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Electric Power Systems Research 126 (2015) 111–120

Contents lists available at ScienceDirect

Electric Power Systems Research

j o ur na l ho mepage: www.elsev ier .com/ locate /epsr

novel VSC-HVDC link model for dynamic power system simulations

uis M. Castroa,∗, Enrique Achaa, Claudio R. Fuerte-Esquivelb

Tampere University of Technology, Department of Electrical Engineering, Tampere, FinlandUniversidad Michoacana de San Nicolás de Hidalgo, Faculty of Electrical Engineering, Morelia, México

r t i c l e i n f o

rticle history:eceived 23 September 2014eceived in revised form 9 April 2015ccepted 7 May 2015

eywords:SCVDCACTSewton–Raphson methodynamic power system simulations

a b s t r a c t

This paper introduces a new RMS model of the VSC-HVDC link. The model is useful for assessing thesteady-state and dynamic responses of large power systems with embedded back-to-back and point-to-point VSC-HVDC links. The VSC-HVDC model comprises two voltage source converters (VSC) linked bya DC cable. Each VSC is modelled as an ideal phase-shifting transformer whose primary and secondarywindings correspond, in a notional sense, to the AC and DC buses of the VSC. The magnitude and phaseangle of the ideal phase-shifting transformer represent the amplitude modulation ratio and the phaseshift that exists in a PWM converter to enable either generation or absorption of reactive power purely byelectronic processing of the voltage and current waveforms within the VSC. The mathematical model isformulated in such a way that the back-to-back VSC-HVDC model is realized by simply setting the DC cableresistance to zero in the point-to-point VSC-HVDC model. The Newton–Raphson method is used to solvethe nonlinear algebraic and discretised differential equations arising from the VSC-HVDC, synchronous

generators and the power grid, in a unified frame-of-reference for efficient, iterative solutions at eachtime step. The dynamic response of the VSC-HVDC model is assessed thoroughly; it is validated againstthe response of a detailed EMT-type model using Simulink®. The solution of a relatively large powersystem shows the ability of the new dynamic model to carry out large-scale power system simulationswith high efficiency.

© 2015 Elsevier B.V. All rights reserved.

. Introduction

Continuous increases in electrical energy consumption havencouraged a great deal of technological development in thelectrical power industry. In particular, the development of newquipment for power transmission that enables a more flexibleower grid aimed at achieving higher throughputs, enhancing sys-em stability and reducing transmission power losses, has beenigh on the agenda [1,2]. The VSC-HVDC link is the latest equipmenteveloped in the arena of high-voltage, high-power electronics and

ts intended function is to transport electrical power in DC form, asell as to enable the asynchronous interconnection of otherwise

ndependent AC systems [3], and to provide independent reac-ive power support. The technology employs Insulated Gate Bipolarransistors (IGBTs), driven by pulse width modulation (PWM) con-rol. This valve switching control permits to regulate dynamically,

n an independent manner, the reactive power at either terminal ofhe AC system and the power flow through the DC link [4].

∗ Corresponding author. Tel.: +358 465496289.E-mail address: luis miguel c g@hotmail.com (L.M. Castro).

ttp://dx.doi.org/10.1016/j.epsr.2015.05.003378-7796/© 2015 Elsevier B.V. All rights reserved.

The VSC-HVDC model put forward in this paper comprises twoVSC models linked by a cable on its DC sides. In turn, each VSCmodel is made up of an ideal phase-shifting transformer whichsynthesises the phase-shifting and scaling nature of the PWM con-trol. The ideal phase shifter is taken to be the interface betweenthe AC and DC circuits of the VSC. The model makes provisions forthe representation of conduction losses and switching losses. Sinceboth converters are capable of independently controlling the reac-tive power exchanged with the AC power grid at their respectiveAC nodes, the VSC-HVDC dynamic model uses two independentdynamic voltage regulators. Both control loops are aimed at pro-viding the required reactive power support at their respective ACnodes to maintain pre-set voltages, by regulation of their amplitudemodulation coefficients. Likewise the model correctly accounts forthe dynamics of the DC link. This is carried out by using a controlblock that acts upon the DC current to adjust the DC voltage of theVSC-HVDC link.

