Power System Dynamic State Estimation Based on a New Particle Filter

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Procedia Environmental Sciences 11 (2011) 655 – 661

doi:10.1016/j.proenv.2011.12.102

Procedia

Environmental

Sciences

Procedia Environmental Sciences 00 (2011) 000–000

www.elsevier.com/locate/procedia

Power System Dynamic State Estimation Based on a New

Particle Filter

CHEN Huanyuan

a

_{, LIU Xindong}

b

_{, SHE Caiqi, Yao Cheng}

College of Electrical Information Engineering Jinan University Zhuhai, China

a_{[email protected] ;}b_{[email protected]}

Abstract

In order to improve the performance of power system dynamic state estimation, a new particle filter for nonlinear filtering problems (Mixed Kalman Particle Filter, MKPF) is introduced. The MKPF method which based on the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), can obtain a more accurate approximate expression of the true distribution. Combined with the real-time data of mixed measurement (WAMS/SCADA), a simulation of power system dynamic state estimation is established. Finally, the simulation results show that the method can quickly follow to the real value after the power system is disturbed and obtain higher estimated accuracy and robustness than the EKF and UKF methods.

Keywords: Power system；Dynamic state estimation； Mixed Kalman Particle Filter； mixed measurement； simulation

1. Introduction

Dynamic state estimation is a branch of state estimation. The actual power system is a complex, nonlinear and dynamic system. Dynamic state estimation is more in line with the nature of the power system than the static state estimation. Dynamic state estimation, which has forecasting capabilities, can provide the real-time operational status of the power system. Therefore, it is an important part of the energy management system (EMS)[1-2]_.

Currently, the power system dynamic state estimation method is based on extended Kalman filter (EKF) method. In normal operating conditions, it is comparatively accurate to use EKF method to obtain the power system dynamic state estimation. While in some specific cases, such as load or generator output power mutates, the limitation of EKF method will produce a large error. In order to improve its prediction and filtering performances, Chinese and overseas scholars had made some improvements: On the basis of load forecasting model, reference [3] used a dynamic method which was able to truly predict the trend of the system load. But this load model could not use the original Kalman filter model for iterating. Reference [4] used Adaptive Kalman filtering (AKF) to improve the filtering accuracy. But due to its online estimate

Selection and/or peer-review under responsibility of the Intelligent Information Technology Application Research Association.

Open access under CC BY-NC-ND license.

Selection and/or peer-review under responsibility of the Intelligent Information Technology Application Research Association.

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Author name / Procedia Environmental Sciences 00 (2011) 000–000

model parameters and statistical characteristics of noise, the calculated amount was too large and was difficult to meet the online requirements. Reference [5] used the Unscented Kalman filter (UKF) method for power system dynamic state estimation, and achieved a more accurate estimation than traditional EKF method. However, UKF has certain restrictions on use. It applies only to ordinary Gaussian distribution model. While the actual power system is a nonlinear system, especially after the large disturbances. The load will change and the generators will also appear large oscillation. This change and oscillation are highly nonlinear, and the entire system is a time variant nonlinear system, that using UKF method for dynamic state estimation has certain defects.

Based on the above considerations, a new particle filter for nonlinear filtering problems proposed by reference [6] is introduced in this paper. This method, which mixes the EKF and UKF method as recommended distribution, can achieve a more accurate approximate expression to the real distribution and with more forecasting and filtering accuracy. Combined with the real-time data of mixed-measurement, a power system dynamic state estimation simulation is established. Finally, the simulation results verify the validity of this method.

2. EKF Dynamic State Estimation

The general transfer and measurement equations of the power system dynamic state estimation can be written as: 1 ( ) ( ) k k k k k k x f x w z h x v + = + ⎧ ⎨ = + ⎩  (1)

Where xk and zk are state and measurement vectors at moment k ; f and hare non-linear state transfer function and non-linear measurement function; wk andvkare model and measurement noise;

(0, )

k k

w ∝N Q ,vk∝N(0, )Rk , Qkis the model errors variance, Rkis the measurement errors covariance. Currently, the common power system dynamic state estimation method is the EKF method. Specifications of EKF method see reference [7]. While in practical application of power system, EFK method has certain disadvantages: When the load or generator output power mutates, the entire system is strongly nonlinear. That EKF ignores the second-order and higher-order entries will greatly affect the estimate accuracy, or even causes serious distortion. Moreover, the conditional distribution of the power system is strong non-Gaussian, that EKF method uses Gaussian distribution conditions will give rise to considerable error.

3. UKF Dynamic State Estimation

Unscented Kalman filter (UKF) is also a kind of recursive Bayesian estimation method[8]_{, which} applies unscented transform (UT) method to use a group determine sampling points to approximate a posterior probability. But it does not have to linearism the nonlinear state equation or measurement equation. It directly uses the nonlinear state equation to estimate the probability density function of state vectors. Specifications of UKF method see reference [8]. Reference [5] applied UKF method to the power system dynamic state estimation, which had solved the traditional EKF method’s shortcomings such as slow convergence speed and poor robustness, but still not solved the nonlinear problems of power system.

