Power System State Estimation
A Comparative Study of Kalman Filters and A New Approach
Abstract— The Synchronous machine angle and speed variables availability give us an exact picture of the overall condition of power system networks leading therefore to an improved situational cognizance by system operators. As the enlargement of power systems continue, the challenge of real time monitoring and its control gets bigger. All significant network parameters are measured at various points on the system and that collected data transferred to the control center, where the data is used for Estimation Process through various EMS functions. This paper compares the performance of four filtering approaches in estimating dynamic states of a power system network using PMU data. The four methods are extended Kalman filter, unscented Kalman filter, Particle Filter, and Enhanced unscented Kalman filter. The statistical performance of each algorithm is compared using IEEE bus test system. According to the comparison, this paper makes some recommendations for the proper use of the methods.
Keywords— Extended Kalman filter (EKF), Particle filter (PF), Phasor measurement unit (PMU), power system dynamics, state estimation, unscented Kalman filter (UKF) and The Enhanced UKF.
I.
Introduction
The rapid growth and increasing complexity in recent years makes the monitoring and control of power systems a very significant issue. The Energy Management Systems (EMS) at the control centers are responsible for this task of monitoring and control of the system. The state estimator, which is the backbone of the energy management systems, provides an optimum real time data of the system state based on the available measurements on the assumed system model. The efficiency and accuracy of the state estimator output is very crucial as it forms the basis for the EMS functions such as security analysis, automatic generation control, optimal power flow and load forecasting. Thus, the concept of state estimation plays a major role in ensuring the secure and economic operation of the power systems in large-scale interconnected power grids. Depending on the desired states (static or dynamic), power system state estimation can be formulated as a static or dynamic estimation problem. The power systems are defined as quasi-static systems. This means that they change with time very slowly but steadily. The change of the power system is driven by the continuous variation of the loads. As the loads change, the generators feeding the network need to be adjusted in accordance with the load variations. This results in the change of power
injections and power flows which makes the system dynamic. The resulting dynamic changes need to be monitored continuously and therefore the power system state estimation needs to be performed in short interval of time.
The performance and the accuracy of the state estimators in power systems heavily depends on the measurements
gathered from the network. Traditionally, the measurements
including injections, flows and voltage magnitudes, are collected by the SCADA systems via Remote Terminal Units (RTU) and processed in the state estimation algorithms at the control center. In mid 1980s, a new device has developed which is called as Phasor Measurement Unit (PMU). The main importance of this device lies in the fact that it can measure both the voltage phasor and current phasor at the system buses where the device is present. By the emergency of this synchronized measurement tool, for the worst time, both the bus voltage magnitudes and bus voltage angles can be directly measured which are the state vector elements.
Moreover, PMU measurements are highly accurate compared to the SCADA measurements. In this regard, the PMUs which work in synchronization with Global Positioning
System (GPS) satellites, are superior to the traditional SCADA
systems. The introduction of highly accurate angular measurement data by means of these high updating rate synchronized measurement devices play an important role in the modern day energy management systems. As a result, the PMUs can also be incorporated into the dynamic state estimation studies of power systems such that the dynamic view can be captured in a more accurate and efficiently way. The PMU records fifteen different channels of the real and imaginary parts of the secondary voltages and currents [1]. The phasor reporting rate is 60 phasor/sec for a voltage, 5 currents, 5 watt measurements, 5 VAR measurements, frequency and the rate of change of frequency. The state estimators are used both measurements together or separately for estimating system states.
The progress and improvement in the way of monitoring the power networks by the emergence of these new technologies creates a deep motivation for developing new methodologies in the field of power system dynamic state estimation. In the studies of power system operation and analysis, historically, generator and transmission modeling have received the most attention.
The continually increasing complexity of the power networks and the incorporation of various load components from many sources motivates the researchers to concentrate on system accuracy and load modeling. The accurate and verified load model plays a major role in the power system stability analysis. It is clear that more accurate load models need also be used during the state estimation processes rather than making traditional assumptions.
