a51 1 vrdoljak10 applying optimal sliding mode based load frequency control in power systems with...

8/8/2019 A51 1 Vrdoljak10 Applying Optimal Sliding Mode Based Load Frequency Control in Power Systems With Controllable

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Kreimir Vrdoljak, Nedjeljko Peric, Ivan Petrovic

Applying Optimal Sliding Mode Based Load-Frequency Control

in Power Systems with Controllable Hydro Power Plants

UDK

IFAC

621.311.07681.5375.5.4 Original scientific paper

In this paper an optimal load-frequency controller for a nonlinear power system is proposed. The mathema-tical model of the power system consists of one area with several power plants, a few concentrated loads and atransmission network, along with simplified models of the neighbouring areas. Firstly, a substitute linear modelis derived, with its parameters being identified from the responses of the nonlinear model. That model is used forload-frequency control (LFC) algorithm synthesis, which is based on discrete-time sliding mode control. Due toa non-minimum phase behaviour of hydro power plants, full-state feedback sliding mode controller must be used.

Therefore, an estimation method based on fast output sampling is proposed for estimating the unmeasured systemstates and disturbances. Finally, the controller parameters are optimized using a genetic algorithm. Simulationresults show that the proposed control algorithm with the proposed estimation technique can be used for LFC in anonlinear power system.

Key words: Fast output sampling, Genetic algorithm, Identification, Load-frequency control, Power systemmodel, Sliding mode control

Primjena optimalnog kliznog reima upravljanja u sekundarnoj regulaciji frekvencije i djelatne snage

razmjene regulacijskim hidroelektranama. U radu se predlae optimalna regulacija frekvencije i djelatne snagerazmjene za nelinearni elektroenergetski sustav. Unutar matematickog modela sustava jedno se regulacijsko po-drucje sastoji od nekoliko elektrana, manjeg broja koncentriranih troila i prijenosne mree. Ostala su regulacijskapodrucja u modelu modelirana pojednostavljeno, nadomjesnim linearnim modelom sustava ciji su parametri do-biveni identifikacijom iz odziva nelinearnog sustava. Taj je linearni model zatim primijenjen u sintezi algo-

ritma sekundarne regulacije koji je zasnovan na kliznom reimu upravljanja. Zbog neminimalno-faznog vladanjahidroelektrana primijenjena je struktura regulatora zasnovana na svim varijablama stanja sustava. Estimacija nem-jerljivih stanja i poremecaja zasnovana je na metodi brzog uzorkovanja izlaznih signala sustava. Optimizacijaparametara regulatora provedena je koritenjem genetickog algoritma. Simulacijski rezultati pokazuju kako je pred-loeni upravljacki algoritam, uz predloenu metodu estimacije, moguce koristiti za sekundarnu regulaciju frekven-cije i djelatne snage razmjene u nelinearnom elektroenergetskom sustavu.

Kljucne rijeci: brzo uzorkovanje izlaznog signala, geneticki algoritam, identifikacija, sekundarna regulacijafrekvencije i djelatne snage razmjene, model elektroenergetskog sustava, klizni reim upravljanja

1 INTRODUCTION

Power systems are complex, nonlinear systems that usu-ally consist of several interconnected subsystems or con-

trol areas (CAs). CAs are interconnected by tie-lines. Mostof European countries are members of the "European Net-work of Transmission System Operators for Electricity"(ENTSO-E) interconnection [1].

Deviation of systems frequency from its nominal valueis a measure of imbalance between generated and con-sumed power. When there is more/less generated thanconsumed active power in a system its frequency in-creases/decreases. For power systems safe operation it is

necessary to keep an equilibrium of consumed and genera-ted active power.

The consumption of electrical energy in a power sys-

tem constantly changes. Because the electrical energy can-not be stored in sufficient amounts, the task of the powersystem is to continuously track the amount of consumedpower by changing generation of its power plants. That isroughly carried out by day-ahead consumption predictionand most of the power plants in a CA follow their day-ahead hourly schedule. Since the prediction is not perfect,difference of the actual consumption of active power fromits predicted value must be compensated by load-frequency

ISSN 0005-1144ATKAFF 51(1), 318(2010)

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Applying Optimal Sliding Mode Based Load-Frequency Control in Power Systems K. Vrdoljak, N. Peric, I. Petrovic

control (LFC). LFC within a CA is in charge of keepingareas frequency at reference value and active power ex-changes with all neighbor areas at contracted values. Thatis obtained by changing reference powers of generators en-

gaged in LFC in a CA. For LFC studies it is usually as-sumed that each CA consists of a coherent group of gener-ators and that the frequency is unique within each CA. OneLFC controller is usually responsible only for its CA.

Before applying any new LFC algorithm to real powersystem, its behavior must be thoroughly tested through si-mulations. An example power system is usually modeledas an interconnection of a few CAs. Models for LFC algo-rithms testing purposes are generally linear and they haveall power plants engaged in LFC within a CA replaced withjust one substitute power plant [25]. That power plant canbe of thermal type [69] or hydro type [1013].

Since linearized models are valid only in the vicinity

of the operating point, they trustworthily represent realpower system dynamics only for close-to-nominal systemoperation. Changing of power systems operating point iscaused by current amount and characteristic of power con-sumption, characteristics of all power plants within a CAand current number of power plants engaged in LFC ina CA. Additionally, linear models have constant parame-ters, while parameters of real power systems constantlyvary in time, due to their dependency on the operatingcondition. However, these commonly used models mayhave nonlinearities within turbines and their governors,but generator dynamics and nonlinearities in tie-lines areusually neglected. Although simplified and linear powersystem models are mostly used for testing new LFC al-gorithms, they may not be sufficient to demonstrate algo-rithms adequacy and preparedness for proper functioningin real power system.

As opposed, in this paper a detailed nonlinear powersystem model is used for LFC algorithm testing. In themodel one CA is modeled in detail, while other CAs withinthe interconnection are modeled with several simplifica-tions. Besides, many different nonlinearities are present inthe model.

