Online State Space Model Parameter Estimation in Synchronous ...

Online State Space Model Synchronous Machines

Parameter

Estimation

in

Z. Gallehdari*, M. Dehghani**(C.A.) and S. K. Y. Nikravesh*

Abstract: In this paper a new approach based on the Least Squares Error method for estimating the unknown parameters of the 3rd order nonlinear model of synchronous generators is presented. The proposed approach uses the mathematical relationships between the machine parameters and on-line input/output measurements to estimate the parameters of the nonlinear state space model. The field voltage is considered as the input and the rotor angle and the active power are considered as the generator outputs. In fact, the third order nonlinear state space model is converted to only two linear regression equations. Then, easy-implemented regression equations are used to estimate the unknown parameters of the nonlinear model. The suggested approach is evaluated for a sample synchronous machine model. Estimated parameters are tested for different inputs at different operating conditions. The effect of noise is also considered in this study. Simulation results declare that the efficiency of the proposed approach. Keywords: Identification, Nonlinear Model, Regression Equation, State Space Model, Synchronous Machine.

1 Introduction1 As the power system becomes more complicated and more interconnected, modeling and identification of its elements become more essential. Synchronous generators play an important role in the stability of power systems, so the accurate modeling of synchronous generators is essential for a valid analysis of dynamic and stability performance in power systems [1]. There are many different methods for synchronous machine modeling [2-4], but we can categorize all modeling techniques to three classes: white box [5-6], grey box [7-9] and black box [10-12]. The first category assumes a known structure for the synchronous machine such as the traditional methods, which are specified in IEEE and IEC standards [13]. These approaches are often conducted under off-line condition. The parameters obtained by these methods may not truly characterize the synchronous machine under various loading conditions [8]. According to the disadvantages of off-line methods, on-line parameter estimation approaches are of more

Iranian Journal of Electrical & Electronic Engineering, 2014. Paper first received 16 Jan. 2013 and in revised form 6 Oct. 2013. * The Authors are with Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran. ** The Author is with School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran. E-mails: [email protected], [email protected] and [email protected].

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interest in the recent years. The second and the third categories both use online measurements to estimate synchronous generator parameters, but they act in different ways. In the second type of identification methods, a known mathematical model for synchronous generator is assumed and on-line measurements are used to estimate machine’s physical parameters. In the third one, input data set is mapped to the output data set without considering any known structure for the model. In synchronous generators modeling, we have two kinds of nonlinearities. The first kind of nonlinearities is structured nonlinearities, such as sine and cosine functions of the rotor angle, which are modeled in the well-known nonlinear structures of synchronous models. On the other hand, the unstructured nonlinearities e.g. magnetic saturation in the iron parts of the rotor and stator are not usually considered in the structure of the models and instead are reflected by adjusting the physical parameters of the nonlinear model based on the online measurements [14]. In this paper, an analytical identification procedure for the 3rd order model of synchronous generators is suggested. The proposed method estimates physical parameters of synchronous machine. In [1, 9, 14-15] different algorithms for estimation of physical parameters of synchronous machine are presented. In [9] an approach for synchronous generator parameter estimation is suggested which needs to apply a short circuit on the generator terminals. This short circuit test

Iranian Journal of Electrical & Electronic Engineering, Vol. 10, No. 2, June 2014

t normal operation of the machine. In [14] [ will disturb the a method iss suggested for f the synchhronous machhine parameter esstimation, whhich results inn some nonlinnear set of equatioons. In each teest, the nonlinnear set shouldd be solved numerically. Furtheermore, the allgorithm does not discuss the conditions c in which w the sysstem of equatiions can be solveed. In some situations, thee algorithm may m result in a sinngular set of equations andd no solution can be found. Inn [15], an annalytic approaach is suggessted which needs to perturb the field voltage with the PR RBS signal. Thiss signal is not n harmful for the norm mal operation off the machinee. The suggessted method uses u not only thee mathematiccal relation between b machhine parameters, but also it needs n to idenntify the systtem eigenvalues using the Proony approach.. Here, we tryy to find an algorrithm, which uses u the matheematical relatiions of the param meters and the extra step of Prony P is omitted. In other wordds, we try to extend e the meethod of [15] and find a way too omit its disaddvantage. In this study, we intrroduce an annalytical, straiight forward andd easy-implem mented methhod, uses onlline measurementts to estimatte the unknoown parametters. Here we onlly need to soolve two regreession equatioons. Then, using some simplle relations all a the unknoown parameters of o 3rd order noonlinear state space model are estimated. The papeer is organizedd as follows: the synchronnous generator model m and thhe identificatiion method are described inn Sections 2 and 3, reespectively. The T simulation reesults are proovided in Secttion 4. Sectioon 5 concludes the paper. nerator Model 2 The Syncchronous Gen In this paaper, the thirdd order nonlinnear synchronnous generator connected to ann infinite bus is consideredd as the system under study (Fig. 1). In this model, the stator’s dynnamics and the t effects of o dampers are neglected, soo we have onlly one electrical equation (the ( field dynamiic). This moddel can be ussed for studyying low-frequenccy oscillationns and stabiility analysis of power system ms [1, 14]. The thirdd order nonlinnear model deerived in [16--18] is used in thiis paper. All the t parameterrs are assumedd to be in per unnit values. Thhe model is described d by the following noonlinear equatiions: δ = ω 1 ω = (Tm − Te − Dω ) (1) J 1 eq′ = ( E FDD − eq′ − ( xd − xd′ )id ) Tdo′ where:

