robust dynamic state estimator accounting for load changes... · 2019-02-13 · robust dynamic...

Robust Dynamic State Estimator Accounting

for Load Changes

A Thesis Presented

by

Alireza Rouhani

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in

Electrical Engineering

in the field of

Power Systems

Northeastern University

Boston, Massachusetts

August 2013

i

Abstract

Load models are commonly approximated by fixed impedance type loads when

formulating dynamic simulations. This is an acceptable approximation when using

dynamic simulation for a specific initial operating condition and the study runs for

a brief period typically to check system stability. When implementing an on-line

dynamic estimator such assumptions may not be viable due to the load dynamics

which are not typically known.

Unlike previous studies reported in literature, this work takes into account the

effect of changes in bus loads on dynamic state estimation. Two methods are

proposed to address this problem, in the first method the performance of dynamic

state estimator (DSE) is improved by monitoring the load changes on longer

intervals and strategically varying the variance of the system state error covariance

matrix to compensate for the errors introduced by changing loads[7]. Based on the

second method, by extending the state vector to include network variables, we can

eliminate the need to model loads and also it is possible to implement a distributed

yet synchronized solution algorithm that facilitates simultaneous solutions by

multiple areas which is highly helpful, especially when a large scale system is

considered with several generators[8].

The proposed methods are tested on the New-England 37-bus system where the

Unscented Kalman Filter (UKF) is chosen in formulating the DSE, due to its

advantages over the commonly used Extended Kalman Filter (EKF).

ii

Acknowledgments

I wish to express my gratitude to my research advisor and mentor, Professor Ali

Abur, for the tremendously rewarding educational and personal experience he has

afforded me. I have learned so much from his patient teaching and his tremendous

technical expertise. His constant support, guidance, and encouragement helped me

greatly in successfully completing my thesis. My thesis progressively improved as

a result of Professor Abur’s ability to look at problems from different perspectives,

together with many hours of discussions with me. Being a research assistant to

Professor Abur will greatly enhance my ability to stay at the forefront of my

current research area.

I would also like to express my heartfelt thanks to Professor Bahram Shafai and

Professor Hanoch Lev-Ari for serving as my thesis committee members. During

my graduate studies, they gave me inspiration and immense knowledge of their

areas of expertise.

Finally I would like to express my love and gratitude to my parents. Their

encouragement and support helped me to attend Northeastern University and for

that I will always be grateful.

iii

Contents

1 Introduction 1

1.1 Motivations for the Study ................................................................................................................ 2

1.2 Contribution of the Thesis ............................................................................................................... 3

1.3 Thesis Outline .................................................................................................................................. 3

2 Technical Background 4

2.1 Historical Overview of Power System State Estimation ................................................................ 4

2.2 Review of the Related Works in Dynamic State Estimation .......................................................... 6

3 Improving Performance of Dynamic State Estimators under Unknown Load Changes 9

3.1 Extended Kalman Filter ................................................................................................................ 10

3.2 Unscented Transformation ............................................................................................................ 12

3.3 Unscented Kalamn Filter ............................................................................................................... 14

3.4 Case Studies ................................................................................................................................... 16

3.4.1 Case 1: Comparing the performance of UKF and EKF ........................................................... 18

3.4.2 Case 2: Line switching transients .......................................................................................... 21

3.5 Conclusions .................................................................................................................................... 28

4 Distributed Implementation of an Augmented State Dynamic Estimator 29

4.1 Problem Formulation ..................................................................................................................... 30

4.2 Implementation of Dynamic Estimator .......................................................................................... 31

iv

4.2.1 Conventional Approach ........................................................................................................... 32

4.2.2 Dynamic State Estimation Using Augmented State Vector ..................................................... 35

4.2.3 Distributed Multi-Area Dynamic State Estimation ................................................................. 38

4.3 Conclusions .................................................................................................................................... 59

5 Concluding Remarks and Further Study 60

5.1 Concluding Remarks ...................................................................................................................... 60

5.2 Further Study ................................................................................................................................. 62

A Fundamental Scripts Used for Dynamic State Estimaton 64

A.1 EKF Using Conventional Approach-Assuming Classical Model for Generators ......................... 64

A.2 UKF Using Conventional Approach-Assuming Classical Model for Generators ........................ 67

A.3 UKF Using Augmented State Vector-Assuming Classical Model for Generators ....................... 72

A.4 UKF Using Augmented State Vector-Assuming Two-Axis Model for Generators ..................... 75

References 79

v

List of Figures

3.1 New-England, 37-bus system ....................................................................................................... 18

3.2 Per-unit rotor speed of generator#1 obtained from EKF ............................................................... 20

3.3 Per-unit rotor speed of generator#1obtained from UKF ................................................................ 21

3.4 Rotor angle of generator#1 when σ = 0.0001 ................................................................................ 22

3.5 Per-unit rotor speed of generators #1, 2 and 3 when σ =0.0001 .................................................... 23

3.6 Rotor angle of generator #1 when σ = 0.1 ..................................................................................... 23

3.7 Per-unit rotor speed of generators #1, 2 and 3 when σ = 0.1 ......................................................... 24

3.8 Magnified view of the per-unit rotor speed of generator#1, 2 and 3, for σ = 0.1 .......................... 25

3.9 Rotor angle of generator 1 when σ is gradually increased ............................................................. 26

3.10 Per-unit rotor speed of generators 1, 2 and 3 when σ is gradually increased ................................. 27

3.11 Magnified view of the per-unit rotor speed of generators 1, 2 and 3 when σ is gradually increased

................................................................................................................................................................. 27

4.1 New-England, 37-bus system with sudden load change in HALE69 ............................................ 34

4.2 Rotor angle of generator #1obtained based on conventional approach ......................................... 34

4.3 Per-unit rotor speed of generators 1, 2 and 31obtained based on conventional approach ............. 35

4.4 Rotor angle of generator #1obtained using augmented state vector approach .............................. 37

4.5 Per-unit rotor speed of generators# 1, 2 & 31obtained using augmented state vector approach ... 37

4.6 New-England, 37-bus system divided into four zones .................................................................. 40

4.7 Rotor angles of generators#1&8 inside zone 1 .............................................................................. 41

4.8 Per-unit rotor speeds of generators# 1&8 inside zone 1 ................................................................ 42

vi

4.9 Bus voltage’s magnitudes of generators# 1&8 inside zone 1 ........................................................ 42

4.10 Bus voltage’s angles of generators# 1&8 inside zone 1 ................................................................ 43

4.11 Rotor angles of generators#2-4 inside zone 2 ................................................................................ 43

4.12 Per-unit rotor speeds of generators# 2-4 inside zone 2 .................................................................. 44

4.13 Bus voltage’s magnitudes of generators# 2-4 inside zone 2 .......................................................... 44

4.14 Bus voltage’s angles of generators# 2-4 inside zone 2 .................................................................. 45

4.15 Rotor angles of generators#6-7 inside zone 3 ................................................................................ 45

4.16 Per-unit rotor speed of generators# 6-7 inside zone 3.................................................................... 46

4.17 Bus voltage’s magnitudes of generators# 6-7 inside zone 3 .......................................................... 46

4.18 Bus voltage’s angles of generators# 6-7 inside zone 3 .................................................................. 47

4.19 Rotor angle of generator#5 inside zone 4 ...................................................................................... 47

4.20 Per-unit rotor speed of generator# 5 inside zone 4 ........................................................................ 48

4.21 Bus voltage’s magnitude of generator# 5 inside zone 4 ................................................................ 48

4.22 Bus voltage’s angle of generator# 5 inside zone 4 ......................................................................... 49

4.23 Rotor angle of the generator in example 4.1 .................................................................................. 54

4.24 Per-unit rotor speed of the generator in example 4.1 ..................................................................... 54

4.25 Internal voltage-q axis of the generator in example 4.1 ................................................................. 55

4.26 Internal voltage-d axis of the generator in example 4.1 ................................................................. 55

4.27 Excitation voltage of the generator in example 4.1 ....................................................................... 56

4.28 Rate feedback of the generator in example 4.1 .............................................................................. 56

4.29 Exciter input of the generator in example 4.1 ................................................................................ 57

4.30 Mechanical input power(pu) of the generator in example 4.1 ....................................................... 57

4.31 Terminal bus voltage’s magnitude of the generator in example 4.1 .............................................. 58

4.32 Terminal bus voltage’s angle of the generator in example 4.1 ...................................................... 58

vii

List of Tables

3.1 Error metrics for case 1 .................................................................................................................. 19

3.2 Error metrics for case 2 .................................................................................................................. 24

4.1 CPU-Time associated with dynamic state estimation of the zones ............................................... 50

4.2 Data for example 4.1 ...................................................................................................................... 53

.

viii

Dedicated to my parents

1

Chapter 1

Introduction

Nowadays power system dynamic analysis is greatly important for the power

engineers. This is due largely to the desire to utilize transmission networks for

more flexible interchange transactions. In order to have a more realistic dynamic

analysis of the system it is significantly beneficial to know about the most updated

dynamic state variables in the system. Fortunately by taking advantage of Phasor

Measurement Units (PMU’s) we can make a dynamic state estimator which allows

us to estimate the real-time dynamic states of the system.

Researchers have considered this idea in the past decades and there are various

proposed algorithms regarding power system dynamic state estimation in the

literature. One of the fundamental characteristics of DSE’s algorithms is the usage

of physical time-varying model of the system. In a power system the behavior of

many of bus loads are unpredictable therefore assuming a constant load model for

these bus loads may not reflect their true behavior and as a result the assumed

time-varying model of the system will not reflect a true model of system.

