Stability Analysis of a DC MicroGrid for a Smart Railway

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Stability Analysis of a DC MicroGrid for a SmartRailway Station Integrating Renewable Sources

Filipe Perez , Alessio Iovine , Member, IEEE, Gilney Damm , Member, IEEE, Lilia Galai-Dol,and Paulo F. Ribeiro, Fellow, IEEE

Abstract— A low-level distributed nonlinear controller for aDC MicroGrid integrated in a Smart Railway Station capableto recover trains’ braking energy is introduced in this paper.The DC MicroGrid is composed by a number of elements: twodifferent types of renewable energy sources (regenerative brakingenergy recovery from the trains and photovoltaic panels), twokinds of storages acting at different time scales (a battery anda supercapacitor), a DC load representing an aggregation of allloads in the MicroGrid, and the connection with the main ACgrid. The nonlinear model of the MicroGrid is introduced, and acomplete stability analysis is investigated to the purpose to meetpower balance and grid voltage stability requirements. An Input-to-State Stability (ISS)-like Lyapunov function is obtained witha System-of-Systems approach, and it is utilized to develop thecontrol laws for the converters in order to fulfill the dedicatedobjective each of them has. Simulation results, showing thedesired grid behavior using the proposed nonlinear controllaws, are introduced and compared with classical ProportionalIntegral (PI) linear controllers, with respect to performances andparametric robustness. The DC MicroGrid is shown to be able tooperate braking energy recovery while performing load feedingand renewable energy integration and guaranteeing a proper DCvoltage profile.

Index Terms— DC MicroGrids, grid stability, input-to-statestability (ISS), Lyapunov methods, nonlinear control, railwaystation, renewable source integration.

NOMENCLATURE

η Dynamics related to the DC bus interconnection.μ Controlled dynamics of the supercapacitor sys-

tem.ωr Grid frequency in [rad/s].ξ1 Dynamics controlled by feedback linearization.ξ2 Dynamics controlled by dynamical feedback lin-

earization.ζ Zero dynamics.

Manuscript received March 6, 2019; accepted June 3, 2019. Manuscriptreceived in final form June 17, 2019. This work was supported in part byErasmus Mundus and in part by CNPq. Recommended by Associate EditorG. Papafotiou. (Corresponding author: Filipe Perez.)

F. Perez is with the L2S Laboratory, CentraleSupélec, Paris-SaclayUniversity, 91190 Gif-Sur-Yvette, France, and also with ISEE,Federal University of Itajubá, Itajubá 37500-015, Brazil (e-mail:[email protected]).

A. Iovine and L. Galai-Dol are with the Efficacity, Researchand Development Center, 77420 Champs sur Marne, France (e-mail:[email protected]; [email protected]).

G. Damm is with the IBISC Laboratory, Paris-Saclay University,91190 Courcouronnes, France (e-mail: [email protected]).

P. F. Ribeiro is with ISEE, Federal University of Itajubá, Itajubá 37500-015,Brazil (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2019.2924615

ILm Inductor current in [A], m = {3, 6, 9, 13, 16}.Ild Direct current on AC grid in [p.u.].Ilq Quadratic current on AC grid in [p.u.].IL Current on DC load in [A].Km.n Control gains.L f,g Lie derivative.r Reference vector.ui Control input, i = {1, 2, 3, 4, 5, 6, 7}.vm Additional control inputs.VB Voltage on battery in [V ].VCn Voltage on the capacitor Cn in [V ], where n =

{1, 2, 4, 5, 7, 8, 11, 12, 14, 15, 17}.Vdc Voltage on the DC bus in [V ].Vld Direct voltage on AC grid in [p.u.].Vlq Quadratic voltage on AC grid in [p.u.].VL Voltage on DC load in [V ].VPV Voltage on PV array in [V ].VS Voltage on supercapacitor in [V ].VT Voltage on train system in [V ].W Lyapunov function.x Extended state variable.xe Equilibrium point of x .y Output control.z1,2 Variable transformation for VC14 .

I. INTRODUCTION

THE continuing electric load growth along with modernloads based on power converters remarked the necessity

to suit the system operation in electrical grids. In addition,environmental issues related to the emission of pollution inthe atmosphere and the reduction of fossil fuel reserves hasbrought the use of renewable energy sources. Renewableenergy is now the key for locally producing clean and inex-haustible energy to supply the world’s increasing demand forelectricity. The intermittent characteristic of renewable energyand its application through distributed generation resulted ingreat impacts on power quality [1]–[3]. Direct Current (DC)MicroGrids are attracting interest, thanks to their ability toeasily integrate modern loads, renewable sources, and energystorages [4]–[9]. They also acknowledged the fact that mostrenewable energy sources and storages use DC energy (asphotovoltaics (PVs) and batteries for example), and allowthe reduction in the number of power converters in the gridwith simpler topology. By doing this, they increase energyefficiency and allow fast control of the grid.

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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

DC MicroGrids are an innovative solution to be used alsoin transportation systems to integrate the advantages of energystorage utilization [10]–[15]. Indeed, they can help to reach thetarget of increasing the energy saving [16] and the capabilityto compensate strong perturbations [17]. DC MicroGrids rep-resent also a new possibility of integrating a different kind ofenergy resource, which is due to the trains’ braking energyrecovery systems that regenerate this energy by providingnegative torque to the driven wheels. Then, the motor act asa generator, injecting power to the grid. Since the generatedenergy in regenerative braking is free from pollutant emissionand waste, it can be considered as an alternative energyresource [18]–[20]. In a railway station, the regenerated energyis usually transferred to the third rail in order to let nearbytrains utilize it. In case it cannot be used by other trains,it is dissipated on resistors. The purpose of this paper is tointroduce a model and related control methods for a SmartRailway Station able to store the regenerated energy in abattery, allowing the option to become a market participantand sell the energy (maybe providing also ancillary servicesthrough a connection to the main AC grid) or simply to havethe energy available if needed [13].

Several results are available in the scientific literature aboutan energy management system (EMS) for optimal managementof the regenerative energy [12], [21]–[23], but only few resultsfocus on the physical systems’ interconnection and on therelative stability analysis [13], [24], [25]. Indeed, due to thefact that they are well known in the scientific communitybecause of their simplicity, usually linear control techniquesbased on proportional integral (PI) strategies are used: they donot rely on a rigorous stability analysis [26]. Unfortunately,the linear control is limited to a linearized model, which isrestricted to only one operating point. When dealing withstrong perturbations, as the impact of regenerative energy ina MicroGrid, linear techniques may not be enough to solvethe question of general stability in a power system [27]–[29].Thus, a nonlinear control approach is more suitable tothis purpose due to the nonlinear nature of the electricalmodels [30]–[32].

For example, a complete nonlinear model of a DC distrib-ution system driven by PI cascaded droop-based controllersincluding a damping factor is developed in [33], where avalid nonlinear stability analysis is conducted using Lyapunovtechniques. Also, small-signal stability studies are introducedas in [34], where different DC loads and a supercapacitorcompose the DC networks of aircraft. Then, a large-signal-stabilizing study is proposed to ensure global stability bygenerating proper stabilizing power references for the wholesystem. In [35], a simplified model of a small DC Micro-Grid under droop control is addressed to reduce the com-plexity of the nonlinear stability analysis, which is basedon the bifurcation theory, and a relation among grid para-meters is provided. Several strategies for stability analysisand stabilization techniques for DC MicroGrids are presentedin [28] and [29].

