Optimal Parameters for Sliding Mode Based

Load-Frequency Control in Power Systems

Kresimir Vrdoljak, Student Member, IEEE,

Nedjeljko Peric, Senior member, IEEE

Faculty of Electrical Engineering and Computing

University of Zagreb, Croatia

Email: kresimir.vrdoljak@fer.hr, nedjeljko.peric@fer.hr

Muharem Mehmedovic, Member, IEEE

Faculty of Electrical Engineering

University of Osijek, Croatia

Email: muharem.mehmedovic@etfos.hr

Abstract— One of main obstacles in the usage of sliding modebased load-frequency control in a power system is the difficultyin choosing optimal controller parameters. Those parametersdefine the sliding surface and also dynamics of the reachinglaw. Constraints upon the choice of those parameters comeout from the requirements of system’s stability and no steadystate error. In this paper the parameters are computed usinga genetic algorithm. Obtained optimal parameters are testedon simulations, which are conducted on a power system modelconsisting of three interconnected control areas.

I. INTRODUCTION

”Union for the Co-ordination of Transmission of Electricity”

(UCTE) is the association of transmission system operators in

continental Europe and it consists of 25 countries, therefore

being the largest power system in Europe. According to the

UCTE rules and recommendations [1], goals of load-frequency

control (LFC) in each control area (CA) are to restore the

frequency to its nominal value in case of a deviation and to

maintain area’s power flow interchange with the neighbor CAs.

CAs are subsystems of an interconnected power system. They

consist of coherent group of generators and are connected with

other CAs by tie-lines.

Reasons for frequency and active power flow deviations

are differences between generation and consumption in a

CA. When modeling a CA those deviations can be seen as

external disturbances. Another problem for control of a CA is

that system parameters constantly vary in time. The reasons

for that can be the size and characteristic of consumption,

characteristics of power plants and number of power plants

engaged in LFC in a CA.

Load-frequency controllers implemented in real power sys-

tems are usually PI type [2], and they have many drawbacks,

such as long settling time and relatively large overshoots

[3]. Also, PI controllers with conventionally tuned parameters

are not robust to system parameters deviations. Therefore, a

robust control algorithm that ensures system’s good dynamical

performance for a wide range of operating conditions is needed

for LFC.

Many different approaches addressing LFC problem have

appeared in the literature [4]. In the recent years those ap-

proaches are based on H∞ control [5], fuzzy logic control [6]-

[8], neural networks [9]-[11], model predictive control [12],

maximum peak resonance approach [13], genetic algorithms

[14] and adaptive control [15].

Sliding mode control (SMC) imposes as a possible choice

for overcoming the drawbacks of classical LFC algorithm, due

to its very good behavior in controlling systems with external

disturbances and parameter variations [16]. A few researches

about sliding mode based LFC can be found [17]-[20].

In SMC, system closed-loop behavior is determined by

a sub-manifold in the state space which is called a sliding

surface. The goal of the sliding mode control is to drive the

system state to reach the sliding surface (reaching phase)

and then to stay on it (sliding phase). When the state is

on the surface, system’s invariance to some uncertainties and

parameter variations is guaranteed.

The main problem in applying sliding mode to load-

frequency control is choosing the appropriate sliding surface.

Also, an adequate reaching law must be chosen to ensure

convergence of a system’s trajectory towards the surface [21].

Unappropriate choice of those parameters can cause steady

state error or even system’s instability. Genetic algorithm (GA)

is a robust search technique, therefore being appropriate for

finding the optimal values of the parameters.

In continuous time SMC system it is possible to achieve

trajectory’s sliding along the surface. Continuous time SMC is

not appropriate for LFC because real power plants are unable

to respond to fast changes in control signal. Therefore discrete

time sliding mode control is used here, where control signal

can switch only every one or few seconds. In discrete time

SMC, the trajectory is unable to stay on the surface, but it is

being kept in a small band around the surface. That behavior

is known as quasi sliding mode (QSM) [22].

The outline of the paper is as follows. In Section II power

system model is described, whereas Section III gives an

overview of discrete-time SMC. Section IV presents a genetic

algorithm used for the purpose of finding the optimal sliding

mode algorithm parameters. Section V presents simulation

results with controller’s parameters computed for the proposed

power system model.

II. MATHEMATICAL MODEL OF A POWER SYSTEM

A mathematical model of a power system used in this paper

consists of three interconnected CAs, as it is shown in Fig. 1.

