Online State Estimation of a Synchronous Generator ... - IEEE Xplore

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

1099

Online State Estimation of a Synchronous Generator Using Unscented Kalman Filter From Phasor Measurements Units Esmaeil Ghahremani and Innocent Kamwa, Fellow, IEEE

Abstract—The most important reference quantities for monitoring and controlling transient stability in real time are the rotor angle and speed of the synchronous generators. If these quantities can be estimated with sufficient accuracy, they can be used in global and local control methods. In the classic state estimation methods, such as the extended Kalman filter (EKF) technique, the linear approximations of the system at a given moment in time may introduce errors in the states. In order to overcome the drawbacks of the EKF, the authors of this paper have applied the unscented Kalman filter (UKF) to estimating and predicting the states of a synchronous machine, including rotor angle and rotor speed, using phasor measurement unit (PMU) quantities. The UKF algorithm propagates the pdf of a random variable in a simple and effective way and is accurate up to the second order in estimating the mean and covariance. The overall impression is that the performance of the UKF is better than the EKF in terms of robustness, speed of convergence, and also different levels of noise. Simulation results including saturation effects and grid faults show the accuracy and efficiency of the UKF method in state estimation of the system, especially at higher noise ratios. Index Terms—Phasor measurements units, power system monitoring, power system operation, state estimation, synchronous machine, unscented Kalman filter.

I. INTRODUCTION NCREASED demand for sustainable and reliable energy in support of a massively digital economy stimulates the investigation of new control techniques to enhance power system stability and security. This aim can certainly be achieved by adding bulk equipment such as FACTS [1], [2] but the least expensive strategies are generally based on more intensive and innovative use of global and local controllers which tend to add more “intelligence” to the grid, instead of iron and copper. Indeed, Wide Area Measurement and Control (WAMAC) system are contemplated worldwide to advance system security and stability, while modernizing the power grid technological backbone at the same time. Enhancing system stability through

I

Manuscript received December 23, 2010; revised May 8, 2011; accepted September 7, 2011. Date of publication October 13, 2011; date of current version November 23, 2011. Paper no. TEC-00503-2010. E. Ghahremani is with the Department of Electrical and Computer Engineering, Laval University, Qu´ebec, Canada (e-mail: esmaeil.ghahremani.1@ ulaval.ca). I. Kamwa is with Hydro-Qu´ebec/IREQ, Power Systems and Mathematics, Varennes, Qu´ebec, 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.2011.2168225

a more sophisticated control requires, however, accurate information about synchronous generator rotor speeds and angles in order to carry out online high-sampling rate dynamic security assessments. Unfortunately, the existing Supervisory Control and Data Acquisition (SCADA) system can only provide steady, lowsampling density, and non-synchronous information about the network, and its measurements are too infrequent and nonsynchronous to capture information about the system dynamics [3]–[5]. The WAMAC system using phasor measurement units (PMUs), on the other hand, enables control systems to have an accurate picture of the power system both synchronously and in a precise sample time. Using wide-area measurements from PMUs installed on the generator buses, new dynamic state estimators could generate the dynamic states of a power system, e.g., generator speed and rotor angles, which could next be used in various advanced control methods. Recent literature reveals that there are two different approaches in this area of research: 1) model-free rotor angle or rotor speed estimator [6], [7], and 2) model-based state estimator of machine states [8]. In the first category, artificial intelligence methods, such as neural networks, have usually been adopted for the state estimation. These AI-based model-free speed estimators generate the rotor angle estimation without requiring a mathematical model or any machine parameters [6]. However, it is sometimes preferable to assume a physical model of the synchronous machine and then estimate its states as part of a large power network. Different variants of the latter approach with different levels of complexity can be obtained by selecting a synchronous generator model of various orders. For example, in [4] a gain scheduling scheme is used for the state observer in a third-order single-machine infinite-bus system. In [5], a dynamic state estimation method for the second-order synchronous machine is proposed. Also, in [9], a dynamic state estimation process for a sixth-order power system assuming a third-order model for the synchronous machine was presented. Additionally, [9] assumed that the exciter output voltage Ef d and rotor angle δ were measurable. In [10], a parameter estimation procedure based on the unscented Kalman filter (UKF) was presented for the third-order model of a synchronous generator. In contrast with the present research, the goal of [10] was not to estimate the states but the parameters. Furthermore, the output power, Pe , was assumed in [10] to be a dynamic state and the only measured output, while Ef d was the measured input, and Tm , a constant input signal.

