Fractional Order Load-Frequency Control of Interconnected
Power Systems Using Chaotic Multi-objective Optimization
Indranil Pana,b and Saptarshi Dasb,c,*
a) Centre for Energy Studies, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016,
India.
b) Department of Power Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3,
Kolkata-700098, India.
c) School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ,
United Kingdom.
Authors’ Emails:
indranil.jj@student.iitd.ac.in, indranil@pe.jusl.ac.in (I. Pan)
saptarshi@pe.jusl.ac.in, s.das@soton.ac.uk (S. Das*)
Phone: +44-7448572598

Abstract:
Fractional order proportional-integral-derivative (FOPID) controllers are designed for load
frequency control (LFC) of two interconnected power systems. Conflicting time domain
design objectives are considered in a multi objective optimization (MOO) based design
framework to design the gains and the fractional differ-integral orders of the FOPID
controllers in the two areas. Here, we explore the effect of augmenting two different chaotic
maps along with the uniform random number generator (RNG) in the popular MOO
algorithm – the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Different measures
of quality for MOO e.g. hypervolume indicator, moment of inertia based diversity metric,
total Pareto spread, spacing metric are adopted to select the best set of controller parameters
from multiple runs of all the NSGA-II variants (i.e. nominal and chaotic versions). The
chaotic versions of the NSGA-II algorithm are compared with the standard NSGA-II in terms
of solution quality and computational time. In addition, the Pareto optimal fronts showing the
trade-off between the two conflicting time domain design objectives are compared to show
the advantage of using the FOPID controller over that with simple PID controller. The nature
of fast/slow and high/low noise amplification effects of the FOPID structure or the four
quadrant operation in the two inter-connected areas of the power system is also explored. A
fuzzy logic based method has been adopted next to select the best compromise solution from
the best Pareto fronts corresponding to each MOO comparison criteria. The time domain
system responses are shown for the fuzzy best compromise solutions under nominal operating

1

conditions. Comparative analysis on the merits and de-merits of each controller structure is
reported then. A robustness analysis is also done for the PID and the FOPID controllers.
Keywords: Two area Load frequency control (LFC); power system control; fractional order
PID controller; control trade-off design; chaotic NSGA-II
1. Introduction
Large scale power system networks comprise of several interconnected subsystems
representing particular geographical areas. Each of these subsystems has their own generation
capability and has variable load demand. These sub-systems are connected by tie lines which
control the flow of power between the different areas [1]. A sudden load demand in a certain
area results in a drop in system frequency which is detrimental for connected electrical loads.
To ensure proper power quality, the load frequency controllers in the interconnected power
system regulate the flow of power among the different areas through the tie lines and balance
the load and the drop in frequency. The LFCs balance the mismatch between the frequencies
of the interconnected areas and schedule the flow of power through the tie lies, helping the
interconnected power system to overcome the aberrations introduced due to varying load
demand, generation outage etc. Recently the LFCs are gaining more importance due to the
integration of renewables in the grid which have an inherent stochastic characteristic due to
the vagaries of nature unlike those of the base load thermal power plants [2], [3]. Thus proper
design and operation of the LFCs are very important for the stable and reliable operation of
large scale power systems. Control of interconnected power system considering various
aspects has been a topic of intense research in the recent past. Different type of generating
units and their effects have been studied e.g. thermal with reheat [4], generation rate
constraint (GRC) [5], reheat and battery energy storage both the areas [6], hydro turbine and
hydro-governor in both the areas [7], thermal with reheat turbine along with hydro and gas
turbine plants in both the areas [8], etc.
Traditionally a proportional-integral (PI) or a proportional-integral-derivative (PID)
controller is used for the LFCs [9] and a variety of different methods exist for proper tuning
of the controller parameters. Many robust control design techniques have been applied to the
LFC problem so that the designed controller is able to handle uncertainties of the system.
Some variants of robust designs include an adaptive output feedback based robust control
[10], adaptive robust control [11], decentralized robust control using iterative linear matrix
inequalities (LMIs) [12], decentralized control [13] etc. Optimal control designs using Linear
Quadratic Regulator (LQR) technique has been reported in [14]. Several other popular
control philosophies like the model predictive control (MPC) [15], sliding mode control [16],
and singular value decomposition (SVD) [17], etc. have also been applied in decentralized
LFC of multi-area power systems.
Many computational intelligence based techniques have been employed in the design
of LFCs as well. Global optimization techniques using evolutionary and swarm intelligence
has been used to tune the PID controller parameters in various literatures. Genetic algorithm
(GA) has been used in the design of LFCs in [7]. A variable structure controller has been
designed for LFC’s using GA in [18]. Fuzzy logic based gain scheduling has been done in
2

[19] to obtain improved control strategy for LFC. The application of neural networks in the
problem of load frequency control has been investigated in [20], [21]. There are other
literatures which employ robust control techniques like H ∞ loop shaping [22], µ-synthesis
[23] and LMI approaches [12], [22] with intelligent genetic algorithms to obtain robust
controllers. A detailed review of the existing design methodologies in LFC has been
documented in [24]-[26]. Other intelligent algorithms like Bacterial foraging algorithm [27],
fuzzy logic [28], recurrent fuzzy neural network [29] have also been applied in LFC of multiarea inter-connected power systems.
Fractional order controllers have been gaining popularity in recent years due to added
capability to handle control design specifications [30], [31]. Fractional order PID controllers
have been applied in a wide variety of control systems and have generally proven to be better
than their integer order (IO) counterparts [32]. Recently fractional order controllers have been
applied to power systems and favourable results have been obtained. In [33] a fractional order
controller has been designed for an automatic voltage regulator (AVR) with particle swarm
optimization (PSO) algorithm to show that the FO controllers have more robustness to tackle
uncertainties than the conventional IO-PID controller. Alomoush [34] has applied fractional
order controllers for LFC of a two area interconnected power system and an automatic
generation control for an isolated single area power system. It has been shown in [34] that the
fractional order PID controller has more flexibility in design and can adjust the system
dynamics better than the IO-PID controller. FOPID controllers are also shown to be robust
and competitive to IO-PID controllers [35–38]. In these previous literatures, only a single
objective intelligent optimization has been employed to design the control system. However
it is well known that there exists multiple trade-offs among different design specifications in
control [39] and similar system design [40]. It is not possible to simultaneously minimize all
design objectives using a particular control structure and different controller structure may
yield different trade-offs depending on the choice of the conflicting control objectives [41],
[42]. Thus there is a requirement of multi-objective approach for addressing different
conflicting objectives in the control system design [42]. In [42–44], time and frequency
domain multi-objective formulation have been used to study the design trade-offs among
various conflicting FOPID design objectives in an automatic voltage regulator (AVR) system.
It has been shown in [42–44] that sometimes the FOPID and at other times the PID controller
performs better depending on the choice of the conflicting objective functions. This concept
of MOO for FO controllers has been extended in this paper for two area LFC problem. In this
paper, FOPID controllers in the two area LFC are designed using chaotic multi-objective
NSGA-II algorithm. Set point tracking and low control signal are chosen as the two
conflicting objectives for the MOO based tuning of FOPID/PID controllers and the
performance improvement due to the FOPID compared to the PID is illustrated by numerical
simulations.
The main highlights of the paper includes:
•

