Improving Low Voltage Ride-Through Requirements (LVRT) Based on Hybrid PMU, Conventional Measurements in Wind Power Systems

Förbättra Lågspänning Rider Genom Krav (LVDT) Baserat på Hybrid PMU, Konventionella Mätningar i Vindkraftsystemet

Chinedum John C. Ekechukwu

Faculty of Health, Science and Technology Master’s program in Electrical Engineering

Degree Project of 30 credit points

Internal Supervisor: Arild Moldsvor, Karlstad University, Sweden

Examiner: Jorge Solis, Karlstad University, Sweden

Date:12-02-2014

Serial Number:

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Abstract

Previously, conventional state estimation techniques have been used for state estimation in power

systems. These conventional methods are based on steady state models. As a result of this, power

system dynamics during disturbances or transient conditions are not adequately captured. This

makes it challenging for operators in control centers to perform visual tracking of the system, proper

fault diagnosis and even take adequate preemptive control measures to ensure system stability

during voltage dips. Another challenge is that power systems are nonlinear in nature. There are

multiple power components in operation at any given time making the system highly dynamic in

nature. Consequently, the need to study and implement better dynamic estimation tools that

capture system dynamics during disturbances and transient conditions is necessary.

For this thesis work, we present the Unscented Kalman Filter (UKF) algorithm which integrates

Unscented Transformation (UT) to Kalman filtering. Our algorithm takes as input the output of a

synchronous machine modeled in MATLAB/Simulink as well as data from a PMU device assumed to

be installed at the terminal bus of the synchronous machine, and estimate the dynamic states of the

system using a Kalman Filter. We have presented a detailed and analytical study of our proposed

algorithm in estimating two dynamic states of the synchronous machine, rotor angle and rotor

speed. Our study and results shows that our proposed methodology has better efficiency when

compared to the results of the Extended Kalman Filter (EKF) algorithm in estimating dynamic states

of a power system.

Our results are presented and analyzed on the basis of how accurately the algorithm estimates the

system states following various simulated transient and small-signal disturbances.

Keywords: State estimation, Power systems, Unscented Kalman filters, Phasor Measurement units

(PMUs)

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Acknowledgments

This thesis has been made possible because of the invaluable support, guidance and encouragement

that I have received from a number of individuals.

Without listing such persons in any particular order, I thank you my supervisor, Professor Arild

Moldsvor who has been supportive through the period during which I have been writing this thesis. I

also want to thank my examiner, Professor Jorge Solis, for his contributions toward successful

completion of this work. Thanks to all the professors, staff and students of Karlstad University who

have influenced me in one way or another while I have been studying at Karlstad University.

Special thanks and profound gratitude to the following people: my parents, Engr. and Mrs. Lucius

Ekechukwu who have given me immeasurable support and encouragement beyond words; my

siblings, my adorable Loretta and my brother in-law; Engr. Chibike Nnorom. You all have encouraged

and supported me in every way possible. Your various roles in my life, before and during my studies

as well as through the development of my career thus far, continue to humble me. Words fail me in

expressing my gratitude.

I would also love to thank my friends, both in Karlstad and around the world. For those in Karlstad,

thank you all for making my transition a smooth one when I arrived to Sweden initially. You all

remained very supportive.

Above all, I will forever bless God and continue to thank Him for my life.

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Contents

Abstract .................................................................................................................................................... i

Acknowledgments ....................................................................................................................................ii

List of Figures ............................................................................................................................................ v

List of Abbreviations ................................................................................................................................ vi

1 Introduction ..................................................................................................................................... 1

1.1 Project Motivation ................................................................................................................... 1

1.2 Problem definition ................................................................................................................... 2

1.3 Kalman Filtering ....................................................................................................................... 2

1.4 Main Contribution and objective ............................................................................................ 3

1.5 Thesis outline........................................................................................................................... 4

2 Wind Power Systems ....................................................................................................................... 5

2.1 Background .............................................................................................................................. 5

2.2 Wind power systems requirements and limitations ............................................................... 5

2.2.1 Low Voltage Ride through (LVRT) .................................................................................... 5

2.2.2 Power System Stability .................................................................................................... 7

2.3 Modeling the wind power system ........................................................................................... 9

3 Measurement acquisition methods in power systems ................................................................. 17

3.1 Conventional/SCADA measurements .................................................................................... 18

3.1.1 Measurement model and assumptions ......................................................................... 19

3.2 Synchronized Phasor Measurements Units (PMUs) .............................................................. 21

3.2.1 Measurement model and assumptions ......................................................................... 21

3.2 Mixed (PMU and Conventional/SCADA) measurements ...................................................... 24

4 State Estimation ............................................................................................................................ 26

4.1 Static State Estimation (SSE) ................................................................................................. 27

4.2 Forecasting Aided State Estimation (FASE) ........................................................................... 28

4.3 Incorporating measurements to FASE ................................................................................... 31

5 Proposed methodology ................................................................................................................. 32

5.1 Unscented Transformation (UT) ............................................................................................ 32

5.2 The Unscented Kalman Filter (UKF) ....................................................................................... 34

5.3 UKF based state estimation in power systems (UKF/SE) ...................................................... 36

6 Simulations and Results ................................................................................................................. 39

6.1 Implementation of our proposed methodology ................................................................... 39

6.2 Results ................................................................................................................................... 45

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6.3 Discussion .............................................................................................................................. 49

7 Conclusions and future work ......................................................................................................... 52

7.1 Conclusions ............................................................................................................................ 52

7.2 Future work ........................................................................................................................... 52

References ............................................................................................................................................. 54

Appendix A: Holt’s two initialization ..................................................................................................... 57

Appendix B: Gradient calculation for EKF algorithm method ............................................................... 58

Appendix C: Machine parameters used and states to be estimated .................................................... 60

Appendix D: UKF algorithm ................................................................................................................... 61

v

List of Figures

Fig. 2.1: Required ride through capability of wind farms for Super grid Voltage dips of

duration greater than 140ms ................................................................................................................ 6

Fig. 2.2: Single machine infinite bus system ..................................................................................... 10

Fig. 2.3: Schematic description of the powers and torque in synchronous machines ........................ 11

Fig. 3.1: Elements of the EMS/SCADA system ..................................................................................... 18

Fig. 3.2: Two-port π-model of transmission line ................................................................................. 20

Fig. 3.3: Phasor illustration .................................................................................................................. 21

Fig. 3.4: Hierarchical placement of phasor measurement units. ........................................................ 22

Fig. 3.5: Two-port π network. .............................................................................................................. 23

Fig. 5.1: Unscented Transformation, showing sigma points being transformed.......................... 32

Fig. 5.2: Unscented Transform (UT) block diagram ............................................................................. 34

Fig. 6.1: Proposed block diagram of UKF estimator ............................................................................ 39

Fig. 6.2: Top level of the Implmentation block in Simulink ............................................................ 40

Fig. 6.3: Layout of the synchronous machine connected to the transmission line ............................. 39

Fig. 6.4: Mechanical part Sub-model of the synchronous machine .................................................... 40

Fig. 6.5: Electrical part Sub-model of the synchronous machine ........................................................ 42

Fig. 6.6: Powergui machine initialization tool ..................................................................................... 44

Fig. 6.7: Output power Pt measured at the terminal bus.................................................................... 45

Fig. 6.8: Generator rotor angle ............................................................................................................ 46

Fig. 6.9: Generator rotor speed .......................................................................................................... 46

Fig. 6.10(a): Real rotor angle Vs estimated angle using the UKF algorithm ........................................ 47

Fig. 6.10(b): Real rotor angle Vs estimated angle using the EKF algorithm ........................................ 47

Fig. 6.11(a): Real rotor speed Vs estimated speed using the UKF algorithm ...................................... 48

Fig. 6.11(b): Real rotor speed Vs estimated speed using the EKF algorithm ....................................... 48

Fig. 6.12(a): Rotor angle response with simulated fault cleared in 0.07s for the UKF algorithm ....... 48

Fig. 6.12(b): Rotor angle response with simulated fault cleared in 0.07s for the EKF algorithm ........ 49

Fig. 6.13: Efd Step input ........................................................................................................................ 50

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List of Abbreviations

AMI Advanced Metering Infrastructure

CPU Central Processing Unit

DGs Distributed Generators

DS Dynamic State

DSE Dynamic State Estimators

DR Distributed Resources

DFIG Doubly Fed Induction Generator

EMS Energy Management Systems

EMS Energy Management Systems

EKF Extended Kalman Filter

FASE Forecasting-Aided State Estimation

GPS Global Positioning System

GLFS Generation and Load Forecast System

HVDC High Voltage Direct Current

LVRT Low Voltage Ride through

MASE Multi Area State Estimation

OPF Optimal Power-Flow

ODEs Ordinary Differential Equations

PMUs Phasor Measuring Units

PDF Probability Density Function

RTUs Remote Terminal Units

SMIB Single Machine Infinite Bus

SV State Variable

SCADA Supervisory Control and Data Acquisition

SEA State Estimation Algorithms

SSE Static State Estimation

SR-UKF Square-Root Unscented Kalman Filter

UKF Unscented Kalman Filter

UT Unscented Transformation

WLS Weighted Least Squared

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1 Introduction

With the emergence of distributed generation and the need to tie distributed generators (DGs) to the

smart grid while maintaining power systems stability, reliability and security continues to be of great

concern in today’s energy development across the world. An important angle to this requires

adequate and continuous knowledge of the power systems state. Energy management systems

(EMS) play significant roles in obtaining system state through monitoring and state estimation. State

estimations in the past have been performed by static approach which is based on the weighted least

squared (WLS) method. Although this approach is simple to implement and has fast convergence

since it uses a single set of measurement, its ability to predict future operating points of the system

remains limited. These have led to several cases of severe system instability and in some cases

complete system collapse or blackout, as operators could not foresee impending system

contingencies and risks to the system. Secondly, such static estimators must be reinitialized for every

new measurement without using predictions from previous state estimators [1]. In addition, the

quasi steady-state nature of the system network due to slow dynamic load changes makes state

estimation quite a challenging task. For this reason, estimators which take into account the changing

dynamics of the system are required and have been developed for state estimation in recent times.

Such estimators are called Dynamic State Estimators (DSE). DSEs have the ability to predict the

systems state vector one step ahead, providing a foresight to potential contingencies and security

risks. They also take multiple samples of measurements which provide fast and accurate estimation

for each time sample [1]. In addition, they provide state estimates at a particular time instant based

on past and present measurements.

1.1 Project Motivation

The motivation for this thesis is to investigate how these DSE are used for power system estimation

on a broad view and more closely for the estimation of the synchronous generator states, which is

presented in this thesis as a part of the power system.

State estimation requires adequate and accurate measurements of system parameters and or states

that can be measured so as to facilitate estimation of states which cannot be directly measured.

Measuring devices such as Phasor Measuring Units (PMUs), which were developed in the early 80s to

supplement measurements provided by conventional means, have greatly improved state estimation

over time considering they offer near real-time monitoring of the system. Their measurements are

synchronized through the GPS space-based satellite navigation system. They have added a new

dimension to state estimation.

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We are also motivated to investigate and present how hybrid PMUs, Conventional measurements

can be integrated to our proposed forecasted estimation scheme, the Unscented Kalman Filter (UKF),

for state estimation.

1.2 Problem definition

The main task to be accomplished is to estimate the rotor angle and rotor speed of the synchronous

generator modeled in MATLAB/Simulink with a single machine infinite bus system (SMIB).

Considering that all states cannot be directly measured, we are saddled with the responsibility of

developing a state estimation algorithm as well as filtering out the synchronous machine’s rotor

angle and rotor speed in the presence of simulated small disturbances and transient faults. Thus,

given a number of control inputs and measurable observations, this thesis work focuses on

estimating the dynamic states of the synchronous machine during simulated system disturbances.

1.3 Kalman Filtering

In this section, we present an overview of the Kalman filter so as to lay a foundation for better

understanding of the proposed filtering method and to show the motivation for the choice of the

proposed methodology.

Kalman filters have remained one of the best filtering methods for estimating dynamic states in a

dynamic system using noisy measurements. They are considered to be recursive filters. They are

known as linear quadratic estimators and when mixed with linear quadratic regulator, they solve

linear quadratic Gaussian control problems. The Kalman filter is named after Rudolph E. Kalman, who

published his famous paper describing a recursive solution to the discrete-data linear filtering

problem in 1960. Kalman filtering was then used in the 70’s when the term “dynamic state

estimation” was first introduced [2], [3] in combination with a steady state power system, to track

the static states, bus voltages and phase angle. Kalman filters were used to improve the

computational performance of the traditional steady state estimation process used for power system

applications.

