**4. Flow chart and implementation**

The proposed WAMS based dynamic state estimator during electromechanical process is described as in Fig. 1.

The problems which should be noted in the implementation of the proposed estimator are described as follows:

#### **4.1 Startup criterion**

The steady state estimation aims at the power system steady state, whereas the proposed dynamic state estimator aims at the power system electromechanical transient process. Therefore, the startup criterion is needed for the proposed dynamic state estimator. The general startup criterion of the micro-computer based protection can be adopted as the startup criterion in the proposed dynamic estimator. The criterion triggers startup if three sequential instantaneous sampling values of generator terminal voltage and current all exceed the preset threshold.

After startup, a appropriate length of historical data of rotor angle, electrical angular velocity and electrical output power are backdated to ensure the convergence of Kalman filter before the startup time (power system fault occurring time).

#### **4.2 Bad data identification and elimination**

The rotor angle and electrical angular velocity have no discontinuous variation; therefore, the absolute residuals related with the bad measurement data will be increased anomaly. This feature can be used to identify and eliminate the bad data.

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

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

0 0

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

value should be employed to calculate the instantaneous electrical active power *Pe*.

0

**Q** (18)

2 2 *P P I R ui ui ui I R e t aa bb cc t* (19)

0 0.0004 0.0001 *Pe* 

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

To avoid the impact of the time delay of phasor calculation, the instantaneous sampling

The proposed WAMS based dynamic state estimator during electromechanical process is

The problems which should be noted in the implementation of the proposed estimator are

The steady state estimation aims at the power system steady state, whereas the proposed dynamic state estimator aims at the power system electromechanical transient process. Therefore, the startup criterion is needed for the proposed dynamic state estimator. The general startup criterion of the micro-computer based protection can be adopted as the startup criterion in the proposed dynamic estimator. The criterion triggers startup if three sequential instantaneous sampling values of generator terminal voltage and current all

After startup, a appropriate length of historical data of rotor angle, electrical angular velocity and electrical output power are backdated to ensure the convergence of Kalman

The rotor angle and electrical angular velocity have no discontinuous variation; therefore, the absolute residuals related with the bad measurement data will be increased anomaly.

filter before the startup time (power system fault occurring time).

This feature can be used to identify and eliminate the bad data.

**4.2 Bad data identification and elimination** 

generator reject and fast valving are not triggered.

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

**4. Flow chart and implementation** 

covariance matrix *Q* is set as:

requirement generally.

described as in Fig. 1.

described as follows:

**4.1 Startup criterion** 

exceed the preset threshold.

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

Dynamic State Estimator Based on Wide Area

Fig. 2. IEEE 9-bus test system.

state variable

successfully.

standard deviations which are given in Section III.B.

Fig. 3 gives the dynamic estimation effect of Gen 2 with *δ* and

eliminates the measurement noise and bad data effectively for *δ* and

Measurement System During Power System Electromechanical Transient Process 333

of both ends trip to clear fault at the 56th cycles. The true values of rotor angles, electrical angular velocities and electrical output power are acquired by commercial power system simulation software BPA, and the time step length is 2 cycles. The measurement values consist of true values, additional measurement errors and bad data. The errors of all types of measurements are assumed to follow the normal distribution with zero mean and the

Since the measurement precision of rotor angle in steady state is relative high, the steady measurement value can be set as the initial value of state variable *δ*, and the initial value of

covariance matrix P0 is set to zero matrix for convenience. The startup time step is the 50th cycle, the initial time step to estimate is the 26th cycle and the end time step is the 600th cycle.

It can be seen form Fig.3 that the proposed dynamic estimator has favorable filter effect and

The subgraph (c) which zooms in the interested part of estimation effect for *δ* shows clearly that the proposed estimator eliminates the adverse impact of serial bad data points

shows the dynamic estimation effect of Gen 2 when only *δ* can be measured indirectly.

is set to 1 (p.u.). the reliability of initial value is relative high, the initial

direct measurement. Fig. 4

direct measurement.

For the example of rotor angle direct measurement, *Δδ*(k+1) denotes the predictive residual at time step k+1, and it can be calculated as follows:

$$
\Delta \delta(k+1) = \delta\_m(k+1) - \overline{\delta}(k+1) \tag{20}
$$

where, *δm*(k+1) denotes the measurement value at time step k+1, ( 1) *k* denotes the predictive value at time step k+1.