In this paper, a DC MicroGrid for a Smart Railway Stationequipped with renewables, as PVs and regenerative system,storage devices, and loads is proposed, which is able to

connect or disconnect to the AC main grid, and the low-levelcontrol laws needed to let the MicroGrid correctly operate areintroduced, together with a complete stability analysis. In theconsidered framework, a device dedicated to train brakingenergy recovery is added together with the conventional PVsource: the targets are to merge regenerate energy from thetrains (that can be very significant) to the one produced byPV and to keep a desired voltage level for the DC bus.The combination of the two renewable sources stresses thesystem with respect to any kind of perturbation can takeplace. As appropriate hypothesis of power availability forthe DC MicroGrid is taken into account, the AC networkdoes not contribute to the DC MicroGrid stability: it isseen as a controllable load to be fed selecting the neededamount of active power and reactive power. The stabilityof the DC MicroGrid is ensured by different time scalestorage devices utilization (batteries and supercapacitors),in order to obtain a flexible and reliable system in responseto the intermittent nature of the renewables: the batterieshave the duty to provide energy when it is missing fromthe renewable sources, while the supercapacitors act to com-pensate the power transient variations in power production orconsumption [36]–[38].

A System-of-Systems approach like the one introducedin [27] is utilized to develop the controllers for letting eachconverter carry out its task at the local level and for performingwhole system stability, similar to [32], [36], and [39]. Here,the topology of some key converters has been reconsidered:since low voltage and high energy density is desired for thefast storage device due to usual standards and economicalreasons, a bidirectional boost converter (presenting nonmin-imum phase characteristics in the control target variables)is connected to the storage in charge of controlling the DCbus [33]. Also, according to the application, reconsiderationson the DC voltage bus value brought to the necessity of theutilization of buck converters for the loads and the regenerativesystem. The emerging model results to be more complexto control with respect to the one in [32], and with thenecessity to use together several control tools as feedbacklinearization [40], dynamic feedback linearization [41], andinput-to-stable stability (ISS) [42]–[44] to prove the stabilityof the system as a whole. Furthermore, with respect to thestability analysis in [36], the here used ISS-like Lyapunovfunction produces less complex to implement control laws.The proposed distributed nonlinear control technique is shownto ensure better performances of the interconnected nonlinearsystem with respect to linear control techniques, impactingpositively the power quality and allowing for the possibilityto perform a system’s operating region analysis [29].

This paper is organized as follows. In Section II,the complete model of the DC MicroGrid grid-connectedis introduced, according to average models and Pulse-WidthModulation (PWM) methods [30]. Then, in Section III,the adopted analysis is carried out to satisfy stability require-ments and to find the dedicated distributed nonlinear con-trol law. Section IV provides the simulation results, with aparagraph dedicated to explanation about how to physicallyunderstand the meaning of the proposed control law and a


PEREZ et al.: STABILITY ANALYSIS OF A DC MICROGRID FOR A SMART RAILWAY STATION INTEGRATING RENEWABLE SOURCES 3

Fig. 1. Considered MicroGrid framework.

comparison with classical linear control including parametricrobustness tests. In Section V, conclusions are provided.

II. DC MICROGRID

The considered DC MicroGrid is depicted in Fig. 1: it iscomposed by two different types of renewable energy sources(braking energy recovery from the trains and PV panels), twokinds of storages acting at different time scales (a batteryand a supercapacitor), a DC load and the connection withthe main AC grid. The target is to assure voltage stabilityin the DC grid and correctly feed power to the load whileabsorbing power from the PV array and the braking energyrecovery system. To each component of the MicroGrid (PVarray, energy recovery system, battery, supercapacitor, andload). a DC/DC converter is used, and a DC/AC one for theAC grid connection. By applying the Kirchhoff law in theMicroGrid circuit in Fig. 1, the dynamical equations for thestate space modeling of the whole system are obtained in (1)–(19)

VC1 = 1

R1C1VS − 1

R1C1VC1 − 1

C1IL3 (1)

VC2 = 1

R2C2Vdc − 1

R2C2VC2 + 1

C2IL3(1 − u1) (2)

IL3 = 1

L3VC1 − 1

L3VC2(1 − u1) − R01

L3IL3 (3)

VC4 = 1

R4C4VB − 1

R4C4VC4 − 1

C4IL6 (4)

VC5 = 1

R5C5Vdc − 1

R5C5VC5 + 1

C5IL6(1 − u2) (5)

IL6 = 1

L6VC4 − 1

L6VC5(1 − u2) − R04

L6IL6 (6)

VC7 = 1

R7C7VPV − 1

R7C7VC7 − 1

C7IL9 (7)

VC8 = 1

R8C8Vdc − 1

R8C8VC8 + 1

C8IL9(1 − u3) (8)

IL9 = 1

L9VC7 − 1

L9VC8(1 − u3) − R08

L9IL9

+ 1

L9(R08 − R07)IL9u3 (9)

VC11 = 1

R11C11VL − 1

R11C11VC11 + 1

C11IL13 (10)

VC12 = 1

R12C12Vdc − 1

R12C12VC12 − 1

C12IL13 u4 (11)

IL13 = − 1

L13VC11 + 1

L13VC12u4 − R011

L13IL13

− 1

L13(R012 − R011)IL13u4 (12)

VC14 = 1

R14C14VT − 1

R14C14VC14 − IL16

C14u5 (13)

VC15 = 1

R15C15Vdc − 1

R15C15VC15 + 1

C15IL16 (14)

IL16 = 1

L16VC14u5 − 1

L16VC15 − R014

L16IL16 (15)

Ild = − Rl

LlIld + ωr Ilq + 1

2LlVC17u6 − Vld

Ll(16)

Ilq = − Rl

LlIlq − ωr Ild + 1

2LlVC17u7 − Vlq

Ll(17)

VC17 = Vdc − VC17

R17C17− 3

2C17

Ild Vld + Ilq Vlq

VC17

(18)

Vdc = 1

C10

[VC2 − Vdc

R2+ VC5 − Vdc

R5+ VC8 − Vdc

R8

+ VC12 − Vdc

R12+ VC15 − Vdc

R15+ VC17 − Vdc

R17

].

(19)



Another important target to be accomplished is to save thebattery lifetime. Since the regenerative system introduces avery high peak current in a very short time period, the batterywould be stressed if in charge of absorbing it and the lifetimewould be significantly reduced. The supercapacitor is chosento absorb this amount of power; indeed, thanks to its differentcharacteristics, its lifetime will not be affected as the batteryone. Power coming from the battery will then be modifiedaccording to the new level of energy in the supercapacitor,and according to a desired charge/discharge rate [45], [46].

Finally, this DC grid is used to provide ancillary servicesof active and reactive powers to an AC grid. Two assumptionsare made: a higher level controller is supposed to provide ref-erences to be accomplished by the local controllers [26], [47];the second one is about a proper sizing of PV array, battery,and supercapacitor in order to have feasible power balancewith respect to the sizing of the load and of the power comingfrom the braking recovery system.

A. Supercapacitor Subsystem

The supercapacitor is an energy storage device, which isused to improve power quality, due to its capability to providea fast response to grid oscillations. It has a high powerdensity and an increased life cycle while having a considerablylow energy density. The combination of such device with aslower one allows to ensure proper control and managementstrategies of the power flow, dealing with the multiple-time-scale characteristic of DC MicroGrids [48]. Indeed, combiningthe battery characteristics with the supercapacitor ones, it ispossible to have the supercapacitor acting to counteract to gridtransient variations, while the battery deals with the powerflow [49]. The considered supercapacitor is composed of 4parallel and 18 series cells with 9 m� of equivalent DCseries resistance, resulting in 50 F of total capacitance, 420 Vof nominal voltage, and has 1.225 kWh of nominal energycapacity.

A bidirectional-boost converter connects the supercapacitorto the DC link. The average state-space model is introducedin (1)–(3): here, VS is the supercapacitor’s voltage, VC1 is thevoltage of capacitor C1, VC2 is the voltage on capacitor C2,and IL3 is the current on inductor L3. R1 and R2 are theresistances representing the cable losses, while R01 and R02are the switch losses of the semiconductors, where R01 = R02,and u1 is the duty cycle of the converter.