Kalman filters are a set of mathematical equations used to implement a predictor-corrector type

estimator which is optimal considering that it minimizes the estimated error covariance when some

assumed conditions are met [4]. Kalman filtering is generally divided into two steps: Predict and

Update steps. The predict step, also known as time update, uses previous state estimate to predict

present estimates of the state variable. It is responsible for forward projection in time of the state

estimate so as to obtain a priori estimate for the next time step. It is considered the predictor

equation in the predictor-corrector relationship. The update step which is also called the

measurement update, utilizes measurements obtained at the present time step to correct the

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estimated state variables. They are responsible for the feedback, i.e., incorporating the new

measurements into the a priori estimate to obtain an improved a posteriori estimate. It is considered

the corrector equation.

Unlike the Weiner filter which operates on all data directly for each estimate, the Kalman filter

repeats the process after each time and measurement update, using the previous a posteriori

estimates used to predict the new a priori estimate. This recursive nature of the Kalman filter makes

it a unique filtering method.

Considering that most dynamic processes in power systems are non- linear in nature, there is the

need for a filter which estimates these non-linear processes. The Extended Kaman filter (EKF) can be

used in place of the discrete Kalman filter considering its ability to linearize about the current mean.

Assuming now that the process is no longer governed by linear equations but by non-linear

stochastic differential equations [4], the nonlinear functions thus relates the states at time instant

say � − 1 to states at the present time �. The nonlinear measurement functions relate the states at

time instant � to measurements at time instant �. In real time estimation or practice, the value of

the noise is of course not known. As such, approximate values of the state and measurement vector

can be computed.

However, the Jacobian matrices of EKF filters are different for each time step and as such should be

recomputed at each time step. Of particular interest is the fact that the EKF losses the distributions of

its random variables after transformation due to linearization. Although the EKF is still a useful tool

for filtering purposes, the novel Unscented Kalman filter (UKF) which is a more iteratively accurate

tool for filtering has been proposed to be used for this thesis work. It is known for the fact that it

preserves the distributions throughout the nonlinear transformation. The UKF is an efficient discrete-

time recursive filter. It is based on unscented transformation (UT). Unlike the EKF, the UKF does not

linearize the non-linear equations giving it a major advantage over the EKF.

1.4 Main Contribution and objective

We have modeled the power system in MATLAB/Simulink. As we have just previously established,

the proposed UKF algorithm is built upon the Kalman filtering process. We have made modifications

to the standard UKF algorithm to perform dynamic state estimation in power systems as is the focus

in this thesis work. Our algorithm tracks the changes in the state vector of an SMIB system while we

simulate disturbances in the system. In contrast to a similar prior publication in [6], we have provided

a second order model of the synchronous machine using MATLAB/Simulink. We have also presented

a detailed description of the implementation of the UKF algorithm using MATLAB’s embedded

function block. A rigorous description of the algorithm’s codes is presented and two dynamic states

are estimated based on the second order model.

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Incidentally, many papers have treated power systems state estimation but not much emphasis have

been placed on the dynamic states. We thus seek to apply the proposed algorithm to state

estimation in dynamical systems and compare its efficiency to the EKF filtering algorithm [7]. Our

results will show that the proposed algorithm is very relevant for stability analysis in wind energy

systems and today’s smart grid developments.

1.5 Thesis outline

The remaining chapters of the thesis are as follows: In chapter 2, we discuss wind power systems

with emphasis on conditions and requirements necessary for as well as limitations in integrating

wind driven systems to the smart grid. We also define and discuss stability conditions and finally

model the wind power system. In chapter 3, we will discuss some measuring techniques and devices.

In chapter 4, we will present state estimation, giving background knowledge on the evolution of state

estimation. We will also introduce the FASE system. In chapter 5, we will discuss our proposed

methodology. In Chapter 6 we will present our simulations and results as well as discussions. We

conclude in chapter 7 with future works and conclusions.

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2 Wind Power Systems

2.1 Background

The need to meet the challenging demands for sustainable and reliable energy across the globe has

kept engineers and scientist on continuous research for new and improved energy resources. This

constant search led to the discovery of Distributed Resources (DR) such as wind, solar photo-voltaic,

solar thermal, small hydro, micro-turbines etc and energy storage systems. With the emergence of

such improved energy generation technologies came the challenge of improving control measures to

maintain power system stability and security during interconnections. This necessitated the need for

stringent installation and connection requirements and conditions for the wind farms.

2.2 Wind power systems requirements and limitations

Wind energy installation requirements are summarized in [8], where the impact of the wind

installations on the power grid is discussed. Recommendations for power quality limits that need not

be exceeded during the operation of wind installations are also discussed in that literature. However,

we will dive a little into such requirements and demands for the purpose of the background

knowledge needed for our work.

Another important aspect of wind power system involves proper synchronization of these wind

installations to the grid without compromising system stability and security. In [9], a proposed

definition for power system stability states “Power system stability is the ability of an electric power

system, for a given initial operating condition, to regain a state of operating equilibrium after being

subjected to a physical disturbance, with most system variables bounded so that practically the

entire system remains intact.” We will give further discussions on stability in a later paragraph.

2.2.1 Low Voltage Ride through (LVRT)

Recent grid codes require that wind farms remain connected to the grid during severe grid

disturbances, ensuring fast restoration of active power to the pre-fault level as soon as the fault is

cleared and in certain cases produce reactive current in order to support grid voltage during

disturbances [10]. During such disturbances or faults on the grid, there are usually voltage dips,

depending on the severity of such faults. These codes require wind farms, especially those connected

to HV grids, to be able to withstand voltage dips to a certain percentage of the nominal voltage for a

specified duration. Such fault ride through requirements is known as Low Voltage Ride through

(LVRT) requirements.

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Voltage dips resulting from transient faults inevitably cause torque and power transient stability

problems in the wind turbine, for instance that of the DFIG, due to the asynchronous nature of the

turbines. These problems do not depend on the rotor angle but on voltage issues and fault ride

through capability of the generators. During these faults, the short circuit current contributions of

the wind generators are low. In the past, it was acceptable to have wind turbines disconnect during

faults in the network. However, the impact of such disconnections on system stability has remained

adverse. The required fault ride through behavior of a wind farm can be summarized into four

requirements [10]:

1) For system faults that last up to 140 ms the wind farm has to remain connected to the network.

For super grid voltage dips of duration greater than 140 ms the wind farm has to remain

connected to the system for any dip-duration on or above the heavy black line shown in Figure

2.1.

2) During system faults and voltage sags, a wind farm has to supply maximum reactive current to

the Grid System without exceeding the transient rating of the plant.

3) For system faults that last up to 140ms, upon the restoration of voltage to 90% of nominal, a

wind farm has to supply active power to at least 90% of its pre-fault value within 0.5 sec. For

voltage dips of duration greater than 140 ms, a wind farm has to supply active power to at least

90% of its pre-fault value within 1 sec of restoration of voltage to 90% of nominal.

4) During voltage dips lasting more than 140ms the active power output of a wind farm has to be

retained at least in proportion to the retained balanced super grid voltage.

Fig. 2.1: Required ride through capability of wind farms for Super grid

Voltage dips of duration greater than 140ms

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However, for scenarios where less than 5% of the turbines are running or under very high wind speed

conditions where more than 50% of the turbines have been shut down, a wind farm is permitted to

trip or go off the grid.

2.2.2 Power System Stability

The major analytical tools for supervisory, control and planning of the electric power grid are the

power-flow analysis and stability programs. Power-flow programs include conventional power-flow

analysis and optimal power-flow (OPF) analysis, to mention a few. However, in this sub-chapter we

will concentrate on the stability programs. The need to maintain power system stability, during or off

fault conditions, remains an integral part of the power systems generation, transmission and

distribution schemes. The behaviour of the system following a disturbance is of interest in stability

analysis programs. Studies of this behaviour are called transient stability analysis.

To achieve a better overview and structure of stability analysis of power systems, it is important to

categorise stability into two main categories [11]: small signal (or small-disturbance) stability and

transient stability. Small signal stability is the ability of the system to remain in synchronism under

small disturbances. These disturbances are as a result of small variations in the loads and

generations. Instability that may result in this case is: 1) steady increase in the rotor angle as a result

of insufficient synchronizing torque, or 2) increased amplitude of rotor oscillations as a result of

insufficient damping torque. Transient stability is the ability of the system to remain in synchronism

under severe transient disturbances. Instability that may occur involves large excursions of the

generator rotor angles resulting from the nonlinear power-angle relationship of the rotors of the

synchronous machine. Thus, we can define stability in this sense as the ability of the system

machines to recover from disturbances and still maintain synchronism.

In analysing disturbances, the contingencies used are the various types of short circuits: phase-to-

ground, phase-to-phase-to-ground, or three-phase. These contingencies will be used to analyse the

estimation method proposed to be used for this in a later chapter. Categorising stability into the two

mentioned categories makes analysis easier as well as understanding the nature of stability

challenges or problems.

As earlier mentioned, the dynamic condition of the power system can be initiated by disturbances in

the system. Considering the nature of the parameters, such disturbances could be as a result of

changes in the line impedance during short circuit on a transmission line, switching on of a large

block of loads or opening of a line. However, the dynamic behavior of the system after disturbance

depends on how large such disturbances are, considering that the power system is nonlinear. The

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following comments from [10] give further insights to some crucial aspects of the definition of

stability earlier given:

1) It is not necessary that the system re-gains the same steady state operating equilibrium as

prior to the disturbance. This would be the case when for instance; the disturbance has

caused any power system component (line, generator, etc.) to trip. Voltages and power flows

will not remain the same after the disturbance due to changes in the system topology or

structure.

2) It is important that the final steady state operating equilibrium after the fault is acceptable.

Else, protection or control actions could introduce new disturbances that might further

influence the stability of the system. Acceptable operating conditions must be clearly defined

for the power system under study.

Power system stability can thus be discussed under three main classifications: Rotor Angular or

Synchronous Stability, Frequency Stability and Voltage Stability.

2.2.2.1 Rotor angular or synchronous stability

The sum of the total power output fed into the power system (or infinite bus system as is used to

model the power system in this work) is equal to the sum of the active power consumed by the

loads, including losses in the system. However, the power fed into the generators by their prime

movers, such as the wind, steam, gas or hydro turbines are not always equal to the power consumed

by the loads. If such an imbalance occurs, perhaps due to a three phase fault, the rotating parts of

the generator act as energy buffers and thus increase or decrease their kinetic energy. This increase

or decrease could cause the generators to fall out of synchrony since they have been accelerated

during the fault, depending on how close the fault is to the generator or how severe the disturbance

or fault is. Rotor angle stability is thus the ability for the synchronous machines of a power system to

maintain synchrony after a disturbance to the system.

2.2.2.2 Frequency stability

In comparison to the previously described scenario for rotor angular instability, if the power fed from

the prime movers is less that that consumed by the loads, including losses, such imbalance results in

frequency instability in the system. The kinetic energy stored in the rotating parts of the synchronous

machine and other rotating electrical machines will compensate for the frequency imbalance by

regulating the active power input from the prime movers for cases where the imbalance is not too

large. For large deviations in the frequency imbalance, the generators in most cases are cut off

because such imbalances cause detrimental oscillations in the turbines. This is why in most control

rooms in power generating stations, the system frequency readings are usually displayed boldly and

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are closely monitored. This form of instability is termed frequency instability and it plays an

important role in power systems generation and transmission.

2.2.2.3 Voltage stability

We have just stated in the previous subchapter that the generators are very important for the

analysis of angular instability. It is often times said that these are the driving force in instability.

However, more detailed studies and analysis show that the loads on the system are more often the

driving force in terms of voltage instability.

2.3 Modeling the wind power system

In modeling the wind power systems, it is almost impossible to develop models that perfectly

describe all dynamics of a power system and be sure that they can be used practically. This is true for

two reasons: first, such models will require so much uniquely defined parameter data for the whole

system. Hence, only models that capture adequately the specific system dynamics and interaction to

be analyzed or investigated are used. Secondly, analysis of such a system would result in great or

huge amount of computational work and may be almost impossible. The ability to make informed

analysis, review and take necessary actions is an integral part of power systems engineering. It is

important to have as precise as possible, input information for power system analysis softwares so as

to ensure little or no errors in system modeling and design. When modeling the power system or any

other system, care should be taken such that the models chosen are not too simple, as such,

misrepresenting the interactions and processes within the system. It is also very detrimental if wrong

parameter data are used in modeling the system. In principle, these design parameter data

describing the system should be available for soft ware developers. But in reality, there are

parameters that may be “genuinely” unknown, for instance, the ground resistivity under a power

line. The importance of having adequate and sufficient parameter data for engineering systems

modeling and design cannot be overemphasized and as such it could be very important, though

costly in practice, to keep and update data bases for these parameters. Today, such practice is called

data engineering.