Nowadays, proportional-integral (PI) algorithms withconstant parameters are mostly used in real power sys-tems [1417]. The reasons to replace them with some

advanced control algorithm are the following: 1) systemswith PI control have long settling time and relatively largeovershoots in frequencys transient responses [18]; 2) PIcontrol algorithm provides required behavior only in thevicinity of the nominal operating point for which it is de-signed; 3) future power systems will rely on large amountsof distributed generation with large percentage of renew-able energy based sources and that will further increasesystem uncertainties [19]; 4) the shortening of time peri-

ods in which each level of frequency regulation must befinished is also expected in the future [20].

Therefore, throughout the recent decades many diffe-rent LFC algorithms were studied [21,22] and they showed

very good results on linearized power system models.Those algorithms are based on various methods, such asrobust control theory [14, 23], fuzzy logic [12, 24], neu-ral networks [25, 26], model predictive control [27, 28],optimal control [29, 30], adaptive control [31,32], slidingmode control (SMC) [33,34] and others. Despite their ad-vantages, new and innovative LFC algorithms have hardlymanaged to replace classical and proven PI algorithm withconstant parameters, which is still prevailing in real powersystems.

What is the reason for the prevalent usage of PI al-gorithm in real power systems despite the developmentof many advanced control techniques for LFC purpose?

Could it be that simplified and linearized models used inthe studies of advanced methods are not adequate, andgood results obtained on simulations do not replicate inreal power systems? In this paper well try to find out ifthats true by testing the behavior of an advanced LFC al-gorithm (whose behavior is already proven on linear powersystem model [35]) on a nonlinear power system model.

The nonlinear model used in this paper is created usingMATLABs SimPowerSystems Toolbox [36]. Althoughthe detailed nonlinear model is used here for testing the al-gorithm, its synthesis is based on the substitute linearizedmodel, as it is common in control applications. The sub-stitute linear model parameters are obtained by analyzingresponse signals recorded from the nonlinear model [37].

An advanced controller based on discrete-time SMCfrom [35] is tested in this paper. When compared to PIcontroller, this controller shows improvement of systemsbehavior regarding: better disturbance rejection, maintain-ing required control quality in the wider operating range,simultaneous shortening of frequencys transient responsesand avoidance of the overshoots and also robustness to un-certainties present in the system.

In SMC, system closed-loop behavior is determined bya sub-manifold in the state space, which is called a slidingsurface. The goal of the sliding mode control is to drivethe system trajectory to reach the sliding surface and thento stay on it. When the trajectory is on the surface, system

invariance to particular uncertainties and parameter vari-ations is guaranteed. The trajectorys convergence fromany point in state space towards the surface can be ensuredwith an appropriate choice of a reaching law [38]. Theideal sliding along the sliding surface can only be achievedwhen continuous-time SMC with very high switching fre-quency of the control signal is used. Since in real powersystem LFC control signal is sent to power plants in dis-crete time, a discrete-time sliding mode controller is used

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here. In discrete-time SMC system the trajectory couldonly be kept inside a small band around the sliding sur-face. Optimal parameters of the sliding surface and of thereaching law are obtained by an optimization procedure,

which is conducted using genetic algorithm (GA).If only thermal power plants are used for LFC in a CA

then stable sliding mode controller can be designed usingonly measured output signals. But in some power sys-tem there are either both, hydro and thermal power plants,or only hydro power plants that are engaged in LFC (e.g.Croatian power system [39]). Sliding mode based on fullstate feedback must be used in those systems because non-minimum phase behavior is present within a hydro powerplant. Therefore, from systems measured output and state,all unmeasured system state must be estimated. State esti-mation method used here is based on fast output sampling(FOS) [40]. That method is appropriate for estimations in

LFC due to the availability of multiple measurements ofoutput signals in each sampling period of the controller.FOS estimation method is also used for the estimation ofexternal disturbance, what additionally improves overallsystem behavior.

The brief outline of the paper is as follows. Section 2presents nonlinear power system model and Section 3 de-scribes an identification procedure which results in its sub-stitute linear model. Section 4 describes state and distur-bance estimation technique for the substitute model. Sec-tion 5 presents discrete-time SMC algorithm, while in Sec-tion 6 that algorithm is applied for LFC. Finally, Section7 contains simulation results obtained on nonlinear powersystem model.

2 NONLINEAR POWER SYSTEM MODEL

An interconnection of four CAs, as shown in Fig. 1,is used to test the proposed LFC controller. CA1 is mod-eled in high detail, with many nonlinearities present, whileall other CAs are partly simplified. The model is built us-ing SimPowerSystem, which is one of MATLABs tool-boxes. A nonlinear model of CA1 is shown in Fig. 2. CA1consists of seven power plants, which are hydro power

Control Area 3

(thermal)

Control Area 4

(thermal)

Control Area 1

(hydrothermal)

Control Area 2

(hydro)

Fig. 1. Four control areas interconnection

plants (HPPs) or thermal power plants (TPPs). Three hy-dro power plants (HPPs 1, 2 and 3) are engaged in LFC.All loads within the CA are modeled as three concentratedinstances (Loads 1, 2 and 3). Breakers are used to switch

parts of loads on or off, or to switch off thermal powerplant TPP 2. These switchings are done at different timeinstances. Transmission lines connecting plants and loads,as well as CAs of the interconnection, are modeled as sections of various lengths. Buses are used to measure ac-tive powers on important power lines. They measure pro-duction of each HPP engaged in LFC and total interchangewith neighbor CAs. In each of neighbor CAs total powergeneration is represented with a single power plant. Thepower plants in CA3 and CA4 are thermal power plants,while the power plant in CA2 is of hydro type. All valuesin the model are expressed in per units of their base values,which is common in LFC studies. Although, wherever it isneeded, those values are converted into real SI values.

There are several inputs to LFC block in Fig. 2: 1)reference value of systems frequency (50 Hz or 1 p.u.),2) measured value of systems frequency from TPP 1, 3)reference value of total tie line active power, 4) measuredvalue of total tie line active power, and 5) total generatedpower from all HPPs engaged in LFC. Based on those va-lues, LFC block computes a change in total generation ofall power plants engaged in LFC. That generation is thendivided among the power plants engaged in LFC and sentto them as control signals u1, u2 and u3.

SMC based LFC algorithm has already been designedfor linear power system with hydro power plants engagedin LFC [35]. In order to apply that algorithm to the pro-

posed nonlinear model, a linearization of the nonlinearmodel must be done firstly.