Fig. 1 structurre of the study system. s

id =

eq′ − V B cos δ x d′

iq =

V B sin δ xq

Te ≅ Pe =

(2))

VB V2 1 1 eq′ sin s δ + B ( − ) sin(2δ ) ′ 2 x q x d′ xd

In this model, the influencce of magneticc saturation iss neg glected, so x d , x q and xd′ are asssumed to bee con nstant. x d , x q and xd′ aree the augmentted reactance,, i.e.. the line andd transformer reactances arre added withh theem [15]. The definition d of tthe variables and constantss is given g in the Appendix. A In our studyy, it is requirred to find th he state spacee mo odel of the sysstem, so the abbove model iss converted too thee state spacee one. The ssystem statess, inputs andd outtputs are definned as followss: ⎡x1 ⎤ ⎡δ ⎤ ⎡u ⎤ ⎡ E ⎤ ⎢ ⎥ X = ⎢⎢ x 2 ⎥⎥ = ⎢ ω ⎥ , U = ⎢ 1 ⎥ = ⎢ FDD ⎥ , (3)) ⎣u 2 ⎦ ⎣ T m ⎦ ⎢⎣ x 3 ⎥⎦ ⎢⎣e q′ ⎥⎦ ⎡y ⎤ ⎡δ ⎤ Y = ⎢ 1⎥ = ⎢ ⎥ ⎣ y 2 ⎦ ⎣ Pe ⎦ The state sppace nonlineaar model of the t system iss giv ven in the folloowing [15]: ⎡ ⎤ ⎡ ⎤ 0 ⎥ ⎢0 1 ⎢ 0 0⎥ ⎥ ⎡x1 ⎤ ⎢ ⎥ ⎡ x 1 ⎤ ⎢ 1 ⎥ ⎡u 1 ⎤ D ⎥ ⎢ ⎢ ⎢x ⎥ + 0 ⎢ x 2 ⎥ = 0 − 0 ⎥⎢ 2⎥ ⎢ J J ⎥ ⎢⎣u 2 ⎥⎦ ⎢ x ⎥ ⎢ ⎣ 3⎦ ⎢ ⎥ ⎣x 3 ⎦ ⎢ ⎥ ⎢0 0 − 1 ( x d )⎥ ⎢ 1 0⎥ ⎢⎣ ⎢⎣ T d′0 ⎥⎦ T d′0 x d′ ⎦⎥

⎡ ⎤ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 VB V B2 1 1 + ⎢− ( x 3 sin x 1 + ( ) sinn 2 x 1 )) ⎥ − 2 xq x d′ ⎥ ⎢ J x d′ ⎥ ⎢ 1 x d − x d′ ( )V B cos x 1 ⎥ ⎢ ′ ′ T x d 0 d ⎦ ⎣ (4)) From [15], the t linearizedd model of Eq. (4), in thee viccinity of an operating o poiint ‘o’, will conclude thee kno own Heffron--Philips modell, which is giv ven below: ⎡ ⎢ 0 ⎡ x1 ⎤ ⎢ ⎢ x ⎥ = ⎢ − K 1 ⎢ 2⎥ ⎢ J ⎢⎣ x 3 ⎥⎦ ⎢ ⎢− K 4 ⎢⎣ T do′

⎤ ⎡ ⎢ 0 ⎥ ⎥ ⎡x1 ⎤ ⎢ D K ⎥ ⎢ − − 2 ⎥ ⎢⎢ x 2 ⎥⎥ + ⎢ 0 J J ⎥ ⎢x ⎥ ⎢ 1 ⎥⎣ 3⎦ ⎢ 1 − 0 ⎢⎣T do′ K 3T do′ ⎦⎥ ⎡x1 ⎤ ⎡ y 1 ⎤ ⎡ Δδ ⎤ ⎡ 1 0 0 ⎤ ⎢ ⎥ Y =⎢ ⎥=⎢ ⎥ ⎢x 2 ⎥ ⎥=⎢ ⎣ y 2 ⎦ ⎣ ΔPe ⎦ ⎣ K 1 0 K 2 ⎦ ⎢ x ⎥ ⎣ 3⎦ 1

