ABSTRACT The need for satisfactory operation of power stations running in parallel and the relation between system frequency and the speed of the motors has led to the requirement of close regulation of power system frequency. Power systems are frequently subjected to varying load demands. The perturbation in generated power must match the load perturbations if exact nominal state is to be maintained. A mismatch in the real power affects primarily the system frequency. For an efficient and successful power system operation in the wake of area load changes and abnormal conditions, such mismatches have to be corrected via supplementary control. In this project work, a detailed investigation on load frequency control problem for both the isolated power system and interconnected power system has been carried out. In case of interconnected power system, a two area system model is taken into consideration for simplicity. Conventional Transfer function approach and State Space approach are adopted to analyze the dynamic performance of the system. The response obtained by the two approaches are verified by using MATLAB. 1
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. Load Frequency Control for Interconnected Systems Using Optimal Controller
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ABSTRACT
The need for satisfactory operation of power stations running in parallel and
the relation between system frequency and the speed of the motors has led to the
requirement of close regulation of power system frequency. Power systems are
frequently subjected to varying load demands. The perturbation in generated power
must match the load perturbations if exact nominal state is to be maintained. A
mismatch in the real power affects primarily the system frequency. For an efficient
and successful power system operation in the wake of area load changes and
abnormal conditions, such mismatches have to be corrected via supplementary
control.
In this project work, a detailed investigation on load frequency control problem for
both the isolated power system and interconnected power system has been carried
out. In case of interconnected power system, a two area system model is taken into
consideration for simplicity. Conventional Transfer function approach and State
Space approach are adopted to analyze the dynamic performance of the system.
The response obtained by the two approaches are verified by using MATLAB.
Firstly the system studies have been carried without proportional feed
back controllers, later the proportional plus integral strategy is implemented to
obtain an improved response for the system. Also the effect of + 50% variation in
system parameters from their nominal values on the dynamic performance of the
system has been studied by obtaining the response plots of frequency deviation of
disturbed area.
Finally, the techniques of Optimal control theory are applied to develop an optimal
feed back controller for enhancing the system dynamic performance of both
Isolated and Interconnected power systems. Numerical examples have been
considered to demonstrate the effectiveness of optimal controller over the PI
controller and the results are presented and duly discussed. ka
1
CONTENTSAbstract
1. Introduction1.1 Introduction ……11.2 Load Frequency Problem ……21.3 Literary review ……3
2. Load Frequency Control of Isolated Power Systems2.1 Introduction …….42.2 Modeling of Power system components ..…..4 2.2.1 Modeling of speed governing system …….4
2.2.2 Modeling of turbine ..…..82.2.3 Modeling of Generator-Load ..…..9 2.2.4 Block Diagram of an Isolated Power system ……11
2.3 Dynamic Response without feedback PI Control ……12 2.3.1 Transfer Function Approach ……13
2.3.2 State Space Approach ……16 2.4 Dynamic Response with PI Control ……25 2.4.1 Control strategy ……25
2.4.2 Transfer Function Approach ……27 2.4.3 State Space Approach ……30
2.5 Case Studies ……36 2.6 Discussions ……46 3. Load Frequency Control of Interconnected Power Systems3.1 Introduction …….473.2 Modeling of Multi Area Power Systems ……48
2
3.3 Modeling of Two Area Systems ……503.4 Dynamic Response of Two Area Systems
with PI control …….52 3.4.1 Area Control Error …….52 3.4.2 State Space Approach …….553.5 Case Studies …….643.6 Discussions ……..734. Load Frequency Control for Interconnected Systems using Optimal Controller4.1 Introduction …….74 4.2 Optimal Control Theory …….74 4.2.1 System State x …….75 4.2.2 System Cost C …….75
4.2.3 Optimal Controller .……76 4.2.4 Calculation of the Optimal controller K .……76 4.2.5 Snag of Optimal Control
…….78 4.3 Application of Optimal Control to an
Isolated Power System …….79 4.3.1 Isolated Power System with
Reheater constraint …….83 4.3.2 Isolated Power System with
Reheater constraint using Optimal controller …….85
4.4 Application of Optimal Control toInterconnected Systems ……89
4.5 Discussions …….95 5. Conclusions
…….96
3
References …….99
4
INTRODUCTION The continuous growth in size and complexity of electric power
systems along with increase in power demands has motivated the power
control engineers to put their best efforts in the area of Power System
Control. The operation of an interconnected power system usually leads to
improved system security and economy of operation. In addition, the
interconnection permits the utilities to take the advantage of the most
economical transfer of power. The benefits have been recognized from
beginning and interconnections continue to grow. The various areas or
power pools are interconnected through tie-lines. These tie-lines are utilized
5
for contractual energy exchange between areas and provide inter-area
support in case of abnormal conditions.
