978-1-4799-5022-5/14/$31.00 ©2014 IEEE

Abstract—The accuracy of the models used to predict the

dynamic behavior of an electric power system is vital to ensure the stability and reliability of electricity supply. Parameter estimation of these models is essential for power system planners and operators to maintain a consistent and secure operating environment.

In this work, the whole set of parameters of the synchronous generator coupled to the exciter and governor are estimated using an Unscented Kalman Filter, with input data from phasor measurements obtained by Phasor Measurement Units (PMU). Unlike other estimation methods previously used, excitation voltage and mechanical torque are estimated as states and only the measurements of the electrical power output, current phasor and voltage phasor at the generator terminals are required. A fast estimation algorithm is obtained that allows the estimation of all machine as well as velocity and voltage regulator parameters. Estimation is possible under different levels of measurement noise and initial operating conditions.

Index Terms—Phasor Measurement Units, Kalman Filter,

Synchronous machine, Exciter, Governor

I. INTRODUCTION Many tasks of planning, operation and analysis such as

network expansion, dynamic stability, and the establishment of operational limits of the transmission lines are based on the results of model simulations of the power system [1]. In this process the quality of the models used in power systems is essential for safe and reliable operation. The models include parameters that must be calibrated or estimated at certain intervals of time mainly due to changes in the operating conditions of synchronous machines. Incorporat-ing Phasor Measurements Units (PMU) technology makes possible to obtain synchronized phasor measurements using GPS (Global Positioning System) devices, so that real time data of current and voltage phasors, active and reactive power, and frequency on different nodes of the power system can be recorded.

With higher sampling rates than those obtained in SCADA (Supervisory Control and Data Acquisition), PMU’s can capture the dynamics of the power system more accurately. This can be advantageous when parameter estimation and model validation is required.

This work was supported by the National Program of Research in Energy and Mining of Colciencias.

Juan Carlos Niño, Hernando Diaz, and Andres Olarte are with the Department of Electrical Engineering, Universidad Nacional de Colombia, Bogota (e-mail: jcninopa@unal.edu.co; hdiazmo@unal.edu.co; faolarted@ unal.edu.co).

The estimation of the parameters is usually done by off-

line testing with the generator disconnected from the network, and introducing disturbances using step-like or binary pseudorandom signals [2], [3]. However, this repre-sents economic loss and deterioration of the generator, so it is desirable to perform the parameter estimation using event data using the off-line tests as last resort.

Moreover, there are procedures to estimate the parameters of the generator and excitation system, in which online tests are made by introducing a disturbance, and measuring the frequency response of the system [4], [5]. The drawback of these calibration methods is that the machine can be damaged due to the external disturbances, and as a rule, this type of tests are not permitted by the owners of the machines. Additionally, it is difficult in practice to obtain adequate signals to achieve the estimation of all parameters of the synchronous machine.

Generally, once the required signals of the field tests are obtained, the estimation process proceeds, using different optimization techniques: least squares [6], [7], curve fitting methods such as polynomial fitting [8], trajectory sensitivity [9], [10], series based techniques, such as the Harley’s series [11], and evolutionary algorithms [5], [12]. Other tech-niques for estimating parameters of generators and its voltage and velocity regulators, include Kalman filter varia-tions [13]–[16].

However, these estimation methodologies require knowing in advance several parameters, and require that measure-ments such as field voltage and the mechanical torque be available; if such measurements are not available, then they are considered constant magnitudes. Decoupling the exciter and governor from the generator to calibrate the parameters of each component individually, is also required. In the literature an approach for the simultaneous estimation of all the parameters of the generator coupled to the exciter and governor is not readily available.

Although there are estimation methods using phasor measurements [17]–[20], they do not consider all signals that can be obtained by PMU’s; such is the case of current phasor measured in generator terminals, which currently has not been used in parameter estimation.