0

Gallehdari et al: Online State S Space Moodel Parameteer Estimation in Synchronoous Machines

⎤ 0⎥ ⎥ 1 ⎥ ⎡u 1 ⎤ J ⎥ ⎢⎣u 2 ⎥⎦ ⎥ 0⎥ ⎥⎦

(5))

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where: x V 1 1 (6) K 1 = 30 B cos x10 + V B2 ( − ) cos 2 x10 x d′ x q x d′ V sin x10 (7) K2 = B x d′ x′ (8) K3 = d xd V sin x10 (9) K4 = B ( x d − x d′ ) x d′ To avoid complexity in equations, we define unknown parameters in Eq. (5) as p1, p2, …, p7. The linear dynamical model can be written as follows: 1 0⎤ 0⎤ ⎡0 ⎡0 ⎢ ⎥ ⎢  X = ⎢ p1 p2 p3 ⎥ X + ⎢ 0 p7 ⎥⎥ U ⎢⎣ p4 0 p5 ⎥⎦ ⎢⎣ p6 0 ⎥⎦  

 

A

⎡ ⎢ 1 Y =⎢ ⎢ p1 ⎢− p ⎣ 7

(10)

B

⎡ 1 ⎡ y1 (k ) ⎤ ⎢⎢ ⎢ ⎥= ⎣ y2 (k ) ⎦ ⎢ − p1 ⎢ p ⎣ 7

⎤ 0 ⎥ ⎡ x1 (k ) ⎤ ⎥ ⎢ x2 (k ) ⎥ ⎥ p3 ⎥ ⎢ 0 − ⎥ ⎢⎣ x3 (k ) ⎥⎦ p7 ⎦

3 Identification Method In the identification methods, the signals are sampled and the samples are used to identify the model. On the other hand, the discrete equations are difference not differential type; this causes the discrete equations to be easy for implementation. Because of these two reasons, we discretize the synchronous generator model with sample time ‘h’ and use the discrete model instead of the continuous one. The discrete model of the system in Eq. (10) is given in the following:

(13)

Since our goal is to find the relation between the machine’s parameters and its outputs, we write the states in terms of the machine’s outputs. (14) x 1 (k ) = y 1 (k )

x 2 (k ) =

⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ (11) 2 VB VB 1 1 ⎢1 ⎥ ⎢ J (T m − x ′ x 3 sin x 1 − 2 ( x − x ′ ) sin 2 x 1 )) ⎥ d q d ⎢ ⎥ ⎢ ⎥ − ( p 5 + p 6 )V cos x 1 ⎢ ⎥ ⎣ ⎦ x1 ⎡ ⎤ ⎥ V 2⎛ 1 1 ⎞ y = ⎢V ⎢ x 3 sin x 1 + − ⎜⎜ ⎟⎟ sin 2 x 1 ⎥ 2 ⎝ x q x d′ ⎠ ⎥⎦ ⎢⎣ x d′ Comparing the above equations, it is obvious that the linear and the nonlinear models have some common parameters, which are independent from the operating point condition. These common parameters are used to identify the nonlinear model.

(12)

0

y 1 ( k + 1) − y 1 ( k ) h p p x 3 ( k ) = − 7 ( y 2 ( k ) + 1 y 1 ( k )) p3 p7

⎤ 0 ⎥ ⎥X p3 ⎥ 0 − ⎥ p7 ⎦ 0

The corresponding nonlinear model will be: 0 ⎤ ⎡x1 ⎤ ⎡ 0 0⎤ ⎡ x1 ⎤ ⎡ 0 1 u ⎢ x 2 ⎥ = ⎢ 0 p 2 0 ⎥ ⎢ x 2 ⎥ + ⎢ 0 p 7 ⎥ ⎡ 1 ⎤ + u ⎢ ⎣ ⎢⎣ x 3 ⎥⎦ ⎢⎣ 0 0 p 5 ⎥⎦ ⎢⎣ x 3 ⎥⎦ ⎢⎣ p 6 0 ⎥⎦ 2 ⎦⎥

126

h 0 ⎤ ⎡ x 1 (k ) ⎤ ⎡ x 1 ( k + 1) ⎤ ⎡ 1 ⎢ x ( k + 1) ⎥ = ⎢ hp 1 + hp hp 3 ⎥⎥ ⎢⎢ x 2 ( k ) ⎥⎥ 2 ⎢ 2 ⎥ ⎢ 1 ⎢⎣ x 3 ( k + 1) ⎥⎦ ⎢⎣ hp 4 0 1 + hp 5 ⎥⎦ ⎢⎣ x 3 ( k ) ⎥⎦ 0 ⎤ ⎡ 0 ⎡u ( k ) ⎤ + ⎢⎢ 0 hp 7 ⎥⎥ ⎢ 1 u ( k ) ⎥⎦ 0 ⎦⎥ ⎣ 2 ⎣⎢ hp 6