978-1-4244-2200-5/08/$25.00 ©2008 IEEE 331

SlidingMode LFC

SlidingMode LFC

PI LFC

GA

Hydro PowerPlant

Thermal PowerPlant

Thermal PowerPlant

Control Area 3

Control Area 1 Control Area 2

∞H

Fig. 1. Three interconnected control areas

Every CA from Fig. 1 is represented with one substitute

power plant and one load-frequency controller. Power plant in

CA 1 is hydro power plant and power plants in CA 2 and

CA 3 are thermal power plants. Controller in CA 1 is based

on SMC with parameters calculated using GA and it will be

described in detail in Section IV. Controller in CA 2 is also

based on SMC, but its parameters are calculated using H∞

norm, as it is described in [23]. Controller in CA 3 is PI type.

Any of three CAs can be described with the following

equations:

xi(t) = Aiixi(t) +N∑

j=1,j 6=i

Aijxj(t)

+ Biui(t) + Fidi(t),

yi(t) = Cixi(t).

(1)

A simplified linearized continuous-time model of CA 1 is

shown in Fig. 2. The reason for the usage of linearized model

is small changes in load which are expected during normal

system operation. For this model, state, output and disturbance

vectors from (1) are:

xi(t) =

∆fi(t)∆Ptiei(t)∆Pgi(t)∆xgi(t)∆xghi(t)

,

yi(t) = ACEi(t),

di(t) = ∆Pdi(t),(2)

while matrices in (1) are:

Aii =

− 1

TP i−KPi

TP i

KPi

TP i0 0

N∑

j = 1,

j 6= i

KSij 0 0 0 0

2α 0 −2

TW i2γ 2β

−α 0 0 −1

T2i−β

− 1

T1iRi0 0 0 − 1

T1i

,

Aij =

0 0 0 0−KSij 0 0 0

0 0 0 00 0 0 00 0 0 0

, Bi =

00

−2TRi

T1iT2i

TRi

T1iT2i

1

T1i

,

Fi =

−KPi

TP i

0000

, Ci =[

KBi 1 0 0 0]

,

(3)

with coefficients in matrix Aii:

α =TRi

T1iT2iRi

, β =TRi − T1i

T1iT2i

, γ =T2i + TWi

T2iTWi

.

Details about modeling CAs with thermal power units can

be found in [24].

1

1

1 isT+ 2

1

1

Ri

i

sT

sT

+

+ 1

Pi

Pi

K

sT+

2

s

π

1

iR

-

-+

iu

diP∆

if∆

iδ∆

+

- ghix∆giP∆

1

( )2

Nsij

i j

jj i

Kδ δ

π=

≠

∆ − ∆∑tieiP∆

1

1 0.5

Wi

Wi

sT

sT

−

+

gix∆

LFC

Fig. 2. Block diagram of a control area represented with hydro power plant

332

A numerical measure of CA’s deviation from the proposed

behavior is an area control error (ACE) signal. ACE is a

combination of frequency deviation in a CA and active power

flow deviation in tie lines connecting CA with the neighbor

areas:

ACEi(t) = KBi · ∆fi(t) + ∆Ptiei(t). (4)

Signals and parameters of the ith area subsystem (shown in

Fig. 2), described with (1)-(3) are:∆fi(t) frequency deviation (Hz),

∆Pgi(t) generator output power deviation

(p.u. MW),

∆xgi(t) governor valve position deviation,

∆xghi(t) governor valve servomotor position

deviation,

∆Ptiei(t) tie line power deviation (p.u. MW),

∆Pdi(t) load disturbance (p.u. MW),

T1i, T2i, TRi governor time constants (s),

TWi water starting time (s),

TPi power system time constant (s),

KPi power system gain (Hz / p.u. MW),

KSij interconnection gain between the ith

and the jth area (p.u. MW),

KBi frequency bias factor (p.u. MW / Hz),

Ri speed droop due to governor action

(Hz / p.u. MW).In LFC, control signal is sent to the power plants in

discrete time, therefore control algorithm will be synthesized

for discrete time equivalent model of a CA. A zero-order-

hold (ZOH) discrete time approximation of system (1), with

sampling period T , is:

xi(k + 1) = Giixi(k) +N∑

j=1,j 6=i

Gijxj(k)

+Hiui(k) + Widi(k),

yi(k) = Cixi(k),

(5)

where matrices are:

Gii = eAiiT ,

Gij =T∫

0

eAiiηAijdη,

Hi =T∫

0

eAiiηBidη,

Wi =T∫

0

eAiiηFidη.