For model validation and parameter estimation purposes, a number of ways of representing a network equivalent have been proposed. Models of generators controlled by fast-responding exciters and governors, whose reference signals are the voltage and frequency, measured by a PMU, have been used as network equivalents [21]. Another efficient technique referred to by some authors as event playback [1] or hybrid dynamic simulation [22], [23], also has been used.

Simultaneous Estimation of Exciter, Governor and Synchronous Generator Parameters Using

Phasor Measurements Juan Carlos Niño Pantoja, Andrés Olarte, and Hernando Díaz

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It consists of using recorded actual (or simulated) signals as input to the generator model, considered as a network sub-system, in order to compare the model simulation results against measured event responses.

In practice, the field voltage of the generator and the output mechanical torque of the turbine are not measura-ble. However, these variables may be estimated using a generator model coupled to two control systems: the excita-tion system, or voltage regulator and the velocity regulation system or governor. This paper presents a methodology for estimation of all parameters of a synchronous generator and its control systems, using the Unscented Kalman Filter (UKF) [14], [16], [25], and synchronized phasor measure-ments from a PMU installed at the generator terminals. For this purpose the next measures available from a PMU were used: current and voltage phasors, active and reactive powers delivered by the generator to the network.

This paper is organized as follows. In Section II, the Unscented Kalman Filter (UKF) is described. In Section III, the general algorithm used for simultaneous parameter estimation is presented. In Section IV the general algorithm for the parameter estimation is applied to a system, which includes a generator, a governor and an exciter. In Section V, the estimation results with different initial parameters values and noise level in the measurements are compared. Section VI concludes the paper.

II. UNSCENTED KALMAN FILTER This Kalman filter variant was originally conceived for

state estimation of non-linear systems. Unlike Extended Kalman Filter, the unscented Kalman Filter (UKF) does not require the linearization of the non-linear system as a previ-ous step. This implies that it is not necessary to calculate Jacobian or Hessian matrices, and for that reason it is a method with less computational load, and less prone to errors due to the number of calculations [25]. UKF does not make a linear approximation of the non-linear system, instead it deterministically chooses a set of sample points which have all the information of mean and covariance of the state vector, and are evaluated in the non-linear function [26].

The functionality of UKF can be extended to joint estima-tion of state variables and unknown parameters of the system. The non-linear model for this joint estimation can be expressed through (1) and (2). = = ( , , ) + (1)

= ( , , ) + (2)

where is the parameter vector and is the system state vector, both estimated in iteration k. These two vectors form the -dimensional vector . represents the inputs to the system, represents the non-linear model and is the measurement function. and are the process noise and the measurements noise, both defined as Gaussian variables or normally distributed with covariance and respec-tively. That is, ~ (0, ) and ~ (0, ).

UKF algorithm is described below: Step I. The initial values of state variables and system

parameters , as well as its covariance matrix , are estimated.

Step II. In each iteration , the state variables, the parameters and the outputs are predicted using the infor-mation obtained in the previous iteration, and then updated using the new measurements in the present iteration. This involves the next steps:

1): 2 + 1 points denominated sigma points are determin-istically chosen which capture exactly the important moments (at least the mean and covariance) of the original distribution of : ( ) = ( ) = [( ) ] (3)

( ) = ( ) + ( + )( ) , = 1, … , (4)

( ) = ( ) ( + )( ) , = 1, … , (5)

where and are the estimated means of and at step k 1, respectively. is a scaling parameter defined as: = ( + ) (6)

The positive constants and are used as calibration parameters of the filter.

denotes the ith column of the matrix which is defined as

= (7)

Now the weights related to sigma points are determined: ( ) = + (8)

( ) = + + (1 + ) (9)

( ) = 12( + ) , = 1, … ,2 (10)

( ) = 12( + ) , = 1, … ,2 (11)

The positive constant is also used as calibration parame-ter of the filter.

2): The predicted means of the state variables and parame-ters, and are calculated as

= = ( ) ( )( ) (12)

where ( ) and ( ) are obtained by propagating each sigma point through (1):

( ) = ( )( ) = ( ( ), ( ), ) ( ) + ( ) , i = 0, ..., 2n (13)

Here, ( ), ( ) and ( ) are the components of the ith sigma point, which correspond to the state variables, parame-ters and process noise, respectively.