4. Mixed Kalman Particle Filter Dynamic State Estimation

On the basis of EKF and UKF methods, preference [6] introduced a new type of particle filter, called mixed Kalman particle filter (MKPF). It mixed the EKF and UKF methods as the recommended distribution. At moment k, UKF method is used to produce the system state estimation first, and then EKF

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method is used to repeat this process and obtain the final state estimation value of k moment. Specific algorithms are as following:

Assuming that at moment k-1, the state estimation and corresponding covariance estimation are 1

k

x∧

− andPk∧−1. At the next moment k, UKF method is used to update particles first. The selection of the Sigma points in the process according to the equation:

, 1 1 1 ( ) , 1 i k xk xk nx Pi k χ ₋ ∧ ∧ λ ∧ − − − ⎡ ⎤ =_⎣ ± + _⎦ (2) Then forward deliver the Sigma points through the system and measurement model, the mean value of the state and measurement prediction are:

2 2 ( ) ( ) , , 1 0 0 ( ) UKF n n m m k i i k i i k i i x W χ W f χ ₋ = = =

∑

=

∑

∼ _(3) 2 2 ( ) ( ) , , 0 0 ( ) UKF n n m m k i i k i i k i i z W Z W h χ = = =

∑

=

∑

∼ _{ (4)} The measurement estimate variance and the covariance between state and measurement prediction are:

2 ( ) . . 0 ( _UKF)( _UKF) k n c T i i k k i k k k z i P∧ W Z z Z z R = ⎡ ⎤ =

∑

_⎣ − ∼ − ∼ _⎦+ _{ (5)} 2 ( ) . . , 0 ( UKF)( UKF) k k n c T i i k k i k k x z i P∧ ∧ W χ x Z z = ⎡ ⎤ =

∑

_⎣ − ∼ − ∼ _⎦ _{ (6)} From equation (5) and (6), the gain matrix of UKF is:

1 ,

UKF k k k

k x z z

K =P∧ ∧P∧− (7)

Then EKF method is used to update the particles to obtain the state prediction and covariance: 1 ( ) ( ) EKF UKF k k k x∼ = f x∼− = f x∼ (8)

( )

1 EKF T T k k k k K k K P F P∧ F G Q G − = + ∼ ₍₉₎ Next the prediction value is amended to obtain the final estimates:

1

( ) ( ) ( )

EKF EKF EKF

T T T

k k k k k k k k k

K =P∼ H ⎡_⎣U R U +H P∼ H ⎤_⎦− (10)

EKF EKF EKF EKF

k k k k k

P∧ ₌P∼ ₋K H P∼ ₍₁₁₎

1

( ) ( ( ))

EKF EKF EKF EKF

T

k k k k k k k

x∧ ₌x∼ ₊P∧ H R z− ₋h x∼ ₍₁₂₎ Where Qk is the system noise covariance matrix, Rk is the measurement noise covariance matrix, k

F and GKare the Jacobian matrix of system model, HkandUkare the Jacobian matrix of measurement model. Ultimately, xk∧_EKFand PkEKF

∧ _{are the estimates at moment k.} Then samples are taken from the approximate Gaussian distribution:

(

0: 1, 1:

)

( EKF, EKF)

k k k k k k

x q x x z N x∧ P∧

− =

∼ (13) According to the importance density q x x z

(

k 0:k, 1:k

)

,the weights of the particles are updated as following:

(

)

1 1 1 ( ) ( ) , k k k k k k k k k p z x p x x w w q x x z − − − = ∝ (14) Finally is the further sampling procedure. In this paper, residual sampling algorithm is applied. Standard particle filter algorithms see reference [9].

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5. Mixed Measurement Dynamic State Estimation

Wide-Area Measurement System (WAMS) can provide phasor information, which has high precision, strict synchronization throughout the network and other advantages. That WAMS combines with supervisory control and data acquisition system (SCADA), can improve the data redundancy of single SCADA, and provide an effective way to the power system dynamic state estimation[10]_.

For power system dynamic state estimation, corresponding state transfer and measurement equations are needed. State transfer equation can use the Holt double parameter exponential smoothing[11]_:

1 1 1 1 1 (1 ) (1 )(1 ) (1 ) (1 ) ( ) (1 ) k k k k k k k k k k k k k k k k x F x u F u x a b a x x b a a b ϕ γ γ ϕ γ γ ϕ ϕ γ γ + − − ∧ − − = + ⎧ ⎪ ₌ ₊ ⎪⎪ ₌ ₊ ₋ ₋ _{+ −} ⎨ ⎪ ₌ _{+ −} ⎪ = − + − ⎪⎩ ∼ ∼ (15)

Where ϕand γare smooth parameters, and their value interval are [0, 1]. In this paper, ϕ= 0.8 and γ=0.5.

Specifications of the mixed measurement for dynamic state estimation see reference [10, 12], here will not repeat.

