Efficient Power System

State Estimation

Zafirah Baksh

Expected BS, Department of Electrical Engineering

May 2013

ELEN E4511 Power Systems Analysis

Professor Javad Lavaeiyanesi

1. Introduction Power systems are supervised and maintained using SCADA. Measurements are taken at

various points in the system and the state of the system is estimated. There are varying errors in

the measurements so an appropriate estimation model is required to account for this. Erroneous

measurements can be detected using a Chi-Squared Distribution test and can then be discarded

from the state estimation. Accurate state estimation enables active monitoring of power systems.

2. Background Power systems have four main components: transmission, sub-transmission, distribution

and generation systems. This entire system can operate in three states: normal, emergency or

restorative. There are two categorizations to the normal state, the normal secure state and the

normal insecure state. System operators monitor daily to ensure the system stays in the normal

secure state. In the normal secure state, a system can handle the impact of certain occurrences

such as partial equipment failure or generator outages. In the normal insecure state, a system may

not be able to handle these occurrences and can thus move into the emergency state which may

result in the system collapsing. Thus it is critical that the system is always monitored and actions

are taken in a timely manner to ensure that it stays in the normal secure state. Measurements

from all parts of the system are collected and then an estimate of the system state is created. This

state is then assessed to see if actions are required. Measurements include values of various

components of the system such as the bus voltage and line current magnitudes, and switchable

capacitor bank values. The accuracy of the estimate of the system state depends on the accuracy

of these measurements. Furthermore it is not efficient for every single component to be measured

and reported. The state estimator determines the optimal estimate for the system state and also

provides estimates for unmeasured components of the system. It also determines if there are

erroneous measurements as well. [1]

3. Measurement of System Parameters There are typical error measurements such as erroneous meter readings, and then there

are errors that can be caused by a malicious attack. The state estimator has filters to catch the

typical errors and then there is the Bad Data Detection test to find errors. However a malicious

attack may design the errors they cause to be passed through undetected. If enough errors are

passed through to the state estimator, the state estimation would then be inaccurate. In the worst

case scenario, data could be fabricated and passed through so that the system state always

appears to be normal, when in reality it is not. Then the employees at the control center would

not be able to act preemptively to alleviate any situations on the ground. They might find out too

late, when there are already several points of failure on the grid. [2]

There are several vulnerabilities in the overall monitoring system, SCADA, Supervisory

Control and Data Acquisition. The remote terminal units (RTUs) actually measure the data from

the system components such as transmission line power flows, bus power injections and bus

voltages [2]. Basically measurements would be taken from the two endpoints of the branch, as

follows:

Figure 1: Diagram of branch connecting two buses [1].

This branch of the system is Branch j. The voltage and phase angle measurements are obtained

for Bus k and Bus m, and then stored as j and j, respectively, for various branches in the system. The equation j is defined as ln(Vk) ln(Vm) and j is defined as k - m. These become part of branch variable vectors defined as X and X that can be combined to form matrix X.

Now a scaled measurement vector can be obtained, z, which is defined as a nonlinear vector

equation, . From this we can obtain a linear measurement equation, as follows:

The scaled measurement vector, z, was approximated at some arbitrary value , so we did the first order Taylor series approximation to obtain an equation for . This can further be developed into a model including measurement errors, e, [1]. This can be further implemented as a linear regression model as follows, [3].

4. State Estimation There are many different models to choose from to perform the regression. The model chosen for

this regression is the Weighted Least Squares (WLS) model. It is preferable to use this model for

this situation instead of other more common models such as the Ordinary Least Squares (OLS)

model. The WLS model is preferable over the basic OLS model because the WLS model

accounts for the case where the error variances of the observations are unequal, so it is not

constant. This condition is called heteroskedasticity. This is different from homoskedasticity,

where the error has the same variance given any value of the regressor [4]. Homoskedasticity is a

necessary condition to perform the OLS regression, therefore because this scenario is

heteroskedastic by nature, the OLS regression would not provide optimal results. The weighted

least squares model is more appropriate because it is a modified version of the OLS model that

accounts for the non-constant error variances [5]. It applies weights to the model depending on

the variances, which can be defined as

[6]. SSR is the sum of squared residuals, which

accounts for positive and negative differences.

Applied to the case at hand, the WLS model can be utilized as follows:

However there is also a disadvantage to this method as well. This method is based on the

assumption that all the weights are known, however they are not known usually so they need to

be estimated, and then these weights are used in the state estimation. However, for the results to

still be accurate, it is assumed that the weights can be estimated precisely relative to each other

[7].

5. Error Detection The Bad Data Detection (BDD) test can be performed on this to detect erroneous

measurements. It would be performed on the residual, which is [2]:

Once this residual is too large, the BDD test will result in an error found. However a great

concern is that attackers can create errors that bypass such tests, therefore they can introduce a

flood of arbitrary errors to the state estimator [3].

A popular BDD method is to perform a Chi-Squares Test which is based on the Chi-

Squared Distribution which is the distribution of the sum of m squared independent standard

normal random variables, with N being the degree of freedom [8]. The largest degree of freedom

possible is the difference between the total number of measurements and the system states [1].

The following function, , is computed and compared to a error detection level of high probability from the Chi-Squared distribution table.

The estimated measurement is defined by , the actual measured value is , m is the number of measurements and

is the variance of error in measurement i as seen previously. If the value

of this function is greater than or equal to the Chi-Squared table value, then it is likely that an

error exists. The percent likeliness of an error in the set would be selected by the value of the

error detection level chosen before. Different percentages can be tested.

6. Further Development Effective error recognition will enable better monitoring of power systems due to better state

estimation, however time-lags within the SCADA network should be considered in the

estimation models as well. Other model applications can be considered for better state

estimation, such as the time series regression which would provide information needed to create

forecasting models. Conditions in power systems change spontaneously, for example due to a

sudden fallen tree, however sometimes forecasting models used to forecast the state could be

useful. For example, on a hot summer day when power usage is high, previous records exist with

the measurements, so these could be used to forecast the state of the system for the next heat

wave.

References:

[1] A. Abur and A. Expsito, Power System State Estimation. New York: Marcel Dekker, Inc.,

2004.

[2] K. C. Sou, H. Sandberg, and K. H. Johansson, On the Exact Solution to a Smart Grid Cyber- Security Analysis Problem, IEEE Trans. Smart Grid, vol. PP, Issue: 99, 2013. [3] Y. Liu, M. Reiter, and P. Ning, False Data Injection Attacks against State Estimation in Electric Power Grids, in the 16th ACM Conference on Computer and Communication Security, New York, 2009, pp. 21-32.

[4] J. Wooldridge, Introductory Econometrics: A Modern Approach. Mason, OH: South-Western

Cengage Learning, 2009.

[5] C. Tofallis, Least Squares Percentage Regression, Journal of Modern Applied Statistical Methods, 2009.

[6] T. Tong, Weighted Least Squares, Department of Applied Mathematics, University of Colorado, 2010. < http://amath.colorado.edu/courses/7400/2010Spr/lecture23.pdf>.

[7] Weighted Least Squares Regressions, NIST/SEMATECH e-Handbook of Statistical

Methods, 2012. < http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd143.htm>.

[8] J. Stock and M. Watson, Introduction to Econometrics. Boston, MA: Addison-Wesley, 2010.