In the following, we will consider IL3 as the output of thesubsystem (1)–(3) (relative degree equal to 1), and VC1 andVC2 the zero dynamics. The control target will be to properlylet the voltage VC2 track a desired trajectory, which is V e

C2,

in order to accomplish the aforementioned duty of ensuringvoltage stability. The choice of controlling directly the currentis done according to a common practice in power systemscommunity and the physics of the device. Indeed, to directlycontrol the voltage would bring several difficulties that comefrom the characteristics of the system. These difficulties canbe dealt with by creating a multi-time-scale behavior on thesystem by the control scheme, since the current dynamics canbe designed to be faster than the voltage one [40], [50]–[52].

To directly control the voltage would result in an oscillatingbehavior because of the nonminimum phase characteristicsof the subsystem [40], [53]. Then, the dynamic feedback lin-earization technique is used (see [41], [54]), as in [52].

B. Battery Subsystem

The battery is a storage device which is usually usedas energy reservoir. Here, an ion-lithium battery bank isconsidered. Compared to other battery technologies, this kindof battery has a higher energy density, longer life cycle, andabsence of memory effect [5], [55]. The target of the batteryis to regulate the power flow in the system according to areference given by the secondary control level. A proper sizingis mandatory for the battery to be able to inject or absorb theneeded amount of power. A piecewise constant power supplyis demanded for maximizing the lifetime. The chosen batteryhas 380 V of nominal voltage, current capacity of 1000 Ah,and nominal discharge current of 434 A, resulting in 380 kWhof energy capacity and 165 kW of nominal power.

A bidirectional-boost converter is used to integrate thebattery in the MicroGrid. The average state-space model ofthe battery converter is introduced in (4)-(6): where VB is thebattery’s voltage, VC4 and VC5 are the voltages on capacitorsC4 and C5, respectively, and IL6 is the current on the inductorL6. R4 and R5 are the resistances representing the cable losses,while R04 and R05 are the switch losses of the semiconductors,where R04 = R05. u2 is the duty cycle of the converter.

In the following, we will consider IL6 as the output of thesubsystem (4)–(6), and VC4 and VC5 the zero dynamics. As forthe battery, the relative degree of the system is 1, but since thecontrol target is to properly reach a desired current referencefor IL6 , i.e., I ∗

L6, the classical feedback linearization technique

can be used [40].

C. PV Subsystem

The PV is the main generation in the considered MicroGrid,and it is obviously scaled according to the MicroGrid’s load.The incremental conductance algorithm is applied to track thereference for the PV array in the maximum power point. Thismethod is one of the classical maximum power point track-ing (MPPT) algorithms and is characterized by fast responseand simple design [56], [57]. The MPPT algorithm is chosento be used as the reference to PV’s control strategy optimizingthe power generation in the MicroGrid. The PV array modelis composed by 15 modules in series and 100 modulesin parallel of Kyocera SM48-KSM model, which generates72 kW in nominal conditions (1000 W/m2 irradiation and25 ◦C temperature). Its open-circuit voltage is 342 V.

A boost converter connects the PV to the DC bus. Thestate-space model of the PV converter is presented in (7)–(9):where VPV is the panel’s voltage, VC7 is the voltage oncapacitor C7, VC8 is the voltage on capacitor C8, while IL9 isthe current on the inductor L9. R7 and R8 are the resistancesrepresenting the cable losses, and R07 and R08 are the switchlosses of the semiconductors. u3 is the duty cycle of theconverter.



In the following, IL9 is considered as the output of the sub-system (7)–(9). Since the control target is to follow the currentreference I ∗

L9, similar considerations to the case of the battery

subsystem can be done.

D. Load Subsystem

The general DC load in the MicroGrid represents lights,ventilation, heating, and electrical vehicles, as an aggregationof the power demand in the smart train station. The loadis piecewise time variant, shifting the power according tothe required demand [58], where the voltage deviation is notallowed to exceed the grid limits. The DC load connected via aDC/DC converter is an adequate representation for modern DCdistribution systems, since it represents a significant portion ofthe total load in DC networks [33]. The main load requirementis to maintain its voltage inside of the grid requirements(normally ±5%). The nominal load voltage is set to be 500 Vand the maximum power consumption is 62.5 kW.

The load converter is a buck one. The state-space model isintroduced in (10)–(12): where VL is the load voltage, VC11 andVC12 are the voltages on capacitor C11 and C12, respectively,and IL13 is the current on the inductor L13. R11 and R12 are theresistances representing the cable losses, and R011 and R012are the switch losses of the semiconductors. u4 is the dutycycle of the converter. The load is represented as a variablecurrent source IL with a fixed resistance in parallel RL . Thevariations in the current source represent the load demandvariations and are related to the load voltage as follows:

VL = RL

R11 + RL(VC11 − R11 IL). (20)

In the following, VC11 is the output of the subsystem(10)–(12). According to its structure, the relative degree ofthe subsystem is 2, and there is only a zero dynamics, whichis VC12 . Feedback linearization can be used to properly controlthe voltage VC11 to follow a desired reference V ∗

C11.

E. Train’s Recovery Braking Energy Subsystem

A train line can recover energy through regenerative brakingwhen the train motor occasionally becomes a generator byproducing a counter-torque in the electrical motor [19]. Thebraking energy recovery system is a renewable energy source,and the perturbation it introduces is different with respectto the usually considered ones. Indeed, it introduces a highlevel of current in a short-time period [14], [15], [59] in apredictable way. However, this intermittent high power peakcan cause instability in a MicroGrid context; to avoid it,the supercapacitor has the duty to absorb the transient peak ofenergy [17], [24], [49], [60]. As a consequence, the battery isless stressed.

A voltage source VT around 750 V models the train DC bus;when the regenerative braking takes place, the voltage of thetrain line increases to reach the values around 900 V. A buckconverter is exclusively dedicated to the energy recovery,connecting the train line to the DC MicroGrid. The load ofthe train is fed by one or more others converters and they areindependent: train’s load is not included in the regenerative

braking system. This system recovers the braking energyinstead of wasting it, and also helps the train’s grid to keep itsvoltage inside desired operational margins. The target of theconverter is to keep its voltage to the constant value of thetrain line, resulting in a power injection of approximately 0.5up to 1.0 MW in a few seconds when the regenerative energyis recovered.

The state-space model of the regenerative energy subsystemincluding the buck converter is presented in (13)–(15). Here,VT is the train’s line voltage source, VC14 is the voltage oncapacitor C14, VC15 is the voltage on capacitor C15, and IL16

is the current on inductor L16. R14 and R15 are the resistancesrepresenting the cable losses, and R014 and R015 are the switchlosses of the semiconductors, considering: R014 = R015. u5 isthe duty cycle of the converter.

In the following, we will consider IL16 as the output ofthe subsystem (13)–(15), and the two voltages are the zerodynamics. Since the control target is to track a constantreference for VC14 , similar considerations for the case of thesupercapacitor subsystem are done, and dynamic feedbacklinearization is used [52].

F. Connection to the AC Grid

A connection with a main AC grid is included in theconsidered MicroGrid, with the possibility to supply activeand reactive powers according to the grid demand. From theMicroGrid point of view, it results as to supply an AC load,ensuring power supply to the network. The connection is donethrough a three-phase AC bus of 400 V rms and ωr = 50 Hzof the fundamental frequency. The equivalent impedance ofthe grid is Ll = 0.5 mH and Rl = 2 m�, and the short-circuitpower is Psc = 50 MVA.