1.1 Introduction
Normally, the power systems operate in nominal system state which is
characterized by constant system frequency and voltage profile with certain
specified system reliability. The change in frequency and voltage from their
nominal values change when there is any mismatch in real and reactive
power generations and demands. It can be proved by sensitivity analysis that
a mismatch in the real power balance affects primarily the system frequency,
but leaves the bus voltage essentially unaffected. Also a mismatch in the
reactive power balance affects only the bus voltage magnitudes, but leaves
the system frequency essentially unaffected.
Automatic generation control (AGC) of interconnected power systems is
defined as the regulation of power output of generators within a prescribed
area, in response to change in system frequency, tie line loading, or the
relation of these to each other, so as to maintain scheduled system frequency
and/or established interchange with other areas within predetermined limits.
Over the years, many automatic generation control (AGC) schemes have
been suggested to deal this problem efficiently and effectively. The main
requirement of AGC is to ensure that:
1) Frequency of various bus voltages and currents are maintained at near
specified nominal values.
2) Tie line power flows among the interconnected areas are maintained at
specified levels
3) Total power requirement on the system as a whole is shared by
individual generators economically in optimum fashion.
1.2 Load Frequency Problem
6
~ ~S1 S2
Load Load
Consider two machines S1 and S2 running in parallel .
Fig 1 Two Plants connected through a tie-lineThe possibility of sharing the load by the two machines is as follows: Say,
there are two stations S1 and S2 interconnected through a tie-line. If the
change in load is either at S1 or S2 and if the generation of S1 alone is
regulated to adjust this change so as to have constant frequency, the method
of regulation is called Flat Frequency Regulation. Under such situations
station S2 is said to be operating on base load. the major draw back of flat
frequency regulation is that S1 must absorb all load changes for entire system
thereby the tie line between the two stations would have to absorb all load
changes at station S2 since the generator at S2 would maintain its output
constant.
The other possibility of sharing the change in load is that both S1 and S2
would regulate their generations to maintain the frequency constant. This is
known as Parallel frequency Regulation.
The third possibility is that the change in a particular area is taken care of by
the generator in that area thereby the tie-line loading remains constant.
This method of regulating the generation for keeping constant frequency is
known as Flat-Tie line loading Control. This arrangement has the advantage
that load swings on station S1 and the tie line would be reduced as compared
with the flat frequency regulation.
The application of modern control theory to AGC problem of
interconnected power system has been the subject wide range of applications
over the past three and half decades. Among the various types of automatic
7
generation controllers, the most widely employed are the conventional
proportional integral (PI) controller and the state feedback controllers based
on optimal control theory to achieve better system dynamic performance.
8
LOAD FREQUENCY CONTROL OF AN ISOLATED POWER SYSTEM
2.1 Introduction
The main objective for the load frequency control is to exert
control of frequency and at the same time exchange of real power via the
Tie-lines. The change in frequency and tie-line real power are sensed which
is a measure of the change in rotor angle δ, i.e. the error ∆δ to be corrected.
The error signals i.e. ∆f and ∆Ptie are amplified mixed and transformed into a
9
real power command signal ∆PC which is sent to the prime mover to call for
an increment in the torque. The prime mover therefore brings in the
generator output by an amount ∆PG which will change the values of ∆f and
∆Ptie. The process continues till deviation ∆f and ∆Ptie are well below the
specified tolerances.
2.2 Modeling of Power System Components
Modeling of different power system components i.e. Speed
governing system, Turbine, Generator-load are described and the various
block diagrams representing the components are presented in this section.
2.2.1 Modeling of Speed Governing System
The schematic diagram of speed governing system which controls the real
power flow in the power system is shown in fig 2.1.
