MIT International Journal of Electrical and Instrumentation Engineering Vol. 1, No. 1, Jan. 2011, pp.

1-5 1
ISSN No. 2230-7656 ©MIT Publications

Optimal Load Frequency Control of an

Interconnected Power System
K.P. Singh Parmar S. Majhi, Member, IEEE, D.P. Kothari, Senior Member,
IEEE
Assistant Director (Technical) Professor and Head of the Department of Former Director (in charge) of
with Centre for Advanced Electronics and Communication Engineering, IIT, Delhi, former Vice
Management & Power Studies, Indian Institute of Technology, Chancellor of VIT University,
National Power Training Guwahati, Assam, India Vellore. Currently Advisor to
Institute, Faridabad, Chancellor of VIT University,
Haryana,India Velore (TN), India
email:kpsingh_jss@rediffmail.co
m

Abstract—Load frequency control is one of the important

issues in electrical power system design/operation and is II. INTRODUCTION
becoming much more significant recently with increasing size,
In recent years, major changes have been introduced into
changing structure and complexity in interconnected power
system. This paper deals with load frequency control of an
the structure of electric power utilities all around the world.
interconnected power system considering reheat type turbine The successful operation of interconnected power system
in one of the thermal areas. The dynamical response of the load requires the matching of total generation with total load
frequency control problem in an interconnected power system demand and associated system losses. As the demand
under consideration is improved with a practical viewpoint by deviates from its normal value with an unpredictable small
designing the output feedback controller. The optimal control amount, the operating point of power system changes, and
methods are proposed and their dynamic responses are hence, system may experience deviations in nominal system
compared. The results indicate that the proposed controllers frequency (which is due to drooping characteristics of the
exhibit better performance. In fact, the control systems
governor-turbine system) and scheduled power exchanges to
designed on these methods satisfy the load frequency control
requirements with a reasonable dynamic response.
other areas, which may yield undesirable effects. There are
two variables of interest, namely, frequency and tie line
Keywords—Automatic generation control, area control error,
power exchanges which must remain fixed to their nominal
optimal control, reheat turbine.
values [19, 20].
A literature survey shows that most of the earlier works in
I. NOMENCLATURE the area of AGC pertain to interconnected thermal systems
with non reheat type turbines and relatively lesser attention
has been devoted to the AGC of interconnected thermal
AGC Automatic gain Control system with reheat type turbines [10]-[14]. In reheat
ACE i area control error of area i
turbines, the reheating may be in a single stage or in multi-
Prti rated power capacity of area i stage .A two area interconnected Thermal-Thermal Power
f nominal system frequency system incorporating reheat type turbine in one of the areas
Di system damping of area i has been considered for study in this paper. The linearised
TSG speed governor time constant models of governors, reheat turbines and non-reheat turbines
TT steam turbine time constant are taken for simulation of the system [9].
TPS power system time constant Probably the most important contribution modern optimal
R governor speed regulation parameter control theory has made to the control engineer is the ability
to handle a large multivariate control problem with ease.
K PS power system gain
The engineer has only to represent the control system in
KR steam turbine reheat constant
state variable form and specify the desired performance
TR steam turbine reheat time constant
mathematically in terms of a cost to be minimized. A unique
B frequency bias parameter or best controller in the sense of minimizing the cost may
be designed by applying well proven theories and
i ith area frequency response characteristics
techniques [1]. Application of the optimal control theory to
P
a12  rt1 power system has shown that an optimal load frequency
Prt 2
T12 synchronizing coefficient controller can improve the dynamic stability of a power
w i  PDi incremental load change in area i system [10].
MIT International Journal of Electrical and Instrumentation Engineering Vol. 1, No. 1, Jan. 2011, pp. 1-5 2
ISSN No. 2230-7656 ©MIT Publications

In this paper, the dynamical response of the LFC 

problem is improved with a practical point of view. Because x  Ax  Bu (1)
practically access to all of the state variables of system is With x(0)  x0
limited and measurement of all of them is not feasible, an
output feedback controller design is also presented in this
y  Cx (2)
paper to overcome this problem. where
The proposed optimal feedback methods are evaluated x is a state vector of dimension n1, n no. of states
on a two-area power system. The results of the proposed u is a control vector of dimension m1, m no. of control
controllers are compared by means of computer simulations. variables
The result indicates that proposed methods improved the y is a output vector of dimension p 1, p no of output
dynamic response of system considered and provides a variables
control system that satisfied the LFC requirements. A, B and C are constant matrices with appropriate
dimensions.
The performance of the system is specified in terms of a cost
III. MATHEMATICAL BACKGROUND that is to be minimized by the optimal controller.

