Advanced Vector Control for Voltage Source Converters Connected to Weak Grids

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Advanced Vector Control for Voltage SourceConverters Connected to Weak Grids

Agustí Egea-Alvarez , Graduate Student Member, IEEE, Sajjad Fekriasl , Member, IEEE,Fainan Hassan , Senior Member, IEEE, and Oriol Gomis-Bellmunt , Senior Member, IEEE

Abstract—This paper addresses an advanced vector currentcontrol for a voltage source converter (VSC) connected to a weakgrid. The proposed control methodology permits high-perfor-mance regulation of the active power and the voltage for thefeasible VSC range of operation. First, the steady state charac-teristics for a power converter connected to a very weak systemwith a short circuit ratio (SCR) of 1 are discussed in order toidentify the system limits. Then, the conventional vector control(inner loop) and the conventional power/voltage control (outerloop) stability and frequency responses are analyzed. From theanalysis of the classic structure, an enhanced outer loop based ona decoupled and gain-scheduling controller is presented and itsstability is analyzed. The proposed control is validated by meansof dynamic simulations and it is compared with classic vectorcurrent control. Simulation results illustrate that the proposedcontrol system could provide a promising approach to tackle thechallenging problem of VSC in connection with weak AC grids.

Index Terms—Gain-scheduling control, stability, vector currentcontrol, VSC-HVDC, weak grid.

I. INTRODUCTION

H IGH voltage direct current (HVDC) systems based onvoltage source converters (VSCs) are emerging as the

main technology to connect remote renewable energy sources(RES), as offshore wind power plants, to the existing power sys-tems. VSC-based technology has been used in several HVDCpoint-to-point and back-to back projects in the last 15 years [1].The connection point between the VSC and the AC system

may be located remotely, leading to a low or very low SCR( 2). For HVDC systems based on line commuted converters

Manuscript received March 13, 2014; revised April 17, 2014, July 18, 2014,and October 24, 2014; accepted December 07, 2014. This work was supported inpart by the Spanish Ministerio de Economia y competitividad under the projectENE2012-33043 and in part by the EIT-KIC under the SmartPower project.Paper no. TPWRS-00363-2014.A. Egea-Alvarez is with Centre d'Innovacio Tecnològica en Convertidors

Estatics i Accionaments (CITCEA-UPC), Departament d'Enginyeria Electrica,Universitat Politecnica de Catalunya, ETS d'Enginyeria Industrial de Barcelona,08028 Barcelona, Spain (e-mail: [email protected]).S. Fekriasl and F. Hassan are with Advanced Research and Technology

Centre, Alstom Grid, Stafford, U.K.O. Gomis-Bellmunt is with Centre d'Innovacio Tecnològica en Convertidors

Estatics i Accionaments (CITCEA-UPC), Departament d'Enginyeria Electrica,Universitat Politecnica de Catalunya. ETS d'Enginyeria Industrial de Barcelona,08028 Barcelona, Spain. He is also with the Catalonia Institute for Energy Re-search (IREC), Electrical Engineering Area, 08019 Barcelona, Spain.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2014.2384596

(LCC) there is a limitation of the minimum required SCR (it issuggested to be higher than 2 in order to avoid instabilities [2]),but such a theoretical limit does not exist for VSC based systems[3]. This means that a VSC-HVDC converter is capable to inter-face with any kind of electrical grid and can potentially create agrid without synchronous generators (e.g., offshore wind powerplants).There are several proposed control techniques in order to in-

ject power to anAC system using aVSC. One of themost widelyused is the vector current control [4]. The vector current con-trol is based on the control of two independent current com-ponents, -axis and -axis in the synchronous reference frame(SRF) while the synchronization is provided by a phase lockedloop (PLL) [5]. This control technique permits an independentcontrol of active and reactive powers [6] with an fast dynamicresponse. Typically, the vector current control is considered asthe inner control loop, and an outer control loop is added tomanage the active power and the voltage/reactive power [7] forgrid connected converters.While this advantage fromVSC over LCC is often mentioned