The average of three previous sequential predictive absolute residual Δδabsmean=(|*Δδ*(k)|+|*Δδ*(k-1)|+|*Δδ*(k-2)|)/3, so, the bad data identification criterion at time step k+1 is described below:

$$|\Delta\delta(k+1)| \geqslant \dots \geq \Delta\delta\_{\text{abscmen}}\tag{21}$$

If equation (21) holds, the corresponding measurement is identified as bad data. To eliminate its adverse impact, the absolute predictive residual |*Δδ*(k+1)| at time step *k*+1 is replaced by *Δδabsmean*, but its sign is reserved. After that, the filtering step of Kalman filter and the estimation at next time step are continued.

Equation (21) aims at the single bad data point, but does not work for the serial bad data points. For instance, there are serial bad data points with negative residuals in measurement form time step *k*+1~*k*+*n*, if the criterion is applied, the absolute predictive residuals at time step *k*+1~*k*+*n* will be all replaced by *Δδabsmea*n, and the negative sign will be reserved, thus, the predictive residuals at time step *k*+1~*k*+*n* tend to reach a negative constant, which breaks the randomness of predictive residuals. In this situation, the divergence of Kalman filter may be resulted in and the sequent normal measurement points may be identified the bad data wrongly.

To avid this, the function sin(k) is used to recovery the randomness of predictive residuals at bad data points. In general, The value of function sin(*k*), *k*=1,···*n* (rad) obey the random distribution rule in the range of [–1,+1], thereby, if there is bad data at time step *k*+1, the substitute of the predictive residuals at the time step is

$$
\Delta\delta(k+1) = \sin(k+1) \cdot \Delta\delta\_{\text{absmear}}\tag{22}
$$

As mentioned before, the inferred rotor angle using equation (13) and (14) will produce an obvious discontinuity in fault duration. But in fact, the real rotor angle has no discontinuous variation; therefore, the discontinuity during fault stage can be regarded as a series of bad data. Applying the above bad data identification and elimination method, the more accurate rotor angle will be estimated.

#### **5. Numerical study**

IEEE 9-bus test system shown in Fig.2 and a real generator in North China power grid are chosen to carry out the numerical simulation.

In IEEE 9-bus test system, Gen1 and Gen2 are chosen as the generator to estimate. Gen1 and Gen2 are both equipped with governors and voltage regulators, and Gen2 is equipped with PSS. All the generators adopt the sixth-order detail model. A three-phase metal short-circuit fault is set on the beginning end of line BusB-Bus1 at the 50th cycles, and the circuit breakers

Fig. 2. IEEE 9-bus test system.

For the example of rotor angle direct measurement, *Δδ*(k+1) denotes the predictive residual

The average of three previous sequential predictive absolute residual Δδabsmean=(|*Δδ*(k)|+|*Δδ*(k-1)|+|*Δδ*(k-2)|)/3, so, the bad data identification criterion at

 |*Δδ*(*k*+1)|>5×*Δδ*absmean (21) If equation (21) holds, the corresponding measurement is identified as bad data. To eliminate its adverse impact, the absolute predictive residual |*Δδ*(k+1)| at time step *k*+1 is replaced by *Δδabsmean*, but its sign is reserved. After that, the filtering step of Kalman filter

Equation (21) aims at the single bad data point, but does not work for the serial bad data points. For instance, there are serial bad data points with negative residuals in measurement form time step *k*+1~*k*+*n*, if the criterion is applied, the absolute predictive residuals at time step *k*+1~*k*+*n* will be all replaced by *Δδabsmea*n, and the negative sign will be reserved, thus, the predictive residuals at time step *k*+1~*k*+*n* tend to reach a negative constant, which breaks the randomness of predictive residuals. In this situation, the divergence of Kalman filter may be resulted in and the sequent normal measurement points may be identified the

To avid this, the function sin(k) is used to recovery the randomness of predictive residuals at bad data points. In general, The value of function sin(*k*), *k*=1,···*n* (rad) obey the random distribution rule in the range of [–1,+1], thereby, if there is bad data at time step *k*+1, the

As mentioned before, the inferred rotor angle using equation (13) and (14) will produce an obvious discontinuity in fault duration. But in fact, the real rotor angle has no discontinuous variation; therefore, the discontinuity during fault stage can be regarded as a series of bad data. Applying the above bad data identification and elimination method, the more accurate