II. STATE ESTIMATION
The state estimation process for a power network is determining the best estimate of the present state of the system by collecting the real time measurements from the sensors monitoring the grid. The vector consisting of bus voltage magnitudes and bus voltage phase angles is called the static vector of an electric power system. The real time measurement data gathered from the network and used in the estimation process includes power injections, power flows on the transmission lines and voltage magnitudes at each bus of the system. The telemetered measurement data is received through the Supervisory Control and Data Acquisition Systems (SCADA) and the state vector is estimated by using the predetermined state estimation algorithm and power system model.
A. Static State Estimation
SSE is a very useful tool for the economic and secure operation of transmission networks. From early days of Scheweppe [2]–[4], developments of SSE are done as a notion of robust estimation, hierarchical estimation, with and without inclusion of current measurements, etc. The SSE uses only voltage magnitude real and reactive power flow injections and SCADA measurements. If the state vector is obtained for the current instant of time k from the set of measurement data received at the same instant of time k, then such an estimation method is called as Static State Estimation (SSE).
In static state estimation, the snap-shot of the measurements are taken, processed and the estimate of the state vector variables is obtained at the same point of time.
B. Dynamic State Estimation
The static state estimators cannot efficiently and accurately capture dynamic behavior of the expanded large power networks. Consequently, another method is developed in order to monitor the continuous dynamic changes in power systems which is called as Dynamic State Estimation (DSE). By using the actual physical modeling of the time varying nature of the power system, DSE algorithm predicts the system state at the next instant of time k + 1 along with the state estimates obtained at the previous instant of time k. The DSE method has a vital advantage such that it allows the prediction of the system state at one time step ahead. Hence, the forecasting ability of the DSE algorithm plays a major role in the improvement of the overall energy management system operation and control.
DSE has two objectives: 1) prediction of power system state at the next time period and 2) SE based on both sets of predicted
and measured data. The first feature provides the power
system operator an additional time for making control decisions and analyzing the security of operating system. Talking specifically about WAMS, this feature is less important since the measurement time instants are very close. The second attribute would significantly improve the performance of DSE. The consideration of predicted values in the estimation process would enhance the data redundancy and
make the DSE more robust as compared with SSE and TSE, which use real measurements alone.
Two consecutive stages are recognized in DSE which are state prediction and SE (it is referred to as state filtering in some documents). The power system state at the next time
period is predicted at the first stage, and upon receiving the measurement set, the system state is determined by the estimation process.
III. PROBLEM FORMULATION AND SOLUTIONS
The nonlinear state estimation problem
To define the state estimation problem first we consider the general form of the discrete nonlinear process model, with m input signals, p output signals and the order of system is n. The discretization was made with sampling time Ts and we use the following notation for signal sequence
. In this reason the nonlinear discrete model is
where is the state vector, is the system input vector and is the noisy output vector of the system. The functions F and G are nonlinear and they need to be continuous. The is one n dimensional process noise sequence and is p dimensional observation (measurement) noise sequence. Both noises are Gaussian (normal distribution), independent random processes with zero means and known time invariant covariance matrices. If the E{} is the expected value operator we can write:
The objective of the estimation problem is to recursively estimate xk from the output measurements yk. In accordance with Bayes theory this mean, that recursively calculate the estimation of xk at time k given the dates
y1,…,yk up to time k. This is required calculation of the
probability distribution function pdf (xk | y1:k ) . We suppose that the initial pdf function of the state vector (pdf (x0 | y0) ) is known and the pdf (xk | y1:k ) is obtained recursively in two sections: prediction step and update (correction) step.
IV. KALMAN FILTER AND ITS BASED TECHNIQUES
The Kalman filter is the most widely used Bayesian- based method. It was named after Rudolf Kalman, who published his famous recursive method to estimate dynamic states [5]. Assuming Gaussian noise and a linear system, the Kalman filter provides minimum variance estimates of states through a recursive approach.
In addition to its original successful applications in linear systems, there are many publications adapting the Kalman filter to nonlinear systems. These nonlinear methods include but are not limited to EK, UKF, EnKF, PF and EUKF. One major difference among these nonlinear Kalman-filter methods is their approaches to propagating the mean and
covariance of the dynamic states. The EKF [6], [7] linearizes the state space model using a first-order approximation. The mean and covariance of states are propagated using Jacobian matrices.
