1

Electric boat modelling:energy sources, energy storage system and electric

motor control.Tiago M. Freire, MSc. degree, IST

Abstract—This thesis is an initial study with a scope fo-cused on electric propulsed ships.

The electric vehicles are nowadays one of the most impor-tant ways to overcome the modern energy paradigm. Thereare a continuous growing number of electrical propulsedships available on market mainly due to legally imposedrestrictions, environment concerns and increased on-boardcomfort. This opens new challenges on engineering, side-by-side with electric automotive applications, but with anotherlevel of needs.

Battery modelling is one of the cores of this work; it’sone of the most concerning bottlenecks on this scope, turn-ing difficult improvements on dynamics and performance inelectric vehicles. In this work were used Ni-MH and Lead-Acid batteries.

Energy converters, its control and propulsion motor’sspeed control were also focused on this thesis.

It was made a great effort to optimize the dynamic re-sponse of the entire energy chain, its control and the modelsof the batteries.

The simulations performed on Matlab Simulink softwareseems promising and could contribute to further develop-ment in this topic.

Index Terms—Electric boat, ni-mh battery, lead-acid bat-tery, state of charge, variable structure control, slidingmode, current-mode control, quasi-linear converter.

I. Introduction

SINCE the beginning of Time the mankind felt the needfor transportation, creating a strong value chain capa-

ble of generate economic progress and development.

The stress around fossil fuels availability strikes againstthe lifestyle within modern societies and its long term sus-tainability. Urges a social and technological revolution ca-pable of bringing a new, stable and durable balance be-tween two heavy interests: progress versus environmentalcosts.

The goals and the advantages of the electric power arewell-known and established within modern societies, im-proving energy efficiency and promoting its use on a grow-ing number of systems almost in exclusive holding of pri-mary energy sources, some of then highly harmful to theenvironment.

The electric vehicle is taking some terrain on this scopeand, slowly, also the electric boat. The electric boat showsan easy way for convertion to electric propulsion and mayuse some infrastructures already available, tax free andwithout the nowadays problems of chaotic traffic on maincities. It is a growing tendency to use this kind of trans-portation in urban and tourism context.

Besides that, the new developments on electric powersources like fuel cells – hydrogen and other biosustainablefuels – together with the new generation of electric boats,

it’s expected to see an improvement on this transporta-tion technology and find solutions to its engineering andeconomical challenges.

There is a highly potential source of social developmentand a good way to find a solution to the modern energeticparadigm.

II. Summary

This paper is a short version of the main document andits structure may be summarized as follows:

Introduction: The first section is dedicated to a compactintroduction, exploring the scope of this thesis’s work.

Vessel model: The derivation for a resistance forcemodel (vessel) and lumped-parameter dynamic modelof a propeller is taken on Section III.

Ni-MH model: Section IV is dedicated to Ni-MH bat-tery modelling and experimental work for determiningthe parameters of a RCL+E circuit model, includingestimation of SOC and discharge evolution modelledby electromotive force (open circuit voltage).

Lead-acid model: Lead-acid modelling is presented onSection V where is available a model for SOC esti-mation and typical full discharge dynamics.

DC/DC converter: On Section VI is proposed twoquasi-linear DC/DC converters with current-modePWM control (CMC) scheme based on IC MaximMAX668. One converter is buck-boost and the otheris a boost type one.

VSS control: The propulsion motor speed control iscomposed by a Variable Structure control System(VSS) based on a first approach also shown in Sec-tion VII. Was developed a Simulink model to proofthe concept and show the advantages of this type ofcontroller.

Conclusions: Section VIII is the final one, where areconceived the conclusions of this entire work.

Clearly, there is some appendix not included in this pa-per and they are available on the main document.

III. Vessel model

One of the main goals of this thesis is modelling theboat’s vessel. It was taken a hybrid approach from a alge-braic model plus empirical model based on.

Then, the total resistance is based on two main sources:friction and residual, functions of Reynolds and Froudenumber, respectively.

The forward total resistance is composed on three lon-gitudinal components:• Skin effect (RVskin

);

2

• Wave (residual) drag (RW);• Air drag (RVair ).Then:

RT = RVskin+RW +RVair (1)

A. Water viscous drag

RVskin= Cfriction ·A|w| · ρ · Vx · |Vx| (2)

With A|w| the submersed area in the platform [m2],ρ fluid’s density [kg/m3], Vx tangential speed [m/s] andCfriction friction coefficient of the vessel defined by [3]:

Cfriction =0.075

(log(Re)− 2)2(3)

With Re Reynolds number [4]:

Re =Vx · L|w|

ν(4)

And L|w| the length of line-of-water [meters], ν fluidkinematic viscosity given in m2/s.

B. Residual resistance (wave)

The wave resistance is given by an empirical model basedon [8]:

RW = c1 · c2 · c5 · ∇ · ρ · g · exp(m1·Fdn+m2·cos(λ·F−2

n )) (5)

Where Fn is the Froude number:

Fn =Vx√g · L|w|

(6)

C. Aerodynamic drag

This force is proportional to the air stream exposed area.It may be, in a first approach, given by (7):

RVair=

1

2· Cair · ρair · Sair · V 2

air (7)

Where Cair is the air friction coefficient, ρair the airdensity [kg/m3], Sair the vessel surface exposed to the airstream [m2] and Vair the air velocity relative to the vessel’sone [m/s].

This drag force may be neglected, as air limited densityreduces the weight of this force between the others appliedto the vessel. This is only valid if the air stream is notstrong enough to increase its force to a significant level.

D. Dynamic propeller model

In this subsection is presented a lumped-parameter dy-namic model for the propeller to install on the boat. Mostsmall-to-medium-sized over-water vehicles are powered byelectric motors driving propellers mounted in ducts. Ingeneral, the propeller is mounted in a duct or shroud whichincreases the static and dynamic efficiency of the thruster,like the setup shown on Fig. 1:

Figure 1

Classic setup formed by propeller + shroud. [11].

The derivation of a lumped-parameter dynamical hydro-dynamic model for propellers operating in incompressiblefluids found in most introductory fluid dynamics texts [9][12], take us to Eq. 8 [10]:

Tinst = (ρ · l · γ) · vp + (∆β · ρ · a) · vp · |vp| (8)

With the control volume (ρ · l · γ), γ and ∆β two empiri-cally determined added mass and flux coefficients. Tinst isthe thrust generated and vp the axial fluid velocity at thepropeller.

It is well known, exempli gratia [9] [12], that underbollard-pull conditions a symmetrical propeller’s steady-state axial thrust, T, is proportional to the square of thepropeller’s rotational velocity, np.

Tinst = ρ ·A · r2 · η2p · tan(p) · (np)2 (9)

where ηp is the propeller efficiency coefficient, p is thepropeller pitch (rad), A is the propeller area (m2), ρ is thefluid density (kg/m3)and r the radius of propeller (m).