0885-8969/$26.00 © 2011 IEEE

1100

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

Equation (1) can be rewritten by determining the state variables and the inputs in the following form: x = [δ u = [Tm

Fig. 1. lines.

Synchronous machine connected to infinite bus via the transmission

In this paper, the proposed online state estimator in the modelbased category according to above classification, uses the UKF algorithm to generate the estimated states from the available signals obtained from a PMU, which is assumed to beinstalled in the substation of a power plant. The benefits of the UKF over the extended Kalman filter (EKF) result from the fact that in the UKF method, there is no linearization step in the algorithm [16]. The paper is organized as follows. The fourth-order nonlinear model structure considered for the synchronous machine is presented in Section II. Section III describes the UKF mathematical algorithm while Section IV presents the online state estimation of the synchronous generator using UKF. Section V compares the EKF and UKF methods. Section VI is the discussion section and Section VII concludes the paper.

Δω

eq

ed ]T = [x˙ 1

Ef d ]T = [u1

x˙ 2

x˙ 3

x˙ 4 ]T

u2 ]T

x˙ 1 = ωo x2 1 x˙ 2 = (u1 − Te − D x2 ) J 1 x˙ 3 =  (u2 − x3 − (xd − xd ) id ) Tdo x˙ 4 =

1 (−x4 + (xq − xq ) iq ) Tq o

(2)

where ω o = 2πfo is the nominal synchronous speed (elec. rad/s), ω the rotor speed (pu), Tm the mechanical input torque (pu), Te the air-gap torque or electrical output power (pu), Ef d the exciter output voltage or the field voltage, as seen from the armature (pu) and δ the rotor angle in (elec.rad). Other variables and constants are defined in Appendix. Based on the phasor diagram of Fig. 1 in the dqo domain, the air-gap torque Te will be equal to the terminal electrical power Pt (or Pe = ω r Te ) by neglecting the stator resistance (Ra = 0) and assuming ω r = 1.0 p.u.: R =0

a Te = Pt + Ra It2 −−− − − −−−→ Te ∼ = Pt = ed id + eq iq

(3)

where the voltages (ed , eq ) can be expressed as (4): II. SINGLE-MACHINE INFINITE-BUS POWER SYSTEM Compared with higher-order nonlinear structures, the effect of damper windings and stator dynamics are neglected in the fourth-order nonlinear model of a synchronous machine. This is possible when very fast dynamic (subtransient) are not of interest. However, the effect of damper windings is considered approximately in the rotor-damping factor, D [1] (see Appendix for symbol nomenclature). The single-machine infinitebus (SMIB) power system, shown in Fig. 1, is considered here as the benchmark system. Giving a classical model for the synchronous generator and neglecting the transmission line resistance (Re = 0), all the active power produced by generator Pt , is delivered to the infinite bus (Pt = PB ). Also, δ is the angle by which eq , the q-axis component of the voltage behind transient reactance xd , leads the terminal bus of machine Et (or Vt ). With Vt as the reference phasor, the SMIB power system in Fig. 1, can be represented in per unit (pu) by the fourth-order nonlinear equation in the dqo domain as (1): δ˙ = ωo Δω 1 Δω˙ = (Tm − Te − DΔω) J 1 e˙ q =  (Ef d − eq − (xd − xd ) id ) Tdo e˙ d =

1 (−ed + (xq − xq ) iq ). Tq o

(1)

ed = Vt sin δ eq = Vt cos δ ⇒ Et = V t =



e2d + e2q .

(4)

Also, the d-axis and q-axis currents (id , iq ) are id = It sin(δ + Φ) =

eq − Vt cos δ xd

iq = It cos(δ + Φ) =

Vt sin δ xq

⇒ It =



i2d + i2q .