Consequently, the usage of this wrong model will lower the performance of DSE.

Thus in order to obtain a more reliable and efficient state estimation result from a

DSE, it will be extremely beneficial to reduce or even eliminate the effect of the

usage of wrong load models within the DSE’s algorithm.

2

1.1 Motivations for the Study

One of the main goals of the project was to perform a robust dynamic state

estimation on a very large-scale power system in order to facilitate on-line

dynamic security analysis which will require the current dynamic state of the

system. However after evaluating the proposed approaches in the area of power

system dynamic state estimation some obstacles were found with these methods

when they were attempted to be implemented on a real large-scale system.

As mentioned earlier one of the main difficulties was to obtain a solution which

will take into account the effect of load changes on DSE and the other main issue

was related to the computational complexity which grows significantly with

system size.

Thus the main motivation of this work was to address these important issues in

order to make the implementation of DSE on the real systems more robust and

practical.

3

1.2 Contribution of the Thesis

This thesis describes two improvements that are related to the implementation of

dynamic state estimation in power systems. One of them is the way bus loads are

represented so that changes in bus loads can be taken into account during dynamic

state estimation. The other is related to the computational performance where the

computational burden per time step is reduced and made independent of system

size via strategically distributing the computational load. This is accomplished by

formulating the dynamic state estimation problem as a multi-area distributed

estimation problem and solving it in parallel by each area. Individual area

solutions are kept synchronized via the sue of phasor measurements that are

assumed to be available in each designated area.

1.3 Thesis Outline

The thesis is composed of five chapters. The first chapter states the motivation for

the presented work and outlines the contributions of the thesis. The next chapter

reviews the state of the art and provides historical perspective by considering

relevant work in the area.

Chapter 3 is about the proposed method of improving the dynamic state estimation

performance under unknown load changes.

The next chapter describes the multi-area distributed implementation of the

dynamic state estimator in order to address the problem of computational

complexity. The final chapter will conclude the thesis and also mention some of

the future work.

4

Chapter 2

Technical Background

This chapter provides a brief review of the static and dynamic state estimation

problems, their formulation and solution. Given the large volume of available

literature on these topics, this chapter refers to a subset of the most relevant work

done in the recent years.

2.1 Historical Overview of Power System State Estimation

As power systems grow larger and more intricate, real time monitoring becomes

significantly important in order to achieve reliable operation of the power systems.

The lack of appropriate real time monitoring of the system may lead to the

occurrence of drastic problems such as the Northeast Blackout of 1965.

The Energy Management System (EMS) functions are accountable for the

monitoring and control of power systems. On the other hand the state estimation

constitutes the foundation of EMS by providing real time state of the system to be

used by all other EMS functions [9]. Thus, an accurate and efficient state

estimation is necessary for a reliable and efficient operation of the power system.

5

In the early 1970s, Fred Schweppe [10,11,12] introduced the concept of static state

estimation to the area of power systems where static state of an electric power

system was defined as the voltage magnitudes and angles at all network buses.

Then assuming high number of measurements (more than state variables) these

state variables were estimated by obtaining the optimized solution for specific

equations which were related to load-flow equations. Since then, numerous

approaches have been proposed to estimate the state vector of the power systems.

Most of these proposed methods were mainly about the static state estimation

where the state vector is estimated based on the available measurements taken at a

point in time.

Since the loads in the power systems are continuously varying in time the power

delivered by the synchronous generators also have to be adjusted accordingly.

Thus, the dynamic state variables associated with system generators will have to be

tracked as the system load and/or topology change. Thus, in order to have a more

accurate continuous monitoring of the power system, state estimation should be

able to estimate the dynamic state variables of the system. This lead to the

development of the so called “Dynamic State Estimaton” solution, where in

addition to the static network model, generator dynamics need to be modeled as

well.

Dynamic state estimator tracks the dynamic state of the system and allows

prediction of the one step ahead dynamic states using the dynamic models. This

forecasting ability has tremendous advantages. As security analysis can now be

performed one time step ahead and hence may facilitate certain control actions,

particularly in cases of any emergency. Thus, dynamic state estimation algorithms

for power systems form an important branch of power system state estimation

6

techniques, with a potential to impact the very nature of operation of the real time

monitoring and control of power systems[9].

2.2 Review of the Related Works in Dynamic State Estimation

One of the first papers published on this topic is by A. S. Debs and R. E.

Larson[13], where kalman filtering techniques were first applied to the traditional

steady-state estimation process in power system applications. Since then, there

have been various works in this area. For instance, in the paper written by J.

Chang, G. N. Taranto and J. Chow[14], a gain-scheduling scheme was used for

state observer design in a single-machine infinite-bus (SMIB) system while

constant voltages were assumed in the dynamic modeling, which significantly

reduces the ability to represent the full dynamics of a power system. L. Lin,

Linawati, L.Jasa and E. Ambikairajah[15] propose a dynamic state estimation

scheme for a sixth-order synchronous machine where they assume some of the

state variables such as excitation voltage fdE and rotor angle are measurable.

Such measurements are unfortunately not commonly available and may be

obtained at additional cost. In the paper published by M. Huanga W. Li and W.

Yana[16], authors proposed a parameter estimation procedure based on square-root

UKF to estimate the parameters of the third order model of a synchronous

generator assuming injected active power by machine eP as one of the states with

the excitation voltage fdE as a input signal of the machine. V. Venkatasubram anian

7

and R. G. Kavasseri in [17] assumed a third-order synchronous machine and they

used approximate algebraic equilibrium equations to derive the quasi-steady states

of the generator, assuming the field current fdi was available as a measurement in

addition to the terminal quantities.

W. Gao and S. Wang in[18]used UKF to estimate the rotor angle and rotor speed

of a classical synchronous generator in a multi-machine system assuming loads

are modeled as constant impedances, where this assumption is not true in the real

systems and as we will see in this work the performance of DSE will suffer with

this assumption.

In a similar work by Z. Huang, K. Schneider, and J. Nieplocha [19] the loads are

considered to be modeled as constant impedances and the EKF is implemented to

estimate the state variables of the classical synchronous machines . A. Jain and

N. R. Shivakumar [20] investigated the effect of the number of PMUs and the

weightage given to their measurements on the accuracy of dynamic state

estimation. Based on their results with increase in the number of PMUs the error in

the estimates reduces drastically for small numbers but seem to saturate as the

PMUs are increased beyond a point. The weighting given to the PMU

measurements also affects the accuracy of the estimates. In their test cases studied,

a weighting of around 100 to 200 seems to give higher quality results.

Two papers are published by E. Ghahremani and I. Kamwa in this area. In the first

paper [21], the authors use EKF to estimate the state variables of a two-axis

synchronous generator connected to an infinite bus, here one parameter of the

machine is considered to be unknown and the authors mention that we can consider

the unknown parameter as a state variable and therefore estimate it beside the other

state variables of the machine. In the second paper [22], they use UKF to estimate

8

again the state variables of a two axis synchronous machine which is connected to

an infinite bus, they also talk about the benefits of UKF in comparison with EKF

such as the robustness of UKF when underlying functions of the system are highly

nonlinear.

A paper by G. S. Wang, W. Gao, A. P. Meliopoulos [23] describes how UKF can

be used to estimate the state variables of synchronous generators in multi-machine

systems. As an example in their nine bus test system with constant impedance

loads, they estimate the state variables of the two-axis synchronous machines with

IEEE-Type I exciters. They also show that in their simulations UKF has a better

performance than EKF. The work by G. Valverde and V. Terzija [24] also shows

that UKF can be used to estimate the voltage magnitudes and angles of the system.

They also show that the UKF performs better compared to EKF and static WLS.

In the work by G. Valverde, E. Kyriakides, G. T. Heydt and V. Terzija[25], UKF

has been used for on-line parameter estimation of a synchronous machines during

steady state conditions. The proposed methodology uses voltage and current

measurements taken from the stator and the field windings to update the

parameters of the synchronous machine.

In a nutshell, most of the papers published in this area try to estimate the dynamic

state variables associated with a single synchronous machine or several

synchronous machines in a power system where the loads are modeled by constant

impedances without considering load changes which occur in the real power

systems. The following chapters will address the issue of accurately estimating

dynamic state of the system while accounting for these load changes.

9

Chapter 3

Improving Performance of Dynamic State

Estimators under Unknown Load Changes

Availability of wide-area synchronized measurements at a rate which is

commensurate with the requirements of capturing the state dynamics of the

transmission grid allows new ways of estimating system states. One possibility is

to use these measurements and the assumed or known model of the transmission

system and devise a dynamic state estimator. This idea is certainly not new and has

been considered by several researchers in the past four decades. Initial attempts

ignored machine dynamics mainly due to the technological limitations of acquiring

synchronized measurements at the desired sampling rate. These resulted in tracking

estimators which used conventional measurements and no specific model for the

system states. Such tracking estimators were later improved via the use of load

forecasting. Due to the assumed sampling intervals in the order of seconds, these

estimators do not account for generator or load dynamics.

Given the recent proliferation of synchronized phasor measurements in

transmission substations, it is possible to formulate the dynamic state estimation

problem by incorporating both generator dynamics and network equations. The

main motivation of this study is to develop such a dynamic estimator which can

provide the best estimate for the real-time dynamic state of the system.