The UKF [8] propagates the mean and covariance of states using a deterministic-sampling approach to pass the sigma points through the nonlinear system. The EnKF propagates the mean and covariance of states using a Monte Carlo sampling approach [9]. In the EnKF, the distribution of the states is represented by a collection of samples, referred to as ensembles. All the above Kalman filters assume the joint Gaussian distribution of both measurements and states, and use the Bayesian approach to derive the Kalman gain. In contrast, the PF [10] is a more general Bayesian approach, which does not rely on Gaussian noise assumption. Similar to the EnKF, the PF also uses the samples (also known as particles) to represent the probability distribution of random variables. Different from the EnKF, the PF directly corrects the states without assuming Gaussian distribution.
For a linear system with additive Gaussian noise, it is well known that the KF is an optimal estimator in the sense of obtaining minimum mean square error (MSE) estimate. In addition, the KF is a recursive estimator and can be implemented efficiently for many real time applications. Yet, for a nonlinear system, all available algorithms (such as EKF, UKF, EnKF, PF, and EUKF) are only suboptimal estimators. Each algorithm has its advantages and disadvantages. The EKF is probably the most widely used estimation algorithm for nonlinear systems and is often considered a standard algorithm because of its high computational efficiency and high accuracy for quasi-linear systems [11]. Yet, when a system is highly nonlinear, the EKF tends to have poor estimation accuracy and even diverge. This is because first order Taylor approximation used in the EKF introduces too much error. In contrast, the UKF can achieve second or third order Taylor approximation for a nonlinear system. Therefore, the UKF tends to produce more accurate estimates than the EKF when system nonlinearity is severe and noise is additive. In addition, the UKF has the same order of the computational complexity as the EKF [8]. The EnKF was mainly motivated by the needs of solving a system with a large number of states. For a large system, the EnKF was shown to have overcome the unbounded error growth problem of the EKF and to require less computation time [9]. The applications of EKF, UKF, and EnKF are restricted by their additive Gaussian noise assumption.
As a result, they are not suitable for analyzing probabilistic distributions with multiple modes. In contrast, the PF is more general by not making the restrictive assumption and, therefore, is more applicable to highly nonlinear systems. It was shown that the PF can give more accurate estimates than the EKF for the two particular nonlinear models in [12]. Yet, the PF usually requires a very large number of samples to represent a probabilistic distribution and, therefore, is not suitable for a large system. Due to this feature here we are representing a new approach that is Enhanced UKF which includes the properties of nonlinear equations of taylor series with noise and with particle filter property to make more
accurate value of estimation. Considering features of the EKF, UKF, EnKF, PF, and EUKF users often need to examine and test filtering algorithms to select appropriate algorithms according to the requirement of their specific applications. Under a Bayesian framework, the implementations of theses algorithms have a similar structure. After initialization, all the filtering algorithms assimilate one snap shot of data at every time step. For one snap shot of data, there are two steps: a prediction step and a correction step. In the prediction step, the mean and covariance of states at time step k are predicted based on the states at step k-1. In the correction step, the predicted mean and covariance are corrected based on new measurements obtained at time step k. The algorithms for implementing these filtering methods are detailed as follows.
A. EKF
The EKF linearizes the system at the current operating point using the jacobian matrices as in (13-15).
EKF Prediction
EKF Correction
Where and are known as the a priori mean and covariance, respectively. They are estimated from the data up to time step k-1. The symbols and are known as a posteriori mean and covariance of the states respectively, which are derived by adding the information from to and . The symbol is the Kalman Gain. The symbol is the residual between estimates and measurement . and are Jacobian matrices defined by given equations. A perturbation approach is used to numerally derive the Jacobian matrices in this paper
B. UKF
The UKF uses an unscented transform to pick a set of samples to represent the probability distribution of states and propagates these samples through the nonlinear functions f and h to reconstruct the mean and covariance. The UKF estimation method is summarized as (16) and (17) [18].
UKF Prediction
UKF Correction
Where enKF is the total number of samples, which are used to
+ represent the distribution. The variable is a sample generated according to the Qd to simulate process noise. The symbol stands for the samples of a posteriori states.
D. PF Approach
The PF can be applied to systems with Gaussian and other distributions. A basic approximates a probability distribution function by a set of weighted discrete samples, as shown in
= )
Where and Wi are 2n + 1 sigma points and their corresponding weights. K is a scaling parameter that controls the positions of the sigma points.