(5)

Replacing the variables δ and eq by the state variables x1 and x3 , we will have id =

x3 − Vt cos x1 xd

iq =

Vt sin x1 . xq

(6)

Replacing (4) and (5) in (3) and after mathematical simplification, the electrical output power Pe at the terminal bus (Pe = Pt ) can be presented as   Vt  Vt2 1 1 ∼ −  sin 2δ. (7) Pt =  eq sin δ + xd 2 xq xd

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

In terms of states x1 and x3 we will have   Vt2 1 1 Vt ∼ −  sin 2x1 . Pt =  x3 sin x1 + xd 2 xq xd

1101

(8)

To summarize, using (6) and (8) in (2), the fourth-order nonlinear synchronous machine state-space model in (2), can be presented as (9) x = [δ Δω eq ed ]T = [x˙ 1 u = [Tm Ef d Vt ]T = [u1

x˙ 2 u2

x˙ 3

x˙ 4 ]T

u 3 ]T

x˙ 1 = ωo x2      Vt 1 Vt2 1 1 x˙ 2 = x3 sin(x1 ) + −  sin(2x1 ) u1 − J xd 2 xq xd  − D x2    x3 − Vt cos x1 1  − x − (x − x ) u 2 3 d d  Tdo xd    Vt sin x1 1 x˙ 4 =  −x4 + (xq − xq ) Tq o xq   2 1 Vt V 1 y1 =  (x3 ) sin(x1 ) + t −  sin(2x1 ). xd 2 xq xd

x˙ 3 =

(9)

In (9), the electrical output power Pt selected as the measurable output and all the parameters and input-output quantities are (or assumed to be) known and measurable. Only the states are unknown. This structure is suitable for nonlinear state estimation with the UKF approach. We can, therefore, represent (9) in the following global structure: ⎧ x˙ 1 = f1 (x, u) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎨ x˙ 2 = f2 (x, u) x˙ = f (x, u, w) (10) x˙ 3 = f3 (x, u) ⇒ ⎪ y = h(x, u, v) ⎪ ⎪ ⎪ x˙ 4 = f4 (x, u) ⎪ ⎪ ⎩ y1 = h1 (x, u) where x is the state variable vector defined in (9), u is the input variable vector defined in (9), w is the process (state) noise, v is the measurement noise, f is the system function, h is the output function, and y is the measurable output. For online state estimation of the system presented in (9) or in (10), the available terminal bus signals like Vt , Pt , and Qt are used. These signals are accessible using a PMU device, which is assumed to be installed at the terminal bus of the synchronous generator. The PMU is a power system device that provides measurements of the real-time phasors of the bus voltage and line currents. Basically, it samples input voltage and current waveforms using a common synchronizing signal from the GPS, and calculates the phasors (magnitude and angle) using the discrete Fourier transform (DFT) [11]–[13]. The measurement set is composed of the bus voltage magnitude and angle, as well as the line and injection currents magnitudes and angles. By measuring the quantities of the three-phase voltages and currents Vabc , Iabc , the signals Vt , Pt , and Qt can

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

be extracted using known formulas for state estimation purposes [7]. The overall plan of the estimation process is illustrated in Fig. 2. After receiving the signals Vt , Pt , and Qt , obtained from a PMU, the UKF state estimator generates the dynamic states estimates. The principles of the UKF algorithm [14] are presented in the next section. III. UNSCENTED KALMAN FILTER ALGORITHM The extended Kalman filter is probably the most widely used estimation algorithm for nonlinear systems. However, previous experience in the estimation community has shown that the EKF is often difficult to implement, difficult to tune, and reliable only for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, unscented transformation (UT) was applied in [15] to propagate mean and covariance information by nonlinear transformation. It is more accurate, easier to implement, and uses the same order of calculations as linearization [15]. In this section, the state estimation of the system will be presented using the UKF algorithm. Considering a system in the form of y = g(x), the question is, given the pdf of x, how the ) of a UKF computes the mean (¯ yUKF ) and covariance (PUKF y random variable (y). The unscented transformation is founded on the intuition that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation [16]. The basic idea of the UKF approach is illustrated in Fig. 3. In Fig. 3, a set of points x(i) (i = 1, . . . , 2n + 1), termed sigma points, is chosen so that its mean and covariance are y ¯UKF UKF and Py . The nonlinear function is applied to each point, in turn, to yield a cloud of transformed points. The statistics of the transformed points can then be calculated to form an estimate of the nonlinearly transformed mean and covariance [16]. In unscented transformation, each sigma point is associated with a weight W(i) . The following steps are then involved in approximating the mean and covariance [17]: 1) Propagate each sigma point through the nonlinear function: y(i) = g(x(i) ).