(15) (16)

Substitute the values of x2(k) and x3(k) from Eqs. (15) and (16) in Eq. (12) to conclude the followings: y 1 ( k + 2 ) − y 1 ( k + 1) h 1 + hp 2 + ( y 1 ( k + 1) − h p + h p 3 ( − 7 )( y 2 ( k ) + p3

−

= h p 1 y 1 ( k ) + h p 7u 2 ( k ) y 1 ( k ))

(17)

p1 y 1 ( k )) p7

p7 p ( y 2 ( k + 1) + 1 y 1 ( k + 1)) = hp 4 y 1 ( k ) − p3 p7

(18)

p p (1 + hp 5 ) 7 ( y 2 ( k ) + 1 y 1 ( k )) + hp 6u 1 ( k ) p3 p7

The Eq. (17) can be written as follows: y 1 ( k + 2 ) − y 1 ( k + 1) = h (19) ⎡ y 1 ( k + 1) − y 1 ( k ) ⎤ h ( u ( k ) y ( k ) ) − × 2 2 ⎢⎣ ⎥⎦ h ⎡1 + h p 2 ⎤ ⎢ p ⎥ 7 ⎣ ⎦

Since Eq. (19) is a linear regression equation, we can use the Least Square Error (LSE) method to identify the unknown parameters p2 , p7 . If we apply N different time points to the above formula, the following equation, which is a proper format for LSE method will be obtained [19].

Iranian Journal of Electrical & Electronic Engineering, Vol. 10, No. 2, June 2014

y 1 (3) − y 1 (2 ) ⎡ ⎤ ⎢ ⎥ h ⎢ ⎥ # ⎢ ⎥ ⎢ y ( k + 2) − y ( k + 1) ⎥ 1 ⎢ 1 ⎥= h ⎢ ⎥ ⎢ ⎥ # ⎢ ⎥ y ( N ) − y ( N − 1) 1 ⎢ 1 ⎥ ⎥ ⎣⎢    h 

⎦

⎡ hu1 (1) ⎤ ⎢ ⎥ # ⎢ ⎥ ⎢ hu1 (k ) ⎥ = ⎢ ⎥ # ⎢ ⎥ ⎢⎣hu1 (N − 1)⎥⎦  

Y2

Y1

y 1 ( 2) − y 1 (1) ⎡ ⎤ h (u 2 (1) − y 2 (1)) ⎢ ⎥ h ⎢ ⎥ # # ⎢ ⎥ ⎢ y ( k + 1) − y ( k ) ⎥ 1 1 h (u 2 ( k ) − y 2 ( k )) ⎢ ⎥ h ⎢ ⎥ ⎢ ⎥ # # ⎢ ⎥ y ( N 1) y ( N 2 ) − − − 1 ⎢ 1 h (u 2 ( N − 2 ) − y 2 ( N − 2 )) ⎥ ⎥ ⎣⎢    h               

⎦ H1

⎡1 + h p 2 ⎤ ×⎢ p 7 ⎥⎦ ⎣   

θ1

(20) where from [20], we can estimate the parameters as follows: θ 1 = ( H 1T H 1 ) − 1 H 1T Y1 (21) From Eqs. (20) and (21), the values of D and J can be calculated as follows: 1 (22) J = θ1 (2) − θ 1 (1) + 1 (23) D = hθ1(2) After estimating the mechanical parameters (D and J), consider Eq. (18) and divide it by p6 , then rewrite it in a suitable form for the LSE method, as follows: hu 1 ( k ) =

[ y 2 (k ) − y 2 (k

+ 1)

y 1 ( k ) − y 1 ( k + 1) hy 2 ( k ) hy 1 ( k ) ]

p7 ⎡ ⎤ ⎢ p p ⎥ 3 6 ⎢ ⎥ ⎢ ⎥ p1 ⎢ p p ⎥ 3 6 ⎥ ×⎢ ⎢ p5 p7 ⎥ ⎢ ⎥ ⎢ p3 p6 ⎥ ⎢ p1 p 5 p 4 ⎥ − ⎥ ⎢ ⎣ p3 p6 p6 ⎦

(24) To reduce the effect of measurement noise, the formula is considered for N different time points (see Eq. (25). Equation (25) is a linear regression equation. We can write: (26) θ 2 = ( H 2T H 2 ) − 1 H 2T Y 2 where (see Eq. (27)):

y 1 (1) − y 1 (2) hy 2 (1) hy 1 (1) ⎤ ⎡ y 2 (1) − y 2 (2) ⎢ ⎥ # # # # ⎢ ⎥ ⎢ y 2 (k ) − y 2 (k + 1) y 1 (k ) − y 1 (k + 1) hy 2 (k ) hy 1 (k ) ⎥ ⎢ ⎥ # # # # ⎢ ⎥ ⎢⎣ y 2 (N − 1) − y 2 (N ) y 1 (N − 1) − y 1 (N ) hy 2 (N − 1) hy 1 (N − 1)⎥⎦

