Hybrid-State-Model-Based Time-Domain Identification of Synchronous ...

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 1, MARCH 2008

Hybrid-State-Model-Based Time-Domain Identification of Synchronous Machine Parameters From Saturated Load Rejection Test Records Ren´e Wamkeue, Member, IEEE, Frederic Baetscher, and Innocent Kamwa, Fellow, IEEE

Abstract—This paper presents a suitable method for timedomain identification of synchronous machine parameters from the hybrid state model recently introduced by the authors in a compact matrix form. The saturated version of this model is developed in terms of generator equivalent-circuit parameters. The load rejection test of a combined resistive/inductive load is performed for the parameter identification while the online symmetrical threephase short-circuit test is carried out for the model cross-validation. For weak power factor initial loads connected to the generator, the rotor speed is quite constant during the full load rejection test. Thus, the mechanical transients do not have any influence on estimated electrical parameters since they are decoupled from the electrical model of the machine. The method is successfully applied for the parameter identification of 380 V, 3 kV·A, four-pole, 50 Hz saturated synchronous generator. Index Terms—Cross-validation, identification, load rejection, nonlinear hybrid state model, saturation, synchronous generator.

I. INTRODUCTION ARGE-SCALE integration of wind farms and others types of embedded generation into current power systems, as observed in many developed countries, gives rise to new challenges in building reliable programs for stability analysis and control [1], [2]. The generator models used in these programs must, therefore, be reliable and suitable for representing the machine performances for large, severe disturbances as well as for small perturbations. This could explain the strong interest in parameter identification using online tests, although most parameter estimation algorithms are based on voltage-controlled models of synchronous machines (SMs) [3]–[6]. To the best of our knowledge, there is no available state–model structure other than the voltage-controlled model that is able to perform large-disturbance tests for the modern identification of electrical parameters of the SM. Accordingly, it appears necessary to

L

Manuscript received September 19, 2006; revised February 27, 2007. This work was supported in part by the National Sciences and Engineering Research Council of Canada. Paper no. TEC-00438-2006. R. Wamkeue is with the Universit´e du Qu´ebec en Abitibi-T´emiscamingue (UQAT), Rouyn-Noranda, QC J9X 5E4, Canada. He is also with Laval University, Quebec City, QC G1K7P4, Canada (e-mail: [email protected]). F. Baetscher is with Laval University, Quebec City, QC G1K7P4, Canada (e-mail: [email protected]). I. Kamwa is with the Power System Analysis, Operation, and Control Department, Hydro-Qu´ebec Research Institute (IREQ), Varennes, QC J3X 1S1, Canada. He is also with Laval University, Quebec City, QC G1K7P4, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEC.2007.914663

explore new models and test procedures in order to be able to compute machine parameters. Short-circuit tests strongly stress machine windings and could damage the machine when operating around the rated voltage. The load rejection test is a particular type of decrement test with a constant field voltage first introduced in [7] and [8], followed by some experimental works proposed in [9]–[12]. The Electric Power Research Institute (EPRI) project EL-5736 had performed rotating time-domain response (RTDR) tests including the load rejection test to compute machine parameters [13]. Comparisons between load rejection and standstill frequencydomain response (SSFR) tests based models can be found in [14]. The impacts of the automatic voltage regulation AVR on recorder data following the load rejection tests are discussed in [15]. In [16] and [17], modern identification techniques based on the arbitrary axis load rejection test are proposed to compute d- and q-axis transient and subtransient reactances and opencircuit time constants. A graphical approach to computing generator parameters from a d-axis load rejection test is proposed in IEEE Std 1110 [18]. A suitable method to predict the load rejection test from a so-called hybrid model using equivalentcircuit parameters has also been introduced [19], [20]. A recent paper proposed by nine Japanese researchers in their load rejection study illustrates new interests raised by this test [21]. The purpose of this paper is to present a suitable approach for time-domain identification of SM electrical parameters from saturated load rejection data. The main objectives are: 1) to develop a hybrid model of the SM in terms of equivalentcircuit parameters for the identification process; 2) to present a voltage-controlled model (admittance model) of the SM used for cross-validation; 3) to adapt the cross-saturation method proposed in [19] in both the hybrid and the admittance models, respectively, for identification and cross-validation aims; 4) to perform the rejection test of a resistive/inductive load for the identification process; 5) to provide a practical prediction-error identification procedure based on an asymptotic weighted least-squares estimator (AWLSE) and a Newton-type finite difference constrained optimization algorithm; 6) to illustrate the practicability of the proposed scheme in the identification of electrical parameters of an SM; 7) to prove the reliability of the estimated parameters with a cross-validation of the generator model using online symmetrical sudden three-phase short-circuit test data.