(11)

1102

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

(i)

ˆ+ ˜ (i) i = 1, . . . , 2n ˆ k −1 = x x k −1 + x  T

˜ (i) = i = 1, . . . , n nP+ x k −1 i T

 ˜ (n +i) = − nP+ x i = 1, . . . , n. k −1 i

(17)

(b) Use the known nonlinear system equation f (.) to trans(i) ˆ k −1 vectors as shown in (11), with form the sigma points into x appropriate changes since the nonlinear transformation is f (.): (i)

(i)

ˆ k = f (ˆ xk −1 , uk , tk ). x Fig. 3.

(i)

Principle of the unscented transformation (UT).

ˆ k −1 vectors to obtain the a priori state (c) Combine the x estimate at time k. This is based on (12):

2) The mean is approximated by the weighted average of the transformed points (Fig. 3): y ¯UKF =

p

W(i) y(i)

(12)

i=0

where the weighting coefficients W(i) are defined as follows: p W(i) = 1. (13) i=0

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

p

W(i) (y(i) − y ¯)(y(i) − y ¯)T .

(18)

(14)

i=0

1 (i) ˆ . x 2n i=1 k 2n

ˆ− x k =

(19)

(d) Estimate the a priori error covariance as shown in (14). However, we should add Qk −1 to the end of the equation to take the process noise into account: 1 (i) (i) T ˆ− ˆ− (ˆ x −x xk − x k )(ˆ k ) + Qk −1 . 2n i=1 k 2n

P− k =

(20)

Step III: Now that the time update equations are done, we implement the measurement-update equations. (i) ˆ k −1 as specified in (17), with ap(a) Choose sigma points x propriate changes since the current best guess for the mean and − ˆ− covariance of xk is x k andPk : ˆ− ˜ (i) i = 1, . . . , 2n ˆk = x x k +x

 − T ˜ (i) = nPk i i = 1, . . . , n x  T

˜ (n +i) = − nP− i = 1, . . . , n. x k i (i)

Both the sigma points and the weights are computed deterministically using a set of conditions given in [16]. The UKF algorithm is presented briefly below. For more details and the background theory, readers are referred to [16]. Let the n-state discrete-time nonlinear system be represented by (15): xk +1 = f (xk , uk , tk ) + wk yk = h(xk , uk , tk ) + vk wk ∼ (0, Qk ) vk ∼ (0, Rk ).

(15)

The UKF algorithm for this system can be expressed as [15] Step I: Initialization of the filter at k = 0 as follows:  + ˆ 0 = E(x0 ) x (16) T ˆ+ ˆ+ P+ 0 = E[(x0 − x 0 )(x0 − x 0 ) ] where “E” indicates the expected value and the “+” subscript denotes that the estimate is an a posteriori estimate. Step II: The following time update equations are used to propagate the state estimate and covariance from one measurement time to the next. (a) To propagate from time step (k − 1) to k, first choose (i) sigma points xk −1 as specified in (17), with appropriate changes since the current best guesses for the mean and covariance of + ˆ+ xk are x k −1 and Pk −1 :

(21)

This step can be omitted if desired. Instead of generating new sigma points, we can reuse the sigma points that were obtained from the time update. This will save computational effort if we are willing to sacrifice performance. (b) Use the known nonlinear measurement equation h(.) to (i) ˆ k vectors (predicted measuretransform the sigma points into y ments) as shown in (11): (i)

(i)

ˆ k = h(ˆ xk , tk ). y

(22)

(i)

ˆ k vectors to obtain the predicted measure(c) Combine the y ment at time k. This is based on (12): 1 (i) ˆ . y 2n i=1 k 2n

ˆk = y

(23)

(d) Estimate the covariance of the predicted measurement as shown in (14). However, we should add Rk to the end of the equation to take the measurement noise into account: 1 (i) (i) ˆ k )(ˆ ˆ k )T + Rk . (ˆ y −y yk − y 2n i=1 k 2n

Py =

(24)

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

1103

ˆ− ˆk : (e) Estimate the cross covariance between x k and y 1 (i) (i) ˆ− ˆ k )T . (ˆ x −x yk − y k )(ˆ 2n i=1 k 2n

Pxy =

(25)

(f) The measurement update of the state estimate can be performed using the normal Kalman filter equations: Kk = Pxy P−1 y ˆ− ˆk ] ˆ+ x k + Kk [yk − y k =x − T P+ k = Pk − Kk Py Kk .