   H2

⎡ p7 ⎤ ⎢ pp ⎥ 3 6 ⎢ ⎥ ⎢ p1 ⎥ ⎢ pp ⎥ 3 6 ⎥ ×⎢ ⎢ p5 p 7 ⎥ ⎢ ⎥ ⎢ p3 p 6 ⎥ ⎢ p1 p5 p4 ⎥ − ⎥ ⎢ p3 p 6 p 6 ⎦ ⎣ 

θ2

(25)

p p pp pp p θ2T = [ 7 , 1 , 5 7 , 1 5 − 4 ] p3 p6 p3 p6 p3 p6 p3 p6 p6

(27)

Now we use θ2 to estimate the unknown parameters of synchronous machine. Due to Eq. (22) and Eq. (27), we have: θ (2) (28) p3 p6 = 1 θ 2 (1) θ (3) (29) p5 = 2 θ 2 (1) From Eq. (5), Eq. (7), Eq. (8) and Eq. (10), p5 and p3p6 can be written in terms of the machine’s parameters. We can write: −1 x′ 1 T d′0 d − p5 K 3T d′0 xd J = = = xd K V x 1 sin p3 p6 − 2 V B sin y 10 10 B J T do′ x d′ −1 T do′ J (30) From the above equation we can estimate xd as follows:

xd =

p5 VB sin y10 p3 p6 J

(31)

It should be noted that we can substitute the values of p5, p3p6 and J from Eq. (28), Eq. (29) and Eq. (22), respectively. Moreover, looking at θ2(2), θ2(1) and θ2(4) in Eq. (27), p1 and p4/p6 can be calculated as follows:

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127

θ 2 ( 2 )θ 1 ( 2 ) (32) θ 2 (1) p 4 θ 2 (3)θ 2 ( 2 ) (33) = − θ 2 (4) p6 θ 2 (1) Using Eq. (33) and Eqs. (9)-(10), p4 and p6 can be written in terms of the machine’s parameters. Their ratio is given below: x − x d′ p4 (34) )V B sin x 10 = −( d p6 x d′ p1 =

In the above equation, V B sin x 10 , x d are known and x d and p4/p6 are estimated before, so it can be used to

estimate xd′ . x d′ =

xd p4 1 1 − ( )( ) p 6 V B sin y10

(35)

Since xd and xd′ are known, Eq. (5)-(10) can be used to calculate Tdo′ as follows:

Tdo′ = −

xd p 5 x d′

(36)

The only unknown remained parameter is xq From Eq. (2), we have:

Pe =

VB V2 1 1 eq′ sin δ + B ( − ) sin(2δ ) xd′ 2 xq xd′

(37)

eq′ = VB cos δ + xd′ id Substituting eq′ into Pe and rewrite it into the following form:

Pe − V B id sin δ =

V B2 1 sin(2 δ ) 2 xq

(38)

Thus we will achieve the following formula: V B2 s in ( 2 δ ) xq = (39) 2 ( Pe − V B i d sin δ ) Writing Eq. (39) in operating point o, xq is calculated as follows: V B2 sin(2 x10 ) (40) xq = 2( P0 − V B id 0 sin x10 ) In the above equation, Po is the active power, ido is the direct axis current and x10 is the value of rotor angle at the operating point. It is assumed that the value of Po and x10 are measured and ido is calculated from the following relations [16, 17]: P 2 +Q 2 Q ϕ = arctan( ), I = , i d = I sin(δ + ϕ ) (41) P VB P0 2 + Q 0 2 Q (42) sin( x 10 + tan −1 ( 0 )) VB P0 In the above equation, Qo is the reactive power at the operating point. Since the equations in Eq. (20) and Eq. (25) can be updated online, we can use the whole algorithm for id 0 =