3 SUBSTITUTE LINEAR POWER SYSTEM

MODEL

A linear model of a CA represented with single hydropower plant is shown in Fig. 3, while signals and parame-ters of that model are given in Tab. 1. The linear modelfrom Fig. 3 can be described with the following equations:

xi(t) = Aixi(t) +NCAj=1j=i

Aijxj(t) + Biui(t) + Fidi(t),

yi(t) = Cixi(t), (1)where xi 4 n is the systems state vector, xj 4 p is astate vector of a neighbor system, ui 4 m is the controlsignal vector, di 4 k is the disturbance vector and y 4

l is the output vector. Matrices in (1) have appropriatedimensions: Ai 4 nn, Aij 4 np, Bi 4 nm,Fi 4

nk and Ci 4 ln. NCA is a number of CAs inthe interconnection.

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==> Tie line

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _||

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C

3 B

2 A

1

Planned_Gen_2

Planned_Ptie

Planned_Gen_6

Planned_Gen_3

Planned_Gen_7

Planned_Gen_4

Planned_Gen_5

Planned_Gen_1

1

TPP 2

Pref

A

B

C

TPP 1

Pref

W11

A

B

C

Load 3B

ABC

Load 3A

ABC

Load 2B

ABC

Load 2A

ABC

Load 1B

ABC

Load 1A

ABC

Line 9 (33 km)

A

B

C

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C

Line 8 (43 km)

A

B

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A

B

C

Line 7 (15 km)

A

B

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C

Line 6 (20 km)

A

B

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B

C

Line 5 (70 km)

A

B

C

A

B

C

Line 4 (55 km)

A

B

C

A

B

C

Line 3 (5 km)

A

B

C

A

B

C

Line 2 (40 km)

A

B

C

A

B

C

Line 10 (17 km)

A

B

C

A

B

C

Line 1 (35 km)

A

B

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C

LFC

F_meas

Ptie_meas

F_ref

Ptie_ref

P_lfc

u1

u2

u3

HPP 5

Pref

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B

C

HPP 4

Pref

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HPP 3

Pref

A

B

C

HPP 2

Pref

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HPP 1

Pref

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C

Bus 4

Bus 3

Bus 2

Bus 1

Breaker 5

A BC

a b c Breaker 4

A

B

C

a

b

c

Breaker 3

A

B

C

a

b

c

Breaker 2

A

B

C

a

b

c

Breaker 1

A

B

C

a

b

c

Fig. 2. A nonlinear model of CA1

1

1 1isT+

1

1

Ri

2i

sT

sT

+

+ 1

Pi

Pi

K

sT+

2

s

1

iR

-

-+iu

diP

if

i

+

- gix giP

1

2BiK

++ t ie iP ( )N sij i j

j 1

K =

iACE

1

1 0.5

Wi

Wi

sT

sT

+

gh ix

Fig. 3. A linear model of i-th CA represented with hydro power plant

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Table 1. Hydro power system variables and parameters

Parameter/variable Description Unit

f(t) Frequency HzPg(t) Generator output power p.u.MW

xg(t) Governor valve position p.u.xgh(t) Governor valve servomotor position p.u.Ptie(t) Tie-line active power p.u.MWPd(t) Load disturbance p.u.MW(t) Rotor angle radKP Power system gain Hz/ p.u.MWTP Power system time constant sTW Water starting time s

T1, T2, TR Hydro governor time constants sKS Interconnection gain between CAs p.u.MWKB Frequency bias factor p.u.MW/ Hz

R Speed droop due to governor action Hz / p.u.MWACE Area control error p.u.MW

For the model shown in Fig. 3 state and disturbancevectors from (1) are:

xi(t) =

fi(t)Ptiei(t)Pgi(t)xgi(t)

xghi(t)

,

di(t) = Pdi(t).

(2)

System matrix from (1) is:

Ai =

1

TPi

KPi

TPi

KPi

TPi 0 0NCAj=1

KSij 0 0 0 0

2 0 2TWi

2 2

0 0 1T2i

1T1iRi

0 0 0 1T1i

,

(3)and the coefficients in (3) are:

=TRi

T1iT2iRi, =

TRi T1i

T1iT2i, =

T2i + TWiT2iTWi

. (4)

Input and disturbance matrices from (1) are:

Bi =

00

2Ri

Ri1

T1i

, Fi =

KPiTPi0000

. (5)

In linearized model of a CA represented with thermalpower plant, there are only four states, since there is no

state xgh, (which is present in hydro CA state vector(2)), due to simpler governor used in thermal power plants.Therefore, matrices Aij in (1) have dimensions 5 4 or5 5, depending on whether they are used to describehydro-thermal (e.g. CA1-CA3 in the interconnection fromFig. 1) or hydro-hydro (e.g. CA1-CA2 in the intercon-nection from Fig. 1) connection. All elements of matricesAij are equal to zero, except the element at position (1, 2),which is equal to KSij .

The output of the model from Fig. 3 is area control error(ACE) signal. It is introduced as a measure of imbalancebetween total generated and consumed active power (in-

cluding power import/export) in each CA. The goal of LFCin each CA is to compensate for ACE deviations. ACE isdefined as a combination of frequency deviation in a CAand of total active power flow deviation in tie-lines con-necting the CA with its neighbor areas:

ACE(t) = KBf(t) + Ptie(t). (6)

If system output is chosen to be yi(t) = ACEi(t), thenthe system output matrix in (1):

Ci =

KBi 1 0 0 0

. (7)

Ideally, parameter KB in (6) should ensure that thevalue ofACE in a CA is equal to zero when disturbancesin other CAs appear, and as close to the negative value ofdisturbance just after it appears within the CA, before LFCstarts to take effect.

Parameters of the substitute linear model can be iden-tified from the responses of the nonlinear model. Sinceidentification procedure presented in [41] can only be usedfor CAs that have only one power plant engaged in LFC, it

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can not be used here because LFC in CA1 from Fig. 2 isconducted using three HPPs. Therefore, different identifi-cation method must be used.

3.1 Identification of substitute linear model parame-ters

In this paper, a procedure based on GA is used for para-meters identification. GA is a random search optimizationalgorithm which is appropriate for finding global optimalsolution inside a multidimensional searching space [42].