(6)

Based on (5), a discrete-time sliding mode controller will

be designed.

III. DISCRETE-TIME SLIDING MODE CONTROL

Equations (5) and (6) describe the ideal discrete-time model

of a CA. But, in real power systems, many uncertainties

affect the system behavior, such as unmodeled dynamics or

variations of system parameters. They can be decoupled into

one of two types of uncertainties: matched or unmatched [25].

In continuous SMC, the influence of matched uncertainties can

be fully compensated, but that is not the case for discrete-time

SMC.

To explain sliding mode based controller design, let us

start with general, discrete-time linear system, with matched

and unmatched additive uncertainties. It is described with the

following equation:

x(k + 1) = Gx(k) + Hu(k) + Wd(k) + ξm(x, k) + ξu(x, k),(7)

where x ∈ Rn is the system state vector, u ∈ R

m is the control

input vector, d is measurable disturbance, while ξm and ξu are

matched and unmatched uncertainties, respectively. Matched

uncertainties are those that fulfill the following condition:

ξm ∈ R(H), (8)

and therefore can be written as ξm = Hγ, where γ ∈ Rm. All

other uncertainties are unmatched.

For better understanding of sliding mode control, let the

system (7) be transformed into regular form [16] using trans-

formation matrix Tr:[

x1(k + 1)x2(k + 1)

]

=

[

G11 G12

G21 G22

][

x1(k)x2(k)

]

+

[

0H2

]

u(k)+

+

[

W1

W2

]

d(k) +

[

ξu(x, k)ξm(x, k)

]

.

(9)

where are:[

x1(k) x2(k)]T

= Trx(k).Design of sliding mode controller for the system (9) consists

of two major steps.

The first step is choosing a sliding surface, σ(x) = 0, which

will define desired system dynamics. The aim of sliding mode

control is to force the state, firstly to reach, and then to stay

on the surface. Let us assume the sliding surface is chosen as

σ(k) = Sx(k), where S is a switching vector. For the system

in the regular form (9), switching function σ(k) becomes:

σ(k) = S1x1(k) + S2x2(k). (10)

System (9) can be transformed using (10) into:[

x1(k + 1)σ(k + 1)

]

=

[

G11 G12

G21 G22

][

x1(k)σ(k)

]

+

[

0

H2

]

u(k)+

+

[

W1

W2

]

d(k) +

[

ξu(x, k)

ξm(x, k)

]

,

(11)

where are:G11 = G11 − G12S

−1

2S1,

G12 = G12S−1

2,

G21 = S1G11 − S1G12S−1

2S1 + S2G21 − S2G22S

−1

2,

G22 = S1G12S−1

2+ S2G22S

−1

2,

H2 = S2H2,

W2 = S1W1 + S2W2,

ξm(x, k) = S1ξu(x, k) + S2ξm(x, k).

For the system in sliding mode it is σ(k+1) = σ(k) = 0. If

that condition is included into system (11), two improvements

333

in system behavior arise. One is that the order of system

dynamics is reduced in comparison to the original system

(9). The other improvement is system’s invariance to matched

uncertainties. Ideally, those are both true only at the sampling

instants, when the trajectory is on the surface. Between two

samples system trajectory will be inside a narrow band around

sliding surface.

Matrix S must be chosen such that the system dynamics

in sliding mode are stable, i.e. all eigenvalues of matrix G11

must be within the unit circle. Apart from stability issues, the

chosen matrix S must ensure minimal ACE deviation during

QSM [23].

The second step of sliding mode controller design is the

choice of a reaching law and its parameters. The reaching

law is in charge of driving system trajectory to the sliding

surface σ = 0. There are several reaching laws proposed in

the literature ([21],[22]).

In this paper a linear reaching law from [26] is used:

σ(k + 1) = λσ(k), (12)

where 0 ≤ λ < 1.

The reasons for its usage are control law’s straightforward

design and no intentional switching, which would cause wear

and tear of plant’s actuators. In an ideal case, with reaching law

(12) system trajectory would be forced to linearly approach the

sliding surface.