Equations (16)–(18) represent the state-space model of theVoltage Source Converter (VSC) connecting the MicroGridwith the main grid. The synchronous dq reference frameis chosen such that the d-axis is fixed to the AC voltage,i.e., Vld = Vl,rms and Vlq = 0. Ild and Ilq are the directand quadratic currents in the AC line, respectively, and Vld

and Vlq are the related direct and quadratic voltages on theAC grid using Park’s transformation [61]. VC17 is the voltageon capacitor C17, and R17 is the cable losses in the VSC. Themodulation indexes u6 and u7 are the control inputs: they arebounded by (u2

6 + u27)

1/2 ≤ 1.In the following, Ild and Ilq will be the outputs of the sub-

systems, and VC17 the zero dynamics. Feedback linearization isused to meet the control target, i.e., to let the outputs reach twodecoupled current references I ∗

ld and I ∗lq . They are obtained by

given power references P∗l and Q∗

l , as in [51]

I ∗ld = 2

3

P∗l

Vld, I ∗

lq = −2

3

Q∗l

Vld. (21)

G. DC Bus

The DC bus is the point of common coupling (PCC) amongthe devices, and it represents the interconnection of differentsubsystems. This node is represented by the capacitor C10in Fig. 1, and its dynamical behavior Vdc is introduced in (19).



It is not directly controllable by any control inputs; then,we will consider it as a zero dynamics of the whole system.The control target is to fix a certain reference voltage V ∗

dc forthe DC bus; the control law of the supercapacitor subsystemwill be developed to this purpose.

H. Hierarchical Control Structure

A hierarchical control structure is considered in thiswork [26], [28]. While the low-level controllers for the con-verters are here introduced, their needed references are sup-posed to be provided by a higher secondary level controller,which development is out of the scope of this work. Thereferences for the voltages of DC bus, DC load, and regen-erative system are fixed and chosen a priori: they are V ∗

dc,V ∗

C11, and V ∗

C14, respectively. The current reference I ∗

L9for the

PV is given by the incremental conductance MPPT algorithm,in order to address the maximum power point for the PV. Thereference of active and reactive powers to be provided to theAC grid are I ∗

ld and I ∗lq , while the one for the battery is I ∗

L6.

These references are supposed to be calculated with an optimalpower flow controller, considering grid codes, the physicalconstraints of the devices given by their sizes and the needsof the MicroGrids, as in [47] and [62]. The reference vector is

r = [I ∗

L6I ∗

L9V ∗

dc V ∗C11

V ∗C14

I ∗ld I ∗

lq

]. (22)

Part of the contribution of this paper relies on the develop-ment of control laws for stabilizing the dynamics introducedin (1)–(19) with respect to given references, introduced in (22).Furthermore, a complete rigorous analysis of the DC Micro-Grid is implemented, proving the stability of the whole systemand answering to the following problem.

Problem 1: Given the equations in (1)–(19), and supposingthe given references in (22) to fulfill the steady-state stabilityconditions (i.e., to correctly satisfy power balance in steady-state), the development of dedicated control law u1, . . . , u7such to ensure grid stability is needed.

III. CONTROL STRATEGY

This section describes the control design for each subsystemcomposing the entire system and provides a stabilizationanalysis for the whole grid.

A. Preliminaries

Let us consider the extended state x , the respective equilib-

rium point xe, the output y and the error x�= x − xe

x = [ξ1 ξ2 η μ ζ ]T (23)

xe = [ξ e1 ξ e

2 ηe μe ζ e ]T (24)

y = [yμ yξ1 yξ2 ]T (25)

where

ξ1 = [IL6 α6 IL9 α9 Ild αd Ilq αq ] (26)

ξ2 = [VC11 α11 IL13α13 VC14 VC14 IL16 α16] (27)

η = [VC5 VC8 VC12 VC15 VC17 Vdc] (28)

μ = [VC2 VC2 IL3 α3 ], ζ = [VC1 VC4 VC7] (29)

ξ e1 = [

I ∗L6

0 I ∗L9

0 I ∗ld 0 I ∗

lq 0]

(30)

ξ e2 = [

V ∗C11

0 I eL13

0 V ∗C14

0 I eL16

0]

(31)

ηe = [V ∗

C5V ∗

C8V ∗

C12V ∗

C15V ∗

C17V ∗

dc

](32)

μe = [V e

C2V e

C2I e

L30

], ζ e = [

V ∗C1

V ∗C4

V ∗C7

](33)

yμ = [y1] = [IL3], yξ2 = [y6 y7] = [VC11 IL16

](34)

yξ1 = [y2 y3 y4 y5] = [IL6 IL9 Ild Ilq]. (35)

The equations of IL J , with J = {3, 6, 9, 13, 16, d, q},and VCI , with I = {1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17},are introduced in (1)–(19), while the terms αH , with H ={3, 6, 9, 11, 13, 16, d, q}, are integral terms ensuring zero errorin steady state and are governed by the following equations:

α3 = IL3 − I eL3

; α6 = IL6 − I ∗L6

(36)

α9 = IL9 − I ∗L9

; α11 = VC11 − V ∗C11

(37)

α13 = IL13 − I eL13

; α16 = IL16 − I eL16

(38)

αd = Ild − I ∗ld ; αq = Ilq − I ∗

lq . (39)

Here, ξ1 is composed by the directly controllable dynamicsthat have a fixed reference and are the outputs of their subsys-tems, together with their related integral error terms. ξ2 is com-posed by the controllable dynamics of subsystems (10)–(12)(using feedback linearization) and (13)–(15) (using dynamicalfeedback linearization), the related integral error terms andthe time derivative of VC14 , which is needed for the controlpurposes. Moreover, η is composed by the zero dynamicsdealing with the system interconnection. Then, μ is composedby the controllable dynamics of the subsystem (1)–(3) (usingdynamical feedback linearization), an integral error term andthe time derivative of VC2 . Finally, ζ is composed by theremaining zero dynamics.

The values of the equilibrium point are either providedby the higher level controller in (22), either obtained bysolving (1)–(19), for VCI , I = {1, 4, 5, 7, 8, 12, 15, 17}, and(36)–(39) in steady state, given (22). Finally, the trajecto-ries V e

C2, I e

L3, I e

L13, and I e

L16are obtained by backstepping

(V eC2

), feedback linearization (I eL13

), and dynamical feedbacklinearization (I e

L3and I e

L16) techniques to meet the control

purpose

V eC2

= Vdc + K10

Vdc

(V ∗

dc2 − V 2

dc

) − R2�dc

− R2

Vdc[�5 + �8 + �12 + �15 + �17 + �W ]

(40)

I eL3

=∫

1

Lg2 L1f2

(VC2

) [θd − L2

f2

(VC2

)]dt (41)

I eL13

= VC11 − VL

R11− C11

[K11

(VC11 − V ∗

C11

) + K α11α11

](42)

I eL16

=∫

1

Lg14 L1f14

(VC14)

[θt − L2

f14

(VC14

)]dt (43)

with

�5 = (VC5 − V e

C5

) [Vdc − VC5

R5+ IL6(1 − u2)

](44)



�8 = (VC8 − V e

C8

) [Vdc − VC8

R8+ IL9(1 − u3)

](45)

�12 = (VC12 − V e

C12

) [Vdc − VC12

R12− IL13u4

](46)

�15 = (VC15 − V e

C15

) [Vdc − VC15

R15+ IL16

](47)

�17 = (VC17 − V e

C17

) [Vdc − VC17

R17− 3(Ild Vld + Ilq Vlq )

2VC17

]

(48)

�dc = VC5 − Vdc

R5+ VC8 − Vdc

R8+ VC12 − Vdc

R12

+ VC15 − Vdc

R15+ VC17 − Vdc

R17(49)

�W = −(VC5 − V e

C5

)2

R5−

(VC8 − V e

C8

)2

R8−

(VC12 − V e

C12

)2

R12

−(VC15 − V e

C15

)2

R15−

(VC17 − V e

C17

)2

R17(50)

where

Lg2 L1f2

(VC2

) = 1

C2VC2

(VC1 − 2R01 I e

L3

)(51)

Lg14 L1f14

(VC14

) = 1

C14VC14

(VC15 + 2R014 I e

L16

)(52)

L2f2(VC2) = I e

L3

C2VC2

VC1 + Vdc

R2C2

−[

1

R2C2+ VC1 I e

L3− R01 I e

L3

2

C2VC22

]VC2

(53)

L2f14

(VC14

) = I eL16

C14VC14

VC15

−[

1

R14C14− VC15I

eL16

+ R014 I eL16

2

C14VC142

]VC14

(54)

θd = V eC2

− K2(VC2 − V e

C2

) − K α2 (VC2 − V e

C2)

(55)

θt = −K14(VC14

) − K α14

(VC14 − V ∗

C14

). (56)

For lack of space, no further details will be provided on theselection process of these trajectories. An explanation will begiven in the formal stability proof, which is introduced in aconstructive way. However, details on the used techniques canbe found in [40] and [41].

