**1. Introduction**

322 Advanced Topics in Measurements

Shahraeini, M. Javidi, M. H. & Ghazizadeh, M. S. (2011). Comparison between

Taylor, C. W. (2006). Wide Area Measurement, Monitoring and Control in Power Systems,

Thomas M. S., Nanda D. & Ali I. (2006). Development of a Data. Warehouse for Non-

Thomas, M. S. Nanda, D. & Ali, I. (2006). Development of a Data Warehouse for Non-

Vaishnav, R. Gopalakrishnan P. & Thomas, J. (2008). Using public mobile phone networks

Xiaorong, X., Yaozhong, X. Jinyu, X. Jingtao W. & Yingdao, H. (2006). WAMS applications in Chinese power systems, *IEEE Power Energy Magazine*, Vol. 4, No. 1, pp. 54–63 Yan, D. (2006). Wide-area Protection and Control System with WAMS Based, *2006* 

Zhang, P. (2006). Meeting the Protection and Control Challenges of 21st Century, *EPRI* 

Stallings, W. (1997). Data and Computer Communications, fifth edition, Prentice-Hall Inc Stouffer, K.; Falco, J.; & Kent, K. (2008). Guide to Industrial Control Systems (ICS) Security

Special Publication, vol. 800

India

India

2006, pp. 1-5

*Technical Report* 

Synchrony (2001). Trends in SCADA for Automated Water Systems

Power Systems, Imperial College, London, 16–17 Mar. 2006

*and Delivery of Electrical Energy in the 21st Century*, pp. 1-5

Communication Infrastructures of Centralized and Decentralized Wide Area Measurement Systems, IEEE Transaction on Smart Grid, Vol. 2, No. 1, pp. 206-211

Recommendations of the National Institute of Standards and Technology, NIST

Presented at Workshop on Wide Area Measurement, Monitoring and Control in

operational Data in Power Utilities, *Proceeding Power India Conference*, New Dehli,

Operational Data in Power Utilities, Proc. Power India Conference, New Dehli,

for distribution automation, *Power and Energy Society General Meeting-Conversion* 

*International Conference on Power System Technology (PowerCon2006)*, China, Oct.

The wide area measurement system (WAMS) developed rapidly in recent years [1-3]. It has been applied to the studies on many topics in monitoring and control of power systems. But as a kind of measurement system, WAMS has the measurement error and bad data unavoidably. The steady measurement errors of WAMS have been prescribed in corresponding IEEE standard [4], but the dynamic measurement errors now become the focus of discussion [5-6] and attract the attention of PSRC workgroup H11. If the dynamic raw data is applied directly, the unpredictable consequence will be resulted in, which will do a lot of damage to power systems. Therefore, the dynamic estimation for the state variables during electromechanical transient process is the backbone for WAMS based dynamic applications and real-time control.

There was no effective means to measure the power system dynamic process before WAMS come forth; therefore, the dynamic estimation for power system state variables during electromechanical transient process was not feasible. In reference [7], a dynamic estimator for generator flux state variables during transient process is proposed, but the dimension of the flux state variables is relative high, and the accurate values of parameters can not be achieved easily. In reference [8], a non-linear dynamic state observer for generator rotor angle during electromechanical transient process is proposed, but the method is only applicable to one machine infinite bus system (OMIB), and the fault scenarios is required to satisfy the preset mode.

After WAMS come forth, many references focused on the steady state estimation with PMU measurements and had many achievements [9-11]. Comparing with the steady state estimation, the traditional dynamic state estimation [12-15] aims at the relative slow load fluctuation, which is different with the proposed dynamic state estimation during electromechanical transient process. The traditional dynamic state estimation employs the measurement equations based on the network constraints, and predicts the state variables using exponential smoothing techniques. But during the power system fault stage and consequent dynamic process, the network topology is changed and can not be acquired in time; the bus voltage phase angles has jump discontinuities and are not easy to be predicted. Thereby, the centralized dynamic estimation which adopts the measurement equations

Dynamic State Estimator Based on Wide Area

steps: prediction step and filtering step.

The prediction step is:

The filtering step is:

measurement covariance matrix ).

sequences and is Gaussian with mean covariance:

and *Pk* is the estimated error matrix at time step *k*.

**3. The proposed dynamic estimator model** 

The rotor motion equation is written as follows:

*d dt*

  matrix.

