Multirate Output Sampling Based Load-Frequency Control

A thesis submitted in partial fulfillment ofthe requirements for the degree of

Bachelor of Technology

Submitted by Supervised byPoonam Chand (2012UEE1378) Dr. Satyanarayana NeeliRam Raj (2012UEE1216) Assistant Professor Dharmpal kumar (2012UEE1488) Department of EE Shubham Gupta (2012UEE1618) MNIT Jaipur

Department of Electrical EngineeringMalaviya National Institute of Technology Jaipur

May, 2016

Malaviya National Institute of Technology, Jaipur

Certificate

This is to certify that following students of Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur, have undergone a project titled “ Multirate Output Sampling Based Load-Frequency Control ” under my guidance. During this project their performance is found

Poonam Chand (2012UEE1378) satisfactory/unsatisfactory Ram Raj (2012UEE1216) satisfactory/unsatisfactory Dharmpal kumar (2012UEE1488) satisfactory/unsatisfactory Shubham Gupta (2012UEE1618) satisfactory/unsatisfactory

Dr. Satyanarayana Neeli Assistant professor Department of EE MNIT Jaipur

Malaviya National Institute of Technology, Jaipur

Candidate’s Declaration

We hereby declare that this project report on “Multirate Output Sampling Based Load-Frequency Control” which is being submitted in partial fulfillment of the award of degree Bachelor of Technology in Electrical Engineering, is the result of work carried out by us under the guidance of Dr. Satyanarayana Neeli, Assistant Professor of Malaviya National Institute of Technology, Jaipur. We further admit that this project work has not been submitted to MNIT before or for other purpose.

Poonam Chand Ram Raj(2012UEE1378) (2012UEE1216)

Dharmpal kumar Shubham Gupta(2012UEE1488) (2012UEE1618)

Date: Place:

Malaviya National Institute of Technology, Jaipur

Acknowledgement

We are highly indebted to our supervisor Dr. Satyanarayana Neeli, Assistant Professor, Department of Electrical Engineering and would like to express our special gratitude for his guidance and constant supervision as well as for providing necessary information regarding the project and also for his support in completing the project.We would like to thank Dr. Vikas Gupta, Head of Electrical Engineering Department for giving us this opportunity to do this project work.We would also like to thank Project Coordinator, Mr. Vinod Sahai Pareek, Associate Professor, Department of Electrical Engineering and Ms. Nikita Jhanjariya, Associate Professor, Department of Electrical Engineering for giving us this opportunity to explore new area and enhance our knowledge.We would also like to thank our institution and faculty members, without whom this project would have been a distant.

Poonam Chand Ram Raj(2012UEE1378) (2012UEE1216)

Dharmpal kumar Shubham Gupta(2012UEE1488) (2012UEE1618)

ABSTRACT

In an interconnected power system, if a load demand changes randomly, both frequency and tie line power varies. The main aim of load frequency control is to minimize the transient variations in these variables and also to make sure that their steady state errors is zero. Many modern control techniques are used to implement a reliable controller. The objective of these control techniques is to produce and deliver power reliably by maintaining both voltage and frequency within permissible range. When real power changes, system frequency gets affected while reactive power is dependent on variation in voltage value. That’s why real and reactive power are controlled separately. Control of load frequency controls the active power. The role of automatic generation control (AGC) in power system operations with reference to tie line power under normal operating conditions is analyzed. Future power systems will rely on large amounts of distributed generation with large percentage of renewable energy based sources, what will further increase system uncertainties and thereby induce new requirements to the LFC system. Power systems consist of control areas representing a coherent group of generators i.e. generators which swing in unison characterized by equal frequency deviations. In addition to their own generations and to eliminate mismatch between generation and demand these control areas are interconnected through tie-lines for providing contractual exchange of power under normal operating conditions. One of the control problems in power system operation is to maintain the frequency and power interchange between the areas at their rated values. Automatic generation control is to provide control signals to regulate the real power output of various electric generators within a prescribed area in response to changes in system frequency and tie-line loading so as to maintain the scheduled system frequency and established interchange with other areas. The report presents a full state feedback controller for load-frequency control (LFC) in control areas (CAs) of a power system. As it uses full-state feedback it can be applied for LFC not only in CAs with thermal power plants but also in CAs with hydro power plants, in spite of their non-minimum phase behaviours. To enable full-state feedback we have proposed a state estimation method based on fast sampling of measured output variables, which are frequency, active power flow interchange and generated power from power plants engaged in LFC in the CA. The same estimation method is also used forth estimation of external disturbances in the CA, what additionally improves the overall system behaviour.

Table of Content

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s1. Introduction 3

1.1 Load Frequency Control 31.2 Need of Maintaining Constant Frequency 21.3 Frequency power characteristic of synchronous generator 2

1.3.1 Concept of Load-Frequency Control 41.4 Power Swing Equation 41.5 Automatic Load Frequency Control 5

2. Modeling of ALFC 72.1 Introduction 72.2 Discussion on Speed Governor Model 72.3 Turbine Model 112.4 Power System Model 112.5 State Space Model for single Area System 14

3. State Space and Multirate Output Sampling 153.1 Introduction to state space 15

3.1.1 General state-space model 163.2 Solution of Continuous Time State Space Model 173.3 Importance of Discrete Time Controller 183.4 Discretization of the continuous time system 183.5 Numerical Examples 19

3.5.1 Example 1 193.5.2 Example 2 20Discretization of continuous system when input is exponential without disturbance 20

3.5 Multirate output sampling 224. Controller Design 24

4.1 Discretization of Continuous Time System with Disturbance 244.1.1 Example of Discretization of the System With Disturbance 25

4.2 Fast Output Sampling Method 274.3 Design Procedure 29

4.3.1 State Feedback Controller 294.3.2 Numerical Example 31

5. Conclusions 33Appendix 34References 37

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1. Introduction

1.1 Load Frequency Control

Electricity is a form of energy and we need it for just about everything. Almost all of our modern conveniences are electrically powered. Along with the growth in demand for electric power, sustainable development, environmental issues, and power quality and reliability have become concerns. Electric utilities are becoming more and more stressed since existing transmission and distribution systems are facing their operating constraints with growing load.

A quality of power system can be judged by maintain of voltage and frequency at desired values irrespective of change in loads. It is in fact impossible to maintain both active and reactive power without control, which would result in variation of voltage and frequency levels. To cancel the effect of load variation and to keep frequency and voltage level constant, a control system is required. Though the active and reactive powers have a combined effect on the frequency and voltage, the control problem of the frequency and voltage can be separated. Frequency is mostly dependent on the active power and voltage is mostly dependent on the reactive power. The successful operation of interconnected power systems requires the matching of total active power generation with load demand and associated power system losses. As the demand deviate from its nominal value with an unpredictable small amount, the operating point of power system changes, and hence system may experience deviations in system frequency and scheduled tie line power exchanges, which may yield undesirable affects [14]. Thus, the issue of controlling power systems can be separated into two independent problems. The active power and frequency control is called as load frequency control (LFC). It has gained the importance with change in power system structure and the growth of size and complexity of interconnected systems [15].

The most important task of LFC is to maintain the frequency constant against the varying active power loads, which is also referred as unknown external disturbance. This is done by minimizing transient deviations of frequency in addition to tie-line power exchange and also making the steady state error to zero [8]. It has been shown that for small changes active power is dependent on internal machine angle δ and is independent of bus voltage: while bus voltage is dependent on machine excitation (therefore on reactive generation Q) and is independent of machine angle δ . Change in angle δ is caused by momentary change in generator speed. Therefore, load frequency and excitation voltage controls are non-interactive for small changes and can be modeled and analyzed independently. Changes in load demand can be identified as: (i) slow varying changes in mean demand, and (ii) fast random variations around the mean. The regulators must be designed to be insensitive to fast random changes, otherwise the system will be prone to haunting resulting in excessive wear and tear of rotating machines and control equipment.