B. Control Inputs

The control inputs u1, u2, u3, u4, u5, u6, u7 are designedby feedback linearization, dynamical feedback linearization,and Lyapunov technique as follows. For the lack of space,the design procedure is not introduced in detail (see [40]);however, some calculations are provided together with thecomplete stability analysis. The developed control laws are

u1 = 1 + 1

VC2

[L3v3 − VC1 + R01 IL3] (57)

u2 = 1 + 1

VC5

[L6v6 − VC4 + R04 IL6] (58)

u3 = L9v9 − VC7 + VC8 + R08 IL9

VC8 − (R08 − R07)IL9

(59)

u4 = L13v4 + VC11 + R011 IL13

VC12 − (R012 − R011)IL13

(60)

u5 = 1

VC14

[L16v16 + VC15 + R014 IL16] (61)

u6 = 2

VC17

[Llvid + Rl Ild − ωr Ll Ilq + Vld ] (62)

u7 = 2

VC17

[Llviq + Rl Ilq + ωr Ll Ild + Vlq ] (63)

with

v3 = I eL3

− K3(IL3 − I e

L3

) − K α3 α3 (64)

v6 = −K6(IL6 − I ∗

L6

) − K α6 α6 (65)

v9 = −K9(IL9 − I ∗

L9

) − K α9 α9 (66)

v4 = v13 +(

1

R11− C11 K11

)v11

−C11K α11

(VC11 − V ∗

C11

)(67)

v11 = −K11(VC11 − V ∗

C11

) − K α11α11 (68)

v13 = I eL3

− K13(IL13 − I e

L13

) − K α13α13 (69)

v16 = I eL16

−K16(IL16 − I e

L16

) − K α16α16 (70)

vid = −Kd(Ild − I ∗

ld

) − K αd αd (71)

viq = −Kq(Ilq − I ∗

lq

) − K αq αq . (72)

The control inputs are bounded: ui ∈ [0, 1], i ∈{1, 2, 3, 4, 5}, and (u2

6 + u27)

1/2 ≤ 1. Let us consider the set�xe of all possible values of the equilibrium point xe andthe set �K of all possible positive values of the gains Ki ,K α

i , i ∈ {3, 6, 9, 10, 11, 13, 16, d, q}, such that the physicalconstraints of the converters are not violate. Given xe ∈ �xe ,let us consider the set �RL of all the possible values of RL

and IL such that the condition of balance of power is satisfied.Such condition can be expressed by

V eC2

− V ∗dc

R2+ V ∗

C5− V ∗

dc

R5+ V ∗

C8− V ∗

dc

R8

+ V ∗C12

− V ∗dc

R12+ V ∗

C15− V ∗

dc

R15+ V ∗

C17− V ∗

dc

R17= 0. (73)

Let us, furthermore, consider the set �x of any evolutionof x satisfying for each t the conditions

VC2 > 0; VC5 > 0; VC8 − (R08 − R07)IL9 > 0

VC14 > 0; VC17 > 0; VC12 − (R012 − R011)IL13 > 0. (74)

C. Stability Analysis

The main result can now be formulated as follows.Theorem 1: Given xe ∈ �xe , RL , IL ∈ �RL , Ki , K α

i ∈�K , the control laws u1, u2, u3, u4, u5, u6, u7 introducedin (57)–(63) solve the Problem 1, since there exist suitablefunctions ω ∈ KL and γ ∈ K such that

�x(t)� ≤ ω(x(0), t) + γ(V ∗

dc

)(75)

provided that x ∈ �x .



Proof: The proof is based on the developmentof a Lyapunov function W for the whole system x(see [27], [32], [36]). Such Lyapunov function is composedby several Lyapunov functions

W (x(t)) = Wx = Wξ1+ Wξ2

+ Wη + Wμ + Wζ > 0. (76)

In the following, the proof is structured according tothe introduction of the extended states ξ1, ξ2, η, μ, andζ . For each of them, a Lyapunov function is provided,aiming at showing stability for the overall system. Wξ1and Wξ2

will be the first pieces, since their dynamics aredirectly controllable with respect to a fixed given reference.The focus will move on the zero dynamics involved inthe systems’ interconnection, with Wη, and how to controlsuch interconnection, with Wμ. Finally, the stability analy-sis is provided for the remaining zero dynamics consideredby Wζ .

Classical input–output feedback linearization techniqueis used to calculate the control inputs in (58), (59), (62),and (63), according to the output yξ1 (see [40]). Consequently,the following Lyapunov function Wξ1

can be used to provestability of ξ1:

Wξ1= W6 + W9 + Wd,q > 0 (77)

where

W j = 1

2

(IL j − I ∗

L j

)2 + K αj

2α2

j > 0, j ∈ {6, 9} (78)

Wd,q = 1

2

((Ild − I ∗

ld

)2 + (Ilq − I ∗

lq

)2

+ K αd α2

d + K αq α2

q

)> 0. (79)

Using the control inputs in (58), (59), (62), and (63),the related time derivatives are calculated as

W j = −K j(IL j − I ∗

L j

)2 ≤ 0, j ∈ {6, 9} (80)

Wd,q = −Kd(Ild − I ∗

ld

)2 − Kq(Ilq − I ∗

lq

)2 ≤ 0. (81)

From W6 ≤ 0, it results W6(t) ≤ W6(0), which impliesthat IL6 − I ∗

L6and α6 are bounded, thanks to Lyapunov

theorem. Consequently, it can be shown that W6 is boundedas well. Then, W6 is uniformly continuous in time. At thispoint, by applying Barbalat’s lemma, it is possible to establishthat W6 → 0 as t → ∞: ergo (IL6 − I ∗

L6) → 0 and

α6 → 0 [40]. The same is valid for W9 and Wd,q with respectto W9 and Wd,q .

Asymptotic stability is then proved for the errors con-sidered in (77), thanks to (80), (81) and Barbalat’s lemma.We will continue in a similar way for the remaining Lyapunovfunctions.

We continue the analysis by focusing on Wξ2

Wξ2= W11,13 + W14 + W16. (82)

First, classical input–output feedback linearization is usedfor obtaining the control input in (60) that leads to a positivedefinite W11,13 with a negative semidefinite W11,13, ensuring

stability according to the same argumentations used for (78),(79), (80), and (81)

W11,13 = 1

2

(VC11 − V ∗

C11

)2 + K α11

2α2

11

+1

2

(IL13 − I e

L13

)2 + K α13

2α2

13 > 0 (83)

W11,13 = −K11(VC11 − V ∗

C11

)2 − K13(IL13 − I e

L13

)2 ≤ 0.