Augmenting the NSGA-II algorithm with different chaotic maps (like Logistic and
Henon maps) to obtain better Pareto optimal solutions.
3

•

Using Pareto metrics like hyper-volume indicator, spacing metric, Pareto spread and
diversity metric [45]-[47], to assess the performance of the chaotic MOO algorithms.

•

Use of FOPID controller to obtain better system performance for the two area LFC.

•

Demonstration of conflicting time domain trade-offs in the controller performances
for the FOPID and the PID controller and use of a fuzzy based mechanism for
selecting the best compromise solution.

•

Robustness study of FOPID as LFC over that with PID, under system parametric
uncertainty and random change in load patterns.

The rest of the paper is organised as follows. Section 2 gives a brief description of the
two area LFC problem in the interconnected power system. Section 3 introduces the basics of
fractional calculus, FOPID controller and its flexibility over PID. Section 4 outlines the
conflicting time domain criteria based MOO for the design of the LFC system. Section 5
introduces the chaotic versions of the multi-objective NSGA-II algorithm. Different MOO
measure based selection of the best optimizer and the controller are enunciated in Section 6,
along with the respective time domain responses. In section 7, the effect of uncertainty in the
synchronizing coefficient and the effect of randomly changing load patterns in both areas are
explored. The paper ends with the conclusions in Section 8 followed by the references.
2. Load frequency control of interconnected two area power system
The main functionalities of the LFC are
a) To keep the operating power system frequency within specified tolerance limits to
ensure power quality
b) To ensure proper load sharing between the generators of the interconnected system
c) To honour the pre-specified load exchange constraints by controlling the power flow
between the interconnected areas.

Figure 1: Block diagram representation of a two area AGC system with secondary LFC loop

4

Table 1: Data for the two area system considered in the present simulation
Area

KPS (Hz/pu)

TPS (s)

R (Hz/pu MW)

B (pu
MW/Hz)

TG (s)

TT (s)

TR (s)

K1

K2

T12

Area 1

120

20

2.4

0.425

0.1

0.3

10

0.5

0.5

0.0707

Area 2

120

20

2.4

0.425

0.1

0.3

10

0.5

0.5

A schematic diagram for the two area LFC is depicted in Figure 1. The parameters of
the various units of the system are shown in Table 1 [23]. The area control error (ACE) in
each area is a function of the frequency deviation ( ∆f ) and the inter area tie-line power flow
( ∆Ptie ). The ACE is fed as an input to the FOPID controller where it calculates the
appropriate control signal to be applied. This control signal is fed into the amplifier and then
the actuator to produce an appropriate change in the mechanical torque of the turbine (prime
mover). This produces a change in the active power output of the generator to compensate the
power flow in the system and thus ∆f and ∆Ptie are kept within desired limits. The tasks of
the PID/FOPID controller in each area are to ensure faster damping of individual ACEs and
also damping the inter-area power oscillation ∆Ptie .
Each area includes steam turbine, governor, reheater stages along with GRC
nonlinearity in the turbine and dead-band in the governor. The power systems (PS) are
represented by first order transfer functions. The tie-line power is also affected by the choice
of synchronizing coefficient (T12). There are different configurations like primary and
secondary LFC as described in [34]. In the primary LFC control, load change in one area is
not corrected by a controller in that area or in other areas. For the secondary LFC loop, the
tie-line is represented by the accelerating power coefficient ( Ps = 2π T12 ). Any change in the
demand-load ( ∆PL ) will result in deviation of frequency in both the areas and the tie-line
power flow. The ACE for the i th area can be expressed as (1).
M

ACEi = ∑ ∆Pij + Bi ∆fi

(1)

j =1

where ∆Pij is the deviation in tie-line power flow from its scheduled values between the

i th area and the j th area, ∆fi is the frequency aberration in the i th area and M is the number of
areas connected to area i. The frequency bias factor ( Bi ) can be expressed as a combination
of the speed regulation ( Ri ) and the damping coefficient ( Di ) and is given by (2).
Bi = (1 Ri ) + Di

(2)

In the present model, significant effect of nonlinearity is studied in the form of deadzone in the governors and GRC in the turbines. The GRC keeps the rate of change of power
within a specified limit of δ = ±0.005 which is implemented by replacing the linear model of