Then, we employ an energy balance analysis equatingaxial power expended at the propeller disk (T ·vp) to powerexpended on the propeller shaft (Qp ·np):

T · vp = Qp · np (10)

Applying this assumption directly to Eq. 8, we obtain adifferential equation of motion with independent variableof propeller angular velocity and with an input – the shafttorque:

np =1

(ηp)2 · r2 · tan(p)2 · ρ · V·Qp −

ηp · r · tan(p) ·A2 · V

· (np)2 (11)

This approach omits transient momentum balance termsto write instantaneous propeller thrust [10] using thesteady-state equation 9.

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.3

IV. Ni-MH modelling

On this study is proposed a Thevenin circuit for mode-lling the dynamics from milliseconds to several hours timescale. It is based on a RLC+E equivalent circuit, as shownon Fig. 2:

Figure 2

Equivalent circuit proposed for the Ni-MH battery.

UBat and IBat are the voltage and current at the batteryterminals, Em the open circuit voltage (Voc), Ri the inter-nal resistance (terminals, electrodes and electrolyte), RD eCD represent the effects on the surface of electrodes (dou-ble layer capacity) and RK e CK describes internal phe-nomenons due to diffusion processes on electrolyte core.The Em source is a controlled one, depending on the dy-namics between OCV and discharging of the battery.

The passive (Rj) and reactive (L e Cj) parametersare taken as constants throughout all discharging process,while Voc is defined by Eq. 12:

Voc = [VH(IBat) + Vφ(SOC, T )] (12)

Where VH(IBat) represents the voltage hysteresis – func-tion of discharge current of the battery (and not studiedon this document) – and Vφ(SOC, T) the part based onNernst modified equation [32]:

Vφ(SOC,T ) =N ·[U0 +

Rg ·Trefne ·F

· ln(SOC −Ξ

100−SOC

)−γ ·SOC + (T −Tref ) ·

∆S

ne ·F

](13)

Knowing the expression that gives the OCV (Voc), wemust show the relation that opens a way to estimate Emthrough the battery output voltage (Vout), modelled by thecircuit on Fig. 2:

Vout(t) = Em(t)− Iout ·[Ri +RL · e(−αL·t)

+RD ·(

1− e(−αD·t))

+RK ·(

1− e(−αK ·t))]

(14)

Trivially,

Voc = Em ; Iout = 0 and t→∞ (15)

So,

Em =

Vout(t)− Iout · [Ri +RD +RK ] ;

N ·[U0 +

Rg ·Trefne ·F

· ln(SOC −Ξ

100−SOC

)−γ ·SOC + (T −Tref ) ·

∆S

ne ·F

].

A. Algorithm for SOC estimation

The estimation of SOC is composed by: Coulomb-Accumulation method (CAM) and by Open Circuit Voltagemethod (OCVM). The first one is based on the historicalevolution of output current – including self-discharging andlifetime decay (both not studied here) – and the second onefollows the relationship between OCV and SOC evolution.

The output of this two methods are named as SOCce SOCv, respectively, and both are combined on a singlevalue defined by Eq. 16 [30] [32]:

SOC = w · (SOCv) + (1− w) · (SOCc) (16)

Where w is a weighting factor.

B. Inicial SOC calculation

As initial condition to estimate SOC, we must defineSOC0 as [30]:

SOC0 =

SOClast 0≤ t≤ 86400 [s]SOClast−SOCselfdischarge t≥ 86400 [s]

(17)

With SOClast the last value recorded from state-of-charge and SOCselfdischarge the self-discharging lost SOC.Both not taken on this study approach.

C. SOCc estimated algorithm

The CAM contribution (SOCc) is determined throughthe following Eq. 18:

SOCc = SOC0 −∫ t

0i(τ) dτ

Cnom(Ioutavg)· 100 (18)

With SOC0 the initial state-of-charge, i the evolution ofbattery output current (positive convention), t the integra-tion time lapse and Cnom the nominal capacity considered.

D. SOCv estimated algorithm

The state-of-charge estimated through the open circuitvoltage is based on the Nernst modified equation:

Voc(SOCv ,T ) =N ·[U0 +

Rg ·Trefne ·F

· ln(SOCv −Ξ

100−SOCv

)−γ ·SOCv + (T −Tref ) ·

∆S

ne ·F

](19)

Voc is determined taking into consideration the circuitlaws and Eq. 15. Solving Eq. 19 in order to SOCv, ispossible to estimate it.

E. The weighting factor w

The SOC of the battery is jointly determined by SOCve SOCc, so the weighting factor w should be calculatedcarefully. Considering that the battery is a dynamic andtime-varying system, if we can get an accurate current mea-surement from the battery parameter sample system, theweighting factor of SOCc should dominate the compositeSOC of the battery. In this case, w is given by:

4

0 ≤ w ≤ 0.5

When the OCV is under the steady-state, the SOCv hasa higher accuracy. Therefore, we should consider both thevalue of OCV and the time of Voc goes to steady, and thencalculate the weighting factor as below.

Considering the range of Voc like [Vα, Vβ ], we assumethat the voltage corresponding to the SOCv has a higheraccuracy near both lower and higher end of that range.Therefore, we divide it into 100 sections, each one weight-ing value w1 calculated as Eq. 20:

w1 = 2 · |k − 50| · 0.5/100, 1 ≤ k ≤ 100 (20)

On the other hand, the average time of Voc going tosteady-state needs about 1800 sec according to the exper-iments performed and [31].

Then, if the time between two sample points is largerthan 1800 sec, since Voc goes to steady state:

w = w1

For less than 1800 sec:

w = w1 · t/1800

So, w is determined by Eq. 21 [30]:

w =

2 · |k− 50| · 0.5/100 if 1≤ k ≤ 100 & t≥ 1800[s]

(2 · |k− 50| · 0.5/100) · t/1800 if 1≤ k ≤ 100 & t < 1800[s]

(21)

Were performed several tests on lab and Simulink model.The lab part is composed by ON/OFF tests, done at SOCequal to 50% and 100%. The idea was to proof the in-variance of battery dynamics for repetitive and sequencialON/OFF states, different discharge currents and state-of-charge. For example, in Fig. 3 are represented six essaysand overlayed by one simulation with the same parame-ters extracted by curve fitting, using cftool() in Matlabsoftware.

Figure 3

Experimental trials on ON/OFF tests for 50% of SOC with

a resistive load of 0.2 Ohm, for a current slope with tup =

4.4 ms.

Figure 4

Experimental trials on ON/OFF tests for 100% of SOC

with a resistive load of 0.2 Ohm, for a current slope with

tup = 4.4 ms.

The results shows that in the thesis model it is possibleto simulate with high accuracy both dynamics for 50% and100% for the range of discharge currents imposed to thebattery.

E.1 Simulation of SOC evolution for standard discharges

After determining the parameters of Eq. 19 throughcurve fitting methods (cftool() in Matlab software), weredone several discharge simulations to find the evolution ofSOC by solving Eq. 19 (SOCv) and through CAM. Wasalso found a relationship between the average discharge

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.5

current and the nominal capacity of the battery, as (22).