(26)

The algorithm above assumes that the process and measurement equations are linear with respect to the noise, as shown in (15). In general, the process and measurement equations may have noise that enters the process and measurement equations nonlinearly. That is, xk +1 = f (xk , uk , wk , tk ) yk = h(xk , uk , vk , tk ).

(27)

In this case, the UKF algorithm presented above is not rigorous because it treats the noise as additive, as seen in (20) and (24). To handle this situation, we augment the noise on the state vector as shown in (28) [16]: xa+ k = [xk

wk

vk ]T .

(28)

We then use the UKF to estimate the augmented state xa+ k . The UKF is initialized as: ˆ a+ x 0 = [E(x0 )

0

0]T

(29)

  ˆ 0 )(x0 − x ˆ 0 )T , Q0 , R0 ). (30) Pa+ 0 = diag(E (x0 − x Then we use the UKF algorithm presented above but, since we are estimating the augmented state with mean and covariance, we can remove Qk −1 and Rk from (20) and (24). IV. MATLAB IMPLEMENTATION AND SIMULATION RESULTS The overall structure of the MATLAB implementation based on the Simulink embedded function block is shown in Fig. 4. For time-step simulation in MATLAB, we need to convert the continuous-time equations into discrete-time equations in order to be compatible with the Simulink structure. From the basic definition of the time derivative of a variable x, we have: x˙ =

x(k) − x(k − 1) Δt

⇒ x(k) = x.Δt ˙ + x(k − 1) (31)

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

(32)

x(k) = Δt × f (x, u, w) + x(k − 1)

(33)

Fig. 4. Implementation of UKF algorithm using the embedded MATLAB function block as the UKF estimator with T m , E f d , and P t as its input signals.

which can be rewritten in the familiar structure presented in (15) or (27) in Section III as  xk = fk −1 (xk −1 , uk −1 , wk −1 ) (34) yk = hk (xk , uk , vk ). The embedded MATLAB function was used to develop the UKF approach because this block enables us to predict the dynamic state (bottom of Fig. 4) while simulating the synchronous machine (top of Fig. 4). The embedded function block creates an “m-file” page in the Simulink model, which means we can easily set and run more or less complicated algorithms in the Simulink file. As shown in Fig. 4, in the embedded MATLAB function block, the signals Tm and Ef d and the system observation signal Pe ( = y 1 ), taken from the machine model, have been used as inputs of the UKF block. The latter thus has access to these input-output signals at each time step in order to generate the state estimation based on the algorithm described in the previous section. Further details about MATLAB implementation and initial values of some parameters are described in the Appendix. The noise-free results of the state estimation of the UKF method are depicted in Fig. 5(a), based on excitation voltage step responses. The corresponding estimated output signal yˆ k compared with the actual output signal is shown in Fig. 5(b). Details of the Ef d step signal are given in the Appendix. Next, the effectiveness of the UKF method was checked in the presence of a network fault. The general configuration of the power system can be reduced to the form of Fig. 6 using Th´evenin’s equivalent theory [1]. Based on this equivalent circuit, we will consider that the fault occurred in the mid-point of the transmission line, as shown in

1104

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

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

Fig. 6.

Mid-point short-circuit fault.

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

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

The results of the state estimation process during and following the contingency in this case are shown in Fig. 7. From these results, it is clear that broadly speaking, the UKF state estimator generates very accurate states right after fault clearing. However, some variables, notably the angle and speed, are temporarily wrong during the fault due to the abrupt change in the external reactance. For the second scenario, the fault was cleared after 0.3 (s), at T = 20.3 (s) and the system transitioned into an unstable condition. In this case also, the UKF state estimator generates the estimated states with an appropriate accuracy once the fault is cleared (Fig. 8). Generally, it takes several milliseconds for the state estimator to track the real signal after a sudden change in the network reactance and there is a large error between the

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

1105

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

V. COMPARISON OF THE UKF AND EKF METHODS ON NOISY DATA AND PROCESSES The difference in the principles of state estimation using the UKF approach and the EKF method is illustrated in Fig. 9. This figure shows 300 points randomly generated by (35) with r uniformly distributed between ±0.1 and θ uniformly distributed between ±0.35 rad:  y1 = r cos θ (35) y2 = r sin θ.