5

the turbine ( ∆PR ∆X G ) by the nonlinear one as shown in Figure 1. The governor dead-band
affects the speed control under disturbances and has been chosen as 0.06% in each area. For
the present simulation study, both the areas are subjected to step-load disturbance of
PL1 = 0.02 pu and PL 2 = 0.008 pu respectively. In contemporary literature there were several
studies on the dynamics of nonlinearities like GRC [5], [29], [37], [36] and dead-zone [15]
along with reheat turbine [48], but their MOO based FO control design has not been
investigated yet, which is the main motivation of the present paper.
3. Fractional calculus and Fractional order PIλDµ (FOPID) controller
The generalized fractional differentiation and integration has mainly three definitions,
the Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. The
Grunwald-Letnikov formula is basically an extension of the backward finite difference
formula for successive differentiation. This formula is widely used for the numerical solution
of fractional differentiation or integration of a function. The Riemann-Liouville definition is
an extension of n-fold successive integration and is widely used for analytically finding
fractional differ-integrals. In the FO systems and control related literatures, mostly the
Caputo’s definition of fractional differ-integration is referred. This typical definition of
fractional derivative is generally used to derive fractional order transfer function models from
fractional order ordinary differential equations with zero initial conditions. According to
Caputo’s definition, the α th order derivative of a function f ( t ) with respect to time is given
by (3) which is used in the present paper for realizing the fractional integro-differential
operators of the FOPID controllers.
C
0

Dtα f ( t ) =

t
D m f (τ )
1
dτ , α ∈ ℝ + , m ∈ ℤ + , m − 1 < α < m.
Γ ( m − α ) ∫0 ( t − τ )α +1− m

(3)

Figure 2: Four quadrant operation of FOPID with different choice of speed level and noise amplification, compared to
the conventional integer order PID controller

6

The FOPID controller is an extension of the IO-PID controller with non-integer choice of
integro-differential orders along with the conventional PID controller gains. The transfer
function representation of a FOPID controller is given in (4).
C ( s ) = K p + ( Ki s λ ) + K d s µ

(4)

This typical controller structure has five independent tuning knobs i.e. the three

controller gains { K p , K i , K d } and two fractional order integro-differential operators {λ , µ} .

For λ = 1 and µ = 1 , the controller structure (4) reduces to the classical PID controller in
parallel structure. Figure 2 shows the schematic representation of the FOPID controller in the
integro-differential (λ-µ) plane and its relation with the conventional integer order P, PI, PD
and PID controllers. Several other special cases of the FO controller structure can also be
defined in the λ-µ plane like PIλ, PDµ, PIλD, PIDµ etc. It has been shown in [31] that
increasing the integral order above λ > 1 amplifies the low frequency components thus
making the closed-loop system oscillatory. Similarly, an increase in the derivative
order µ > 1 increases the high frequency gain, thus amplifying the measurement noise and
high frequency random components. The extension of ‘points to plane’ has led to the concept
of FOPID or PIλDµ controllers as shown in Figure 2 which can be further refined to define
regions of low/high noise amplification and slow/fast time response depending on the range
of values for µ and λ, below/above one. Figure 2 shows how different trade-offs could be
achieved by selecting the range of operation in the four quadrants of the λ-µ plane (instead of
only one quadrant with {λ , µ} < 1 ), for different application of FOPID controller. For the
present power system control application, the controller selection problem is divided in two
parts – fast FOPID (λ>1 or 1st/2nd quadrant operation) and slow FOPID (λ<1 or 3rd/4th
quadrant operation). Additionally, for choice of speed (λ), four other combinations of the
controllers in the two areas are explored, depending on the derivative order µ>1 or µ<1.
Within the present MOO framework, considering similar or different controller structure in
both the areas, the best non-dominated solution generated by the four controller combinations
is selected. The choice of the controller within the MOO framework has been shown in (5).

Select Slow FOPID ( λ1 , λ2 < 1) ,in both the areas
choice 1: µ1 < 1,in area 1 and µ2 < 1,in area 2 
choice 2 : µ1 < 1,in area 1 and µ2 > 1,in area 2 
 selection by MOO
choice 3 : µ1 > 1,in area 1 and µ2 < 1,in area 2 
choice 4 : µ1 > 1,in area 1 and µ2 > 1,in area 2 
Select Fast FOPID ( λ1 , λ2 > 1) ,in both the areas
choice 1: µ1 < 1,in area 1 and µ2 < 1,in area 2 
choice 2 : µ1 < 1,in area 1 and µ2 > 1,in area 2 
 selection by MOO
choice 3 : µ1 > 1,in area 1 and µ2 < 1,in area 2 
choice 4 : µ1 > 1,in area 1 and µ2 > 1,in area 2 

7

(5)

Therefore, within the slow and fast family of FOPID controllers, different levels of
noise amplification in the respective areas are selected automatically by the MOO which
results in non-dominated Pareto front.
Few recent research results show that band-limited implementation of FOPID
controllers using higher order rational transfer function approximation of the integrodifferential operators gives satisfactory performance in industrial automation. Here, the
Oustaloup’s recursive approximation has been used to implement the integro-differential
operators in frequency domain is given by (6), which represents a higher order analog filter.
N

sα ≃ K ∏

k =− N

s + ωk′
s + ωk

(6)

where, the poles, zeros, and gain of the filter can be recursively evaluated as (7).

ωk = ωb (ωh ωb )

k + N + (1+α ) 2
2 N +1

, ωk′ = ωb (ωh ωb )

k + N + (1−α ) 2
2 N +1

, K = ωhα

(7)

Thus, the area control errors (ACEs) can be passed through the filter (6) and the output of the
filter can be regarded as an approximation to the fractionally differentiated or integrated
signal Dα f ( t ) . These FO differentiated or integrated signals are weighted by the respective
gains to form the final control signal which goes to the governor in Figure 1. In (6)-(7), α is
the order of the differ-integration, ( 2 N + 1) is the order of the filter and (ωb , ωh ) is the
expected frequency fitting range. In the present study, 5th order Oustaloup’s recursive
approximation is implemented to approximate the integro-differential operators within the
frequency band of ω ∈ {10−2 ,102 } rad/sec for the constant phase elements (CPEs) of the
FOPID controller.