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1.2 Need of Maintaining Constant Frequency

The speed of the A.C motors depends on the frequency of the power supply. There are situations where speed consistency is expected to be of high order. In the case of turbine, if the normal operating frequency is 50 Hz and the turbines run at speeds corresponding to frequencies less than 49.9 Hz or above 50.1 Hz, then the blades of the turbines may be damaged. The operation of a transformer below the rated frequency is not desirable. When frequency goes below rated frequency at constant system voltage then the flux in the core increases and then the transformer core goes into the saturation region. Due to that power transformer results in low efficiency and over-heating of the transformer windings.

Change in frequency causes change in speed of the consumer’s plant affecting production processes. Further, it is necessary to maintain network frequency constant so that power stations run satisfactorily in parallel, the various motors operating on the system run at the desired speed [1]. The electric clocks are driven by the synchronous motors. The accuracy of the clocks are not only dependent on the frequency but also is an integral of the frequency error The most serious effect of subnormal frequency operation is observed in the case of Thermal Power Plants. Due to the subnormal frequency operation the blast of the ID (induced draft) and FD (forced draft) fans in the power stations get reduced and thereby reduce the generation power in the thermal plants. This phenomenon has a cumulative effect and in turn is able to make complete shutdown of the power plant if proper steps of load shedding technique is not engaged. It is pertinent to mention that, in load shedding technique a sizable chunk of load from the power system is disconnected from the generating units to restore the frequency to the desired level.

1.3 Frequency power characteristic of synchronous generator

Since synchronous generators are the most common type of machines used in the generation of electrical power, its characteristics can be used to describe the relationship between frequency and power during load changes. The most common type of prime mover is a steam turbine, but other types include diesel engines, gas turbines, water turbines, and even wind turbines. Regardless of the original power source, all prime movers tend to behave in a similar manner. Any imbalance of power between generation and consumption the speed of generator will vary corresponding. The decrease in speed is in general non-linear, but some form of governor mechanism is usually included to make the decrease in speed linear with an increase in power demand.

The Speed Droop (SD) of a prime mover is defined by

SD =N nl−N fl

N fl100 % (1.1)

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Where,N nl=¿ No-load prime-mover speed N fl=¿the full-load prime-mover speed.Most generator prime movers have a speed droop of 2 to 4 percent, as defined in equation (1.1).

In addition, most governors have some type of set point adjustment to allow the no-load speed of the turbine to be varied. A typical speed-versus-power plot (known as the house curve)

Fig1.1 Speed-power and frequency-power curves (house curves)

Although the house curves are only used for studying the parallel operation of two generators or that of a single generator connected to a certain network, it helps understanding the variations of electrical frequency as the power demanded is changed, since the shaft speed is related to the resulting electrical frequency by the equation (1.2).

f e=N m P120

(1.2)

Then the power output of a synchronous generator is related to its frequency and this is clear in fig 1.1.The relationship between frequency and power can be described quantitatively by the equation P=S p [ Fnl−FSYS ] (1.3)where, P = power output of the generator Fnl= no-load frequency of the generator FSYS = operating frequency of system Sp= slope of curve, in kW/Hz or MW/ HzBut this equation is not accurate for multi-area power systems. A similar relationship can be derived for the reactive power Q and terminal voltageV T , for which the AVR control loop is used.

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1.3.1 Concept of Load-Frequency Control

In the steady state operation of power system, the load demand is increased or decreased in the form of Kinetic Energy stored in generator prime mover set, which results the variation of speed and frequency accordingly. Therefore, the control of load frequency is essential to have safe operation of the power system.Neglecting resistances

P= EVX

sin δ (1.4)

Where E is the excitation voltage, V is the terminal voltage, X is the effective reactance and δ is the power angle.If δ changes to +∆ δ , then P changes to P+∆ P

P+∆ P=EVX

sin ( δ+∆ δ )

¿EVX

¿ (1.5)

Since ∆ δ is very small. cos ∆ δ ≅ 1 and sin ∆ δ ≅∆ δ

P+∆ P=EVX

sin δ+ EVX

cosδ . ∆ δ

So ∆ P = EVX

cosδ . ∆ δ (1.6)

Or ∆ P∝∆ δSmall power changes mainly depends on ∆ δ or ∆ f .

1.4 Power Swing Equation

Transient stability in power system are done over a very small period of time equal to the time required for one swing, which approximates to around 1 sec or even less [1]. When the synchronous generator is fed with a supply from one end and a constant load is applied to the other, there is some relative angular displacement between the rotor axis and the stator magnetic field, known as the load angle δ which is directly proportional to the loading of the machine. The machine at this instance is considered to be running under stable condition. Now if we suddenly add or remove load from the machine the rotor decelerates or accelerates accordingly with respect to the stator magnetic field. The operating condition of the machine now becomes unstable and the rotor is now said to be swinging with respect to the stator field and the equation we so obtain giving the relative motion of the load angle δ with respect to the stator magnetic field is known as the swing equation for transient stability of power system. Here for the sake of understanding we consider the case where a synchronous generator is suddenly applied with an increased amount of electromagnetic load, which leads to instability by making Pe less than Pm as the rotor undergoes deceleration. Now the increased amount of the accelerating power required to bring the machine back to stable condition is given by,

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Accelerating power Pa=Pm−Pe

Accelerating torque is given by T a=T m−T e

Pa=M .αwhere, M=I .ωM is angular momentum. I is moment of inertia andωis angular displacement .

θ=ωS+dδdt

Angular acceleration is given by

α=d2θd2 t

=d2 δd2t

Now, Swing Equation is written as

M d2 δd2t

=Pm−Pe (1.7)

where, electric power supplied by generator is given by

Pe=E VX

sinδ= Pmax sinδ (1.8)

Putting the value ofPe in equation (1.7)

M d2 δd2t

=Pm−Pmax sinδ (1.9)

where, Pm= mechanical power input in MW Pe= electrical power output in MW M = also called the inertia constant in MJ/MVA. δ= rotor angular displacement from synchronously rotating reference frame (called torque angle/power angle)

Fig 1.2 Flow of mechanical and electrical powers in a synchronous machine

1.5 Automatic Load Frequency Control

The main purpose of operating the load frequency control is to keep control the frequency during the load changes. During the power system operation rotor angle, frequency and power are the subjected

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parameters to variable. Changes in real power mainly affect the system frequency; in reactive power mainly depend on variable of voltage magnitude. Hence, control of frequency and voltage can be achieved separately Thus, real and reactive powers can be controlled separately.

The Automatic Load Frequency Control (ALFC) controls the real power and the Automatic Voltage Regulator (AVR) regulates the voltage magnitude and hence the reactive power. The basic requirement is to take care of megawatt power output of a generator matching with the changing load and appropriate value of exchange of power linking to the control areas.it aims to facilitate control of the frequency for larger interconnection.

A power system may be an interconnected system of multiple areas or an isolated system comprising of single service area. The LFC plays an important role in both types of power systems. A single area power system is the one, which comprises of a single generator supplying power to a single service area. The function of LFC in a single area power system is to restore the frequency to the specified nominal value in case of any fluctuation. However, in case of an interconnected power system, two or more independently controlled areas are connected together. In such systems, along with frequency, generation within each area also has to be controlled. This is required to maintain the scheduled power interchange. So, the main aim of the load frequency control in multi area power systems is to regulate the frequency to the specified nominal value and to maintain the interchange power between areas at the scheduled values. However, in case of an interconnected power system, two or more independently controlled areas are connected together. In such systems, along with frequency, generation within each area also has to be controlled. This is required to maintain the scheduled power interchange. So, the main aim of the load frequency control in multi area power systems is to regulate the frequency to the specified nominal value and to maintain the interchange power between areas at the scheduled values [5].