The working principle of the wind farm requires that the wind is harnessed through the generator’s

rotor blades, which are connected to the wind turbine through a shaft. This mechanical energy is

subsequently converted to electrical energy using electrical generators. Energy is transferred through

the grid transmission lines and subsequently distributed for consumption. For the sake of

simplification, we will assume a variable speed wind turbine generator model based on the DFIG

technology and assume its state estimates to be a part of those of a larger power system. The order

of the synchronous generator model used determines the number of states to be estimated and of

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course the level of complexity of the estimation process. Several orders of a generator model have

been proposed and discussed by various authors. In [12], a third order SMIB model is used. In [13], a

sixth-order power system assuming a third order model for the synchronous machine was used. In

[14], a fourth-order nonlinear model of the synchronous machine was used. In [6], a fourth-order

nonlinear model for the synchronous generator was presented. Our contribution to this thesis is

observed in our choice of a second-order classical model for the synchronous generator, in contrast

to the previously proposed model orders. Compared to the model order presented in [6] and other

higher order models presented in other publications, the effect of damper windings and stator

dynamics are neglected in our work considering that we are not interested in a very fast dynamic

(sub-transient) system. The choice of a second-order model and hence the number of states to be

estimated is motivated by the fact that an increase in the order of the overall system model indicates

that the size of the system that can be simulated is limited. In other words, in increasing the order of

the overall system model, the size of the system that can be simulated is limited [11]. So with a

minimal order, we do not place a limit on the size of the overall system that can be simulated.

Power systems analysis in recent times has been based on two kinds of machine bus systems. They

include the Single Machine Infinite Bus (SMIB) system and the Multi machine infinite Bus (MMIB)

system. The SMIB is subdivided into classical and detailed models. For the purpose of simplification,

we will use the classical model to represent the generator connected to an infinite bus in an SMIB

power system. The concepts developed for the SMIB system can be easily applied to a two machines

or Multi machine power system as well when they are reduced to a SMIB system. Considering the

SMIB system described with figure 2.2, we will assume that we are not interested in very fast

dynamic (sub-transient) conditions and as such, neglect the damper winding and stator dynamics of

the synchronous generator. But the effect of the damper winding is considered in the rotor-damping

factor �[11]. Hence for this study, modeling of the generator will be done considering transient

conditions.

Fig. 2.2: Single machine infinite bus system

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Power system transient stability assessment involves evaluating the system’s ability to remain

synchronized after undergoing disturbances as well as proposing adequate remedial actions when

there is the need for such [15]. In transient stability assessment, state variables (SV) associated with

transient stability can be divided into

• Machine state variables, which has a minimum of two mechanical SVs, machine rotor angle �

and machine rotor speed �, and at least six electrical SVs, among other variables;

• Load state variables, which include static loads and dynamic loads SVs and;

• Special devices such as FACTS,HVDC links ,SVCs, SVs

As such, for real-time transient stability monitoring and control, the rotor angle and speed SVs play

very important roles. If they can be considerably and accurately estimated, they can be exploited to

monitor real time loss of synchrony and devise automatic closed loop stabilization schemes [16].

We will begin modeling the synchronous machine by first defining a few important parameters of the

machine model. The total moment of inertia of the synchronous machine, , is the sum of all

moments of inertia of all rotating parts of a synchronous machine, i.e., moments of inertia of the

rotor, turbines, shafts etc. The inertia constant , of a synchronous machine [17] is thus described as

(2.1)

= 0.5����� (2.1) Where the numerator is the total kinetic energy stored in the synchronous machine at steady state

and � is the MVA rating of the machine. The inertia constant is measured in seconds. It gives us the

time taken to bring the machine from its synchronous speed to a standstill if the rated power is

extracted from it with no mechanical power fed in. Its value varies within a much smaller range than

the value of for different machines.

As we mentioned earlier, in steady state the synchronous machines rotate with similar electrical

angular velocities but during disturbances, the generators kinetic energy tend to increase or decrease

and as such they lose synchrony. Figure 2.3 [17] illustrates the electromechanical description of the

synchronous machines:

Fig. 2.3: Schematic description of the powers and torque in synchronous machines

12

The rotor dynamics for the synchronous machine can be described with [17] the second order

ordinary differential equation (ODE) presented in (2.2)

������� = �� − ��

(2.2) �� ������� = �� − ��

where

�� = ���� is the mechanical power acting on the rotor (W) �� = ���� is the electrical power acting on the rotor (W) �� = = Inertia constant.

Both equations of (2.2) are alternative forms of each other where the second equation of (2.2), the

power form, is obtained by multiplying both sides of the first part of (2.2) by ��. Higher order ODEs

can be written as first order ODEs and are used for describing multi machine systems.

Using the classical model for the synchronous machine (generator), which ignores the saliency of the

round rotor, only the quadrature transient reactance ��� is considered, assuming that the direct and

quadrature components are equal. We will also assume the transmission line resistance,� is zero (0).

Hence the total active power from the generator is delivered to the infinite bus, i.e. �� = � . The

generator’s voltage is represented by !� while the infinite bus voltage is represented with ! . � is

the rotor angle which represents the angle by which generator voltage !� leads the infinite bus

voltage ! . It can also be described as the angle by which the "-axis component of the internal

generator voltage !�� leads the terminal bus voltage of the machine !#($%&#). For the purpose of

clarity, we will use &# from here on.

�� = '()* !�+,-� (2.3)

Whenever the system experiences disturbance, the magnitude of the generator voltage !� remains

constant at its pre-disturbance value while the rotor angle � changes as the generator rotor speed �

deviates from synchronous speed �.. Thus we can introduce the relation �/ = ∆�. If a detailed

model is to be considered, the field coil on the direct axis (d-axis) and damper coil on the quadrature

axis (q-axis) are used. �1� is the d-axis transient reactance. Taking &# to be the reference phasor, a

third order nonlinear differential equation [18] can be used to describe the synchronous machine

connected to an infinite bus through two parallel transmission lines, each with impedance � as

described in figure 2.3.

13

However, since we are considering the classical model, we can thus introduce the relation �/ = ∆�

to (2.2) and re-write it as (2.4): �/ = � ∆�

(2.4) ∆�/ = 12 (�� − �� −�∆�)

where � is the damping factor. �/ = � ∆� shows the relationship between the relative angular

velocity and the synchronous rotating system �. This relative angular velocity is of interest when

rotor oscillations are being studied while the absolute angular velocity is of interest when studying

frequency stability. We can rewrite (2.4) in terms of system states and inputs as (2.5a) and (2.5b):

2 = [�∆�]5 = [2/62/�]5

(2.5a)

7 = [�� !81]5 = [767�]5

as such,

2/6 = � 2�

(2.5b)

2/� = 6�9 (76 − �� −�2�) where

� is the generator rotor angle (first state),

� is the machine nominal or base synchronous speed

∆� is the generator rotor speed in pu (second state),

is inertia constant

�� is the electrical output torque in pu,

�� is the mechanical input torque from the turbine in pu,

!81 is the control voltage or exciter output voltage as seen from the armature. It is in

pu.

The electrical power output �� will be equal to the terminal electric power �# measured at the

generator bus if we assume the stator resistance to be zero (�: = 0) and �; = 1pu. Thus,

�� = �# + �:=#�

(2.6)

�: = 0, �� ≅�# = !1,1 + !�,�

where !1 @-�!� are the internal �-and "-axis voltages of the machine respectively.

14

The rotor angle � and the rotor speed � are available measurements. However the dynamics of the

internal voltages !1 @-�!� cannot be measured because they are “lumped variables that

aggregates many voltage dynamics around the machine principal axis” [19]. However, pseudo

measurements technology can be used to acquire such variables. For instance the output voltage &# measured at the terminal bus and the injected current to the generator are used to solve some

observation equations which are nonlinear functions of these variables. Thus using measurable rotor

angle �, output voltage &# can be obtained[6]:

!1 = &#+,-�

(2.7a)

!� = &#A$+�

Solving further, we get

!# =&# = B!1� + !��. (2.7b)

Also, the �-and "-axis currents are given as:

,1 = !�′ − &# cos ��1′

(2.8a) ,� = &# sin ���

Where !�� is assumed to be measureable. After further calculations, we get

=# = B,1� + ,��. (2.8b)

Substituting the variables !�� and � in (2.7) with the states as described in (2.5a), we obtain

,1 = 2H − &# cos 26�1�

(2.9) ,� = &# sin 26��

Substituting &# and =# in (2.6) and solving, we obtain the electrical output power �# at the terminal

bus as (2.10)

�# ≅ &#�1� !�� +,-� + &#�2 I 1�� − 1�1� J sin 2�(2.10)

In terms of the states, �# is re-written as

15

�# ≅ &#�1� 2H+,-26 + &#�2 I 1�� − 1�1� J sin 226 (2.11)

Using (2.5a), (2.5b), (2.9) and (2.11), the vectors for the state and input parameters are described in

equation (2.12a) as:

2 = K262�L = M�/�/ N ,7 = K767�L = M��!81N (2.12a)

The second order dynamic model for the synchronous machine can is thus described with (2.12b) as 2/6 = � 2�

2/� = 12 (76 − (&#�1� 2H+,-26 + &#�2 I 1�� − 1�1� J sin 226) − �2�) O = &#�1� 2H+,-26 + &#�2 I 1�� − 1�1� J sin 226 (2.12P)

A compact representation for the dynamic model described above can be summarized as,

Q 2/6 = R6(2, 7)2/� = R�(2, 7)O6 =ℎ6(2, 7)T T−−U V2/ = R(2, 7, ")W = ℎ(2, 7, %) T (2.13)

The terminal bus signals �#, X# and &#, used for online state estimation can be measured directly

from the PMUs placed at the generator bus terminal. Considering that the PMU measures the

voltage magnitude and angle as well as the line injection and flow current magnitudes and angles,

these quantities are thus used to calculate the terminal bus signals which in turn are used as inputs

for the UKF algorithm proposed to be used for this thesis. As will be seen later, the output

measurement �# is directly fed to our embedded MATLAB function block as an input for the UKF

algorithm.

We had earlier said that another synchronous machine model used for power network studies was

the multi-machine classical model. In this model the generators are represented by constant voltage

behind transient reactance, constant impedance loads etc. When uniform damping is considered for

this model, the generators motion is described with the following set of equations [20]:

YZ�[/ Z = −�Z�[Z + �Z − ��Z − \]\^ �_`a

(2.14a)

b/Z = �[Z , , = 1,2, … , -

16

where the acronym COI means center of inertia. When damping is not considered, the motion of the

system is described as

YZ�[/ Z = �Z − ��Z − \]\^ �_`a

(2.14b)

b/Z = �[Z , , = 1,2,… , -

where bZ is the displacement angle and �[Z is the angular velocity described as bZ = �Z − �.

and

�[Z = �Z − �. respectively,

where

�Z is angle of voltage behind transient reactance, indicative of generator rotor

position,

�Z is rotor speed,

YZ is generator inertia constant,

�Z is damping coefficient.

As mentioned earlier, for the purpose of this thesis, we will be using the classical model of the single-

machine infinite bus (SMIB) system to represent the power system. As such, detailed study of other

models, which we have briefly described in this work, can be done with the given references.

When we discuss dynamic state estimation in power system, the dynamic states of the system are

considered rather than the static states such as the voltage magnitude and angle at the buses. There

are of course several dynamic states in a power system. The choice of the states to be estimated also

depends on the scope or extent of the period of study and analysis to be done. The states are

represented by a set of non-linear differential equations and an increase in the number of equations

increases the order of the overall system model. Recalling that an increase in the order of the overall

system model limits the size of the system that can be simulated, we are hence motivated to limit

our studies to two dynamic states; the generator rotor angle and generator speed. Our objective in

dynamic state estimation is to estimate the system states using a second order, highly non-linear

differential-algebraic equation representation of the system, which has been earlier presented in this

chapter.

In chapter four, we give an overview of state estimation and then in chapter five we present our

proposed methodology. But before these, it is imperative that we discuss some measurement

acquisition methods in power system.

17

3 Measurement acquisition methods in power systems

Monitoring the power system is done by obtaining and processing measurements from the system.

These measurements are used to obtain measurement models. Measurement models show the

relationship between the measurements obtained from the system and the estimated or calculated

variables. These variables are divided into state variables and dependent variables. State variables

are a part of the set of problem variables which describe the network. These variables are connected

by a set of network equations, called network model. The network model expresses the relationship

between branch active power flows and bus voltage angle. They also show the active power balance

at network busses. State variables describe the system completely, i.e., if the states are known, the

remaining dependent variables can be calculated using the network model equations. These set of

state variables is minimum. This implies that if they are removed, the other dependent variables

cannot be calculated. In other words, when operators have correct network model gotten via

measurements obtained, the system is monitored adequately.