128

online identification of the machine parameters. It means that we can estimate the machine parameters when it is in service and we do not need to turn it off. 4 Simulation Results In this section, the proposed approach is evaluated by simulating the system presented in Fig. 1. The estimated parameters are compared to the simulated ones and the model is validated under different operating conditions. Finally, the effect of noise on the parameters is examined. 4.1 Model Simulation Our goal is to find a practical approach for synchronous generator model identification, so the mechanical torque is assumed to be constant and only the field voltage, which can be perturbed more easily, is perturbed. To evaluate the proposed method, a PseudoRandom Binary Sequence (PRBS) with amplitude of 5% of the nominal value is applied to the field voltage [18]. Since the change in the field voltage is very small compared to the nominal value, it will not disturb the normal operation of the system too much. When the field voltage is perturbed with the PRBS, the active output power, the rotor angle and the field voltage are sampled with sample time h=2 millisecond. The operating point is considered to be P=0.9, Q=0.1, vt=0.98. The input/output data collected from the system model in this operating point are shown in Fig. 2. After simulating the model at operating point P=0.9, Q=0.1 and vt=0.98, the field voltage, the rotor angle and the active power signals are sampled, then matrixes in Eq. (20) and Eq. (25) are obtained. Afterward, θ1 and θ2 are estimated using LSE method. Finally, using the procedure of the previous section, the machine parameters can be calculated. The results of this procedure are given in Table 1. According to the third column of the Table 1, we can conclude that the estimation error is acceptable. In reality, we don’t know the real value of the parameters. To validate the results, the outputs of real model and the identified one are compared. In the next part, the procedure of model validation is discussed. 4.2 Model Validation In this section the state space model, achieved in the previous section is validated. For this purpose, several different input signals are applied to the field voltage and the model performance in different operating conditions is observed. The first operating point is considered as P=0.8 pu, Q=0, vt=1.05 pu. The system is excited by a PRBS input and the power and the rotor angle are measured. The results are presented in Fig. 3. To calculate the estimation error, we use the following indexes [20]: I =

1 N

N

∑ ( y (k ) − yˆ (k ))

2

(43)

k =1

Iranian Journal of Electrical & Electronic Engineering, Vol. 10, No. 2, June 2014

Fig. 2 data colllected at operating point P=0.9, Q=0.1, vt=0.98. Fig g. 4 autocorrelattion function off Fig. 3.

The values of o I, FPE and PR

ee

FPE fo or PRBS inputt

ould be: wo

Iδ = 1.8521e-0005, I P = 4.10009e-006 FP PEδ = 1.8576e-005, FPE EP = 4.1132e-006 PR _ δ = 100% %,

PR _ P = 100%

ee

Fig. 3 data colllected at operating point P=0.8, Q=0, vt=1.055. Table 1 Synchhronous Machinne Parameters. Estimaated Parameterrs Real Vallues Valuees J 0.02522 0.02552 0.5000 0.507 D

xd xq xd′

′ T pdo FPE =

Error [%] 0 1.400

ee

In order too investigate the propossed approachh cap pability in covering the syystem non-lineearities, moree stu udies were carried c out. The perform mance of thee ideentified modell and the systeem at P=1.1 pu, p Q=0.2 pu,, vt=1.02 = pu, wheen the system is excited by y a step signall in the t generator field voltage is compared in i Fig. 5. Thee auttocorrelation functions f for the collected data in Fig. 5 aree given in Fig.. 6. The value of o I , FPE annd PR for th he step inputt ee

2.0722

2.11332

1.988

wo ould be: I δ = 2.4416e-0006,

1.5599

1.55889

0.006

FP PE δ = 2.46377e-006, FPE E P = 4.2142e-0006

0.5688

0.5922

4.22

PR _ δ = 100%, PR _ P = 1000%

0.131

0.13004

N +n N ∑ ( y (k ) − yˆ (k ))2 N (N − n ) k =1

ee

0.06

( (44)

⎛ 2 2 ⎞ ( (45) , PR = % R( y− yˆ )(( y− yˆ ) in ⎜ − ⎟ ee ⎝ N N⎠ where, n is thhe number of estimated parrameters, N is the number of samples and R is the autocorrelattion function. Inn the PR index, wee calculate the ee

autocorrelation function of o error. If more m than 95% % of data remain in the rangee ⎛ − 2 , 2 ⎞ , the estimattion ⎟ ⎜ ⎝

N

N⎠

procedure acccuracy is called satisfactoryy. The autocorrelation fuunctions for thhe collected data d in Fig. 3 are given in Fig. 4.

(46))

I P = 4.1764e-006 (47))

ee

As mentioneed in the begginning of this section, wee app ply the PRBS to the input oof the system.. Although, inn som me papers [144, 21], it is shhown that PRB BS is appliedd to the input of real r power pllants, it is nott necessary too usee this kind off signal. In another examplee, we apply a triaangular signall, which is thhe integral of PRBS, to thee field voltage. The T collected data are show wn in Fig. 7.. he identificatiion proceduree is done ag gain and thee Th ressults are presented in Table 2. The estimatioon error in thee Table 2 is saatisfactory, soo onee can concludde that the triaangular signaal can be usedd in practice. p In order to perform furrther, the fielld voltage iss exccited by the pulse input with period T=4 sec. att opeerating poinnt P=1.2 puu, Q=0.15 pu, vt=0.9.. Co omparison of the t performannce of the ideentified modell and d the system at a P=1.2 pu, Q Q=0.15 pu, vt=0.9 = when thee sysstem is excitedd by a pulse siignal is shown n in Fig. 8.