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2. Modeling of ALFC

2.1 Introduction

The control mechanism needed to maintain the system frequency. The maintaining of system frequency constant is commonly known as AUTOMATIC LOAD FREQUENCY CONTROL (ALFC). It has other nomenclatures such as Load Frequency Control, Power Frequency Control, Real Power Frequency Control and Automatic Generation Control. To maintain the desired megawatt output power of a generator matching with the changing load and it assist in controlling the frequency of larger interconnection. In order to keep the net interchange power between pool members, at the predetermined values. The ALFC loop will maintain control only during small and slow changes in load and frequency. It will not provide adequate control during emergency when large megawatt imbalances occur.Figure 2.1 shows schematically the speed governing system of a steam turbine. The system consists: speed changer, speed governor, hydraulic amplifier, and control valve.

2.2 Discussion on Speed Governor Model

Fly Ball Speed Governor:

This is the heart of the system, which senses the change in speed (frequency). As the speed increases, the fly balls move outwards and the point B on linkage mechanism moves downwards. The reverse happens when the speed decreases.

Hydraulic Amplifier:

It comprises a pilot valve and main piston arrangement. Low power level pilot valve movement is converted into high power level piston valve movement. This is necessary in order to open or close the steam valve against high-pressure steam.

Linkage Mechanism:

ABC is a rigid link pivoted at B and CDE is another rigid link pivoted at D. This link mechanism provides a movement to the control valve in proportion to change in speed. It also provides a feedback form the steam valve movement (link 4).

Speed Changer:

It provides a steady state power output setting for the turbine. Its downward movement opens the upper pilot valve so that more steam is admitted to the turbine under steady conditions (hence more steady power output). The reverse happens for the upward movement of speed changer.

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Their incremental movements are in vertical direction. In reality these movements are measured in millimeters; however analysis we shall rather express them as power increments expressed in MW or pu MW. The movements are assumed positive in the directions of arrows. Corresponding to “raise” command, linkage movements will be: “A” moves downwards; “C” moves upwards; “D” moves upwards; “E” moves downwards. This allows more steam or water flow into the turbine resulting incremental increase in generator output power. When the speed drops, linkage point “B” moves upwards and again generator output power will increase [2].

Fig. 2.1 Turbine Speed Governing System

Speed Governor:

When the electrical load is suddenly increased then the electrical power exceeds the mechanical power input. Because of this, the deficiency of power in the load side is extracted from the rotating energy of the turbine. Due to this reason, the kinetic energy of the turbine i.e. the energy stored in the machine is reduced and the governor sends a signal to supply more volumes of water or steam or gas to increase the speed of the prime mover to compensate speed deficiency.

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Assume that the system is initially operating under steady conditions-i.e., linkage mechanism stationary and pilot valve closed, steam valve opened by a definite magnitude, turbine running at constant speed with turbine power output balancing the generator load. The operating conditions be characterized byf 0, PG

0, Y E0.

where, f 0=¿System frequency (speed) PG

0= generator output = turbine output (neglecting generator loss) Y E

0= steam valve setting We have to obtain a linear incremental model around these operating conditions.Let the point A on the linkage mechanism be moved downwards by a small amount∆ Y A. It is a command, which causes the turbine power output to change to mathematically, represented as∆ Y A=KC ∆ PC (2.1)Where ∆ PC is the commanded increase in the power. KC is the turbine constant.The command signal ∆ PC sets into motion a sequence of events-the pilot valve moves upwards, high pressure oil flows on to the top of the main piston moving it downwards; the steam valve opening consequently increases, the turbine generator speed increases. i. e. the frequency goes up. We can model these events mathematically.The two factors contribute to the movement of C are

(i) ∆ Y Acontributes -( l2

l1)∆ Y A or -k1 ∆ Y A (i.e. upwards) of -k1 KC ∆ PC.

(ii) Increase in frequency ∆ f causes the fly balls to move outwards so that B moves downwards by a proportional amountk 2

' ∆ f . The consequent movement of C with A remaining fixed.

∆ Y A=¿ ( l1+l2

l1)k 2

' ∆ f = k 2∆ f

The net movement of C is ∆ Y C=−k1 KC ∆ PC+k2 ∆ f (2.2) The movement of D (∆ Y D ¿ , is the amount by which the pilot valve opens. It is contributed by ∆ Y Cand ∆ Y E and can be written as

∆ Y D=( l4

l4+l3)∆ Y C+( l3

l4+ l3)∆Y E

=k3 ∆ Y C+K 4 ∆ Y E (2.3)

The movement ∆ Y Ddepending upon its sign opens one of the ports of the pilot valve admitting high-pressure oil into the cylinder thereby moving the main piston and opening the steam valve by ∆ Y E. Certain justifiable simplifying assumptions, which can be made at this stage, are –

(i) Inertial reaction forces of main piston and steam valve are negligible compared to the forces exerted on the piston by high-pressure oil.(ii) Because of (i) above the rate of oil admitted to the cylinder is proportional to port opening∆ Y D.

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The volume of oil admitted to the cylinder is thus proportional to the time integral of ∆ Y D. The movement ∆ Y Eis obtained by dividing the oil volume by the area of the cross-section of the-piston. Thus

∆ Y E=k5∫0

t

(−∆ Y D ) dt (2.4)

It can be verified from the schematic diagram that a positive movement∆ Y D, causes negative (upward) movement ∆ Y E accounting for negative sign used in equation (2.4).Taking Laplace transform of equations (2.2), (2.3) and (2.4), we get∆ Y C(s)=−k1 KC ∆ PC (s)+k2 ∆ F (s) (2.5)∆ Y D (s )=k3 ∆ Y C(s)+k 4 ∆ Y E(s ) (2.6)

∆ Y E (s )=−k51s

∆Y D(s) (2.7)

Eliminating∆ Y C(s) and∆ Y D(s), we can write

∆ Y E (s )=

k1 k3 KC ∆ PC ( s )−k2 k3 ∆ F (s)

(k 4+sk5 )

= [∆ PC (s )− 1R

∆ F (s)]∗( K sg

1+T sgs ) (2.8)

Where

R= k1 K C

k2=¿ speed regulation of the governor (in Hz/Mw)

K sg=k1k 3 KC

k4=¿gain of speed governor

T sg=1

k4 k 5=¿time constant of speed governor

We are considering hydraulic valve actuator as a part of Governor model. Therefore, the block diagram of Governor will be as shown in Fig 2.2

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Fig 2.2

where, G H=K sg

1+T sg s

2.3 Turbine Model

In normal steady state, the turbine power PT keeps balance with the electromechanical air-gap power Pg resulting in zero acceleration and a constant speed and frequency. During transient state, let the change in turbine power be ∆ PT and the corresponding change in generator power be ∆ Pg. The accelerating power in turbine generator unit = ∆ PT−∆ Pg. The turbine power increment ∆ PTdepends entirely upon the valve power increment ∆ PV and the characteristic of the turbine. Different type of turbines will have different characteristics. Taking transfer function with single time constant for the turbine, we can write

∆ PT ( s )=GT ∆ PV ( s)= 11+T T s

∆ PV (s )

Where, the turbine time constant T Tis in the range of 0.2 to 2.0 seconds.

The generator power increment ∆ Pg depends entirely upon the change ∆ PD in the load PD being fed from the generator. The generator always adjusts its output so as to meet the demand changes ∆ PD. These adjustments are essentially instantaneous, certainly in comparison with the slow changes in PT . The help of previous three equations updates the block diagram developed updated as shown in Fig. 2.2. This corresponds to the linear model of primary ALFC loop excluding the power system response.