Voltage magnitudes and phase angles enable us effectively calculate all other quantities such as

power flows, loads, generations etc. As such, voltage magnitudes and their phase angles are

considered state variables. The measurements used to estimate these state variables could be

corrupted as a result of faulty sensors or measuring devices. As such, there is the need to make the

“best” guess for the state variables given the noisy measurements. Traditionally, for static systems,

the Weighted Least square (WLS) technique is used for this task where the system is linear (non-

iterative). In order to run the state estimator, we must know how the transmission lines are

connected to the load and generation buses. We call this information the network topology or

configuration. Since the breakers and disconnect switches in any substation can cause the network

topology to change, a program must be provided that reads the telemetric breaker/switch status

indications and restructures the electrical model of the system. Proper knowledge of the systems

network topology improves power system monitoring, as operating conditions of the system at any

given time is known when the network model and complex phasor voltages of every system bus are

known [21]. The set of complex voltages is called the static state of the system since it fully specifies

the system. The state estimation function is thus responsible for monitoring the system state. It does

this by processing redundant measurements to get “best” estimates of the systems current operating

state. State estimators also serve the purpose of filtering incorrect or corrupted measurements, data

etc., received by the Supervisory Control and Data Acquisition (SCADA).

18

3.1 Conventional/SCADA measurements

Various tools/elements such as the Energy Management System (EMS) and Supervisory Control and

Data Acquisition (SCADA) systems have been put in place for monitoring, supervisory control, data

acquisition and delivery among other things. They also ensure that power network regulatory

conditions are met. A few of such elements are listed below:

• Energy Management Systems (EMS)

• State Estimation Algorithms (SEA)

• Supervisory Control and Data Acquisition(SCADA)

• Remote Terminal Units (RTUs)

• Advanced Metering Infrastructure (AMI)

• Generation and Load Forecast System(GLFS)

These tools, in addition to planning and analysis functions, form the EMS/SCADA system. A basic

understanding of the relationship between the EMS/SCADA system and state estimation is illustrated

with figure 3.1 [22]. The data acquisition function receives real-time measurement from remote

terminal units (RTUs) and phasor data concentrator elements installed at various parts in the

network.

Fig. 3.1: Elements of the EMS/SCADA system

19

3.1.1 Measurement model and assumptions

Most commonly used measurements for state estimation are line real and reactive power flows, real

and reactive bus power injection, line current flow magnitudes and bus voltage magnitudes. These

measurements can be expressed in terms of state variables given the state vector 2 ∈ ℝ�fg6i.e.,

for h buses, there are h voltage magnitudes & and (h − 1) phase angle b. The state vector for a

static system can thus be described as

2 = [b�, bH, … , bf , |&6|, … , |&f|]5 (3.1)

where |&j| is voltage magnitude and bj is phase angle at bus -. It should be noted that one of the

buses id selected as a reference or slack bus resulting in a (h − 1) phase angle. It is safe to assume

that the complex phasor voltage measured, &k = & cos b + ,+,-b, has a magnitude 1 p.u. and voltage

phase angle between the adjacent busses is arbitrarily small.

At this juncture, it is imperative that we remind ourselves of a few fundamentals to enable us

understand power expressions in a transmission line. One of such is Ohm’s law which states that “the

current through a conductor between two points , andl are directly proportional to the potential

difference across the two points.” It is mathematically expressed as

&Zm = &Z − &m = WZm=Zm (3.2)

where &Zm@-�=Zm are complex voltage and current phasors. WZm is complex impedance expressed as WZm = %Zm + l2Zm. R is resistance to flow of direct current (DC) and 2Zm is reactance, i.e., resistance to

the flow of alternating current. Re-writing equation (4.2) in, we have

=Zm = OZm(&Z − &m) (3.3)

where admittance OZm (inverse of impedance) is expressed as OZm = nZm + lPZm. nZm is conductance and PZm is susceptance. The flow of current within the network is described by Gustav Kirchhoff’s law

which states that “the algebraic sum of currents in a network of conductors meeting at a point is

zero.” It is expressed mathematically as

o =pqpr6 = 0(3.4)

If we represent the transmission line with the two-port u-model described in figure 3.2 [21] for an N

bus system, the set of nodal equations expressing

20

Fig. 3.2: Two-port v-model of transmission line connected between bus w and x

Kirchhoff’s current law at each bus, is given as

y̅ = {=6=�⋮=f} = {~66~�6⋮~f6~66~�6⋮~f6

⋯⋯⋮⋯~66~�6⋮~f6} {

&6&�⋮&f} = ~�̅ (3.5)

Where y̅ ∈ ℂf�6 is the vector of net current injections at each bus

�̅ ∈ ℂf�6 is voltage phasor at each bus, with �̅j = |&j|�m��

Y is admittance matrix with each entry given as

~Zm = �Zm + l�Zm = Q 0 ,R-$���,+-$�A$--�A����$-$��-O�+ +∑ O���∈�� ,R� = -−OZm $�ℎ�%�,+� T(3.6)

where OZm is admittance of the line from node ,�$l, �Z is the set of all nodes connected to node ,, OZ� is the sum of all shunt admittances connected to node ,. The conventional measurements for static estimations are given by the real power �Z and reactive

power XZ injection at bus,, real power line flow �Zm and reactive power line flow XZm between bus , andl . They are expressed as

• Power injection at bus ,: �Z = |&Z| o|&�|(�Z� cos bZ� +�Z�+,-bZ�)m∈f] (3.7)

21

XZ = |&Z| o|&�|(�Z� cos bZ� −PZ�+,-bZm)m∈f] (3.8) • Power flow from bus ,�$l:

�Zm = |&Z|��n�Z +nZm) − |&Z|�&m�(nZm cos bZm +PZm +,-bZm� (3.9)

XZm =−|&Z|��P�Z +PZm) − |&Z|�&m�(nZm sin bZm +PZm A$+bZm� (3.10)

These conventional measurements are obtained from SCADA systems and are related to the state

vector described in equation (3.1) by the over determined system of non linear equations given as

(3.11)

W = ℎ(2) + - (3.11)

with - describing the mean Gaussian measurement noise vector with covariance matrix ∁j∈ ℝ���.

3.2 Synchronized Phasor Measurements Units (PMUs)

PMUs are measurement devices capable of providing near real-time measurements of positive

sequence voltage and current phasors at monitored and adjacent buses, utilizing the satellite-based

GPS system. The measured current and voltage signals are oversampled and conditioned using

analogue and digital anti-aliasing filters. The sampling clock is phase-locked with the GPS system,

providing synchronized measurement up to 0.2 micro seconds. The IEEE 1344 standard requires that

the PMU’s reliability exceeds 99.87 percent [23]. The phasor representation is illustrated with figure

3.3

Fig. 3.3: Phasor illustration

3.2.1 Measurement model and assumptions

In wide area monitoring, measurement data is collected at various part of the power system and

transmitted to a central location for analysis and control of the systems operating conditions. The

22

SCADA system transmits these measurement data provided by RTUs which are placed at various

parts of the network. However, the SCADA system captures only steady state conditions of the

power system and is limited by its time skew making it difficult to get real-time condition of the

power network. To cushion these effects, PMUs are installed at strategic places in the network such

as at substations and they provide time-stamped measurements. Figure 3.4 [22] illustrates the

placement of PMUs within the system and how the measurements gotten from the PMUs are

transmitted to the SCADA system.

Fig 3.4: Hierarchical placement of phasor measurement units.

Let us consider the two-port pie network figure 3.5 [22] of a transmission line. We would assume

that we have placed a PMU at bus ,. The complex voltage phasors at buses ,@-�l is �Z@-��m respectively. Let the state vector be denoted as x. We thus have for this system

� = �ℛ��Z�ℛ��m�ℐ��Z�ℐ��m��5 (3.12)

23

Fig. 3.5: Two-port π network.

The voltage measurement from the PMU at bus , is given as

W�] = Mℛ��Z�ℐ��Z� N +�6

= K1 0 0 00 0 1 0L 2 +�6 (3.13)

= �2 + �6

For the current measurements at bus ,, using the PMU, we have

WZ] = Mℛ�=Z�ℐ�=Z� N +�� (3.14)

Where W�] and WZ] are the vector of PMU voltage and current measurements respectively while �6

and �� are their corresponding zero-mean Gaussian measurement noise.

Applying Kirchhoff’s law, we derive the relationship between the current =Z and the state vector

(3.15):

=Z = =Zm + =�Z = �nZm + ℐPZm���Z − �m� + (n�Z + ℐP�Z)�Z (3.15)

Simplifying and separating the real and imaginary parts, we obtain (3.16) and (3.17)

ℛ�=Z� = �nZm + n�Z�ℛ��m� − nZmℛ��Z� − �PZm + P�Z�ℐ��Z� + PZmℐ�&m� (3.16) ℐ�=Z� = �PZm + P�Z�ℛ��Z� − PZmℛ��m� + �nZm + n�m�ℐ��Z� − nZmℐ&m (3.17)

Combining equations (3.16) and (3.17), we obtain (3.18)

24

Mℛ�=Z�ℐ�=Z� N = �(nZm + n�Z) −nZm −(PZm + P�Z) PZm(PZm + P�Z) −PZm (nZm + n�Z) −nZm  � (3.18)

= ~� (3.19)

Substituting (3.19) into (3.14), we obtain (3.20)

WZ] = ~� +�� (3.20)

Combining (4.13) and (4.20), we get (3.21)

W = K�~L � + � (3.21)

= X� + � (3.22)

Where W is the measurement vector of PMU measurements with ¡Z voltage measurements and ¡�

current measurements, � ∈ ℝ�¢)(�fg6) is matrix containing rows of vectors with zeros but having a

one in the column where representing the bus where a PMU is installed. ~ ∈ ℝ�£)(�fg6) is the

matrix of admittances while e is the zero-mean Gaussian measurement noise which occurs when

taking measurements.

3.2 Mixed (PMU and Conventional/SCADA) measurements

In networks where sufficient numbers of PMUs are placed, the system is usually completely visible

and as such estimation can be done strictly with PMU measurements. They do not need iterative

solutions as the PMU readings are direct readings of all system buses. Such systems are regarded as

linear systems. However, this is a challenge today due to the cost implications of installing sufficient

PMUs for this purpose. Today, state estimators are used which make use of SCADA and PMU

measurements without necessarily changing the already existing SCADA structure in place in within

the network. Various literatures have considered various methods of combining both measurements

for state estimation. An optimal method/algorithm for PMU placement for hybrid SCADA/PMU based

state estimation will be considered later on in this thesis work.

There are two methods [24] of integrating PMU measurements to the state estimation process. One

method is to use an estimator which uses a mixture of PMU and traditional power flow

measurements. This method has an advantage of relatively better performance in terms of accuracy

and redundancy. The other method uses a two-stage scheme. It is called hierarchical state estimator.

State estimation is first done using the SCADA measurements. The estimates are then improved in

the second stage by using another estimator that uses only PMU measurements. This method of

estimation has the advantage of leaving the existing SCADA software/structure in place. In this

method, the state estimate x¥ (as given by equation 3.23) is converted to voltage phasors �¥6 = �(2¥) and subsequently added as an additional measurement to a linear measurement model [22] as

described below

25

M�¥6W�N = M�¦~N � +M�̃6��N (3.23)

Where �¦ sifts out the needed phasors and �̃6 is the noise vector associated with the transformed

conventional measurements. The weighted least square approach can be used to solve for the

unknown phasor �¥6.

26

4 State Estimation

A background understanding of the evolution of state estimation is important in understanding the

goals of this thesis. This chapter starts by discussing briefly the evolution of some state estimation

techniques, with an understanding that they are built basically on conventional power-flow, power-

injection and bus-bar voltage-magnitude measurements. Thereafter, we will go on to give a brief

discus on maximum likelihood estimation before we proceed to our main focus of estimation in

power systems in the next chapter.

State estimation (SE) schemes can be classified into three distinct frameworks [22]. They include

• Static state estimation (SSE)

• Forecasting-aided state estimation (FASE)

• Multi area state estimation (MASE)

Conventional measuring and monitoring technologies, such as implemented by SCADA systems, can

only take non-synchronized measurements every two to four seconds. As such, state estimation in

previous years could only be assumed to be static which is based on the WLS method. The state

estimates of the SSE are updated once every few minutes so as to reduce computational complexity

during SE. Consequently, the SSE process cannot be considered an optimal process for real time

monitoring of the network. Also, the SSE depends on single set of measurements taken at one snap

shot of the system. This means that the states of previous measurement instances are disregarded

when estimating a new state. Hence, even when the system is fully observable and the state

estimates are within appreciable limits, the SSE cannot predict future operating point for the system.