4. Need of multi-objective optimization and conflicting time domain control
objectives
It is well known that a single controller structure cannot give good results for all
design specifications. For example, a fuzzy logic controller is good at coping with uncertainty
in the loop, whereas a model predictive controller is good for tackling large time delays in
process control. For specific applications, different controller structures would give a tradeoff solution among conflicting design specifications. Hence for effective comparison of
different controller structures it is essential to know the limits of performance of each of the
individual controllers for conflicting design specifications. In [42]-[44], a similar approach
has been taken to compare the efficacy of the FOPID controller vis-à-vis the IOPID one for
several conflicting time and frequency domain objectives respectively. In the present case,
the disturbance rejection and controller effort are considered as the conflicting objectives and
expressed in (8)-(9).

8

M

M ∞

i =1

i =1 0

J1 = ∑ ITSEi = ∑ ∫ t ⋅ ei 2 (t )dt

M

M ∞

i =1

i =1 0

(8)

J 2 = ∑ ISDCOi = ∑ ∫ ( ∆ui (t ) ) dt
2

(9)

where, ITSE represents the Integral of the Time multiplied Squared Error, ISDCO represents
Integral of the Squared Deviation in Controller Output, ei ( t ) represents the error signal
(ACE) in area i , ui ( t ) represents the control signal and M represents the total number of
areas.
The reason for considering these two as conflicting objective functions can be briefly
explained as follows. To achieve a faster damping of ACEs and the grid frequency
oscillation, it is essential that the controller gains should be higher. In other words, the
controller must be able to exert much more control action on the power generating system so
that the frequency oscillation settles within a short amount of time. However, the control
signal should ideally be smaller to prevent actuator saturation and minimize the cost
associated with sizing of a larger actuator. It can be inferred that both these objectives of
small control signal, as well as faster damping of load-disturbances cannot be ideally
obtained by a fixed set of parameters of the PID/FOPID controller. Thus there would exist a
range of values for the tuning parameters of the controller { K p , K i , K d , λ , µ} , where the
controller would show good load disturbance rejection at the cost of higher control signal and
vice-versa. After a large number of trade-off solutions between the two chosen objectives are
obtained, a compromise solution could be selected next for deciding the most optimal
controller setting [42], [49].
5. Multi-objective controller design using chaotic maps
5.1.

Chaotic multi-objective optimization
A generalized multi-objective optimization framework can be defined as follows:

Minimize F ( x) = ( f1 ( x), f 2 ( x),..., f m ( x))

F ( x) = ( f1 ( x), f 2 ( x),..., f m ( x))

(10)

such that x ∈Ω ; where Ω is the decision space, ℝ m is the objective space, and F : Ω → ℝ m
consists of m real valued objective functions.
Let, u = {u1 ,..., um } , v = {v1 ,..., vm } ∈ ℝ m be two vectors.

u is said to dominate

v

if

ui < vi ∀ i ∈ {1, 2,..., m} and u ≠ v . A point x* ∈ Ω is called Pareto optimal if ∃ x | x ∈Ω
such that F ( x) dominates F ( x* ) . The set of all Pareto optimal points, denoted by PSMOO is

9

{

}

called the Pareto set. The set of all Pareto objective vectors, PF = F ( x) ∈ ℝ m , x ∈ PS MOO , is
called the Pareto Front or the set of non-dominated solutions. This implies that no other
feasible objective vector exists which can improve one objective function without
simultaneous worsening of some other objective function.
Multi-objective Evolutionary Algorithms (MOEAs) which use non-dominated sorting
and sharing, have higher computational complexity. They use a non-elitist approach and
require the specification of a sharing parameter. The non-dominated sorting genetic algorithm
(NSGA-II) removes these problems and is able to find a better spread of solutions and better
convergence near the actual Pareto optimal front [50], [51].
The NSGA-II algorithm converts different objectives into one fitness measure by
composing distinct fronts which are sorted based on the principle of non-domination. In the
process of fitness assignment, the solution set not dominated by any other solutions in the
population is designated as the first front F1 and the solutions are given the highest fitness
value. These solutions are then excluded and the second non-dominated front from the
remaining population F2 is created and ascribed the second highest fitness. This method is
iterated until all the solutions are assigned a fitness value. Crowding distances are the
normalized distances between a solution vector and its closest neighbouring solution vectors
in each of the fronts. All the constituent elements of the front are assigned crowding distances
to be later used for niching. The selection is achieved in tournaments of size 2 according to
the following logic.
a) If the solution vector lies on a lower front than its opponent, then it is selected.
b) If both the solution vectors are on the same front, then the solution with the highest
crowding distance wins. This is done to retain the solution vectors in those regions of
the front which are scarcely populated.
The optimization variables for the fractional order PID controller are the proportionalintegral-derivative gains and the differ-integral orders, i.e.

{K

p

, K i , K d , λ , µ} for both the

areas. For the IO-PID controller the optimization variables are the gains only i.e.

{K

p

, K i , K d } for both the areas. In other words, the dimension of the decision space Ω is

five for the FOPID controller and three for the PID controller. The population size is taken as
15 × nvar and the algorithm is run until the cumulative change in fitness function value is less
than the function tolerance of 10-6 or the maximum generations ( 200 × nvar ) are exceeded.
Here, nvar of parameters to be tuned by the MOO algorithm in both the areas and given
PID
FOPID
by nvar
= 6 and nvar
= 10 for the two controller structures. The crossover fraction is taken as

0.8 and an intermediate crossover scheme is adopted. The mutation fraction is chosen as 0.2
and the Gaussian scheme is adopted. For choosing the parent vectors based on their scaled
fitness values, the algorithm uses a tournament selection method with a tournament size of 2.
The Pareto front population fraction is taken as 0.7. This parameter indicates the fraction of
10

population that the solver tries to limit on the Pareto front [51]. For the MOO problem, the
limits of

{λ , µ}

{K

p

, K i , K d } are chosen as [ 0,10] and the bounds of the differ-integral orders

are chosen within the range [ 0, 2] which is described in (5).

