Differential game approach for the analysis of two area load frequency control

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 02 | Feb -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1044
DIFFERENTIAL GAME APPROACH FOR THE ANALYSIS OF TWO AREA
LOAD FREQUENCY CONTROL
S. Karthick1, M. Sangeetha2
1 PG student, Department of Electrical and Electronics Engineering, The Kavery Engineering College,
Tamil Nadu, India
2 Assistant Professor, Department of Electrical and Electronics Engineering, The Kavery Engineering College,
Tamil Nadu, India
------------------------------------------------------------***-----------------------------------------------------------------
Abstract-In the traditional interconnected power
systems, the LFC plays an important role, which
constantly requires different Control Areas (CA) to
share the regulation burden of the control area that
lacks regulation capacity by providing the power
supports via the tie-line. Such process results in extra
regulation costs due to wear and tear of generating
units which may result in unfairness and CA’s deviate
from the LFC command. This situation becomes even
more serious with the integration of intermittent
renewable energy such as wind and solar power. The
main objective of this paper is to control the load and
frequency of the power system by using the
Differential Game approach. In previous research, the
PI controller with Genetic Algorithm (GA) was
implemented. But the disadvantage associated with
the GA is that it requires the repeated fitness function
evolution for complex problem and also the stop
criterion is not clear. In the proposed method, the
Differential Game (DG) approach is implemented.
Despite the challenge associated with the
computational complexity, the DG theory is a powerful
mathematical tool for solving the multiplayer multi
objective optimization problems. With two-area LFC
model, non-co-operative equilibrium solutions and
two cooperative equilibrium solutions with different
time consistency properties are derived. The proposed
strategies are then applied to a two-area power
system and compared with the traditional
Proportional–Integral (PI) controller and optimal
controller. The proposed method implemented in
MATLAB/Simulink and maintains the stability of the
system frequency by balancing the total generation
and total loads.
Key Words: Differential Game (DG), Control Area (CA),
Load Frequency Control (LFC), Cooperative Solution
1. INTRODUCTION
Electrical power systems are evolving toward the smart
grids all over the world, which are generally regarded as
the next-generation power grids. More and more
advanced techniques of control, communications and
computation will be employed. Electric power systems
are fundamentally reliant on control, communications,
and computation to ensure stable, reliable, and efficient
operation. Generators rely on governor and automatic
voltage regulator to counter the effects of disturbances
that continually affect power systems.
At a higher level, energy management systems use
supervisory control and data acquisition to collect data
from expansive power systems and then use
sophisticated analysis tools to ensure secure and
economic operational conditions. Load Frequency
Control (LFC) is a distributed closed-loop control
scheme that optimally reschedules generator power set
points to maintain frequency and tie-line flows at their
specified values. With the development of renewable
energy such as wind and solar power, power system
must overcome a number of technical difficulties to
deliver renewable energy in significant quantities.
Control is one of the key technologies for
accommodation of renewable energy into power
systems.
A game-theoretic approach to cooperative control by
highlighting the connection between cooperative control
problems and potential games is proposed. A new class
of games and enhanced existing learning algorithms is
introduced to broaden the applicability of game-
theoretic methods to cooperative control setting. The
cooperative control of multiple agents under different
communication scenarios is discussed. Cooperative
control laws are proposed with the aid of suitable
transformations and results from graph theory. Stable
control laws robust to communication delays are also
developed and studied. The different control trajectories
are compared for the two area system.
2. LOAD FREQUENCY CONTROL
The power systems is the interconnection of more than
one control areas through tie lines. The generators in a
control area always vary their speed together (speed up
or slow down) for maintenance of frequency and the
relative power angles to the predefined values in both
static and dynamic conditions. If there is any sudden