(84)

Next, the dynamical feedback linearization is used to calcu-late the control input (61) based on (43); indeed, the referenceI e

L16in (43) is calculated in a way such there exists W14 > 0

with a W14 < 0, where

W14 = 1

2

(VC14 − V ∗

C14

)2 + 1

2V 2

C14> 0. (85)

This can be shown using a change of variable, wherez1 = VC14 and z2 = VC14 . The resulting system would be⎧⎨

⎩z1 = z2

z2 = L2f14

+ Lg14 L1f14

d I eL16

dt

(86)

and, considering (d I eL16

/dt) as a control input, the chosenvalue in (43) allows to obtain a linear stable system withrespect to the desired equilibrium{

z1 = z2

z2 = −K14z2 − K α14

(z1 − z∗

1

).

(87)

From (87), it is possible to calculate the proper timederivative of the Lyapunov function in (85) such that

W14 < 0. (88)

Then, the control input in (61) correctly operates the track-ing of the calculated reference in (43), as it can be shownby calculating the time derivative of the following Lyapunovfunction W16:

W16 = 1

2

(IL16 − I e

L16

)2 + 1

2K α

16α216 > 0 (89)

W16 = −K16(IL16 − I e

L16

)2 ≤ 0. (90)

In the same way as for (78) and (80), in the case ofW16, asymptotic stability can be proven by Barbalat’s lemmautilization with the calculation of W16.

The control input u1 represents the remaining degree offreedom of the system: it is then designed with the purpose tocontrol the interconnection of the whole system. To analyzethe stability of such interconnection, the Lyapunov functionWη is introduced to perform calculations of the reference forVC2 to be applied by u1 in order to control the interconnections

Wη = C10

2V 2

dc + C5

2

(VC5 − V ∗

C5

)2

+C8

2

(VC8 − V ∗

C8

)2 + C12

2

(VC12 − V ∗

C12

)2

+C15

2

(VC15 − V ∗

C15

)2 + C17

2

(VC17 − V ∗

C17

)2> 0.

(91)



The time derivative can be calculated as

Wη = Vdc

(1

R2(VC2 − Vdc) + �dc

)

+ �5 + �8 + �12 + �15 + �17 (92)

where the term representing the available degree of freedom,i.e., VC2 , is clearly indicated, and the others are definedin (44)–(49). Then, by backstepping, supposing the dynamicsof VC2 matches its reference calculated in (40), the functionWη can be written as

Wη = K10

R2V ∗

dc2 − K10

R2V 2

dc −(VC5 − V ∗

C5

)2

R5

−(VC8 − V ∗

C8

)2

R8−

(VC12 − V ∗

C12

)2

R12

−(VC15 − V ∗

C15

)2

R15−

(VC17 − V ∗

C17

)2

R17. (93)

The Lyapunov function in (91) results to be a ISS-likeLyapunov function [42], [63], [64], as explained at the end ofthe proof.

Here, VC2 is considered as the control input, and by trackingits reference trajectory V e

C2in (40), it can regulate Vdc and

assure stability for the considered dynamics. The extendedstate μ is studied to obtain convergence of VC2 to V e

C2. Then,

using the control input in (57), the trajectory in (41) anddefining a Lyapunov function Wμ as

Wμ = W2 + W3 (94)

where

W2 = 1

2

(VC2 − V e

C2

)2 + 1

2

(dVC2

dt− dV e

C2

dt

)2

> 0 (95)

W3 = 1

2

(IL3 −

∫ d I eL3

dt

)2

+ 1

2K α

3 α23 > 0. (96)

Stability is proven with arguments similar to the ones usedfor (85), (86), (87), and (88), considering a state-dependenttrajectory and its time derivatives V e

C2and V e

C2, obtaining

W2 < 0; W3 = −(

IL3 −∫ d I e

L3

dt

)2

≤ 0. (97)

Let us finally focus on the states that represent the remainingzero dynamics of the system. To show their stability, the Lya-punov function Wζ is used and the controlled dynamics areconsidered already on their equilibrium point

Wζ = W1 + W4 + W7 (98)

W j = 1

2C j

(VC j − V ∗

C j

)2> 0, j ∈ {1, 4, 7}. (99)

By (1), (4), and (7), it is easy to show that

W j = − 1

R j

(VC j − V ∗

C j

)2< 0, j ∈ {1, 4, 7} (100)

assuring asymptotic stability for these zero dynamics.From (77), (82), (91), (94), and (98), we can state that there

exist functions α, α ∈ K∞ such that α(|x |) ≤ Wx (η) ≤ α(|x |)

TABLE I

MICROGRID PARAMETERS

for Wx in (76). From (80), (81), (84), (88), (90), (93), (97),and (100), we can state that

Wx(x, V ∗

dc

) ≤ −βx(|x |) + γx(|V ∗

dc|)

(101)

with βx , γx ∈ K∞, and V ∗dc playing the role of a virtual

input [39], [42], [63]. From (101), inequality in (75) follows.�

Remark 1: ISS stability is practically accepted for the pro-posed application domain [65], [66]. However, the strongerstability results can be derived from the proposed analysisby the passivity-based considerations, as in [33] and [67].Also, the domain of attraction and the convergence to theequilibrium can be better defined as depending on the pro-posed Lyapunov function and the control gains, accordingto [43], [44]. Numerical methods as the one proposed in [68]can be used for computing the reachable set.

IV. SIMULATION RESULTS

The DC MicroGrid model for a Smart Railway Stationproposed in Fig. 1 has been implemented with the SimPower-Systems toolbox of MATLAB/Simulink. The used parameters’values are introduced in Table I. The switching frequency ofthe DC/DC converters is 10 kHz while the VSC converteris 20 kHz. Here, a simulation of the introduced nonlinearcontrol laws is presented, and a comparison with the classicallinear technique is proposed. The simulation time is 30 s:during this time, both the conditions of having or not havingregenerative energy from the braking recovery system isproposed.

A. Proposed Nonlinear Control

Here, the proposed nonlinear control is shown to be capableto stabilize the DC MicroGrid. Fig. 2 introduces the incidentirradiation on the PV system and the current variation inthe DC load, respectively. They represent, in a realistic way,the time-varying disturbances impacting the system. Thesevariations are used to test the control system performancewhen several nonlinearities characteristics take place. Paneltemperature is kept constant during the simulation time; thecurrent values are negative since the load is absorbing powerfrom the MicroGrid.

The voltages VS , VB , VPV , and VT , of supercapacitor,battery, PV array, and train, respectively, are depicted in Fig. 3.While VS has important variations, highlighting the charge and



Fig. 2. PV incident irradiance and the demanded dc load current, respectively.

Fig. 3. Voltages VS , VB , VPV , VT , of the supercapacitor, battery, PV array,and train, respectively.

Fig. 4. Currents IL3 , IL6 , and IL9 and their references I eL3

, I∗L6

, and I∗L9

.

discharge of the supercapacitor, VB is not affected by the sameproblems as expected, since the references provided by thehigher level controller allow for a lower charge/discharge rate.The voltage VPV varies according to the incident irradiationin Fig. 2. The voltage VT produces a high-voltage peak in ashort-time period due to the regenerative braking.

The currents IL3 , IL6 , and IL9 , respectively, related to thesupercapacitor, battery, and PV array subsystem, are depictedin Fig. 4. They are shown to correctly follow their referencesgiven by the higher level controller, as for I ∗

L6from the power

flow model or I ∗L9

by the MPPT algorithm, or calculated suchthat to keep DC bus voltage stability, as for I e

L3, which depends

on the desired value of V eC2

for VC2 (see Fig. 5). The referenceV e

C2is calculated for having a proper regulation of the voltage

in the DC bus, and it is a function of several state variables,as shown in (40), and on the way it is controlled, as shownin (41).

The obtained reference V eC2

, together with the consequentdynamics of IL3 , is shown to correctly be capable to perform

Fig. 5. Voltage VC2 and its nominal reference V eC2

.

Fig. 6. DC bus voltage Vdc and its reference V ∗dc.