J    xT Qx  uT Ru dt
Modern control theory is to be applied to design an optimal 1 (3)
load frequency controller for a two-area system. In 20
accordance with modern control terminology Pref 1 and
Where
Pref 2 will be referred to as control inputs u1 and u2 . In the Q is nn positive semi definite symmetric state cost
weighting matrix
conventional approach u1 and u2 were provided by the R is mm positive semi definite symmetric control cost
integral of ACEs. In modern control theory approach u1 weighting matrix
The matrices Q and R are chosen appropriately.
and u2 will be created by a linear combination of all the
The optimal controller that minimizes the cost of the system
system states (full state feedback approach) or a linear
in state variable form is a function of the present states of
combination of states to be controlled/measurable states
the system weighted by the components of a constant gain
(output feedback approach) [1].
matrix K1 of dimension mn and can be defined by
u  K x 1
(4)

K1 can be obtained from the solution of the reduced matrix

Riccati equation given below.
1 T
AT P1  P1 A  PBR
1 B P1  Q  0 (5)
1 T
K1  R B P1 (6)
The acceptable solution of K1 is that for which the system
remains stable. For stability all the eigen values of the
matrix  A  BK1  should have negative real parts. From
equation (4), we get the optimal control of our choice.
So for it was assumed that all the states are available for
feedback. Practically it is very difficult and costly to
measure and to have readily available information of all
the states in most of the large power systems. Usually
reduced number of state variables or a linear combination
thereof is available. The output feedback controller is as
described below

u   Ky (7)
Fig. 1. State space model of a two-area power system.
where K is an output feedback gain matrix of dimension
(mp). In the optimal control scheme the control inputs are
The Generalized linear model of the power system may be generated by means of feedbacks from all the controlled
described in state space form as output states with feedback constants to be determined in
MIT International Journal of Electrical and Instrumentation Engineering Vol. 1, No. 1, Jan. 2011, pp. 1-5 3
ISSN No. 2230-7656 ©MIT Publications

accordance with optimality criterion. The linear model given x9   ACE1dt , x10   ACE2 dt etc as depicted in Fig.1.State
by equation (1) and (2) can be arranged as

variables x1  f1 , x5  f 2 , x9   ACE1dt and x10   ACE2 dt
x  ( A  BKC ) x  AC x (8)
are taken as controlled output feedback signal for the design
The PI may be expressed in terms of K as of PI output feedback controller and state variables

J
1
2 0
 xT (Q  CT K T RKC  xdt (9) x9   ACE1dt and x10   ACE2 dt are taken as controlled
output feedback signal for obtaining the Integral optimal
The design problem is now to select the gain K so that J is
output controller gain. The optimum gains of full state
minimized subject to dynamical constraint

feedback controller, Integral output feedback controller and
x  ( A  BKC ) x (10) PI output feedback controller have been obtained through
This dynamical optimization problem may be converted into MATLAB codes generated on the basis of methods
an equivalent static one that is easier to solve. After applying described in mathematical background. The computer
the suitable optimization techniques, we obtain the following simulations are carried out with the optimum gains obtained.
Optimal Gain Design Equations:
0  AcT P  PAc  CT K T RKC  Q (11) V. SIMULATION RESULTS
0  Ac S  SA  X T
c
(12) The optimum values of controller gains for full state
1
K  R B PSC (CSC )T T T 1
(13) feedback, Integral optimal output feedback and PI Optimal
output feedback controller are obtained by minimizing the
Where
Performance Index. Dynamic responses of the system are
Ac  A  BKC
obtained for 1% step load perturbations in area-1 and area-2
X  E  x(0) xT (0) through computer simulation. The dynamic responses for
If initial states are assumed to be uniformly distributed on 1% load perturbation in area-1 are depicted in Fig. 2, Fig. 3
the unit sphere, then X=I, X is a n×n symmetric matrix. In and Fig. 4 and the dynamic responses for 1% load
many applications x(0) may not be known, this dependence perturbation in area-2 are depicted in Fig. 5, Fig. 6 and Fig.
is typical of output feedback design. It is usual to sidestep 7. Dynamic responses obtained by optimal output PI and
this problem by minimizing not the PI but its expected value Integral feedback controllers are better than full state
[17], X  E J  feedback controller and satisfy the requirements of load
frequency control problem.
E  xT (0) Px(0)  tr ( PX ) (14)
1 1
E J   0.015
2 2 0.01
full state feedback
Integral output feedback
PI output feedback
Frequency Deviation (Hz),area 1