Cnom(Ioutavg ) = 928 · (Ioutavg )2−4400 · Ioutavg + 14200 [C] (22)

The unavailable technical data from manufacturer waspartially solved by determining the output currents forstandard discharge rates, like C2, C4 and C6. This is illus-trated in Table I.

TABLE I

Discharge Rates for Ni-MH battery.

Parameter C2 C4 C6

Unit

Tempavg 26,10 24,50 24,15[C]Iavg 1,50 0,80 0,47

[Ampere]Exp. time 1:50 3:40 6:25

[hours]Nominal capacity 9688 11275 12355

[Coulomb]

As an example from one of the discharges simulated onSimulink model, on Fig 5 is shown the evolution of a C4

discharge.

Figure 5

Evolution of SOCc, SOC and SOCv during a C4 discharge

rate on Simulink model.

We could see that the error between the SOC and SOCcis lower than 1%, from the begining to the end of simulationat C4. The SOCv also is very similar to the other ones butaround SOC equal to 50%, it shows some ripple inherentto the solving method applied to (19). Although, SOCvshows a saturation zone, near SOC equal to 0 %, from alimiting block submited on Voc value to avoid an overflowduring the simulation.

V. Lead-Acid modelling

This work was also focused on lead-acid battery mod-elling. In the section is presented a RC+E model, basedon [70]. The intent is to modelling the battery dynamicsduring complete discharges taken from dozens of minutesto several hours. The model is shown on Fig. 6.

Figure 6

Electric model of Lead-Acid battery on study.

UBat e IBat are the output voltage and current of thebattery. Em is the open-circuit voltage (Voc), R0 is theinternal resistance (terminals, electrodes and electrolyte)and R1 e C1 represents the effects of the diffusion dynam-ics on electrolyte core and at electrodes surface. The Emvoltage source is a controlled one by the evolution of thebattery state-of-charge, along of its discharge process.Rj and Cj as passive and reactive electrical elements,

respectively, are taken as variable parameters throughoutthe discharge [70]:

R0 =R00 · [1 +A0 · (1−SOC)]R1 =−R10 · ln(DOC)C1 = τ1

R1

(23)

While R00, A0 e R10 are empirical parameters. τ1 is thetime constant of RC circuit loop.

Em = Em0 −KE · (273 + Θ)·(1− SOC) (24)

A. State-of-charge estimation

The block used to model the thermal dynamic of elec-trolyte, is based on a function of Joule losses (PJoule), am-bient temperature and battery thermal properties. All ofthis is done with a first order differential model, whoseparameters are the thermal resistance (RΘ) and thermalcapacity (CΘ) – see Eq. 25.

CΘ

(∂Θ

∂t

)=

Θ−Θe

RΘ+ PJoule (25)

Θinicial, Θbat, Θamb are, respectively, the initial mea-sured temperature, instantaneous temperature and ambi-ent one.

The SOC block outputs the state-of-charge and depht-of-charge (DOC) of battery based on the electric charge

6

extracted (Qe), electrolyte temperature (Eq. 25) and twoother capacities determined with Eq. 26 [70].

C(Ibat; Θ)I,Θ=const =Kc · C0∗ ·

(1 + Θ

−Θf

)ε1 + (Kc − 1) ·

(Ibat

Iref

)δ (26)

And,

SOC = 1− Qe

C(0;Θ)

DOC = 1− Qe

C(Iavg ;Θ)

(27)

Where Kc, ε e δ are fitting parameters, Θf is the meltingpoint of electrolyte, Iref is a reference current for the fittingparameters and Iavg is the average current at the terminalsof battery. So:

Qe(t) =

∫ t

0

Ibat(τ) dτ (28)

With positive convention of current flowing out of bat-tery.C0∗ is a parameter determined through (29):

C(Iref ;Θref ) = C0∗ ·(

1 + Θ−Θf

)εε= α · (Θref −Θf )

(29)

with α taken as a constant, as [70]:

α =1

C·(∂C

∂Θ

)B. Simulation of SOC evolution for standard discharges

To find the SOC evolution were done several dischargesimulations through CAM. Was also found a relationshipbetween the average discharge current and the nominalcapacity of the battery, as (30).

Cnom(Ioutavg ) = 36993 · (Ioutavg )2 − 45971 · Ioutavg + 19695 [C](30)

This is a regression based on data collected for the rangeIoutavg

= [0.40; 0.76] Ampere, with a coefficient of deter-mination R2 = 1.

The unavailable technical data from manufacturer waspartially solved by determining the output currents forstandard discharge rates, like C2, C4 and C6. This is illus-trated in Table II.

In Table III is written the result of curve fitting andparameter estimation for the ones on Fig. 6 (Eq. 23, Eq.24 and Eq. 26).

TABLE II

Discharge Rates for Lead-Acid battery.

Parameter C2 C2 C4 C6

Unit

Tempavg 26,05 25,00 25,40 23,70[C]Iavg 0,76 0,76 0,46 0,40

[Ampere]Exp. time 2:15 2:27 3:51 6:20

[hours]Nominal capacity 6124 6694 6376 7226

[Coulomb]

TABLE III

Parameters for the Lead-Acid model.

Parameter Mean value Unit

Em 6,337 VoltKe 1,566E-3 Volt · K−1

R00 1,264E-1 OhmA0 0,4209 AdimR10 0,0370 Ohmτ1 1,281 s−1

B.1 Conclusion notes

It is important to emphasize that the work presentedgives a complementary approach to the usually ones foundin recommended literature focused on this scope, especiallyon:

• Obtaining voltage curves for constant current dis-charge for standard times, like 2 hours, 4 hours and 6hours (C2, C4 and C6);

• Estimation of the parameters for the equivalent cir-cuit, based on curve fitting methods;

• Thermal model constructed on Matlab Simulink forre-enforcing side-by-side the electric model;

• Electric model with variable parameters, function ofstate-of-charge (SOC);

• Possibility for estimation of nominal capacity, within adischarge current rated between C2 and C6 (Eq. 30).

VI. DC/DC converter

The main limitations of nowadays DC/DC convertersare, namely, optimization of its reactive filters (LC) to tryapproaching a zero ripple value at the output, small di-mensions, reducing power losses and high performance inload and source transients.

Besides that, boost converters and buck-boost DC/DCconverters suffer from slow dynamic response due tothe presence of a characteristic right-half-plane zero [36],whose location shifts with the operation point. The conse-quence is the evolution of output states tending, on initialmoments, in the opposite direction of the final value. Thisforces the designers to limit the overall closed-loop band-width [27], making difficult its control with a closed-loop

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.7

system [28].

The main goal of the approach proposed in this work isto implement a system with the following advantages, forachieving a high performance and high power converters:

• Non-linear voltage control with current mode controlCMC with high impedance output [37]. The CMCalso solves the problem of resonance between the maininductor and the capacitor at the output [37], mini-mizing electric oscillations.