A
BC D
El1 l3l2 l4
Speed changer
Lower
Raise
Speed governor
High pressure oil
Pilot valve
XDirection of positive movement
steam
To turbine
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Fig:2.1 Speed Governing System
The Speed Governing System consists of the following parts:-
1. Speed Governor: This is a fly-ball type of speed governor and
constitutes the heart of the system as it senses the change in speed or
frequency. With the increase in speed the fly ball move outwards and the
point B on linkage mechanism moves downwards and vice versa.
2. Linkage Mechanism: ABC and CDE are the rigid links pivoted at B
and D respectively. The mechanism provides a movement to the control
valve in the proportion to change in speed. Link4 (l4) provides a feed back
from the steam valve movement.
3. Hydraulic Amplifier: This consists of the main piston and pilot valve.
Low power level pilot valve movement is converted into high power level
piston valve movement which is necessary to open or close the steam valve
against high pressure steam.
4. Speed Changer: The speed changer provides a steady state power
output setting for the turbine. The downward movement of the speed
changer opens the upper pilot valve so that more steam is admitted to the
turbine under steady condition. The reverse happens when the speed changer
moves upward.
Consider the steady state condition by assuming that linkage
mechanism is stationary, pilot valve closed, steam valve opened by definite
magnitude, the turbine output balances the generator output and the
turbine/generator is running at a normal speed or at a normal frequency
f° ,the generator output PGO and let the steam valve setting corresponding to
these conditions be XE.
11
Let the point A of the speed changer lower down by an amount
∆XA as a result the commanded increase in power ∆PC then ∆XA = K1∆PC.
The movement of linkage point A causes small position changes ∆XC and
∆X D of the linkage points C and D. With the movement of D upwards by
∆X D high pressure oil flows into the hydraulic amplifier from the top of the
main piston thereby the steam valve will move downwards a small distance
∆XE which results in increased turbine torque and hence power increase,
∆PG. This further results in increase in speed and hence the frequency of
generation. This increase in frequency ∆f causes the link point B to move
downward a small distance ∆XB proportional to ∆f. Assume the
movements are positive if the points move downwards.
Two factors contribute to the movement of C:
i)Increase in frequency causes B to move by ∆XB when the frequency
changes by ∆f as then the fly-ball moves outward and B is lowered by ∆XB .
Therefore, this contribution is positive and is given by K1∆f.
ii) The lowering of the speed changer by an amount ∆XA lifts the
point C upwards by an amount proportional to ∆XA, i.e. K2∆PC.
∆XC = K1∆f - K2∆PC …………………..2.1
Where K1 and K2 are the positive constants depends upon the
length of the linkage arms AB and BC and upon the proportional constants
of the speed changer and the speed governor.
The movement of D is contributed by the movement of C and
E. Since C and E move downwards when D moves upwards,
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therefore, ∆X D= K3∆XC + K4∆XE ……………………2.2
Where K3 and K4 are positive constants depend upon the length of the
linkage CD and DE
Let the oil flow into the hydraulic cylinder is proportional to position
∆X D
of the pilot valve, the value of ∆XE is given by
∆XE=K5 -(∆X D )dt ……………………2.3
Where the constant K5 depends upon the fluid pressure and the
geometries of the orifice and the cylinder.
Taking Laplace transforms to equations 2.1, 2.2 & 2.3
∆XC (s)= K1∆F(s) - K2∆PC(s)
∆X D (s)= K3∆XC (s)+ K4∆XE (s)
∆XE (s)= -K5∆X D (s) /s
= -K5 (K3.( K1∆F(s) - K2∆PC (s))+ K4∆XE (s)) s
Eliminating the variables ∆XC and ∆X D ,
∆XE (s) = K2K3 ∆PC(s) - - K3 K1∆F(s) K4+s/K5
∆XE(s)= K G [∆PC (s) -∆F(s)/R] ……………….2.4 1+sTG
Where R = K2/K1 → speed regulation of governor.
K G = K2K3/K4 → gain of speed governor.
TG = 1/K4K5 → time constant of speed governor.