Measurement System During Power System Electromechanical Transient Process 325

where, *Q* is the plant noise covariance matrix, *R* is the measurement noise covariance

The initial state *x*(0) is assumed to be uncorrelated with the plant and measurement noise

*E*[*x*(0)]= *x*ˆ(0) ; *E*[(*x*(0)– *x*ˆ(0) )(*x*(0)– *x*ˆ(0) )*T*]=*P***<sup>0</sup>**

The Kalman filter formulas for time-invariant linear model can be largely divided into two

( 1) ( ) ( )

where, **x**( 1) *k* is the predictive value of state variables *x* at time step *k*+1, *k*<sup>1</sup> **P** is the predictive error covariance matrix (pre-measurement covariance matrix ) at time step *k*+1,

1

*k*

where, *Kk*+1 is the Kalman gain matrix at time step *k*, **x**ˆ( 1) *k* is the estimated value of state variables *x* at time step *k*+1, and *Pk*+1 is the estimated error matrix at time step *k+*1 (post-

During the power system fault stage and consequent dynamic process, the network topology is changed and can not be acquired in time; the bus voltage phase angles has discontinuities and are not easy to be predicted. Therefore, the generator rotor angle and electrical angular velocity are regarded as the estimated state variables which can not mutate suddenly and obey the rotor motion equation. Moreover, the generator angle trajectories implicate abundant dynamic information; thereby, acquiring accurate generator

1 1 ( )( ) *m e m e*

  *J J*

*dt T T*

*<sup>d</sup> P P T TD <sup>D</sup>*

*kk k k*

( ) ˆ( 1) ( 1) ( ( 1) ( 1))

*T T*

*k kk*

*T T*

1

**<sup>P</sup> <sup>Φ</sup><sup>P</sup> Φ ΓQΓ** (4)

(5)

(6)

1

11 1

**K P C CP C R x x K y Cx**

*kk k*

**3.1 The proposed dynamic estimator model for generator variables** 

rotor trajectories is of great importance to power system real time control.

<sup>0</sup> ( 1)

 

1 11

( )

*k kk*

**P I K CP**

*k k*

 **x Φx Γu**

based on network constraints and regards the bus complex voltages as the state variables is not feasible for the electromechanical dynamic process.

In this paper, a novel WAMS based dynamic state estimator during power system electromechanical transient process is proposed. The estimator chooses the generator rotor angle and electrical angular velocity as the state variables to estimate. The generator output power measured by PMU is used to decouple the generator rotor movement equation and the outer network equation. And the linear Kalman filter based dynamic state estimator mathematical model is presented. The WAMS measurement noise and dynamic model noise are analyzed in detail. The total flow chart and the bad data detection and elimination approach are given as well. The numerical simulation is carried out on the IEEE 9-bus test system and a real generator in North China power grid. The simulation results indicate that the proposed real time dynamic estimator can estimate the generator rotor angles accurately; therefore, which can serve the power system dynamic monitoring and control system better.

#### **2. Kalman filter**

The time-invariant linear system model can be written as [16]

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} \\ \mathbf{y} = \mathbf{C}\mathbf{x} \end{cases} \tag{1}$$

The discrete form of equation (1) is

$$\begin{cases} \mathbf{x}(t+T) = \mathbf{O}(T)\mathbf{x}(t) + \Gamma(T)\mathbf{u}(t) \\ \mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) \end{cases} \tag{2}$$

where

*T* is the sampling period, *t=kT*, *k*=0, 1, 2, …, *Φ*(*T*) is the state transition matrix, and

$$\mathbf{O}(T) = \exp(AT) = \sum\_{i=0}^{n} \mathbf{A}^{i} T^{i} \text{ / (i \text{ }!) }, \quad A^{0} = \mathbf{I}; \quad \mathbf{T}(T) = \left[ \int\_{0}^{T} \mathbf{O}(a) d(a) \right] \mathbf{B} \text{, and } \mathbf{I} \text{ is the identity matrix.} $$

When subject to a random plant and measurement noise, the sampled-data system can be expressed as

$$\begin{cases} \mathbf{x}(k+1) = \mathbf{Opx}(k) + \Gamma \mathbf{u}(k) + \Gamma \mathbf{w}(k) \\ \mathbf{y}(k) = \mathbf{Cx}(k) + \mathbf{v}(k) \end{cases} \tag{3}$$

The plant noise sequence *w*(*k*) and the measurement noise sequence *v*(*k*) are assumed to be Gaussian stationary white-noise sequence with zero means and covariance:

$$\begin{aligned} E[\mathbf{v}(k)] &= E[\mathbf{w}(k)] = \mathbf{0} \\ E[\mathbf{w}(k)\mathbf{w}^T(j)] &= \mathbf{Q}\mathbf{\tilde{6}}\_{kj} \quad \mathbf{\tilde{6}}\_{kj} = \begin{cases} 1 & k=j \\ 0 & k \neq j \end{cases} \\ E[\mathbf{v}(k)\mathbf{v}^T(j)] &= \mathbf{R}\mathbf{\tilde{6}}\_{kj} \end{aligned}$$

where, *Q* is the plant noise covariance matrix, *R* is the measurement noise covariance matrix.