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inal angular frequency, and ωm is the rotor speed (per unit). Relationships between the reactances of (3) are given in the Appendix. I5 in (6) is the five-order identity matrix. Flux equations:      −xdd id ψd 0 xdf xdD 0    ψq   0 −xq q 0 0 xq Q   iq           0 xff xf D 0   if   ψf  =  −xf d       ψD   −xD d 0 xD f xD D 0   iD  0

ψQ

0

0

0

xQ Q

iQ (3)

or equivalently 

Fig. 1.

Equivalent circuits of SM. (a) d-axis. (b) q-axis.

II. SATURATED STATE MODELS OF SM A. Parameter Network Models of SM The equivalent circuits of an SM (Fig. 1) reduced to a two-port and a bipolar networks, respectively, with regard to the electric circuit analysis leads to four principal parameter networks in the Laplace domain using the superposition principle of d- and q-axis: impedance, admittance, hybrid, and inverse-hybrid parameter networks [19], [20], [22]. The hybrid and the admittance parameter networks concerned can be defined in terms of input– output models as given by (1) and (2), respectively, for a given constant rotor speed ωm:      0 if (s) vf (s) hff (s) hf d (s)      0   id (s)   vd (s)  =  hdf (s) hdd (s) vq (s) 0 0 hq q (s) iq (s) ⇔ Y (s) = [C (sI − A)−1 B + D]U (s)      0 if (s) vf (s) yff (s) yf d (s)      0   vd (s)   id (s)  =  ydf (s) ydd (s) iq (s)

0

0

yq q (s)

(1)



 =

Voltage equations:    −ra vd  vq   0        vf  =  0     0   0 0 0

Xs T Xsr

Xsr Xr

0

0

−ra 0

0

O2,3 Rr

 

(4)



0

0 0

id

0 0

      

(5)

0

ψQ or equivalently    Vs Rs = Vr O3,2  1 + ωn

 Is . Ir

  iq     if  0 rD 0   iD 0 0 rQ iQ    −ψq   ψd         + ωm  0       0  0 rf

0 0  ψd  ψq 1 d   +  ψf ωn dt   ψD



Is



Ir 

ωm Ξ O2,3 d I5 + dt O3,2 O3,3

 

Ψs



Ψr

. (6)

Motion equations: d 1 (ωm ) = [(Tm − Te ) − dωm ], dt 2H

vq (s)

⇔ Ya (s) = [Ca (sI − Aa )−1 Ba + Da ]Ua (s).

Ψs Ψr

(2)

The variable s in (1) and (2) is the familiar Laplace operator, U and Y are, respectively, the controlled input and the output variables of a given parameter network, and matrices A, B, C, and D are related state matrices of the so-called hybrid state space model of the SM. Variables with subscript a indicate the voltage-controlled (admittance) model (2) that will be used for the cross-validation of the identified model. B. Hybrid State Model for Load Rejection Test Prediction (Predictor for Identification Process) The electrical and mechanical equations of the SM in Fig. 1 are defined by (3)–(7), ωn (in radians per second) is the nom-

d (δ) = ωn (ωm − 1) dt (7)

where ΨS = [ ψd

ψq ]t ,

IS = [ id

iq ]t ,

V S = [ vd

vq ]t (8)

t

Ψr = [ ψf ψD ψQ ] ,   0 −1 Ξ= . 1 0

V r = [ vf

t

0 0] , (9)