The above equation 2.4 can be represented as a block diagram shown in
fig2.2
13
Fig. 2.2 Block diagram of speed governing system for steam turbine
2.2.2 Modeling of Turbine
The turbine power increment ∆P T depends entirely upon the
valve power increment ∆Pv and the response characteristics of the turbine. A
non-reheat turbine with a single gain factor K T and a single time constant T T
is considered and in the crudest model representation of the turbine the
transfer function is given as
G T (s)= ∆P T (s) = K T ….……………..2.5
∆XE (s) 1+sT T
The above transfer function is represented in the form of Block diagram
along with the governor as shown in fig 2.3
Fig.2.3 Block diagram of power control mechanism of turbine
KG
1+sTG
+-
1/R
∆PC ∆XE(s)
∆F(s)
KG 1+sTG
+-
1/R
∆PC ∆XE(s)
∆F(s)
KT 1+sTT
14
∆PT(s)
2
2.2.3 Modeling of Generator-Load
The model gives relation between the change in
frequency as a result of change in generation when the load changes by a
small amount.
Let ∆PD be the change in load demand, as a result the generation also swings
by an amount ∆PG. The net power surplus at the busbar is ∆PG-∆PD and this
power will be absorbed by the system in two ways
a) Rate of increase of stored kinetic energy in generator rotor
Let Wo be the Kinetic Energy before change in load occurs when
the frequency is f o.
Let ∆f be the change in frequency.
Let W be the Kinetic Energy when the frequency is ∆f+ f o.
As K.E. is proportional to square of the speed of the generator
ENTER THE PARAMETER IN WHCICH THE VARIATION IS TO BE SOUGHT:
1.CHANGE IN SPEED REGULATION(R)\n
42
2.CHANGE IN GOVERNOR TIME CONSTANT(Tg)\n
3.CHANGE IN TURBINE TIME CONSTANT(Tt)\n
4.CHANGE IN INTEGRAL GAIN(Ki)\n
5.CHANGE IN LOAD DEMAND(dP)\n\n
ENTER A VALUE :1
ENTER THE TIME CONSTANT OF GOVERNOR IN SEC :0.2
ENTER THE TIME CONSTANT OF TURBINE in sec :0.5
ENTER THE INTEGRAL GAIN :7
ENTER THE CHANGE IN LOAD DEMAND IN TERMS OF % INCREASE:20
ENTER THE SPEED REGULATION GOVERNOR(R) in puMW/Hz:3
43
Fig2.13 Response plot of an Isolated Power System with PI controller with variation in Governor Speed Regulation
ENTER A VALUE :2
ENTER THE TIME CONSTANT OF TURBINE in sec :0.5
ENTER THE INTEGRAL GAIN :7
ENTER THE CHANGE IN LOAD DEMAND IN TERMS OF % INCREASE:20
ENTER THE SPEED REGULATION GOVERNOR(R) in puMW/Hz:3
ENTER THE TIME CONSTANT OF GOVERNOR in sec :0.2
44
Fig2.14 Response plot of an Isolated Power System with PI controller with variation in Time Constant of Governor
ENTER A VALUE :3
ENTER THE TIME CONSTANT OF GOVERNOR in sec :0.2
ENTER THE SPEED REGULATION GOVERNOR(R) in Hz/puMW :3
ENTER THE INTEGRAL GAIN :7
ENTER THE CHANGE IN LOAD DEMAND IN TERMS OF % INCREASE:20
ENTER THE TIME CONSTANT OF TURBINE in sec :0.5
45
Fig2.15 Response plot of an Isolated Power System with PI controller with variation in Time Constant of Turbine
ENTER A VALUE :4
ENTER THE SPEED REGULATION GOVERNOR(R) in Hz/puMW:3
ENTER THE TIME CONSTANT OF GOVERNOR in sec :0.2
ENTER THE TIME CONSTANT OF TURBINE in sec :0.5
ENTER THE CHANGE IN LOAD DEMAND IN TERMS OF % INCREASE:20
ENTER THE INTEGRAL GAIN :7
46
Fig2.16 Response plot of an Isolated Power System with PI controller with variation in Integral Gain
ENTER A VALUE :5
ENTER THE SPEED REGULATION GOVERNOR(R) in Hz/puMW :3
ENTER THE TIME CONSTANT OF GOVERNOR in sec :0.2
ENTER THE TIME CONSTANT OF TURBINE in sec :0.5
ENTER THE INTEGRAL GAIN :7
ENTER THE CHANGE IN LOAD DEMAND IN TERMS OF % INCREASE:20
47
Fig2.17 Response plot of an Isolated Power System with PI controller with variation in Load Disturbance
2.4 Discussions
48
The response of an isolated power system without integral feedback control when subjected to load change are shown in figs 2.6&2.8. From these figs. it is clear that response for the isolated power system obtained by the transfer function and state space approaches. Also the settling time is 8 secs and the steady state frequency deviation is .0095Hz/pu.