The uniformly distributed RNG is normally used for the crossover and mutation
operations in the standard version of the NSGA-II algorithm [50][51]. However since the
strength of evolutionary algorithms lies in the randomness of the crossover and mutation
operators, many contemporary researchers have focussed on increasing the efficiency of these
algorithms by incorporating different random behaviours through various techniques like
stochastic resonance and noise [52], chaotic maps [53] etc. These different strategies can be
classified as special cases of a broader principle of diversification which essentially entails a trade-off
between exploration and exploitation [54]. In [55] it has been shown that the performance of

single objective evolutionary algorithms increase if different types of chaotic maps are
introduced instead of the uniform RNG for the crossover and mutation operations. It has also
been demonstrated in [55] that, in general, using chaotic systems for the RNG in the
crossover and mutation operations may yield better result than using RNG from a noisy
sequence in terms of convergence and effectiveness of the algorithms in finding global
minima. In [56]-[58] it has been shown that the multi-objective NSGA-II algorithm can be
improved by using chaotic maps and gives better result than the original NSGA-II algorithm
in terms of convergence and efficiency. This is due to the fact that the chaotic process
introduces diversity in the solutions. In this paper, we adopt similar policy and use chaotic
logistic map and chaotic Henon map to obtain comparable solutions and convergence with
respect to the standard NSGA-II algorithm. The logistic map is one of the simplest discrete
time dynamical systems exhibiting chaos. The equation for the logistic map is given in (11).
xk +1 = axk +1 (1 − xk )

(11)

The Henon map is a discrete time dynamical system that exhibits chaotic behaviour. Given a
point with co-ordinates { xn , yn } , the Henon map transforms it to a new point { xn +1 , yn +1} using
the set of equations in (12).
xn +1 = yn + 1 − axn2 ,
yn +1 = bxn

(12)

The map is chaotic for the parameters a = 1.4 and b = 0.3 . It is actually a simplified model of
the Poincare section of the Lorenz system. The output yn +1 varies in different ranges

depending on the initial seed { x0 , y0 } . Since the Henon map is used here as a random number
generator, it must produce a random number in the range [ 0,1] . Hence the output of the map

for different initial conditions have been scaled in the range [ 0,1] as also done in [59].

11

Figure 3: Random number generation by multiplying chaotic Henon map and Logistic map, normalized in the range [0,1]
with uniform RNG. Top: first 500 samples of the sequences. Bottom: histogram of 10000 samples.

The initial seed of the Logistic map ( x0 ) in (11) has also been chosen randomly for
the fixed parameter a = 4 exhibiting chaos, similar to that studied in [60]. For the
implementation of the chaotic versions of the MOO algorithms, the outputs of the chaotic
maps are scaled and multiplied with a uniform RNG which generates numbers in the
range [ 0,1] . The first 500 samples of uniform RNG, the scaled output of the Henon map and
the Logistic map and the outputs obtained by multiplying them with the uniform RNG are
shown in Figure 3. It also shows the histogram of the respective cases with 10000 samples. It
can be seen that even though the Logistic map and the Henon map have different
distributions, the corresponding distributions obtained by multiplying them with a uniform
RNG is highly skewed with lower values occurring more often than higher values. However,
it is to be noted that even though both the histograms show similar skewed characteristics, it
does not capture the time domain evolution like randomness and autocorrelation of the
numbers. Therefore this generation mechanism with chaotic maps is different than simply
drawing a random number from such skewed probability distributions each time.

5.2.

Quantification of Pareto fronts and choosing the best compromise
controller parameters

Due to the stochastic nature of the NSGA-II algorithm and also its two chaotic
versions, the final Pareto fronts obtained at the end of each independent run would be slightly
different with respect to one another. Therefore multiple runs are conducted to assess the
convergence characteristics of the algorithms. There are several measures to compare
multiple Pareto fronts apart from the non-domination criteria. In most realistic problems, two
Pareto fronts under comparison could intersect with each other which indicates that in one
region one set of solutions are more non-dominated in terms of one objective whereas for the
12

other conflicting objective the other Pareto front would be more non-dominated. Especially in
such cases of weak dominance, it could be difficult to judge the quality of the MOO solutions
from a global perspective. In the case of strong Pareto dominance where one front dominates
the other in all objectives, such problem is avoided due no intersection between multiple
Pareto fronts. In order to avoid such case dependent strategy formulation, several generic
metrics have been proposed in [45], which may indicate towards the quality of MOO
solutions. In this paper, four different Pareto measures viz. minimum hypervolume indicator,
maximum diversity metric, maximum Pareto spread, minimum spacing metric are explored.
Since none of the measures can capture all necessary properties of the best possible Pareto
front for the controller parameters [39], [61] a composite criteria is used to choose the best
Pareto front.
The hypervolume indicator is the given by the total area/volume/hypervolume under
the Pareto front with respect to the reference point as the origin. Therefore, a Pareto front
closest to the origin will have the minimum hypervolume indicator showing a strong
nondominance over other set of solutions [62]. The hypervolume indicator for two
dimensions with reference to the origin is given by
N

HI = ∑ ( xi − xi −1 )( yi − yi −1 )

(13)

i =1

where, N is the number of points on the Pareto front, x, y are the two dimensions, x0 , y0
correspond to the projections of the end points of the Pareto front on the two axes
respectively. This has to be minimized to obtain a better non-dominated Pareto front.
The spacing metric measures the distance between the variance of neighbouring datapoints on the Pareto-set [63], [64], which is given by (14).
SP =

(

1 S
∑ d − di
S − 1 i =1

)

2

(14)

 m

where, the distance di = min  ∑ xki − xkj  , {i, j} = 1, 2,⋯, S and d is the mean value of d i .
j
 k =1

Here, S is the number of non-dominated solutions and m is the number of objectives.
The total Pareto spread is given by the Euclidean distance between the extreme points
of the Pareto front [65]. The total Pareto spread helps in understanding the span of the
solutions along different conflicting objectives. For two dimensions, if ( x1 , y1 ) and ( x2 , y2 )
are the coordinates of the end points of the Pareto front, then the Pareto spread is given by
Pspread =

( x1 − x2 ) + ( y1 − y2 )
2

13

2

(15)

The moment of inertia based diversity metric is another popular measure to judge the
quality of the Pareto fronts [66]. Let there be Spt number of points in the m-dimensional
objective function space. Then the centroid for ith dimension is given by (16).