Availability of this dynamic state will facilitate execution of various contingency

10

scenarios such as sudden switching of certain lines, loss of system loads or

generation.

Previous dynamic estimator investigations commonly ignore load changes and use

constant impedance approximation for the loads [13-25]. While the load changes

are indeed much slower compared to the system dynamics, their dynamic behavior

will impact operating state of the system. Hence, such changes need to be

accounted for in the dynamic state estimation formulation. A simple and effective

approach to accomplish this with minimal extra computational burden is proposed

in this chapter. Implementation of the approach on the New-England 37-bus

system with 8 generators will be described and simulation results will be presented.

The well-known and documented [3] Unscented Kalman Filter (UKF) is used in

developing the dynamic state estimator which estimates the dynamic states namely

the rotor angle and speed of the synchronous generators in the system.

Comparative results to illustrate benefits of using UKF over the more commonly

known Extended Kalman Filter (EKF) are also presented for the same system and

operating conditions.

3.1 Extended Kalman Filter

EKF which was originally proposed by Stanley Schmidt so that the Kalman filter

could be applied to nonlinear spacecraft navigation problems[26] is undoubtedly

the most widely used nonlinear state estimation technique that has been applied in

the past few decades. However, the EKF can be difficult to tune and often gives

unreliable estimates if the system nonlinearities are severe. This is because the

EKF relies on linearization to propagate the mean and covariance of the state[3].

11

The discrete-time EKF can be summarized as follows:

Assume the system dynamics and measurement equations are given as follows:

kkk wkxfx ),(1 (3.1)

kkk vkxhz ),( (3.2)

where:

nx is a discrete state vector.

mz is a discrete measurement vector.

),0(~ kk QNw Gaussian process noise at time step k

),0(~ kk RNv Gaussian measurement noise at time step k

kQ and kR are covariance matrices of kw and kv respectively.

The EKF is initialized as follows:

)(ˆ00 xEx

, ])ˆ)(ˆ[( 00000

TxxxxEP (3.3)

Now for ,...,2,1k perform the following.

(a) Compute the following partial derivative matrices:

1ˆ

11 |

kx

kk

x

fF

1ˆ

11 |

kx

kk

fL

(3.4)

12

(b) Perform the time update of the state estimate and estimation-error

covariance as follows:

T

kkk

T

kkkk LQLFPFP 111111

)1,ˆ(ˆ11

kxfx kkk (3.5)

(c) Compute the following partial derivative matrices:

kx

kk

x

hH

ˆ|

kx

kk

v

hM

ˆ| (3.6)

(d) Perform the measurement update of the state estimate and estimation-error

covariance as follows:

1 T

kkk

T

kkk

T

kkk MRMHPHHPK

),ˆ(ˆˆ kxhzKxx kkkkk

kkkk PHKIP (3.7)

3.2 UNSCENTED TRANSFORMATION

Given the nonlinearity of the measurement and generator dynamic equations, it is

quite cumbersome if not impossible to transform the entire probability density

function (pdf) for the modeling and measurement errors. Use of first order

approximation as done in Extended Kalman Filter will introduce errors which may

at times cause significant biases and even divergence.

13

As an alternative, an approximation to the true pdf can be generated by

transforming individual points in state space and calculating a sample pdf based on

them. This is essentially what an unscented transformation (UT) does. To illustrate

the idea, consider a random vector x with mean x and covariance P . One can

find a set of deterministic vectors called sigma points whose ensemble mean and

covariance are equal to x and P . Let us also consider the nonlinear measurement

function )(xhz which can be used to obtain measurement points by

substituting the sigma points for x in the measurement equations. The ensemble

mean and covariance of the transformed vectors will give a good estimate of the

true mean and covariance of z .

Suppose that x is an 1n vector that is transformed by a nonlinear function

)(xhz ,The unscented transformation base on Choosing n2 sigma points is

mentioned by following steps:

)(

*

)( ii xxx ni 2,...,1

T

i

i nPx )()(

* ni ,...,1

T

i

in nPx )()(

* ni ,...,1 (3.8)

Please note that inP)( is the i th row of nP . Transform the sigma points as

follows:

)( )()( ii xhz ni 2,...,1 (3.9)

The approximated mean and covariance of z can then be obtained as follows:

nzu

2

1

n

i

iz2

1

)( (3.10)

14

nPu

2

1 T

u

i

u

n

i

i zzzz ))(( )(2

1

)(

(3.11)

The unscented transformation mentioned above (which is based on choosing n2

sigma points ) is not the only one that exists. As an example, it can be shown that

general unscented transformation which uses 12 n sigma points to obtain the

propagated mean and covariance, gives the same order of mean and covariance

estimation accuracy as given by unscented transformation base on n2 sigma

points[3]. For computational saving purposes, we can use other type of

transformations which use less number of sigma points. It can be shown

[27,28]that minimum number of sigma points which makes it possible to obtain the

propagated mean and covariance is equal to 1n .

In this work we will consider the standard unscented kalman filtet or (UKF) which

uses the n2 sigma points UT and we evaluate the performance of UKF in compare

with EKF. In the next part we summarize the algorithm of UKF.

3.3 UNSCENTED KALMAN FILTER

The UKF uses UT(here with n2 sigma points) for solving nonlinear problems by

considering system dynamics and measurement given by (3.1)&(3.2).

where:

nx is a discrete state vector.

mz is a discrete measurement vector.

15

),0(~ kk QNw Gaussian process noise at time step k

),0(~ kk RNv Gaussian measurement noise at time step k

kQ and kR are covariance matrices of kw and kv respectively.

The UKF is initialized as follows:

)(ˆ00 xEx

, ])ˆ)(ˆ[( 00000

TxxxxEP (3.12)

Time update equations are:

(a) Calculation of sigma points:

)(

*1

)(

1ˆˆ

i

k

i

k xxx

ni 2,...,1

T

ik

i nPx )( 1

)(

*

ni ,...,1

T

ik

in nPx )( 1

)(

*

ni ,...,1 (3.13)

(b) )1,ˆ(ˆ )(

1

)( kxfx i

k

i

k ,n

xk2

1ˆ

n

i

i

kx2

1

)(ˆ

n

Pk2

1

n

i

T

k

i

kk

i

k xxxx2

1

)()( )ˆˆ)(ˆˆ(1 kQ (3.14)

The measurement update equations are as follows:

(a) Calculation of sigma points:

)(

*

)( ˆˆi

k

i

k xxx ni 2,...,1

16

T

ik

i nPx )()(

*

ni ,...,1

T

ik

in nPx )()(

*

ni ,...,1 (3.15)

In order to save computational effort step (3.15) can be omitted [3] with a slight

degradation in filter performance.

(b) ),ˆ(ˆ )()( kxhz i

k

i

k , n

zk2

1ˆ

n

i

i

kz2

1

)(ˆ

n

Pz2

1

k

n

i

T

k

i

kk

i

k Rzzzz

2

1

)()( )ˆˆ)(ˆˆ(

n

Pxz2

1

n

i

T

k

i

kk

i

k zzxx2

1

)()( )ˆˆ)(ˆˆ(

1 zxzk PPK

)ˆ(ˆˆkkkkk zzKxx

T

kzkkk KPKPP (3.16)

3.4 CASE STUDIES

In this chapter the following discretized swing and measurement equations will be

used:

),(1 kxfx kk (3.17)

),( kxhz kk (3.18)

17

where:

][ T

k

T

k

T

kx

T

mkkk

T

k ][ 21

ki :rotor angle for generator i at time step k

T

mkkk

T

k ][ 21

ki :rotor speed for generator i at time step k

m2

n : # of generators

),(1 kxfx kk and ),( kxhz kk are obtained based on the discretized (using

second-order Runga-Kutta method) swing and measurement equations given

below:

00

0

/)(

iiGiMiii DPPf

H

0 ii (3.19)

EYIG exp

)Re( *

GiiGi IEP mi ,...,1 (3.20)

Please note that Hi, PMi, PGi, and Di are the inertia constant, mechanical input

power, injected active power and damping coefficient for the ith generator

respectively.

18

Classical machine model is assumed for the synchronous generators, and based on

this assumption E is an (mx1) matrix with entries:

iiEiE

where: |E| is the back-emf of generator i.

Yexp is an (m x m) matrix which is obtained using the bus admittance matrix Ybus,

equivalent constant admittances of loads and subtransient - reactances of the

generators. Further details can be found in [2,4,5].

3.4.1 Case 1: Comparing the performance of UKF and EKF

Dynamic estimators are implemented using UKF and EKF algorithms and they are

tested on New-England 37-bus, 8 generator system(Figure 3.1).

Fig.3.1. New-England, 37-bus system.

19

The algorithms are initialized by the same values of state variables

0x̂ (which are

different from actual values),

0P , kR and kQ . Assuming Time-step=0.02sec, we

run the simulation for 10 sec where at t=5sec the indicated transmission line in

Figure3.1which is the line between LAUF138 and MORO138(here MORO 138 is

heavily loaded bus)has been switched out. The following error metrics are

calculated for rotor angles and rotor speeds separately in order to compare the

performances:

Estimation-error for rotor angles =

m

ii

t

kik xxm

E1

2

ˆ1

(3.21)

Estimation-error for rotor speeds = E

m

mii

t

kik xxm

2

1

2

ˆ1

(3.22)

where:

ikxˆ and i

t

kx are the i th elements of

kx̂ and t

kx (true state variable at time k ).