The initial state *x*(0) is assumed to be uncorrelated with the plant and measurement noise sequences and is Gaussian with mean covariance:

$$E[\mathbf{x}(0)] = \hat{\mathbf{x}}(0) \; ; \; E[(\mathbf{x}(0) - \hat{\mathbf{x}}(0))(\mathbf{x}(0) - \hat{\mathbf{x}}(0))^T] = \mathbf{P}\_0$$

The Kalman filter formulas for time-invariant linear model can be largely divided into two steps: prediction step and filtering step.

The prediction step is:

324 Advanced Topics in Measurements

based on network constraints and regards the bus complex voltages as the state variables is

In this paper, a novel WAMS based dynamic state estimator during power system electromechanical transient process is proposed. The estimator chooses the generator rotor angle and electrical angular velocity as the state variables to estimate. The generator output power measured by PMU is used to decouple the generator rotor movement equation and the outer network equation. And the linear Kalman filter based dynamic state estimator mathematical model is presented. The WAMS measurement noise and dynamic model noise are analyzed in detail. The total flow chart and the bad data detection and elimination approach are given as well. The numerical simulation is carried out on the IEEE 9-bus test system and a real generator in North China power grid. The simulation results indicate that the proposed real time dynamic estimator can estimate the generator rotor angles accurately; therefore, which can serve the power system dynamic monitoring and control

> **x Ax Bu y Cx**

() ()

*t t* **x Φ x Γ u**

**<sup>A</sup>** ,*A*0=*I*; <sup>0</sup> ( ) ( )( )

Gaussian stationary white-noise sequence with zero means and covariance:

[ ( )] [ ( )] [ ( ) ( )] [ ( ) ( )]

*Ek E k Ek j Ek j*

*T*

*T*

( ) ( )() ( ) ()

*T T d* 

When subject to a random plant and measurement noise, the sampled-data system can be

( 1) ( ) ( ) ( )

The plant noise sequence *w*(*k*) and the measurement noise sequence *v*(*k*) are assumed to be

*kj*

**δ**

*kj*

*k kk k*

() () ()

 

**v w0 ww Qδ vv Rδ**

*k kk* **x Φx Γu Γw**

*tT T t T t*

(1)

**y Cx** (2)

**Γ Φ <sup>B</sup>** ,and I is the identity matrix.

**y Cx v** (3)

*k j k j* 

1 0 *kj*

not feasible for the electromechanical dynamic process.

The time-invariant linear system model can be written as [16]

system better.

where

*Φ*(*T*)=exp(*AT*)=

expressed as

**2. Kalman filter** 

The discrete form of equation (1) is

*T* is the sampling period, *t=kT*, *k*=0, 1, 2, …, *Φ*(*T*) is the state transition matrix, and

/( !) *i i*

*T i*

0

*i*

$$\begin{cases} \overline{\mathbf{x}}(k+1) = \boldsymbol{\upPhi} \mathbf{x}(k) + \Gamma \mathbf{u}(k) \\ \qquad \mathbf{P}'\_{k+1} = \boldsymbol{\upPhi} \mathbf{P}\_k \boldsymbol{\upPhi}^T + \Gamma \mathbf{Q} \Gamma^T \end{cases} \tag{4}$$

where, **x**( 1) *k* is the predictive value of state variables *x* at time step *k*+1, *k*<sup>1</sup> **P** is the predictive error covariance matrix (pre-measurement covariance matrix ) at time step *k*+1, and *Pk* is the estimated error matrix at time step *k*.

The filtering step is:

$$\begin{cases} \mathbf{K}\_{k+1} = \mathbf{P}'\_{k+1} \mathbf{C}^T (\mathbf{C} \mathbf{P}'\_{k+1} \mathbf{C}^T + \mathbf{R})^{-1} \\ \hat{\mathbf{x}}(k+1) = \overline{\mathbf{x}}(k+1) + \mathbf{K}\_{k+1} (\mathbf{y}(k+1) - \mathbf{C} \overline{\mathbf{x}}(k+1)) \\ \mathbf{P}\_{k+1} = (\mathbf{I} - \mathbf{K}\_{k+1} \mathbf{C}) \mathbf{P}'\_{k+1} \end{cases} \tag{5}$$

where, *Kk*+1 is the Kalman gain matrix at time step *k*, **x**ˆ( 1) *k* is the estimated value of state variables *x* at time step *k*+1, and *Pk*+1 is the estimated error matrix at time step *k+*1 (postmeasurement covariance matrix ).