Gallehdari et al: Online State S Space Moodel Parameteer Estimation in Synchronoous Machines

129

Fig. 8 data collected at operating point P=1.2, Q=0.15, vt=0.9, when the input signal is a pulse signal. Fig. 5 data collected at operating point P=1.1, Q=0.2, vt=1.02, when the input is step signal.

4.3 Noise Effect In this part, the effect of white noise on the proposed approach is examined by adding the white noise with different SNR (Signal to Noise Ratio) to the measured outputs. The estimation procedure is done for noisy measurements and the results are presented in Table 3. According to Table 3, we conclude that for SNR more than 110 the estimated parameters are close to the real values, but for less than it, the estimation error is increased especially for parameters D and J. In addition, we see that the value of xq is robust with respect to the noise.

Table 2 Synchronous machine parameters when the input signal is triangular. Estimated Error Parameters Real Values Values [%] J 0.0252 0.0252 0 0.500 0.511 2.2 D

Fig. 6 autocorrelation function of Fig. 5.

xd xq

Fig. 7 data collected at operating point P=1, Q=0.4, vt=0.95, when the input signal is triangular.

The value of I and FPE for pulse input would be: I _ δ = 9.2595e-007, FPE _ δ = 9.2817e-007,

I _ P = 1.6253e-007 FPE _ P = 1.6292e-007

2.0117

2.91

1.559

1.5592

0.0012

xd′

0.568

0.5544

2.39

′ T pdo

0.131

0.1344

2.59

Table 3 estimated parameters for noisy measurements with different SNR. SNR 500 150 120 100 90 Parameters 0.0511 0.0511 0.0523 0.134 0.26 D 0.0252 0.0252 0.0250 0.0174 0.0048 J 1.5588 1.5588 1.5588 1.5588 1.5588 x q

(48)

Since in most real cases the measurements are noisy, in the next Section the effect of noise is studied.

130

2.072

xd′ xd

0.6068

0.6068

0.6068

0.6091

0.631

2.1164

2.116

2.116

2.1207

2.1575

T d′o

0.1299

0.1299

0.1299

0.1283

0.115

Iranian Journal of Electrical & Electronic Engineering, Vol. 10, No. 2, June 2014

5 Conclusion In this paper, the theoretical relation based approach, which uses the known least square error method, has been presented to identify the nonlinear 3rd order synchronous generator state space model parameters. The most important point about the method is to use the linear least square error method to estimate the nonlinear model. In this paper, the field voltage is considered as the machine’s input and the rotor angle and active powers are as the outputs of the machine. Simulation results show that the proposed method has good accuracy for estimation of state space synchronous generator parameters. Appendix The main variables of the model in Eq. (1) are: J , D : rotor inertia and damping factor

Tdo′ : direct-axis transient time constant xd : direct axis reactance xd′ : direct axis transient reactance xq : quadrature axis reactance δ : rotor angle with respect to the machine terminals ω : rotor speed Tm : input mechanical torque Te : output electric torque EFD : The equivalent EMF in the excitation coil eq′ : transient internal voltage of armature

[5]

[6]

[7]

[8]

[9]

[10]

[11]

P, Q : terminal active and reactive power per phase

vt : generator terminal voltage VB : Infinite bus voltage id , iq : direct and quadrature axis stator currents [12] References [1] M. Karrari M and O. P. Malik, “Identification of Heffron-Phillips model parameters for synchronous generators using online measurements”, IEE Proceedings-Generation, Transmission and Distribution, Vol. 151, No. 3, pp. 313-320, 2004. [2] A. Damaki Aliabad, M. Mirsalim and M. Fazli Aghdaei, “A Simple Analytic Method to Model and Detect Non-Uniform Air-Gaps in synchronous generators”, Iranian Journal of Electrical and Electronic Engineering, Vol. 6, No. 1, pp. 29-35, 2010. [3] M. Shahnazari and A. Vahedi, “Accurate Average Value Modeling of Synchronous MachineRectifier System Considering Stator Resistance”, Iranian Journal of Electrical and Electronic Engineering, Vol. 5, No. 4, pp. 253-260, 2009. [4] M. Kalantar and M. Sedighizadeh, “Reduced order model for doubly output induction generator in wind park using integral manifold

[13]

[14]

[15]