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3): The covariance is computed from the weighted outer product of the transformed points:

= ( )( ( ) ) ( ( ) ) (14)

This covariance is used to generate new sigma points as specified in (3), (4) and (5). However, this step can be omitted and the sigma points calculated previously can be reused.

4): The predicted mean of the measurement variables, denoted by are computed by

= ( ) ( ) (15)

where ( ) is computed through (2) as ( ) = ( ( ), ( ), ) + ( ), = 0, … , 2 (16)

Here, ( ) represents the component of the ith sigma point corresponding to the measurement noise.

5): The covariance of is computed by

= ( )( ( ) ) ( ( ) ) (17)

6): The means of state variables and parameters at step k are updated using the predicted means and the measure-ments at the current step k: = = + ( ) (18)

where is the Kalman gain, which is calculated by = (19)

Here, is the cross-covariance matrix of the state variables, parameters and measurements, and is computed by: = ( )( ( ) ) ( ( ) ) (20)

Step III. The process given in Steps I and II proceeds iteratively until the difference between two time steps for each parameter is sufficiently small.

III. GENERAL ESTIMATION ALGORITHM The estimation algorithm can be summarized through the following pseudocode: 1. Construction of the non-linear model ( , , ) 2. Inclusion of the measurement model = ( , , ) 3. Selection of , and 4. Estimation of , and 5. Injection of voltage phasor in the model (hybrid dynamic

simulation) 6. m = 1

Repeat 7. = selection of initial parameters 8. = ( ) estimation of initial states, as a function of

the parameters.

9. Execution of UKF for k = 1 to #steps

9.1. Transformation of = at sigma points

9.2. Estimation of mean and covariance of

9.3. Estimation of mean = through the gain

and the measurement vector 9.4. k = k +1 end

10. m = m + 1 until | | < & <

The variable #steps, indicates the number of discrete steps choosen for the algorithm implementation. The variable m indicates the number of the estimation.

IV. IMPLEMENTATION OF THE PARAMETER ESTIMATION ALGORITHM

This section presents the application of the general estimation algorithm to a particular system. The algorithm is used to estimate simultaneously the parameters of a genera-tor model coupled to its velocity regulation system or governor, and to its excitation system.

The measurements of the system response to a disturb-ance, including voltage and current phasors, active and reactive powers, are required. Because of the higher sample rate and accuracy, and the possibility of obtaining phase data, the measurements obtained by PMU’s are ideal for this purpose. Additionally the measurement of the rotor velocity is needed, and although it is not provided by a PMU, generally that measurement is available.

A. Construction of the Non-Linear Model ( , , ) Fig. 1 shows a synchronous generator and its voltage

regulation system (exciter) and its velocity regulation system (governor), connected to an infinite bus. , are voltage and current at generator terminals and , the active and reactive powers delivered from the generator to the infinite bus. is the reactance in p.u. connecting the synchronous machine to the infinite bus. The measurements are taken at the generator terminals by a PMU. The model of the excitation system or voltage regulator is shown in Fig. 2 [24]. is the field voltage in p.u., is the exciter gain in p.u./p.u., is the time constant of the transducer in seconds. is the setpoint of the controller in p.u. is the initial voltage in terminals in p.u.

Fig. 3 shows the block diagram of the governor-turbine system, with a type A steam turbine (non-reheat) [27]. is the reference value of the controller in p.u. is the open-ing percentage in p.u. of the turbine steam inlet valve. is the mechanical torque in p.u. delivered from the turbine to the generator. is the turbine time constant in seconds, and is the regulation constant of the governor in p.u.

It is assumed that the controller’s settings and are known from the operating conditions when the measure-ments were taken with the PMU.

A number of works dedicated to parameter estimation for generators have used the classic fourth order generator model (or two axis model) [14], [24] and they consider it adequate for transitory stability studies. In this work the fourth order model was used to demonstrate the effective-ness of the methodology proposed.