when comparing both technologies, some studies have identi-fied relevant drawbacks when vector current control is used in aweak or a very weak grid [8]–[11].First problem is the low frequency resonances that can interactwith the vector current control [12]. Second problem is due tothe PLL dynamics when the power converter is synchronizedto a weak grid [11], [12]. Zhang et al. [11], [13], [14] proposedan alternative technique referred to as power synchronizationcontrol (PSC), which does not require synchronization with aPLL via emulating the behavior of a synchronous machine. Itis reported that PSC provides a good performance and fast dy-namics for low SCR values. However, the main disadvantageof this topology is in dealing with faults in the AC grid, PSCswitches to classic vector current control when the power con-verter current limit is reached [13]. In addition, in [15] a controlsystem for microgrids and railway electrical weak grids is pre-sented.The viability of vector control (composed by an inner cur-

rent loop and an outer power-voltage loop) in an extremelyweak network is demonstrated in the present paper. The presentpaper does not consider the system performance when the gridstrength changes, which is a very important topic and will bestudied in further works. The outer control scheme is basedon the gain-scheduled multi-variable controller [12], such gain-scheduling approach allows ensuring stable operation of thewhole VSC operating range. A grid with is utilized as

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2 IEEE TRANSACTIONS ON POWER SYSTEMS

Fig. 1. Model of the analyzed system.

a benchmark. Please note that this value is related to the worstcase scenario, for which the system stability and performancemust be analyzed and validated.This paper is structured as follows. In Section II and III, the

steady state characteristics of a VSC-HVDC connected to aweak grid are presented. In Section IV, the control structurebased on conventional vector current control is presented andits stability and frequency response are discussed in Section V.Using the benchmark grid with , it is shown inSection V that the system becomes unstable at higher powerdemands. It is demonstrated that the outer loop control isresponsible for resolving such instability. In Section VI, an en-hanced outer control is proposed and discussed. In Section VII,the stability and performance of the system with the proposedcontrol is analyzed, the dynamic behavior is validated throughnumerical simulations, and finally the results are comparedand justified with those obtained using the conventional vectorcurrent control.

II. ANALYZED SYSTEM

The analyzed system is a VSC-HVDC power converter con-nected to a weak grid by means of an inductance+capacitor cou-pling filter as shown in Fig. 1 (the use of the capacitor is justifiedin Section III). The grid is represented using a Thevenin equiva-lent and the overall system is to be modelled in the synchronousreference frame in the form of

(1)

where and are

(2)

(3)

and the state and input vectors are

(4)

(5)

Fig. 2. Active power exchanged between the VSC-HVDC terninals and PCCfor a .

where and are the resistance and inductance of the induc-tive filter between the power converter and the electrical grid,

and are the grid Thevenin equivalent resistance and in-ductance, is the filter capacitance, is the grid frequency,

is the voltage applied by the power converter, is thevoltage at the point of common coupling (PCC), is the gridvoltage, is the current flowing through the coupling filter,and is the current flowing through the grid equivalent. Sub-script refers to the -axis and the subscript refers to the-axis.For the phasor analysis presented in Section III, the voltage

of the grid Thevenin equivalent is considered as theslack and its angle is 0, is the power converter voltage,

is the voltage at the PCC, is the current throughthe grid, and is the current through the coupling filter.

III. STEADY STATE CAPABILITY

VSC-HVDC connected to strong grids present some limita-tions due to physical restrictions of the power converter, par-ticularly maximum current and voltage. When VSC-HVDC isconnected to a weak grid, the stability limit is also relevant. Thesteady state stability will determine the maximum amount ofactive and reactive powers that can be exchanged between thegrid and the power converter. The results that are presented inthis section have been calculated using the parameters from thesimulation results section (see Section VII).Fig. 2 shows the active power as a function of the angle

keeping the amplitude of constant. Therelation between the active and reactive power and voltagesand are defined by

(6)

(7)

(8)

where , , and. As it can be seen, the maximum power that can be

inverted is different from the maximum power that can be recti-fied. When the power converter is in rectifier mode, the 90 are


EGEA-ALVAREZ et al.: ADVANCED VECTOR CONTROL FOR VOLTAGE SOURCE CONVERTERS CONNECTED TO WEAK GRIDS 3

Fig. 3. V/U curves as function of different power factors and active power.

reached below and when it is operating in invertermode, the are reached above the active power unity. Thisrestriction is caused due to the effect of the resistances [16].From the voltage point of view, Fig. 3 shows the relation be-

tween the amplitude of the voltage at the PCC, U, and the mag-nitude of the voltage at the power converter terminals, V, fordifferent power factors and active power. As depicted in Fig. 3,if 1 pu of active power has to be inverted, a large amount ofcapacitive reactive power is needed. Therefore, the power con-verter must be oversized (or alternatively an external reactivepower source should be installed). For this reason, a shunt ca-pacitor is added in order to provide reactive power support atthe PCC.