0.005
The Optimal cost can be given by 0
J O  tr ( PX ) (15) -0.005

The equations (11) and (12) are Lyapunov equations and the -0.01

equation (13) is an equation for the gain K. To obtain the -0.015

output feedback gain K minimizing the JO, these -0.02

three-coupled equations may be solved by some iterative -0.025

technique [17]. -0.03

0 5 10 15 20 25 30
time, s

IV. SYSTEM INVESTIGATED Fig. 2. Frequency deviation response of area-1 to 1% step load
perturbation in area-1
A two area interconnected Thermal-Thermal Power system 0.01
incorporating reheat type turbine in one of the areas has full state feedback
Integral output feedback
0.005
been considered for study in this paper. The linearised PI output feedback
Frequency Deviation (Hz),area 2

models of governors, reheat turbines and non-reheat turbines 0

are taken for simulation of the system [9]. Fig.1 shows the
-0.005
AGC model with state variables. A bias setting Bi=βi is
considered in both areas. MATLAB version 7.0 has been -0.01

used to obtain dynamic response for Δf1, Δf2, ΔPtie for 1% -0.015

step load perturbation in either area. The typical system

-0.02
parameters considered are given in appendix.The system has 0 5 10 15 20 25 30
time, s
n=10 state variables, x1  f1 , x5  f 2 , Fig. 3. Frequency deviation response of area-2 to 1% step load
perturbation in area-1
MIT International Journal of Electrical and Instrumentation Engineering Vol. 1, No. 1, Jan. 2011, pp. 1-5 4
ISSN No. 2230-7656 ©MIT Publications

-3
VI. CONCLUSION
x 10
4

2
full state feedback
Integral output feedback
An attempt has been made to design the optimal output
0
PI output feedback feedback controller for a two area interconnected Thermal-
Deviation in Ptie (P.U.MW)

-2
Thermal Power System incorporating reheat type turbine in
-4
one of the areas. In this paper the proposed controllers are
-6
tested and their dynamic responses are compared. Frequency
-8 deviation response of area-1 and area-2 and Tie line power
-10 deviation response to 1% step load perturbations in either
-12
0 5 10 15 20 25 30
areas with full state feedback, Integral optimal output
time, s
feedback and PI Optimal output feedback controller have
Fig. 4. Tie line power deviation response of two-area system to 1% been obtained. It is observed that optimal state feedback
step load perturbation in area-1
controllers give good dynamic responses that satisfy the
0.02
full state feedback
requirements of LFC. The controller design is simple and
systematic.
Frequency deviation (Hz),area-1

0.015 Integral output feedback

PI output feedback
0.01

0.005 APPENDIX
0

-0.005 System Parameters

-0.01 Prt1  Prt 2  2000 MW
-0.015 f=60Hz
-0.02
D1  D 2  0.00833 puMW / Hz
TSG1  TSG 2  0.08s
-0.025
0 5 10 15 20 25 30
TT 1  TT 2  0.3s
time, s
TR  10 s
Fig. 5. Frequency deviation response of area-1 to 1% step load
K R  0.5
perturbation in area-2
TPS 1  TPS 2  20 s
0.015
full state feedback R1  R2  2.4 Hz / puMW
Frequency deviation (Hz),area-2