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Fig. 1. Generator connected to an infinite bus trough a reactance .

Fig. 2. Block diagram of the excitation system used.

Fig. 3. Block diagram of the governor-turbine system used.

The non-linear system ( , , ) that includes the dynamic equations of generator, governor and exciter is: = (21) = (22) = ( ) (23) = + ( ) (24) = (25) = + 1

(26) = + (27)

where is the rotor angle in radians, is the deviation in p.u. of the rotor velocity with respect to synchronous velocity , that is, - . is the mechanical torque in p.u., is the electrical torque in p.u. and are the components in the d-q axis of transient voltage , ex-pressed in p.u. and are the components in the d-q axis of generator output current, in p.u: = ( )

(28)

= ( + ) (29)

Since the output electric power is: = (30)

it is assumed that the angular velocity of the rotor is 1, then = and can be calculated as: = = + . (31)

The components of voltage terminal and are: = sin( )– cos ( ) (32) = cos( ) + sin ( ) (33) The real and imaginary components of the voltage phasor and can be found by: = cos( ) (34) = sin( ) (35) Here, is the angle in radians of the voltage phasor with

respect to a reference, for the case study, the angle of the infinite bus that is equal to zero radians. The terminal voltage can be introduced directly from the PMU measurements or can be calculated with (36). = + (36)

B. Inclusion of the Measurement Model The measurement model proposed = ( , , ),

which is independent of the generator model order, is the following: = = sin( ) + cos ( ) (37) = = cos( ) + sin ( ) (38) = = + (39) = = (40) = (41) where , are the real and imaginary components of the current measured at the generator terminals; , are also obtained from the PMU. It is assumed that the measurement of rotor speed is available, to compute .

C. Selection of , and UKF requires the introduction of the values of , and

which are used in the conformation of sigma points. These parameters can be considered as calibration parameters; the values used in this paper are: = 0.0001, = 2, = 0, and its choice was made according to typical values proposed in [16] and [26].

D. Estimation of , and Additionally, the matrices , and used in (7),

must be estimated. In this work, the process noise was not taken into account therefore the matrix corresponding to the covariance of the process noise was established as = (2 × 10 ), a diagonal matrix with a principal diagonal formed by elements with a value of 2 × 10 . The matrix is introduced as = ( ) where depends of the noise level in the signals measured by the PMU. In this work, the values = 2 × 10 and = 2 × 10 were used. The matrix is indicative of the degree of confidence in the values of vector z. It was ob-served that the values of the principal diagonal in matrix must be changed when the measurement noise increases. In this case when = 2 × 10 , = (1 × 10 ) was used, and when = 2 × 10 , = (1 × 10 ) was used.

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E. Injection of Voltage Phasor in the Model (Hybrid Dynamic Simulation)

This basically means that in each iteration k the non-linear functions ( , , ) and ( , , ) must be evaluated using the magnitude and phase of the voltage phasor measured in generator terminals as inputs.

Using hybrid dynamic simulation, with the actual parame-ters of the system, the non-linear model proposed is able to reproduce very accurately the PMU measurements. Fig. 4 shows active power of generator simulated (Measured) and the response of the differential equations model after the injection of voltage phasor (Simulated). The same applies to reactive power in Fig. 5.

F. Selection of Initial Parameters Table I shows the 12 parameters to be estimated. In the

generator model, is the friction coefficient and was not estimated because its typical value is considered to be negligible. In this work a value of = 5 × 10 was considered.

G. Estimation of Initial States as a Function of the Initial Parameters

A number of successive estimations can be required to find the values of the parameters, and in each estimation the initial values of state variables are updated from the parame ters of the previous estimation. For example, the state , initial rotor angle in radians is calculated using the follow-ing equations:

Fig. 4. Comparison between the measurement of active power ("Measured"), and that obtained by the model after injecting the voltage phasor as measured by the PMU ("Simulated").