To compensate for this, a dynamic-based estimation approach called the Forecasting-Aided State

Estimation (FASE) has been introduced. This approach provides recursive (almost certain) updates of

the state estimates. This tracks changes during normal system operating conditions. The FASE

approach also takes care of the problem of missing data considering that the predicted states could

be used pending the availability of such data. Although we will be assuming close similarities

between the FASE platform and the true dynamical SE, it should be noted that they are somewhat

different. In [25] a state transition model that uses Kalman filtering and an exponential smoothing

algorithm for forecasting, was developed. This algorithm was developed so as to follow the changing

dynamics of the system. The MASE platform on the other hand is based on the premise that for large

networks, a central computational system increases the computational complexity. Thus, the power

network can be divided into smaller networks and the state estimates for these smaller networks are

computed independently. This reduces amount of data used by each estimator and as such reduced

computational complexity and also increasing system robustness. However, it requires additional

27

communication overhead together with time-skewness challenges resulting from unsynchronized

measurements [22].

In recent years, the need to improve SE as well as capture near dynamics of the power system

network paved way for the introduction of PMUs which has been described in the previous chapter.

PMUs provide synchronized measurements, thus provided more accurate and timely measurements

with more time samples as compared to SCADA systems that use non-synchronized measurements.

Hence SSE can be made to use measurements from both PMUs and conventional or traditional

SCADA systems. In general, two ways have been described in various literatures for including PMU

measurements to the SE process. One method is to use a single estimator where the PMU

measurements are mixed with traditional power flow measurements and the other method makes is

a two-stage scheme where an estimator which uses only PMU measurements is used to improve the

state estimates obtained from the conventional SCADA measurements.

In [24], SSE is done with the two stage method of adding PMU measurements. This method uses the

Gauss-Newton weighted least-square approach. But it has a drawback in that it does several

iterations at one time instant, increasing computational stress on the EMS/SCADA system. In [20], a

method that addresses the computational challenge is discussed. It discusses the use of the two-

stage scheme in the FASE approach. This approach is computationally resource efficient as a single

iteration is performed at each time instant. An added advantage is that it assumes the state vector

contains constantly changing state variables due to constant changes in the power system network.

In that work, a mixed measurement extended kalman filter (EKF) estimation algorithm is derived on a

FASE platform, using a dynamic mixed measurement for the measurement model. A drawback to the

mixed measurement estimation method is the fact that the measurements are of different qualities

and as such combining them in a single estimator can cause the covariance matrix of the combined

noise vector to be ill conditioned. Also, the dimensions of the vector and matrices used in the SE

process increases the computational complexity as a result of the presence of PMU measurements.

To address this challenge, a reduced order EKF (RO/EKF) estimator is proposed and used in the same

work in [22]. The RO/EKF algorithm works by estimating the PMU observable state and the PMU-

unobservable states differently. Although the challenge of computational complexity is solved

somewhat, this comes at the expense of the cost of accuracy. For this thesis work, it is proposed to

use the FASE platform for incorporating the conventional and PMU measurements with the

Unscented Kalman Filter (UKF) for improved filtering.

4.1 Static State Estimation (SSE)

Statistical estimation is concerned with estimating the “best” estimate of the unknown state

variables or parameters using samples of imperfect measurements. Thus the statistical criterion or

28

procedure needed for this becomes the problem to be solved [26]. Several state estimators differ in

the sense that they have deferent objective functions for solving for the state variables. A few

commonly used criteria are

• The weighted least-square criterion (WLS)

• The maximum likelihood criterion(ML)

While some solution methodologies are

• Weighted Least square (WLS) method

• Least absolute value (LAV) method

• Weighted least absolute value (WLAV) method

• Least median of squares(LMS) method

• Non-quadratic method

• Minimum variance method(MV)

In most cases today, the WLS method is the most used because of its advantage in its statistical

properties, less computational work and the fact that it uses simple model. It should be noted at this

point that the use of normally distributed errors can be assumed. However, it has a draw back in its

robustness in the sense that its estimates are affected by bad data and thus may not be real

estimates. Other solution methods such as the Non-quadratic, WLAV and LMS, are robust but have

the disadvantage of much computational work. The WLS, minimum variance and maximum criterion

eventually give the same estimator [26].

4.2 Forecasting Aided State Estimation (FASE)

The state space representation, for a discrete time-variant dynamic system, is described with the

state transition (prediction) model below:

2(� + 1) = ¨(�)2(�) + n(�) + "(�) (4.1)

With a measurement model at time instant �, given as

W = ℎ[2(�)] + � (4.2)

where 2(�) is the state vector containing bus/nodal voltage magnitudes and phase angles, ¨(�) ∈ ℛ(2h − 1)x (2h − 1) is the state transition matrix and vector n(�) relates to the trend

behavior of state trajectory [21]. "(�) represents modeling uncertainties and is given as white

Gaussian noise with zero mean and covariance matrix Xp, vector ℎ[∙] is a non-linear load-flow

function for the current network configuration, relating the state vector to the measurement vector, � represent white Gaussian measurement noise, with diagonal covariance matrix expressed as

29

�g6 = Mªg� 00 ª«\¬g6 N (4.3)

where ª is standard deviation of the conventional SCADA measurements and ª«\¬ is standard

deviation for PMU measurements. The error standard deviation for the PMU measurements is in

general considered to be far less than that of the SCADA measurements since PMUs have more

accurate measurements.

Let 2̅Z(�) and 2̅Z(� + 1)be predictions at time instants � and � + 1. It should be noted that model

parameters ¨(2), n(2) and Xp are not known a priori and are to be calculated. It is considerably

safe to assume (4.1) as the memory of the system state time evolution while (4.2) serves as its

refreshment for every new measurement. In other words, (4.1) represents the model of transition of

states from 2(�) to 2(� + 1). For the dynamic state estimation process, three basic steps are considered. They are parameter

identification, state forecasting and state filtering [25].

Parameter identification

Considering the dynamic model described in (4.1), the parameters ¨(2) and n(x) are calculated using

the Holt’s two parameter linear exponential smoothing technique of forecasting [27]. This method

gives decreasing weights to past observations. Appendix A further illustrates the Holt’s initialization

technique. The noise covariance matrix Xp, is determined in other to optimize the estimation process

in terms of accuracy. The elements of the Xp matrix, which are assumed diagonal, are obtained

through examining the maximum rate of change of the state variables, considering their behavior

through historical data.

If, for instance, the historical behavior of the state vector is given as 2p­6 = 2p + ∆p, where 2

represents the state vector, we can assume [2] that this change ∆p can be replaced by a random

variable "(�) with a Gaussian distribution. The resulting dynamic equation is then given as in (4.1).

State forecasting

With all parameters rightly calculated, the model described in (4.1) is now ready to make a forecast.

If at time instant �, the state estimated vector value is 2¥(�) and the true value is 2(�) with its error

covariance matrix as ∑(�), the statistical characteristic of the estimation error is given as

[2(�) − 2¥(�)] which approximates to [Y(0, ∑(�)]. Using measurement information on the system

behavior up to time instant � and performing the conditional expectation operator on (4.1), a one

step ahead predicted (forecasted ) state vector and its covariance matrix is given by (4.4) and (4.5)

respectively;

2̅(� + 1) = ¨(�)2¥(�) + n(�) (4.4)

30

h(� + 1) = ¨(�)∑(�)¨5(�) + X(k) (4.5)

where

¨(�) = ®p(1 + p̄). = (4.6)

where ®p(1 + p̄) is the �th diagonal element of ¨(�)

n(�) = (1 + p̄)(1 − ®p)2̅(�) − p̄@(� − 1) + (1 − p̄)P(� − 1), (4.7)

where

@(�) = ®p2¥(�) + (1 − ®p)2̅(�)

(4.8)

P(�) = p̄[@(�) − @(� − 1)] + (1 − p̄)P(� − 1)

The forecasted state vector 2̅(� + 1) and its forecasted covariance matrix h(� + 1) is then used to

obtain the forecasted measurement vector W̅(� + 1) and its error covariance matrix �(� + 1)

expressed as

W̅(� + 1) = ℎ(� + 1)[2̅(� + 1)] (4.9a)

�(� + 1) = (� + 1)Y(� + 1)5(� + 1),

where (� + 1) is the Jacobian matrix expressed as

(� + 1) = °±(p­6)°� , at 2 = 2̅(� + 1) (4.9b)

State Filtering

When new sets of measurements W(� + 1) arrive at � + 1 time instant eventually, the predicted

state vector 2̅(� + 1) is then filtered/ updated to obtain a new filtered estimate, 2¥(� + 1) and its

corresponding error covariance ∑(� + 1). The optimization/objective function for the filtering

process at time instant � + 1 is given as

(2) = [W̅ − ℎ(2̅)]5�g6[W̅ − ℎ(2̅)] + [2 − 2̅]5hg6[2 − 2̅] (4.10)

Minimizing this objective function gives us the estimate for the state vector 2¥(� + 1). This estimate

takes in to account the predicted estimate 2̅(� + 1) and the measurements W(� + 1) at � + 1.

31

4.3 Incorporating measurements to FASE

In general, the main objectives of power system state estimation is to provide knowledge of real-

time power system conditions through filtering, filling and smoothening real-time measurement

acquired from the power system network. State estimation also provide representation for power

system security analysis through contingency analysis, on-line power flow and load frequency

control.

The input to the state estimator include but are not restricted to measurements from PMUs,

Network parameters such as generator parameters, line impedances and system configuration given

by the topological processor etc. These known input parameter and/state measurements are not

always perfect and usually noisy or contain redundant measurements. Hence, the state estimator

estimates the “best estimates” for the unknown states. Such unknown states could be voltage

magnitude and phase angle for static systems or the dynamic states of the synchronous machine(s).

The state estimator also filters measurement errors and errors from model approximation. It also

detects and identifies bad data and provides estimates for metered and unmetered quantities. The

idea behind estimation is to formulate the network model using imperfect real-time telemetry

measurements, to make raw-detection on real time measurement data so as to eliminate bad data

and supplement deficient measuring points, also known as unobservable islands, with pseudo

measurements. Hence, system state estimation and power flow calculations are done, thus

improving complete network observability.

To understand the method of using a single estimator, and subsequently process mixed

measurements from PMU devices and conventional SCADA system for static state estimation, we

need to relate the conventional state vector (which is in polar coordinates) to the PMU state vector

(in Cartesian coordinates) through a non-linear transformation defined as � = �(2). Hence, a single

state estimator incorporating both measurements [22] is given as

KW6W�L = M ℎ(2)²�(2)N + K�6��L, (4.11)

where W6, W� represent the conventional and PMU measurements while �6,�6 represent noise

vectors from both measurements respectively.

32

5 Proposed methodology

Earlier in chapter one, we established that the UKF is an efficient discrete-time recursive filter of the

family of Kalman filters and that it is based on unscented transformation (UT). We also noted that

unlike the EKF, it does not linearize the non-linear equation. Rather, it propagates the statistical

distributions of the estimated states through non-linear equations leading to better estimates of the

state and a posterior covariance matrix. The use of statistical distribution (mean) to represent the

state offers the following advantages:

1) Mean and covariance of unknown distribution requires the maintenance of only a small

amount of information which is sufficient for all kinds of operational activities. It gives a good

compromise between computational complexity and representational flexibility

2) Mean and covariance (or covariance square root) are linear transformable quantities. i.e., the

mean and covariance estimates are maintained after being subjected to linear and quasi

linear transformation.

3) Sets of mean and covariance estimates can be used to characterize additional features of

distribution.

5.1 Unscented Transformation (UT)

An important fact in non linear estimation is the quality of the approximation of the non-linear

function around the operating point. If this is not properly done, estimation errors occur and the

efficiency of the estimator is questioned. To mitigate this problem, the unscented transformation

(UT) assumes that “It is easier to approximate a probability distribution than it is to approximate an

arbitrary nonlinear function or transformation” [28]. The UT [29] could be illustrated with figure 5.3

below.