The so-called hybrid state model of the SM given in (10) is derived from (4) and (6) (see Appendix for details) [19]: d (Ψr ) = AΨr + BU, dt y = CΨr + DU

Ψr 0 (10)

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with  −rf xD D N    rD xf D A = ωn   N  0  1

   B = ωn  0   0

rf xf D N −rD xff N

0 −

0

rQ xQ Q

 O1,2 , ωm Ξ

1 Υ= O2,1

0 rQ xq Q xQ Q

−xf D N xdD xff − xdf xf D N 0

0





−



0 0 x2Q q − xq xQ Q  

if   y =  vd  , vq

      

(12)

 0

   0    xQ q  xQ Q (13)



xf d xD D − xD f xD d N d22 N

    D = Υ0    0

(11)

0

0

xD D  N   xD D xdf − xdD xf D C = Υ  N   0

0

     

rf (−xD D xdf + xdD xf D ) N rD (−xdD xff + xdf xf D ) N







0

0

t

t



       

vt =

ωn (ωm (ς) − 1)dς + δ0 =

vd2 + vq2 ,

ωn P t2 4H

(17)

va (t) = vd cos(ωn t) + vq sin(ωn t).

(14)

(18) C. Admittance Model for Short-Circuit Test Simulation (Cross-Validation Process)



vf   U =  id  iq

1 1 (P )dζ + ωm 0 = P t + ωm 0 2H 2H

δ(t) = 0

(15)

where Ψr 0 is the initial steady-state value of the state vector for the hybrid model. The armature flux vector derivative p(Ψs ) in (6) provokes a peak terminal voltage only during the sudden load change and the quantity Rs Is in (6) is very small. These two terms have been neglected when deriving the hybrid model output matrices C and D [16], [19]. The effect of neglecting armature flux derivative is illustrated in Fig. 2. The analytical formulations (11)–(14) were derived using symbolic software. The transient torque is equal to zero (Te = 0) following a 100% (full) load rejection test. The rotor speed and internal rotor angle of the generator are given by (16) and (17), respectively. P is the active power consumed by the initial load before its rejection, H is the equivalent of the inertia constant, and d is the damping friction. For a weak power factor and reactive loads (P ≈ 0), the generator maintains the synchronism ωm = ωm 0 = 1 p.u. in (15), and the rotor internal angle is δ(t) = δ0 ≈ 0 rad. Under such operating conditions, the mechanical transients do not have any influence on the model obtained in (10). Accordingly, the terminal and phase A voltage are computed from (18): ωm (t) =

Fig. 2. Effects of neglecting the armature flux derivative on the dynamic performance of the generator following a load rejection test of a combined resistive/inductive load [19].

(16)

In order to prove the validity of estimated parameters in a wide range of test simulations, the classical admittance model controlled by machine voltages is used for the cross-validation. The admittance model with the machine flux vector as state variable is chosen to avoid the incremental inductance introduced by the current model when the saturation phenomenon is taken into account [19]. Organizing (4) and (6) yields (19) and (20). Substituting (19) into (20) and solving the latter for p(Ψ) yields the admittance state equation (22). The output vector is given by (23): (19) Ψ = XI ⇒ I = X −1 Ψ V = RI + [Π + Ω]Ψ with Π=

1 d I5 , ωn dt

 Ω = ωm

Ξ O3,2

(20) O2,3 O3,3

d (Ψ) = −ωn (RX −1 + Ω)Ψ + (ωn I5 )V dt = Aa Ψ + Ba Ua and   O3,2 I3 ya = [ id iq if ]t = I O2,3 O2,2   O3,2 I3 X −1 Ψ = O2,3 O2,2 = Ca Ψ.