IEEE 9-bus test system shown in Fig.2 and a real generator in North China power grid are

In IEEE 9-bus test system, Gen1 and Gen2 are chosen as the generator to estimate. Gen1 and Gen2 are both equipped with governors and voltage regulators, and Gen2 is equipped with PSS. All the generators adopt the sixth-order detail model. A three-phase metal short-circuit fault is set on the beginning end of line BusB-Bus1 at the 50th cycles, and the circuit breakers

 

( 1) ( 1) ( 1) *k kk <sup>m</sup>* (20)

*Δδ*(*k*+1)=sin(*k*+1)·*Δδ*absmean (22)

( 1) *k* denotes the

at time step k+1, and it can be calculated as follows:

and the estimation at next time step are continued.

substitute of the predictive residuals at the time step is

predictive value at time step k+1.

time step k+1 is described below:

bad data wrongly.

rotor angle will be estimated.

chosen to carry out the numerical simulation.

**5. Numerical study** 

where, *δm*(k+1) denotes the measurement value at time step k+1,

of both ends trip to clear fault at the 56th cycles. The true values of rotor angles, electrical angular velocities and electrical output power are acquired by commercial power system simulation software BPA, and the time step length is 2 cycles. The measurement values consist of true values, additional measurement errors and bad data. The errors of all types of measurements are assumed to follow the normal distribution with zero mean and the standard deviations which are given in Section III.B.

Since the measurement precision of rotor angle in steady state is relative high, the steady measurement value can be set as the initial value of state variable *δ*, and the initial value of state variable is set to 1 (p.u.). the reliability of initial value is relative high, the initial covariance matrix P0 is set to zero matrix for convenience. The startup time step is the 50th cycle, the initial time step to estimate is the 26th cycle and the end time step is the 600th cycle. Fig. 3 gives the dynamic estimation effect of Gen 2 with *δ* and direct measurement. Fig. 4 shows the dynamic estimation effect of Gen 2 when only *δ* can be measured indirectly.

It can be seen form Fig.3 that the proposed dynamic estimator has favorable filter effect and eliminates the measurement noise and bad data effectively for *δ* and direct measurement. The subgraph (c) which zooms in the interested part of estimation effect for *δ* shows clearly that the proposed estimator eliminates the adverse impact of serial bad data points successfully.

Dynamic State Estimator Based on Wide Area

good as that in Fig.3, but is still acceptable.

Whether the state variables *δ* and

performance.

steps. ˆ*<sup>i</sup> x* ,*xi*

*<sup>M</sup>* and *xi*

immune to the dynamic plant noise.

Measurement System During Power System Electromechanical Transient Process 335

The similar conclusion can be drawn form Fig. 4, Moreover, the subgraph (a) and (c) show that the indirect measurement of rotor angle *δ* has an obvious sag in fault duration. Applying the proposed bad data elimination method for serial bad data, the sag is made up well, which smoothes the estimated curve for *δ* and achieves excellent filter effect. Since

measurement noises and bad data are added on the electrical output active power introduced as the controlling variable (Fig. 4(d)), but the estimation effect is almost not affected. The reason is that the proposed dynamic estimator is an integral process of the electrical output active power substantially, and the integral itself has better antinoise

To acquire the quantified estimation index, the filter effect ρ and index ε[13] are defined as:

1

*i <sup>n</sup> <sup>M</sup> i i*

*n*

1

*<sup>n</sup> i i i i*

*x x x n*

where, *i* indicates the sequence number of time step, n indicate the total number of time

variable *x* (*δ* or *ω*) at time step *i* respectively. The filter effect *ρ* and index *ε* of *δ* and *ω* are

Table 1 gives the estimation indices of Gen1 and Gen2 with different measurement modes, and all results in Table I are the average values over 100 runs of Monte Carlo. It can be seen from Table I that the proposed dynamic estimator achieves good estimation effects with different measurement modes, and the estimation time is about 0.1 ms, which can satisfy the requirements of real-time applications. The estimation results of Gen 1 and Gen 2 with direct measurement of *δ* and *ω* is higher than those with only indirect measurement of *δ*. Moreover, the estimation precisions of Gen1 are higher than those of Gen2, because that the inertia constant of Gen1 is bigger than of Gen2, which makes the estimation of Gen1 is more

A real generator in North China power grid is also chosen as the estimated generator, a three-phase metal short-circuit fault is put on a 500 kV line in the grid to excite the electromechanical transient, and the corresponding circuit breakers are tripped to clear fault after 5 cycles. All the simulation conditions are same as those of IEEE-9 test system except

*i*

1

calculated respectively for the difference between their quantities is very large.

that the time step length is 1 cycle. The simulation results is given in Table 2.