### **3. The proposed dynamic estimator model**

#### **3.1 The proposed dynamic estimator model for generator variables**

During the power system fault stage and consequent dynamic process, the network topology is changed and can not be acquired in time; the bus voltage phase angles has discontinuities and are not easy to be predicted. Therefore, the generator rotor angle and electrical angular velocity are regarded as the estimated state variables which can not mutate suddenly and obey the rotor motion equation. Moreover, the generator angle trajectories implicate abundant dynamic information; thereby, acquiring accurate generator rotor trajectories is of great importance to power system real time control.

The rotor motion equation is written as follows:

$$\begin{cases} \frac{d\mathcal{S}}{dt} = (\alpha - 1)\alpha\_0\\ \frac{d\alpha}{dt} = \frac{1}{T\_f}(T\_m - T\_e - D\alpha) = \frac{1}{T\_f}(\frac{P\_m}{\alpha} - \frac{P\_e}{\alpha} - D\alpha) \end{cases} \tag{6}$$

Dynamic State Estimator Based on Wide Area

equation (8) that the impact of

If only *δ* can be measured by PMU, then:

that can be covered by the dynamic plant noise.

synchronized by GPS, the electrical angular velocity

nominal range of

Therefore,

Measurement System During Power System Electromechanical Transient Process 327

2. During electromechanical transient process, the value of is about 1 (p.u.) and the off

*δ* can be measured by PMU synchronistically (direct measurement or inferred by generator terminal electrical variables). If the pulse sequences of the rotor angular velocity meter are

Calculating the state transition matrix (equation (2)) in engineering, only the three former

*T J J*

0 2

Now, the generator dynamic equation qualified to Kalman filter (equation (2)) is built up, but to realize the predictive step (equation (4)) and filtering step (equation (5)) of Kalman filter algorithm, the exact value of plant noise covariance Q and measurement noise R. the

The measurement noise (error) covariance matrix *R* and plant noise covariance matrix *Q* are analyzed respectively in terms of the rotor angle measurement mode (direct measurement

The direct measurement method of generator rotor angle using rotor position sensor has relative higher accuracy. The method assumes that the electrical angular velocity is constant

**3.2.1 The direct measurement mode of rotor angle and electrical angular velocity** 

**y C** 

2 2

terms are taken into account, which can achieve sufficient accuracy.

( ) / 2

*T TT*

**Φ IA A**

( ) ( )( )

**Γ Φ <sup>B</sup>**

 

*T d*

following error analysis analyzes the problem in detail.

**3.2 Error and noise analysis** 

mode or indirect measurement mode).

is about from parts per thousand to 2 percent. It can be seen form

0

<sup>180</sup> 1 (1 ) <sup>2</sup>

0 1 (1 ) <sup>2</sup>

2 0

<sup>180</sup> (1 ) 2 3

2

*T DT DT T T T*

*J J J*

<sup>0</sup> (1 ) <sup>2</sup> <sup>3</sup>

*T DT <sup>T</sup>*

*J J*

*T T*

*DT DT T T*

*DT <sup>T</sup> T*

fluctuation upon the controlling variable is so small

**y C** (10)

1 0 (11)

*J*

also can be measured by PMU.

where, *δ* is the generator rotor angle (rad), *ω* is the generator electrical angular velocity; *Tm* and *Te* are the mechanical torque and electrical torque on generator shaft respectively; *Pm* and *Pe* are the mechanical input power and electrical output powert respectively; *TJ* is the moment of inertia of the machine rotor and *D* is the damping coefficient.

It can be seen form equation (6) that if the electrical torque (power) and mechanical torque (power) at any time step are known, the generator motion equation will be decoupled from outer network [17]. Therefore, the generator rotor motion becomes single rigid body motion in two-dimension state space (displacement and velocity). If the generator mechanical torque is assumed constant, when only the electrical torque curve in time domain is known, the rotor motion equation is decoupled from outer network.

Equation (6) is written as follow form:

$$
\begin{bmatrix}
\dot{\delta} \\
\dot{\alpha}
\end{bmatrix} = \begin{bmatrix}
0 & \alpha\_0 \\
0 & -\frac{D}{T\_f}
\end{bmatrix} \begin{bmatrix}
\delta \\
\alpha
\end{bmatrix} + \begin{bmatrix}
T\_m - T\_e \\
T\_f
\end{bmatrix} \tag{7}
$$

It is noted that equation (7) is a standard time-invariant linear system which is qualified to linear Kalman filter well.