Fig. 7. DC load voltage VC11 and its reference V ∗C11

.

Fig. 8. Currents IL13 , IL16 , and IR17 with their respective references I eL13

and ILe16

.

DC bus voltage regulation to the desired value V ∗dc in Fig. 6.

The effects of such fast control response to the grid variationsare such that the transient peaks are lower than 2.4% of thenominal value.

To better highlight the quality of the proposed controlaction, a more detailed description of the involved powersources and power load are needed. The DC load profile hasbeen introduced in Fig. 2. Fig. 7 shows the effectiveness ofthe proposed control action for DC load voltage regulation,resulting in the requested current IL13 depicted in Fig. 8.It is calculated according to the current demanded by DCload. The obtained load voltage successfully meets the gridcode requirements (±5%). Fig. 8 also describes the renewablepower profile of the braking regenerative system, IL16 , andthe current flowing to the AC grid through the VSC, IR17 .The profiles of the current IL16 and its reference I e

L16are a

consequence of the desired reference for VC14 , i.e., V ∗C14

, andthe proper control of such dynamics (see Fig. 9). Indeed, sinceVT increases very fast during brake recovery peaks, to control



Fig. 9. Voltage VC14 and its reference V ∗C14

ensuring the injection of powerfrom the regenerative system.

Fig. 10. Direct and quadrature currents Ild and Ilq with their references I∗ld

and I∗lq , respectively.

Fig. 11. AC bus voltage and the injected current into AC grid in [p.u.].

the voltage VC14 to a constant reference V ∗C14

equal to the samevalue of VT when there is no regenerative energy means toabsorb the whole injected power, obtaining a desired currentprofile as the one of I e

L16in Fig. 8. To better understand the

possible risks of grid instability due to it, it is important toremark that the injected regenerative power is much higherthan the one normally provided by the PV array.

The current IR17 in Fig. 8 is given by the required direct(Ild ) and quadratic (Ilq ) current to supply the AC grid. Fig. 10introduces the references I ∗

ld and I ∗lq given by the second-level

controller and shows how they are perfectly tracked by therelated dynamics. Fig. 11 depicts the rms voltage and currentin the AC side of the main grid in per unit (Sbase = 100 kVAand Vbase = 400 V). The current injected in AC grid is directlyrelated to the desired active power injected in the grid as shownin (21) . The AC voltage is kept constant since the AC bus isconsidered to be a strong grid.

Finally, the zero dynamics are introduced to verify theirstability. The voltage dynamics VC1 , VC4 , and VC7 are depictedin Fig. 12. Their variations are related to the dynamics of VS ,VB , VPV , and IL3 , IL6 , IL9 in Figs. 3 and 4, according to thepower injected/absorbed by the devices. The remaining zerodynamics are depicted in Fig. 13. They are the output voltagesof the converters connecting the devices (battery, PV, DC load,train, and AC grid) into the MicroGrid by the DC bus.As for the ones previously described, their dynamical behavior

Fig. 12. Dynamics of VC1 , VC4 , and VC7 , respectively.

Fig. 13. Output voltages VC5 , VC8 , VC12 , VC15 , and VC17 on the MicroGridconverters (zero dynamics).

depends on the ongoing power flow. Fig. 13 characterizes theirstable behavior.

Once that the whole dynamics are introduced, and a fullknowledge of the disturbances acting on the MicroGrid hasbeen acquired, we would like to better highlight the success ofthe proposed control action. Indeed, Fig. 6 shows how the pro-posed nonlinear control performs voltage stability accordingto the grid codes and with a proper margin of error under thewhole set of adverse circumstances, as the regenerate powerinput, AC and DC load power variations, PV and batteryoperating point variations. In the next, a further remark aboutthe importance of dealing with nonlinear control is proposed.

B. Control Comparison: Linear Versus Nonlinear

A very common control strategy applied to power convertersand to MicroGrids in general, both in academia and in industry,is the linear PI technique. However, in this work, the impor-tance of using nonlinear techniques is addressed. This isbecause the PI technique works around a linearized operationpoint, which suffers of great limitations when compared tothe nonlinear control techniques. In this section, a comparisonbetween the two control strategies is done, with the objectiveto compare the control performance and the related limitations.

For the lack of space, a full comparison concerning thewhole set of variable is not proposed. However, since DC busvoltage stability is the most important requirement, the com-parison in Fig. 14 will focus on this dynamics, which is thekey for the good interconnection of the subsystems. Fig. 14shows the comparison of different voltage behaviors in the DCbus when using the proposed nonlinear controller (in red) andthe classical PI controller (in blue). As it is possible to state,when implementing the introduced nonlinear control, the DCbus dynamics presents smaller variations with respect to the



Fig. 14. Comparison of the DC bus dynamics when the whole system iscontrolled by simple PI (blue curve) and by the introduced nonlinear technique(red curve).

Fig. 15. Zoomed-in view on Vdc to compare PI and nonlinear control in themost critical transients.

ones with the simpler PI control: indeed, the PI control hashigher overshoots. This is better highlighted in Fig. 15, wherea zoom of the comparison for the two most critical transientsis depicted. It is clearly shown that the nonlinear control hasa smaller overshoot and a faster convergence rate than the PIcontrol. Also, since the nonlinear control considers differentnonlinearities, it has a larger operating region and there isno need to tune the gains continuously according to differentoperating points.

Therefore, the proposed nonlinear control is shown tobetter perform when dealing with the system interconnection.A better controller is very important for improving energyefficiency, which is relevant for renewable energy systemsintegration in a MicroGrid. However, the most interestingpoint on this comparison is the fact that in an extensive setof simulations carried out exploring the effects of severaldisturbances, it was necessary to readjust the PI parametersfor each case. While in the same simulations, the nonlinearcontrol has always kept the same tunning. In the same way,the nonlinear tunning is a trivial pole placement problem forall elements, while for the PI, it is necessary to use oneof the standard methods like root locus, but with complexinterferences between one controller setting to the others.Indeed, this is much easier tunning procedure, and the factthat it is valid for the whole operation region is a part of themost interesting advantages of the nonlinear control.

C. Robustness

Simulations to the purpose to test the robustness of theproposed control laws are here introduced. The presence oferrors for the value of the grid parameters as resistances,inductors, and capacitors is treated. Two cases are described:in the first one, the parameters have a value of −20%smaller than nominal one, while in the second case, the valueis +20% larger than the nominal value. The consideredgrid parameters are: R2, C1, C2, L3, R5, C4, C5, L6,

Fig. 16. Voltage Vdc when comparing PI and nonlinear control withparametric errors of +20% and −20%.

Fig. 17. Zoomed-in view on Vdc in Fig. 16 to compare PI and nonlinearcontrol in the most critical transients.

R8, C7, C8, L9, R12, C11, C12, L13, R15, C14, C15, L16,R17 and C17.

The simulations are performed for both the PI and theproposed nonlinear control, in order to better introduce aproper comparison. Fig. 16 introduces the behavior of theDC bus voltage Vdc considering the parametric errors: as itis possible to state, the transients present higher peaks forboth control techniques, but the nonlinear approach has betterperformances compared with the PI one because of the smallerovershoots.

A zoom on the highest transients in the proposed robustnesstest for both control strategies is depicted in Fig. 17. It isclearly possible to see that the proposed nonlinear strategypresents smaller overshoots and a faster response than thetraditional PI strategy even with considerable parametricalerrors. We can conclude that the nonlinear controller showsgood robustness properties in these simulations.