  2

( ) ˆ

*x x*

*i i*

( )

<sup>ˆ</sup> 100%

*x x*

 

2

*<sup>+</sup>* indicate the estimated value, measured value and true value of state

(24)

are measured directly or indirectly, the considerable

is not as

(23)

there is no measurements of electrical angular velocity, the estimation effect for

Fig. 3. The estimation effect of Gen2 with δ and ω direct measurement.

Fig. 4. The estimation effect of Gen2 when only δ can be measured indirectly.

Fig. 3. The estimation effect of Gen2 with δ and ω direct measurement.

Fig. 4. The estimation effect of Gen2 when only δ can be measured indirectly.

The similar conclusion can be drawn form Fig. 4, Moreover, the subgraph (a) and (c) show that the indirect measurement of rotor angle *δ* has an obvious sag in fault duration. Applying the proposed bad data elimination method for serial bad data, the sag is made up well, which smoothes the estimated curve for *δ* and achieves excellent filter effect. Since there is no measurements of electrical angular velocity, the estimation effect for is not as good as that in Fig.3, but is still acceptable.

Whether the state variables *δ* and are measured directly or indirectly, the considerable measurement noises and bad data are added on the electrical output active power introduced as the controlling variable (Fig. 4(d)), but the estimation effect is almost not affected. The reason is that the proposed dynamic estimator is an integral process of the electrical output active power substantially, and the integral itself has better antinoise performance.

To acquire the quantified estimation index, the filter effect ρ and index ε[13] are defined as:

$$
\rho = \frac{\sum\_{i=1}^{n} (\hat{\mathbf{x}}\_i - \mathbf{x}\_i^+)^2}{\sum\_{i=1}^{n} (\mathbf{x}\_i^M - \mathbf{x}\_i^+)^2} \tag{23}
$$

$$
\varepsilon = \frac{\sum\_{i=1}^{n} \left| \frac{\hat{\mathbf{x}}\_i - \mathbf{x}\_i^+}{\mathbf{x}\_i^+} \right| \times 100\,\% \tag{24}
$$

where, *i* indicates the sequence number of time step, n indicate the total number of time steps. ˆ*<sup>i</sup> x* ,*xi <sup>M</sup>* and *xi <sup>+</sup>* indicate the estimated value, measured value and true value of state variable *x* (*δ* or *ω*) at time step *i* respectively. The filter effect *ρ* and index *ε* of *δ* and *ω* are calculated respectively for the difference between their quantities is very large.

Table 1 gives the estimation indices of Gen1 and Gen2 with different measurement modes, and all results in Table I are the average values over 100 runs of Monte Carlo. It can be seen from Table I that the proposed dynamic estimator achieves good estimation effects with different measurement modes, and the estimation time is about 0.1 ms, which can satisfy the requirements of real-time applications. The estimation results of Gen 1 and Gen 2 with direct measurement of *δ* and *ω* is higher than those with only indirect measurement of *δ*. Moreover, the estimation precisions of Gen1 are higher than those of Gen2, because that the inertia constant of Gen1 is bigger than of Gen2, which makes the estimation of Gen1 is more immune to the dynamic plant noise.

A real generator in North China power grid is also chosen as the estimated generator, a three-phase metal short-circuit fault is put on a 500 kV line in the grid to excite the electromechanical transient, and the corresponding circuit breakers are tripped to clear fault after 5 cycles. All the simulation conditions are same as those of IEEE-9 test system except that the time step length is 1 cycle. The simulation results is given in Table 2.

Dynamic State Estimator Based on Wide Area

0-7381-4820-2 SH95382

pp. 1-6

0272-1724

0018-9510

1987, ISSN : 0885-8950

**7. References** 

Measurement System During Power System Electromechanical Transient Process 337

[1] A. G. Phadke, "Synchronized phasor measurements—a historical overview," in *Proc*.