• Unlike some approaches using high capacitance at theoutput to stabilize the voltage and reducing ripple,this solution applies [24]:

An output linear voltage regulator stage thatminimizes load voltage ripple without the use of largeoutput capacitance (cost reduction);

An independent current injection stage to limitvariations on output voltage that eliminates voltagedeviation during fast transient response. Also, thisstage suppress the right-half-plane zero influence inthe boost and buck-boost converters, thus signifi-cantly improving the transient response bandwidth.

On Fig. 7 are shown both quasi-linear converters, boostand buck-boost ones.

MOSFETs Q2 and Q3 and resistor Rs are added to aconventional boost converter, along with a tapped induc-tor L1 and an auxiliary storage capacitor Caux. Q2 is con-trolled using an op-amp OP1 to realize a voltage followerfunction, with Vref being the output reference voltage.Under steady state conditions, Q2 operates in the linearzone with the op-amp modulating the gate voltage to keepthe output virtually ripple free.

If the load current decreases suddenly, the output volt-age increases suddenly as the converter control attemptsto reduce the inductor current. This forces the voltageacross Q2 to be higher, moving it into a saturation zone,wherein Q2 sees higher dissipation momentarily, but herethe output voltage remain constant and equal to Vref .

Under a step increasing in load current, the output volt-age would decrease. This would in turn cause the controllerto increase the duty cycle, allowing the inductor currentto build up. If the system operates in a continuous modeunder full load with low current ripple, the current incre-ment could take 3 to 10 cycles [24], causing a significanttransient in the output voltage. Smaller values of capac-itance exacerbate the problem of voltage deviation underthis conditions.

The issue of large voltage deviation during step changein load is addressed through a current injection stage. Thisis done with Q3 ON/OFF state – is also operated as a volt-age follower using op-amp OP2. Under normal operatingconditions, the diode D3 is reverse biased but, within asudden load transient, the boost converter output drops,forcing D3 to be forward biased and clamping converteroutput voltage using a current injection from the capaci-tor Caux into the output capacitor C.

(a) Quasi-linear boost DC/DC converter. [24].

(b) Quasi-linear buck-boost DC/DC converter. [24].

Figure 7

Quasi-linear DC/DC converters. [24].

A. Current-Mode Control

A current-mode control was implemented in the MatlabSimulink model for these converters, which is representedin Fig. 8.

This method regulates the output current and, with infi-nite loop gain, the output is a high-impedance source [37].In CMC, the current loop is nested with a voltage loop,as shown in Fig. 8; a ramp is generated by the slope ofthe inductor current (Vsense) and compared with the er-ror signal (Verror). So, when the output voltage sags, theCMC supplies more current to the load.

In CMC, the duty cycle is determined by the number oftimes which inductor current reaches the maximum limitdefined by the voltage control loop signal (vide Fig 9).

The current-mode scheme has several advantages [33]

8

Figure 8

Current mode controller, with a typical boost DC/DC

converter. [33].

Figure 9

Time-lapse on the evolution of inductor current and

trigger signal (PWM) in a DC/DC converter. [33].

over the conventional voltage-mode scheme (VMC):1 Several switching converters can be operated in pa-

rallel without a load-sharing problem because all theswitching converters receive the same PWM controlsignal from the feedback circuit and carrying the samecurrent.

2 During current-mode operation, the average inductorcurrent follows a reference voltage. As such, the in-ductor acts as a current source. Thus, the inductorbehaves as a voltage-controlled current source thatsupplies the output capacitor and the load, therebyreducing the order of the system by one. This simpli-fies its feedback compensation considerably.

The major drawback of the CMC is its instability. Anoscillation generally occurs whenever the duty cycle ex-ceeds 50%, regardless of the type of switching converter.However, this instability can be eliminated by the additionof a cyclic artificial ramp [35] either to the sample of theinductor current (Vsense) or to the voltage control signal(Verror).

This control scheme was implemented on MatlabSimulink based on the Maxim MAX668 integrated circuit

– Fig. 10.

Figure 10

Internal circuit scheme from Maxim MAX668. [34].

The model is an approach for fixed frequency PWMcycle-by-cycle current model control. Additional informa-tion about this implementation can be seen in [34].

A.1 Step response to RL load type

The quasi-linear converter model and its control weresubmitted to a limited number of steps on its input refer-ences (voltage and current), for two different types of loads:pure resistance (R) and series resistance+inductance (RL),as shown in Fig. 11 and Fig. 12.

Figure 11

Output voltage (Vout) evolution for a step up on voltage

and current input references, to a series R = 1 Ω and L = 1

mH load.

On figures above was applied a step with 25 Volt and25 Ampere slope, for both voltage and current input ref-erences. It is visible on output voltage evolution that thesettling time is 45 ms and 32.5 ms, for start-up and step updynamics, respectively. This times are equal to a resistiveload only (R) too. The settling time for steeping down andstart-up are 75 ms e 45 ms, respectively, like in the R typeload situation. No overshoot was observed on anyone ofthe simulations done.

The error on steady condition is contained between 1%and 2% of reference values for output voltage and current.

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.9

Figure 12

Output voltage (Vout) evolution for a step down on voltage

and current input references, to a series R = 1 Ω and L = 1

mH load.

It is also important to notice that the response dynamics(time and evolution) seems to be independent of load typefor the parameters submitted in simulation.

A.2 Current injection stage

On figures below is shown the evolution of output voltageof a buck-boost quasi-linear converter, for a R = 1 Ω typeload. It was submitted with a step up with slope of 25Ampere for the current reference (ton = 350ms).

The Fig. 13 illustrates the transient response on Vout

for a step up in reference current.

Figure 13

Output voltage evolution (Vout) for a step up on current

input reference with current injection stage off-line. Load

at R = 1 Ω.

It is clearly visible the influence of the right-hand-planezero; the output initially changes in the wrong direction.

As an example of this current injection stage, in the sameconditions of Fig. 13, it is shown on Fig. 14 a zoom-in overoutput voltage (Vout) evolution.

There are evident the advantages of this stage: fasterresponse (rising time about half on off-line state) and lowervoltage dip. The influence of right-hand-plane zero is al-most suppressed on voltage load.

Figure 14

Output voltage evolution (Vout) for a step up on current

input reference with current injection stage on-line. Load

at R = 1 Ω.

On Fig. 15 is illustrated the dynamic of the currentinjected by its own stage, in the same conditions of Fig 14.

Figure 15

Output voltage evolution (Vout) and injected current

(Iaux) for a step up on current input reference with

current injection stage on-line. Load at R = 1 Ω. (Detail

from Fig 14.)

This current flows by a pulsing way, so that the out-put voltage (Vout) stays constrained to a a priori definedminimum limit, as a turn-on parameter of this stage. Theamplitude of this current varies in a range from 160 Am-pere to 105 Ampere, depending only on the dischargingprocess of capacitor Caux.

For the situation with a series RL load, and for the sameconditions above, on Fig. 16 is shown a zoom-in detail ofoutput voltage Vout and the injected current Iaux.