GA is used here to find an optimal set of linear modelparameters for CA1 (Fig. 3), within the interconnectionfrom Fig. 1. That parameter set is defined as:

=

T1 T2 TR TW R KP TP KS12 KS13

,

(8)where KS12 and KS13 are interconnection gains betweenCA1 and its neighbor areas, CA2 and CA3, respectively.

The optimal set is the one that minimizes deviation be-tween responses of the linear model and the measured sig-nals from the nonlinear model, when identical control anddisturbance signals are applied in both models. For theidentification procedure it is essential to firstly bring thesystem into steady state operating point by using LFC.Then any further LFC action is disabled and small step sig-nals are applied as control and disturbance signals, becauselinear models are valid only in the vicinity of the operatingpoint.

For identification purposes, the deviation is evaluatedby the following fitness function:

i()=Ts0

(fL(t)fN(t))

2 + (PgL(t)PgN(t))2+

+(PtieL(t)PtieN(t))2

dt,(9)

where Ts is identification time horizon, is given in (8), and are weighting factors, while indices L and N denotelinear and nonlinear model, respectively.

A detailed flow chart of GA used for parameters iden-tification is shown in Fig. 4. From random initial popula-tion GA computes candidate sets of parameters and com-putes the fitness function of each set based on simulationresults. Then it starts a loop of evolution processes, con-sisting of selection, crossover and mutation, in order to im-prove the average fitness function of the whole population.Besides the evolved chromosomes, each new generationalso includes the best chromosome from the previous gen-eration (elitism) and one random new chromosome. Theoptimization ends when maximal number of generations isproduced. Parameters of the used GA are shown in Tab. 2.

Identification results are shown in Fig. 5. The optimalparameters of the substitute linearized model of CA1 are

Initial population

Genotype Phenotype

Fitness function computation

Elitism

Selection

Crossover (pc)

Mutation (pm)

The next generation

StopnoMaximum

generations?

yes

Random newchromosome

Simulation

Fig. 4. Flow chart of GA algorithm

Table 2. Parameters of GA used for identification

Parameter Description Value

r Size of population 150g Genes per chromosome 9b Bits per gene 15

pc Probability of crossover 0.7pm Probability of mutation 0.005Ng Number of generations 150

Pg weighting factor 0.1 Ptie weighting factor 0.05

obtained as: T1 = 314.97 s, T2 = 0.92 s, TR = 17.94s, TW = 0.15 s, R = 4.39 Hz/p.u.MW, KP = 5.73Hz/p.u.MW, TP = 2.65 s, KS12 = 4.42 p.u.MW andKS13 = 0.12 p.u.MW.

From Fig. 5 it can be seen that frequencies in both mod-els are practically identical, while there are significant dif-ferences in generated power and tie line power deviationsignals. The first reason for that is because three power

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0 10 20 30 40 50

-10

0

10

time [min]

f[Hz]

0 10 20 30 40 50

0

5

10

time [min]

Pg

[p.u.MW]

nonlinear

linear

nonlinear

linear

x 10-3

x 10-3

0 10 20 30 40 50-10

-5

0

time [min]

Ptie

[p.u.M

W]

nonlinear

linear

x 10-3

Fig. 5. Comparison between nonlinear and linear model

plants engaged in LFC in the nonlinear model are substi-tuted with just one power plant in the linear model. Thesecond reason is oversimplification of tie line dynamics de-

scription in the linear model. Nevertheless, the results aresatisfactorily, since robust algorithm will be used for LFC.

Once the identification of the substitute linear model isdone, areas state dynamics can be described as:

xi(t) = Aixi(t) +NCAj=1j=i

Aijxj(t) + Biui(t)+

+Fidi(t) + i(x, u, t),

(10)

where i is a vector of uncertainties. Among other uncer-tainties, the uncertainty term i(x, u, t) in (10) includes alllinearization errors.

In this paper, the value of parameter KB from (6) isnot found using GA, but it is chosen such that the minimaldeviation of steady state ACE signal from its ideal behav-ior is ensured, as it is shown in Fig. 6. Those results areobtained with LFC turned off. It is impossible to achieveideal behavior of ACE signal, since that would require KBto constantly change its value. In fact, in ENTSO-E in-terconnection KB is set yearly for each control area andits value is based on portion of areas yearly generation in

0 10 20 30 40 50

-10

-5

0

time [min]

ACE1

[p.u.MW]

nonlinear model

ideal

0 10 20 30 40 45 50

-10

0

10

time [min]

Pd[p.u.MW]

CA1CA2

CA3

CA4

x 10-3

x 10-3

Fig. 6. Ideal and true ACE signals

total yearly generation of the whole interconnection [43].Nevertheless, with LFC turned on, the difference of KBstrue value from its ideal value is noticeable only during thetransient period, since steady state ACE value is indepen-dent of the value ofKB.

Additional advantage of the proposed identification pro-cedure is that it could also be used for identifying parame-ters of substitute linear model representing real power sys-tem, since it uses only signals that are commonly measuredin real power systems.

Once the substitute linear model is obtained and its pa-rameters found, it can be used for LFC algorithm synthe-sis. Since most of advanced control algorithms need theknowledge of full system state, to apply one of them forLFC, unmeasured system state of the linear model must beestimated.

4 SYSTEM STATE AND DISTURBANCE ESTIMA-

TION

In real power system the measured signals are: fre-quency, tie line powers and each plants generated power,so those signals are considered measurable in both linearand nonlinear model used in this paper. Therefore, signalsof governor valve position deviation (xg) and governorvalve servomotor position deviation (xgh) in the substi-tute linear model of hydro power plant (Fig. 3) must be es-timated. Further improvement in system performance canbe achieved if load disturbance (Pd) and the influence ofneighbor CAs (

Aijxj) are also estimated.

To simplify state and disturbance estimation proce-dures, uncertainties are hereby neglected, and their influ-ence to system behavior is addressed in Section 5.

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4.1 State estimation

To estimate unmeasured state of the substitute linearmodel of CA1, a subsystem shown in Fig. 7 is used.