The response of an Isolated Power System with PI controller when subjected to a load change are shown in figs 2.10&,2.12. From these figures, it is clear that the settling time is 15 seconds and the steady state frequency deviation is zero. Hence it can concluded that the feed back control reduces the steady state deviation to zero.
The response for the variation in different parameters are obtained in section 2.5. The response for + 50% variation in speed regulation of governor is shown in figure 2.13. From this figure, it can be observed that when R is increased the settling time decreases but frequency deviations increases and vice versa.
The response for + 50% variation in Time constant of governor is shown in figure 2.14. From this figure, it is clear that when TG is decreased the settling time decreases and frequency deviations decreases and vice versa.
The response for + 50% variation in Time Constant of Turbine is shown in figure 2.15. From this figure, it is evident that when TT is increased the settling time decreases and frequency deviations decreases and vice versa.
The response for + 50% variation in Integral Gain is shown in figure 2.16. From this figure, it is evident that when Ki is increased the settling time increases and frequency deviations increases.
The response for + 50% variation in Load disturbance is shown in figure 2.17. From this figure, it can be observed that when dP is increased the frequency deviation increases with no considerable change in settling time.
49
CHAPTER-3
50
LOAD FREQUENCY CONTROL OF INTERCONNECTED SYSTEMS
3.1 Introduction
All power systems today are tied together with neighboring
areas and the problem of load-frequency control becomes a joint
undertaking. By considering a practical system with a number of generator
stations and loads, it is possible to divide an extended power system into sub
areas in which the generators are tightly coupled together so as to form a
coherent group. Such a coherent area is called a control area in which the
frequency is assumed to be the same throughout in static as well as dynamic
conditions. The important advantages can be derived by pool operation are
a) to improve system security and economy of operation
b) the interconnection permits the utilities to make economy transfers
The basic operating principles for interconnecting systems are
a) under normal operating conditions each pool member or control area
should strive to carry its own load, except such scheduled portions of the
other members’ loads as have been mutually agreed upon.
b) Each control area must agree upon adopting regulating and control
strategies and equipment that are mutually beneficial under both normal and
abnormal situations. The advantages belonging to a pool are particularly
evident under emergency conditions.
51
The Problems of frequency control of interconnected areas are more
important than those of isolated areas. The objective of load frequency
control of interconnected power systems is two fold: minimizing the
transient error deviations in both frequency and tie line power and ensuring
zero steady state errors of these two quantities. By a simple proportional
integral control law the above mentioned objective is achieved. This chapter
presents proportional control for minimizing the transient error and the
integral control for zero steady state error.
3.2 Modeling of Multi area Power Systems
In an isolated control area case, the incremental power (∆PG-∆PD) was
accounted by the rate of increase of stored kinetic energy in or out of an
area. Changes in the Tie-line power flows also affect the power balance in
corresponding areas.
Consider any area i in an n area power system corresponding to
change in load demand ∆PD.