It is clearly visible the reduction of amplitude on injectedcurrent, in a range from 160 Ampere to 133 Ampere. Thisis also due to the fast overshoot visible on Vout with originon the dynamic imposed by the inductive load.

There is also visible a steady-state error on Fig. 13 toFig. 16 due to errors introduced by the use of classicalequations for this type of converter [28] and on calculationof α and β (voltage and current gains) [34], ignoring losses

10

Figure 16

Output voltage evolution (Vout) and injected current

(Iaux) for a step up on current input reference with

current injection stage on-line. Load at R = 1 Ω and L = 1

mH. (Zoom-in detail from a simulation).

on semiconductors and others minor approximations.

VII. VSS control

DC motor drives have been widely used in industry forspeed control because of their excellent control characteris-tics. One of the commonly used techniques for controllinga separately excited DC motor is closed-loop integral con-trol, where the speed is controlled by varying the voltageapplied either to the armature terminals or to the fieldterminals. It is capable of taking the steady-state error atzero, e.g., on motor speed, however it shows a week dy-namic performance clearly visible in prominent overshootsor very long settling time.

The Variable Structure control System with SlidingMode (VSS-SM) was developed on mid 50’s of 20th cen-tury in USSR by the hand of Emelyanov [60] and followedby others [58] [59] [63]. This pioneer researches tried to ap-proach 2nd order linear systems control by phase-shifting.Since then, the VSS have been developed as a general ap-plication method, applied in a variety of other types ofnon-linear, discrete and stochastic systems. Moreover, themain goals of this control method is its robustness andit could be insensitive to uncertainty on plant parametersand external disturbances.

A. DC motor drive model

Consider a separately excited DC motor, as shown, inFig. 17. The equations describing the dynamic behaviourof the motor are as following:

Va =Ra · ia +La · diadt +Kf · if ·ωVf =Rf · if +Lf · difdtdωdt =

Kf ·if ·ia−Tload

J − fJ ·ω

(31)

Where Kf is a constant.

Figure 17

Equivalent DC motor circuit. [38].

Linearising Eq. 31 about the operating point x0 includ-ing the integral controller, we obtain the linearised stateequation of the DC motor drive system as (32):

x=A ·x+B ·u+ Γ · zx(0) = 0

(32)

Which the state vector, x, is:

x = [x1 x2 x3]T

(33)

Where:

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.11

x1 =∫

(∆ωref −∆ω) · dtx2 = ∆ωx3 = ∆ia

And:

u= [0 0 ∆Va]z = [∆ωref ∆Tload 0]

The system matrix:

A =

0 −1 0

0 −f/J Kf ·If0J

0 −Kf ·If0La

−Ra

La

(34)

B = [0 01

La]T (35)

Γ = [1−1

J0]T (36)

Quod Erat Demonstrandum.

On Fig. 18 is illustrated the block diagram of the controlsystem above presented. It has a 3rd order dynamics, witha single integrator at the output of speed error, ∆ωe. Theblock SMC includes the controller described below.

Figure 18

Block diagram control system with VSS controller

embedded on it. [66].

B. Unit vector sliding mode control

B.1 Introduction

Among the various VSS, unit vector sliding mode controlis the most attractive approach for electrical drive systemsbecause of its smooth manipulation.

The control signal u consists of two independent func-tions: a linear state feedback control function Ul(x) and aunit vector control function Uv(x) [67]:

u = ∆Va = Ul + Uv

Where:

Ul = L · x (37)

Uv = −ρ · cT · x|cT · x|+ δ

, ρ > 0 e δ > 0 (38)

With L is a linear state feedback matrix:

L = [l1 l2 l3] (39)

and c is the switching vector defined as:

c = [c1 c2 1]T (40)

The elements of L are defined as [63] [67]:

l1 = c1 ·La ·Φ∗ (41)

l2 = La ·[c1 + c2 · (Φ∗+

f

J)

]+Km (42)

l3 = La ·[

1

Ta+ Φ∗− c2 ·

Km

J

](43)

Where Φ∗ is a non positive scalar [69].The function Ul should always be able to bring the sys-

tem state trajectory from anywhere in state space into themanifold in which the sliding mode occurs. The unit vec-tor control function Uv switches dynamically to force thetrajectory to remain in the manifold and slide towards theorigin of the state space. On the other hand, the magnitudeof switching control function Uv is relatively small [67], dis-tinguishing this sliding mode control from the bang-bangone.

Additionally, the smoothing factor, δ, in Eq. 38 anddiscussed on [68] [69], lowers the ripple on the system evo-lution during sliding mode, as also the empirical parameterρ in (38).

B.2 Robustness of the drive to a step load disturbance

Considering the ideal trajectory in sliding mode is ex-pressed by:

c1 · x1 + c2 · x2 + c3 · x3 = 0 (44)

The transfer function of the drive speed deviation to astep load disturbance, during sliding mode, is given by (32)and (44):

∆ωr(s)

∆ωref (s)=

−c1 · Km

J

s2 + (c2 · Km

J + fJ )·s− c1 · Km

J

(45)

Under sliding mode, the drive speed response to a stepload disturbance is reduced from a 3rd order to a 2nd orderimpulse-type response. Re-writing (45) in a well-knowncanonical 2nd-order form (46):

W (s) =ω2n

s2 + 2 · ς · ωn · s+ ω2n

(46)

it is possible to write c1 and c2 in function of ωn and ςparameters [66]:

12

c1 =−ω2n · JKm

(47)

c2 =2 · ς ·ωn · J − f

Km(48)

And the following condition must hold [63]:

ς >

√1

2·(

f2

2 · J2 · ω2n

+ 1

)(49)

In [53] was proved that are needed high values of ωn andς to lowering the steady-state error in angular velocity andsettling time, during step-load disturbances, on this VSScontroller. This is unusual, namely in conventional designprinciple based on linear control theory; there is always acompromise between ωn and ς, due to the intrinsic conflictto achieve a good balance on large overshoot versus longsettling time. The region of ς > 1 is almost always aban-doned because sluggish response will result. This situationis very favourable for the sliding mode control system, be-cause the strategy of two-stage state motion (Ul and Uv)is very effective for resolving this conflict [67]. Moreover,the independent selection of the values of ς and ωn basedon the freedom in the selection of the switching vector cprovides a new level of performance, capable of achievingboth fast response and robust performance, impossible inapproaches based on linear control theory.

B.3 Simulations tests – Start-up

On Fig. 19 and Fig. 20 are illustrated the time evo-lutions of DC motor speed (∆ωr) and its voltage controlsignal (u= ∆Va).

Figure 19

Evolution of motor speed ∆ωr for the simulation

parameters on Table IV.

For the curves 1 to 4 the dynamic of those does not ex-hibit overshoot and has high range of settling times, vary-ing from slow to high speed evolution performance. ∆Vahas a smooth evolution, without a switching evolution ty-pical of Bang-Bang VSS controllers, as proved in the maindocument.

Figure 20

Evolution of voltage control signal u for the simulation

parameters on Table IV.