load change occurs in a control area of an interconnected
power system then there will be frequency deviation as
well as tie line power deviation.
The two main objective of Load Frequency
Control (LFC) are
 To maintain the real frequency and the desired
power output (megawatt) in the interconnected
power system.
 To control the change in tie line power between
control areas.
If there is a small change in load power in a
single area power system operating at set value of
frequency then it creates mismatch in power both for
generation and demand. This mismatch problem is
initially solved by kinetic energy extraction from the
system, as a result declining of system frequency occurs.
As the frequency gradually decreases, power consumed
by the old load also decreases. In case of large power
systems the equilibrium can be obtained by them at a
single point when the newly added load is distracted by
reducing the power consumed by the old load and power
related to kinetic energy removed from the system.
Definitely at a cost of frequency reduction we are getting
this equilibrium. The system creates some control action
to maintain this equilibrium and no governor action is
required for this. Each output frequency finds the
information about its own area and the tie line power
deviation finds the information about the other areas.
Thus the load frequency control of a multi area power
system generally incorporates proper control system, by
which the area frequencies could brought back to its
predefined value or very nearer to its predefined value
so as the tie line power, when the is sudden change in
load occurs.
2.1 INTERCONNECTED SYSTEM
According to practical point of view, the load frequency
control problem of interconnected power system is
much more important than the isolated (single area)
power systems. Whereas the theory and knowledge of an
isolated power system is equally important for
understanding the overall view of interconnected power
system.
Generally, now days all power systems are tied with
their neighboring areas and the Load Frequency Control
Problem become a joint undertaking. Some basic
operating principle of an interconnected power system
is written below:
 The loads should strive to be carried by their
own control areas under normal operating
conditions, except the scheduled portion of the
loads of other members, as mutually agreed
upon.
 Each area must have to agree upon adopting,
regulating, control strategies and equipment
which are beneficial for both normal and
abnormal conditions.
2.2 Two Area Power System
If there is interconnection exists between two control
areas through tie line than that is called a two area
interconnected power system. Fig-1 shows a two area
power system where each area supplies to its own area
and the power flow between the areas are allowed by
the tie line.
Fig- 1: Two Area Interconnected Power System
In this case of two area power system an assumption is
taken that the individual areas are strong and the tie line
which connects the two area is weak. Here a single
frequency is characterized throughout a single area;
means the network area is ‘strong’ or ‘rigid’. There may
be any numbers of control areas in an interconnected
power system.
2.3 Model of A Two Area LFC
The block diagram model of two area LFC is shown in
Figure 3.3.
where, the control command inputs are changed from
the requested deviation of generator output Pci to its
adjusting speed ui. Using ui, it is more convenient to
limit the damping rate of the units and to reflect the
wear and tear of the units in the control objective
function. We have the following linear model to describe
the system dynamics
x(t) = Axd(t) + Bud(t) + E Pd (t)

Fig-2: Two Area LFC
A=
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11
With subscript i representing the CA, , ,
and are the deviation of frequency, generator
mechanical output, valve position, tie-line power, and
requested generator output, respectively as
are the time constant of the
governor, the time constant of the turbine, the electric
system gain, the electric system time constant, the tie-
line synchronizing coefficient, and the speed drop,
respectively. Considering the intermittent energy in the
system, the power perturbation Pdi is considered as
the external perturbation input, consists of two parts as
= -
where, and are the perturbations of loads and
intermittent energy output, respectively. We assume that
both as and are known constants during the
control period obtained by some short-term forecasting
technology. Hence is a known step perturbation.
3. CLASSIFICATION OF DIFFERENTIAL GAME
The cooperative differential game is used to control the
control areas in the multi area power system. Based on
the stability criterion the games can be classified as
dynamic games, static games and state space games. The
main aim of this game is to attain the stability of the
system in a shorter duration, during the frequency
deviation.
3.1 Cooperative Games
A fundamental element in the theory of cooperative
games is the formulation of the optimal behavior for the
players satisfying specific optimality principles. From
the solution of the cooperative games, the control action
can be performed among the players. In particular, the
solution of a cooperative game is produced by a set of
optimality principles [for instance, the Nash bargaining
solution].
A cooperative solution is sub game consistent if an
extension of the solution policy to a situation with a later
starting time and any possible state brought about by the
prior optimal behavior of the players remains optimal. In
the field of cooperative stochastic differential games,
little research has been published to date due to the
inherent difficulties in deriving tractable sub game
consistent solutions. A sub game consistent solution is
developed for a class of cooperative stochastic
differential games with nontransferable payoffs.
3.2 Dynamic Games
In the dynamic games a stringent condition on their
solutions is required: the specific optimality principle
must remain optimal at any instant of time throughout
the game along the optimal state trajectory chosen at the
outset. This condition is known as dynamic stability or
time consistency. The issue of dynamic stability in
differential games has been explored rigorously in the
past three decades; In the presence of stochastic