Fig. 5.1: Unscented transformation, showing sigma points being transformed

33

The idea behind the UT is to obtain deterministically chosen sigma points which capture exactly the

mean and covariance of the original distribution of the measurement set W. Let us consider the

random variables 2 and a system in the form of O:

2~�(�, �) (5.1)

O = R(2) 2 is a an n x 1 vector of state variables described by the probability distribution � with mean � and

covariance �, R(2) is a non linear function, Ois m x 1 vector of variables resulting from R(2). UT aims

at obtaining a set of deterministically chosen vectors called sigma points. These vectors (or sigma

points), containing real numbers, capture the exact mean and covariance of the original distribution

of 2. The sigma points are then propagated through the non linear function R(2) to obtain the mean

and covariance of O. The UT process can be summarized as below:

1) Select sigma points: Form an - x 2- + 1 matrix containing a set of 2- + 1 column vector

sigma points which has the state mean vector � and state covariance matrix � as follows

�. = �

�Z = � + �´(- + µ)��Z , , = 1, … , - (5.2)

�Z = � − �´(- + µ)��Z , , = 1, … , -

�´(- + µ��Z is the ,th column of the matrix ´(- + µ�, parameter µ is scaling parameter described

as

µ = ®�(- + �) − - (5.3) ® is a constant, usually between 0.0001 and 1 i.e.,10g¶ ≤ ® ≤ 1 . It determines the spread of the

sigma points away from or around the state mean� . Hence, it can be used to vary the amount of

higher order nonlinearities around �. The closer the sigma points are specified to m, the more the

higher order nonlinearities tend to be ignored. Constant � is a secondary scaling value usually set to 0 or 3 − - . It can be used to reduce the higher order errors of the mean and covariance

approximations. Note, if the square root matrix of � is ², then � = ²². If � is positive definite as

assumed in the UKF, it can be rewritten as � = ²²5. The Cholesky factorization of � gives ², where ² is a lower triangular matrix. Cholesky factorization ensures sigma points are not complex numbers.

2) Nonlinear Transformation of the sigma points through the dynamic function: the sigma

points set �Z are propagated through the non linear function R to obtain the propagated

sigma points

�̧Z = R(�Z , ¹), , = 0,1,… ,2- (5.4)

The predicted mean vector ¹̂ and predicted covariance matrix �̧ for the propagated sigma points is

computed as

34

¹̅ =o»Z��jZr. �̧Z(5.5)

�̧ =o»Z¼�jZr. [(�̧Z − ¹̅)(�̧Z − ¹̅)5](5.6)

where » are the weights and can be calculated using the formulas:

».� = ¾j­¾ , ».¼ = ¾j­¾ + (1 − ®� + ¯)

(5.7)

»Z� = 6�(j­¾) ,»Z¼ = 6�(j­¾)

The variable ¯ typically has a value of 2 for Gaussian distributions. It is used to incorporate prior

knowledge of the distribution of 2.

The UT approach provides approximations that are accurate to a third order for Gaussian inputs for

all nonlinearities [30]. The approximations for non-Gaussian inputs are accurate to a second order.

For higher order moments, their accuracy depends on the choice of ® and�. A block diagram

representation of the UT is presented in figure 5.4 below [30]

Fig. 5.2: Unscented Transform (UT) block diagram

5.2 The Unscented Kalman Filter (UKF)

As previously mentioned, the UKF is a recursive estimation algorithm. It uses the UT method in its

recursive prediction and update structure. It is somewhat similar to the EKF however it is

computationally easier to implement as the Jacobian matrix is not calculated. Let us consider the

non-linear functions of state transition and observation models [1] representing a system as given by

(5.8), 2p = R(2pg6) + "pg6

35

Wp = ℎ(2p) + %p (5.8)

where 2 is the state vector and W is the measurmrnt vector,"pg6 and %p are system and

measurement Gaussian noise with covariance matrices X and �, non-linear functions R and ℎ

represent the system and measurements . The UKF algorithm performs estimation in three major

steps

1) Selection of the sigma points

2) Kalman filter state prediction

3) Kalman filter state correction

Sigma point selection: As described in (5.2), the UT procedure creates the set of 2- + 1 sigma

points, using the a posteriori estimate for the state mean 2 and covariance �, at time instant� − 1.

The expression for the sigma points [1] is given as (5.9)

�. =2pg6

�Z = 2pg6 + ¿´�pg6, , = 1,2,… ,2- (5.9)

�Z = 2pg6 + ¿´�pg6, , = 1,2,… ,2-

where ¿ = √- + µ is a scaling factor that determines the spread of the sigma points around 2¥. To

initialize the estimation procedure, that is at time instant � = 0, the initial state and covariance

ought to be defined based on priori information of the system. The estimate at � = 0 is an a

posteriori estimate.

Prediction step: The sigma points calculated are propagated one after the other through the systems

non- linear function R, propagating the state estimate and covariance from one time instant to the

next and thus forming the predicted a priori state vector estimate at time � as;

2̅pZ = R��Z , 7pg6� (5.10)

We then compute the a priori covariance matrix �̧p and the state mean vector �Áp by the weighted

average of the transformed points as

2̅p_� = o »Z��j

Zr.2̅pZ (5.11)

�̧p = o »Z¼ K�2̅pZ − 2̅p_���2̅pZ − 2̅p_��5L + Xpg6�j

Zr. (5.12)

(5.12) can also be written as

�̧p = 2̅pZ »Z¼ [2̅pZ ]5 + Xpg6 (5.13)

Update step: Next, we update the sigma points with the predicted state mean vector 2̅p_� and

covariance matrix �̧p :

�̧. = 2̅p_�

36

�̧Z = 2̅p_� + ¿´�̧p, , = 1,2,… ,2- (5.14)

�̧Z = 2̅p_� + ¿´�̧p , , = 1,2,… ,2-

We then implement the measurement update. As described in (5.8), the sigma points are propagated

through the known non-linear measurement function ℎ(∙) as shown in (5.15):

Ã̅p = ℎ(�̧Z) (5.15)

This is the predicted measurement at time instant k. The mean ¹p of the predicted measurement is

computed as:

¹p =o»Z�Ã̅p �jZr. (5.16)

We also compute the covariance matrix �p of the predicted measurements:

�p =o»Z¼[(Ã̅p − ¹̅p)(Ã̅p − ¹̅p)5] + �p�jZr. (5.17)

also expressed as

�p = Ã̅p»¼ [Ã̅p]5 + �p (5.18)

The cross covariance matrix Äp between the measurements and state is:

Äp =o»Z¼���̧Z − 2̅p_��(Ã̅p − ¹p)5��jZr. (5.19@)

also expressed as

Äp = �̧Z»¼ [Ãp]5 + �p (5.19b)

The filter gain Æp, the updated state mean �p and the covariance�p ,are computed as

Æp = Äp�pg6

2p = 2̅p_� + Æp[Wp − ¹p] (5.20)

�p = �̧p + Æp�pÆp5

where 2p is the corrected and thus generator state.

5.3 UKF based state estimation in power systems (UKF/SE)

In power systems, the UKF remains a very useful technique for state estimation. Its ability to

estimate dynamic system states without calculating the Jacobian matrix makes it a unique algorithm.

This enables the application of real and complex system model without fear of the errors that result

from model linearization.

A few examples of the application of the UKF to power systems are seen in [14] and [6], where it is

used for parameter estimation of synchronous machines. In [31] the UKF is used to estimate

frequency and amplitude of power signals by filtering the noise measurements. The power system

can be modeled in different forms. In [1], a linear model is used to represent the smooth dynamics of

37

the system determined by slow load variations. However a linear model might not be sufficient to

capture the true dynamics of the system introduced by the generator dynamic parameters.

In general, the power system may be modeled using a set of nonlinear differential-algebraic

equations with unknown initial values expressed in (5.21) below 2/ = R(2, O) 0 = n(2, O) (5.21)

z= ℎ(2, O) where R(∙) is the nonlinear state functions of the state transition, n is the set of algebraic equations

representing passive network of the power system. It is formed as a nodal equation with a nodal

admittance. ℎ(∙) is contains the nonlinear measurement equations.

To use these equations in solving for x given y, we need to convert them from continuous-time

equations to discrete-time equations using numerical integration methods such as Euler methods,

modified Euler methods, Runge-Kutta (R-K) methods, implicit integration methods, etc.

The time derivative of a variable 2 is given as:

2/ = �(p)g�(pg6)∆# (5.22a)

2(�) = 2/∆� + 2(� − 1) (5.22b)

where ∆� is the time step, � and � − 1 indicate time at � = �∆� and � = (� − 1)∆� respectively.

Substituting (5.21) in (5.22b), we get (5.23):

2(�) = ∆�R(2, 7) + 2(� − 1) (5.23)

It can be re-written as (5.24),

2p = Rpg6(2pg6) + "pg6

(5.24)

Wp = ℎp(2p) + %p,

Let us assume that for an h bus system, the initial state vector and its corresponding covariance

matrix are represented as 2¥. and ��¥Ç respectively, where the subscript represent the time instant � = 0. The dimension of the state vector is given by - = 2h − 1, which corresponds to the number

of unknown state variables, assuming that the reference bus angle has a value of zero. The UKF steps

for the power system can be summarized as follows:

Sigma point selection

As described in (5.9), we use the a posteriori estimate for the state mean 2 and covariance �, at time

instant� − 1 to calculate the set of 2- + 1 sigma points:

�. =2pg6

�Z = 2pg6 + ¿´�pg6, , = 1,2, … ,2- (5.25)

�Z = 2pg6 + ¿´�pg6, , = 1,2, … ,2-

38

State prediction (forecasting)

According to (5.22), each column of �Z representing the sigma points at instant � − 1 are propagated

one after the other through the state update function to form a matrix of propagated sigma points at

the next time instant �:

�̧p = �Z + (∆�)R��Z , 7pg6� (5.26)

The predicted state mean vector and the corresponding predicted covariance matrix are then

calculated using (5.11) and (5.12).

State update

We compute or update the existing set of sigma points as described by (5.21) which captures the

distribution of the predicted state. As described in (5.22), the sigma points are then propagated

through the measurement update function such as (5.27);

~̧6Z = ℎ(�̧6Z) (5.27)

Where the superscript , represents the ,-th column of the respective matrices. We then calculate the

mean and measurement covariance matrices using (5.23) and (5.24) respectively while the filter gain,

state mean and covariance matrix are calculated using (5.26). This procedure is repeated for every

time instant �.

To effectively estimating the two dynamic states we have chosen, we have done a few modifications

to the standard UKF algorithm described in (5.8) to (5.20). We have randomly chosen the initial

covariance matrix variable and two random numbers, each representing the initial values of each

state. These variables are used for the state mean x which is in turn used for calculating the sigma

points as described in equation (5.25).

The model equations describing the synchronous machine as described in (2.12b) were used for the

propagation of the sigma points in the prediction process as described in (5.10) and the update

process as described in (5.15). It should be noted that the continuous time model equations (2.12b)

were first converted to the discrete form before being implemented in (5.26). The discrete form

equations will be presented in chapter 6.

39

6 Simulations and Results

This chapter presents simulations of our proposed algorithm which has been described in chapter

five. We have modeled our test system in MATLAB Simulink. Simulations of system disturbances are

done to reflect the dynamic nature of our modeled system. Our proposed UKF algorithm along with

the synchronous machine differential algebraic equations presented in chapter two, are used for the

estimation of the dynamic states of the system. Results are then presented and discussions are done

on the simulations.

6.1 Implementation of our proposed methodology

The proposed methodology for this thesis is tested on a single machine infinite bus (SMIB) system

described in chapter two. The choice to use this simple system configuration for our analysis and

simulations is because it is very useful in understanding basic concepts and effects which can be used

for further studies in large complex networks. We have assumed the generator, whose state

dynamics we seek to estimate, to be a part of the whole dynamic system, though separated from the

other parts of the system by the transmission network. The schematic representation for the

proposed estimation of the rotor angle � and rotor speed � is shown below in figure 6.1.

Figure 6.2 illustrates the Simulink implementation block diagram for the proposed method. It shows

the connection between the synchronous machine and the embedded MATLAB function block used

for implementing the UKF algorithm.

Pe

Fig. 6.1: Proposed block diagram of UKF estimator

Gen terminal bus t

Vtb

��!R�Pt

�

� UKF est.

Synch.

Machine

PMU

Power System

=@PA&@PA

Qt I

40

Fig. 6.2: Top level of the Implementation block in Simulink

The inputs to the UKF function block are: measurable electrical output power ��(�#) from the

machine, assumed to be measured at the terminal bus of the generator by a PMU device, the

mechanical power �� given by the prime mover (i.e. wind turbine in our case) and the internal

voltage dynamics of the exciter system represented with the variable !81 where it is assumed to be

directly measured from the generator winding with negligible measurement noise. The terminal

voltage &#∠b(or !#∠b) and frequency are assumed to be measured by the PMU installed on the

generator terminal bus. The machine parameters (see appendix for details) are assumed to be known

or measured with little or no errors. Only the dynamic states to be estimated are unknown. These

input output parameters described are needed to initialize estimation using the proposed UKF

algorithm.