 (21)

(22)

(23)

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D. K-Factor Cross-Saturation Theory of SM Interesting studies on the saturation analysis of SM have been proposed [23]–[27], although many of them are difficult to adapt to resolving a practical problem such as SM parameter identification. In the present paper, the cross-saturation theory developed in [23], and used in [19], is adapted for the estimation frame study reported here. Armature fluxes ψd and ψq from (3) can be expressed by (24) and (25) in terms of magnetizing reactances xsm d and xsm q where the subscript s indicates that they are related to saturation. In the cross-saturation theory, the dand q-axis magnetizing currents im d and im q and the magnetizing fluxes ψm d and ψm q can be replaced by their equivalent cross-effects defined in (26):

Fig. 3.

Matlab/Simulink diagram of saturated hybrid state model of SM.

E. Saturated Hybrid and Admittance State Models of SM ψd = −xa id + xsm d (−id + if + iD ) = −xa id +

xsm d im d

= −xa id + ψm d

(24)

Knowing the machine electrical parameters of Fig. 1 in vector θt (33) allows numerical implementation of the hybrid model (10) and the admittance model (22)–(23):

(25)

  θt = ra rf rD rQ xa xf xD xQ xsm d xsm q xk f 1 .

ψq = −xa iq + xsm q (−iq + iQ )

Im

= −xa iq + xsm q im q = −xa iq + ψm q

1 2 + ∆2 ψ 2 = i2m d + 2 i2m q , Ψm = ψm mq d ∆

x0 d xsm d ∆2 = m = x0m q xsm q Kψ (Im ) = xsm d =

1 Ψm x0m d Im

(33) (26) (27) (28)

ψm = Kψ (Im )x0m d , Im

xsm d = Kψ ∆2 x0m q = ∆2 xsm q

(29)

xsm q = Kψ (Im )x0m q

(30)

1 Ψm 1 vt (if ) = 0 Kψ (Im ) = 0 xm d Im xm d if

(31)

2 z Kψ (Im ) = p1 + p2 Im + p3 Im + · · · + pz Im ,

Kψ (0) = p1 = 1.

(32)

The variable ∆2 in (27) is the saliency ratio, which is assumed to remain constant whether the machine is saturated or not [23], [24]. This assumption is realistic, as confirmed in [24], for machine ratings up to 3 kV·A. The variables x0m d and x0m q are unsaturated values of magnetizing reactances. By introducing the saturation factor KΨ (Im ) in (28), the saturated reactance xsm d and xsm q are computed from (29) and (30). Since a loaded and an unloaded machine saturate identically in the steady and transient states, the saturation factor in (28) can be formulated by (31) using the unloaded saturated characteristic curve vt (if ) of the SM. Equation (31) can be approximated with the zth-order polynomial function (32). The constraint Kψ (0) = p1 = 1 ensures a saturation factor equal to 1 for the linear region of the saturation curve.

Unloaded data vt (if ) is used to compute coefficients pi . For a given value Im (tk ) at each time instant tk from (26), KΨ (tk ) can be computed from (32). The magnetizing level of the machine at tk +1 can be adjusted using (29) and (30) in the parameter vector (33), which is introduced instant by instant in state models (10) and (22)–(23) to account for saturation.

III. ESTIMATION METHOD: ALGORITHMS AND SOFTWARE A. SM Nonlinear Predictor and Model Parameterization To understand the behavior of the system subject to a constant field voltage and forced armature currents (armature currents are forced to zero during the load rejection test), the predicted output yp (t) is sampled using an experimental test at intervals {tk = t0 + kT ; k = 0, 1, . . . , N }. The predicted output and measurement vector y(k) are related by (35), where ξ(k) is the innovation sequence vector (residual). N is the number of test points. It, thus, appears that the hybrid state model (10) should be replaced in practice by the discrete representation (34). ξ is assumed to be zero mean spectrally colored noise, which includes measurement and modeling errors. At the true parameter value θ = θopt obtained with the estimation process, ξ(tk , θopt ) becomes white. The predicted or simulated variable vector yp (tk ) is computed from the Matlab/Simulink software (Fig. 3):   

x0 = x(0) x(tk + 1, θ) = A (θ(tk )) x(tk ) + B (θ(tk )) U (tk )   yp (tk , θ) = C (θ(tk )) x(tk ) + D (θ(tk )) U (tk )

(34)

y(tk , θ) = yp (tk ) + ξ(tk , θ).