elements, an even more stringent condition that of sub
game consistency is required.
The previously intractable problem of obtaining sub
game consistent cooperative solution has now been
rendered tractable for the first time. If the players in a
game wish to make an agreement to share the benefits of
cooperative behavior, the axiom states that no player is
willing to accept an agreement that will give her less
payoff than what she could obtain by rejecting to
participate in the cooperative solution.
3.3 Static Games
In a static game, the issue of individual rationality can be
resolved since one can determine what each player will
be offered as her part of an agreed solution and what
amount of payoff she could secure for herself in case of
no agreement. Comparing these payoffs provides an
answer to the individual rationality question. In games
evolving overtime, resolving the problem of individual
rationality may not be easy. The reason is that individual
rationality may fail to apply when the game has reached
a certain position, despite the fact that it was satisfied at
the outset. This phenomenon is notable in differential
games, and instate-space games as such. Studies in
repeated games have developed conditions under which
individually rational outcomes can be sustained as
equilibrium over time.
An important aspect of this is that cooperative out comes
may be supported by non cooperative equilibrium
strategies, but it raises the question if individually
rational outcomes exist. In repeated games with
complete information and perfect monitoring, the
answer is yes since the players face the same game at
every stage.
3.4 State-Space Games
In state-space games the situation is different. In a
discrete-time setup, a stochastic game includes a state
variable that evolves over time, as a product of the initial
conditions, the players' actions, and a transition law.
Particular results exist for situations in which Pareto-
optimal outcomes are supported by trigger strategies
Note that the punishments underlying trigger strategies
are effective only if cooperation is individually rational
throughout the game. For only in this case punishments
will hurt a defector. Thus, one reason for studying inter
temporal individual rationality is that it is an important
component of a self-enforcing agreement.
Such instabilities arise from the simple reason that in
general the position of the game at an intermediate
instant of time will differ from the initial position.
Therefore, the original solution may not be the solution
of the sub game that starts out in an intermediate
position of the game. The problem is that although
individual rationality was satisfied initially, it may not be
satisfied at intermediate positions of the game.
4. PROPOSED METHOD BASED COOPERATIVE
CONTROL MODEL OF LFC
Consider a two-player differential game played on a
prescribed time interval [ ] where the time horizon T
may be infinite. The state equation is,
̇( ) ( ( ) ( ) ( ) ) ( ) (1)
In which, x(t) ( ) Here X denotes the
state space and is the set of admissible controls of
player i. we refer to the pair (t,x) , - as the
position of the game at time t. If the game is played non
cooperatively, players use markovian strategies given by
the pair s(t,x) ( ( ) ( )) For each
, - the resulting control strategies are
represented.
The payoff functional to be maximized by player i is
( ( ) ) ∫ ( ( ) ( ) ) ( ( ) )
(2)
In equation(2), the function represents
player’s instantaneous payoff and function is terminal
payoff. If the horizon is infinite, put the terminal payoffs
equal to zero. The initial position ( ) is fixed.
The players negotiate at the initial position to
establish an agreement on how to play the cooperative
dynamic game (and how to distribute the resulting
payoff) using equation (2). An agreement must satisfy
group rationality and thus we consider cooperative
outcomes belonging to the Pareto optimal frontier. Such
outcomes can be found by solving at (to, xo) the optimal
control problem given by,
*∑ , ( ) -+ (3)
The problem of time consistency arises due to the
following fact Clearly, the problem disappears if
agreements are binding or if the players have pre
committed.
Players bargain at the initial instant of time and
arrive at an agreement which is individually rational at
this particular position. To present the scenario in a
transparent way, we introduce three assumptions.

Assumption 1
Reconsidering the game at any instant ,
the players have cooperated from time till time T.
Assumption 2
At any position ( ( )) one of the two
situations may occur:
 1f the players agree, they reopen negotiations
and may choose to replace the original
agreement with a new one. 1f implemented, the
new agreement is supposed to be used
throughout the time interval [t,T].
 Any player can reconsider her participation in
the agreement, and decide to deviate from it. 1f
a player deviates, she abandons her cooperative
strategy throughout the time interval [t, T].
Assumption 3
If a player at time T deviates from her
cooperative control path, she switches to her
disagreement strategy.
In the Cooperative control model of LFC, the different
types of controllers are used to control the control areas
of the LFC. The conventional controllers are PI
controllers and the optimal controllers. These
conventional controllers are combined with the
Differential Game based controller to achieve the best
control action as shown in Figure-1.
4.1 PI Controller
As the name suggests it is a combination of proportional
and an integral controller the output (also called the
actuating signal) is equal to the summation of
proportional and integral of the error signal. Now let us
analyze proportional and integral controller
mathematically. As we know in a proportional and
integral controller output is directly proportional to the
summation of proportional of error and integration of
the error signal, writing this mathematically we have,
( ) ∫ ( ) ( ) ( ) (4)
Removing the sign of proportionality we have,
( ) ∫ ( ) ( ) (5)
Where Ki and are proportional constant and integral
constant respectively.
Fig-3: Block Diagrams of Three Kinds of Controllers
(a) PI Controller
(b) Controller Based on DGs or Optimal
Control
Using the conventional control strategy, we can control
the dynamic frequency response and also make the
steady state error to zero with changes in load. An
integral controller is added to the un-controlled system
which actuates the speed changer by real power
command signal ΔPc.
∫ (6)
The negative polarity must be chosen so as the
frequency error will give a negative control to reduce the
value of command signal.
Area Control Error (ACE), is
(7)
Taking Laplace transform of equation 4.6, we get Δ PC
( ) ( ) (8)
for step input load
( ) (9)