Figure 6.3 is the schematic drawing representing the power system. It shows the synchronous

machine connected to the transmission line through a transformer. A 150kV voltage source

connected to the transmission line represents the aggregate of the loads on the infinite system. In

power systems, the load can be defined to be as the sum of the continuous power ratings of all load

consuming apparatus connected to the grid. The voltage source has constant voltage and constant

41

frequency and as such, is referred to as an infinite bus [11]. This characteristic of an infinite bus is

seen in its unlimited power capability and thus can be represented by a voltage source with zero

internal impedance. The magnitude of the infinite bus voltage remains constant for any given system

condition when the machine is disturbed. In real events, the loads on the power system are actually

varying constantly. But the assumption that the loads are constant is a relevant approximation for

this thesis work and many other power systems studies. This condition or state is generally referred

to as steady state. As the system steady state conditions changes, the magnitude of the infinite bus

voltage changes too and this affects the operating conditions of the entire system network.

Fig. 6.3: Layout of the synchronous machine connected to the transmission line

The importance of system loads, represented by the voltage source, cannot be over emphasized.

During our simulations and analysis, we observed that when the voltage source was disconnected,

there were swings in the frequency plot of the rotor angle. The swings observed in the rotor angle

plot are as a result of the load change in the system, typified by the disconnected voltage source. If

such swings of the rotor angle are extreme or excessive, the synchronous machine or generator can

trip off resulting in an unstable system.

The choice of using Simulink library tools to design the model for a simple single machine system was

necessary so as to simulate various operating conditions of the synchronous machine. We have used

the embedded MATLAB function block considering that it allows us implement dynamic state

estimation simultaneously while simulating different operating conditions of the machine.

Voltage

source

42

Figure 6.4 illustrates the subsystem model representing the mechanical part of the machine. The

subsystem shows the two inputs; the steady state value of the mechanical torque and the

instantaneous value of the electrical torque.

Figure 6.4: Mechanical part Sub-model of the synchronous machine

Figure 6.5 illustrates the subsystem model representing the electrical part of the machine.

Figure 6.5: Electrical part Sub-model of the synchronous machine

The initial value for the dynamic states were gotten from the Machine but were not used as initial

states for initiating the UKF algorithm. We ran the simulations for various states values. Our

simulations showed the need to have the initializing states as close to the real states as possible to

43

improve the accuracy of the estimates. A list of the synchronous machine parameters and the states

to be estimated can be found in appendix C.

The motivation for the choice of the UKF method over the EKF, even though they both have relatively

good estimations, is due to the fact that the UKF gives better approximation for the R(2, 7) and ℎ(2) functions with respect to the Extended Kalman Filter (EKF). This is because the model equations

are not linearized during the estimation process as is done in the EKF. Although there is higher

computational complexity when using the UKF, its advantage in providing more accurate result is

compensation. As we have shown in the algorithm, only a few sigma points are evaluated to cope

with the non-linear equations. The number of sigma points is limited to 2n+1 where for this work,

number of states n is 2, giving a total of six sigma points to be transformed. With respect to the

particle filter where considerably amount of non-deterministic random points are evaluated, the

UKF’s number of sigma points is small.

We have fixed the UKF parameters as α = 1, υ = 3 - n and β = 2. We also varied these parameters but

discovered they did not have so much effect on the estimation results. The implementation codes for

the UKF algorithm are given in the appendix.

Before simulation, we have converted the continuous time equations describing the power system

model to their discrete form using the time derivative equations which we have described earlier.

The discrete forms of the state equation are presented in (6.1) and (6.2) as

26(� + 1) = 26(�) + � 2�(�) ∗ ����@_� (6.1)

2�(� + 1) = 2�(�) + 6�9 �76 − I'()Ë* !�� (�)+,-26(�) + '(£� Ì 6)Í − 6)Ë*Î sin 226 (�)J − �2�(�)  ∗ ����@_� (6.2)

����@_� is the time step which was set to be 0.001s in the configuration parameter dropdown of

Simulink file. We did set the time step in the embedded MATLAB function block to be the same with

that used in the Simulink file for the purpose of simulation and to provide an acceptable base for

comparison. We also set the solver to be ode3 (Bogacki-shampine) .

Before simulation, the “Powergui” tool of our model, found at the topmost layer of the model as

seen in figure 6.2 is opened to reveal the “machine initialization” tool. The “Powergui Machine

Initialization Tool” shows the machine information as we can see in figure 6.6. It is used to set the

machine’s initial steady state settings. To do this, we initiated the “compute and apply” button. This

computes and automatically loads the Simulink file during simulation with the initial steady state

values of the machine.

44

Fig 6.6: Powergui machine initialization tool

To initialize the UKF algorithm, we set the initial covariance matrix P with the following command:

% Error covariance matrix; P0=diag([10,10]); P=P0;

This command initiates the algorithm by loading the initial covariance matrix variable P using P0. As

observed in figure 6.2, the output power �# measured by the PMU is fed into the algorithm as an

input. It is used in the in the UKF algorithm to correct the estimated states.

With our simulations, we desire to test and analyze the robustness and effectiveness of the proposed

power system state estimator. These analyses provide in-depth insights to power system transient

performance as well as the nature of the machine. Through this analysis and the results presented in

this thesis, it would be easy to make approximations necessary for large scale studies as well as

control studies for improved system performance. We will simulate:

• Small signal disturbance and

• Three phase fault disturbance

45

To simulate these disturbances, we have used matlab’s fault block. Keeping the input factors Tm

(mechanical torque supplied by prime mover) and !81 (internal voltage) constant, the fault block,

fault1, as described in figure 7.3 is used to program and implement a three phase fault. The fault

clearing time is defined and varied within the fault block to illustrate different disturbance scenarios.

The fault is introduced to the system closer to the generator terminals and cleared almost

immediately, depending on the clearing interval. We implemented a stable fault condition with a

clearing time that lasts for 0.07 seconds. With these, we have shown that the proposed estimator

estimates the system states even in unpredictable and unprecedented system conditions.

6.2 Results

The noise free results of our simulation using the UKF and EKF estimation methods are thus

presented. While analyzing the power system, we have decided to present the results of the two

filters while simulating the power system. This will enable us make proper comparison between both

estimation methods and subsequently show the advantage of the UKF over the EKF. Both filters

differ in the ability to track the system during system disturbances. In the EKF, linearization was done

by evaluating a Jacobian in the neighborhood of the estimate.

Figure 6.7 presents the machine output power �# measured at the terminal bus. As we illustrated in

figure 6.2, it is observed from figure 6.7 that the constant input power drawn from the turbine Tm is

0.8pu. When the stator resistance is neglected as we have assumed in this work, the steady state

Fig. 6.7: Output power Pt measured at the terminal bus

terminal power �# , which also represents the air-gap power, is equal to the turbine mechanical input �� . In figure 6.8 the effect on the rotor angle as the generator responds to small signal disturbance

is presented. As we have said, these small disturbances usually occur continually in the system due

to small variations in the loads and generation.

0 2 4 6 8 10 12

x 104

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Active p

ow

er

(pu)

46

Fig. 6.8: Generator rotor angle

Fig 6.9: Generator rotor speed

While figure 6.9 shows simulation results for the synchronous generator rotor speed, we have

presented in figure 6.10a and 6.10b, the response of the rotor angle to a step input in the presence

of small signal disturbance using both the UKF and EKF algorithms respectively. The true value and

the estimated value of the state variables are presented and compared. The grey color represents

the true system data while the blue color represents the estimated data. We can observe that both

methods estimate the rotor angle dynamics properly judging by their rise time. A closer look at both

estimation curves show that the UKF method gives a better estimation judging by the overshoot at

0 2000 4000 6000 8000 10000 12000-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Time (seconds)

Ele

c. R

ad (pu)

0 2000 4000 6000 8000 10000 120000.994

0.996

0.998

1

1.002

1.004

1.006

1.008

Time (seconds)

Ele

c. R

ad/s

(pu)

47

the 0.1 second of the EKF curve. Also note that during the steady state, the EKF estimate tends to

deviate from the real value as opposed to the estimates when using the UKF algorithm.

Fig 6.10 (a): Real rotor angle Vs estimated angle using the UKF algorithm

Fig 6.10 (b): Real rotor angle Vs estimated angle using the EKF algorithm

In figure 6.11, the real and estimated rotor speed is presented. A close look at the response plots

show that the UKF estimator gives a better tracking performance of the machine dynamics better

than the EKF estimator.

48

Fig 6.11 (a): Real rotor speed Vs estimated speed using the UKF algorithm

Fig 6.11 (b): Real rotor speed Vs estimated speed using the EKF algorithm

Note that both estimators follow the form of the output power described with figure 6.7 indicating

that they use the non linear dynamic model of the system. In general, we can deduce from these

results that the UKF gives a better approximation of the system dynamics than the EKF method.

Next we introduce transient fault conditions to the system. The transient response with fault clearing

time of 0.07 seconds is presented in figure 6.12a and 6.12b, applying both estimators to the system.

49

Fig 6.12 (a): Rotor angle response with simulated fault cleared in 0.07s for the UKF estimator

Fig 6.12 (b): Rotor angle response with simulated fault cleared in 0.07s for the EKF estimator.

We observe in both estimators that they tend to follow, relatively, the initial state conditions of the

system before the fault is introduced. During the duration of 0.07 seconds for which the fault stays,

the estimate tends to degrade slightly and does not accurately estimate the states. Whereas the EKF

fails to adequately follow the system dynamics properly during and after the fault, the estimate in

the case of the UKF is temporarily wrong during the fault but tends to follow the real value soon after

the fault. This is as a result of abrupt change in the external reactance. In general, it usually takes a

few milliseconds for the state estimators to track the system dynamics after such external reactance

change. Observe that in figure 6.12 (a), the UKF estimator begins to track the systems dynamics just

after the 7000 second. We can also observe that the swing after the fault is properly reduced soon

after the fault is cleared. This is as a result of proper damping in the system.

6.3 Discussion

For transient stability studies, it is generally important to consider study periods of three to five

seconds after a disturbance or fault may have occurred. Studies could however be extended to ten

seconds for considerably large networks. Thus we have analyzed our estimators based on this

50

following simulated small signal and transient faults. In equation (2.3), it is observed that the power

output is a function of the rotor angle. When there is zero rotor angle, there is subsequently no

power output. Increasing the exciter and wind for the turbine increases the rotor angle and hence

the power output into the power system.

Figure 6.8 describes the deviation between the rotor’s internal voltage angle and the voltage angle

on the infinite bus as seen from the transmission line. As we can observe, the plot is an oscillatory

motion or swing. This swing is results from the rotor inertia. This inertia affects the instant change of

the rotor angle and thus the mechanical power becomes in excess of the electrical power. The

resulting accelerating torque makes the rotor to accelerate from the initial operating point, say “a” as

illustrated in figure 6.13 to the new operating point, say “b”, represented by the new step value of

the input signal , at a rate determined by the swing equation described in equation (2.2).

Fig 6.13: Efd step input

When point b is reached, the accelerating power is zero. Although this pint is reached, the rotor

speed is now higher than the synchronous speed �., which is the frequency of the infinite bus of the

infinite system and as such, the rotor angle continues to increase. For rotor angles more than the

rotor angle at the operating point “b”, the rotor decelerates and at some peak value of the rotor

angle, the rotor speed then realigns with its synchronous value of �.. The rotor speed continues to

decelerate as the rotor decelerates to speeds below �.. This causes the indefinite oscillation of the

rotor angle about the new equilibrium angle until a steady state oscillation is obtained and eventually

damps out as we have shown in figure 6.8. As shown in figures 6.8, 6.9, 6.10a, 6.10b, 6.11a, 6.11b,

6.12a and 6.12b, a well stable system is one whose oscillations or swings are well damped out after a

small signal or transient fault without affecting the first- swing stability of the system.

Figures 6.12a and 6.12b illustrates the response of the rotor angle to a transient disturbance

simulated by a three phase fault applied to the system. This form of instability is greatly pronounced

when there is insufficient damping of the oscillations during or after the fault among other factors.

The use of the classical model to represent the synchronous machine further neglects all sources of

damping. The effect of damping can be seen clearly when the results gotten in [6] is compared with

our results. In [6], a damping factor D of 0.05 was used whereas we have used a damping factor of

1.2, thus the difference in our plots of the rotor angle. The value of the damping determines how fast

a

b

51

a stable solution will converge to its equilibrium point. For real time power system stability, power

system stabilizers are used to enhance system stability by controlling the generator excitation system

to obtain better stability in the presence of transient or fault disturbances. Such controls are also

expected to improve small signal disturbance stability. The power system is considered to be a closed

loop system. The estimated values and measured power output are used as feedback controls in the

algorithm during the update process to correct the estimated states and to ensure that the

estimated data is as close as possible to the real data.