Let ∆ Ptie, i be the tie line schedule deviation
∆ Ptie, , i = ∆ Ptie, , ij ………………..3.1
where ∆ Ptie,, ij is the change in tie line power flow over the line connecting
ENTER THE GOVERNOR SPEED REGULATION OF AREA 1 in Hz/puMw :2.4
ENTER THE GOVERNOR SPEED REGULATION OF AREA 2 in hz/puMw :2.4
ENTER THE INERTIA CONSTANT OF AREA 1 in secs :5
ENTER THE INERTIA CONSTANT OF AREA 2:in secs :5
ENTER THE TIME CONSTANT OF GOVERNOR OF AREA 1 in sec :0.08
ENTER THE TIME CONSTANT OF GOVERNOR OF AREA 2 in sec :0.08
ENTER THE TIME CONSTANT OF TURBINE OF AREA 1 in sec :0.3
ENTER THE TIME CONSTANT OF TURBINE OF AREA 2 in sec :0.3
ENTER THE LOAD FREQUENCY CONSTANT OF AREA 1 in puMw/Hz :0.00833
ENTER THE LOAD FREQUENCY CONSTANT OF AREA 2 in puMw/Hz :0.00833
ENTER THE FREQUENCY BIAS CONSTANT OF AREA 1 in puMw/Hz :0.425
ENTER THE FREQUENCY BIAS CONSTANT OF AREA 2 in puMw/Hz :0.425
ENTER THE INTEGRAL GAIN OF AREA 1 :1
ENTER THE INTEGRAL GAIN OF AREA 2 :1
ENTER THE SYNCHRONISING POWER COEFFICIENT in puMw :0.545
ENTER THE CHANGE IN LOAD IN TERMS OF % INCREASE:20
ENTER THE PARAMETER IN WHCICH THE VARIATION IS TO BE SOUGHT:
1.CHANGE IN FREQUENCY BIAS CONSTANT(b)
71
2.CHANGE IN SYNCHRONISING COEFFICIENT (T12)
3.CHANGE IN LOAD DEMAND(dP)
ENTER A VALUE :1
Fig 3.5 Response plot in Area-1 with variation in Frequency Bias Constant
72
fig 3.12 response plot of in area-2 with variation in frequency bias constant
Fig 3.6 Response plot in Area-2 with variation in Frequency Bias Constant
>> ENTER A VALUE :2;
73
Fig 3.7 Response plot in Area-1 with variation in Synchronizing Coefficient
Fig 3.8 Response plot in Area-2 with variation in Frequency Bias Constant
ENTER A VALUE :3
74
Fig 3.9 Response plot in Area-2 with variation in Load Disturbance in Area-1
Fig 3.10 Response plot in Area-2 with variation in Load Disturbance in Area-1
75
Discussions
The response of a Two Area Power System with PI controller when subjected to a load change in area-1 is shown in figs 3.4. From these figures, it is clear that the settling time for both the areas is 250 seconds and the steady state frequency deviation is zero. But the peak value of the transient response in area-1 is0.0095 Hz/pu and in area-2 is 0.0062 Hz/pu. Hence it can be concluded that the feed back control reduces the steady state deviation to zero.
The response for the variation in different parameters is obtained in section 3.5. The response for + 50% variation in Frequency Bias Constant is shown in figure 3.5 for area 1 and in fig 3.6 for area-2 with a load disturbance in area-1. From these figures, it can be observed that when b is increased the settling time decreases and frequency deviations increases for both the areas and vice versa.
The response for + 50% variation in Synchronizing Coefficient is shown in figure 3.7 for area-1 and in fig 3.8 for area-2 with a load disturbance area-1. From these figures, it is evident that when T12 is increased the settling time decreases where as the frequency deviations increases and vice versa.
The response for + 50% variation in Load disturbance in area-1is shown in figure 3.9 for area-1 and in fig 3.10 for area-2. From these figures, it is evident that when dP is increased the frequency deviation increases with no considerable change in settling time for both the areas.
76
77
LOAD FREQUENCY CONTROL OF INTERCONNECTED SYSTEMS USING OPTIMAL CONTROLLER
4.1 Introduction
The classical design techniques used so far utilize the plant
output for feed back with a dynamic controller. In this chapter, Modern
control designs that require the use of all state variables to form a linear
static controller are employed. Optimal Control is a branch of modern
control theory that deals with designing controls for dynamic systems by
minimizing a performance index. This is also called as linear quadratic
regulator (LQR) problem. The object of the optimal controller design is to
determine the optimal control vector uopt(x,t) which can transfer the system
from its initial state to the final state such that a given performance index is
minimized. The performance index used in optimal control design is known
as the Quadratic Performance Index and is based on minimum-error and
minimum-energy criteria.
4.2 Optimal Control Theory
The problem in optimal control theory can be explained as follows:
Given the linear time invariant system represented by the state
variable differential equation
(t)=Ax(t)+Bu(t) …………………4.1
where x→ n X 1 state vector
u→ m X 1 control vector
78
A→ n X n State Distribution matrix
B→ n X m Control Distribution Matrix
The objective is to find the control vector uopt which minimizes the
cost function
C= ½ ∫ (x´Qx+u´Ru) dt …………………4.2
where Q → n X n positive semi-definite symmetric state cost weighting
matrix
R → m X m positive definite symmetric control cost weighting matrix
x´ and u´ → transpose of x and u respectively.
4.2.1 System State x:
If the system is linear and time invariant , the state x can be
represented in the form of equation 4.1 with the state variables x1,x2,…..,xn
are the components of the state vector x. these state variables are the
minimum number of variables containing sufficient information about the
previous state of the system, assuming the control inputs are known. The
state variables are not purely mathematical but have true physical meaning.