It should be mentioned that in an early model of the VSC-HVDCsystem, the two VSC are emulated by idealised voltage sources

[5–8]. Alternatively, the VSCs have also been represented by equiv-alent controlled current sources [9,10], where the currents to beinjected into the AC grids are computed by the existing differencebetween the complex voltages of the VSC terminals and the AC

1 Systems Research 126 (2015) 111–120

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12 L.M. Castro et al. / Electric Power

ystem nodes at which the VSC-HVDC is embedded. More recently,he concept of dynamic average modelling has caught the atten-ion of the power system community since it allows the modellingf VSCs in a more detailed manner [11]. In the dynamic averageodelling approach, the average value of the output voltage wave-

orm is calculated at each switching interval, a value that changesynamically depending on the value of the reference waveform. TheSC is represented by a three-phase controlled voltage source on

ts AC sides and as a controlled current source on its DC sides [12].owever, this approach may be time consuming when repetitive

imulations studies are required, such as in power grid expansionlanning and in operation planning. The solution time is always an

mportant point to keep in mind and in this method, increasing timeteps is always a temptation but caution needs to be exercised whensing the dynamic averaging method since, as reported in [12], these of large time steps may affect the accuracy of the results. It isorth mentioning that if harmonics or electromagnetic transients

re the study subject, such a high level of modelling detail is nec-ssary, where, for instance, the PWM control needs to be modelledxplicitly to achieve meaningful results.

On the other hand, in large-scale power system applications,t looks attractive to represent each VSC as a controlled voltageource owing to its much reduced complexity. However, its internalariables may not be readily available. In contrast, the new modelntroduced here captures very well the key operational character-stics of the VSCs making up the HVDC link. This is done by usingxplicit state variables that encapsulate the actual performance ofhe AC and DC circuits for both, the steady-state and dynamic oper-ting regimes. Furthermore, the new VSC-HVDC model possesseshe four degrees of freedom found in actual VSC-HVDC installa-ions, characterised by having simultaneous voltage support at itswo AC terminals, DC voltage control at the inverter converter andegulated DC power at the rectifier converter.

The numerical implementation of the VSC-HVDC model is car-ied out using a unified framework which suitably combines thelgebraic and discretised differential equations of the VSC-HVDCink model, the synchronous generators and the non-linear alge-raic equations of the power grid. This iterative solution takesdvantage of the Newton–Raphson (NR) method thus facilitatinghe efficient solution of the non-linear equations. The discretisationf the differential equations is carried out using the implicit trape-oidal rule of integration which has been proven to be numericallytable and accurate [13,14]. In this paper, special attention is paido the new dynamic VSC-HVDC model, emphasising how the alge-raic and discretised differential equations are assembled together

n this framework.

. VSC-HVDC model for dynamic analysis

.1. Key physical characteristics

If two VSC stations are linked as shown in Fig. 1, a VSC-HVDC

ystem is formed and termed point-to-point configuration. In thisrrangement, electric power is taken from one point of the AC net-ork, converted to DC in the rectifier station, transmitted through

he DC link and then converted back to AC in the inverter station

Fig. 1. Schematic representation of a VSC-HVDC link.

Fig. 2. VSC equivalent circuit for the inverter station.

and injected into the receiving AC network. In addition to transportpower in DC form, this combined system is also capable of supplyingreactive power and providing independent dynamic voltage con-trol at its two AC terminals. It is worth mentioning that by settingthe cable resistance RDC to zero, the representation reduces to thatof the so-called back-to-back VSC-HVDC configuration. Please referto the Appendix A for the symbols used in all equations and figures.

2.2. VSC-HVDC steady-state model

Fig. 2 depicts the equivalent circuit of the VSC corresponding tothe inverter station; a similar topology can be formulated for therectifier station. Its steady-state representation relies on an idealphase-shifting transformer with complex taps, a series impedanceon its AC side as well as an equivalent variable shunt susceptanceBeqI, and a shunt resistor on its DC side [15].