1

1

1 sT+ 2

1

1

RsT

sT

+

+

1

iR

u

f

+

- ghxgP

1

1 0.5

W

W

sT

sT

+

gx

Estimator

g

gh

x

x

T

T

Fig. 7. A dynamical subsystem for state estimation

As it can be seen there are two input signals to the sub-system from Fig. 7: frequency deviation f (sampledwith period T), and control signal u (sampled with pe-riod ). The sampling periods of input signals are diffe-rent because in ENTSO-E interconnection control signalu is sent to the power plants every 1-5 seconds [16, 43],while within each period of that signal, there can be mul-tiple measurements of systems frequency f. The subsys-tems output signal is total generated output power of allpower plants engaged in LFC (it is assumed here that allof them are of hydro type). The output signal can also bemeasured with the same sampling period as the frequencysignal. Since period T is few times smaller than period ,

estimation procedure based on FOS can be used for stateestimation [44].FOS is an estimation technique appropriate for

continuous-time systems with discrete-time input signalsand where the output signals can be sampled several timesduring one period of the input signals [40]. Therefore, FOScould be used here for state estimation, but slight modifi-cation are needed, due to different sampling periods of twoinput signals in the subsystem from Fig. 7.

FOS shows better performance than standard estimationtechniques, because it reduces the estimation error to zeroafter just one sampling period [44], while standard esti-mators need several sampling periods to achieve errorlessestimation in high order systems [40].

The principle of using FOS in systems control loop isshown in Fig. 8. Firstly, last N subsamples of the outputsignal y, measured in the most recent sampling period ,are used to estimate the system state x. Then, that esti-mated state is used to compute the control signal u for thenext sampling period .

For simplicity it is assumed here that it is:

T =

N, (11)

y

t

u

x

t

( 1)k k ( 1)k + t

T

Controller

Fig. 8. FOS estimation method in system control

where is N N .

To apply FOS, a discrete-time system model must beused. Therefore, the discrete-time approximation of sub-system from Fig. 7, obtained using Zero-Order-Hold(ZOH) discretization method with sampling period T isgiven as:

xH((k + 1)T) = GxH(kT) + H(f(kT) + u(k)),

Pg(kT) = CxH(kT),

(12)where is xH =

Pg xg xgh

T.

Values of state variable of subsystem from Fig. 7 in Nconsecutive subsamples, during constant value of controlsignal u, can be obtained by iteratively using state equationfrom (12):

xH(k) = xH(k)

xH(k + T) = GxH(k) + H(f(k)+u(k)),

xH(k+2T) = G2xH(k)+(GH + H)u(k)+

+GHf(k) + Hf(k + T),

...

xH(k+(N1)T) = GN1xH(k)+

N2i=0

(GiH)u(k)+

+N2i=0

GiHf(k + (N2i)T).

(13)

From (12) and (13) it follows that N consecutive sub-samples of output signal Pg , measured during constant

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value of control signal u, can be computed as:

Pgk

= GxH(k) + Hu(k) + Qfk , (14)

where are:

Pgk

=

Pg(k)Pg(k + T)

Pg(k + 2T)...

Pg(k + (N 2)T)Pg(k + (N 1)T)

, G =

C

CG

CG2

...CGN2

CGN1

,

H =

0

CH

C(GH + H)...

C

N3i=0 GiHCN2

i=0 GiH

, fk =

f(k)f(k + T)

f(k + 2T)...

f(k + (N 3)T)f(k + (N 2)T)

,

Q =

0 0 0 0

CH 0 0 0

CGH CH 0 0...

.... . .

......

CGN3H CGN4H CH 0

CGN2H CGN3H CGH CH

.

(15)

Since matrix G might not be invertible, instead ofits regular inverse, Moore-Penrose matrix pseudoinverse,

G+

, is used to compute the estimated state. The pseudoin-verse is calculated as [45]:

G+ = (GTG)1GT. (16)

Although, if only equation (14) is used for state esti-mation, estimated values would be one sampling perioddelayed from the true values. Therefore, equation (14) iscombined with the discrete-time approximation of systemdynamics to obtain the estimated state as:

xH((k + 1)) = GG+(P

gkQfk ) + Hu(k) + H

fk ,

(17)where is:

H =

GN1H GN2H GH H

, (18)

while matrices G and H are obtained by ZOH dis-cretization of the system from Fig. 7 (just like system (12),but with sampling period instead ofT).

The results of the proposed state estimation method areshown in Fig. 9. Those results are obtained on the substi-tute linear model.

0 0.1 0.2 0.3 0.4 0.5

0

5

10

time [min]

xg

[p.u.]

0 0.1 0.2 0.3 0.4 0.5

0

5

10

15

20

time [min]

xgh

[p.u.]

true

estimated

true

estimated

x 10-3

x 10-3

Fig. 9. Estimation of system state in CA1

From the figure it can be seen that FOS estimation givesvery good results at sampling instants. Additionally, it canbe seen that FOS estimation is independent of the choseninitial conditions, since the estimation error is very small,even in the first few seconds of the simulation.

4.2 Disturbance estimation

Total disturbance signal in CA1 consists of load distur-bance, Pd1, and the influence of neighbor CAs, e1, whichis:

e1(t) =

3j=2

(Ks1jfj(t)) . (19)

Total disturbance could also be estimated using FOS,but due to unknown disturbance dynamics the estimationwould be seconds delayed [35]. Therefore, simpler andmore accurate methods for disturbances estimation are pre-sented here.

4.2.1 Load disturbance estimation

To estimate load disturbance two facts observed on thelinear system model are utilized: 1) immediately after a

step load disturbance occurs within a CA, the change inareas ACE signal is equal to the negative value of that dis-turbance; 2) after that load disturbance is compensated byLFCand ACE returned to zero, total change in power gene-rated by power plants engaged in LFC is equal to the valueof the disturbance. Therefore, the following equation canbe used for load disturbance estimation:

Pd(k) = Pg(k)ACE(k). (20)

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4.2.2 Estimation of the influence of neighbor CAs

A subsystem for estimating the influence of neighborCAs is shown in Fig. 10.

+

+

1f

1tieP

1eEstimator

T

T

N

S1 j

j 2

K=

2f

kf

.

.

.

.

.

.