 S pt 
Ci =  ∑ xij  S pt , for i = 1, 2, ⋯, m
 j =1 

(16)

where, xij denotes the ith dimension of the jth point. Then, the diversity metric can be
calculated as (17).
I = ∑∑ ( xij − Ci )
N

S

2

(17)

i =1 j =1

Generally, the best, worst, mean and standard deviation of the diversity metric and other
measure are compared for different MOO algorithms [46], [47]. In the present paper, we also
report the box-plots of each of the four measures to ascertain the best Pareto front amongst 30
independent runs of each variant of the NSGA-II – nominal and chaotic. Additionally we
report the comparison of the convergence times for different controller settings and
optimizers.
After selecting the best Pareto front out 30 independent runs depending each of the
four criteria i.e. minimum hypervolume indicator, maximum diversity metric, maximum
Pareto spread, minimum spacing metric, a best compromise solution on the Pareto front has
been obtained using a fuzzy based mechanism. In [43], [44] the median solution has been
reported amongst all solutions on the Pareto front which has been improved here with a fuzzy
based systematic choice of the best compromise solution [46], [47]. The designer may have
imprecise goals for each objective function which can be encoded in the form of a fuzzy
membership function µ F . Here, µ Fi for each objective function i is taken to be a strictly
monotonic and decreasing continuous function expressed as (18).
1

 max
µ Fi = ( Fi − F ) ( Fi max − Fi min )

0


if

Fi ≤ Fi min

if Fi min ≤ Fi ≤ Fi max
if

Fi ≥ Fi

(18)

max

The value of µ Fi represents the degree to which a particular solution has satisfied the
objective Fi . The membership function lies between zero and one implying worst and best
satisfaction of the objective respectively. The degree of satisfaction of each objectives by
each solution can be represented as in (19).


m

 

S

 

j =1 i =1

m



µ k =  ∑ µik   ∑∑ µi j 
 i =1

14



(19)

where, m is the number of objectives and S is the number of solutions on the Pareto front.
The best compromise solution on the Pareto front has been chosen in such a way for which
(19) reaches its maximum.
6. Simulation and Results
6.1.

Multi-objective criteria for the best controller selection
Similar to [23], the system in Figure 1 is simulated with a step input ∆PL1 of 0.02 pu

in the first area and the ∆PL 2 of 0.008 pu in the second area. The PID/FOPID controllers have
been tuned using time domain performance indices in (8)-(9) under a MOO framework to
handle these load inputs. Three different controller structures are considered – traditional PID
controller, slow FOPID controller ( 0 < λ < 1 ) and the fast FOPID controller ( 1 < λ < 2 ). Each
of these cases are run with three different optimization algorithms – the standard NSGA-II,
the Logistic map adapted NSGA-II and the Henon map adapted NSGA-II. These 9 cases (3
controllers × 3 MOOs) are run for 30 times each and the statistics of the simulation time and
Pareto metrics (like hypervolume indicator, diversity metric, Pareto spread and spacing
metric) are calculated. The corresponding statistics are shown in the box-plots in Figure 4 to
Figure 8. From Figure 4 it can be observed that the median time taken by the chaotic versions
of the NSGA-II is higher than that of the normal NSGA-II for all the controllers. This is due
to the additional computational time taken by the chaotic maps in the various crossover and
mutation operators which employ RNG.

Figure 4: Box-plot of the simulation times with different MOO algorithms and controller structures.

15

Figure 5: Box-plot of the hypervolume indicator with different MOO algorithms and controller structures.

Figure 5 shows the box plots of the hypervolume indicator for the three different
algorithms and controller structures. From the minimum and median values, it is evident that
the performance of the FOPID controller is better than the PID controller. The hypervolume
indicator shows that the median values obtained using the traditional NSGA-II algorithm is
better than the chaotic versions for the FOPID controllers but is worse for the PID structure.

Figure 6: Box-plot of the moment of inertia based diversity metric with different MOO algorithms and controller
structures.

Figure 6 shows the box plot of the moment of inertia based diversity metric for 30
runs of all the cases. A higher value of diversity metric indicates a better Pareto front. It can
16

be observed that the Pareto fronts obtained for the FOPID controllers using the chaotic
NSGA-II algorithms have a higher value of the diversity metric. But for PID controllers, the
NSGA-II gives better results. It can also be seen that the slow FOPID controller has a higher
range of the diversity metric than the PID controller. This is possibly due to the extra degrees
of freedom of the FO integro-differential orders of the FOPID controller which has a larger
search space and thus allows more diverse solutions. The fast FOPID has a very small value
of diversity metric, indicating that most of the stable solutions belong to a small region of the
search space. For all the other cases, the solutions are either unstable or dominated.

Figure 7: Box-plot of the total Pareto spread with different MOO algorithms and controller structures.

Figure 8: Box-plot of the spacing metric with different MOO algorithms and controller structures.

17

Figure 7 shows the box plots of the total Pareto spread for 30 runs of all the cases. It is
observed that the chaotic versions of the NSGA-II are able to obtain a wider Pareto spread for
the FOPID controllers, indicating a more diverse set of solutions. However, for the PID
structure, the traditional NSGA-II gives a better Pareto spread. Figure 8 shows the box plots
of the spacing metric for 30 runs of all the cases. The original NSGA-II is found to give a
more uniform distribution of different solutions on the Pareto front.

Figure 9: Comparison of the Pareto fronts for slow FOPID controller using different MOO algorithms and selection
criteria.

Figure 10: Comparison of the Pareto fronts for fast FOPID controller using different MOO algorithms and selection
criteria.