In table 3.1 we show the error metrics for both algorithms.

Table 3.1: Error metrics for case 1

Algorithm EKF UKF

Estimation-error for rotor angles

0.093

0.046

Estimation-error for rotor speeds

0.087

0.028

20

It is evident that estimation errors of UKF are much less than those of EKF under

the same initialization conditions.

As an example we also show the estimated rotor speed of generator#1 which are

obtained from both dynamic state estimator in Figure3.2&Figure3.3. In all plots

the dashed lines indicate the estimated state variable and the solid lines show the

true state trajectories.

Fig.3.2. Per-unit rotor speed of generator#1 obtained from EKF.

0 1 2 3 4 5 6 7 8 9 100.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

time [sec]

Ro

tor

spe

ed

/ S

ync.

sp

ee

d

1/

sync

+1/

sync

21

Fig.3.3. Per-unit rotor speed of generator#1obtained from UKF.

3.4.2 Case 2: Line switching transients

As mentioned earlier, while the loads are modeled as constant impedances, they

indeed vary in time, albeit much slowly compared to the system states. Slow

changes in bus loads can be captured via SCADA system which scans system loads

every 3-5 seconds. Hence, system dynamic equations can then be updated each

time a load update is received. However, errors will inadvertently accumulate

during the interval in between load updates. Two approaches are proposed below

in order to address this issue:

0 1 2 3 4 5 6 7 8 9 100.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

time [sec]

Ro

tor

spe

ed

/ S

ync.

sp

ee

d

1/

sync

+1/

sync

22

I. Increasing standard deviation ( ) for kQ

In order to evaluate this approach, bus loads are changed in the 37 bus system

during a time-interval of 180 seconds. Changes are implemented as discrete

movements occurring 10 times within this interval. As the reference, true state

trajectories are simulated using this data. Also, for simplicity we just show the

rotor angle of generator 1 and the per-unit rotor speeds of generators 1, 2, and 3.

Figure 3.4 and 3.5 show the simulation results for =0.0001 and Figure 3.6 and

3.7 show the results for =0.1. The corresponding estimation-errors are given in

Table 3.2.

Fig.3.4. Rotor angle of generator#1 when σ = 0.0001 .

20 40 60 80 100 120 140 160 180

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

time [sec]

Ro

tor a

ng

le [

De

g]

1

1

+

23

Fig.3.5. Per-unit rotor speed of generators #1, 2 and 3 when σ =0.0001 .

Fig.3.6. Rotor angle of generator #1 when σ = 0.1 .

0 20 40 60 80 100 120 140 160 1800.98

0.985

0.99

0.995

1

1.005

time [sec]

Ro

tor

sp

ee

d/

Syn

c.

sp

eed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

0 20 40 60 80 100 120 140 160 180-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

time [sec]

Ro

tor

an

gle

[D

eg]

1

1

+

24

Fig.3.7. Per-unit rotor speed of generators #1, 2 and 3 when σ = 0.1 .

Table 3.2: Error metrics for case 2

0.0001 0.1

Estimation-error for rotor angles

148.2

2.54

Estimation-error for rotor speeds

2.91

0.16

As evident from the plots, estimation error is reduced by increasing the value of σ

within the time-interval. However, estimation errors show significant variation

within the interval.

0 20 40 60 80 100 120 140 160 1800.975

0.98

0.985

0.99

0.995

1

1.005

time [sec]

Roto

r speed/ S

ync. speed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

25

This can be illustrated by opening one of the lines in the 37 bus system 5 seconds

after the beginning of the interval. Estimation of the dynamic state for the

following 5 seconds using σ = 0.1 yields the per-unit rotor speeds of generators 1,

2 and 3 as given in Figure 3.8. Large mismatches between the estimated and true

trajectories imply less than satisfactory performance.

Fig.3.8. Magnified view of the per-unit rotor speed of generator#1, 2 and 3, for σ = 0.1 .

II. Gradual increase of σ within time intervals

In this approach, σ values are increased gradually within the time interval. This is

accomplished via the expression given below:

nnk Im

kQ

1

2

1

(3.23)

where:

0 1 2 3 4 5 6 7 8 9 100.99

0.992

0.994

0.996

0.998

1

1.002

time [sec]

Ro

tor

sp

ee

d/

Syn

c.

sp

eed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

26

x represents the next integer larger than x .

m is the number of time-steps within the time-interval.

σ1 and σ2 are the lower and upper limits chosen for σ.

The same line switching transients are simulated using 0.0001 and 0.1 for the

lower and upper limits of σ. The results are shown in Figure 3.9-3.11. Estimation-

errors for rotor angles and rotor speeds in this case are calculated as 0.268 and

0.022 respectively, which is much less than the previous approach. Dynamic state

estimator performance is considered very satisfactory using this approach.

Fig.3.9. Rotor angle of generator 1 when σ is gradually increased.

0 20 40 60 80 100 120 140 160 180-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

time [sec]

Ro

tor

an

gle

s [

De

g]

1

1

+

27

Fig.3.10. Per-unit rotor speed of generators 1, 2 and 3 when σ is gradually increased.

Fig.3.11. Magnified view of the per-unit rotor speed of generators 1, 2 and 3 when σ is gradually

increased.

0 20 40 60 80 100 120 140 160 1800.98

0.985

0.99

0.995

1

1.005

time [sec]

Ro

tor

sp

ee

d/

Syn

c.

sp

eed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

0 1 2 3 4 5 6 7 8 9 100.99

0.992

0.994

0.996

0.998

1

1.002

time [sec]

Roto

r speed/ S

ync. speed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

28

3.5 CONCLUSIONS

In this chapter, a simple yet effective approach to improve the performance of the

dynamic state estimation (using UKF formulation) is presented. The main idea is

to compensate for the changing load and maintain acceptable estimation accuracy

during intervals when loads are not updated. This is accomplished by gradually

increasing the σ values within the intervals. Simulation results for line switching

scenarios validate effectiveness of this approach in reducing estimation errors to

satisfactory levels. However, this approach does not give us a precise solution for

dynamic state estimation of systems where load are continuously being changed. In

the next chapter we will talk about an alternative approach in order to obtain an

accurate solution for these systems.

29

Chapter 4

Distributed Implementation of an

Augmented State Dynamic Estimator

Dynamic security assessment involves dynamic simulation of the system’s

response to expected contingencies such as sudden load changes, line switching,

short circuit faults, etc. This is a challenging problem due to the large number of

machine and network variables typically used in the formulation. However, since

the main concern is system stability, it is usually sufficient to run these simulations

not longer than a few seconds by which time system stability under the considered

contingency can be assessed. Given the relatively slow response of most loads to

changes in system frequency it has been customary to model loads as fixed

impedances and thus reducing the problem size down to generator buses. While

this approximation works well for dynamic studies carried out periodically each

time approximating the loads based on the current operating point, it may not be

suitable if the model is to be used continuously in real-time. This is indeed the

case when a dynamic state estimator (DSE) is to be implemented on-line since it

has to run continuously and has to track load changes in real-time.

As we mentioned in the previous chapter, one way to improve the performance of

the DSE is to monitor the load changes on longer intervals and strategically vary

the variance of the system state error covariance matrix to compensate for the

errors introduced by changing loads. This approach is simple to implement yet it

30

still yields an approximate solution. So, in this chapter an alternative and accurate

approach will be proposed where the dynamic state vector will be augmented by

the algebraic variables representing the complex bus voltages and the dynamic

state and measurement equations will be accordingly reformulated.

The downside of this proposal is that given the large number of buses compared to

the number of generating units connected to a typical transmission grid this will

lead to a significant increase in the problem size and complexity. In order to

overcome this limitation the dynamic state estimation formulation needs to be

scalable, i.e. complexity of the problem should remain minimally affected with

increasing system size. Thus, a multi-area approach to dynamic state estimation

will be used for this purpose. It will be shown that through the use of phasor

measurement unit (PMU) measurements, this can be achieved easily and with little

change in the existing dynamic state estimation formulation.

4.1 PROBLEM FORMULATION

Power system dynamics can be modeled as a set of differential algebraic equations

(DAE) where the expressions contain two sets of variables:

Dynamic state variables: denoted by vector x , which includes the rotor angle

and the rotor angular speed of each generator (assuming the classical

generator model only).

Algebraic variables: voltage magnitude V and voltage phase angle of each

bus.

Thus, the dynamic state variables and algebraic variables are estimated and

updated simultaneously at each time step as described in [6], [23].

31

The proposed approach which addresses both the computational complexity as

well as the load tracking is implemented on the 37-bus New England system with 8

generators. Here we have used Unscented Kalman Filter (UKF) in developing the

dynamic state estimator. For more details about UKF please refer to Chapter 3.

4.2 IMPLEMENTATION OF DYNAMIC ESTIMATOR

Consider the dynamic equations for synchronous generators using the classical

generator model:

00

0

/)(

iiGiMiii DPPf

H

0 ii (4.1)

where

id

iiii

Gixj

VEI

)Re( *

GiiiGi IEP (4.2)

mi ,...,1

m is total number of generators.

Please note that iH , MiP , GiP , iD , iE , dix are the inertia constant, mechanical

input power, injected active power, damping coefficient, back emf and subtransient

32

reactance for the i th generator respectively. iV and i are also the voltage

magnitude and voltage phase angle of the corresponding machine’s terminal

voltage.