52

7 Conclusions and future work

7.1 Conclusions

This work has considered dynamic states estimation of a synchronous machine using the UKF

algorithm for the filtering process. We were saddled with the task of estimating two dynamic states

of a power system represented by a modeled SMIB system. We considered the fact that during

transient conditions, the dynamic states of the power system are not adequately captured by the

regular observer methods. We also considered that the static states of the power system, such as

voltage magnitude and angle, which are often used for power system stability studies and analysis,

do not give adequate representation of the system especially during transient conditions. We then

presented the UKF filtering algorithm as a better filter for the estimation of dynamic states. We

focused on estimating the generator rotor angle and the generator rotor speed, considering that

these dynamic states of the synchronous machine are only a few of the many dynamic states of the

power system network which can be estimated. We also considered that it would take considerable

amount of work and a huge amount of variables and data to estimate all states of the power system.

The proposed algorithm utilized the measured field voltage available from the generator as well as

the terminal measurements obtained from a PMU assumed to be installed at the terminal bus of the

synchronous machine. The proposed methodology was implemented using MATLAB embedded

function block and has been tested on an SMIB system, modeled in Simulink. We evaluated its

performance based on how well the estimated states tracked the real system states. We also

compared it to the simulations done with the EKF method as well as similar simulations and analysis

performed by other authors using the EKF.

Our results showed that our proposed filtering algorithm tracked the states better than the EKF

algorithm. Through our simulations, analysis and results, we have clearly shown that for practical

power systems network, the addition of PMU technologies to the UKF based dynamic state estimator

provides very good accuracy when estimating dynamic states of a power system.

7.2 Future work

For this thesis work, we have considered state estimation in power systems by narrowing our

estimation to two states of the synchronous machine. We have assumed that the PMU is installed at

the terminal bus closer to the synchronous machine, allowing us to take measurements of the

magnitude and phasor of the three phase voltage and current (&:ϼ and =:ϼ) required for the

calculation of active power �#, &# =# needed by the UKF estimator. It has also been assumed that the

53

exciter voltage !81, and mechanical torque �� as well as the transient voltage of the q axis is !�� are

measureable.

• Perhaps, further studies can be done on the possibility of estimating the states of the

synchronous machine using readings from PMUs installed at various buses along the

transmission lines of the system. This would be important for contingency analysis should there

be loss of the PMU located at the terminal bus of the synchronous machine.

• Secondly, we have considered a SMIB system for this work. It could of interest to study the

performance of the UKF on our modeled system but with more than one synchronous machine.

Perhaps, this algorithm could also be implemented on the nine bus system or the 14 bus system.

• In addition, considering that the function R of the system is nonlinear, conversion from the

continuous form to a discrete form is needed. An approach to solving this is using the 4th

order

Runge-kutta method which numerically integrates the state-space equations. We have

approached with a slightly different method. The effect of the Runge-kutta method on the

estimation algorithm could be investigated to ascertain if the estimates can be improved using

the Runge-Kutta method for numerical integration of the state-space equations.

54

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57

Appendix A: Holt’s two initialization

The Holt’s two method was initialized by the first two samples, at time instant � = 0 and � = 1, of

voltage magnitude and angles, estimated from previous estimation. It is thus assumed that 2. and 26

are known. It is now our desire to estimate the state vector 2 at instant � = 2. Given that

@(�) = ®p2¥(�) + (1 − ®p)2̅(�) P(�) = p̄[@(�) − @(� − 1)] + (1 − p̄)P(� − 1) (A.1)

we calculate @(�) and P(�) for instants � = 0 and � = 1. We assume that 2̅(1) = 2¥(1), i.e., the

prediction at time instant � = 1 was as accurate as needed. We also set the initial values as @. = 2.@-�P. = 0. We then had the task of calculating @6@-�P6 as

@(1) = ®2¥(1) + (1 − ®)2̅(1) P(1) = ¯[@(1) − @(0)] + (1 − ¯)P(0) (A.2)

Haven obtained these values, we can then proceed to calculate 6̈and n6 as (8.3);

¨(1) = ®(1 + ¯)= n(1) = (1 + ¯)(1 − ®)2̅(1) − (1 − ¯)P(0) (A.3)

The prediction at time instant � = 2 is then given as

2̅(2) = ¨(1)2(1) + n(1) (A.4)

The updated state vector is subsequently obtained using a UKF or EKF filtering process. When new

measurements eventually arrive at instant � = 3, @(�) and P(�) needs to be calculated for instant

k=2 as

(2) = ®2¥(2) + (1 − ®)2̅(2) P(2) = ¯[@(2) − @(1)] + (1 − ¯)P(1) (A.5)

It should be noted that the value for ® and ¯ are kept constant for all successive calculations. Also,

for time instants � = 2 and above, the prediction vector 2̅(�) is calculated by the UKF or EKF as the

case may be.

58

Appendix B: Gradient calculation for EKF algorithm

method 26 =�.2� (B.1) R6(2, 7) = �.2� 26(�) = ∆� x R6(2, 7, �) +26(� − 1) = ∆� x �.2� +26(� − 1) Ð26(�)Ð2 = [ 6̈6 6̈�]

6̈6 =Ð26(�)Ð26 = 1

6̈� = °�¢(p)°�£ = ∆�xω.

2� = 6Ò x M�� − Ì&���′ !"′ +,-21 + &�22 Ì 1�" − 1��′ Î sin 221Î − �22N (B.2)

R�(2, 7) = 1 x ��� − I&���′ !"′ +,-21 + &�22 I 1�" − 1��′ J sin 221J − �22  2�(�) = ∆� x R�(2, 7, �) +2�(� − 1) Ð2�(�)Ð2 = [ �̈6 �̈�] �̈6 = Ð2�(�)Ð26 = −∆� I&���′ !"′ A$+21 + &�2 I 1�" − 1��′ J cos 221J

�̈� = Ð2�(�)Ð26 = −∆� [– �] + 1

p̈g6 = M 6̈6 6̈��̈6 �̈�N p̈g6 = Õ 1 ∆�xω.g∆#Ò Ì&���′ !"′ A$+21 + &�2 Ì 1�" − 1��′ Î cos 221Î g∆#Ò �– �� + 1Ö (B.3)

The output equation of the system is given as:

Op = ℎp(2p , 7p , �p) (B.4)

O6 = ℎ6 = Ì'()Ë* !�� +,-26 + '(£� Ì 6)Í − 6)Ë*Î sin 226Î (B.5)

p is calculated as: p = [666�] 66 =Ðℎ6(�)Ð26 = I&���′ !"′ A$+21 + &�2 I 1�" − 1��′ J cos 221J

59

6� = Ðℎ6(�)Ð2� = 0

Therefore p is

p = [Ì&���′ !"′ A$+21 + &�2 Ì 1�" − 1��′ Î cos 221Î 0] (B.6)

60

Appendix C: Machine parameters used and states to be

estimated

D,J Damping factor, Inertia constant in per unit �1�� , ���� d and q transient open circuit time constant �:, 2� Stator resistance and stator leakage reactance 21 , 2� Direct and quadratic-axis reactance 21� , 2�� Direct and quadratic-axis transient reactance �� , �� Line reactance and resistance 2:1 , 2:� Direct and quadratic-axis mutual reactance 2:1× , 2:�× Unsaturated d and q axis mutual reactance � 1st state, rotor angle with respect to machine

terminals(load angle)

∆�,�. 2nd

state, rotor speed and nominal synchronous

speed �� , �� Mechanical input and electric torque !1 , !� Direct and quadratic axis voltage =1 , =� Direct and quadratic axis current &#(!#), =# Terminal bus voltage and current & (! ) Infinite bus voltage �# , X# Terminal bus active and reactive power ∅ Power factor angle !81 Steady state internal voltage of armature !1� , !�� Transient voltage of d and q axis ÆÙ1 , ÆÙ� Direct and quadratic saturation factors 281 Field circuit reactance

61

Appendix D: UKF algorithm function xest = UKF(Pt, Tm, Efd)

% Input % Pt Active power at synchronous machine terminal bus (in pu) % Tm Mechanical input % Ef Armature steady state internal voltage(exciter voltage) % % Output % xest State estimate x = [rotor angle; rotor speed]

y=Pt; lx = 2; % length x ly = length(y); % length y

% r1 = 0.01^2*eye(1); % r2 = 0.01^2*eye(1);

u=[Tm;Efd]; Xq=1.214; Xprime_d = 0.8808; Eprime_q = 1.0566; Vt = 1; angular_0 =377; H=13; D = 1.2;

% UKF Parameters/variables delta_t=0.001; alpha = 0.24; kappa = 0; beta = 2;

lambda = alpha^2*(lx+kappa)-lx; gamma = sqrt(lx+lambda);

xm = zeros(lx,1); ym = zeros(ly,1); Pm = zeros(lx); xsi = zeros(lx,2*lx+1); XSI = zeros(lx,2*lx+1); Y = zeros(ly,2*lx+1); Pxy = zeros(lx,ly); Pyy = zeros(ly);

Wc = lambda/(lx+lambda)+(1-alpha^2+beta); Wm = lambda/(lx+lambda); Wmi = 1/(2*(lx+lambda)); Wci = 1/(2*(lx+lambda));

% state vector x=[0.6;0.5];

% lemgth of state vector lx=length(x);

62

% Error covariance matrix; P0=diag([10,10]); P=P0;

% Standard deviations std_x = 1e-3; std_y = 1e-2; Q = (std_x)^2*eye(lx); R = (std_y)^2;

Qk = Q; % Process noise covariance Rk = R; % Measurement noise covariance

% Calculates sigma points xsi(:,1) = x;

for i = 1:lx A = chol(P); % R = chol(A) where R'*R = A xsi(1:lx,i+1) = x+gamma*A(:,i); xsi(1:lx,i+1+lx) = x-gamma*A(:,i); end

% Machine(process) differential functions (continious form) % f(1) = angular_0*x(2); % f(2) = (1/2*H)*(u(1)-((Eprime_q*Vt)/Xprime_d)*sin(x(1))+

((((Vt)^2)/2)*((1/Xq)-(1/Xprime_d))*sin(2*(x(1)))-D*(x(2));

% Nonlinear system(process) equation (discrete form) % x(k+1) = x(1)+ angular_0*(x(2))* dt +r1*randn; % x(k+1) = x(1)+ ((1/2*H)*(u(1)-((Eprime_q*Vt)/Xprime_d)*sin(x(1))+

((((Vt)^2)/2)*((1/Xq)-(1/Xprime_d))*sin(2*(x(1)))-D*(x(2)));

% Time Update for i = 1:2*lx+1 XSI(1:lx,i) = [xsi(1,i)+angular_0*(xsi(2,i))*delta_t ;

xsi(2,i)+(1/2*H)*(u(1)-

(((Eprime_q*Vt)/Xprime_d)*sin(xsi(1,i))+(((Vt)^2)/2)*((1/Xq)-

(1/Xprime_d))*sin(2*(xsi(1,i))))-D*(xsi(2,i)))*delta_t ]; end

% Calculates mean for i = 1:2*lx+1 if i == 1 W = Wm; else W = Wmi; end xm = xm+ W*XSI(:,i); end

% Calculates covariance for i = 1:2*lx+1 if i == 1 W = Wc; else W = Wci;

63

end Pm = W*(XSI(:,i)-xm)*(XSI(:,i)-xm)'+Qk; end

xsi(:,1) = xm;

for i = 1:lx xsi(1:lx,i+1) = xm+gamma*Pm(:,i); xsi(1:lx,i+1+lx) = xm-gamma*Pm(:,i); end

% Output/measurement equation(function)

h =((Eprime_q*Vt)/Xprime_d)*sin(xsi(1)) + ((Vt)^2*((1/Xq)-

(1/Xprime_d))*sin(2*(xsi(1)))); % z = ((Eprime_q*Vt)/Xprime_d)*sin(x(1))+((Vt)^2*((1/Xq)-

(1/Xprime_d))*sin(2*(x(1))));

% Measurement Update for i = 1:2*lx+1 Y(1:ly,i)=((Eprime_q*Vt)/Xprime_d)*sin(xsi(1,i))+(((Vt)^2)/2)*((1/Xq)-

(1/Xprime_d))*sin(2*(xsi(1,i))); end

for i = 1:2*lx+1 if i == 1 W = Wm; else W = Wmi; end ym = ym+ W*Y(:,i); end

for i = 1:2*lx+1 if i == 1 W = Wc; else W = Wci; end Pyy = Pyy+ W*(Y(:,i)-ym(:,ones(1,size(XSI,2))))*(Y(:,i)-

ym(:,ones(1,size(XSI,2))))'+Rk; Pxy = Pxy+ W*(XSI(:,i)-xm(:,ones(1,size(x,2))))*(Y(:,i)-

ym(:,ones(1,size(x,2))))';

end

K = Pxy/Pyy; x_k = xm+K*(y-h); Pest = Pm-K*Pyy*K';

xest = x_k;