4.2.2 System Cost C:
The performance of the system is specified in terms of a cost that is to
be minimized by the optimal controller. The components Q and R can be
chosen mathematically by the way the system is to be performed. There are
two cases in choosing Q and R which are
1) If R=0 but require Q≠0 then it means there is no charge for the control
effort used but the state for being nonzero is penalized. Here the best control
79
strategy would be in the form of infinite impulses. This control would drive
the state to zero in the shortest possible time with the greatest effort.
2) If Q=0 and R≠0 then the control effort is penalized but do not charge for
the trajectory the state x follows. In this case , the best control is to use u=0
i.e. not to provide any control effort at all.
These two cases are the extreme cases, but they emphasize the
importance in choosing the components of Q and R.
4.2.3 Optimal Controller:
The optimal controller that minimizes the cost C of the system
in state variable form is a function of the present states of the system
weighted by the components of a constant gain matrix K of dimension m
X n:
u(t)= -Kx(t) …………………4.3
This optimal gain matrix is determined by solving the
differential equation, the matrix Riccati equation. For the infinite time
problem, the Riccati equation has a steady solution. Since the gain matrix is
a constant, the optimally controlled system can be expressed in the closed
form. x=Acx
Ac ~ A-BK
4.2.4 Calculating the value of optimal controller K:
Consider the plant described from the equations 4.1 and 4.2
The objective function is to: minimize C= ½ ∫ (x´Qx+u´Ru) dt
with the constraint: Ax(t)+Bu(t)=
To obtain the formal solution, Lagrange
80
multipliers method is applied. The constraint problem is solved by
augmenting the equation 4.1 into equation 4.2 using an n-vector of Lagrange
multipliers, λ . The problem reduces to the minimization of the following
unconstrained function. L (x,λ,u,t) = [x´Qx+u´Ru]+λ[Ax+Bu-
] …………………4.4
The optimal values(denoted by opt) are found by partial differentiating
the Lagrangian function w.r.t. λ, u, x and equating them to zero
i.e. ∂L /∂λ = Axopt+Buopt- opt=0
→ opt = Axopt+Buopt ………………………..4.5
∂L/∂u = 2Ruopt+λ´ B =0
→ uopt= ½R-1λ B ………………………..4.6
∂L/∂x = 2xoptQ+λ’´ +λ´ A=0
→ λ’ = 2Qxopt – A´λ ………………………..4.7
Assuming there exists a symmetric, time-varying positive
definite matrix P(t) satisfying
λ = 2P(t)xopt ……………………….4.8
substituting equation 4.3 into equation 4.6 gives the optimal closed-loop control law
Fig 4.4 Response plot of area-1 with a step load disturbance in area-1
with PI and Optimal controller
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Fig 4.5 Response plot of area-2 with a step load disturbance in area-1
with PI and Optimal controller
o Discussions
The response of an Isolated Power System when subjected to load change with PI and optimal controller is shown in fig 4.1 From this fig it is clear that the settling time without optimal controller is 14 secs and with optimal controller is just 6 secs. It can also be observed that the peak value of transient frequency with and without optimal controller is 0.0145Hz/pu and 0.0016 Hz/pu.
The response of an Isolated Power System with Reheater Constraint when subjected to load change with PI and optimal controller is shown in fig 4.2 From this figure, it is evident that a steady state value of frequency deviation without optimal control is not reached where as when optimal control is applied, a steady state value is reached within 2.2 seconds.
The response of an Two Area Power System when subjected to a load change in area-1 is shown in figs 4.4 and 4.5. From these figures, it is clear that there is no considerable change the settling time for both the areas for with PI and optimal controller (i.e. 250 secs) but a vast difference in the peak value of transient frequency deviation is observed. i.e. the value for without optimal controller is 0.009 Hz/pu and for with optimal control is 0.045Hz/pu. for area-1.
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CONCLUSIONS
An exhaustive study on load frequency control problem of both
isolated and interconnected power systems has been carried out. Necessary
computer programs have been developed to carry out these studies by
Transfer function approach, State space approach and finally verified by
using Matlab.