C. EnKF
The Enkf uses samples to represent and propagate the probability distributions of the states. By using a large number of samples, the probability density can be approximated with high accuracy. The EnKF can be summarized by given Equations.
EnKF Prediction
After processing several data snapshots, a PF often suffers from a degeneracy problem. To reduce the degeneracy problem, a resampling step is often added to redispose the discrete samples by generating a new set of particles according to the discrete distribution of above equation. To detect degeneracy, the effective sampling size Neff is defines by given equation.
PF Prediction
PF Correction
PF resampling if degeneracy is detected using
EnKF Correction
Here is prior weights of the ith state sample.
is the likelihood of zk given the prior states and . The likelihood function is
determined by the measurement noise model. is the total
number of samples that are used to represent a state which is in probability distribution.
E. Enhanced UKF
This is new technique that includes the properties of UKF and PF it means this system deals with the system with Gaussian and other distribution and after that this will pick up a sample to present the probability distributions of states and propagates these samples through the nonlinear functions to reconstruct the mean and covariance. This technique is also called UKF+ P filter or Enhanced unscented Kalman filter (EUKF).
V. STUDY APPROACHES
Because each nonlinear filtering algorithm often has its advantages and drawbacks, it is important to set up a specific case in a power system for comparison. In the addition of this
-12.7
-12.705
-12.71
-12.715
-12.72
-12.725
-12.73
-12.735
Actual Reading
EKF UKF
P- Filter
The Enhanced UKF/UKF+PF
we are considering standard IEEE 14 and IEEE 30 bus system bus data and apply these models and take the state estimation output.
A. IEEE 14 Bus System
By using the IEEE 14 bus system test data of bus and line data with shunt data we get some results that shows the output of all Kalman filter output. Here 2 figures shown with the name of figure 1 and figure 2. Here we are taking 20 line data values and 14 bus data values from previous available data.
-12.74
0 10 20 30 40 50 60
time,k
Fig. 2. Error in different state variable estimation from actual value
B. IEEE 30 Bus System
Figure 3 shows the all four type of Kalman filter output and figure 4 shoes the error in different state variable estimation from actual values calculated.
0.5
0.5
0.45
0.4
0.45
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.15
0.2
0.15
0.1
0.1
0.05
0
5 10 15 20 25 30 35 40 45 50 time,k
0.05
Fig. 3. Different state variable estimation
0
5 10 15 20 25 30 35 40 45 50
time,k
-4.655
-4.66
Actual Reading
EKF
Fig. 1. Different state variable estimation
In figure 1 we are considering all four methods for comparison but when we consider the comparison with actual value then figure 2 gives the result.
-4.665
-4.67
-4.675
UKF
P- Filter
The Enhanced UKF
-4.68
-4.685
-4.69
-4.695
0 10 20 30 40 50 60 time,k
Fig. 4. Error in different state variable estimation from actual value
TABLE 1
PERFORMANCE COMPARISON AMONG EKF, ENKF, UKF, PF
AND UKF+PF
EKF
UKF
EnKF
PF
UKF+PF
Efficiency of
Interpolation
High
High
Low
High
High
Number of
Sample
Needed
Not
Applicable
Small
Medium
Large
Large
Sensitivity of the missing Data
Low
Low
Low
Low
Low
Computation
Time
Shortest Same
order
as EKF
Longer than EKF Same
order
as EnKF Longer
than
EnKF
Conclusion
Accurate information about dynamic states is critical to efficient control of a power system, especially with the increasing complexity resulting from uncertainties and stochastic variations introduced by intermittent renewable energy sources, responsive loads, mobile consumption of plugin vehicles, and new market designs. Using a statistical framework, this paper compares the performance of an EKF, UKF, EnKF, PF and UKF+PF for the purpose of estimating dynamic states from real-time phasor measurements. To summarize the observations from the simulation using a two- area-four-machine test system, Table I is constructed for quick comparison.
Ongoing and future work includes dynamic state estimation methods at system levels and sensitivity studies to determine how parameter errors may influence the state estimation, as
well as efficient, accurate, and flexible methods for estimating
the states in both real time and offline environments.
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