TABLE IV

Table with the parameters submitted on simulation (Fig. 19

and Fig. 20). [66].

B.4 Evolution on output parameters of the model

To obtain the evolutions of the output parameters onthis controller were done multiple tests with this simulationmodel. All the tests were done with ωn = 90rad/s e ς = 1.

Two steps were applied to the model: ∆ωref = 500RPM(ton = 1s) and ∆Tload = 25N ·m (ton = 5s), shown on Fig.21.

The motor speed ∆ωr shows a settling time about 130ms, without any visible overshoot. At 5 secs of simulationtime, was applied a step-up on load torque input of 25 Nm,responding the system with a small down-dip on ∆ωr lowerthan 2.97% of its steady-state value. The intrinsic settlingtime of this disturbance is about 95 ms.

The highest overshoots on ∆Va and ∆ia are visible at thestart-up of the simulation, due to the fast dynamic imposedby the controller. This was only done as an example of itscapacity to achieve that, howsoever this situation is beyondthe ability that a typical DC/DC converter could operatewithin bearable cost and dimensions.

At the torque step-up disturbance, both electrical quan-tities shows an overshoot around +4.35%. The system res-ponse gives an almost unnoticeble dip on motor speed, dueto the fast dynamics of the controller.

Therefore, we can conclude that the system presents athe lower insensibility to step disturbances on load torque.

B.5 External disturbances response

To analyse possible disturbances, was applied one step-up on ∆ωref = 500RPM (ton = 0s) and extracted the evo-

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.13

(a) Evolution on output current signal ∆ia.

(b) Evolution on output voltage signal ∆Va.

(c) Evolution on output motor speed ∆ωr.

Figure 21

Evolutions on the mechanical and electrical outputs on

the VSS controller.

lutions of ∆ωr on Fig. 22, for some different values ofsystem parameters.

This VSS controller shows a virtual insensibility to sys-tem parameter disturbances, namely on the armature cir-cuit time constant Ta (Fig. 22(a)). On the Km situation,the controller can tolerate, without significant impact onits performance until -80% of its nominal value. With Km

at 0.5 p.u. it is visible a small overshoot with a faster risingtime, but at the cost of a worst dynamic performance.

Therefore, we can assume a lower insensibility for ex-ternal parameters disturbances on this controller perfor-mance.

(a) Evolution of ∆ωr for 3 different values of Ta.

(b) Evolution of ∆ωr for 3 different values of Km.

Figure 22

Evolution of angular motor speed ∆ωr for some values of

Ta and Km.

VIII. Conclusion

This paper shows a first approach to modelling a shipwith electrical propulsion and were taken in considerationseveral models, namely, the vessel propeller turbine torquemodel, battery modelling (both Lead-Acid and Ni-MH),DC/DC converters and its control system, and speed con-trol for the DC motor model.

The most evident contribution given to this work couldbe found in battery modelling of Ni-MH one. Is was expe-rimentally confirmed its micro-second dynamic (ON/OFFtransients), its slow discharging dynamics (Em evolutionmodelling) and SOC estimation. In ON/OFF transient,was verified that it could be modelled by a constant para-meter model along SOC evolution, for the discharge cur-rents imposed. The curve fitting done to the experimentalpoints proved a high performance modelling within thisapproach. Moreover, this work accomplished a good ba-sis structure to future studies and could be improved onmany ways, mostly due to the Em hysteresis, influence ofdischarge current on battery output voltage dynamic andother internal/external disturbances.

It is important to emphasize that the work presented onLead-Acid modelling gives a complementary approach tothe usually ones found in recommended literature focusedon this scope, especially on:• Obtaining voltage curves for constant current dis-

charge for standard times, like 2 hours, 4 hours and 6hours (C2, C4 and C6);

14

• Estimation of the parameters for the equivalent cir-cuit, based on curve fitting methods;

• Thermal model constructed on Matlab Simulink forre-enforcing side-by-side the electric model;

• Electric model with variable parameters, function ofstate-of-charge (SOC);

• Possibility for estimation of nominal capacity, within adischarge current rated between C2 and C6 (Eq. 30).

On this work were also implemented models for opti-mization on DC/DC converters, especially on transientresponse, for a auxiliary injection current block, reducingthen the presence of a characteristic right-half-plane zerofor both converters (boost and buck-boost). Was also pro-posed a ripple rejection model, capable of reducing subs-tantially the capacitance needed at the converter output.This could reduce also the cost, dimensions and wearingon DC motor due to ripple on its output torque. Thecontroller for both converters can be built only with digi-tal components, capable also of PWM control at constantfrequency, with current-mode control done cycle-by-cycle.This approach is based on Maxim MAX668 integrated cir-cuit, which is a current-mode controller highly efficient,independent of the load type (R, RL, etc) and with virtu-ally unlimited output voltage range for both converters.

Finally, the motor speed control was approached bya non-classic way. A Variable Structure control System(VSS) with unit vector sliding mode was tested in MatlabSimulink, being an easy way to test its performance, basedon industrial parameters (ωn e ς), and with wide rangeand high performance transient response, without difficultcontrol compensation methods for settling time or othercritical parameters.

References

[1] Dudhia, Anu; Basic Physics of Rowing; Atmospheric, Oceanicand Planetary Physics, Oxford University, 2001, http : //www.atm . ox . ac . uk / rowing / physics/;

[2] Pulman, Chris; Physics of Rowing; Gonville & Caius College,University of Cambridge;

[3] SNAME, Testing and Extrapolation Methods – Resistance Test,International Towing Tank Conference, 1957;

[4] Takinaci, Ali Can; On the fairing of resistance test data: Asoftware-based approach, Ocean Engineering, Volume 33, Issues17-18, pp. 2260-2269, Dec. 2006;

[5] Tuck, E.O.; Lazauskas, L.; Unsconstrained Ships of MinimumTotal Drag – Form Effects, University of Adelaide, Australia,1996, http : / / www . cyberiad . net / library / multihulls /multipep / basics . htm;

[6] Savitsky, Daniel; On the subject of high speed monohulls,Stevens Institute of Technology, Athens, Oct 2nd 2003;

[7] Crowe, C. T.; Robertson, J. A.; Elger, D. F.; Engineering FluidMechanics, New York, NY: John Wiley & Sons, Inc., 7th ed.,2001;

[8] Holtrop, J.; Mennen, G.G.J.; An approximate power predictionmethod, International Shipbuilding Progress, Vol. 29, pp. 166-170, 1982;

[9] Van Manen, J. D.; Van Ossanen, P.; Principles of Naval Ar-chitecture, Second Revision, Volume II: Resistance, Propulsion,and Vibration, Society of Naval Architects and Marine Engin-ners, Jersey, EUA, 1988;

[10] Whitcomb, Louis L.; Yoerger, Dana R.; Preliminary experi-ments in the model-based dynamic control of marine thrusters,pp. 2160-2165, 1996;

