power Syst. 26 (4) (Nov 2011) 2556–2566.

[3] A.G. Phadke, J.S. Thorp, Synchronized Phasor Measurements and Their applica-tions, Springer, New York, 2008.

[4] W. Gao, S. Wang, On-line Dynamic State Estimation of Power Systems, NorthAmerican Power Symposium (NAPS), USA, Sep 2010, pp. 1–6.

[5] S. Wang, W. Gao, A.P.S. Meliopoulos, An alternative method for power systemdynamic state estimation based on unscented transform, IEEE Trans. PowerSyst. 27 (2) (May 2012) 942–950.

[6] E. Ghahremani, I. Kamwa, Online state estimation of a synchronous generatorusing unscented Kalman filter from phasor measurements units, IEEE Trans.Energy Convers. 26 (4) (Dec 2011) 1099–1108.

[7] P. Tripathy, S.C. Srivastava, S.N. Singh, A divide-by-difference-filter basedalgorithm for estimation of generator rotor angle utilizing synchrophasor mea-surements, IEEE Trans. Instrum. Meas. 59 (6) (June 2010) 1562–1570.

[8] N. Zhou, D. Meng, S. Lu, Estimation of the dynamic states of synchronousmachines using an extended particle filter, IEEE Trans. Power Syst. 28 (4) (Nov2013) 4152–4161.

[9] J.D. Glover, M.S. Sarma, T.J. Overbye, Power System Analysis and Design,Thomson-Engineering, Stamford, 2011.

10] M. Pavella, P.G. Murthy, Transient Stability of Power Systems, John Wiley &Sons, West Sussex, 1994.

11] G. Welch, G. Bishop, An Introduction to The Kalman Filter, 2006, Published July.

12] D. Simon, Optimal State estimation; Kalman, H∞, and Nonlinear Approaches,

John Wiley & Sons, New Jersey, 2006.13] PowerWorld Simulator, ver. 17.0, PowerWorld Corporation, Champaign, IL,

USA.14] Matlab, ver. 7.14.0.739 (R2012a), MathWorks, Natick, USA.