IV. CLASSIC CONTROL APPROACH

A classic control structure of the power converter for grid in-tegration purposes based on vector current control is developedin two control levels, the lower level control (inner loop) and theupper level control (outer loop). The inner control is in chargeof the regulation of the components of the current throughthe coupling filter. The outer control is in charge of the controlof the active power and the magnitude of the voltage at the PCC.A sketch of the general structure is presented in Fig. 4.

A. Inner Loop

The lower level control is based on the vector current con-trol. The control structure is formed by proportional-integralcontroller (PI) that regulates the current through inductor.In addition, there are decoupling terms to allow independent-axis and -axis current control. The current control can beimplemented, assuming synchronization with the component

and positive current for the rectifying operation, as

(9)

(10)

For more details on the vector current control concept, thereader is referred to [6] and other references therein. In Fig. 4 thevector current control is shown inside the inner loop box. The PI

Fig. 4. Classic control structure for VSC.

controllers are expressed as, . The con-troller gains have been tuned using the internal model control[17] technique resulting in

(11)

(12)

where is the desired closed loop time constant. A generalrule-of-thumb is to choose between 5 and 10 times slowerthan the power converter switching frequency. Another impor-tant component of the lower level control is the PLL. The PLLis required to extract the angle needed to synchronize the con-trol system under the SRF. A classic scheme of a PLL is basedon the -axis voltage feedback by a PI controller to obtain thegrid angular velocity and an extra integrator to obtain the angle.Fig. 4 shows a representative scheme of the PLL, which can bedescribed as follows:

(13)

where is the PLL proportional gain and is the PLLintegral gain. These gains have been tuned according to [5] andare listed in Appendix A. The used Park transformation, ,is outlined in Appendix B.

B. Outer Loop

The outer loop calculates the current reference in orderto obtain the desired active power (P) and the amplitude of thevoltage at the PCC (U). A classic approach to the upper levelcontrol consists of two independent PI controllers, one for eachcurrent component [7]. A scheme of the aforementioned methodis depicted in Fig. 4.The upper level controls are

(14)

(15)



Fig. 5. Poles and zeros map of the current loop dynamic system subject tovariation of .

where and are defined asand where and are theproportional gains and and are the integral gains ofthe power and voltage controllers. These gains are thoroughlydetailed in Appendix B.

V. DYNAMIC ANALYSIS WITH CLASSICVECTOR CURRENT CONTROL

Since the analyzed system presents nonlinearities, the dy-namic equations are to be linearized in order to obtain and re-alize the stability studies. Such nonlinearities are related to theactive power (P), the magnitude of the voltage at the PCC (U)and the effect of the angle on the Park transformation and theinverse Park transformation. The linearized equations are pro-vided in Appendix C.

A. Frequency Response and Stability Analysis of the InnerLoop

Fig. 5 shows the pole and zero configuration of the VSC-HVDC system using the inner loop only for different powervalues. Please note in this figure that the direction of the arrowsstart from (inverting) to (rectifying). Thesystem inputs are the current references and . The plantdynamics is linearized around the desired equilibrium points, inassociation with the operating points that permit injecting thedesired amount of power, yet retaining the voltage amplitudeconstant. As illustrated in Fig. 5, the system poles are all locatedat the Left Hand Plane (LHP). Consequently, the vector currentcontrol and the PLL tuned using classic control approaches arestable for the full range of operation.From the generic frequency analysis of the system, when a

VSC-HVDC is connected to a strong grid the variation effectof the is mainly observed on the active power response andthe variation effect of the is mainly observed on the voltageresponse. Consequently, it is assumed that the active power andthe voltage amplitude at the PCC can be controlled indepen-dently.

Fig. 6. Bode frequency response of the system at different active power oper-ation points.