The series reactance X1I represents de VSC’s interface magnet-ics whereas the series resistor R1I is associated to the ohmic losseswhich are proportional to the AC terminal current squared. Theshunt resistor (with a conductance value of GswI) produces powerloss to account for the switching action of the converter valves.This conductance is calculated according to rated conditions andensures that the operating conditions on switching losses are repre-sented by scaling the quadratic ratio of the actual terminal current

II to the nominal current Inom : GswI = G0I

(II/Inom

)2. Note that the

squaring of this ratio is to give the switching conductance term anoverall power performance. The following assumptions are madein the model: (a) the complex voltage V1 = k2maI EDCej�I is the volt-age relative to the system phase reference; (b) the tap magnitudemaI of the ideal phase-shifting transformer corresponds to the VSC’samplitude modulation coefficient where the following relationship

holds for a two-level, three-phase VSC : k2 =√

3⁄8; (c) the angle�I is the phase angle of voltage V1; (d) EDC is the DC bus ampli-tude voltage which is a real scalar. Bearing this in mind, the nodalpower flow equations for the series branch of the VSC representingthe inverter station are derived from the nodal admittance matrixdeveloped in Appendix B. After some arduous algebra, the activeand reactive powers expressions for the powers injected at bothends of the VSC, nodes vI and 0vI, are arrived at:

PvI = V2vIG1I − k2maIVvIEDCI

[G1I cos

(�vI − �I

)+ B1I sin

(�vI − �I

)](1)

QvI = −V2vIB1I − k2maIVvIEDCI

[G1I sin

(�vI − �I

)− B1I cos

(�vI − �I

)](2)

P0vI = k22m2

aIE2DCIG1I − k2maIVvIEDCI

[G1I cos

(�I − �vI

)+B1I sin

(�I − �vI

)]+ PswI (3)

Q0vI = −k22m2

aIE2DCIB1I − k2maIVvIEDCI

[G1I sin

(�I − �vI

)−B1I cos

(�I − �vI

)]+ QeqI (4)

Systems Research 126 (2015) 111–120 113

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L.M. Castro et al. / Electric Power

here,

swI = E2DCIG0I

(IvI/Inom

)2(5)

eqI = −k22m2

aIE2DCIBeqI (6)

Likewise, a similar set of equations may be obtained for the VSCorresponding to the rectifier station. To obtain the steady-statequilibrium point, the set of mismatch power flow equations thatust be solved together with those arising from all the network’s

odes is:

PvR = −PvR − PvR,load − PvRcal = 0 (7)

QvR = −QvR − QvR,load − QvRcal = 0 (8)

PvI = −PvI − PvI,load − PvIcal = 0 (9)

QvI = −QvI − QvI,load − QvIcal = 0 (10)

E0R = E2DCR − EDCIEDCR − PschRDC = 0 (11)

P0vR = −Psch − P0vR = 0 (12)

P0vI = Psch − (EDCR − EDCI)2R−1

DC − P0vI = 0 (13)

Q0vR = −Q0vR = 0 (14)

Q0vI = −Q0vI = 0 (15)

here, in this particular case, PvRcal, QvRcal, PvIcal and QvIcal, stand forhe powers flowing from bus vR to k and vI to m, respectively. Theyre given by

vRcal = V2vRGRR + VvRVk

[GRk cos

(�vR − �k

)+ BRk sin

(�vR − �k

)](16)

vRcal = −V2vRBRR + VvRVk

[GRk sin

(�vR − �k

)− BRk cos

(�vR − �k

)](17)

Similar equations may be obtained for the power flowing fromus I to m by simply exchanging subscripts in (16) and (17). It shoulde remarked that (12) ensures that the power flow leaving the rec-ifier station be kept at the scheduled value Psch. Given that thenverter station is chosen to keep the DC link voltage at a constantalue EDCI then Eq. (11) allows the computation of the DC voltage inhe rectifier’s side, EDCR. Since the objective is to regulate the volt-ge magnitude at both AC sides of the VSC-HVDC while keepinghe DC voltage fixed, VvI, VvR and EDCI are not part of the set of stateariables that need to be computed. Thus, �vR, maR, �vI, maI, EDCR, �R,I, BeqR and BeqI, constitute the set of state variables that must bealculated by solving (7)–(15) with the rest of the equations aris-ng from the network. To guarantee that each VSC operates withineasible operating limits, a limit checking of the modulation rationd terminal current must take place, that is, ma ≤ 1 and I ≤ Inom.urthermore, to enable good starting conditions, the NR algorithms initialised as follows: the amplitude modulation ratios, maI and

aR, and the angles, �I and �R, are set at 1 and 0, respectively.