Fig. 5. Comparison between the measurement of reactive power ("Measured"), and that obtained by the model after injecting the voltage phasor as measured by the PMU ("Simulated").

TABLE I. LIST OF PARAMETERS TO BE ESTIMATED

Parameter Description Real Value Units Inertia constant 20 p.u.

Quadrature-axis transient open circuit time constant 0.6 s

Direct-axis transient open circuit time constant 3.7 s

Quadrature-axis reactance 2 p.u. Direct-axis reactance 1.56 p.u.

Quadrature-axis transient reactance 0.4 p.u.

Direct-axis transient reactance 0.296 p.u.

Excitation system gain 200 p.u.

Time constant of the transducer 0.05 s

Governor time constant 0.05 s Turbine time constant 3 s

Regulation constant of the governor 0.05 p.u.

= = jj (42) = arg ( + ). (43) Here is the initial voltage measured in terminals in

p.u, with its real and imaginary components and . and are initial active and reactive powers in p.u., which form the complex number . All of these magnitudes are acquired through the PMU. The parameter is the initial value in p.u. of the quadrature-axis reactance found in the previous estimation. The new calculated value of is introduced as an initial condition in the next estimation.

V. RESULTS The parameter estimation algorithm was implemented

using simulated data of the system, which consists in a synchronous machine connected to an infinite bus (see Fig. 1). Because sudden load increase is an event that is more likely to occur than a three-phase fault, for the imple-mentation of the algorithm, measurements of the system response to a generator load increase from 0.32 p.u. to 1.4 p.u. were used (see Fig. 4).

All parameters of generator, voltage regulator or exciter, and governor were estimated. The errors in the measure-ments were simulated adding a noisy signal to the meas-urement signals or simulation data sets. The noisy signal consists of a normally distributed pseudorandom variable with a standard deviation , where i = 1, ..., . Two noisy signals were used, the first with = 2 × 10 and the second with = 2 × 10 .

Table II shows the results of successive estimations to calibrate simultaneously all parameters, when the noise in measurements had a variance = 2 × 10 , and with different error percentages in the initial parameters.

Table III shows the results of successive estimations to calibrate simultaneously all parameters, when the noise in measurements had a variance = 2 × 10 . In this case, when the initial error of the parameters was ± 30%, it was not possible for any of the parameters, to obtain values close to the real ones.

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TABLE II. RESULTS FOR THE PARAMETER ESTIMATION WITH A NOISY SIGNAL ( = 2 × 10 ). ERROR PERCENTAGE IN THE ESTIMATED VALUE WITH AN INITIAL PARAMETER ERROR OF ±10%, ±20%, AND ±30%

Parameter Error ±10% Error ±20% Error ±30% 0.46% 0.18% 0.25% 0.19% 0.25% 1.87% 2.20% 3.40% 6.20% 0.00% 0.00% 0.00% 0.42% 0.17% 0.18% 0.13% 0.10% 0.00% 0.52% 0.21% 0.29% 0.03% 0.00% 0.07% 0.06% 0.10% 0.16% 0.02% 0.03% 0.05% 3.00% 0.80% 10.00%

0.01% 0.02% 0.02%

TABLE III. RESULTS FOR THE PARAMETER ESTIMATION WITH A NOISY SIGNAL ( = 2 × 10 ). ERROR PERCENTAGE IN THE ESTIMATED

VALUE WITH AN INITIAL PARAMETER ERROR OF ±10% AND ±20%

Parameter Error ±10% Error ±20% 0.46% 1.13% 1.77% 1.48% 4.80% 11.20% 0.00% 0.20% 0.27% 0.64% 0.48% 0.13% 0.53% 1.29% 0.24% 0.57% 0.44% 0.23% 0.03% 0.10% 13.40% 20.40%

0.04% 0.12%

As can be noted in Table II and Table III, in some cases the larger the initial error of parameters, the lesser the error in the estimated value. Because the noise is a stochastic variable and since all the values of the parameters are chang-ing at same time in each iteration k, there is not a direct correlation between the initial error and the estimation error.