Nf

KS12

KS1k

KS1N

-- -

1e

1

s

Fig. 10. A subsystem for estimating the influence of neigh-bor CAs to CA1

Since the influence of neighbor CAs to CA1 is simpli-fied in the substitute model, it can be estimated by usingdiscrete-time derivator:

e1(k) =Ptie1(k)Ptie1((k1))

Nj=2

Ks1jf1(k).(21)

Measurements of all signals needed to compute esti-mated values using (20) and (21) are available in the con-troller. Naturally, for model with uncertainties and forLFC application in real power system, those measurementsmust firstly be filtered through low pass filters to dampnoise component and rapid changes in the signals.

The results of the proposed disturbance estimationmethods are shown in Fig. 11. Those results are also ob-tained on the substitute linear model, with no uncertaintiespresent in the model.

From the figure it can be seen that the disturbance es-timations are slightly poorer than the state estimation, butstill satisfactorily.

Altogether, estimation results from Figs. 9 and 11 showthat the proposed methods can be used to estimate unmea-sured state and disturbances in power system.

From the measurements and by using the proposed esti-mation equations (17), (20) and (21), full system state anddisturbance of the substitute linear power system model areobtained. Therefore, a state based controller can now bedesigned for LFC. Since estimations are not perfect, theyintroduce estimation errors which can be observed as addi-tional components of uncertainties in (10). Therefore, ahighly robust control algorithm is needed for LFC. That isthe main reason why SMC is used for LFC in this paper.

0 0.2 0.4 0.6 0.8

0

5

10

time [min]

Pd

[p.u.

MW]

0 0.2 0.4 0.6 0.8

0

10

20

time [min]

e1

[p.u.

MW]

true

estimated

true

estimated

x 10-3

x 10-3

Fig. 11. Estimation of disturbances in CA1

5 DISCRETE-TIME SLIDING MODE CONTROL

SMC is a control technique appropriate for controllingtime-variant systems in the presence of system uncertain-ties and external disturbances. If SMC based only on out-put signal is used for systems with non-minimum phasebehavior, it will cause system instability [38]. Since hydropower plant is a system with non-minimum phase behav-ior, as it can be seen from Fig. 3, SMC based on full systemstate must be used for LFC in CA1.

To improve the overall system behavior, disturbance in-formation is also included into controllers synthesis. Be-cause all state and disturbances are not measurable, estima-tion techniques described in the previous section is used toobtain the values of their unmeasured components.

LFC based on SMC will be synthesized for discrete-time representation of system (10), where both distur-bances (load disturbance and the influence of neighborCAs) are gathered into one disturbance signal, dt, whichresults in the following system state equation:

x((k + 1)) = Gcax(k) + Hcau(k)+

+Wcadt(k) + ca(x, u, k).(22)

For better insight into dynamics of system in sliding

mode and for simpler controller synthesis it is more con-venient to transform system (22) into a regular canonicalform using transformation matrix T [38]:

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

=

G11 G12G21 G22

x1(k)x2(k)

+

0

H2

u(k)+

+

W1W2

dt(k) +

u(x, k)

m(x, u, k)

,

(23)

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where x1 and x2 represent state vectors of the decoupledsubsystems of system (22), and it is

x1 x2

T= Tx. In

the regular form, systems uncertainties are decoupled intomatched (m) and unmatched (u) ones.

A sliding surface specifies system behavior in the slid-ing mode and it is defined as:

(x) Sx = 0, (24)

where S 4 mn is a switching matrix.

System (23) can be further transformed into [35]:

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

=

G11 G12

G21 G22

x1(k)(k)

+

0

H2

u(k)+

+

W1

W2

d(k) +

u(x, k)s(x, u, k)

,

(25)

where are:S1 S2

= ST1,

G11 = G11 G12S12 S1,

G12 = G12S12 ,

G21 = S1(G11G12S12 S1) + S2(G21G22S

12 S1),

G22 = S1G12S12 + S2G22S

12 ,

H2 = S2H2,

W2 = S1W1 + S2W2,s(x, u, k) = S1u(x, k) + S2m(x, u, k).

(26)As it can be seen from (25), when the system is in the slid-ing mode ((k) = 0), matched uncertainties do not in-

fluence the systems dynamics. Therefore, SMC is robustto matched uncertainties, although only at the sampling in-stants, since between them the system is not in the slidingmode.

The control law that ensures sliding mode will be com-puted according to (25), by an appropriate choice of areaching law. The reaching law is in charge of drivingsystems trajectory to the sliding surface. Among sev-eral known reaching laws [4650], a linear reaching lawfrom [50] is chosen to be used for LFC, because it exactlydefines the trajectory towards the sliding surface and its us-age results in straightforward controller design [38]. It isdefined as:

((k + 1)) = (k), (27)

where is a diagonal matrix whose elements are con-strained to 0 i < 1.

If reaching law (27) is applied to system (25) and uncer-tainties neglected, the following control law is obtained:

u(k) = H12

( G22)(k)

G21x1(k) W2d(k)

.(28)

The control law (28) will ensure that system trajectory re-sides in a narrow band around the sliding surface. Thewidth of that band primarily depends on the maximal valueof uncertainties and upon the value of matrix from (27).

To reduce its width, uncertainties are also estimated.

5.1 Uncertainties estimation

Generally, uncertainties in systems are mostly com-posed of noise, unmodeled system dynamics and time vari-ations of system parameters. Additionally, the proposedcontroller synthesis introduces a few additional compo-nents of system uncertainties which are caused by iden-tification, linearization and estimation errors.

Uncertainties estimation is obtained from the differenceof true and the ideal system behavior in sliding mode.Therefore, the estimated uncertainties are computed as[38]:

s(k) = s((k 1)) + (k)((k 1)). (29)

If estimated uncertainties from (29) are included intocontrol law (28), the following control law is obtained:

u(k) = H12

( G22)(k) G21x1(k)

W2d(k) s(k)

.

(30)

With control law (30), the width of the band aroundthe sliding surface becomes dependent on the maximalrate of change of uncertainties, instead on their maximal

value [38]. For uncertainties present in power system, thatwould result in reduction of width of the band around thesliding surface.

In this section, discrete-time SMC was derived for ageneral minimum-phase system. In the next section it willbe applied to LFC system.

6 LOAD-FREQUENCY CONTROL BASED ON

DISCRETE-TIME SLIDING MODE CONTROL

If discrete-time SMC is applied in LFC controller, therewill be only one sliding surface, since the control signal inLFC has dimension one (m = 1, u u). Furthermore,switching matrix becomes a switching vector and reachinglaw matrix becomes a scalar ( ).