4.2.1 Conventional Approach

Dynamic state vector for a m -generator system will be:

Tmmx ......11 (4.3)

Assuming terminal power output measurements for each unit as the measurement

set:

TmGmG vPvPz ...11 (4.4)

Second-order Runga-Kutta method can be used to discretize the differential

equations of the generators mentioned above. This will yield two sets of difference

equations [2]:

),(1 kxfx kk state dynamics (4.5)

33

),( kxhz kk measurement equations (4.6)

The algebraic variables, iV and i will also be updated at each time step k by

using the network-equations. The following assumptions will be made:

Simulation time-step=0.025s

Arbitrary initialization of variables in (3.12)

zz nnk IeR 51 and nnk IeQ 61 where, zn and n are the total number of

measurements(here is equal to m ) and total number of state variables in state

vector x respectively.

UKF is implemented and tested using the system shown in Figure 4.1. A sudden

increase in load at bus HALE69 at t=1s followed by disconnection of line LAUF69

to HALE69 (at t=1.1s) is simulated as the disturbance event. The generators are

indicated by numbers 1 to 8, and HALE69 bus is indicated by a (red) dashed circle

in Figure 4.1. The output of DSE for rotor angle of generator #1 and per-unit rotor

speeds of generators #1, 2& 3 are shown in Figure 4.2 and Figure 4.3 respectively.

Note that in all plots the dashed lines indicate the estimated state variables whereas

the solid lines show the true state trajectories.

34

Fig.4.1. New-England, 37-bus system with sudden load change in HALE69.

Fig.4.2. Rotor angle of generator #1obtained based on conventional approach.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

20

40

60

time [sec]

Roto

r an

gle

[Deg]

1

1

+

35

Fig.4.3. Per-unit rotor speed of generators 1, 2 and 31obtained based on conventional approach.

As evident from the above figures, after the disturbance at t=1s the estimated and

true state variables clearly deviate. The reason for this mismatch is the fact that

despite the sudden change in load it is not being incorporated into the network

equations of the dynamic state estimator since the reduced set of equations are

continued to be used based on the initial load assumption.

4.2.2 Dynamic State Estimation Using Augmented State Vector

The formulation is modified to incorporate the algebraic variables and the state

vector is augmented. It includes both the dynamic state variables of the generators

and algebraic variables of the network as shown below:

TNNmm VVx ............ 1111 (4.7)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.996

0.997

0.998

0.999

1

1.001

1.002

1.003

time [sec]

Roto

r sp

eed/ S

ync.

speed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

36

where m is total number of generators and N is total number of buses in the

system.

Here, in order to obtain (4.5) for updating the augmented state vector we consider

the following assumptions:

Dynamic state variables of generators again will be updated by discretizing the

differential equations of the generators(4.1)&(4.2).

The algebraic variables will be updated by:

kkVV

1 , kk 1 (4.8)

Where k

V and k are column vectors of the voltage magnitudes and voltage

phase angles of all buses respectively at time step k .

The set of measurements used for this case include:

Injected active power measurements at all generators.

Voltage magnitude and voltage phase angle of all buses.

UKF is again implemented for this formulation and the same test system and

disturbance scenario are used for simulations. The results for the rotor angle of

generator#1 and per-unit rotor speeds of generators#1,2 and 3 are shown in Figure

4.4 and Figure 4.5 respectively.

37

Fig.4.4. Rotor angle of generator #1obtained using augmented state vector approach.

Fig.4.5. Per-unit rotor speed of generators# 1, 2 & 3obtained using augmented state vector approach.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

20

time [sec]

Roto

r an

gle

[D

eg]

1

1

+

0 1 2 3 4 50.9965

0.997

0.9975

0.998

0.9985

0.999

0.9995

1

1.0005

time [sec]Roto

r sp

eed/ S

ync.

speed

1/

sync

+

1/

sync

2/

sync

+

2/

sync

3/

sync

+

3/

sync

38

The estimated and true trajectories of the state variables match closely as a result

of this revised formulation as seen from the above figures. While these results

appear very satisfactory, they may come at a high computational cost for large

scale systems.

In order to address this issue and make this approach scalable, a multi-area solution

will be presented next.

4.2.3 Distributed Multi-Area Dynamic State Estimation

The augmented state vector formulation also allows easy decoupling of the system

into smaller zones (areas) and performing the DSE independently for each zone.

When working with very large scale systems, it will thus be possible to identify

smaller size zones and perform the DSE simultaneously in parallel for these zones

and therefore obtain the DSE result for the entire system in a much shorter time.

Furthermore, as long as such zones can be found as observable islands, they can be

added to the system model without impacting the total solution time significantly.

In order to identify these specific zones, the following conditions should be

satisfied:

1. Zones are connected by lines or transformers, so each bus belongs to only

one zone.

2. Boundary bus injection measurements (current or power) are disregarded.

3. Each zone is fully observable by its own measurements.

39

4. Each zone has to have at least one phasor measurement to facilitate

synchronization with the rest of the zone estimates.

5. Measurements are available at the required scan rate of the dynamic

estimator.

The formulation for each zone will be identical to the one developed in the

previous section. As an example the augmented state vector by considering simple

classical models for the generators will be:

TNNmm zzzzVVx ............ 1111 (4.9)

where zm is total number of generators and zN is total number of buses inside the

zones which satisfy the above assumptions.

Considering the assumptions mentioned above, the test system of Figure 4.1 is

divided into four zones (as indicated by the dotted lines in Figure 4.6).

The following measurements are assumed for each zone:

Injected active powers by generators which are indicated by red arrows in

Figure 4.6.

Voltage magnitude and voltage phase angles of the buses which are indicated

by green circles (assuming PMUs are placed at these buses) in Figure 4.6.

Combination of power flows on transmission lines that make the zone

observable, Here we consider the active and reactive power through

transmission lines which are indicated by blue arrows in Figure 4.6.

Please note that some of the measurements may be redundant. As an example

the active and reactive power from bus A to bus B (end of arrow) can be

obtained by:

40

jVZVVI AAABBBAAAB */)(

)( *

ABAAAB IVrealP

)( *

ABAAAB IVimagQ (4.10)

Please note that ABZ and j are the impedance and the half of the total line

charging susceptance for the transmission line between A and B respectively.

Fig.4.6. New-England, 37-bus system divided into four zones.

Again, consider the same disturbance scenario as above but this time four separate

UKFs are implemented, one for each zone to estimate the corresponding

41

augmented state vectors of individual zones. In order to evaluate the performance

of DSE of each zone, we show the estimated rotor angle, per-unit rotor speed,

magnitude and angle of terminal voltage corresponding to generators of each zone

in below:

Fig.4.7. Rotor angles of generators#1&8 inside zone 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

20

time [sec]

Ro

tor

an

gle

[D

eg

]

1

1+

8

8+

42

.

Fig.4.8. Per-unit rotor speeds of generators# 1 & 8 inside zone 1.

.

Fig.4.9. Bus voltage’s magnitudes of generators# 1&8 inside zone 1.

0 1 2 3 4 50.997

0.9975

0.998

0.9985

0.999

0.9995

1

1.0005

time [sec]

Rot

or s

peed

/ Syn

c. s

peed

1/

sync

+1/

sync

8/

sync

+8/

sync

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time [sec]

Bu

s V

olta

ge

(pu

)

|V1|

|V+1|

|V8|

|V+8|

43

.

Fig.4.10. Bus voltage’s angles of generators# 1 & 8 inside zone 1.

Fig.4.11. Rotor angles of generators#2-4 inside zone 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

time [sec]

Bus

vol

tag

e an

gle

[Deg

]

1

+1

8

+8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-160

-140

-120

-100

-80

-60

-40

-20

0

20

time [sec]

Roto

r an

gle

[Deg

]

2

2+

3

3+

4

4+

44

Fig.4.12. Per-unit rotor speeds of generators# 2-4 inside zone 2.

.

Fig.4.13. Bus voltage’s magnitudes of generators# 2-4 inside zone 2.

0 1 2 3 4 50.996

0.997

0.998

0.999

1

1.001

1.002

1.003

time [sec]

Rot

or s

peed

/ Syn

c. s

peed

2/

sync

+2/

sync

3/

sync

+3/

sync

4/

sync

+4/

sync

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.75

0.8

0.85

0.9

0.95

1

time [sec]

Bu

s V

olta

ge

(pu

)

|V2|

|V+2|

|V3|

|V+3|

|V4|

|V+4|

45

Fig.4.14. Bus voltage’s angles of generators# 2-4 inside zone 2.

Fig.4.15. Rotor angles of generators#6-7 inside zone 3.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-150

-100

-50

0

time [sec]

Bus

volta

ge a

ngle

[Deg

]

2

+2

3

+3

4

+4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

20

time [sec]

Ro

tor

an

gle

[D

eg

]

6

6+

7

7+

46

Fig.4.16. Per-unit rotor speed of generators # 6-7 inside zone 3.

Fig.4.17. Bus voltage’s magnitudes of generators# 6-7 inside zone 3.

0 1 2 3 4 50.997

0.998

0.999

1

1.001

1.002

1.003

time [sec]

Rot

or s

peed

/ Syn

c. s

peed

6/

sync

+6/

sync

7/

sync

+7/

sync

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

time [sec]

Bus V

oltag

e(p

u)

|V6|

|V+6|

|V7|

|V+7|

47

Fig.4.18. Bus voltage’s angles of generators# 6-7 inside zone 3.