(35)

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Fig. 5.

Diagram of test principle.

IV. IDENTIFICATION EXPERIMENTS

Fig. 4.

Flowchart of the SM identification procedure.

B. Asymptotic Weighted Least-Squares Estimator The prediction-error approach based on the AWLSE is selected for the estimation process [28], [29]. Assuming that N observations are available, a good model is the one that minimizes the discrepancies (ξ) between the model prediction (yp ) and the actual data (y). This is achieved by minimizing a certain norm of covariance of ξ that is defined by N 1  (ξ(tk )t W ξ(tk )) V (θ) = N

(36)

k =1

min V (θ) subject to



θ − θm in ≥ 0 θm ax − θ ≥ 0.

(37)

The function (36) is known as a Markov estimator with a weighted covariance matrix Rθ = E [ξ t ξ] = W/N . When the number of measurements N is very large (N → ∞), the AWLSE provides an unbiased and consistent estimate parameter vector [28], [29]. W is a diagonal weighting matrix that is helpful for scaling purposes. Specifically, parameter estimation is done by a nonlinear programming (NP) procedure defined as the optimization problem (37), which can be solved starting with an initial guess of parameter θ0 using a Newton-type constrained optimization algorithm [28]. The “fmincon” function of the Matlab Optimization Toolbox was used for the present paper [30]. θm ax and θm in are the upper and lower bounds of the parameter vector θ to be estimated. The identification procedure including the saturation phenomenon is defined by the flowchart given in Fig. 4.

Tests are performed using the principle illustrated in Fig. 5, and the experimental setup is given in Fig. 6. The machine under test is a 380 V, 3 kV·A, four-pole, 50 Hz synchronous turbine generator driven by a dc motor at 1500 r/min (Fig. 6). According to the load rejection test principle defined in IEEE Std 1110 [18], from the steady state with the load L connected to the machine terminal and for a given constant field voltage vf (k1 = 1 is closed, k2 = 0 is open, and k3 = 1 is closed, Fig. 5), the load rejection test is performed on the machine by suddenly disconnecting the load L from the generator (k3 = 0 is open, Fig. 5) and recording the field current and terminal voltage transients. The test loads are laboratory modules (Fig. 6). Since the chosen load is a weak power factor load, the generator preserves the synchronism during the test, ωm ≈ ωm 0 = 1 p.u. [see (16)]. Numerically, the hybrid model control input vector is U = [vf 0 0]t . In fact, when the generator steady-state load is suddenly disconnected, the SM armature currents are forced to zero. Fig. 7 showing the phase A current during the load rejection by the generator clearly illustrates the step change in the armature current from given steady-state conditions. The unloaded characteristic of the generator is shown in Fig. 8. Coefficient pi in Table I of the saturation factor KΨ in (32) are computed from the unloaded characteristic vt (if ) using the least-squares algorithm with z = 7. The following per unit base system is used, [18]. Stator: √ Vb = 2VN = 311.13 V, Sb = SN = 3VN IN = 2970 V · A ≈ 3 kV · A, √ Ib = 2IN = 6.36 A, Idb = Iq b = IN = 4.5 A, Vdb = Vq b = VN = 220 V. Field circuit: If b = If b0 x0m d = 1.39 A,

Vf b =

Sb = 2136.69 V If b

where If b0 = 1.805 A is the field current for 1 p.u. (311.13 V) open-load terminal voltage on the air-gap line (Fig. 8). The unsaturated magnetizing reactance x0m d = 0.77 p.u. was computed from the saturation curve (Fig. 8) combined with the classical unloaded short-circuit characteristic Isc (if ). The reactance x0m q = 0.569 p.u. is computed from the slip method given

WAMKEUE et al.: HYBRID-STATE-MODEL-BASED TIME-DOMAIN IDENTIFICATION OF SM PARAMETERS

Fig. 6.