Fig. 2.3

2.4 Power System Model

We observed earlier that the loop in Fig. 2.3 is “open”. We now proceed to “close” it by finding a mathematical link between ∆ PT and Δf. As our generator is supplying power to a conglomeration of loads in its service area, it is necessary in our following analysis to make reasonable assumptions about

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the “lumped” area behavior. We make these assumptions: The system is originally running in its normal state with complete power balance, that is, Pg

0=PD0 +losses. The frequency is at normal value f 0. All

rotating equipment represents a total kinetic energy of W kin0 MW sec. By connecting additional load, load

demand increases by ∆ PD which we shall refer to as “new” load. (If load demand is decreased, new load is negative). Then, generation immediately increases by ∆ Pg to match the new load, that is ∆ Pg=∆ PD. It will take some time for the control valve in the speed governing system to act and increase the turbine power. Until the next steady state is reached, the increase in turbine power will not be equal to ∆ Pg. Thus, there will be power imbalance in the area that equals i.e. ∆ PT−∆ Pg. As a result, the speed and frequency change. This change will be assumed uniform throughout the area. The above said power imbalance gets absorbed in two ways.(i) By the change in the total kinetic K.E.(ii) By the change in the load, due to change in frequency.

W kin=W kin0 ( f

f 0 )2

MW Sec.

We know that load demand change (∆ PD) is met by two changes in the system,1. Increased generation ∆ Pg due to opening of steam conditions.

2. Load decrement due to drop in system frequency (D=dPD

df).

The “old” load is a function of voltage magnitude and frequency. Frequency dependency of load can be written as

D=dPD

df

∆ PT−∆ PD=d (W kin)

dt+D ∆ f (2.9)

By solving kinetic energy equation

d (W kin )

dt=

2W kin0

f 0d(∆ f )

dtSubstituting above value in equation (2.9)

∆ PT−∆ PD=2W kin

0

f 0d (∆ f )

dt+D ∆ f

By dividing this equation by the generator rating Pr and by introducing per-unit inertia constant

H=W kin

0

Pr

Where, H is per unit inertia constant, Pr is generator rating.It takes on the form

∆ PT−∆ PD=2 Hf 0

d (∆ f )dt

+D ∆ f pu MW

Laplace transformation of the above equation yields

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∆ PT ( s )−∆ PD (s )=(2 Hf 0 s+D)∆ f (s)

∆ f (s )=GP ( s )¿] (2.10)

where,

GP ( s )= 1

2 Hf 0 s+D

=K P

1+sT P

K P=1D , T P=

2 Hf 0 D

Equation (2.10) represents the missing link in the control loop of Fig. 2.3. By adding this, block diagram of the primary ALFC loop is obtained as shown in Fig. 2.4

∆ Pg (s )=∆ PT ( s) ∆ PD(s)

+ ∆ XE ( s ) -∆ f (s )

∆ PC (s ) - + +

Fig 2.4. – block diagram of load frequency control (isolated power system)

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K sg

1+T sgs

K t

1+T t sK ps

1+T ps s

1R

The transfer function of ALFC system can be calculated by taking ∆ PC as input and ∆ f as output of the system and assuming ∆ PD zero (means no load disturbance).Therefore, the transfer function of system will be given by

∆ f (s )

∆ PC(s)=

K sg K t K ps

(1+T gs s ) (1+T t s ) ( 1+T Ps s)+K sg K t K ps

R

2.5 State Space Model for single Area System

From the above Fig. 2.4, writing equations and rearranging them to obtain the state space equations of single area as under.

d (∆ f )

dt= 1

T Ps[−∆ f +K Ps ∆ PG−K Ps ∆ PD]

(2.11)

d (∆ X E)

dt= 1

T sg[−∆ X E+∆ Pc−

∆ fR

]

(2.12)

d (∆ PG)

dt= 1

T t[−∆ PG+∆ X E]

(2.13)

Where, ∆ f =¿ Change in frequency ∆ XE=¿ Change in steam valve position ∆ pG=¿ Change in generator output powerFrom above three equations the state space matrix is obtained as under

[∆ X E

∆ pG

∆ f ]=[ x1

x2

x3]=[

−1T sg

0 −1RT sg

1T t

−1T t

0

0KPs

T Ps

−1T Ps

][ x1

x2

x3]+[ 1

T sg

00 ]u+[ 0

0−KPs

T Ps]d

In the above matrix u is system input (∆ Pc) andd is the disturbance (∆ PD) [6].

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3. State Space and Multirate Output Sampling

3.1 Introduction to state space

A state-space model is just a structured form or representation of the differential equations for a system where inputs, outputs and states variables are expressed as vectors. The state-space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. Hence, the use of the state-space representation is more convenient to systems with multi-input and multi-outputs.

The conventional control theory is completely based on the frequency domain approach while the modern control system approach is mostly based on time domain approach. Most of these systems are complex hence it has multiple inputs and multiple outputs. In the modern theory of control system the stability analysis and time response analysis can be done by analytically method very easily. Now state space analysis of control system is based on the modern theory which is applicable to all types of systems like single-input single-output systems (SISO), multiple-input and multiple-output systems (MIMO), linear and non-linear systems, time-varying and time-invariant systems [3]. Let us consider few basic terms related to state space analysis of modern theory of control systems.

State space model is a representation of the dynamics of an nth order system as a first order

differential equation in an n-vector, which is called the state. It converts the nth order differential

equation that governs the dynamics into nfirst-order differential equations. In a state space representation, the equation having state variables is called the state equation. The system output is given in terms of a combination of the current system state, and the current system input, through

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the output equation. These two equations form a system of equations known collectively as state-space equations.

Central to the state-space notation is the idea of a state. A state of a system is the current value of internal elements of the system, that influence the system behavior completely. In essence, the state of a system is an explicit account of the values of the internal system components. State variables refers to smallest set of variables whose knowledge at t=t 0 together with the knowledge of input, t ≥ t 0 gives the complete knowledge of the behavior of the system at any time t ≥ t 0. State variables are defined byx1(t), x2(t).........xn(t). The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system,n, is usually equal to the order of the system's defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. State variables must be linearly independent; that is, no state variable can be written as a linear combination of the other state variables, or else we would not have enough information to solve for all other state variables [4].

Suppose there is a requirement of n state variables in order to describe the complete behavior of the given system, then these n state variables are considered to be n components of a vector x (t) . Such a vector is known as state vector. State space refers to the n-dimensional space which hasx1 axis,x2 axis......xn axis. The state space is the vector space that consists of all the possible internal states of the system. State-space models are useful in many situations: such as Linearization of non-linear models ,Calculation of time-responses — both analytically and numerically, Using simulation tools: MATLAB, LabVIEW, Octave, and Scribal have simulation functions that assumes state-space models, Analysis of dynamic systems, e.g. stability analysis, Analysis and design of advanced controllers and estimators Controllability and observability analysis; Design of LQ optimal controllers, Model-based predictive control, and Feedback linearization control; Design of state estimators.

3.1.1 General state-space model

In a state-space system representation, we have a system of two equations: an equation for determining the state of the system, and another equation for determining the output of the system. We will use the variable y (t ) as the output of the system, x (t) as the state of the system, and u(t ) as the input of the system. We use the notation x (t) for the first derivative of the state vector of the system, as dependent on the current state of the system and the current input. The state equation shows the relationship between the system's current state and its input, and the future state of the system. The output equation shows the relationship between the system state and its input, and the output. These equations show that in a given system, the current output is dependent on the current input and the current state. The future state is also dependent on the current state and the current input The most general state-space representation of a linear system with p inputs, q outputs and n state variables is given by the following two equations [7]- x (t)=Ax (t)+Bu(t)

xviii

y (t )=Cx(t )+ Du( t)The first equation is called the state equation; the second equation is called the output equation. The vectors, (t)∈Rn , u(t )∈R p ,and y (t )∈Rq are state, input and output of the system respectively. The constant matrices, A, B, and C, and D are n× n, n× p , q × n, q× prespectively.

Matrix A is the system matrix, and relates how the current state affects the state change x. If the state change is not dependent on the current state, Awill be the zero matrix. The exponential of the state matrix, e At is called the state transition matrix. Matrix B is the control matrix, and determines how the system input affects the state change. If the state change is not dependent on the system input, then B will be the zero matrix. Matrix C is the output matrix, and determines the relationship between the system state and the system output. Matrix D is the feed-forward matrix, and allows for the system input to affect the system output directly. A basic feedback system like those we have previously considered do not have a feed-forward element, and therefore for most of the systems we have already considered, the D matrix is the zero matrix.