The torque can not be measured easily, hence the second term (controlling variable vector) of the right side of equation (7) is written as the form of *Pm* and *Pe*, then the equation (7) is written as follow:

$$
\begin{bmatrix} \dot{\delta} \\ \dot{\alpha} \end{bmatrix} = \begin{bmatrix} 0 & \frac{180}{\pi} a\_0 \\ 0 & -\frac{D}{T\_f} \end{bmatrix} \begin{bmatrix} \delta \\ \alpha \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{T\_f} \end{bmatrix} \begin{bmatrix} -\frac{180}{\pi} a\_0 \\ \left( P\_m - P\_e \right) / \alpha \end{bmatrix} \tag{8}
$$

where, the unit of *δ* is degree.

Comparing equation (8) with equation (1), we have

$$\mathbf{x} = \begin{bmatrix} \delta \\ \alpha \end{bmatrix} \quad \mathbf{A} = \begin{bmatrix} 0 & \frac{180}{\pi} \alpha\_0 \\ 0 & -\frac{D}{T\_f} \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{T\_f} \end{bmatrix} \quad \mathbf{u} = \begin{bmatrix} -\frac{180}{\pi} \alpha\_0 \\ \frac{P\_m - P\_e}{\alpha} \end{bmatrix} \tag{9}$$

It should be pointed out that even the sate variable ω appears in controlling variable vector u of formula (8), thereby, formula (8) is not a strict state equation, but it does not affect the application of Kalman filter. The reasons are:

1. The state differential equation only adopted in predictive step to calculate the state variable at next time step. Even though *u*(*k*) is the function of (*k*), (*k*) has been estimated by Kalman filter at last time step, so it can be regarded as a known variable to substitute in equation (8)

2. During electromechanical transient process, the value of is about 1 (p.u.) and the off nominal range of is about from parts per thousand to 2 percent. It can be seen form equation (8) that the impact of fluctuation upon the controlling variable is so small that can be covered by the dynamic plant noise.

*δ* can be measured by PMU synchronistically (direct measurement or inferred by generator terminal electrical variables). If the pulse sequences of the rotor angular velocity meter are synchronized by GPS, the electrical angular velocity also can be measured by PMU. Therefore,

$$\mathbf{y} = \begin{bmatrix} \delta \\ \alpha \end{bmatrix} \quad \mathbf{C} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \tag{10}$$

If only *δ* can be measured by PMU, then:

326 Advanced Topics in Measurements

where, *δ* is the generator rotor angle (rad), *ω* is the generator electrical angular velocity; *Tm* and *Te* are the mechanical torque and electrical torque on generator shaft respectively; *Pm* and *Pe* are the mechanical input power and electrical output powert respectively; *TJ* is the

It can be seen form equation (6) that if the electrical torque (power) and mechanical torque (power) at any time step are known, the generator motion equation will be decoupled from outer network [17]. Therefore, the generator rotor motion becomes single rigid body motion in two-dimension state space (displacement and velocity). If the generator mechanical torque is assumed constant, when only the electrical torque curve in time domain is known,

> <sup>0</sup> <sup>0</sup> 0 0 *m e J J D T T T T*

It is noted that equation (7) is a standard time-invariant linear system which is qualified to

The torque can not be measured easily, hence the second term (controlling variable vector) of the right side of equation (7) is written as the form of *Pm* and *Pe*, then the equation (7) is

<sup>180</sup> <sup>0</sup> 1 0 <sup>180</sup>

<sup>1</sup> <sup>0</sup> <sup>0</sup> *<sup>J</sup>* ( )/ *m e <sup>J</sup>*

*<sup>T</sup> P P <sup>T</sup>*

<sup>180</sup> <sup>180</sup> <sup>0</sup> 1 0

It should be pointed out that even the sate variable ω appears in controlling variable vector u of formula (8), thereby, formula (8) is not a strict state equation, but it does not affect the

1. The state differential equation only adopted in predictive step to calculate the state

estimated by Kalman filter at last time step, so it can be regarded as a known variable to

<sup>1</sup> <sup>0</sup> <sup>0</sup> *m e <sup>J</sup> <sup>J</sup> <sup>D</sup> P P <sup>T</sup> <sup>T</sup>*

0

variable at next time step. Even though *u*(*k*) is the function of

*D*

Comparing equation (8) with equation (1), we have

application of Kalman filter. The reasons are:

substitute in equation (8)

(7)

0

> (*k*),

(*k*) has been

(8)

0 0

**xA B u** (9)

moment of inertia of the machine rotor and *D* is the damping coefficient.

the rotor motion equation is decoupled from outer network.