Fig. 6 shows the frequency response of the classiccurrent vector control for three different power levels

. This frequency response isshown in four plots, the first column shows U and P responsesubjected to a variation and the second column showsthe same variables subject to a variation. From the Bodeplot analysis at low frequency region, it is deduced that forsmall power values , the independentcontrol between and U and and P is preserved, butwhen the power demand is higher the cross terms effectsare not negligible and the independent component control islost . In particular when the system isoperating near (inverting mode), the effect ofor changes on the power and voltage are almost identical.In other words, when a VSC-HVDC is connected to a weakgrid there exists system nonlinearities that do not permit anindependent and decoupled control for high power values. Thisphenomenon is due to the large angle when the high activepower values are injected. Usually, as the power system isoperating with below 30 , the system is considered to belinear as well as the voltage control is assumed through thereactive power control, but in the presence of a weak powersystem, this assumption can potentially be violated because thepower and voltage control are mutually coupled.

B. Stability of the Closed-Loop System

Fig. 7 shows the poles and zeros plot of the VSC-HVDCcontrolled by an inner current loop plus an outer current loopfor inverting (upper graph) and rectifying mode (lower graph).The outer current loop is designed for the linear operation area

. The arrow indicates the poles movement fromto for the inverting operation mode and

from to for the rectifying mode. Forlow power values (near the linear area) the system is stable in theinverting operational mode, but for higher values it is unstable,that is, the stability margin is located around .This means that an outer loop designed for low power opera-tions points is not suitable for the high power values due to the



Fig. 7. Poles and zeros map of the system using the classic upper level controlat (upper plot) and (lower plot).

system nonlinearities. A similar instability behavior could resultin the rectifying mode.

VI. PROPOSED ADVANCED VECTOR CURRENT CONTROL

From Section V, it is concluded that the vector current con-trol, using a classic PLL, is stable and can drive the powerconverter in the operational envelop described in Section III.However, the classic outer loop for high power values is un-stable. From this analysis, it can be deduced that the classicouter control is not appropriate for weak grids. To this end, anew upper level control considering the system nonlinearities isintroduced.The proposed upper level control consists of additional four

decoupling gains between the voltage magnitude and power er-rors, and , before being processed by the PI. Furthermore,to overcome the nonlinearities and obtain similar responses, aparameter-varying control scheme based on the gain-schedulingtechnique, is proposed for the decoupling gains and the PI con-trollers. The aim of the proposed control scheme is to robustlyhandle the interactions between the active power and voltagecontrol. A block diagram of the proposed controller is shown inFig. 8.The controller can be described as

(16)

(17)

where , , , and are the decoupling gains(proportional gains) andand are proportional-in-tegral (PI) power and voltage controllers.

Fig. 8. Proposed advanced outer loop control.

With reference to Fig. 8, the proposed advanced controlsystem is indeed a multivariable (two-input two-output) dy-namical system, whose inputs are and and outputsare and . In fact, since the plant dynamics is highlynon-linear, the control system performance get worst and evenbecomes unstable if the nonlinearities are not taken in account.In these regards, a justified number of local controllers are tobe designed accordingly for such operating points to provideweak AC system with robust stability (and robust performance,if any). For the purpose of the presented study, 35 local robustcontrollers have been designed based on the same number ofoperating points of the linearized dynamics that cover activepower transmission distributed between .For the purpose of tuning the above eight design parameters

(i.e., , , , , , , , and ) at any given op-erating point, the so-called H-infinity fixed-structure control de-sign methodology is used. For more details, readers are referredto [18] and [19]. This designing/tuning procedure is repeatedfor all possible operating points leading towards the schedulingcontroller gains of , and resulting on arobust gain-scheduling control system for operating power en-velope. In order to address this challenging control problem, theproposed approach is to utilise the gain scheduling approachwith fixed-structure H-infinity controllers as demonstrated inFig. 9.As depicted in Fig. 9, the design gains are structured in a con-

troller architecture and then the H-infinity tuning of fixed-struc-ture are applied to refine/retune the above given controller ar-chitecture. The design/tuning of such fixed order controller isbased on the fact that first, several randomly selected initialpoints are chosen as multi-start points and then they are tunedvia non-smooth optimization using Clarke subdifferential of theH-infinity objective and an appropriate line search [18]. In fact,the H-infinity norm of the closed-loop transfer functionis minimized using fixed-structure control systems at every op-erating condition. While it is a very cost-effective robust con-trol design, the downside of this approach is that there is no al-ways a guarantee that we will find the global minimum for theH-infinity norm. One possible solution for that is to initialisethe controller gain parameters from different starting points.Nonetheless, in a wide range of application, it is shown that afew runs are typically enough to obtain a satisfactory design.In MATLAB, the command HINFSTRUCT is particularly as-