18

It is therefore clear that none of the algorithms are better than their counterparts in all
the metrics individually. Also each metric represents different characteristics of the Pareto
front and finding the best Pareto front must leverage on some of these criteria taken together.
The diversity metric, Pareto spread and spacing metric reflect the distribution of the solutions
on the Pareto front. The hypervolume criterion which indicates the non-domination of the
different fronts, directly affects the quality of the obtained solutions and a better nondomination implies a better control system performance. Hence non-domination is one of the
most important metric among these. Therefore for comparison, the best Pareto front which is
obtained by each of the different Pareto metrics is found out of each of the 30 runs.

Figure 11: Comparison of the Pareto fronts for PID controller using different MOO algorithms and selection criteria.

It can be observed that for all the different Pareto metrics (hypervolume indicator,
diversity metric, Pareto spread and spacing metric) the chaotic version of the NSGA-II (either
logistic map assisted or Henon map assisted) gives the best non-dominated Pareto front. This
can be verified from the superimposed Pareto fronts in Figure 9-Figure 11 for the three
controller structures respectively. In most of the cases the logistic map assisted NSGA-II
gives better performance than the others. In Figure 9-Figure 11, the best Pareto front (out of
30 runs) according to each of the four Pareto measures are shown. According to each criteria
like the minimum hypervolume indicator, maximum diversity metric, maximum Pareto
spread and minimum spacing metric, the chaotic NSGA-II versions gives wider and nondominated Pareto spreads over that with the standard NSGA-II.
Next, for the sake of comparison amongst the best controller structures, the nondominated Pareto fronts according to each of the Pareto metrics is identified from Figure 9Figure 11 and are superimposed in Figure 12. It can be observed that under the nominal
operating condition of the power system, the slow FOPID controller structure is more nondominated and is therefore capable of resulting in better control performance. Although in all
the four cases in Figure 12, the slow FOPID gives the non-dominant Pareto front but
19

depending on different criteria the best compromise solution may perform better/worse.
Therefore, we here report all the best compromise solution of the three controller structures
for each of the Pareto fronts in Figure 12 (4 metrics × 3 controllers = 12 solutions in total).
The corresponding algorithm which gives the best non-dominated Pareto front and along with
the optimum controller parameters and objective functions have been reported in Table 2.
Table 2: Best compromise solutions of the FOPID and PID controller based on different Pareto metrics

Controller

Slow
FOPID

Fast
FOPID

PID

Criterion
Minimum
hypervolume
indicator
Maximum
diversity
metric
Maximum
Pareto spread
Minimum
spacing metric
Minimum
hypervolume
indicator
Maximum
diversity
metric
Maximum
Pareto spread
Minimum
spacing metric
Minimum
hypervolume
indicator
Maximum
diversity
metric
Maximum
Pareto spread
Minimum
spacing metric

Best
Nondominated
Algorithm

case

ITSE

ISDCO

Henon NSGA-II

1

1.01380

1.00040

Logistic NSGAII
Logistic NSGAII

2

1.01645

1.00053

3

1.01145

1.00053

Henon NSGA-II

4

1.01126

1.00068

Logistic NSGAII
Logistic NSGAII
Logistic NSGAII
Logistic NSGAII
Logistic NSGAII
Logistic NSGAII
Logistic NSGAII
Logistic NSGAII

5

1.01236

1.04872

1.00007

7

1.04872

1.00007

8

1.01392

1.00055

1.00726

Ki1
0.297

Kd1
0.036

λ1
0.869

µ1
0.208

Kp2
0.036

Ki2
0.237

Kd2
0.036

λ2
0.533

µ2
0.585

0.215

0.230

0.026

0.573

0.724

0.160

0.121

0.068

0.354

0.612

0.036

0.388

0.048

0.928

0.398

0.000

0.237

0.135

0.823

0.660

0.010

0.484

0.083

0.570

0.544

0.090

0.165

0.140

0.523

0.715

0.129

0.397

0.107

1.014

0.553

0.244

0.173

0.139

1.155

0.745

0.008

0.100

0.037

1.055

0.203

0.034

0.036

0.014

1.222

0.699

0.008

0.100

0.037

1.055

0.203

0.034

0.036

0.014

1.222

0.699

0.062

0.329

0.099

1.048

0.366

0.078

0.174

0.216

1.053

0.502

0.991

0.570

0.376

-

-

0.485

0.269

0.800

-

-

0.411

0.375

0.005

-

-

0.316

0.149

0.042

-

-

0.411

0.375

0.005

-

-

0.316

0.149

0.042

-

-

0.322

0.250

0.006

-

-

0.300

0.178

0.052

-

-

1.00082

6

9

Kp1
0.090

1.15637

10

1.01164

1.00137

11

1.01164

1.00137

12

1.01378

1.00095

Figure 12: Comparison of non-dominance amongst the controllers using best MOO algorithms for each criterion.

20

6.2.

Time domain performance of the LFC system

The time domain performance of the best compromise controller parameters reported
in Table 2 are now compared for the four Pareto metrics (hypervolume indicator, diversity
metric, Pareto spread and spacing metric). Since depending on the spread of the Pareto front
in Figure 12 the best compromise solution may be have different time domain characteristics.
The time domain responses of the grid frequency oscillation in both the areas, tie line power
flow and the control signals have been reported for each Pareto metrics in Figure 13-Figure
16.

Figure 13: Performance comparison of the best compromise solution of the three type of nondominated controllers
obtained using the minimum hypervolume indicator criterion.

Figure 14: Performance comparison of the best compromise solution of the three type of nondominated controllers
obtained using the maximum diversity metric criterion.

21

The time responses of the power system with the fuzzy based best compromise
solution have been shown in Figure 13 according to the solution obtained with hypervolume
indicator criterion. In other words, these solutions for the three different controller structures
are obtained by calculating the best compromise solution with the fuzzy based mechanism for
each of the Pareto fronts in Figure 12(a). It is observed that the time domain performance of
the PID (for supressing the oscillations in ∆f1 , ∆f 2 and ∆Ptie ) lies between those of the slow
and the fast FOPID, but the control signal required by the PID controller is much higher. The
slow FOPID results in less oscillations and overshoot and also has a smaller control signal.
Hence it outperforms the other two controller structures. The next exploration tries to
understand whether a similar response is observed if the controllers are selected based on a
different Pareto metric.