[2] T. W. Cease and B. Feldhaus, "Real-time monitoring of the TVA power system," *IEEE Comput. Appl. Power*, vol. 7, no. 3, pp. 47–51, Jul. 1999. ISSN : 0895-0156 [3] A. G. Phadke, "Synchronized Phasor Measurements in Power Systems", *Comput. Appl.* 

[4] *Standard for Synchrophasors for Power Systems*, IEEE Standard C37.118-2005m, 2005. ISBN:

[5] Zhenyu Huang, John F. Hauer, and Kenneth E. Martin. "Evaluation of PMU dynamic

[6] J. F. Hauer, K. E. Martin, and Harry Lee, "Evaluating the dynamic performance of

[7] William L. Miller and John B. Lewis. "Dynamic state estimation in power systems," *IEEE Trans. Automatic Control*, vol. 16, no. 6, pp. 841-846, Dec. 1971, ISSN : 0018-9286 [8] Jaewon Chang, Glauco N. Taranto, and Joe H. Chow. "Dynamic State Estimation in

[9] A. G. Phadke, J. S. Thorp, and K. J. Karimi, "State estimation with phasor

[10] J. S. Thorp, A. G. Phadke, and K. J. Karimi, "Real time voltage-phasor measurements

[11] R. Zivanovic and C. Cairns, Implementation of PMU technology in state estimation: an

[12] A. S. Debs and R. E. Larson. "A dynamic estimator for tracking the state of the power

[13] A. Silva, M. Filho, and J. Cautera. "An efficient dynamic state estimation algorithm

[14] Durgaprasad G., and Thakur S.S. "Robust dynamic state estimation of power system

[15] Kuang-Rong Shih and Shyh-Jier Huang. "Application of a Robust Algorithm for

*Power Syst*., vol. 13, no. 4, pp. 1331-1336, 1998, ISSN : 0885-8950

performance in both lab environments and under field operating conditions," *IEEE Power Engineering Society General Meeting*, ISSN: 1932-5517, Tampa, 24-28, Jun, 2007,

phasor measurement units: experience in the western power system," WECC

Power System using a Gain-Scheduled Nonlinear Observer," in *Proceedings of the 4th IEEE Conference on Control Applications*, Print ISBN: 0-7803-2550-8, Albany,

measurements," *IEEE Trans. Power Syst*., vol. 1, no. 1, pp. 233–240, Feb,1986. ISSN :

for static state estimation," *IEEE Trans*. *Power App*. *Syst*., vol. PAS-104, no. 11, pp.

overview, in *Proc. IEEE 4th AFRICON,* Print ISBN: 0-7803-3019-6, Stellenbosch ,

system," *IEEE Trans. Power Apparat*. *Syst*., vol. 89, no. 7, pp. 1670–1678, 1970, ISSN :

including bad data processing". *IEEE Trans*. *Power Syst*. vol 2, no. 4 , pp. 1050–1058,

based on M-estimation and realistic modeling of system dynamics". *IEEE Trans*.

Dynamic State Estimation of a Power System". IEEE Trans. Power Syst., vol 17,

7803-7525-4, Yokohama, Japan, 6–10, Oct., 2002, vol. 1, pp. 476–479.

*Power*, vol. 6, no.2, pp. 10-15, 1993, ISSN : 0895-0156

Disturbance Monitoring Work Group, June 15, 2004.

NY,USA,28-29,Sep, 1995, pp. 221-226

3098–3104, Nov. 1985. ISSN : 0272-1724

no.1, pp. 141-147, 2002, ISSN : 0885-8950

South Africa , 24–27, Sep, 1996, vol. 2, pp. 1006–1011.

*IEEE Power Eng*. *Soc*. *Asia Pacific Transmission Distribution Conf*. *Exhib*, Print ISBN: 0-

The similar conclusion can be drawn form Table II; moreover, the estimation precisions are higher than those in IEEE 9 test-system. Since the generator is decoupled with outer network in the proposed dynamic estimator, the parallel dynamic estimation of different generators in large power system can be executed simultaneously. Therefore, the fast speed of the proposed dynamic estimator is not affected by the increasing numbers of generators in large power systems.


Table 1. Estimation results of G1 and G2 in IEEE 9-bus system.


Table 2. Estimation results of one generator in practical grid.