Fig.4.19. Rotor angle of generator#5 inside zone 4.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-150

-100

-50

0

time [sec]

Bu

s vo

ltag

e a

ng

le [D

eg

]

6

+6

7

+7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-140

-120

-100

-80

-60

-40

-20

0

20

time [sec]

Ro

tor

an

gle

[D

eg

]

5

5+

48

Fig.4.20. Per-unit rotor speed of generator# 5 inside zone 4.

Fig.4.21. Bus voltage’s magnitude of generator# 5 inside zone 4.

0 1 2 3 4 50.996

0.997

0.998

0.999

1

1.001

1.002

1.003

time [sec]

Rot

or s

peed

/ Syn

c. s

peed

5/

sync

+5/

sync

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

time [sec]

Bus V

oltag

e(p

u)

|V5|

|V+5|

49

Fig.4.22. Bus voltage’s angle of generator# 5 inside zone 4.

As seen in Figures 4.7 - 4.22, trajectories estimated by individual zone DSEs

closely match those estimated by the system-wide integrated DSE. These results

imply that large scale DSE implementation will be possible by zone-based DSEs

which are synchronized through phasor measurements existing in each zone.

Zones can exchange their results either with their neighbors or with everyone by

configuring a common data bus structure where all results will be uploaded. Now

we compare the CPU-time/time-step associated with dynamic state estimation of

each zone with CPU-time/time-step of the simulations in the previous part which

was obtained based on augmented state vector approach for entire system.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-150

-100

-50

0

time [sec]

Bus

vol

tage

ang

le [D

eg]

5

+5

50

Table 4.1: CPU-Time associated with dynamic state estimation of the zones.

Approach

K

(Number of time steps)

CPU-Time

(sec)

CPU-Time/K

(sec)

Augmented state vector

(entire system)

201

2.6938

0.0134

Conventional approach

201

0.3305

0.0016

Distributed Multi-Area

(zone 1)

201

0.4036

0.0020


(zone 2)

201

0.4220

0.0021


(zone 3)

201

0.4092

0.0020


(zone 4)

201

0.3092

0.0015

Based on the data in table4.1 the CPU-times associated with dynamic state

estimation of the zones(specially for zone 4) are considerably smaller than CPU-

time of the results obtained based on augmented state vector approach for entire

system.

As we mentioned before, for simplicity we considered classical models for the

generators, now we finish this chapter by the following example where we estimate

the augmented state vector associated with a more complex model of a

synchronous generator.

51

Example 4.1

The differential equations corresponding to a synchronous two-axis generator with

IEEE-Type I exciter are given as follows [6]:

fddddq

q

do EIXXEdt

EdT

qqqdd

qo IXXEdt

EdT

0

dt

d

00

0

/2

DPP

dt

dHGM

RfdfdEE

fd

E VEESKdt

dET

fd

F

Ff

f

F ET

KR

dt

dRT

VVKET

KKRKV

dt

dVT refAfd

F

FAfAR

RA (4.11)

Please note that in the equations mentioned above the state variables associated

with two-axis generators are: rotor angle , rotor speed , internal voltages qE

and dE .

The state variables associated with IEEE-Type I exciter are: excitation voltage fdE ,

rate feedback fR , exciter input RV .

refV is called exciter reference (setpoint) where exciter systematically adjusts fdE in

52

attempt to minimize the difference between terminal voltage and this reference

signal.

Here ES is saturation function( function of fdE ) which is given beside the values of

constant variables in table 4.2.

Assuming the stator resistance is zero( 0aR ), we have[4],[21]:

d

mq

dX

VEI

)cos(

q

m

qX

VI

)sin(

)2sin(11

2)sin(

2

m

dq

mq

d

eXX

VE

X

VP

))(sin)(cos

()cos(

22

2

q

m

d

m

mq

d

eXX

VEX

VQ

m (4.12)

Now we consider the following assumptions:

The generator is connected to an infinite bus.

The mechanical input power MP is unknown, therefore we consider it as a

state variable.

The augmented state vector is: TMRffddq VPVREEEx

Tee vvVvQvPz 4321 , where 4451 IeRk and

101061 IeQk

Arbitrary initialization of variables in (3.12)

We discretize the equations mentioned above using Euler method.

53

Time step=0.02 sec.

Now we estimate the augmented state vector x for this machine where at t=1s we

have changed the value of refV (from 1.095 to 1.25). The results are shown in

figures 4.23-4.32.

Table 4.2: Data for example 4.1

(sec)H 23.64

)( puX d 0.146

)( puX d 0.0608

)( puX q 0.0969

)( puX q 0.0769

(sec)doT 8.96

(sec)qoT 0.31

D 0.0125

)( puV 1.04

(deg) 0

)( puVref 1.095

AK 20

(sec)AT 0.2

EK 1

(sec)ET 0.314

FK 0.063

(sec)FT 0.35

)( fdE ES )555.1exp(0039.0 fdE

54

Fig.4.23. Rotor angle of the generator in example 4.1.

Fig.4.24. Per-unit rotor speed of the generator in example 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 4-6

-5

-4

-3

-2

-1

0

1

2

3

4

time [sec]

Ro

tor

an

gle

[D

eg

]

+

0 0.5 1 1.5 2 2.5 3 3.5 40.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

time [sec]

Roto

r sp

eed/

Syn

c. s

peed

/sync

+/sync

55

Fig. 4.25. Internal voltage-q axis of the generator in example 4.1.

Fig.4.26. Internal voltage-d axis of the generator in example 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

time [sec]

Inte

rna

l vo

ltag

e-q

axi

s (

pu

)

Epq

E+pq

0 0.5 1 1.5 2 2.5 3 3.5 4-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

time [sec]

Inte

rna

l vo

lta

ge-d

axis

(p

u)

Epd

E+pd

56

Fig.4.27. Excitation voltage of the generator in example 4.1.

Fig.4.28. Rate feedback of the generator in example 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.5

0

0.5

1

1.5

2

2.5

3

time [sec]

Excitation v

oltage (

pu)

Efd

E+fd

0 0.5 1 1.5 2 2.5 3 3.5 4-7

-6

-5

-4

-3

-2

-1

0

1

2

time [sec]

Rate

feedback

Rf

R+f

57

Fig.4.29. Exciter input of the generator in example 4.1.

Fig.4.30. Mechanical input power (pu) of the generator in example 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 4-35

-30

-25

-20

-15

-10

-5

0

5

10

time [sec]

Exciter

input

VR

V+R

0 0.5 1 1.5 2 2.5 3 3.5 4-12

-10

-8

-6

-4

-2

0

2

4

time [sec]

Me

ch

. in

put

po

we

r(p

u)

PM

P+M

58

Fig.4.31. Terminal bus voltage’s magnitude of the generator in example 4.1.

Fig.4.32. Terminal bus voltage’s angle of the generator in example 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 41

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

time [sec]

Bus V

oltage (

pu)

V

V+

0 0.5 1 1.5 2 2.5 3 3.5 4-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

time [sec]

Bu

s v

olta

ge a

ngle

[D

eg

]

+

59

4.3 CONCLUSIONS

In this chapter, an alternative method for obtaining a more precise solution for

dynamic state estimation of a system with load changes has been proposed.

Furthermore, an extension of this formulation for large scale systems by a multi-

area implementation is also presented. The system is divided into several non-

overlapping zones which are synchronized via phasor measurements. DSEs are

simultaneously executed by each zone. Each zone uses the proposed augmented

state formulation where the state vector for each zone is estimated accounting for

changes in bus loads. Simulation results for a 37-bus system are used to

demonstrate the effectiveness of this approach.

60

Concluding Remarks and Further Study

5.1 Concluding Remarks

In this thesis, two different effective approaches to improve the performance of the

dynamic state estimation (using UKF formulation) are presented. The main idea of

the first approach is to reduce the error associated with dynamic state estimation

due to considering wrong load models within the network equations of the DSE.

Various type of load models can be implemented within the network equation of

DSE for example ZIP load model or constant impedance load. Conspicuously in

the real power system the future dynamic behavior of loads are unknown, therefore

the considered equivalent load models in DSE will be different than true load

models, thus this mis-modeling may drastically reduce the performance of DSE.

The proposed method mentions that the network equations of DSE can be restored

each time the most updated load models are obtained and during the time intervals

that load updates are not received the effect of mis-modeling on the performance of

DSE can be reduced by gradually increasing the σ values within the intervals. The

effectiveness of this approach has been evaluated by showing the simulation results

for New-England 37 bus system where the duration of time interval is considered

to be 180-sec and the loads have been changed( by 10 steps) during this period.

Chapter 5

61

However, the drawback of this method is that although it reduces the error of the

state estimation, it does not give us a precise solution.

Based on the second approach, using the augmented state vector which beside the

machine’s state variables also includes the algebraic network variables, namely

voltage magnitude and voltage phase angle of all buses, the effect of load changes

on dynamic state estimation can be accurately taken into account. The reason is

that machines state variables will be updated through their dynamic equations

which includes the true estimated algebraic network variables. In other words by

estimating these algebraic variables the differential algebraic equations in DSE will

be independent of load models.

This approach also allows us to perform a multi-area dynamic state estimation for a

large scale system where we split the system into several non-overlapping zones

which are synchronized via phasor measurements. Then we implement DSEs

simultaneously in each zone to estimate the corresponding augmented state vectors

which take into account the changes in bus loads. The simulation results for the

New-England 37 bus system also validated the effectiveness of this approach.