The techniques of PI control and optimal control have been
employed to enhance the dynamic performance of both the isolated and
interconnected power systems. The dynamic response for a step load change
and response plots for variation in the system parameters from their nominal
values have been presented.
The results of the response plots obtained for the isolated power
system without PI controller, with PI and optimal controller are presented in
a tabular form below.
System Maximum Frequency Deviation in Hz/pu
Settling Time in seconds
Steady State error In Hz/pu
Isolated Power System without PI feedback Control 0.0138 7.5 0.01
Isolated Power System with PI controller 0.0142 14 0
Isolated Power System with Optimal controller 0.0012 6.5 0
Isolated Power System with Reheater using PI control ∞ ∞ ∞
Isolated Power System with Reheater using Optimal control 0.0012 3.7 0
Table 5.1 Results of response plots for step load change in isolated power system
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From the table 5.1, It is observed that the dynamic response with PI
controller decreases the steady state frequency deviation to zero compared to
the system without PI controller whereas the settling time increases. With
the optimal control the maximum value of frequency deviation is decreased
for the system in addition to decrease in settling time. In case of Isolated
Power System with Reheater, the frequency deviation is drastically
increasing even with PI controller without any settling time. By adopting
Optimal Controller, the steady state frequency error is zero with a decrease
in settling time.
The results of the response plots for various case studies (i.e.,50%
increase in parameter values) carried out for the isolated power systems are
presented in a tabular form below.
Table 5.2 Results of response plots of Different Case Studies for isolated power system
Case Max Frequency Deviation Settling TimeIncrease in Governor Speed Regulation(R) Increases Decreases
Increase in Time Constant of Governor(TG) Increases Increases
Increase in Time Constant of Turbine (TT) Decreases Decreases
Increase in Proportional Integral Gain (KI)
Increases Increases
From the table 5.2, It is clear that for isolated power systems for
normal operation settling time should be low for which the Governor speed
regulation should be high, the Governor time Constant should be small, the
Turbine time constant should be high and the Proportional Integral gain
should be low.
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The results of the response plots, with PI and optimal controller obtained, for
the two area power system are presented in a tabular form below
System Maximum Frequency Deviation in Hz/pu
Settling Timein seconds
Steady State error in Hz/pu
Two Area Power System with PI controller (Area-1) 0.0095 260 0
Two Area Power System with PI controller (Area-2) 0.0064 260 0
Two Area Power System with Optimal Controller(Area-1) 0.0045 260 0
Two Area Power System with Optimal Controller(Area-2) 0.0035 260 0
Table 5.3 Results of the response plots of the two area system From table 5.3, it is clear that there is no change in settling time in
case of two area-systems with PI controller and Optimal controller, but the
maximum frequency deviation decreases incase of optimal controller.
The results of response plots for various case studies (i.e.,50%
increase in parameter values) carried out for two area systems are presented
in a tabular form below
Table5.4 Results of response plots for Different Case Studies for Two Area system Case Max Frequency Deviation Settling TimeIncrease in Frequency Bias Constant(b) Decreases Decreases
Increase in Synchronizing Torque Coefficient(T12)
Increases No considerable change
From table 5.4, it can be inferred that for normal operation of
two area power systems, the frequency bias constant should be high and the
synchronizing torque coefficient should be low.
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In totality, it can be concluded that optimal controller gives a
better performance in terms of settling time as well as frequency deviation
compared to PI controller not only in the case of isolated system but also for
interconnected system.
REFERENCES
1) O.I.Elgerd and C.E.Fosha, “Optimum Megawatt-Frequency
Control of Multiarea Electric Energy Systems”, and “The Megawatt-
Frequency Problem: A New Approach Via Optimal Control Theory,” IEEE
Trans. Power System Apparatus and Systems, vol. PAS-89, No.4, April
1970 pp. 556-576.
2) N.N.Benjamin and W.C.Chan , “Multilevel Load Frequency Control
of Interconnected Power Systems”, IEE vol. 125, No. 6 , June -1978.
3) M.M.Adibi, J.N.Borkoski, R.J.Kkafka, and T.L.Volkman, “Frequency
Response of Prime Movers during Restoration”, IEEE Trans. Power System
Apparatus and Systems, vol. PAS-14, No.2, May 1999 pp. 751-756.
4) Ibraheem and P Kumar, “Study of Dynamic Performance of Power
Systems with Asynchronous Tie-Lines Considering Parameter