.3. VSC-HVDC dynamic model

The VSC capacitor’s dynamics play an important role in theehaviour of the HVDC link when subjected to voltage and powerariations coming from the external AC network. On the other hand,n this VSC application, DC voltage control is a target in order to

ursue a stable operation of the DC link. The inverter converter ishe one that takes on such a task, where the following relation-hip holds at its DC terminals: ic = − IDCR − IDCI, being IDCR and IDCIhe currents injected at rectifier’s DC bus and at inverter’s DC bus,

Fig. 3. VSC-HVDC dynamic controller for the DC voltage.

respectively. Substituting this current relationship into the expres-sion that allows calculating the capacitor’s currentic = CDC

dEDCdt , we

get the differential equation with which the DC voltage dynamicsare represented.

dEDCI

dt= −IDCR − IDCI

CDC(18)

IDCR = P0vR

EDCR(19)

The value of CDC is estimated from the amount of energy storedin the capacitor: Wc = 1⁄2CDCE2

DC. The electrostatic energy stored inthe DC capacitor can be associated with an equivalent inertia con-stant Hc [s] as Wc = Hc Snom, where Snom would correspond to therated apparent power of the VSC. This time constant is small and itmay be taken to be Hc ≈ 5 ms [16]. Hence, the per-unit value of thecapacitor would be CDC = 2SnomHc/E2

DC.Arguably, the current balance shown in (18) is akin to the power

balance inside the HVDC link for steady-state operating conditionswhen the derivative term becomes zero. Thus, whenever the cur-rent/power balance is disturbed, voltage variations will appear inthe DC link. As shown in Fig. 3, the dynamic control of the HVDC’s DCvoltage is carried out by using the DC current entering the inverterconverter, IDCI, as the control variable. The error between the actualvoltage EDCI and EDCInom is used by a PI controller, with gains Kpedcand Kiedc, to obtain new values of DC current IDCI.

The differential and algebraic equations arising from the DCvoltage dynamic controller are

dIDCIaux

dt= Kiedc (EDCI − EDCInom) (20)

IDCI = Kpedc (EDCI − EDCInom) + IDCIaux (21)

Simultaneously, the rectifier converter must ensure that theactive power leaving this station be kept at the scheduled value Psch.From Fig. 1, it can be inferred that the power entering the inverterstation is Psch minus the power loss incurred by the DC cable resis-tor. If the inverter station is selected to perform the control of theDC link voltage EDCI then the DC voltage at the rectifier’s side, EDCR,can be computed at any time by applying Kirchhoff’s voltage lawin the DC circuit, as follows:

EDCR = EDCI − RDCIDCR (22)

The angular aperture between the phase-shifting angle of therectifier �R and the voltage angle �vR is related to the powerexchange occurring at any time between the network and the rec-tifier’s DC bus. Hence, the angular difference �R = �vR − �R is alsoa key parameter that requires proper regulation with the aim toachieve the scheduled active power transfer Psch from the rectifierstation towards the inverter station. Then, the pursued power bal-ance on the DC side will now be given by the following expression:P0vR + Psch = 0, as shown in Fig. 4.

The equations that allow the assessment of the dynamic

behaviour for the scheduled power controller are

d�Raux

dt= Kipdc (Psch + P0vR) (23)

114 L.M. Castro et al. / Electric Power System

�

cTtieamr

m

3

ttwtsdeoosuk[

0

y

F

Fig. 4. DC-power transfer controller for the VSC-HVDC link.