Fig 3.1 Block diagram representation of the linear state-space equations

3.2 Solution of Continuous Time State Space Model

Continuous time system with state space model given by x (t )=Ax ( t )+B u (t ) (3.1) y ( t )=Cx(t ) (3.2)

where, u(t ) is the control vector, x (t ) is the state vector, y (t ) is the measurements vector and x ( t0 ) is the initial value of the state vector, which usually is assumed to be known. A is the state matrix, B is the input matrix, C is the output matrix.Obtaining solutions of this state space equation-multiplying both sides of the equation (3.1) by e− At

e−At x (t )−e−At Ax (t )=e−At Bu (t) (3.3)

since, e−At x ( t )−e−At Ax ( t )= ddt {e−At x (t ) } (3.4)

Using equation (3.4), equation (3.3) can be rewritten as,

xix

∫t0

t ddt

{e−At x (t ) }dτ=∫t0

t

e−Aτ B u(τ)dτ

Here t 0 ,t are time interval during which we are interested to find the response of the system. Substituting the values of the limits on the left side of the equation

e−At x (t )| tt0

=∫t0

t

e−Aτ B u(τ)dτ

Obtaining e−At x ( t ) in terms of x ( t0 ) and input u(τ )

e−At x (t )=e−A t0 Ax (t0 )+∫t0

t

e−Aτ B u(τ )dτ

Now removing e−At from left half of the equation and obtaining expression for x (t ) in terms of x ( t0 )and u(τ )

x (t )=eA (t−t 0) x (t 0 )+∫t 0

t

e A (t−τ )B u(τ )dτ (3.5)

This is the solution of the state space equation. As we see, the solution consists of two parts. The first part represents the autonomous response (homogenous solution) driven only by initial values different from zero. The second term represents the in homogenous solution driven by the control variable,u(t ). In order to compute the first term, we have to compute the matrix exponential e A (t−t0 ). This matrix exponential is defined as the transition matrix, because it defines the transition of the state from the initial value,x ( t0 ), to the internal state x (t ) in an autonomous system x (t )=Ax ( t ) with known initial state x ( t0 ).

3.3 Importance of Discrete Time Controller

Digital controllers operate only on numbers. Decision-making is one of their important functions. In most of the control system, it is not only to stabilize the system but also involved in the optimal overall operation of industrial plants. Digital controllers are extremely versatile, they can handle nonlinear control equation involving complicated computation or logic operations. A very much wider class of control laws can be used in digital controllers. Also, in the digital controller, by merely issuing a new program the operations being performed can be changed completely. Digital controllers are capable of performing complex computational accuracy at relatively little increase in cost at present because of inexpensive microcomputers, digital controllers are being used in many large and small-scale control systems.

In the digital controllers digital components, such as sample-and-hold circuits, analog to digital (A/D) and digital to analog (D/A) converters, and digital transducers, are rugged in construction, highly reliable, and often compact and lightweight. Moreover, digital components have high sensitivity, are often cheaper and are less sensitive to noise signals and, digital controllers are flexible in allowing programming changes

xx

3.4 Discretization of the continuous time system

Consider the continuous time state space system given in the equations (3.1) and (3.2).Now apply an input that changes only at discrete (equal) sampling intervals. It would be nice if we could find matricesGτ and H τ , independent of t and k so that we could obtain a discrete time model of the system, x ( (k+1 ) τ )=G τ x (kτ )+H τ u ( kτ ) (3.6) y (kτ )=Cx (kτ )+Du(kτ ) (3.7)We will now determine the values of the matrices Gτ andH τ . It will turn out that while they are constant for a particular sampling interval, they depend on the value of the sampling interval, so for that reason we have written them asGτ and H τin (3.6) above.Substituting, t o=0andt=(k+1 ) τ in equation (3.5) we get

x ((k+1)τ )=eA (k +1)τ x (0 )+ ∫0

( k+1 ) τ

e A { (k +1) τ−t } Bu ( t )dt (3.8)

At t o=0 , t=kτ , we have

x (kτ )=e Akτ x (0 )+eAkτ∫0

kτ

e−At Bu (t ) dt (3.9)

In order to write x (( K+1 ) τ )in terms of (kτ ) , multiply all terms of (3.9) by e Aτ and solve for e A ( K +1) τ x (0 ) ,obtaining

e A ( K +1) τ x (0 )=e Aτ x (kτ )−e A ( K +1) τ∫0

kτ

e− At Bu ( t ) dt (3.10)

Substituting for e A ( K +1) τ x (0 ) in (3.5), we obtain

x ( (k+1 ) τ )=eAτ x ( kτ )+e A ( K +1) τ [ ∫0

(k+1)τ

e−At Bu (t )dt−∫0

kτ

e−At Bu (t ) dt ] (3.11)

Which, by linearity of integration, is equivalent to

x ( (k+1 ) τ )=eAτ x ( kτ )+e A ( K +1) τ ∫kτ

(k +1) τ

e−At Bu ( t ) dt (3.12)

Next, we notice that within the interval from kτ to (k+1)τ , u (τ )=u (k τ ) is constant, as is the matrix B , so we can take them out of the integral to obtain

x ( (k+1 ) τ )=eAτ x ( kτ )+e A ( K +1) τ ∫kτ

(k +1) τ

e−At Bu (t ) dt , t∈ [kτ ,(k+1)τ ] (3.13)

x ( (k+1 ) τ )=eAτ x ( kτ )+ ∫kτ

(k +1) τ

eA {( k+1) τ −t } Bu ( t ) dt (3.14)

xxi

Now we see that as τ ranges from k τ to (k+1)τ (the lower to the upper limit of integration) the exponent of e ranges from τ to 0. Accordingly, let’s define a new variable θ = (k+1)τ-t . Then dθ = −dt and θ ranges from τ to 0 as τ ranges from k τ to (k+1)τ . Thus we have

x ( (k+1 ) τ )=eAτ x ( kτ )+∫0

τ

e Aθ Bu ( kτ )dθ θ∈¿ (3.15)

We see that in (3.15) we have written the state update equation exactly in the form of (3.6) where Gτ=e Aτ; H τ=(eAτ−I) A−1 B (3.16)

x (kτ )=(G (τ ) )K x (0 )+∑j=0

k−1

(G (τ ) )K− j−1 H (τ ) u ( jτ ) , k=1,2,3 …… (3.17)

and we can see that at the sampling instants k τ, this has exactly the same value as is obtained using (3.1). Specifically, (G (τ ) )K=(e Aτ )k=eAkτ (3.18)These equations and derivation will follow when input and output sampling period are same (i.e. sampling period=τ ).

3.5 Numerical Examples

In this section two different examples are proposed to illustrate the validity of discretization of continuous time system.

3.5.1 Example 1

Let us now apply the results developed in the previous sections for discretization of the continuous time system. Consider a continuous time system represented in state-space model as x = A x + Bu

where, A = [−1 00 −3], B = [11]

and the input is step signal.If the system is sampled with a sampling timeτ=0.1sec, we obtain following descretized model

x (k+1 )=Gτ x (k )+H τ u (k )

where, Gτ=¿ [0.905 00 0.741], H τ=¿ [0.095

0.086]

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0 50 100 1500

0.2

0.4

0.6

0.8

1

time,t

ampl

itude

0 20 40 600

0.2

0.4

0.6

0.8

1

sampling time,k

ampl

itude

0 50 100 1500

0.1

0.2

0.3

0.4

time,t

ampl

itude

0 20 40 600

0.1

0.2

0.3

0.4

sampling time,k

ampl

itude

x2(t)

x1(t) x1(k)

x2(k)

Fig.3.2 State response of continuous time system and discrete time system.