[11] Yoerger, D.R.; Cooke, J.G.; Slotine, J.-J.E., The influence ofthruster dynamics on underwater vehicle behavior and their in-corporation into control system design, Oceanic Engineering,IEEE Journal of , vol.15, no.3, pp.167-178, Jul 1990;

[12] Newman, J.N.; Marine Hydrodynamics, MIT Press, 1977;[13] Whitcomb, L.L.; Yoerger, D.R., Preliminary experiments in

model-based thruster control for underwater vehicle positioning,Oceanic Engineering, IEEE Journal of, vol.24, no.4, pp. 495-506,Oct 1999;

[14] Loois, G.; Wouters, F.P.H.; Koerts, G.M.; van der Weiden,T.C.J.; Monitoring results of PV for electric propulsion inrecreational boating, Photovoltaic Energy Conversion, vol.1, pp.1157-1160, vol.1, 5-9 Dec 1994;

[15] From 40.7 to 42.8% Solar Cell Efficiency – http : / / www .renewableenergyworld . com / rea / news / article / 2007 / 07 /from-40-7-to-42-8-solar-cell-efficiency-49483;

[16] Castro, Rui M.G.; Introduo Energia Fotovoltaica, EnergiasRenovveis e Produo Descentralizada, 2 Edio, Maio 2007;

[17] Weltrekord: 41,1% Wirkungsgrad fr Mehrfachsolarzellen amFraunhofer ISE, http : / / tiny . cc / bmof1;

[18] Bradley, T.H; et al; Test Results for a Fuel Cell-PoweredDemonstration Aircraft, Georgia Institute of Technology, Geor-gia, 2006;

[19] Levy, Alexander; Vandine, Leslie L.; Stedman, James K.; Re-generative fuel cell study for satellites in GEO orbit, Final Con-tractor Report International Fuel Cells Corp., South Windsor,CT, July 1987;

[20] Smokers, R.T.M.; Energie – en milieuaspecten van elektrischvaren, ECN, 1994;

[21] Linen, D.; Thomas, R.; The Handbook of Batteries, Third Edi-tion, Macgraw Hill, pag. 57-68 and 846, 2004;

[22] Kiehne, H.A.; Battery Technology Handbook, Second Edition,Marcel Dekker Inc, pag. 41, 2003;

[23] Berndt, D.; Voss., E.; Proceedings of the 4th International Sym-posium, Oxford, Pergamon Press, 1965;

[24] Divan, D.M.; Rajagopalan, S., Quasi Linear DC/DC Convert-ers, Industry Applications Conference, 2006. 41st IAS AnnualMeeting. Conference Record of the 2006 IEEE, vol.5, pp. 2537-2545, 8-12 Oct. 2006;

[25] Xu, P.; Wei, J.; Yao, K.; Meng, Y.; Lee, F. C.; Investigation ofcandidate topologies for 12V VRM, in Proc. APEC 2002, pp.686-692, 2002;

[26] Panov, Y.; Jovanovic, M. M.; Design considerations for 12-V/1.5-V, 50-A Voltage Regulator Modules, IEEE Trans. PowerElec., vol. 16, pp. 776-783, 2001;

[27] Sable, D. M.; Cho, B. H.; Ridley, R. B.; Use of leading edgemodulation to transform boost and flyback converters into min-

FREIRE: ELECTRIC BOAT MODELLING:ENERGY SOURCES, ENERGY STORAGE SYSTEM AND ELECTRIC MOTOR CONTROL.15

imum phase zero systems, IEEE Trans. Power Electron., pp.704-711, 1991;

[28] Silva, Fernando A.; Projecto de Conversores Comutados, Insti-tuto Superior Tcnico, Pag. 32, 2006;

[29] Pesaran, A.; Battery Choices and Potential Requirements forPlug-In Hybrids, National Renewable Energy Laboratory, LosAngeles, CA, Pag. 5-8, 2007;

[30] Ma Zi-lin; Mao Xao-jian; Wang Jun-xi; Qiang Jia-xi; Zhuo Bin,Research on SOC estimated strategy of Ni/MH battery used forhybrid electric vehicle, Vehicle Power and Propulsion Confer-ence, VPPC ’08. IEEE, pp. 1-4, 3-5 Sept. 2008;

[31] Schweighofer, B.; Raab, K.M.; Brasseur, G., Modeling of highpower automotive batteries by the use of an automated test sys-tem, Instrumentation and Measurement, IEEE Transactions on, vol.52, no.4, pp. 1087-1091, Aug. 2003;

[32] Verbrugge, M.; Tate, E.; Adaptive state of charge algorithm fornickel metal hydride batteries including hysteresis phenomena,Journal of Power Sources, Volume 126, Issues 1-2, pp. 236-249,2004;

[33] Ang, Simon S.; Oliva, Alexandro; Power-switching converters,Technology & Engineering, Pag. 169-170, 2005;

[34] MAX668, MAX669 – 1.8V to 28V Input, PWM Step-Up Con-trollers in µMAX, http : //www.maxim-ic.com/quick_view2.cfm / qv _ pk / 1901 / t / al;

[35] Brown, Marty; Kularatna, Nihal; A. Mack, Raymond; PowerSources and Supplies, Pag. 63-66, Newnes, 2007;

[36] Erickson, R. W.; Fundamentals of Power Electronics, Pag. 294-301, 1997;

[37] DC-DC Converter Tutorial, Application Note 2031, http : / /www . maxim-ic . com / appnotes . cfm / an _ pk / 2031/;

[38] Hsu, Yuan-Yih; Chan, Wah-Chun; Optimal variable-structurecontroller for DC motor speed control, Control Theory and Ap-plications, IEE Proceedings D, vol.131, no. 6, pp. 233-237, Nov.1984;

[39] Ramaswami, Krishnan; Ramaswami, B.; A fast-response DCmotor speed control system, IEEE Trans. Ind. Applicat., vol.10, pp. 643-651, 1974;

[40] El-Sharkawi, M.; Weerasooriya, S.; Development and implemen-tation of self-tuning tracking controller for DC motors, IEEETrans. Energy Conversion, vol. 5, 1990;

[41] Weerasooriya, S., El-Sharkawi, M.;Identification and control ofDC motor using back-propagation neural networks, IEEE Trans.Energy Conversion, vol. 6, pp. 663-669, 1991;

[42] Toliyat, Hamid A.; Kliman, Gerald B.; Handbook of electricmotors, 2nd edition, Pag. 73-74, 2004;

[43] Fitzgerald, A.; Kingsley, C.; Umans, S.; Electric machinery,McGraw-Hill, New York, 1991;

[44] Bogosyan, S. ; Gokasan, M.; Gurleyen, F.; Adaptive optimumcontrol of a seperately excited DC machine, International Con-ference on Electric Machines, September 12-14, Pisa, Italy, 1988;

[45] Nehnr, M.; Fatehi, F.; Tracking control of DC motors via inputoutput linearization, Electonic Machines & Power Systems, vol24, pp. 237-246, 1996;