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

real and the estimated signals within the fault timeframe. This error may be reduced by blocking the slow-changing states to their pre-fault values during the fault occurrence. Next, we have the analysis of the fault simulations (Figs. 7 and 8) in the presence of noise. The noise sequences were set to have the same magnitudes as in Fig. 10 (see Appendix). We again obtained satisfactory results from the UKF state estimation process in the presence of noise, similarly to Fig. 10. Based on the good performance achieved with these two fault scenarios (not shown for the sake of saving space), it could be concluded that the UKF approach will be able to estimate the state of a synchronous machine whatever the stability condition of the power system to which it is connected, even when the process and measurement noises are significant.

The points spread in Fig. 9 represent the exact points generated by (35), and the bold point at (0,1) represents the linearized mean. The true mean and the approximate unscented mean are so close that they are plotted on top of each other and are both equal to (0, 0.9797) to four significant digits. As a result, this figure shows the improved accuracy of the mean and the covariance estimation when unscented transformations are used instead of linear approximations [15]. To demonstrate the superior performance of UKF under noisy conditions, we have implemented separately the EKF method in order to compare it with the UKF approach. In terms of Simulink implementation, we simply replaced the UKF block in Fig. 4 by an equivalent EKF block, implemented according to [15]. As is clear from Fig. 10, when we compare the state estimates in the presence of noise, the UKF approach definitely outperforms the EKF in terms of accuracy and smoothness. In fact, at the same level of process and measurement noise, the EKF estimator results are noisier, like the rotor speed Δω, and are also biased, like the rotor angle, eq and ed voltages while the UKF results are smoother and do not have bias. VI. DISCUSSION The results presented in Figs. 5 and 10 assume that the input signal Ef d is a step function while Tm is a constant input. To confirm the UKF effectiveness under more general conditions, we tested it with different kinds of input. Hence, the simulation

1106

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

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

was repeated for two other configurations: the first for Tm = ramp and Ef d = constant, and the second for Tm = constant and Ef d = ramp. In both simulation studies, the results of the UKF approach showed that the UKF block estimated the states properly and with enough accuracy. These scenarios were repeated for the system in the presence of noise and the results were accurate again. Also, the state estimation process was repeated for different values of the external line reactance and for different values of the synchronous machine parameters. In all different simulation studies, we obtained results that showed the robustness and effectiveness of the proposed method. Interestingly, it turns out that the UKF method is not so dependent on the initial value of the states. For example, if we run the UKF simulation with δ o = 0.4 (while the nominal value of rotor angle is δ = 0.82), we still obtain accurate results. By contrast, the standard EKF algorithm is more dependent on the initial value of the rotor angle. For the same procedure, with a

nominal value of rotor angle at δ = 0.82, if we run the EKF simulation file with a δ o less than 0.6, the state estimator loses its ability to track the actual states. It could, therefore, be concluded that the UKF method is less dependent on the initial values than the EKF method. As is clear from Figs. 2 and 4, in addition to the phasor measurements of voltages and currents (Vabc , Iabc ), which are required to determine Pt , Vt , and It , the UKF approach needs to access Ef d and Tm as two other input quantities. In order to record and measure Ef d and Tm near the machine phasor signals, the PMU should be located in the power plant. From this point of view, one issue of the proposed UKF approach could reside in the fact that measuring the Ef d signal is essential for this method, while it may not be the case in all other methods. On the other hand, for power system engineers, it would be interesting if the dynamic states of a synchronous machine could be obtained using recorded signals of a PMU installed on the terminal bus or transmission line. We are

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

TABLE I DEFINITIONS OF VARIABLES AND CONSTANTS

1107

TABLE II TEST MACHINE AND EXTERNAL SYSTEM DATA

TABLE III DEFINITIONS OF VARIABLES OF UKF ALGORITHM

therefore working on this idea as a useful and practical procedure to remove these limitations (recording Ef d and Tm ) for dynamic state estimation of a synchronous machine. After that, the PMU could be installed on the transmission line but the transformer reactance would then have to be added to the machine reactance. To develop this idea, the first step would be to estimate the states of the synchronous machine without one of these input quantities, for example, the Ef d signal. After that, we could improve the idea by developing an estimator without the two input Ef d and Tm signals.