3.5.2 Example 2

Discretization of continuous system when input is exponential without disturbance

Let us now apply the results of previous sections to discretize a continuous time system system with a disturbance. Consider a continuous time system represented by

x = A x + Bu

xxiii

where, A=[ 1 2−1 1], B=[12], u=e−0.1 t

If the system is sampled with a sampling timeτ=0.1 sec, we obtain following descretized model x (k+1 )=Gτ x (k )+H τ u(k )

where, Gτ=[ 1.094 0.220−.110 1.094 ], H τ=[0.1262

0.2043]

0 100 200 300 400 500-20

0

20

40

60

time,t

ampl

itude

0 20 40 60-50

0

50

100

150

sampling time,k

ampl

itude

0 100 200 300 400 500-10

0

10

20

30

time,t

ampl

itude

0 20 40 60-50

0

50

100

150

sampling time,k

ampl

itude

x2(k)

x1(k)x1(t)

x2(t)

Fig.3.4 State response of continuous time system and discrete time system.

xxiv

3.5 Multirate output sampling

A multirate sampling (MR) system is defined as a hybrid system composed of continuous time elements, usually the plant, and some discrete time components, usually the controllers or the filters, where two or more variables are sampled or updated at different frequencies. It can be also considered that the discrete actions are not equally spaced on time and/or delayed. Moreover, in a great number of computer control applications the approximation of a regular pattern of sampled signals is assumed. In multirate sampling, the output is sampled more rapidly than the input ie the sampling frequency of the output is more which can be written like this:

∆= τN

where,τ is the input sampling period and ∆is the output-sampling period.Value of N is always greater than 1 for multirate sampling (integer value i.e.- 2, 3, 4….).For N =1 both sampling periods are same, it is called single rate sampling. As we know y (kτ )=Cx (kτ ) (3.19)Then y (kτ+∆ )=Cx (kτ+∆) In general, this equation can be written as y (kτ+(N−1)∆ )=Cx (kτ+(N−1)∆ ) (3.20)But we know that during the period t=kτ to t=(k+1)τvalue of x (t ) equal to the sampled value i.e x (t )=x (kτ)Now equation 2 can be written as x (kτ+∆ )=e A ∆ x ( kτ )+(eA ∆−I ) A−1 Bu (kτ )

x (kτ+∆ )=G∆ x ( kτ )+ H∆ u(kτ ) (3.21)

where, G∆=e A ∆ ; H∆=(eA ∆−I ) A−1 B

Now putting the value of x (kτ+∆ ) in output equation (3.19) y (kτ+∆ )=C G∆ x (kτ )+C H ∆u (kτ ) (3.22) x (kτ+2 ∆ )=G∆ x (kτ+∆ )+H∆ u(kτ+∆)since, u (kτ+∆ )=u (kτ )Putting the values from equation, this equation becomes

x (kτ+2 ∆ )=G∆2 x (kτ )+¿ H∆ ¿u(kτ)

In general

x ( kτ+(N−1)∆ )=G∆N −1 x ( kτ )+{∑i=0

N−2

G∆i H∆ u(kτ)}

and output equation is given by

xxv

y (kτ+(N−1)∆ )=C G∆N−1 x (kτ )+C {∑i=0

N−2

G∆i H∆ u(kτ )}

With input sampling period τ and output sampling period ∆, we obtain following discrete-time system x ( (k+1 ) τ )=G τ x (kτ )+H τ u (kτ ) (3.23) yk +1=C0 x (kτ )+D0 u(kτ ) (3.24)

where, yk +1=[ y (kτ )y (kτ+∆)

⋮y (kτ+ ( N−1 ) ∆)]

(3.25)

C 0=[ CC G∆

⋮C G ∆

N −1] , D0=[0

C H ∆

⋮

C ∑i=0

N −2

G∆i H∆]

The matrices of τ system and ∆ system have the following relation Gτ=G∆

N

xxvi

4. Controller Design

Constant frequency electricity is very important and necessary now-a-days. Load frequency control (LFC) is an important tool to insure the stability and reliability of power systems. The goals of the LFC are to maintain zero steady state errors in a multi-area interconnected power system and to fulfill the requested dispatch conditions. Now-a-days power systems are very complex and interconnected. Because of this, to maintain frequency constant, controller is required. So appropriate control should be designed for the concerned power system.

4.1 Discretization of Continuous Time System with Disturbance

A general continuous-time linear system with added disturbance is described with the following equations: x (t )= Ax ( t )+Bu ( t )+Dd (t ) (4.1) y (t )=Cx(t ) (4.2) Let us assume that control signal u from (4.1) is able to change its value only every τ seconds, where τ is a sampling period.In order to design discrete-time estimator, system (4.1) is discretized using the Zero-Order-Hold (ZOH) discretization method, with sampling period τ . That results in the following discrete-time system:

x ( (k+1 ) τ )=G τ x (kτ )+H τ u ( kτ )+W τ d (kτ) (4.3) y (kτ )=Cx (kτ ) (4.4)

Consider the system given by the equation (4.1) and (4.2).We will now determine the values of the matrices Gτ andH τ . It will turn out that while they are constant for a particular sampling interval, they depend on the value of the sampling interval, so for that reason we have written them asGτ and H τin (4.2) above.

Substituting, t o=kτandt=(k+1)τ in equation (3.5) we get

x ((k+1)τ )=eA (k +1)τ x (0 )+ ∫0

( k+1 ) τ

e A { (k +1) τ−t } Bu (t )dt

+ ∫0

( k+1 )τ

e A { (k+1 ) τ−t } Dd (t ) dt

xxvii

At t o=0 , t=k τ , we have

x (k τ )=eAk τ x (0 )+eAk τ∫0

k τ

e−At Bu ( t ) dt+eAk τ∫0

k τ

e−At Dd ( t ) dt

In order to write x (( K+1 ) τ )in terms of (k τ ), multiply all terms of above equation by e Aτ and solve for e A ( K +1) τ x (0 ) ,obtaining

e A ( K+1) τ x (0 )=e A τ x (kτ )−eA ( K+ 1) τ∫0

k τ

e−At Bu (t )dt−eA ( K +1 )τ∫0

k τ

e−At Dd (t ) dt

Substituting for e A ( K+1) τ x (0 ) in (3.5), we obtain

x ( (k+1 ) τ )=eAτ x ( kτ )+e A ( K +1) τ [ ∫0

(k+1)τ

e−At Bu ( t )dt−∫0

kτ

e−At Bu ( t ) dt ] +e A ( K +1 )τ [ ∫

0

(k+1 )τ

e−At Dd (t )dt−∫0

kτ

e−At Dd (t ) dt ]Which, by linearity of integration, is equivalent to

x ( (k+1 ) τ )=eAτ x ( kτ )+e A ( K +1) τ ∫kτ

(k +1) τ

e−At Bu ( t ) dt

+e A ( K +1 )τ ∫kτ

( k+1 ) τ

e−At Dd ( t )dt

Through zero-order hold (ZOH), u(t )=u(k) and d (t )=d (k) over the time interval k τ to (k+1)τ , where τ is sampling period. The matrix B , is constant so we can take them out of the integral to obtain

x ( (k+1 ) τ )=eA τ x (k τ )+eA ( K+1) τ ∫k τ

(k +1) τ

e−At Bu (t ) dt

+e A ( K +1 )τ ∫k τ

( k+1 ) τ

e−At Dd ( t )dt ; τ∈ [k τ ,(k+1)τ ]

Now we see that as t ranges from kτ to (k+1)τ (the lower to the upper limit of integration) the exponent of e ranges from τ to 0. Accordingly, let’s define a new variable θ = (k+1)τ-t . Then dθ = −dt and θ ranges from τ to 0 as t ranges from kτ to (k+1)τ . Thus we have

x ( (k+1 ) τ )=eA τ x (k τ )+∫0

τ

eAθ Bu (k τ )dθ+∫0

τ

eAθ Dd (k τ ) dθ ; θ∈¿ (4.5)

We see that in (4.5) we have written the state update equation exactly in the form of (4.3), where