Figure 15: Performance comparison of the best compromise solution of the three type of nondominated controllers
obtained using the maximum Pareto spread criterion.

Figure 14 indicates towards a similar conclusion which shows that the time domain
performance of the fuzzy based best compromise solution obtained from the Pareto fronts of
Figure 12(b), which is based on the maximum diversity metric criterion. The PID controller is
found to have larger oscillations and overshoot in the time domain performance of ∆f1 , ∆f 2
and ∆Ptie along with a large value of control signal. The slow FOPID controller is found to
outperform both the other two controller structures.
Figure 15 shows the time domain performance of the fuzzy best compromise solution
obtained from the Pareto fronts of Figure 12(c), which is based on the maximum Pareto
spread criterion. The fast FOPID controller has a smaller control signal but has a very
sluggish time response. The slow FOPID controller has a faster time response than the fast
FOPID but this comes at the cost of a higher control signal. However the slow FOPID is
better than the PID in terms of both the peak overshoot and the control signal.

22

Figure 16: Performance comparison of the best compromise solution of the three type of nondominated controllers
obtained using the minimum spacing metric criterion.

Figure 16 shows the time response of the fuzzy best compromise solution obtained
from the Pareto fronts of Figure 12(d), which is based on the minimum spacing metric
criterion. The slow FOPID controller is found to outperform the other two structures in terms
of both time domain performance and low value of control signal.
7. Robustness analysis of the designed solutions
It is desirable that the designed controllers should work in a wide range of operating
conditions without significant deterioration in performance. In other words, the controllers
should be robust with respect to changes in system parameters. To illustrate this, the fuzzy
best compromise solutions for the slow FOPID, fast FOPID and the PID controllers for two
different Pareto metrics (minimum hypervolume indicator and maximum diversity metric),
are simulated by varying the synchronisation coefficient ( T12 ). The corresponding time
response curves for ∆f1 , ∆f 2 , ∆u1 , ∆u2 and ∆Ptie are plotted in Figure 17 and Figure 18
respectively. A similar study for the single objective FOPID controller has also been done in
[34].
From the time response characteristics of ∆f1 , ∆f 2 and ∆Ptie in Figure 17 with increase
in T12 by a factor of two, it might appear that the PID controller performs better than the
FOPID versions as the latter introduces small oscillations and do not settle to a steady state
value quickly. However the control signals are drastically higher for the PID controller and
have sharp jumps which might be detrimental for the governor.
Figure 18 shows the robustness of the fuzzy best compromise solutions for the case of
the maximum diversity metric criterion with increase in T12 by a factor of two and three. It
can also be observed that the fast FOPID controller is better than the slow FOPID and the
PID controller when T12 is increased gradually. For the latter two controllers the system
23

becomes unstable, while the fast FOPID controller is still able to maintain a time response
almost similar to the nominal case which proves the superiority of the FOPID in LFC over
the PID.

Figure 17: Effect of two times increase in T12 with the best compromise solution of the minimum hypervolume indicator
criterion

Figure 18: Effect of gradual increase in T12 with the best compromise solution of the maximum diversity metric criterion.

Simulation reported so far has been done with the nominal system parameters (in
section 6) and under uncertain parameters of the power system (section 7). The capability of
the three controller structures to damp grid frequency oscillations even in the presence of a
random change in load pattern in both the areas [29] is explored next. Figure 19 shows the
time response of the system obtained by different controllers where the load patterns are
24

randomly changing within PL1 = 0.02 pu and PL 2 = 0.008 pu [37]. In terms of fast settling time
and low controller effort, the slow FOPID controller is found to be better than the other two
controller structures.

Figure 19: Effect of random load change in both the areas with the best compromise solution of the maximum diversity
metric criterion.

Overall, both the designed solutions using the PID and the FOPID controllers show
sufficient robustness to system parameter variations and random load change with FOPID
variants outperforming the PID. Therefore the FOPID as load frequency controller could be
applied in a practical setting where significant uncertainty exists with respect to system
parameters as well as the change in load pattern.
8. Discussions and Conclusions
The following points summarise the main findings of the reported simulations in the paper.
•

Irrespective of the chosen MOO metric (like hypervolume indicator, total Pareto
spread etc.), the Pareto front obtained by the chaotic NSGA-II algorithm always
results in a better set of solutions (in terms of non-domination). In other words, even
though all the algorithms produce a set of non-dominated solutions as the output,
those that are obtained from the chaotic NSGA II are more non-dominated vis-à-vis
those obtained from their non-chaotic counterparts. For the present LFC problem, in
most cases the logistic map assisted NSGA-II works better than the corresponding
Henon map assisted version unlike [43].

•

Under nominal conditions and random load change, slow FOPID performs better in
terms of control system performance as indicated from the fast settling time and
keeping the maximum values of ∆f1 , ∆f 2 , ∆u1 , ∆u2 and ∆Ptie low.

25

•

When the system parameters are perturbed (e.g. synchronizing coefficient is
changed), either the fast or the slow FOPID controller is better depending on the
chosen Pareto metric, but it is always better than the PID controller.

In this paper, multi-objective design of an FOPID controller is done for LFC of a two
area power system with GRC in turbine, reheater stages and dead-band in the governor. The
NSGA-II algorithm and its chaotic versions are employed for the MOO task. Different Pareto
metrics are calculated and the optimization algorithms are evaluated based on multiple Pareto
metrics taken together. Numerical simulations show that the chaotic versions of the NSGA-II
algorithm gives better Pareto solutions over the ordinary NSGA-II algorithm. In general, the
chaotic logistic map assisted NSGA-II performs better over the chaotic Henon map assisted
NSGA-II. It is also shown that the FOPID controller outperforms the PID controller for
multi-objective designs of the two area LFC problem. Thus the FOPID controllers are a
viable alternative to the conventional IO-PID controllers in load frequency control of interconnected power systems.
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