R = Kppdc (Psch + P0vR) + �Raux (24)

The AC voltage dynamic control of the VSC-HVDC calls for twoontrol loops, as shown in the first-order control blocks of Fig. 5.he modulation indices maI and maR are responsible for either, con-rolling the voltage magnitudes at the AC sides of the rectifier andnverter stations of at the scheduled values, VvI0 and VvR0, or toxert the fixed reactive power setpoint: Q ref

I and Q refR . The controls

re designed in such a way that the modulation indices maI andaR are readjusted at every time step according to the voltage or

eactive power commands.The differential equations representing the dynamics of the

odulation indices when voltage control is selected are

d (dmaR)dt

= KmaR (VvR0 − VvR) − dmaR

TmaR(25)

d (dmaI)dt

= KmaI (VvI0 − VvI) − dmaI

TmaI(26)

. Dynamic frame of reference

In this paper the interest is in assessing the effectiveness ofhe new VSC-HVDC model to regulate voltage magnitude at eithererminal of the AC system, following a change in the power net-ork such as a step change in system load or the tripping of a

ransmission line or transformer. Hence, the solution method pre-ented in [13], is selected to implement the VSC-HVDC modeleveloped in Section II. This approach combines the set of algebraicquations (27) representing the power network with the systemf differential equations (28) describing the dynamic behaviourf the synchronous generators and their controls, to obtain theolution as a function of time in a unified frame of reference. Itses the implicit trapezoidal method (see Appendix C) which isnown to be numerically stable, preserving a reasonable accuracy

13,14],

= f (X, Y) (27)

˙ = g (X, Y, t) (28)

ig. 5. AC-bus voltage controllers: (a) rectifier station and (b) inverter station.

s Research 126 (2015) 111–120

where X and Y are vectors of variables that are computed at discretepoints in time.

These equations are efficiently solved using the NR method.In this case, the conventional power flow Jacobian matrix, J, isenlarged to accommodate the partial derivatives that arise fromthe discretised differential equations and its control variables. TheNR method provides an accurate solution to the set of equationsgiven by F(Z) = 0, by solving for �Z in the linearised problem J�Z = −F(Z), in a repetitive fashion. In this case Z is a vector that containsthe network’s state variables and the state variables pertainingto the synchronous generators and their controls or, indeed, anyother control device such as the VSC-HVDC link. In an expandedform,

⎡⎣ �P

�Q

�F(y)

⎤⎦ = −

⎡⎢⎢⎢⎢⎢⎢⎣

∂�P

∂�

∂�P

∂V

∂�P

∂y

∂�Q

∂�

∂�Q

∂V

∂�Q

∂y

∂F(y)∂�

∂F(y)∂V

∂F(y)∂y

⎤⎥⎥⎥⎥⎥⎥⎦⎡⎣��

�V

�y

⎤⎦ (29)

where �P and �Q are the active and the reactive power mis-match vectors, respectively; F(y) is a vector that contains thediscretised differential equations of each machine or controllingdevice; ��, �V and �y represent the vectors of incremen-tal changes in nodal voltage angles and magnitudes, as wellas the state variables arising from each differential equation.The NR method starts from an initial guess for Z0 and updatesthe solution at each iteration i, i.e., Zi+1 = Zi + �Zi, until a pre-defined tolerance is fulfilled. In this unified solution, all of thestate variables are adjusted simultaneously in order to computethe new equilibrium point of the power system at every timestep.

3.1. Discretisation and linearisation of the VSC-HVDC equationsfor dynamic simulations

To enable a suitable representation in this unified frame of ref-erence, the VSC-HVDC differential equations are discretised andexpressed in the form of a mismatch equation in the same form asthat of the network’s active and reactive power mismatch equa-tions:

FEDCI = EDCI,t−�t + �t

2EDCI,t−�t −

(EDCI,t − �t

2EDCI,t

)= 0 (30)

FIDCIaux = IDCIaux,t−�t + �t

2IDCIaux,t−�t

−(

IDCIaux,t − �t

2IDCIaux,t

)= 0 (31)

F�Raux = �Raux,t−�t + �t

2�Raux,t−�t −

(�Raux,t − �t

2�Raux,t

)= 0

(32)