The improvements of the nonlinear controller over the PIcan be better remarked in Fig. 18, where the parametric errorsare extended to be of +25% higher than the nominal value.As it is possible to see, in this case, the voltage on the DC busfor the PI control diverges after the regenerative train brakingtakes place, while the nonlinear controller keeps a goodperformance. This behavior is highlighted in Fig. 19, where afirst zoomed-in view of Fig. 18 in the most critical transient isproposed, still highlighting the better control performance ofthe nonlinear control versus the PI, and with a second zoomon the beginning of the unstable behavior of the PI strategy.An interesting point to be addressed is the fact that the PIneeds to readjust the control gains and parameters to keepfollowing the grid reference and to correctly perform stability,while the nonlinear gains can work in a broader region ofattraction and, consequently, need less tuning. Therefore, it ispossible to conclude that the nonlinear controller can standlarger disturbances, like the braking energy recovery, and thatit is able to operate in a wider region with more robust



Fig. 18. Voltage Vdc in case of parametric error of +25% higher than thenominal value.

Fig. 19. Zoomed-in view of Fig. 18 in the highest transients, showingthe unstable behavior of PI control after the high peak transient due to theregenerative braking.

performances toward parametrical errors when compared withthe standard PI control.

V. CONCLUSION

This paper addresses the problem of controlling agrid-connected DC MicroGrid for a Smart Railway Station.The proposed MicroGrid is used to absorb the train brakingregenerative power, which constitutes a very large and suddenpeak of power that is difficult to address by classical linearcontrollers. This MicroGrid is also used to integrate distributedgeneration as PV arrays. The proposed system is based ontwo energy storages with different time scales, i.e., a batteryand a supercapacitor. The MicroGrid is then completed witha variable DC load (that represents the aggregation of allDC loads as lights, heating, and electric vehicles) and aconnection with the AC grid. The proposed DC MicroGrid iscontrolled by low-level distributed nonlinear controllers. Foreach subsystem composing the MicroGrid and for each relatedtarget, a control strategy is provided according to a hierarchicalcontrol structure and a System-of-Systems approach.

The nonlinear control laws are developed according to arigorous analysis based on an ISS-like Lyapunov function,which provides the possibility to let the MicroGrid operateunder several adverse circumstances, as for example, the highperturbations brought by the train’s braking recovery energysystem or the load variations. Both theoretical and numericalresults show the capability of the proposed approach to handlethe complex scenarios given by the mix of different renewablesources, loads, and storage devices.

As future steps, the proposed algorithms will be first imple-mented in the laboratory test bench, and in the following, theywill be tested in a real subway station.

ACKNOWLEDGMENT

The authors would like to thank Prof. P. Pepe for the fruitfuldiscussion and the anonymous reviewers for their helpful andconstructive comments that greatly contributed to improvingthe final version of this paper.

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Filipe Perez received the B.Sc. degree in electricalengineering from the Federal University of Viçosa,Viçosa, Brazil, in 2014, and the M.Sc. degree in elec-tric power systems from the Federal University ofItajubá, Itajubá, Brazil, in 2015. He is currently pur-suing the double Ph.D. degree in nonlinear controlof microgrids with Paris-Saclay University, Gif-Sur-Yvette, France, and Federal University of Itajubá.

His current research interests include nonlinearcontrol applied to microgrids and stability of powersystems, voltage and frequency regulations, control

and synchronization of power converters, integration of renewable energies,management of energy storage systems, distributed generation, and powerquality in the smart-grid context.

Alessio Iovine (M’18) received the B.Sc. andM.Sc. degrees in electrical engineering and com-puter science from the University of L’Aquila,L’Aquila, Italy, in 2010 and 2012, respectively,and the European Doctorate degree in informationscience and engineering from the University ofL’Aquila, L’Aquila, Italy, in collaboration with Cen-traleSupélec, Paris-Saclay University, Paris, France,in 2016.

He was a Post-Doctoral Researcher with the Uni-versity of L’Aquila, a Visiting Researcher with

Paris-Saclay University, and a Post-Doctoral Researcher with the Efficacity,Research and Development Center, Paris. He currently holds a post-doctoralposition with the University of California at Berkeley, Berkeley, CA, USA.His current research interests include advanced control methods for power andenergy systems and traffic control, with smart-grids, integration of renewablesand storage devices, autonomous vehicles and cooperative intelligent trans-portation systems as core applications.

Gilney Damm (M’10) received the bachelor’sdegree in electronic engineering and the M.Sc.degree from COPPE-Rio de Janeiro Federal Uni-versity (UFRJ), Rio de Janeiro, Brazil, in 1995 and1997, respectively, and the Ph.D. degree from Sup-elec/Université Paris Sud, Orsay, France, in 2001,and the Habilitation à Diriger des Recherches fromUniversité d’Evry Val d’Essonne, Évry, France,in 2010.

He is currently an Associate Professor with theIBISC and L2S Laboratories, Paris-Saclay Univer-

sity, Courcouronnes, France. He is also a Partner of the Energy TransitionInstitute (ITE) Efficacity, where he is a Leading Scientist of the Urban

SmartGrids and MicroGrids Pole, including multi-energy vectors, and theintegration of transportation and electric networks. He is also a Partner of theITE SuperGrid, where he works on the control of large power networks. He isalso Co-Leader of the French research forum on SmartGrids, and a Partner ofthe International Associated Laboratory (LIA-CNRS) SmartGrids with Canadaand PowerGrids with Brazil. His current research interests include nonlinearcontrol and observers applied to power systems (SmartGrids-SuperGrid-MicroGrids), applications are in the field of large scale renewable energyintegration, multi-terminal HVdc systems, variable speed pumped storageplants, mixed AC/DC grids, Smart cities, and smart grids, control of powergenerators (transient stabilization, frequency and voltage stabilities, syntheticinertia), synchronization of power networks, microgrids.

Dr. Damm is a member of the IFAC Technical Committee TC 6.3 Powerand Energy Systems. He has received the French Research Excellence Grant(Prime d’Excellence Scientifique) since 2008. He has been an AssociatedEditor of the European Journal of Control since 2010.

Lilia Galai-Dol received the B.Sc. degree inelectrical engineering from École NationaleSupérieure d’Electrotechnique, d’Electronique,d’Informatique, d’Hydraulique et de Télécommu-nication (ENSEEIHT/INPT), Toulouse, France,in 2009, and the Ph.D. degree in power electronicsfrom ENS Cachan (SATIE Laboratory), Cachan,France, in 2014.

She is currently a Research Engineer with the Pôlegare Team, Efficacity, Research and DevelopmentCenter, Paris, France, where she is developing a

multiphysics smart-grid applied to an urban railway structure. Her currentresearch interests include power electronics, development of an innovateswitches supply and driver for power converters and EMI studies.

Paulo F. Ribeiro (M’78–SM’88–F’03) received theB.S.E.E., M.B.A., and Ph.D. degrees from the Fed-eral University of Itajubá, Itajubá, Brazil, and thePh.D. degree from The University of Manchester,Manchester, U.K., in 1985.

He was a Research Fellow with the NASA GlennResearch Center, Cleveland, OH, USA, ElectricPower Research Institute (EPRI), Palo Alto, CA,USA, an Erskine Fellow with the University of Can-terbury, Christchurch, New Zealand, and a ResearchFellow with the Brazilian Institute of Electric Energy

(INERGE), Rio de Janeiro, Brazil. He worked, taught full-time, and part-time,and worked in the industry in the USA for over 23 years. He also taughtat the Technological University of Eindhoven, Eindhoven, The Netherlands.He spent 5 years of his career as a Transmission System Planning Engineerwith the Hydroelectric Company of São Francisco (CHESF), Recife, Brazil.He is currently a Full Professor of electrical engineering with the FederalUniversity of Itajubá. He has authored or coauthored more than 350 papers,several book chapters, and 3 books. His current research interests includepower systems, power electronics and power quality engineering, transmissionand distribution systems, superconducting magnetic energy storage systems,smart grids, engineering education and philosophy of technology.

Dr. Ribeiro is a fellow of IET.

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Stability Analysis of a DC MicroGrid for a Smart Railway