According to state vector from (2), a switching vectorfor CA1 is:

S =

sf sPtie sPg sxg sxgh

. (31)

Additionally, according to (25), systems stability insliding mode is ensured if the parameters of the switching

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vector are chosen such that matrix G11 has all its eigenval-ues within the unit circle.

Discrete-time SMC ensures that system trajectory onlyresides in a narrow band around the sliding surface, but

unfortunately that doesnt imply zero value of ACE insteady state [35], which is inherent to PI controller. There-fore, in order to ensure system stability and to minimizezero steady-state error of ACE, an appropriate relationbetween the parameters of the switching vector must befound. In [35] it is shown that with uncertainties neglected,steady state ACE will be equal to zero if the followingconditions are satisfied:

sPg + sxg + sxgh = 0,sPtie = 0.

(32)

Furthermore, since system behavior in SMC doesntchange if the switching vector is multiplied by any non-

zero scalar, it can be set sPg = 1.Altogether, SMC controller synthesis reduces down to

finding optimal values of parameters , sf, sPtie and sxg.Again, an appropriate GA is used as search algorithm forthat purpose.

6.1 Optimal sliding mode control algorithm parame-

ters

In controller synthesis, GA is used to find optimal va-lues of the following set of controllers independent para-meters:

=

sf sPtie sxg

. (33)

GA used for finding optimal controller parameters issimilar to GA used for substitute linear model identifica-tion in Section 3. However its flow chart is slightly dif-ferent from the flow chart shown in Fig. 4. The first dif-ference is that the flow chart for controller synthesis hasadditional block for computing the control law (30), whichis placed just above block Simulation in Fig. 4 [37].

The other difference is that the fitness function for con-troller synthesis purposes is given as:

c() =

Ts0

(ACE(t))2 z(t)dt + (u(t))

2dt, (34)

where is given in (33), is a weighting factor and z

is a weighting ramp function which reduces the influenceof variations in ACE in the first moments after the distur-bance.

Parameters of the GA used for controller synthesis areshown in Tab. 3.

By using the proposed GA, the optimal controller pa-rameters in CA1 are obtained as: = 0.990 and S =74 0.057 1 98.096 97.096

.

Table 3. Parameters of GA used for controller synthesis

Parameter Description Value

r Size of population 100g Genes per chromosome 7

b Bits per gene 15pc Probability of crossover 0.7

pm Probability of mutation 0.05Ng Number of generations 200 Input signal weighting factor 0.01

Altogether, the complete scheme of the proposed LFCalgorithm is shown in Fig. 12. The procedure starts fromnonlinear power system model and uses GA and recordedmeasured signals from that model to identify parametersof the substitute linear system model. If the proposed al-gorithm is to be installed in real power system, the used

signals from the nonlinear model should be replaced withtrue measured signals.

Continuous-time SMC is applied to continuous-timesystem model resulting in steady state accuracy condi-tion, given by (32). Discrete-time SMC is then appliedto discrete-time system model to obtain control law (30).By simulations on continuous-time model and by using theaccuracy condition and the proposed control law, GA isused to find the optimal controller parameters. All above-mentioned steps of the algorithm are done off-line, andthey are based on measurement data, recorded through cer-tain time period.

The only on-line computations are performed within

two blocks inside Load frequency controller in Fig. 12.

Genetic algorithm

Real power

system or its

nonlinear model

State and

disturbance

estimator

Ptie

(k)

f(k)

Pg(k)

x (k)

d (k)

Identification

f(k)

Ptie(k)

Pg(k)

Continuous time

power system

model

Discrete time

power system

model

Discrete time

sliding mode

control

Control law

Continuous time

sliding mode

control

Steady state

accuracy

conditions

Optimal sliding

model control

Load frequency controller

u (k)

Genetic algorithm

Optimal controller

parameters

Fig. 12. An algorithm scheme of LFC based on discrete-

time SMC

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Firstly, measured signals and discrete-time system modelare used to estimate unmeasured power system state anddisturbances. Secondly, sliding mode controller (with pre-viously off-line calculated optimal parameters) is used

to compute the control signal, which is then distributedamong power plants engaged in LFC [51].

7 SIMULATION RESULTS ON NONLINEAR

POWER SYSTEM MODEL

In the simulations, the proposed algorithm is used forLFC in CA1 from Fig. 2, while in all other CAs commonPI controllers are used. PI controllers are also derived indiscrete time with parameters obtained using GA in orderto minimize fitness function (34).

In the simulations, the sample times are set to = 1 sand T = 0.2 s.

Several disturbances occur during the simulation: 1)

multiple switching on/off of portions of load in CA1 (Load1A, Load 2A, Load 3A and Load 3B in Fig. 2), 2) turningoff of a power plant in CA1 which is not engaged in LFC(TPP 2 in Fig. 2), 3) single switching on/off of portions ofloads in neighbor CAs, 4) changing generation schedule ofa power plant in CA1 not engaged in LFC and 5) changingexchange schedule between areas CA1 and CA2.

All the disturbances in CA1 are shown in Fig. 13. Thefirst plot of Fig. 13 shows total load (PL) in CA1, mea-sured during the simulation. The second plot shows gen-eration schedules (Pref) of power plants in CA1 that arenot engaged in LFC. It can be seen that power plant TPP2 shuts down after 20 minutes and power plant TPP 1

changes its generation schedule. The third plot shows ex-change schedule between CA1 and CA2 (P12). Generationand exchange schedules are carried out using ramps, whichis common in real power systems [43].

Unlike small disturbances used for controller validationon linear power system models [3,11,14,26,30], some dis-turbances from Fig. 13 are significantly large, especiallythose regarding turning off substantial load component att = 4 min (seen on the first plot of Fig. 13) or turning offthe entire power plant at t = 20 min (seen on the secondplot of Fig. 13).

ACE signal in CA1 is shown in Fig. 14 and the switch-ing function in CA1 is shown in Fig. 15. It can be seen

from Fig. 14 that the proposed controller compensates forevery disturbance in the system by returning ACE signal inthe vicinity of zero after each disturbance. The dynamicsof the compensations are similar for different disturbances,since they are determined by the chosen controller parame-ters. Besides, only large disturbances cause considerablesystems deviation from sliding mode, but the return to thevicinity of the surface is very fast, as it can be seen fromFig. 15.