Integral output feedback

K PS 1  K PS 2  120 s
0.01
PI output feedback

 1 
0.005

B1  B2    D1 
 R1 
0

-0.005 a12  1
-0.01 2 T12  0.545
-0.015

-0.02
REFERENCES
-0.025
0 5 10 15 20 25 30 [1] D.P. Kothari and I.J. Nagrath, Modern Power System Analysis, 3rd ed.
time, s
Singapore, McGraw Hill, 2003.
Fig. 6. Frequency deviation response of area-2 to 1% step load
[2] S. Majhi, Advanced control theory-a relay feedback approach,1st ed.
perturbation in area-2 Cengage Learning Asia, 2009.
x 10
-3
[3] Ibraheem, P. Kumar and D.P. Kothari, “Recent philosophies of
6
full state feedback automatic generation control strategies in power systems,” IEEE Trans
Integral output feedback Power Syst 20 (1) (2005), pp. 346–357.
4 PI output feedback
Deviaton in Ptie (pu MW)

[4] S. Majhi, “Relay based identification of time delay processes,” IFAC

2 Journal of process control, Vol.17, Issue 2, pp. 93-101, Feb. 2007.
[5] S. Majhi, “Relay based identification of a class of non-minimum phases
0 SISO Processes,” IEEE Transactions on automatic control, Vol.52,
No.1, pp.134-139, Jan. 2007.
-2
[6] D.P. Kothari and K.P. Singh Parmar, “A Novel Approach for Eco-
Friendly and Economic Power Dispatch Using MATLAB,” IEEE conf.
-4
proceedings, PEDES 2006, New Delhi, India.
-6
[7] D.P. Kothari and K.P. Singh Parmar, “Environment-Friendly Thermal
0 5 10 15
time, s
20 25 30
Generating Unit Commitment,” National power systems conference -
2006 proceedings, Roorkee, India.
Fig. 7. Tie line power deviation response of two-area system to 1%
step load perturbation in area-2
MIT International Journal of Electrical and Instrumentation Engineering Vol. 1, No. 1, Jan. 2011, pp. 1-5 5
ISSN No. 2230-7656 ©MIT Publications

[8] O.I. Elgerd, Electric energy system theory: an introduction (2nd ed.), [15] Elyas R and Sadeh J, Simulation of two- area AGC system in a
McGraw-Hill, New York (1983), pp. 310-360. competitive environment using reduced order observer method, IEEE
[9] P. Kundur, Power System Stability & Control. Tata McGraw Hill, Trans Power Appl Syst ) (2008). pp. 27–33
New Delhi, Fifth reprint 2008, pp. 581-626. [16] F. Liu, Y.H. Song et al., “Optimal load frequency control in restructured
[10] C.E. Fosha and O.I. Elgerd, The megawatt frequency control problem: a power systems”, IEE Proc. Generation, Transmission and Distribution,
new approach via optimal control theory, IEEE Trans Power Appl Syst, Vol 150, no. 1, pp. 87-95, Jan. 2003.
89 (4) (1970), pp. 563–577. [17] F. Llewis and V.L. Syrmos, Optimal Control, Prentice hall, Englewood
[11] O.I. Elgerd and C. Fosha, Optimum megawatt frequency control of liffs, New Jersy, 1995.
multi-area electric energy systems, IEEE Trans Power Appl Syst 89 (4) [18] M.L. Kothari, J. Nanda, D.P. Kothari and D. Das, “Discrete mode
(1970), pp. 556–563. automatic generation control of a two area reheat thermal system with
[12] M. Aldeen and H. Trinh, Load frequency control of interconnected new area control error,” IEEE Trans Power Appl Syst 4 (2) (1989), pp.
power systems via constrained feedback control schemes, Int J Comput 730–738.
Elect Eng 20 (1) (1994), pp. 71–88. [19] K.P. Singh Parmar, S. Majhi and D. P. Kothari, Multi-Area Load
[13] M. Aldeen and J.F. Marsh, Observability controllability and Frequency Control in a Power System Using Optimal Output Feedback
decentralized control of interconnected power systems, Int J Comput Method, IEEE Conf. proceedings, PEDES 2010 New Delhi, India.
Elect Eng 16 (4) (1990), pp. 207–220. [20] K.P. Singh Parmar, S. Majhi and D.P. Kothari, Automatic Generation
[14] W.S. Levin, M Athans, “On the determination of the optimal constant Control of an Interconnected Hydrothermal Power System, IEEE Conf.
output feedback gains for linear Multivariable systems,” IEEE Trans. proceedings, INDICON 2010, Kolkata, India.
On Automatic control,Vol AC-15, no.1, 1970.