Equation (6) is written as follow form:

linear Kalman filter well.

where, the unit of *δ* is degree.

written as follow:

$$\mathbf{y} = \begin{bmatrix} \boldsymbol{\delta} \end{bmatrix} \quad \mathbf{C} = \begin{bmatrix} 1 & 0 \end{bmatrix} \tag{11}$$

Calculating the state transition matrix (equation (2)) in engineering, only the three former terms are taken into account, which can achieve sufficient accuracy.

$$\begin{aligned} \mathbf{D}(T) & \approx \mathbf{I} + \mathbf{A}T + \mathbf{A}^2 T^2 \;/\; 2 = \begin{bmatrix} 1 & \frac{180}{\pi} a\_0 T (1 - \frac{DT}{2T\_f}) \\\\ 0 & 1 - \frac{DT}{T\_f} (1 - \frac{DT}{2T\_f}) \end{bmatrix} \\\\ \mathbf{T}(T) &= \begin{bmatrix} \int\_0^T \mathbf{D}(\alpha) d(\alpha) \end{bmatrix} \mathbf{B} = \begin{bmatrix} T & \frac{180}{\pi} a\_0 \frac{T^2}{2T\_f} (1 - \frac{DT}{3T\_f}) \\\\ 0 & \frac{T}{T\_f} - \frac{DT^2}{2T\_f} (1 - \frac{DT}{3T\_f}) \end{bmatrix} \end{aligned}$$

Now, the generator dynamic equation qualified to Kalman filter (equation (2)) is built up, but to realize the predictive step (equation (4)) and filtering step (equation (5)) of Kalman filter algorithm, the exact value of plant noise covariance Q and measurement noise R. the following error analysis analyzes the problem in detail.

#### **3.2 Error and noise analysis**

The measurement noise (error) covariance matrix *R* and plant noise covariance matrix *Q* are analyzed respectively in terms of the rotor angle measurement mode (direct measurement mode or indirect measurement mode).

#### **3.2.1 The direct measurement mode of rotor angle and electrical angular velocity**

The direct measurement method of generator rotor angle using rotor position sensor has relative higher accuracy. The method assumes that the electrical angular velocity is constant

Dynamic State Estimator Based on Wide Area

generator rotor angle δ is:

corresponding variance is 9º.

discontinuity.

follows:

output active power and reactive power, respectively.

output active power and reactive power, respectively.

Measurement System During Power System Electromechanical Transient Process 329

where, *δ'* is the rotor angle with reference to terminal voltage phase. *P*, *Q* are the generator

It is assumed that the voltage phase of the generator terminal bus is θ, therefore, the

where, *δ'* is the rotor angle with reference to terminal voltage phase. *P*, *Q* are the generator

It should be kept in mind that equation (13) and (14) are both derived with the assumption that there is no damping current in rotor amortisseur, thereby, the equations have sufficient precision in steady state, but considerable errors may be resulted in during dynamic process. Especially in fault duration, the inferred rotor angle will produce an obvious

Strictly speaking, the error propagation theory should be used to calculate the indirect measurement error variance of the inferred rotor angle accurately, whereas, the formula is very complex and hard to calculate due to the factors such as measurement variation, damping current, time-variant parameters and iron saturation, etc. therefore, according to the errors variance of PMU direct measurements (terminal voltage, terminal current and output power) and considering all factors mentioned above synthetically and simply, the standard deviation of the indirect measurement error of rotor angle can be set as 3º, and the

Therefore, when only the rotor angle can be inferred by the PMU generator terminal measurements indirectly, the corresponding measurement noise covariance matrix is as

Since the phasor calculation has time delay, the sampling value with synchronous time stamp is adopted to calculate the instantaneous output active and reactive power directly.

<sup>1</sup> ( ( ) ( ) ( ))

*Q ui i ui i ui i*

where, *ua*,*ia*; *ub*,*ib* and *uc*,*ic* are the instantaneous sampling value of phase *a*, phase *b* and phase

The dynamic plant noise represents the errors of model and parameters. It can be seen form equation (8) that the involved parameters are generator inertial constant *TJ* and damping coefficient *D*. *TJ* can be acquired exactly normally; *D* is very small and only reflects the mechanical friction and the windage since the electrical damping has been covered by the measured output active power. Therefore, the dynamic plant noise mainly roots in the measurement error of electrical output active power and the variation of mechanical input

*ac b ba c cb a*

3

*c* generator terminal voltage and current, respectively.

**3.2.3 The dynamic plant noise analysis** 

*aa bb cc*

*P ui ui ui*

(15)

**R** 9 (16)

(17)

 

during one cycle, but in fact, the electrical angular velocity varies about parts per thousand during one cycle. According to analysis, the measurement error due to this is about 1~2º. Moreover, the detection precision of rotor position pulse also affects the measurement error. Considering all above factors, the standard deviation of the rotor angle direct measurement error can be set as 2º, and the corresponding variance is 4º.