Gτ=e A τ , H τ=(eA τ−I ) A−1 B , W τ=(e A τ−I ) A−1 D (4.6)

xxviii

These matrices can also be written as Gτ=e A τ ,

H τ=∫0

τ

e At Bdt , (4.7)

W τ=∫0

τ

e At Ddt

4.1.1 Example of Discretization of the System With Disturbance

Let us now apply the results of previous sections to discretize a continuous time system system with a disturbance. Consider a continuous-time system with disturbance is represented by

x=Ax+B u+Dd

where, A=[−1 00 −3], B=[11], D=[54 ]

If the system is sampled with a sampling timeτ=0.1 sec, we obtain following descretized model x (k+1 )=Gτ x (k )+H τ u (k )+W τ(k) where, Gτ=[0.905 0

0 0.741] , H τ=[0.09520.0864 ], W τ=[0.476

0.346]

xxix

0 50 100 1500

1

2

3

4

5

6

time,t

ampl

itude

0 20 40 600

1

2

3

4

5

6

sampling time, k

ampl

itude

0 50 100 1500

0.5

1

1.5

2

time,t

ampl

itude

0 20 40 600

0.5

1

1.5

2

sampling time, k

ampl

itude

x1(k)x1(t)

x1(t)x2(k)

Fig.3.3 State response of continuous time system and discrete time system

As we can analyse from the plots that the states response of both continuous-time system and discrete-time system are almost same so we can say that discretization does not change the states hence we can discretise the continuous time system to design controller.

4.2 Fast Output Sampling Method

Fast output sampling (FOS) is an estimation technique appropriate for continuous time system controlled with discrete-time control signal, where the output signal can be sampled several times during

xxx

one period of the control signal [10]. FOS shows better performance than standard estimation techniques, because it reduces the estimation error to zero after just one sampling period [11]. Standard estimators need at least v sampling periods to achieve errorless estimation, where v is the observability index of the system [12]. To use FOS estimation technique, it must be satisfied N>v [10].

Fig.4.1 The usage of the FOS estimation method in system control.

The principle of using FOS estimation technique in system control is shown in Fig. 4.1. Firstly, the lastN subsamples of the output signal y (t ),measured in the most recent sampling period τ , are used to estimate the system state. Then, that estimated state is used to compute the control signal for the next sampling period.

x ( (k+1 ) τ )=G τ x (k τ )+H τ u (k τ )+W τ d (k τ ) (4.8) y (kτ )=Cx (kτ ) (4.9)

Let us also assume that only system output is measurable, and only at certain time instances, y (kτ), where ∆ is a subsampling period:

∆= τN (4.10)

where N∈N . Those samples can be used as input signals of the appropriate estimator for unmeasured state and disturbance signals in (4.1). LFC applied nowadays in real power systems is an example of a system with multiple sampling periods. In LFC, control signal is sent to the power plants in discrete-time. In UCTE interconnection that period is 1–5 s [9]. Additionally, during one sampling period several measurements of frequency f (k ∆) and tie-line power Ptie(k ∆) signals are gathered. Besides those subsamples, which are inputs to classical PI controller, subsamples of generated power Pg (k ∆) are also gathered for monitoring purposes. Those samples could also be used as inputs to the estimator. Because a substitute power plant is used in modeling a CA and also in controller synthesis, all other state and disturbance signals, that cannot be measured in the real system, must therefore be estimated.

xxxi

The system’s N consecutive subsamples, taken during the sampling period τ , can now be calculated as: y (k τ )=Cx(k τ)

Then y (k τ+∆ )=Cx (k τ+∆)

In general y (k τ+(N−1)∆ )=Cx (k τ+(N−1)∆ )But we know that during the periodt=k τ to t=(k+1)τvalue of x (t ) equal to the sampled value i.e x (t )=x (k τ )Now equation (4.5) can be written as x (k τ+∆ )=e A∆ x (k τ )+(e A∆−I ) A−1 Bu (k τ )+( eA ∆−I ) A−1 Dd (k τ )

x (k τ+∆ )=G∆ x ( k τ )+H∆ u (k τ )+W ∆ d (k τ )

where, G∆=e A∆ ; H∆=(eA ∆−I ) A−1 B ; W ∆=(eA ∆−I ) A−1 D (4.11)

Now putting the value of x (k τ+∆ ) in output equation

y (k τ+∆ )=C G∆ x (k τ )+C H ∆u ( k τ )+C W ∆ d(k τ )

Now x (k τ+2∆ )=G∆ x (k τ+∆ )+H∆ u (k τ+∆ )+W ∆ d (k τ+∆ )

since, u (k τ+∆ )=u (k τ )Putting the values from equation, this equation becomes

x (k τ+2∆ )=G∆2 x (k τ )+¿ H∆ ¿u (k τ )+¿ W ∆¿ d (k τ )

In general x ( k τ+(N −1)∆ )=G∆N −1 x ( k τ )+{∑i=0

N −2

G∆i [H ∆u ( k τ )+W ∆ d (k τ )]} (4.12)

And output equation is given by

y (k τ+(N−1)∆ )=C G∆N−1 x (k τ )+C {∑i=0

N−2

G∆i[H∆ u (k τ )+W ∆ d (k τ )]} (4.13)

With input sampling period τ and output sampling period ∆

x ( (k+1 ) τ )=G τ x (k τ )+H τ u (k τ )+W τ d(k τ ) (4.14)

yk +1=C0 x (k )+D0 u (k )+W 0 d (k ) (4.15)

where, yk +1=[ y (k τ)y (k τ+∆)

⋮y (k τ+( N−1 ) ∆)] ; C0=[ C

C G∆

⋮C G ∆

N−1] (4.16)

D0=[0

C H ∆

⋮

C ∑i=0

N −2

G∆i H∆] ; W 0=[

0C W ∆

⋮

C ∑i=0

N−2

G∆i W ∆ ] ; Gτ=G∆

N

xxxii

4.3 Design Procedure

4.3.1 State Feedback Controller

The state of a dynamical system is a collection of variables that permits prediction of the future development of a system. We now explore the idea of designing the dynamics a system through feedback of the state. State feedback, or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane. Full state feedback is utilized by commanding the input vector  u. Consider an input proportional (in the matrix sense) to the state vector [13], u ( t )=−Kx (t) (4.17) Substituting into the state space equations above, x (t )=( A−BK )x (t ) (4.18) y ( t )=(C−DK )x (t)

The roots of the state feedback system are given by the characteristic equation, det [sI−( A−B K ) ]. Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix K  which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation. This works only for Single-Input systems. Multiple input systems will have aK  matrix that is not unique. Choosing, therefore, the bestK values is not trivial.

One should note that although state feedback control is very attractive because of precise computation of the gain matrixK , implementation of a state feedback controller is possible only when all state variables are directly measurable with help of some kind of sensors. Due to the excess number of required sensors or unavailability of states for measurement, in most of the practical situations this requirement is not met. Only a subset of state variables or their combinations may be available for measurements. Sometimes only output y is available for measurement. Hence the need for an estimator or observer is obvious which estimates all state variables while observing input and output. To enable full state feedback we have to estimate the state of the systems [16]. With increase of complexity and optimization of the performance, the discrete controllers are more suitable. Hence need to design a discrete state feedback controller. In the next section, the design of discrete time controller is given.

We consider a continuous time system with added disturbance as

x (t )=Ax (t )+Bu ( t )+Dd (t ) (4.19) y ( t )=Cx(t ) (4.20)

To design discrete time controller, we need to discretize the above system. Assume sampling time period τ seconds. With sampling period , the discretized representation of the above system..

x ( (k+1 ) τ )=G τ x (k τ )+H τ u (k τ )+W τ d(k τ ) (4.21) y (k )=Cx (k )

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where, matrices involved in these equations is given by equation (4.6) and (4.16). The system controlled by a state feedback controller of the form

u (k )=−Kx(k) (4.22)

Where, the matrix K can be obtained by pole-placement method. The biggest challenge in most of the systems is finding the full state vector x (k ). Following will illustrate the design of controller of the form (4.22).