73

Experimental setup for the load rejection test.

TABLE I SATURATION MODEL PARAMETERS OF K ψ = f (Im )

TABLE II INITIAL PARAMETER VALUES OF THE GENERATOR (PER UNIT): θ0

Fig. 7. Phase A terminal and armature currents during the generator load rejection: step change in armature currents.

in IEEE Std 115 [31]. The initial parameter vector θ0 values computed from classical tests are given in Table II.

V. IDENTIFICATION RESULTS AND DISCUSSION A. Computation of Saturated Steady-State Conditions

Fig. 8.

Unloaded characteristic v t (if ) of SM.

As shown in [19], the saturation strongly influences the terminal voltage and the field current waveforms following the rejection of an inductive load. A reduced terminal voltage value was chosen for the load current limitation. The following procedure was used to compute the saturated initial conditions listed in Table III prior to the identification process. 1) For given steady-state data and unsaturated machine parameter values previously computed (Table II), set k = 0, xsm d = x0m d , and xsm q = x0m q .

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TABLE III STEADY-STATE VARIABLES (PER UNIT)

TABLE IV ESTIMATED PARAMETER VALUES OF THE GENERATOR (PER UNIT): θe

2) Compute the rotor angle   (xa + xsm q )it0 cos(ϕ) − ra it0 sin(ϕ) δ0 = a tan . vt0 + ra it0 cos(ϕ) + it0 (xa + xsm q ) sin(ϕ) (38) 3) Compute the d- and q-axis voltages and currents from vd0 = vt0 sin δ0 ,

vq 0 = vt0 cos δ0

(39)

and id0 = it0 sin(ϕ + δ0 ),

iq 0 = it0 cos(ϕ + δ0 ). (40)

4) Compute the field current and field voltage from icf 0 =

vq 0 + ra iq 0 + (xa + xsm d )id0 , xsm d

vf 0 = rf icf 0 .

(41) 5) Compute the steady-state cross-magnetizing current from

 2 1 − id0 + icf 0 + 2 i2q 0 . (42) Im 0 = ∆

Fig. 9. Field current following a full load rejection of combined resistive/inductive load: identified model validation.

6) Compute the saturation factor Kψ (Im 0 ) using (32) and saturation model parameters of Table I. 7) Compute d- and q-axis saturated magnetizing reactances xsm d = Kψ (Im 0 )x0m d ,

xsm q = Kψ (Im 0 )x0m q . im f0

8) Compare the measured and the computed of the field current as c  i − im  f0 f0 × 100 ≤ 5%. im f0

icf 0

(43) values

(44)

9) Return to step 2 and set k = k + 1 until (44) is satisfied. B. Parameter Estimation Results and Model Validation A sample of 50 000 points per channel of the scope was selected. For the identification process, the initial load was wyeconnected resistances of 21 Ω in series with three inductances of 51 mH. The steady-state conditions are given in Table III. The lower and upper bounds of the inequality constraints (37) were θm in = 0.05θ0 and θm ax = 10θ0 , respectively, where θ0 is the initial parameter vector given in Table II. The weighted matrix in (36) was W = diag(5, 3, 3) for the variables scaling. A Newtontype finite-difference constrained optimization algorithm was used for the identification process (fmincon function of Matlab optimization toolbox). The implemented algorithm converges in about 8 min with Pentium IV, 2.6 GHz personal computer. The estimated generator parameters are given in Table IV. Comparisons of the

Fig. 10. Phase A voltage following a full load rejection of combined resistive/inductive load: identified model validation.

estimated output waveforms of the model and actual data show the effectiveness and accuracy of the identification process (Figs. 9–11): C. Cross-Validation of Identified Model An online three-phase short-circuit test was performed on the studied machine in order to valid the reliability of the estimated

WAMKEUE et al.: HYBRID-STATE-MODEL-BASED TIME-DOMAIN IDENTIFICATION OF SM PARAMETERS

Fig. 11.