The measurement of electrical angular velocity is equivalent to the measurement of rotor angular velocity (electrical angular velocity=number of pole pairs× rotor angular velocity). In modern power systems, the rotor angular velocity can be measured by velocity meter for turbine generator and hydro generator. The principle of velocity meter is described below. There is a 60-tooth gear installed in the rotor shaft and the detection circuit of velocity meter detects the pulse generated by each tooth of the gear. Therefore, 6º divided by the time interval between two pluses is the value of instantaneous angular velocity. If the pulse sequences generated by rotor angular velocity meter are synchronized by GPS, the electrical angular velocity can also be measured by PMU. The synchronized accuracy and the pulse detection accuracy both affect the measurement error, thereby, the standard deviation of the direct measurement error of rotor electrical angular velocity can be set as 0.001(p.u., i.e. 0.05~0.06 Hz), and the corresponding variance is1e-6.

It can be seen that when both the rotor angle and the electrical angular velocity can be measured by PMU directly, the corresponding measurement noise covariance matrix is as follows:

$$\mathbf{R} = \begin{bmatrix} 4 & 0 \\ 0 & 10^{-6} \end{bmatrix} \tag{12}$$

#### **3.2.2 The indirect measurement mode of rotor angle**

The direct measurement method of generator rotor angle is required to install the rotor position sensor, but some old style generators do not satisfy the installation condition. The direct synchronized measurement of electrical angular velocity is also required to install the additional GPS receiver and carry out necessary alteration. Thereby, these generators can not adopt the direct measurement mode and have to infer the rotor angle using the generator terminal electrical variables measured by PMU. The inferred rotor angle is regarded as the indirect measurement value of rotor angle and used in Kalman filter to estimate the rotor angle and electrical angular velocity.

If the generator terminal voltage phasor *Ut* and current phasor *<sup>t</sup> <sup>I</sup>* is measured, the virtual internal voltage *EQ* which is used to fix on the q axis position is written as,

$$
\dot{E}\_Q = \dot{\mathcal{U}}\_t + \dot{I}\_t (\mathcal{R} + X\_q) \tag{13}
$$

where, *R* is stator resistance, *Xq* is q axis synchronous reactance, the angle of *EQ* is rotor angle *δ*.

An alternative equation is:

$$\delta' = a \tan \frac{P X\_q - Q R}{\left(L\_t^{'}\right)^2 + P R + Q X\_q} \tag{14}$$

during one cycle, but in fact, the electrical angular velocity varies about parts per thousand during one cycle. According to analysis, the measurement error due to this is about 1~2º. Moreover, the detection precision of rotor position pulse also affects the measurement error. Considering all above factors, the standard deviation of the rotor angle direct measurement

The measurement of electrical angular velocity is equivalent to the measurement of rotor angular velocity (electrical angular velocity=number of pole pairs× rotor angular velocity). In modern power systems, the rotor angular velocity can be measured by velocity meter for turbine generator and hydro generator. The principle of velocity meter is described below. There is a 60-tooth gear installed in the rotor shaft and the detection circuit of velocity meter detects the pulse generated by each tooth of the gear. Therefore, 6º divided by the time interval between two pluses is the value of instantaneous angular velocity. If the pulse sequences generated by rotor angular velocity meter are synchronized by GPS, the electrical angular velocity can also be measured by PMU. The synchronized accuracy and the pulse detection accuracy both affect the measurement error, thereby, the standard deviation of the direct measurement error of rotor electrical angular velocity can be set as 0.001(p.u., i.e.

It can be seen that when both the rotor angle and the electrical angular velocity can be measured by PMU directly, the corresponding measurement noise covariance matrix is as

> 4 0 0 10

The direct measurement method of generator rotor angle is required to install the rotor position sensor, but some old style generators do not satisfy the installation condition. The direct synchronized measurement of electrical angular velocity is also required to install the additional GPS receiver and carry out necessary alteration. Thereby, these generators can not adopt the direct measurement mode and have to infer the rotor angle using the generator terminal electrical variables measured by PMU. The inferred rotor angle is regarded as the indirect measurement value of rotor angle and used in Kalman filter to

which is used to fix on the q axis position is written as,

where, *R* is stator resistance, *Xq* is q axis synchronous reactance, the angle of *EQ*

*a*

<sup>2</sup> tan *<sup>q</sup>*

*t q PX QR*

*U PR QX*

6

and current phasor *<sup>t</sup> <sup>I</sup>*

**R** (12)

*Q tt*( ) *E U IR X <sup>q</sup>* (13)

is measured, the virtual

(14)

is rotor

error can be set as 2º, and the corresponding variance is 4º.

0.05~0.06 Hz), and the corresponding variance is1e-6.