Fig. 9. Illustration of the standard form model used in synthesizing H-infinitytuning of fixed-structure controllers [19], [20].

sociated with the design of H-infinity tuning of fixed-structurecontrollers. A fully automatized synthesise approach to facili-tate the required design tasks we shall refer to [20] and otherreferences therein for more technical discussion.First, the plant dynamics are linearized at any particular op-

erating points, and then, relevant Linear Time Invariant (LTI)models are obtained. After obtaining this set of linearized plants,a set of fixed-structure H-infinity controllers are designed ac-cordingly. Using a scheduling mechanism as a parameter feed-back, the controller dynamics are smoothly changed based onthe variation of the operating condition. The generalized plantdynamics will be also gain-scheduling as a function of oper-ating conditions. For more information, readers are referred to[21] and other references therein. The result of the above controlgains designed for the desired operational envelop are providedin Appendix A.In the next step, the designed global gain-scheduling H-in-

finity control system performance is tested and validatedthrough numerical simulations.

A. Stability of the Proposed Control System

The stability of the proposed control system is analyzed basedon the eigenvalues of the linearized plant, as depicted by thepole-zero map in Fig. 10. The arrows indicate the moving di-rection of the poles from to . As it canbe seen, the proposed control stabilizes the system by retainingall the poles at the LHP.

B. Fault Right Through Strategy

During AC faults, currents demanded by the upper level con-trol may exceed the power converter nominal value. For thisreason the current references must be saturated as

(18)

where is the maximum current permitted through thepower converter in steady state. In addition, the active powerreference is set to zero during the fault and ramped up to the

Fig. 10. System pole-zero map using the proposed advanced control based at.

TABLE IPARAMETERS USED IN THE STUDY

previous value after the fault. These two actions are enough forthe proposed control scheme to remain connected during a faultas it is shown in the simulations presented in Section VII.

VII. SIMULATIONS RESULTS

Three simulation scenarios have been carried out to validatethe proposed control system using MATLAB/Simulink mod-eling packages. The first simulated scenario is an active powerramp change, the second is an active power step change sce-nario and the third one is a three-phase voltage sag. An averageVSC model (valid for low frequencies) is used for the purposeof validation of the proposed concepts throughout our simula-tions [6]. The variable gain controllers are dynamically imple-mented using lookup tables, it means that the parameters changeaccording to the power reference (see Appendix A for the con-troller gain values). Table I summarizes the parameters used inthe simulations.

A. Comparison Between the Convectional Outer Loop and theProposed Control Loop

The first studied case is a comparison between the conven-tional outer loop and the proposed control loop in front of aactive power reference step change. Fig. 11 shows the activepower and the magnitude of the voltage U at the PCC duringthe power step change. From time instant , a stepchange is applied over a period of 200 ms. From the powerpoint of view, it can be seen that the reference is tracked andthe new power point is reached in less than 50 ms. However,



Fig. 11. Active power and U voltage magnitude (the solid line is the currentvalue and the dotted line is the reference) in response to changes in active powerdemand.

Fig. 12. Active power and U voltage magnitude (the solid line is the currentvalue and the dotted line is the reference) in front of step change using the classiccontrol structure.

from the voltage point of view, a small damped oscillation is re-alized around 0.07 pu, as expected.Fig. 12 shows the same step pattern of the active power and

the amplitude of the PCC voltage at the PCC in Fig. 11 butusing the conventional Vector Current Control. As it can beobserved, both controllers result in acceptable behavior forlow powers, approximately below 0.7 pu, but for higher powervalues the convectional control is unstable. This confirms theconclusion drawn from Section V-B, in the validation of thefact that the system is unstable for power values above 0.74 pu.