( )
( ( )
( )( )
)
( )( )
( )( )( ) ( )
(10)
Using final value theorem, we readily obtain from the
above equation the static frequency droops:
, ( )-=0 ie. No error.
Thus, by using the PI controller, the steady state error
can be made to zero.
4.2 Optimal Controller
The theory of optimal control is concerned with
operating a dynamic system at minimum cost. The case
where the system dynamics are described by a set of
linear differential equations and the cost is described by
a quadratic function is called the LQ problem. One of the
main results in the theory is that the solution is provided
by the Linear-Quadratic Regulator (LQR).The dynamics
of a structure is represented in state space form as:
̇ (11)
Where, A and B are the system matrices. The output
equation can be written as,
̇ (12)
Now, assume a static output feedback of the form, U=-Gy
Where, G is the gain matrix.
The objective is to design a controller by choosing a
proper controller gain G, which is also optimal in the
sense that it minimizes a performance index:
( ) *∫ ( ) + (13)
Here,
The optimization results in a set of non-linear coupled
matrix equations which is given by,
=0
Where, M denotes the closed loop system as, M=A+BGC,
L is the liapunov matrix. The optimal gain G may be
obtained by iteratively solving the above equation set. By
tuning the weighting matrices Q and R, a sub-optimal
controller gain G can be achieved using this method.
4.3 Controller Based on DG
The individual payoff function of the ith CA in LQDG
based cooperative control is given by the following
quadratic performance metric:
∫ , ( ) ̅( ) ( ) ̅( )- , (14)
i=1,2
Where, Qi is a 9 × 9 positive definite weight matrix that
determines the penalty associated with the frequency
error and tie-line power error, and Riis a 2×2 positive
definite weight matrix that represents the control costs.
By setting Qi and Ri, each CA specifies the way it wishes
its LFC units to perform.
However, the term E_ .To eliminate this term, we
employ the method by redefining the state, control, and
disturbance vectors by using their final steady state
values,
( ) ̅( ) ̅ ( ) ̅( ) ̅
( ̅̅̅( ) ̅̅̅) . (15)
Under a step power perturbation, the deviations of
steady state frequency, tie-line power, and adjusting
speed of the unit’s power generation are required to be
zero. The cooperative control for the two-area LFC then
takes the following form:
∫ , ( ) ̅( ) ( ) ̅( )- (16)
s.t. ̇( ) ( ) ( ) ( )
The objective of each CA is to minimize its own payoff
function by choosing its appropriate control.
5. ANALYSIS OF RESULT
The simulation of the two area AGC system with
the PI controller, Optimal controller and the control
based on Differential Game is analyzed from the
resulting waveform of frequency deviation. These are
analyzed based on the aspect of the peak overshoot,
settling time and steady state error.

5.1 Frequency Deviation in Area One with PI
Controller
Fig-4: Frequency Deviation in Area One with PI
Controller
5.2 Frequency Deviation in Area Two with PI
Controller
Fig-5: Frequency Deviation in Area Two with PI
Controller
5.3 Tie-Line Frequency Deviation Between
Area One and Two with PI Controller
Fig-6: Tie-Line Frequency Deviation Between Area
One and Two with PI Controller
5.4 Tie-Line Frequency Deviation Between Area
One and Two with Optimal Controller
One and Two with Optimal Controller
5.5 Tie-Line Frequency Deviation Between Area
One and Two with DG Technique Combining PI
and Optimal controller
One and Two with DG Technique Combining PI and
Optimal controller
5.6 Comparison and Evaluation of Results
Figure 5.6 shows the change in tie line power between
area one and two of the power systems with respect to
time, while employing PI controller. From the response
the following observation are obtained.
Peak over shoot = 0.025
Settling time = 35 sec
Steady state error
=
0.5

time, while employing optimal controller. From the
response the following observation are obtained.
Steady state error
=
0.005
time, while employing Differential Game technique by
combining the PI controller and Optimal Controller.
From the response the following observation are
obtained.
Steady state error = 0.01
6. CONCLUSION
The two-area thermal LFC problem is
investigated under the framework of DG-based
cooperative control. Three kinds of equilibrium solutions
have been derived using the DG theory, with several
simplification and approximations to cope with the
computational difficulties. Case studies are provided by
using various LFC schemes. Simulation results show that
unlike the conventional PI controllers and the optimal
controllers, the DG-based cooperative controllers assign
the persuasive amount of LFC regulation to each CA to
guarantee the stable and faithful implementation of
control command. Despite the challenge associated with
the computational complexity, the DG theory is a
powerful mathematical tool for solving the multiplayer
multi objective optimization problems. The system is
more stable, while using this Differential game technique
instead of the conventional PI and Optimal controllers.
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Differential game approach for the analysis of two area load frequency control