62

5.2 Further Study

Obtaining a correct equivalent load model connected to the bus loads can be highly

beneficial in power system analysis. Using wrong load models may lead to

inconsistent results for dynamic stability (as shown in this work) and voltage

collapse studies [29]. In general load modeling approaches are classified into two

categories: Component-based approach [30] and measurement-based approach [31,

32] . The drawback of the first approach is that the load inventory of typical loads

is needed to synthesize loads model therefore the performance of this approach

depends on true inventory of the loads connected to the feeders which is not

always available [33]. The second approach estimates load parameters using

measurements which gives a more precise picture of the real-time loads and their

dynamic characteristics[34].

Aggregate (Composite ) load model which includes static load(ZIP load) in parallel

with induction motor is one of the most accepted models for describing the

dynamic of the loads. Aggregate load model can be generally presented as

follows[35]:

MOTZIPs PPP ** 11

MOTZIPs QQQ ** 22 (5.1)

Where sP , ZIPP and MOTP represent the real power of aggregate load, ZIP load and

induction motor load; and are static and dynamic load percentages.

The ZIP load is expressed as follows:

pppZIP c

V

Vb

V

VaPP )()(

0

2

0

0

63

qqqZIP c

V

Vb

V

VaQQ )()(

0

2

0

0 (5.2)

Where 0P and 0Q represent the real and reactive load power in the steady state;

1 ppp cba and .1 qqq cba

Now the main purpose is to identify the parameters and also estimate the state

variables of the aggregate load which includes:

Load composition parameters: 2121 ,,, .

The parameters of ZIP load: qqqppp cbacba ,,,,, .

The parameters and state variables associated with the induction load motor,

more details are available in [35].

However in real power systems due to large number of bus loads it is not practical

to install a large number of meters such as PMUs on these buses. Fortunately,

estimation of the augmented state vector allows to accurately monitor the systems

state variables under load changes. Based on these estimated state variables the

estimated phasor voltages, injected active and reactive power by the bus loads can

be readily obtained.

Development of novel methods to identify the parameters and estimate the state

variables of the corresponding aggregated load models will be investigated as

future work.

.

64

Appendix A

FUNDAMENTAL SCRIPTS USED FOR DYNAMIC

STATE ESTIMATION

A.1 EKF based on conventional approach assuming classical model for

synchronous generators

**************Extended Kalman Filter(conventional method) **************

for k=2:kmax%k means the kth time step, Kmax is the time step when fault

occurs(here line outage)

%Obtaining partial derivative matrix(F COMPUTATION)

F2=t_step*[eye(no),zeros(no)]; % t_step is time-step, no is total number of

generators

dedx=[zeros(no),diag(j*ABS_E.*exp(j*my_dp(:,k-1)))]; % ABS_E is a column

vector inculing magnitudes of back EMFs, my_dp is the posteriori estimate of

rotor angles

didx=Y*dedx; % Y is the reduced Ybus before the fault(line outage)

E=ABS_E.*exp(j*my_dp(:,k-1));

I=Y*E;

dpgdx=real(conj(diag(I))*dedx+diag(E)*conj(didx));

F1=t_step*diag(M.^-1)*(0-dpgdx); % M is column vector of inertia constants

F_inc=[F1;F2];

F=eye(ns)+F_inc;

%P_M (priori estimation error covariance)computation

65

Q=sig_Q^2*eye(ns); % ns is total number of state variables, sig_Q is std of

state error covariance matrix

P_M=F*P_P*F'+Q;

%x_minus(priori estimation of x) computation


I=Y*E;

my_PG_minus(:,k-1)=real(E.*conj(I));

PM0=my_PG_minus(:,1); %pmech=pelec at t=0

g=M.^-1.*(PM0-my_PG_minus(:,k-1));

my_w_minus=my_xp(1:no,k-1)+t_step*g;

my_delta_minus=my_xp(no+1:ns,k-1)+t_step*my_xp(1:no,k-1);

my_xm=[my_w_minus;my_delta_minus];

%COMPUTATION OF Hk and h=PG @ x_minus

dedx=[zeros(no),diag(j*ABS_E.*exp(j*my_delta_minus))];

didx=Y*dedx;

E=ABS_E.*exp(j*my_delta_minus);

I=Y*E;

h=real(E.*conj(I));

HH=real(conj(diag(I))*dedx+diag(E)*conj(didx));

%COMPUTATION OF Mk

MM=eye(no);

%COMPUTATION of K(Kalman gain)

R=sig_R^2*eye(no);%sig_R is the std of measurement error covariance

matrix

K=P_M*HH'*(HH*P_M*HH'+MM*R*MM')^-1;

y=my_PG(:,k)+sig_R*randn(no,1); %my_PG is the actual injected active

power by the generators

my_xp(:,k)=my_xm+K*(y-h); % my_xp is posteriori estimation of x

my_dp=my_xp(no+1:ns,:);

66

P_P=(eye(ns)-K*HH)*P_M*((eye(ns)-K*HH)).'+K*R*(K).';

end

for k=kmax+1:kend%kend is the last time step of the simulation

%F COMPUTATION

F2=t_step*[eye(no),zeros(no)];

dedx=[zeros(no),diag(j*ABS_E.*exp(j*my_dp(:,k-1)))];

didx=Y_SW*dedx;


I=Y_SW*E; %Y_SW is the reduced Ybus after the fault (line outage)

dpgdx=real(conj(diag(I))*dedx+diag(E)*conj(didx));

F1=t_step*diag(M.^-1)*(0-dpgdx);

F_inc=[F1;F2];

F=eye(ns)+F_inc;

%P_M computation

Q=sig_Q^2*eye(ns);

P_M=F*P_P*F'+Q;

%x_minus computation


I=Y_SW*E;

my_PG_minus(:,k-1)=real(E.*conj(I));

g=M.^-1.*(PM0-my_PG_minus(:,k-1));

my_w_minus=my_xp(1:no,k-1)+t_step*g;

my_delta_minus=my_xp(no+1:ns,k-1)+t_step*my_xp(1:no,k-1);

my_xm=[my_w_minus;my_delta_minus];

%COMPUTATION OF Hk and h=PG @ x_minus

dedx=[zeros(no),diag(j*ABS_E.*exp(j*my_delta_minus))];

67

didx=Y_SW*dedx;

E=ABS_E.*exp(j*my_delta_minus);

I=Y_SW*E;

h =real(E.*conj(I));

HH =real(conj(diag(I))*dedx+diag(E)*conj(didx));%correct

%COMPUTATION OF Mk

MM=eye(no);

%COMPUTATION of K (Kalman gain)

R=sig_R^2*eye(no);

K=P_M*HH'*(HH*P_M*HH'+MM*R*MM')^-1;

y =my_PG(:,k)+sig*randn(no,1);

my_xp(:,k)=my_xm+K*(y-h);


P_P=(eye(ns)-K*HH)*P_M*((eye(ns)-K*HH)).'+K*R*(K).';

end

A.2 Unscented Kalman Filter (UKF) based on conventional approach

assuming classical model for synchronous generators

************Unscented Kalman Filter(conventional method) ************

ABS_E_rep=repmat(ABS_E,1,2*ns);%ABS_E is a column vector including

magtinudes of back EMFs.

M_rep=repmat(M,1,2*ns);% M is column vector of inertia constants

Damp_rep=repmat(Damp,1,2*ns)/(2*pi*f0); Damp is column vector of damping

coefficients

PM0_rep=repmat(PM0,1,2*ns);%pmech=pelec at t=0

68

for k=2:k_kamx%k means the kth time step, Kmax is the time step when fault

occurs(here line outage)

% Obtaining first set of sigma points

A=chol(ns*P_P);

my_x_tilda=[A',-A'];

rep1=repmat(my_xp(:,k-1),1,2*ns);

my_x_sigma=my_x_tilda+rep1;

my_w_sigma=my_x_sigma(1:no,:);

my_del_sigma=my_x_sigma(no+1:ns,:);

%Obtaining the transformed sigma points

E=ABS_E_rep.*exp(j*my_del_sigma);

I=Y*E;

my_PG=real(E.*conj(I));

g=M_rep.^-1.*(PM0_rep-my_PG1-Damp_rep.*my_w_sigma);

ka=g*t_step;

kpa=my_w_sigma*t_step;

E=ABS_E_rep.*exp(j*(my_del_sigma+kpa));

I=Y*E;

my_PGp=real(E.*conj(I));

g=M_rep.^-1.*(PM0_rep- my_PGp-Damp_rep.*(my_w_sigma+ka));

kb=g*t_step;

kpb=(my_w_sigma+ka)*t_step;

my_del_sigma=my_del_sigma+(kpa+kpb)/2;

my_w_sigma=my_w_sigma+(ka+kb)/2;

my_x_sigma=[my_w_sigma;my_del_sigma];

%x_minus(priori estimation of x)

x_minus=(2*ns)^-1*(sum(my_x_sigma'))';

69

%P_M(priori state error covariance matrix)

Q=sig_Q^2*eye(ns);

x_minus_rep=repmat(x_minus,1,2*ns);

P_M=(2*ns)^-1*(my_x_sigma-x_minus_rep)*(my_x_sigma-x_minus_rep)'+Q;

%obtaining new set of sigma points-this step can be omitted

A=chol(ns*P_M);


rep1=repmat(x_minus,1,2*ns);




%y_sigma(predicted measurements)


I=Y*E;

my_PG1=real(E.*conj(I));

y_sigma=my_PG1;

%y predict

y_predict=(2*ns)^-1*(sum(y_sigma'))';