B. Power Ramp Change

Figs. 13 and 14 show the active power, the voltage at the PCCand the -axis and -axis of the current in response to an

active power ramp with a slope of 5 pu/s. Betweenand the system is injecting a power of 0.25 pu andthe voltage is kept constant. At time instant the in-jected power reference is changed and the system is tracking thereference satisfactorily. From the voltage point of view, there is

Fig. 13. Active power and U voltage magnitude (the solid line is the currentvalue and the dotted line is the reference) subject to a ramp change.

Fig. 14. Reference and real value of the current in the -axis.

a small increase of 0.04 pu. As it is presented in Section III themaximum power that can be inverted to the grid is 1 pu andthis level is achieved successfully at . At time instant

the active power reference is changed again and thesystem starts to reduce the inverted power and atthe system achieves the maximum power that can be rectified,i.e., for the studied system. During the tran-sient, the power is followed with a reduced tracking error, butfrom the voltage point of view, a minimum voltage of 0.92 puhas occurred during the power reference change. From the cur-rents point of view, the component variation, during theramp change, follows the active power reference change. Thecomponent is also following the voltage requirements satis-

factorily.

C. AC Three-Phase Fault

Fig. 15 shows the active power (P) and the current magnitudeof the proposed control system in front of a three-phase

80% deepness voltage sag during 500 ms. As it can be observed,the converter was injecting around 0.8 pu of active power be-fore the fault. At time instant the voltage sag is applied



Fig. 15. Active power and evolution during and 80% voltage sag.

and the current is increased in order to maintain the AC voltageconstant but the current limit is reached and the current is sat-urated. Furthermore, the active power reference is reduced to0 in order to make the fault return more smoothly. After somemilliseconds, the power converter operates in a stable operationduring the fault. Immediately after the fault clearance, the activepower has a transient, due to the voltage variation in the powerconverter terminals but the injected current is kept in limits. Fi-nally, the active power reference is increased gradually and thesystems returns to the previous operation point.

VIII. CONCLUSIONS

This paper has addressed an advanced gain-scheduling con-trol system design methodology for VSCs connected to weakAC grids. Each controller can be designed in a way to guaranteerobust stability and performance for any operating condition. Asa result, the outcome of such advanced control is to provide ex-tended operational area of a VSC for a weak grid operation.It was explored and clarified that conventional vector cur-

rent control systems have severe shortcomings in dealing withhigh-power demands at the weak grids. This is mainly due tosystem severe nonlinearities as well as highly-coupled activepower/voltage interactions, which makes the control of the VSCin connection with weak AC grids a very challenging problem.The simulation results illustrate that the proposed advanced con-trol seems a very promising approach to tackle such challengingcontrol application under normal and fault conditions. Further-more, the present control scheme, compared to conventionalcontrol schemes, allows to inject active power in all the feasiblepower converter range.

APPENDIX ATUNED CONTROL GAINS

The current loop time constant has been tuned at. The PLL gains are and. The cut-off frequency of the LPF(s) is 10

kHz. The controller gains for the proposed control loop (all thesimulated cases have been done with the same controller gains)are specified in Fig. 16. Gains are scaled to be plotted together,

Fig. 16. Gain evolution of the proposed control scheme according to the activepower.

TABLE IICONTROLLER GAINS SCALING FACTOR

for this reason they should be multiplied by a scaling factorpresented in Table II:

(19)

where is the specific controller.APPENDIX B

UTILISED PARK TRANSFORMATION

The used Park transformation is defined as

(20)

and its inverse is described as

(21)

where is a vector with the three-phase quantities in theframe and is a vector with the transformed quantities in the

frame.The transformation matrix can be written as

(22)

and its inverse

(23)



Fig. 17. Scheme of the linearized system.