0 10 20 30 40 50900

1000

1100

1200

1300

time [min]

PL[MW]

0 10 20 30 40 50

0

100

200

300

time [min]

Pref

[MW] TPP1

HPP4

HPP5

TPP2

0 10 20 30 40 50-260

-240

-220

-200

time [min]

P12[MW]

Fig. 13. Disturbances in CA1: PL - load, Pref - planned

generation of power plants not engaged in LFC, and P12 -

planned exchange with CA2

0 10 20 30 40 50-0.1

-0.05

0

0.05

0.1

time [min]

ACE

[p.u.

MW]

Fig. 14. ACE signal in CA1

0 10 20 30 40 50-3

-2

-1

0

1

2

3

time [min]

Fig. 15. Switching function in CA1

In Fig. 16 the total control signal of the proposed con-troller and output active powers of power plants engagedin LFC are shown. Total control signal is divided among

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0 10 20 30 50

-500

0

500

time [min]

u[MW]

0 10 20 30 40 500

100

200

300

time [min]

Pg[

MW] 40 45

HPP1HPP2HPP3

40

Fig. 16. Total LFC control signal and output active powers

of power plants engaged in LFC in CA1

the plants engaged in LFC, depending on their participa-tion factors. The participation factors are set to 1 = 0.5,2 = 0.3 and 3 = 0.2. It can be seen from the figure thatHPP 1 takes over the largest share in change of generation,since it has the largest participation factor. Static participa-tion factors are used here for simplicity and their dynamiccalculation and optimization is addressed in [51].

A comparison between PI controller and the proposedcontroller is not addressed here, since in [35] and [37] ithas already been shown that the proposed controller out-performs classical PI controller.

8 CONCLUSIONS

In this paper a discrete-time sliding mode based load-frequency controller is designed based on nonlinear powersystem model. The controller is designed for power sys-tems with only hydro power plants engaged in LFC, sincethose types of models are not sufficiently treated in avail-able LFC studies. Since sliding mode controller for non-minimum phase system needs full-state feedback, all un-measured system states and disturbances are estimated us-ing FOS. The standard FOS estimation method has beenslightly modified so it could also be used for disturbanceestimation. GAs are used for finding optimal parameters

of the substitute linear model and SMC controller.The proposed controller is validated through simulati-ons on power system model consisting of four differentcontrol areas. Simulation results show very good compen-sation of various disturbances in the system. Because goodbehavior of the proposed controller is hereby confirmed ondetailed nonlinear power system model, there are no ob-stacles to keep further development of the algorithm andeventually implement it in real power system.

ACKNOWLEDGEMENTS

This work has been financially supported by The Na-tional Foundation for Science, Higher Education and Tech-

nological Development of the Republic of Croatia andKoncar - Power Plant and Electric Traction Engineering,Zagreb, Croatia.

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dustrial Electronics (ISIE), 2010.

Kreimir Vrdoljak received his B.Sc. and Ph.D.degrees in Electrical Engineering from the Uni-

versity of Zagreb, Croatia in 2003 and 2009, re-spectively. From 2003 to 2009 he was employedat the Department of Control and Computer En-gineering at the Faculty of Electrical Engineer-ing and Computing in Zagreb, Croatia. He iscurrently employed at Koncar - Power Plant andTraction Engineering Inc., Zagreb, Croatia. Hisresearch interests are in advanced control tech-niques applied to power systems.

Nedjeljko Peric (1950.) has been professionallyworking for the last thirty five years as a scientistand researcher in the area of automatic controland automation of complex processes and sys-tems. His scientific and professional work can begrouped into two distinct phases. During the firstphase he was working at the Koncars Institute ofElectrical Engineering (1973-1993) with empha-sis on development of the automation systems forcomplex processes. In particular, he initiated andled the corporate research and development pro-

grams for the microprocessor control of electrical machines and relatedfast processes. The second phase of his work is connected to the employ-ment at the Faculty of Electrical Engineering and Computing in Zagreb(from 1993 onwards), where he has initiated a broad research activity inadvanced control of complex and large scale technical systems. His re-search results are published in scientific journals (more than 30 papers),proceedings of international conferences (more than 200 papers), and innumerous research studies/reports (more than 60 reports). Prof. Perichas particularly excelled in the leadership of national and internationalresearch projects. For his work he has received numerous awards, amongothers, the Croatian national award for science (for year 2007) for im-portant scientific achievement in development of advanced control andestimation strategies for complex technical systems. He has also receivedthe "Fran Bonjakovic" award (in 2009) for the exceptional contributionto the development and promotion of the automatic control research fieldwithin the area of technical sciences.

Ivan Petrovic received B.Sc. degree in 1983,M.Sc. degree in 1989 and Ph.D. degree in 1998,all in Electrical Engineering from the Faculty ofElectrical Engineering and Computing (FER Za-greb), University of Zagreb, Croatia. He hadbeen employed as an R&D engineer at the In-stitute of Electrical Engineering of the KoncarCorporation in Zagreb from 1985 to 1994. Since1994 he has been with FER Zagreb, where he iscurrently the head of the Department of Controland Computer Engineering. He teaches a number

of undergraduate and graduate courses in the field of control systems andmobile robotics. His research interests include various advanced controlstrategies and their applications to control of complex systems and mobilerobots navigation. Results of his research effort have been implementedin several industrial products. He is a member of IEEE, IFAC - TC onRobotics and FIRA - Executive committee. He is a collaborating memberof the Croatian Academy of Engineering.

AUTHORS ADDRESSES

Kreimir Vrdoljak, Ph.D.

KONCAR - Power Plant and Traction Engineering Inc.,

Fallerovo etalite 22, HR-10000 Zagreb, Croatia,

email: [email protected]

Prof. Nedjeljko Peric, Ph.D.

Prof. Ivan Petrovic, Ph.D.

Department of Control and Computer Engineering,

Faculty of Electrical Engineering and Computing,

University of Zagreb,

Unska 3, HR-10000 Zagreb, Croatia

emails: [email protected], [email protected]

Received: 2009-12-04Accepted: 2010-02-11

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