% cov y predict &R

y_predict_rep=repmat(y_predict,1,2*ns);

R=sig_R^2*eye(no);

P_y=(2*ns)^-1*(y_sigma-y_predict_rep)*(y_sigma-y_predict_rep)'+R;

%cross cov x_minus and y predict

P_xy=(2*ns)^-1*(my_x_sigma-x_minus_rep)*(y_sigma-y_predict_rep)';

%measurement update

K=P_xy*P_y^-1;

70

y =my_PG(:,k)+sig_R*randn(no,1);

my_xp(:,k)=x_minus+K*(y-y_predict);


P_P=P_M-K*P_y*K';

end

for k=kmax+1:kend %kend is the last time step of the simulation


A=chol(ns*P_P);






% Obtaining the transformed sigma points


I=Y_SW*E;



ka=g*t_step;



I=Y_SW*E;

my_PGp=real(E.*conj(I));


kb=g*t_step;




my_x_sigma=[my_w_sigma;my_del_sigma];

% x_minus(priori estimation of x)

71


% P_M(priori state error covariance matrix)

Q=sig_Q^2*eye(ns);



% obtaining new set of sigma points-this step can be omitted

A=chol(ns*P_M);






%y_sigma(predicted measurements)


I=Y_SW*E;


y_sigma=my_PG1;

%y predict


% cov y predict &R


R=sig_R^2*eye(no);




72

%measurement update

K=P_xy*P_y^-1;

y =my_PG(:,k)+sig_R*randn(no,1);



P_P=P_M-K*P_y*K';

end

A.3 Unscented Kalman Filter (UKF) based on augmented state vector

approach assuming classical model for synchronous generators

************Unscented Kalman Filter(augmented state vector) ************

ABS_E_rep=repmat(ABS_E,1,2*ns);%ABS_E is a column vector including magtinudes

of back EMFs.



coefficients


for k=2:kend%k means the kth time step kend is the last time step of the

simulation


A=chol(ns*P_P);





my_del_sigma=my_x_sigma(no+1:2*no,:);

my_v_sigma=my_x_sigma(2*no+1:2*no+nd,:);

my_theta_sigma=my_x_sigma(2*no+nd+1:2*no+2*nd,:);

%Obtaining the transformed sigma points

73


EV=[E;my_v_sigma.*exp(j*my_theta_sigma)];

I=Y_BUS*EV; %Y_BUS is the bus admittance matrix which also includes the

internal buses of the generators

my_PG1=real(E.*conj(I(1:no,:)));


ka=g*t_step;




I=Y_BUS*EV;

my_PGp=real(E.*conj(I(1:no,:)));


kb=g*t_step;




my_x_sigma=[my_w_sigma;my_del_sigma;my_v_sigma;my_theta_sigma];




Q=sig_Q^2*eye(ns);



% obtaining new set of sigma points

A=chol(ns*P_M);



74






% y_sigma(predicted measurements)



I=Y_BUS*EV;

my_PG1=real(E.*conj(I(1:no,:)));

my_absv1=my_v_sigma;

my_theta1=my_theta_sigma;

y_sigma=[my_PG1;my_absv1;my_theta1];

%y predict


% cov y predict &R


R=sig_R^2*eye(2*nd+no); %nd is total number of buses without internal buses

of the generators




%measurement update

K=P_xy*P_y^-1;

y_small=my_PG(:,k)+sig_R*randn(no,1);

y_v=my_abs_v(:,k)+sig_R*randn(nd,1);

75

y_theta=my_theta_v(:,k)+sig_R*randn(nd,1);

y=[y_small;y_v; y_theta];


my_dp=my_xp(no+1:2*no,:);

my_vp=my_xp(2*no+1:2*no+nd,:);

my_thetp=my_xp(2*no+nd+1:2*no+2*nd,:);

P_P=P_M-K*P_y*K';

end

A.4 Unscented Kalman Filter (UKF) based on augmented state vector

approach assuming two axis synchronous generator with IEEE-TypeI Exciter

********Unscented Kalman Filter(augmented state vector-two axis) ********

ABS_E_rep=repmat(ABS_E,1,2*ns);%ABS_E is a column vector including magnitudes

of back EMFs.



coefficients


for k=2:kend%k means the kth time step kend is the last time step of the

simulation


A=chol(ns*P_P);






my_eq_sigma=my_x_sigma(2*no+1:3*no,:);

my_ed_sigma=my_x_sigma(3*no+1:4*no,:);

my_ef_sigma=my_x_sigma(4*no+1:5*no,:);

my_RF_sigma=my_x_sigma(5*no+1:6*no,:);

my_VR_sigma=my_x_sigma(6*no+1:7*no,:);

76

my_v_sigma=my_x_sigma(7*no+1:7*no+nd,:); %nd is total number of buses without

internal buses of the generators


my_PM0_sigma=my_x_sigma(7*no+2*nd+1:8*no+2*nd,:);

% Obtaining the transformed sigma points

my_PGa=(my_v_sigma./xdp).*my_eq_sigma.*sin(my_del_sigma-

my_theta_sigma)+0.5.*(1/xq-1/xdp).*(my_v_sigma.^2).*sin(2.*my_del_sigma-

2.*my_theta_sigma);

g=M_rep.^-1.*(my_PM0_sigma-my_PGa-Damp_rep.*my_w_sigma);

my_w_sigma=my_w_sigma+t_step*g;

my_del_sigma=my_del_sigma+t_step*my_w_sigma;

g=(1/Tdop).*(my_ef_sigma-my_eq_sigma-(xd/xdp-1).*(my_eq_sigma-

my_v_sigma.*cos(my_del_sigma)));

my_eq_sigma=my_eq_sigma+t_step*g;

g=(1/Tqop).*(-my_ed_sigma-(1-xqp/xq).*(my_v_sigma.*sin(my_del_sigma)));

my_ed_sigma=my_ed_sigma+t_step*g;

SE=0.0039*exp(1.555*my_ef_sigma); % Here SE is the saturation function of Efd

g=(1/TE).*(-(KE+SE).*my_ef_sigma+my_VR_sigma );

my_ef_sigma=my_ef_sigma+t_step*g;

g=(1/TF).*(-my_RF_sigma+(KF/TF).*my_ef_sigma);

my_RF_sigma=my_RF_sigma+t_step*g;

g=(1/TA).*(-my_VR_sigma+KA.*my_RF_sigma-(KF*KA/TF).*my_ef_sigma+KA.*(Vref-

my_v_sigma));

my_VR_sigma=my_VR_sigma+t_step*g;

my_x_sigma=[my_w_sigma;my_del_sigma;my_eq_sigma;my_ed_sigma;my_ef_sigma;my_RF

_sigma;my_VR_sigma;my_v_sigma;my_theta_sigma;my_PM0_sigma];




Q=sig_Q^2*eye(ns);



77

% obtaining new set of sigma points

A=chol(ns*P_M);






my_eq_sigma=my_x_sigma(2*no+1:3*no,:);

my_ed_sigma=my_x_sigma(3*no+1:4*no,:);

my_ef_sigma=my_x_sigma(4*no+1:5*no,:);

my_RF_sigma=my_x_sigma(5*no+1:6*no,:);

my_VR_sigma=my_x_sigma(6*no+1:7*no,:);



my_PM0_sigma=my_x_sigma(7*no+2*nd+1:8*no+2*nd,:);

% y_sigma(predicted measurements)

my_PG1=(my_v_sigma./xdp).*my_eq_sigma.*sin(my_del_sigma-

my_theta_sigma)+0.5.*(1/xq-1/xdp).*(my_v_sigma.^2).*sin(2.*my_del_sigma-

2.*my_theta_sigma);

my_QG1=(my_v_sigma./xdp).*my_eq_sigma.*cos(my_del_sigma-my_theta_sigma)-

(my_v_sigma.^2).*((cos(my_del_sigma-

my_theta_sigma)).^2./xdp+(sin(my_del_sigma-my_theta_sigma)).^2./xq);

my_absv1=my_v_sigma;

my_theta1=my_theta_sigma;

my_Ef1=my_ef_sigma;

my_VR1=my_VR_sigma;

my_RF1=my_RF_sigma;

my_w1=my_w_sigma;

my_delta1=my_del_sigma;

y_sigma=[my_PG1;my_absv1;my_theta1;my_QG1];

%y predict

78


% cov y predict &R


R=sig_R^2*eye(2*nd+2*no);




%measurement update

K=P_xy*P_y^-1;

y_small=my_PG(:,k)+sig_R*randn(no,1);

y_smallb=my_QG(:,k)+sig_R*randn(no,1);

y_v=my_abs_v(:,k)+sig_R*randn(nd,1);

y_theta=my_theta_v(:,k)+sig_R*randn(nd,1);

y=[y_small; y_v;y_theta;y_smallb];


my_dp=my_xp(no+1:2*no,:);

my_eqp=my_xp(2*no+1:3*no,:);

my_edp=my_xp(3*no+1:4*no,:);

my_efp=my_xp(4*no+1:5*no,:);

my_RFp=my_xp(5*no+1:6*no,:);

my_VRp=my_xp(6*no+1:7*no,:);

my_vp=my_xp(7*no+1:7*no+nd,:);

my_thetp=my_xp(7*no+nd+1:7*no+2*nd,:);

my_PM0p=my_xp(7*no+2*nd+1:8*no+2*nd,:);

P_P=P_M-K*P_y*K';

end

79

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