[16]

theory”, Iranian Journal of Electrical and Electronic Engineering. Vol. 1, No. 1, pp. 41-48, 2005. S. H. Wright, “Determination of Synchronous Machine Constants by Test”, AIEE Transactions, Vol. 50, pp. 1331-1351, 1931. IEEE Standard 1110-2002, IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analysis, pp. 1-72, 2003. H. B. Karayaka, A. Keyhani, G. T. Heydt, B. L. Agraval and D. A. Selin, “Synchronous Generator Model Identification and Parameter Estimation From Operating Data”, IEEE Transactions on Energy Conversion, Vol. 18, No. 1, pp. 121-126, 2003. M. Hasni, O. Touhami, R. Ibtiouen, M. Fadel and S. Caux, “Estimation of synchronous machine parameters by standstill tests”, Mathematics and Computers in Simulation, Vol. 81, No. 2, pp. 277289, 2010. E. Mouni, S. Tnani and G. Champenois, “Synchronous Machine Modelling and Parameters Estimation Using Least Squares Method”, Simulation Modelling Practice and Theory, Vol. 16, pp. 678-689, 2008. M. Dehghani and M. Karrari, “Nonlinear robust modeling of synchronous generators”, Iranian Journal of Science and Technology, Trans. B, Engineering, Vol. 31 (B6), pp. 629-640, 2007. M. Dehghani, M. Karrari, W. Rosehart and O. P. Malik, “Synchronous Machine Model Parameters Estimation by A Time-Domain Identification Method”, International Journal of Electrical Power and Energy Systems, Vol. 32, No. 5, pp. 524-529, 2010. V. Vakil , M. Karrari, W. Rosehart and O. P. Malik, “Synchronous Generator Model Identification Using Adaptive Pursuit Method”, IEE proc. Generation, Transmission and Distribution, Vol. 153, No. 2, pp. 247-252, 2006. IEEE Standard 115-1995 IEEE Guide: Test Procedures for synchronous machines, Part Iacceptance and performance testing Part II-test procedures and parameter determination for dynamic analysis. M. Karrari and O. P. Malik, “Identification of Physical Parameters of a Synchronous Generator from On-line Measurements”, IEEE Trans. on Energy Conversion, Vol. 19, No. 2, pp. 407-415, 2004. M. Dehghani and S. K. Y. Nikravesh, “Nonlinear state space model identification of synchronous generators”, Electric Power Systems Research, Vol. 78, No. 5, pp. 926-940, 2008. Y. Yu, Electric Power System Dynamics, Academic Press, 1983.

Gallehdari et al: Online State Space Model Parameter Estimation in Synchronous Machines

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[17] P. Kunndur, Power System Stabillity and Conttrol, McGraaw-Hill Inc., 1994. 1 [18] M. Deehghani and S. K. Y. Nikravesh, N “State Space Model Param meter Identifiication in LarrgeScale Power Systems”, IEEE Trans. T on Pow wer System ms, Vol. 23, No. 3, pp. 14499-1457, 2008. [19] M. Mosavi, M A. Akhyani, “P PMU Placem ment Methoods in Poower System ms based on Evoluttionary Algorrithms and GPS G Receiveer”,. Iraniann Journal of Electrical and Electroonic Engineeering, Vol. 9, 9 No. 2, pp. 766-87, 2013. to identificatiion, [20] J. P. Norton, N An Introduction I Acadeemic Press, 1986. [21] M. Rasouli R andd M. Karraari, “Nonlinnear identiffication of a brushless b excittation system via field tests”, IEEE Trans. T on Eneergy Conversiion, Vol. 19, No. 4, pp. 733–740, 7 20044. Zahra Gallehdari received r the B.Sc. B degree and M.Sc. deegree in Electrrical 2 and 2009 2 Engineeering in 2005 respecttively. She is currently pursuuing the Ph.D. P degree in Control at Concorrdia Univerrsity, Montrreal, Canadaa. Her researrch interests are identificattion, controll, system a their applicaations in powerr systems. optimization and

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Maryam D Dehghani receeived her B.Sc.. and M.S Sc. degrees in electricall engineeringg from Shiraaz University,, Shiraz, Irran, in 1999 9 and 2002,, respectivelly and her Ph.D. from m Amirkabir University of Technology,, Tehran, Iraan in 2008. Shee is currently ann assistant prrofessor in scho ool of electricall and com mputer engineering, Shirazz uniiversity, Shiraaz, Iran. Her research intterests includee dyn namics and conntrol of power ssystems, system m identificationn and d phasor measurement units. Seyyed Kamaleddin n Yadavarr B degree inn Nikravesh received the B.Sc. electrical eengineering from Amirkabirr University of Technology (Tehrann Polytechnicc), Tehran, Iran n, in 1966 andd the M.Sc. and Ph.D. deg grees from thee University of Missouri at Rolla andd Columbia and 1973,, in 1971 respectivelyy. Currently, hee is a Professorr in the Electricaal Engineeringg Department at Amirkabirr Un niversity of Teechnology. His research intterests includee dyn namical and biomedical b moddeling, system m stability, andd sysstem optimizatioon.

Irranian Journaal of Electrical & Electronicc Engineeringg, Vol. 10, No. 2, June 20144