75

Zoom of Fig. 10 (phase A voltage). Fig. 13. Phase A current following an online symmetrical three-phase short circuit: identified model cross-validation.

Fig. 12. Field current following an online symmetrical three-phase short circuit: identified model cross-validation.

able for predicting the load rejection test is developed in terms of the equivalent-circuit parameters of the generator. A load rejection test on a combined resistive/inductive load was performed for the identification process, while an online symmetrical short-circuit test was carried out for the cross-validation of the identified model. Comparison of the identified model with experiments shows the pertinence of the proposed estimation approach. The quasi-closeness of simulated and actual data at the model cross-validation stage clearly attests to the suitability of load rejection tests for parameter identification experiments of SM. The present paper provides solutions to some of the frequently cited shortcomings of the load rejection test: 1) it allows the field circuit parameters to be determined; 2) rotor positioning is not necessary to compute the q-axis parameters; and 3) the method can predict the load rejection with the field voltage variation if this voltage can be recorded. APPENDIX

parameters for a wide range of large disturbance tests. The three-phase short-circuit test is simulated using the admittance model (22) with the estimated parameters of Table IV. The control input vector is Ua = [0 0 vf 0 0]t for a symmetrical three-phase short-circuit test using the model (22). It is observed in Figs. 12 and 13 that the simulation results and the actual data are close, confirming the pertinence of the identification method and confidence in the estimated generator parameters. Since the level of the short-circuit test was small (low shortcircuit current, Fig. 13), the rotor speed variation is negligible and ωm ≈ ωm 0 = 1 p.u. in (21). Accordingly, for the crossvalidation analysis, the mechanical transients did not need to be included in the model (22). VI. CONCLUSION An AWLSE method in conjunction with a Newton-type finitedifference constrained optimization algorithm is applied for SM identification. A saturated hybrid state model of the SM suit-

xdd = xd = xm d + xa ,

xq q = xq = xm q + xa

(A1)

xD D = xm d + xD + xk f 1,

xQ Q = xm q + xQ

(A2)

xff = xm d + xf + xk f 1,

xdf = xf d = xm d

(A3)

xD d = xdD = xm d ,

xQ q = xq Q = xm q

(A4)

xD f = xf D = xD D − xD .

(A5)

Stator Ψs = −Xs Is + Xsr Ir Vs = −Rs Is +

1 d (Ψs ) + ωm ωn dt



0

−1

1

0

(A6)

 Ψs .

(A7)

Rotor t Is + Xr Ir Ψr = −Xsr

or

  t Ir = Xr−1 Ψr + Xsr Is (A8)

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 1, MARCH 2008

Vr = Rr Ir + +

  1 d t (Ψr ) = Rr Xr−1 Ψr + Xsr Is ωn dt

1 d (Ψr ) ωn dt

(A9)

d t (Ψr) = −ωn Rr Xr−1 Ψr + ωn Vr − ωn Rr Xr−1 Xsr Is dt t = −ωn Rr Xr−1 Ψr + ωn [ 1 0 0 ]t vf − ωn Rr Xr−1 Xsr Is   −1 t −1 t = −ωn Rr Xr Ψr + ωn [ 1 0 0 ] − ωn Rr Xr Xsr U

= AΨr + BU

(A10)

ψf = F1 Ψr = xff if + F2 U + F3 Ir t = xff if + F2 U + F3 Xr−1 (Ψr + Xsr Is ) t = xff if + F2 U + F3 Xr−1 Ψr + F3 Xr−1 Xsr Is   t U = xff if + F2 U + F3 Xr−1 Ψr + 0 F3 Xr−1 Xsr

= xff if + F3 Xr−1 Ψr + (F2 − F4 )U  1  1 F1 − F3 Xr−1 Ψr + if = [F4 − F2 ]U xff xff = cf Ψr + df U

(A11)

(A12)