When applied to the second row of system (11), and with

uncertainties neglected, reaching law (12) results with the

following control law:

u(k) = −H−1

2

[

G21x1(k) − (λ − G22)σ(k) + W2d(k)]

.

(13)

Because of the discretization issues and because uncer-

tainties are neglected in the computation of the control law

(13), the ideal sliding mode is not guaranteed. Instead of that,

system trajectory will reside in a quasi sliding mode band,

whose width depends upon the value of uncertainties and also

parameter λ from (12). To decrease the width of the band,

control law (13) is modified with uncertainties estimation term

[21]:

u(k) = −H−1

2·

[

G21x1(k) − (λ − G22)σ(k)+

+W2d(k) + ξm(x, k)]

,(14)

where uncertainties estimation dynamics are defined as:

ξm(x, k) = ξm(x, k − 1) + σ(k) − λσ(k − 1). (15)

Ideally, it would be ξm(x, k) = ξm(x, k).Specially, if a disturbance d(k) cannot be measured or

estimated independently, it can be estimated together with the

uncertainties, which yields a following control law:

u(k) = −H−1

2

[

G21x1(k) − (λ − G22)σ(k) + ξd(x, k)]

,

(16)

where disturbance and uncertainties estimation dynamics are

now defined as:

ξd(x, k) = ξd(x, k − 1) + σ(k) − λσ(k − 1). (17)

Now, in the case of ideal estimation, it would be ξd(x, k) =ξm(x, k) + W2d(k).

Finally, the proposed controller acts in the following way:

a) Measured system state is transformed into regular form

using transformation matrix Tr.

b) A switching function σ(k) is computed from the trans-

formed state using (10).

c) Control signal u(k) is computed using either control law

(13), (14) or (16) and then applied in a CA.

Scaling of the switching vector S will not alter the control

laws and system behavior, therefore it is chosen S2 = 1.

As it can be seen from (13)-(17), apart from depending

on system parameters, control laws depend on sliding mode

control parameters S and λ. To determine their optimal values

GAs will be used.

IV. GAS FOR FINDING OF SLIDING MODE PARAMETERS

GA is random search approach which imitates natural

process of evolution. It is appropriate for finding global

optimal solution inside a multidimensional searching space.

GAs are used in the literature to find parameters for different

LFC algorithms ([14],[19]).

From the random initial population, GA starts a loop of

evolution processes, such as selection, crossover and mutation,

to improve performance index of the whole population. Every

chromosome in the population consists of five genes. First four

genes represent the eigenvalues of matrix G11, while the fifth

gene represents λ. Therefore, to ensure system stability, first

four genes are limited to values between -1 and 1, while the

fifth gene can obtain values between 0 and 1.

GA consists of the following steps:

1. A random initial population is created.

2. Firstly, from the given chromosomes, a transformed

switching vector S1 in (10) is calculated. Secondly, pa-

rameters of control law (13), (14) or (16) are calculated.

Thirdly, for every chromosome in the population, fitness

function is evaluated by simulation on the proposed

power system model. The chromosome with the best

fitness function is memorized, to be again used in the

step 6.

3. Using roulette wheel selection [27], individual chromo-

somes are selected for creating the next generation.

4. Offsprings are created by the process of one-point

crossover between selected chromosomes. The proba-

bility of the crossover is pc.

5. Some random bits of offspring chromosomes are mu-

tated. The probability of the mutation is pm.

6. The next generation is composed of the obtained off-

springs and of the best chromosome from the current

generation (calculated in step 2).

7. Steps 2-7 are repeated until predefined number of gene-

ration has been produced.

Fitness function is an integral square of ACE:

J =

∫ ∞

0

(ACE(t))2 dt. (18)

334

Parameters of the used GA are: number of chromosomes

in the population, N = 100; number of genes in every

chromosome, ng = 5; number of bits in every gene, nb = 10;

probability of crossover, pc = 0.7; probability of mutation,

pm = 0.05; and the maximum number of produced genera-

tions NG = 50.

V. SIMULATION RESULTS

To test the proposed sliding mode algorithm design, simula-

tions of interconnected power system consisting of three CAs

were conducted. Parameters of the simulated system are shown

in Table I. Sampling time in the simulations was T = 1 s.

For the computations of control laws (13), (14) and (16),

system state x and even disturbance d are presumed to be

measurable. In real power systems only ∆fi and ∆Ptie are

measured. Therefore, the values of all other required signals

are estimated by an estimation technique described in [24].