FdmaR= dmaR,t−�t + �t

2dmaR,t−�t −

(dmaR,t − �t

2dmaR,t

)= 0

(33)

Fdm = dmaI,t−�t + �tdmaI,t−�t −

(dmaI,t − �t

dmaI,t

)= 0 (34)

aI 2 2

where,

EDCI,t = C−1DC

(−IDCR,t − IDCI,t

)(35)

System

E

I

I

�

�

d

d

d

d

Hcpep(iActemat(

�

�

mecm

�

� �Raux

� �

wmwwaoot

The steady-state conditions that are employed to start thedynamic simulation are calculated through the conventional

L.M. Castro et al. / Electric Power

˙ DCI,t−�t = C−1DC

(−IDCR,t−�t − IDCI,t−�t

)(36)

˙DCIaux,t = Kiedc

(EDCI,t − EDCInom

)(37)

˙DCIaux,t−�t = Kiedc

(EDCI,t−�t − EDCInom

)(38)

˙ Raux,t = Kipdc

(Psch + P0vR,t

)(39)

˙ Raux,t−�t = Kipdc

(Psch + P0vR,t−�t

)(40)

maR,t = T−1maR

[KmaR

(VvR0 − VvR,t

)− dmaR,t

](41)

maR,t−�t = T−1maR

[KmaR

(VvR0 − VvR,t−�t

)− dmaR,t−�t

](42)

maI,t = T−1maI

[KmaI

(VvI0 − VvI,t

)− dmaI,t

](43)

maI,t−�t = T−1maI

[KmaI

(VvI0 − VvI,t−�t

)− dmaI,t−�t

](44)

The Eqs. (30)–(44) govern the dynamic behaviour of the VSC-VDC model. The first two equations capture the DC voltage andurrent performance of the DC link when the energy balance iserturbed owing to a disturbance in the AC network. Likewise, thequation involving the angular aperture �R (32) deals with theower unbalance present in the DC link. Also, the Eqs. (33) and34) enable the computation of the new values of the modulationndices with which the target AC voltages are acquired for the actualC network’s operating conditions. In order to link the VSC-HVDC’sontrol variables with the grid’s state variables at nodes vR and vI,he algebraic power mismatch Eqs. (7)–(10) must be used. How-ver, to complete the model for dynamic simulation purposes, twoore algebraic equations are needed. One for calculating the volt-

ge EDCR at the rectifier station’s DC bus (45) and another to enablehe HVDC to achieve the power balance at the inverter’s DC bus46).

E0R = EDCR − EDCI + RDCIDCR (45)

P0I = EDCIIDCI − P0vI (46)

Eqs. (7)–(10), (45)–(46) and (30)–(34) constitute the set ofismatch equations that must be assembled together with the

quations of the whole network, synchronous generators and theirorresponding controllers. The linearised form of the VSC-HVDCathematical model is given by,

F = −[

J11 J12

J21 J22

]�z (47)

F =[

�PvR �QvR �PvI �QvI �E0R �P0I FEDCI FIDCI,aux F

z =[

��vR �VvR ��vI �VvI �EDCR ��I �EDCI �IDCI,aux

here J11 comprises the first-order partial derivatives of the powerismatch equations and inner VSC-HVDC’s mismatch equationsith respect to the network’s and VSC-HVDC’s state variables. Like-ise, J12 contains the first order partial derivatives arising from the

lgebraic mismatch equations with respect to the control variablesf the VSC-HVDC link. The matrix J21 consists of partial derivativesf the VSC-HVDC’s discretised differential equations with respecto the AC voltages and angles, the phase-shifting angle �I and the

s Research 126 (2015) 111–120 115

FdmaRFdmaI

]T

�Raux �dmaR �dmaI

]T

DC voltage EDCR. Lastly, J22 is a matrix that accommodates the first-order partial derivatives of the VSC-HVDC’s discretised differentialequations with respect to their own control variables.