VII. CONCLUSION In this paper, the UKF approach was applied to online state estimation of a synchronous generator including the rotor angle and rotor speed signals, based on two generally available variables, which are, the electrical power and field voltage. This approach allows to overcome the limitations of the linearization process required by the traditional EKF method, and also to increase the operational range of the system variables around the operating point by not using the linearization in the state estimation algorithm. The implemented UKF-based scheme produced high quality results and also showed greater accuracy of the state estimates in the presence of noise, compared to the traditional EKF method.

B. Test Machine and External System Data The test machine parameters and external system data in per unit (pu) are presented in Table II. C. Definition of Variables of UKF Algorithm The main variables of the UKF algorithm presented in (11)– (30) are presented in Table III. D. MATLAB Implementation

APPENDIX A. Definitions of Variables and Constants The main variables and constants of the SMIB system described in (1) to (9) are presented in Table I.

As mentioned in Section IV, the embedded MATLAB function was used for implementation of the UKF method. By running the Simulink file and receiving the signals from UKF block inputs, the block creates the outputs based on its internal commands. At the beginning of the m-file, we can use the commands

1108

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

“persistent,” “isempty,” and “if-end” for initializing the UKF algorithm variables x0 and P0 . These commands in the form below, allows the algorithm to be initialized once at the start of the simulation: P0 = diag([10,10,10,10]); persistent P . . . if isempty(P) P = P0; end; The above script loads the initial value of matrix P with P0 in the first iteration of the algorithm. In the next iterations, the command isempty(P)will not allow the matrix P to be reloaded again. After the second iteration, the matrix P will always have a value and it will not be null so the program will pass the loop “if-end.” We should mention that the time step Δt defined in the embedded MATLAB function block and the time step of the Simulink file must be equal. In our case, the time step for the Simulink file was Ts = 0.1 ms and the time step for the EKF block, which should be equal to the Simulink file time step, was also Δt = 0.1 ms. To have the same case study signals for an acceptable comparison in Fig. 10, the time step also was defined Δt = 0.1 ms for the UKF simulation file. The following initial values were used in implementing the MATLAB function block for the UKF method algorithm: for the state vector, x0 = [0.4; 0; 0; 0] and for the state covariance matrix, P0 = diag([10, 10, 10, 10]). Also, the values of the noise characteristics were defined as follows: For the process noise, we had wk ∼ (0, Qk ) = (0, 0.0012 × (I4×4 )) and for the measurement noise, vk ∼ (0, Rk ) = (0, 0.012 × I). The size of matrix I4×4 is related to the fact that we have four states in the model. Also, the initial values, used for the EKF results in Fig. 10 were set as follows: x0 = [0.6; 0; 0; 0] for the state variable, and P0 = diag([102 , 102 , 102 , 102 ]) for the state covariance matrix. Finally, the noise covariance matrices were defined as follows: for state noise wk ∼ (0, Qk ) = (0, 0.082 × (I4×4 )) and for output noise vk ∼ (0, Rk ) = (0, 0.12 × I). Lastly, for step response simulation studies in Simulink, the value of mechanical torque input was fixed at Tm = 0.8, and the step function settings for Ef d signal were defined as follows: initial value = 0, final value = 0.2, rise time = 0.1 (s), and this variation was added to a constant value of 2.11. To mimic near real system conditions for both the UKF and the EKF methods in Fig. 10, white noise was added to the state process with the mean and covariance (0, 0.0012 × (I4×4 )). Likewise, the output measurement noise mean and covariance were set to (0, 0.012 × I). REFERENCES [1] P. Kundur, Power System Stability and Control. New York: McGraw Hill, 1994. [2] Y. N. Yu, Electric Power System Dynamic. New York: Academic Press, 1983.