Fig. 6 shows the results of the estimation for one of the exciter parameters, using successive estimations. The initial parameters error is ±10%, with a noisy signal ( = 2 × 10 ). Fig. 7 shows the results of the estimation for the same exciter parameter, using successive estimations and a noisy signal with covariance = 2 × 10 .

The initial parameters error is ±10%. It is clear that in the estimation process using a noisy signal with a smaller noise level, 17 estimations was required to stabilize near to the real value of the parameter. Incrementing the noise level, also increments the number of estimations required to 25, as well as the computing time.

It can be seen in Fig. 7 that despite the fact that the algorithm gets very close to the true value of the parameter on iteration 11, the estimates keep varying. Due to the error level, some parameters do not reach his final value in a few iterations. This affects directly the estimation of the other parameters. Only after iteration 20, do all the parameters simultaneously converge to an estimated value.

VI. CONCLUSION An algorithm for parameter estimation of a generator

coupled to a voltage regulation system and a governor using measurements from a PMU connected to generator terminals

Fig. 6. Number of successive estimations required for the parameter , when the initial parameters values had an error of ±10%, and with a noisy PMU signal with variance = 2 × 10 .

Fig. 7. Number of successive estimations required for the parameter , when the initial parameters values had an error of ±10%, and with a noisy PMU signal with variance = 2 × 10 . was designed. This algorithm uses a variant of Kalman filter, the Unscented Kalman Filter (UKF), to estimate simul-taneously all variable states and parameters of the system mentioned above. The algorithm was tested on a simulated fourth order generator model, connected through a reactance to an infinite bus. Noisy measurements were created adding a random variable to the exact values obtained from the dynamic simulation. Two different noise levels were used and the algorithm made a very accurately parameter estima-tion when the lower noise level was used.

The hybrid dynamic simulation technique was imple-mented successfully allowing the isolation of the generator from the power system, to estimate the parameters based on the simulated data and the measurements. The use of this technique could allow the generalization of the estimation, to include multi-machines systems using only the infor-mation obtained from the target generator after a disturb-ance or event, which can occur not only at a point near the generator but on any power system node. The application of the methodology to a complex power system is currently the subject of additional work and it will be reported elsewhere.

Since some difficulties arose in the estimation process when higher noise levels were used in the measurements, particularly when the error in the initial values of all the parameters was ±30%, to achieve the correct values of parameters with any level of noise, the use of a larger set of measurements obtained from the PMU might be necessary.

Through multiple estimations with different reference measurements, the estimated values of parameters could be updated to obtain more accurately results.

When the higher noise level was used, although some parameters values were not estimated correctly – some of them with estimation errors above 10%, it should be noted that this error does not affect the dynamic response of the

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non-linear model. This is reflected in a good estimation of the state variables behavior and in an almost perfect tracking of the PMU measurements, despite the errors in the parame-ter estimation.

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BIOGRAPHIES Juan Carlos Niño Pantoja received a B.S. degree in Electrical

Engineering from the National University of Colombia in 2010. Currently, he is a student of the Master’s program in Industrial Automation at the same institution. Mr. Niño’s main research interest is in power systems operation and control.

Hernando Diaz received a B.S. in Electrical Engineering from the National University of Colombia in 1978. He was awarded an M.S. Degree in Electric Power Engineering and a Ph.D. in Electrical Engineering from Rensselaer Polytechnic Institute, Troy, NY in 1984 and 1986, respectively. He has been with the Department of Electrical Engineering at the National University of Colombia, Bogota, where he is a Professor and Director of the graduate program.

Andres Olarte received a B.S. in Electronics Engineering from the Universidad Distrital, Bogota, Colombia in 2005. He was awarded an M.S. Degree in Industrial Automation and a Ph.D. in Electrical Engineering from National University of Colombia in 2007, and 2011, respectively. He has been with the Department of Electrical Engineering at the National University of Colombia, Bogota, where he is an Assistant Professor of the graduate program.

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