During the simulation, four step disturbances occurred. Two

disturbances occurred in CA 1: ∆Pd11 = −5% p.u. MW, at

t = 1 s, and ∆Pd12 = 5% p.u. MW, at t = 450 s. In CA 2

there was disturbance ∆Pd2 = 1% p.u. MW, at t = 300 s,

while in CA 3 it was ∆Pd3 = −1% p.u. MW, at t = 150s. The absolute value of the disturbances in CA 2 and CA 3

were 5 times lower than of those in CA 1. Therefore, they

had little influence on controller parameters selection in CA 1

using GA, while their presence was needed to test the overall

power system behavior.

System behavior was tested for three different control laws

in CA 1:

A) Control law (13),

B) Control law defined with (14) and (15),

C) Control law defined with (16) and (17).

TABLE I

PARAMETERS OF THE INTERCONNECTED POWER SYSTEM MODEL

Area 1 Area 2 Area 3

KPi [s] 80 120 120

TPi [s] 13 20 25

Ri [Hz p.u.MW−1] 2.4 2.4 2.7

KBi [p.u.MWHz−1] 0.43 0.425 0.38

KSij [p.u.MW−1] 0.5 0.5 0.5

TR [s] 6 – –

T1 [s] 5 – –

T2 [s] 48.7 – –

TW [s] 1 – –

TGi [s] – 0.08 0.072

TTi [s] – 0.3 0.33

TABLE II

CONTROLLER PARAMETERS IN CA 1

Case S λ

A) [-0.8315 -2.2195 3.0052 -3.9318 1] 0.8281

B) [-4.3280 1.8166 2.5023 -3.5535 1] 0.9971

C) [0.0127 5.4615 -2.1205 1.1754 1] 0.1230

Controller parameters obtained with GAs for the proposed

cases are shown in Table II. The control algorithms used in

CA 2 and CA 3 were taken from [23].

Performance indices of the optimal solutions, calculated

using fitness function (18), are for all three cases: JA =5.42904·10−4, JB = 8.51558·10−4 and JC = 10.42995·10−4.

The best performance index is achieved in the Case A, while

the worst was achieved in the Case C.

ACE signals for the optimal controller parameters are shown

in Figs. 3 and 4. Fig. 3 shows ACE in CA 1 after the

disturbance in CA 1, while Fig. 4 shows ACE in CA 1 after

the disturbance in CA 3. The best transient performance after

the disturbance in CA 1 is achieved for the Case A, which is

significantly superior to common PI controller.

From the figures, it can be observed that, in steady state,

0 0.5 1 1.5 2 2.5

-2

0

2

4

6

8

10

12x 10

-3

time [min]

AC

E [

p.u

. M

W]

case A

case B

case CPI

Fig. 3. ACE in CA 1 after the disturbance ∆Pd11 at t = 1 s

5.5 6 6.5 7

-1

-0.5

0

0.5

1

1.5

2

2.5

x 10-3

time [min]

AC

E [

p.u

. M

W]

case A

case B

case C

Fig. 4. ACE in CA 1 after the disturbance ∆Pd3 at t = 300 s

335

ACE signals are not equal to zero. It is caused by the influence

of uncertainties to system behavior, so trajectory stays inside

a narrow band around the sliding surface. The width of quasi

sliding mode band depends upon the value of parameter λ;

lower λ causes narrower band. That can be seen from Figs. 3

and 4, where the band is the narrowest for the Case C, and

the widest for the Case B.

VI. CONCLUSION

In this paper, a method for designing sliding mode based

load-frequency controller for power system is presented. Para-

meters of the controller are chosen using genetic algorithm in a

way to minimize integral square of area’s control error signal.

The proposed controller is validated through simulations for

three different control laws. The choice of the reaching law

depends on whether disturbance is measurable (or can be esti-

mated) or not. Simulation results show good system behavior

for all three proposed reaching laws. The width of the quasi

sliding mode band also depends upon the chosen reaching

law. The purpose of this research was to obtain guidance for

the use of analytical methods in search of optimal controller

parameters.

ACKNOWLEDGMENTS

This research was supported by Koncar - Power Plant

and Electric Traction Engineering Inc., Zagreb, Croatia and

Ministry of Science, Education and Sports of the Republic of

Croatia.

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