6. Simulation Example and Analysis

The simulation example is the four machines two regions system, as shown in Fig.1. The SCADA measurement values include active and reactive power of all lines of the whole network, all bus voltage amplitudes, and active and reactive power of load and generators. The SCADA measurement values obey a normal distribution, which the mean values of deviation and error are 0.02 and 0. PMU (Phasor Measurement Units) are configured on bus 3, bus101, and bus13. PMU measure the bus voltage phases and all the current phases flow out of the bus. The PMU amplitude measurement values obey a normal distribution, which the mean value of standard deviation and error are 0.005 and 0. The PMU phase angle measurement values obey a normal distribution, which the mean value of standard deviation and error are 0.002 and 0.

Figure 1 Simulation system

6.1 Comparison of Three Methods in Steady-state System

In order to verify and compare the calculation performances of the filtering methods in steady state estimation, the estimation results take the following indicators:

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Author name / Procedia Environmental Sciences 00 (2011) 000–000 * , , * 1 ,

1

N i k i k i k i k

x

N

x

α

=

−

=

∑

∼ _{ (16)} * , , * 1 , 1 N i k i k i k i k x x N x β ∧ = − =

∑

(17) Where Nis the total measurement sequence, *

, i k

x ,xi k∼, andxi k∧, are the real value, predicted value and estimate value of the state vector

i

at moment k.

Equation (16) is the mean value of relative error of state vectors at prediction step, the lower the numerical value is, the more accurate the prediction will be. Equation (17) is the mean value of relative error of the state vectors at filtering step, the lower the numerical value is, the more accurate the estimation will be.

The general evaluate functions of the data can be written as: 1 1 n i i n α α = =

∑

(18) 1 1 m i i m β β = =

∑

(19) Where nand mare the dimensions of state and measurement vectors, αand β are the mean value of relative error of general prediction and estimation.

The power system is a steady-state system in most of the time. The changes of state variables are small or slow. Simulation conditions: The power system is not being disturbed; generators and load power do no mutate, dynamic state estimation step is 1 second.

Table 1 lists the simulation results based on the three different filtering methods: EKF, UKF and MKP methods. As can be seen from the table, when the power system is in a steady state, regardless of the prediction or estimation, all the three methods can obtain satisfactory results, and MKPF method works better than the other two methods. The values β of the three methods are far less than 1, which show that the three filtering methods can play a very good role in filtering out the noise in steady-state power system dynamic state estimation.

Table 1 Performance indices of the three methods

Variables EKF UKF MKPF

Voltage Amplitude α_β 0.01718 0.00386 0.00122 0.00671 0.00215 0.00024 Phase Angle α_β 0.02804 0.01892 0.00831 0.01985 0.01297 0.00541

6.2 Comparison of Three Methods in dynamic System

In strongly nonlinear power system, system state variables change rapidly. In this case, the filtering capacities of the three filter methods will be clearly identified, and MKPF method outgoes other two methods. Simulation conditions: At 2 second, three phase fault occurs between bus 3 and bus 101. At 2.1 second, the fault is removed, and dynamic state estimation step is 0.1 second. In order to investigate the impact of the fault data to the estimate results and whether the estimation methods can accurately describe the transition process before and after the fault, the calculating data including the measurement data before and after the fault.

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Take bus 3 as an example, Fig.2 and Fig.3 show its voltage amplitude and phase angle estimation results by the three methods. The performances of the three methods are evidently different at voltage amplitude estimation, as is shown in Fig.2. At 2 second, the fault occurs and the voltage amplitude of bus 3 decline sharply. MKPF method describes this trend very well, while neither UKF nor EKF well follow the mutation of the voltage amplitude, especially the EKF method, almost loses this information. Likewise, in the subsequent voltage amplitude fluctuation, the estimate result of MKPF method is superior to EKF and UKF methods.

In the voltage phase angle estimation, MKPF method works even better than other two methods, as is shown in Fig.3. After the fault, the estimate result of EKF method fluctuates greatly and results in higher estimate error. While UKF and MKPF methods can approximately follow up the real value of the phase angle within one second after the fault, and the estimation result of MKPF method is more accurate.

Figure 2 Comparison of amplitudes

Figure 3 Comparison of angles

7. Conclusion

A new particle filter for nonlinear filtering problems (MKPF) is introduced into power system dynamic state estimation. The method, which utilizes the EKF and UKF methods, can avoid the linearization errors of EKF and the limitation on unsuited to non-gaussian model of UKF, and get a more accurate approximate expression of the true distribution. Finally, the simulation results show that the method can quickly follow up the real value after the power system is disturbed and obtain higher estimate accuracy and robustness than EKF and UKF methods. Therefore, it can reach the online requirements of an accurate

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estimation, and is fit for the actual power system non-linear and non-Gaussian model dynamic state estimation.

Acknowledgment

This project was funded by the National Natural Science Foundation of China (51007030) and National College Students’ Creative Experiment Plan of China (101055937).

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