[3] M. Anjia, Y. Jiaxi, and G. Zhizhong, “PMU placement and data processing in WAMS that complement SCADA,” in Proc. 2005 IEEE/PES General Meeting, San Francisco, CA, MA, pp. 780–783. [4] J. Chang, G. N. Taranto, and J. Chow, “Dynamic state estimation in power system using gain-scheduled nonlinear observer,” in Proc. 1995 IEEE Control Appl. Conf., Albany, NY, pp. 221–226. [5] Z. Huang, K. Schneider, and J. Neplocha, “Feasibility studies of applying Kalman filter techniques to power system dynamic state estimation,” in Proc. 2007 Int. Power Eng. Conf. (IPEC), Singapore, pp. 376–382. [6] I. Kamwa, B. Baraboi, and R. Wamkeue, “Sensorless ANN-based speed estimation of synchronous generator: improved performance through physically motivated pre-filters,” in Proc. 2006 IEEE Neural Network Conf. (IJCNN), Vancouver, BC, pp. 1710–1718. [7] A. Del Angel, P. Geurts, D. Ernst, M. Glavic, and L. Wehenkel, “Estimation of rotor angle of synchronous machines using artificial neural networks and local PMU-based quantities,” Neurocomputing, vol. 70, pp. 2668– 2678, Oct. 2007. [8] V. Venkatasubramanian and R. G. Kavasseri, “Direct computation of generator internal dynamic states from terminal measurements,” in Proc. 37th Hawaii Int. Conf. Syst. Sci., 2004, Washington, DC, pp. 1–6. [9] L. Lin, Linawati, L. Jasa, and E. Ambikairajah, “A hybrid state estimation scheme for power system,” in Proc. 2002 IEEE Circuits Syst. Conf. (APCCAS’02), Bali, Indonesia, Oct. 28–31, vol. 1, pp. 555–558. [10] M. Huanga, W. Li, and W. Yana, “Estimating parameters of synchronous generators using square-root unscented Kalman filter,” Int. J. Electric Power Syst. Res., vol. 80, pp. 1137–1144, Sep. 2010. [11] I. Kamwa, M. Leclerc, and D. McNabb, “Performance of demodulationbased frequency measurement algorithms used in typical PMUs,” IEEE Trans. Power Del., vol. 19, no. 2, pp. 505–514, Apr. 2004. [12] J. Warichet, T. Sezi, and J. C. Maun, “Considerations about synchrophasors measurements in dynamic system conditions,” Int. J. Elect. Power Energy Syst., vol. 31, pp. 452–464, Oct. 2009. [13] I. Kamwa, K. Pradhan, and G. Joos, “Adaptive phasor and frequencytracking schemes for wide-area protection and control,” IEEE Trans. Power Del., vol. 26, no. 2, pp. 744–753, Feb. 2011. [14] G. Valverde and V. Terzija, “Unscented Kalman filter for power system dynamic state estimation,” IET Gener. Transm. Distrib., vol. 5, no. 1, pp. 29–37, 2011. [15] D. Simon, Optimal State estimation; Kalman, H∞ , and Nonlinear Approaches. New Jersey: John Wiley & Sons, 2006. [16] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE, vol. 92, no. 3, pp. 401–422, Mar. 2004. [17] R. Kandepu, B. Foss, and L. Imsland, “Applying the unscented Kalman filter for nonlinear state estimation,” Int. J. Process Control, vol. 18, pp. 753–768, Aug. 2008.

Esmaeil Ghahremani received the B.Sc. and M.Sc. degrees from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2003 and 2007 respectively. He has been pursuing the Ph.D. degree under Prof. Kamwa’s supervision, studying in Power System Stability, at Laval University, Laval, QC, Canada, since 2008. His research interests include power system stability studies, dynamic state estimation of power system, FACTS placement and wide area control of power system. Innocent Kamwa (S’83–M’88–SM’98–F’05) received the Ph.D. degree in electrical engineering from Laval University, Qu´ebec, Canada, in 1988. Since 1988, he has been with the Hydro-Qu´ebec Research Institute (IREQ), Power System Analysis, Operation, and Control, Varennes, Qu´ebec, where he is currently a Project Manager for Automation and Control Systems and also Principal Scientist for smart grids. He has also been an Associate Professor of electrical engineering at Laval University, since 1990. Dr. Kamwa has been active for the last 13 years on the IEEE Electric Machinery Committee, where he is presently the Standards Coordinator. A member of CIGRE´ and a registered Professional Engineer, he is a recipient of the 1998, 2003, and 2009 IEEE Power Engineering Society Prize Paper Awards and is currently serving the IEEE/PES Standards Coordinating and System Dynamic Performance Committees.