We choose a suitable value of N such that N>n. With output sampled at N times faster than the input, the stacked output will be

yk +1=C0 x (k )+D0 u (k )+W 0 d (k ) (4.23)

Where, matrices C0, D0 and W 0 are given by (4.16). From (4.23), we have

yk +1=[C0 W 0 ] [ x (k )d (k)]+ D0u (k ) (4.24)

Let inverse of matrix [C0 W 0 ] is given by matrix [CD ].Multiply (4.24) with [CD ] and rearranging the equation, we will get

[ x (k )d (k )]=[C

D ] [ yk+ 1−D0u ( k ) ] (4.25)

By substituting controller (4.22) into (4.21), we get

x ( (k+1 ) τ )=G τ x (k τ )−H τ Kx (k τ )+W τ d (k τ)

¿ [Gτ−H τ K ] x (k τ )+W τ d (k τ) (4.26)This can be rewritten as

x ( (k+1 ) τ )= [(Gτ−H τ K ) W τ ] [ x (k )d (k )] (4.27)

Substituting (4.25) into (4.27), the close loop system will be

x ( (k+1 ) τ )= [(Gτ−H τ K ) W τ ] [CD ] [ yk+1−D 0u (k ) ]

¿ [ (G τ−H τ K ) C+W τ D ] [ yk +1−D0 u (k ) ] (4.28)

From equation (4.28) it is evident that the close loop system is free from the disturbance. Hence, the controller will nullify the effect of disturbance on the system.

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4.3.2 Controller Algorithm

Step 1: Consider the continuous time system of the form (4.19) and (4.20).

Step 2: Choose sampling period τ.

Step 3: Obtain discrete time system matrices Gτ, H τ and W τ using (4.7).

Step 4: Obtain state feedback matrix K based on pole-placement design.

Step 5: Choose N>¿order of the system(n).

Step 6: Obtain the matrices C0,D0 and W 0 using (4.16).

Step 7: Obtain [CD ], is the inverse of [C0 W 0 ].

Step 8: The state and disturbance vectors are obtained using (4.25).

Step 9: Implement controller using (4.28).

4.3.3 Numerical Example Example of Discretization of the System With Disturbance by the use of multirate sampling:

Let us now apply the results of previous sections to discretize a continuous time system system with a disturbance. Consider a continuous-time system with disturbance is represented by

x=Ax+B u+Dd

where, A=[−1 00 −3], B=[11], D=[54 ] d

If the system is sampled with a sampling timeτ=0.2 sec, we obtain following descretized model

x (k+1 )=Gτ x (k )+H τ u (k )+W τ(k)

where, Gτ=[0.819 00 0.549] , H τ=[0.181

0.150], W τ=[0.9060.602]

Now taking N=2 and obtaining multirate output sampling system output equation, as per described in above section, is given by

yk +1=C0 x (k )+D0 u (k )+W 0 d (k )

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where, C0=[1.0000 1.00000.9408 0.7408], D0=[ 0

0.1816], W 0=[ 00.8214 ]

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

3

4

5

6comparision of true state and its multirate output estimate

x1(k)

x2(k)

MOS of x1(k)

MOS of x2(k)

Fig.4.2 Comparison of true state and its multirate output sampled estimate

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0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Error between true states and its MOS estimate

Fig.4.3 error between the true states and their MOS estimate

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5. Conclusions

From this we can conclude that the discretization of the system does not change its original or true states.so discretization is valid for controller design purposes as we know that mostly controllers are of discrete type. Also we don’t know the system states in most of the control systems or system may not be a full system but we need system states for state feedback controller. For this we have used multirate output sampling estimation of the system states as we have output variable, which we have sampled, using MOS. From graphs we can also conclude that the state estimation done using MOS is very well accurate as the error between the true state reduces to zero after just one sampling period and also estimate state FOS shows better performance than standard estimation technique

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Appendix

3.5.1 Discretization of continuous system without disturbances when input is unit step

clear allclca=[-1 0; 0 -3];b=[1; 1];c=[1 0];d=0;% for continous time systemsys=ss(a,b,c,d);[y,t,x]=step(sys,10);% for discrete time system[ad, bd]=c2d(a,b,0.1);m(:,1)=[0;0];u=1;for k=1:51 m(:,k+1)=ad*m(:,k)+bd*u;endk=1:length(m);subplot(2,2,1)plot(x(:,1),'-r')hold onsubplot(2,2,2)plot(k,m(1,:),'-b');hold onsubplot(2,2,3)plot(x(:,2),'+b');hold onsubplot(2,2,4)plot(k,m(2,:),'--g');hold off;

4.1.1 Discretization of the continuous System With Disturbance when input is unit step:

clear allclca=[-1 0; 0 -3];

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b=[1; 1];c=[1 0];d=0;p=[5;4];[ad,bd]=c2d(a,b,0.1)[dummy, dd]=c2d(a,p,0.1);sys=ss(a,b+p,c,d);[y,t,x]=step(sys,10); m(:,1)=[0;0];u=1;for k=1:51 m(:,k+1)=ad*m(:,k)+bd*u+dd;endk=1:length(m);subplot(2,2,1);plot(x(:,1),'--r');subplot(2,2,2);plot(k,m(1,:),'+b');hold onsubplot(2,2,3)plot(x(:,2),'g');subplot(2,2,4)plot(k,m(2,:),'.r');hold off

4.3.2 Discretization of the System with Disturbance by the use of multirate sampling:

clear allclc%------continuous time system---------a=[-1 0;0 -3]; b=[1;1];c=[1 1];d=[5;4];%multirate sampling matricesT=0.2;[aT bT]=c2d(a,b,T);[dummy cT]=c2d(a,d,T);x(:,1)=[-2;0];u=1;g=1; for k=1:50 x(:,k+1)=aT*x(:,k)+bT*u+cT*g; end%---one output is given for second order system---c=[1 1];y1=c*x(:,1);%observer design parameters J_tan=place(aT',c',[0.3;0.4]);

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J=J_tan';xhat(:,1)=[-1;1]; for k1=1:50 xhat(:,k1+1)=(aT-J*c)*xhat(:,k1)+bT*u+J*y1; y1=c*x(:,k1+1); endk=0:length(x)-1;plot(k,x(1,:),'r:',k,x(2,:),'k-','LineWidth',1.5)hold onk1=0:length(xhat)-1;plot(k1,xhat(1,:),'b.',k1,xhat(2,:),'g-','LineWidth',1.5)hold offtitle('comparision of state and its estimate:stable system')hold onlegend('x_{1}(k)','x_{2}(k)','estimate of x_{1}(k)','estimate ofx_{2}(k)')%---plot of error between actual state and estimated state--plot(k,x(1,:)-xhat(1,:),k,x(2,:)-xhat(2,:))%------Multirate output sampling---------[aD bD]=c2d(a,b,T/2);[dd cD]=c2d(a,d,T/2);co=[c;c*aD];do=[0;c*bD];wo=[0;c*cD];yo(:,1)=[-1;0]; for k2=1:50 xmos(:,k2)=aT*inv(co)*yo(:,k2)+(bT-(aT*inv(co)*do))*u+(cT- (aT*inv(co)*wo))*g; yo(:,k2+1)=co*x(:,k2)+do*u+wo*g; endk2=0:length(xmos)-1;k3=0:length(x)-2;%---plot to compare true state and its estimate based on MOS---%plot(k2,xmos(1,:),'r:',k2,xmos(2,:),'k',k3,x(1,1:50),'b.',k3,x(2,1:50),'g+','LineWidth',2)% title('comparision of true state and its multirate output estimate')%legend('x_{1}(k)','x_{2}(k)','MOS of x_{1}(k)','MOS of x_{2}(k)')% ---Error between true state and its MOS estimate----plot(k2,xmos(1,:)-x(1,1:50),'r:',k2,xmos(2,:)-x(2,1:50),'k:','LineWidth',2)title('Error between true states and its MOS estimate')

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