**3.2.2 The indirect measurement mode of rotor angle** 

estimate the rotor angle and electrical angular velocity.

If the generator terminal voltage phasor *Ut*

internal voltage *EQ*

An alternative equation is:

angle *δ*.

follows:

where, *δ'* is the rotor angle with reference to terminal voltage phase. *P*, *Q* are the generator output active power and reactive power, respectively.

It is assumed that the voltage phase of the generator terminal bus is θ, therefore, the generator rotor angle δ is:

$$
\delta = \theta + \delta'\tag{15}
$$

where, *δ'* is the rotor angle with reference to terminal voltage phase. *P*, *Q* are the generator output active power and reactive power, respectively.

It should be kept in mind that equation (13) and (14) are both derived with the assumption that there is no damping current in rotor amortisseur, thereby, the equations have sufficient precision in steady state, but considerable errors may be resulted in during dynamic process. Especially in fault duration, the inferred rotor angle will produce an obvious discontinuity.

Strictly speaking, the error propagation theory should be used to calculate the indirect measurement error variance of the inferred rotor angle accurately, whereas, the formula is very complex and hard to calculate due to the factors such as measurement variation, damping current, time-variant parameters and iron saturation, etc. therefore, according to the errors variance of PMU direct measurements (terminal voltage, terminal current and output power) and considering all factors mentioned above synthetically and simply, the standard deviation of the indirect measurement error of rotor angle can be set as 3º, and the corresponding variance is 9º.

Therefore, when only the rotor angle can be inferred by the PMU generator terminal measurements indirectly, the corresponding measurement noise covariance matrix is as follows:

$$\mathbf{R} = \begin{bmatrix} \mathbf{\mathcal{G}} \end{bmatrix} \tag{16}$$

Since the phasor calculation has time delay, the sampling value with synchronous time stamp is adopted to calculate the instantaneous output active and reactive power directly.

$$\begin{aligned} P &= \mu\_a \dot{\mathbf{i}}\_a + \mu\_b \dot{\mathbf{i}}\_b + \mu\_c \dot{\mathbf{i}}\_c \\ Q &= \frac{1}{\sqrt{3}} (\mu\_a (\mathbf{i}\_c - \mathbf{i}\_b) + \mu\_b (\mathbf{i}\_a - \mathbf{i}\_c) + \mu\_c (\mathbf{i}\_b - \mathbf{i}\_a)) \end{aligned} \tag{17}$$

where, *ua*,*ia*; *ub*,*ib* and *uc*,*ic* are the instantaneous sampling value of phase *a*, phase *b* and phase *c* generator terminal voltage and current, respectively.

#### **3.2.3 The dynamic plant noise analysis**

The dynamic plant noise represents the errors of model and parameters. It can be seen form equation (8) that the involved parameters are generator inertial constant *TJ* and damping coefficient *D*. *TJ* can be acquired exactly normally; *D* is very small and only reflects the mechanical friction and the windage since the electrical damping has been covered by the measured output active power. Therefore, the dynamic plant noise mainly roots in the measurement error of electrical output active power and the variation of mechanical input

Dynamic State Estimator Based on Wide Area

Measurement System During Power System Electromechanical Transient Process 331

*x*( 1) *k*

*x*ˆ( 1) *k*

*<sup>k</sup>* <sup>1</sup> *P*

*y Cx* ( 1) ( 1) *k k*

Fig. 1. Flow chart of the proposed dynamic state estimator.

power *Pm*. Since *Pm* is hard to measure accurately, it is assumed constant and its variation due to governor action is regarded as dynamic plant noise when only governor operates and generator reject and fast valving are not triggered.

In terms of relative standards [4], the standard deviation of electrical active power measurement error varies between 1%~2%, considering the variation of *Pm*, the plant noise covariance matrix *Q* is set as:

$$\mathbf{Q} = \begin{bmatrix} 0 & 0 \\ 0 & 0.0004 P\_{e0} + 0.0001 \end{bmatrix} \tag{18}$$

where, *Pe0* is the generator electrical output active power in pre-fault steady state.

It should be pointed out that *Q* varies with the measured active power *Pe*, but considering the impact of computational cost and bad data, equation (18) can satisfy the precision requirement generally.

To avoid the impact of the time delay of phasor calculation, the instantaneous sampling value should be employed to calculate the instantaneous electrical active power *Pe*.

$$P\_e = P + I\_t^{\ 2}R = \mu\_a \mathbf{i}\_a + \mu\_b \mathbf{i}\_b + \mu\_c \mathbf{i}\_c + I\_t^{\ 2}R \tag{19}$$

where, *It2R* is the copper loss of generator stator armature.