APPENDIX CLINEARIZED DYNAMIC EQUATIONS

The dynamic system equations are linearized as independentsystems and they are connected according to Fig. 17 for the innerloop and outer loop control. The subscript indicates the valueat the linearization point, indicates an average of the variablequantity and the superscript means that the variables havebeen transformed by means of the linearized park transforma-tion.1) Linearized Electrical System Equations: The electrical

system is composed by the coupling filter and the electrical grid(see Fig. 1). The state space representation of the linearizedsystem is defined by

(24)

(25)

where the state variables, inputs and outputs are

(26)

(27)

where the matrix and have been defined in (2) and (3)and is

(28)

2) Linearized PLL Equations: The PLL is used in order toorient a control with the electrical grid angle. In the linearizedmodel the PLL introduces the angle deviation when the lin-earized system is moved from the linearization point. The PLLhas been linearized following [11]. The PLL linearised transferfunction representation is

(29)

3) Linearized Park Transformation and Inverse-Transfor-mation Equations: The linearized Park transformation (seeAppendix B) expressed is given by

(30)

where is

(31)and the linearized inverse transformation is

(32)

where is

(33)4) Inner Loop Equations: The vector current control equa-

tions are

(34)

(35)

where the state variables, inputs, and outputs are

(36)

(37)

(38)

is the current error, defined as the difference betweenand . The matrix gains are

(39)

(40)

(41)

5) Measures Filter: Where is a first order filter usedto filter the measured signals. This filter is implemented as a firstorder in the natural reference frame as

(42)

where is the filter time constant.6) Classic Outer Loop Equations: For the study of the whole

classic system, the outer loop equations are

(43)

(44)

where the state variables, inputs, and outputs are

(45)

(46)

(47)

and are the active power and voltage error.Where the matrix gains are defined as

(48)

(49)



(50)

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Agustí Egea-Alvarez (S'12) received the B.S.,M.Sc., and Ph.D degrees from the Technical Univer-sity of Catalonia (UPC), Barcelona, Spain, in 2008,2010, and 2014, respectively.Since 2008, he has been with the Centre d'In-

novació Tecnolàgica en Convertidors Estàtics iAccionaments, UPC. His current research interestsinclude control and operation of high-voltage directcurrent systems, renewable generation systems,electrical machines, electrical railway systems, andpower converter control.

Sajjad (Fekri) Fekriasl (S’02–M’06) received theB.S.(Hons) and M.S.(Hons) degrees in electrical en-gineering from the University of Tabriz, Iran, in 1995and 1997, respectively, and the Ph.D. degree in elec-trical engineering and computer science from Insti-tuto Superior Técnico, Lisbon, Portugal, in 2006.He has held a number of technical leadership po-

sitions in control systems design for power systems,automotive, and aerospace applications. He was apost-doc research associate with the Department ofElectrical Engineering, University of Leicester, U.K.

(2006–2009) and lecturer of advanced control and optimization in the Schoolof Engineering, Cranfield University, U.K. (2010–2011). He is now PrincipalScientist with Smart Grids Research and Technology Center, Alstom Grid,Stafford, U.K. His current research focus is on the design and development ofmodern control systems, optimization, and estimation algorithms for powerelectronics applications and/in power systems.

Fainan Hassan (M’01–SM’13) received the Ph.D.degree in electrical engineering from the Departmentof Energy and Environment, Chalmers University ofTechnology, Gothenburg, Sweden, in 2007.She worked as a senior engineer for STRI AB,

Ludvika, Sweden, during 2008. Since the start of2009, she has been with Alstom Grid (then ArevaTD), Research and Technology Centre, Stafford,U.K. She is currently a principle developmentengineer of HVDC grids at Alstom.

Oriol Gomis-Bellmunt (S'05–A'07–M'07–SM'12)received the degree in industrial engineering fromthe School of Industrial Engineering of Barcelona(ETSEIB), Technical University of Catalonia (UPC),Barcelona, Spain, in 2001, and the Ph.D. degree inelectrical engineering from the UPC in 2007.In 2003, he developed part of his Ph.D. thesis

in the German Aerospace Center (DLR), Braun-schweig, Germany. In 1999, he was with EngitrolS.L. as a Project Engineer in the Automation andControl Industry. Since 2004, he has been with

the Department of Electrical Engineering, UPC, where he is a Lecturer andparticipates in the Centre d'Innovació Tecnològica en Convertidors Estàtics iAccionaments (CITCEA) research group. Since 2009, he has also been withthe Catalonia Institute for Energy Research (IREC), Barcelona. His researchinterests include the fields linked with smart actuators, electrical machines,power electronics, renewable energy integration in power systems, industrialautomation, and engineering education.

Documents

Advanced Vector Control for Voltage Source Converters Connected to Weak Grids