[46] Miyamoto, S.; Annaga, S.; Design of motion control system viaH∞ partial state feedback, IEEE Conf On Control Application,August 24-26, Glasgow, Scotland, 1994;

[47] Ahmed, F.; El-Tobshy, A.; Mahfouz, A.; Ibrahim, M.; P-I an I-P controllers in a closed loop for DC motor drives, Proceedingsof the Power Conversion Conference, Nagaoka, Japan, pp 61-68,1997;

[48] Janardanan, E.; Gajendran, F.; Nirmalkumar, A.; Artificialneural network based state feedback control scheme for a sep-arately excited dc motor, Advances in Modeling & Analysis A,vol 68, pp 11-20, 1997;

[49] Chevrel, P.; Siala, S.; Robust DC motor speed control withoutany mechanical sensor, IEEE lnternahonal Conference on Con-trol Applications, Hartford, USA, pp 244-246, 1997;

[50] Baek, S.M.; Kuc, T.Y.; Adaptive PID learning control of DCmotors, IEEE International Conference on Systems, Man, andCybernetics, Orlando, USA, pp 2877-2882, 1997;

[51] Aggoune, M.E.; Boudjemaa, F.; Bensenouci, A.; Hellal, A.;Elmesai, M.R.; Vadari, S.V.; Design of variable structure volt-age regulator using pole assignment technique, Automatic Con-trol, IEEE Transactions on , vol.39, no.10, pp.2106-2110, Oct1994;

[52] Silva, J. Fernando; Sliding mode control design of drive andregulation electronics for power converters, J. Circuits, Syst.,Comput., vol. 5, no. 3, pp. 355-371, Sept. 1995;

[53] Kim, Kyung-Soo; Sliding mode control design in state space,Digital Mechatronics & Control Lab., Dept. of Mechanical En-gineering – KAIST, Page 10, April 15th 2008;

[54] Chan, W.C.; Hsu, Y.Y.; Automatic generation control of in-terconnected power systems using variable-structure controllers,IEE Proc. C, Gen. Trans. & Distrib., 1981, 128, (5), pp. 269-279;

[55] Sabanovic, A.; Izosimov, D.B.; Application of sliding modes toinduction motor control, IEEE Trans., IA-17, pp. 41-49, 1981;

[56] Chang-Ming Liaw; Yeong-May Lin; Kuei-Hsiang Chao; A VSSspeed controller with model reference response for induction mo-tor drive, Industrial Electronics, IEEE Transactions on , vol.48,no.6, pp.1136-1147, Dec 2001;

[57] Matthews, G.; Decarlo, R.; Hawley, P.; Lefebvre, S.; Towarda feasible variable structure control design for a synchronousmachine connected to an infinite bus, Automatic Control, IEEETransactions on, vol.31, no.12, pp. 1159-1163, Dec 1986;

[58] Itkis, U.; Control systems of variable structure, Wiley, NewYork, 1976;

[59] Utkin, V.I.; Variable structure systems with sliding modes,IEEE Trans., AC-22, pp. 212-222, 1977;

[60] Emelyanov, S. V.; Variable Structure Control System, Moscow:Nauka (in Russian), 1967;

[61] Hung, J.Y.; Gao, W.; Hung, J.C.; Variable structure control:a survey, Industrial Electronics, IEEE Transactions on, vol.40,no.1, pp.2-22, Feb 1993;

[62] Lemos, J. Miranda; Modelo de Estado de Sistemas LinearesContnuos, IST, Pag. 40-43, 2001;

[63] Utkin, V.I.; Yang, K.D.; Methods for constructing discontinuityplanes in multidimensional variable structure systems, Autom.& Remote Control, 39, pp. 1466-1470, 1978;

[64] Franklin, Powell; et al.; Digital Control of Dynamic Systems,3rd Edition, Addison Wesley Longman, pp. 280-281, 1998;

[65] Utkin, V.I.; Sliding modes and their applications in variablestructure systems, MIR Publishers, Moscow, pp. 122-123, 1978;

[66] Zhang, J.; Chan, W.C.; Wang, A.; Barton, T.H.; Synthesis ofoptimal sliding mode control for robust DC drive, Industry Ap-plications Society Annual Meeting, 1988, Conference Record ofthe 1988 IEEE , pp. 535-542, vol. 1, 2-7 Oct 1988;

[67] Zhang, J.; Barton, T.H.; Robustness enhancement of DC driveswith a smooth optimal sliding mode control, Industry Applica-tions Society Annual Meeting, 1990., Conference Record of the1990 IEEE, vol. 1, pp. 435-442, 7-12 Oct 1990;

[68] Ambrosino, G.; Celentano, G.; Garofalo, F.; Variable structuremodel reference adaptive control system, Int. J. Control, vol. 39,No. 6, pp. 1339-1349, 1984;

[69] Dorling, C.M.; Zinober, A.S.; Two apporaches to hyperplanedesign in multivariable structure control system, Int. J. Contro.,vol. 44, no. 1, pp. 65-82, 1986;

[70] Barsali, S.; Ceraolo, M.; Dynamical Models of Lead-Acid Batter-ies: Implementation Issues, Power Engineering Review, IEEE,vol.22, no.2, pp.63-63, Feb. 2002;

[71] Shuo Pang; Farrell, J.; Jie Du; Barth, M.; Battery state-of-charge estimation, American Control Conference, 2001. Pro-ceedings of the 2001, vol.2, pp.1644-1649 vol.2, 2001;

[72] Chiasson, J.; Vairamohan, B.; Estimating the state of charge ofa battery, Control Systems Technology, IEEE Transactions on,vol.13, no.3, pp. 465-470, May 2005;

[73] McIntyre, M.; Burg, T.; Dawson, D.; Xian, B.; Adaptive state ofcharge (SOC) estimator for a battery, American Control Con-ference, 2006, vol., no., pp.5 pp.-, 14-16 June 2006;

[74] CMOS Monostable by 4011, http : / / www . elecfree . com /electronic / cmos-monostable-by-4011/;

[75] The 555 Bistable Circuit, http : / / www . eleinmec . com /article . asp ? 5;

[76] Santana, J.; Sousa, D.M.; Laboratrio de Electrnica de EnergiaII – Trabalho n 4, rea Cientfica de Energia, IST, 2009;

16

Tiago M. Freire was born in Lisbon, in 1984.He received the MSc. degree in electrotechni-cal engineering from Instituto Superior Tecnico– Universidade Tecnica de Lisboa, Lisbon, in2009.

IX. Acronyms

CAM Coulomb-Accumulation methodCMC Current-Mode ControlDC Direct CurrentDOC Depth-of-ChargeIC Integrated CircuitMOSFET Metal-Oxide Semiconductor Field-Effect

TransistorNi-MH Nickel-Metal HydrideOCV Open Circuit VoltageOCVM Open Circuit Voltage methodPWM Pulse Width ModulationSOC State-of-ChargeVMC Voltage Mode Control Pulse Width ModulationVSS Variable Structure control SystemVSS-SM Variable Structure control System with Sliding

Mode