F2 = [ 0 −xm d 0 ]   t F3 = −F2 , F4 = 0 F2 Xsr Xr−1     v −ψq Vs = d ≈ ΞΨs = vq ψd    cos(−π/2) sin(−π/2) ψd = ψq − sin(−π/2) cos(−π/2)   t Is Ψs = Xsr Xr−1 Ψr − Xs − Xsr Xr−1 Xsr F1 = [ 1 0 0 ],

(A13) (A14)

(A15)

= cϕ Ψr + dϕ Is = cϕ Ψr + [0 dϕ ]U (A16)       if cf df   y = vd = U = CΨr + DU. Ψr + cϕ 0 dϕ vq (A17)

REFERENCES [1] N. Jenkins, R. Allan, P. Crossley, D. Kirschen, and G. Strbac, Embedded Generation, Series 31. London, U.K.: Inst. Electr. Eng. Power Energy, 2000. [2] J. M. Rodriguez, D. Alvira, D. Beato, R. Iturbe, J. C. Cuadrado, M. Sanz, A. Lombart, and J. R. Wilhelmi, “Impact of wind energy generation on the safety of the electrical transmission network,” presented at the CIGRE, Paris, France, 2004. [3] R. Wamkeue, I. Kamwa, and X. Dai-Do, “Line-to-line short-circuit test based maximum likelihood estimation of stability model of large generators,” IEEE Trans. Energy Convers., vol. 14, no. 2, pp. 167–174, Jun. 1999. [4] M. Karrari and O. P. Malik, “Identification of physical parameters of a synchronous generator from online measurements,” IEEE Trans. Energy Convers., vol. 19, no. 2, pp. 407–415, Jun. 2004. [5] C. Maun, “New procedure for the identification of dynamic parameters of synchronous machines by three-phase short-circuit test and extension to other tests,” in Proc. ICEM 1984, pp. 121–128.

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WAMKEUE et al.: HYBRID-STATE-MODEL-BASED TIME-DOMAIN IDENTIFICATION OF SM PARAMETERS

[30] A. Grace, Optimisation Toolbox for Use With Matlab. Natic, MA: Math Works, 1992. [31] Test Procedures for Synchronous Machine, IEEE/ANSI Standard 115, 1996.

Ren´e Wamkeue (S’95–M’98) received the Ph.D. degree in electrical engineer´ ing from the Ecole Polytechnique de Montr´eal, Montreal, QC, Canada, in 1998. Since 1998, he has been with the Universit´e du Qu´ebec en AbitibiT´emiscamingue, Rouyn-Noranda, QC, where he is currently a full Professor of electrical engineering. He is also an Associate Professor of electrical engineering at Laval University, Quebec City, QC. His current research interests include control, power electronics, modeling, and identification of electric machines, power system cogeneration with induction generators, and wind energy conversion systems. Prof. Wamkeue is a member of the IEEE Power Engineering Society (PES) Electric Machine Committee and the Secretary of the Working Group 7 for revision of IEEE Std 115.

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Frederic Baetscher received the B.Eng. degree from the University of Applied Sciences of Western Switzerland (HES-SO), Fribourg, Switzerland, in 2001, and the M.Sc. degree from Laval University, Quebec City, QC, Canada, in 2003, both in electrical engineering. During 2004, he was a Lecturer at the HES-SO. He is currently with Laval University. His current research interests include electric machine design, modeling, and parameter identification of electric machines and drive system.

Innocent Kamwa (S’83–M’88–SM’98–F’05) received the B.Eng. and Ph.D. degrees in electrical engineering from Laval University, Quebec City, QC, Canada, in 1984 and 1988, respectively. Since 1988, he has been with the Power System Analysis, Operation, and Control Department, Hydro-Qu´ebec Research Institute (IREQ), Varennes, QC, where he is currently a Senior Researcher. He is also an Associate Professor of electrical engineering at Laval University. Dr. Kamwa is a member of the Synchronous Machine and Stability Controls Subcommittees of the IEEE Power Engineering Society (PES).