J11 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂�PvR

∂�vR

∂�PvR

∂VvR

0 0∂�PvR

∂EDCR0

∂�QR

∂�vR

∂�QvR

∂VvR

0 0∂�QvR

∂EDCR0

0 0∂�PvI

∂�vI

∂DPvI

∂VvI

0∂�PvI

∂�I

0 0∂�QvI

∂�vI

∂�QvI

∂VvI

0∂�QvI

∂�I

∂�E0R

∂�vR

∂�E0R

∂VvR

0 0∂�E0R

∂EDCR0

0 0∂�P0I

∂�vI

∂�P0I

∂VvI

0∂�P0I

∂�I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

J12 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0∂�PvR

∂dmaR

0

0 0 0∂�QR

∂dmaR

0

∂�PvI

∂EDCI0 0 0

∂�PvI

∂dmaI

∂�QvI

∂EDCI0 0 0

∂�QvI

∂dmaI

∂�E0R

∂EDCI0 0

∂�E0R

∂dmaR

0

∂�P0R

∂EDCI0 0 0

∂�P0R

∂dmaI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

J21 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂FEDCI

∂�vR

∂FEDCI

∂VvR

0 0∂FEDCI

∂EDCR0

0 0 0 0 0 0

∂F�Raux

∂�vR

∂F�Raux

∂VvR

0 0∂F�Raux

∂EDCR

0

0∂FdmaR

∂VvR

0 0 0 0

0 0 0∂FdmaI

∂VvI

0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

J22 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂FEDCI

∂EDCI0 0 0

∂FEDCI

∂dmaI

∂FIDCI,aux

∂EDCI

∂FIDCI,aux

∂IDCI,aux0 0 0

0 0∂F�Raux

∂�Raux

∂F�Raux

∂dmaR

0

0 0 0∂FdmaR

∂dmaR

0

0 0 0 0∂FdmaI

∂dmaI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(48)

Newton–Raphson power flow algorithm including the VSC-HVDClink steady-state model, as discussed in Section 2.2. Such a solu-tion will provide adequate starting conditions to ensure reliabledynamic simulations.

116 L.M. Castro et al. / Electric Power Systems Research 126 (2015) 111–120

date the proposed VSC-HVDC model.

4

4

bsttagttCpuona

s(V7atnHo(mss7av

tftprtitci

FDetRtcsb

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.85

0.9

0.95

1

1.05

1.1

DC

vol

tage

[p.u

]

EDCREDCIEDCREDCI

Prop osedmod el

Simulink mod el Step change in DC power

Step change in DC voltage

relatively small variations may be explained by the very differentmodelling and solution approaches used by the two quite differentsimulation techniques used for the comparison, the initial steady-state values of the converters’ indices and most importantly due to

0.4

0.6

0.8

1

1.2

DC

pow

er [p

.u]

SimulinkProposed model

Step change in DC power

Step change in DC voltage

Fig. 6. Test system used to vali

. Study Cases

.1. Validation of the new VSC-HVDC model

The prowess of the new VSC-HVDC link model is demonstratedy carrying out a comparison against the widely-used EMT-typeimulation software Simulink®. It should be mentioned that bothypes of simulation tools enable dynamic assessments of elec-rical power networks but they take a fundamentally differentpproach. Simulink® represents every component of the powerrid by means of RLC circuits and their corresponding differen-ial equations require discretisation at rather small time steps, inhe order of micro-seconds, to ensure a stable numerical solution.onversely, the solution of the RMS-type model introduced in thisaper requires only one phase of the network (positive sequence),sing fundamental-frequency phasors of voltages and currents aspposed to the three-phase representation along with instanta-eous waveforms of voltages and currents used in an EMT tool suchs Simulink®.

The VSC-HVDC model comparison is carried out using a ratherimple power system comprising two independent AC networks2000 MVA, 230 kV, 50 Hz) which are interconnected through aSC-HVDC link (200 MVA, ±100 kV DC) with a DC cable length of5 km, as shown in Fig. 6. Both converter stations comprise each

step-down transformer, AC filters, converter reactor, DC capaci-ors and DC filters, where the changes of the transformers’ tap areot simulated. The model of the power system including the VSC-VDC link together with its parameters can be found in the sectionf ‘demos’ in Simulink® as: VSC-Based HVDC Transmission SystemDetailed Model), whereas the parameters of the new VSC-HVDC

odel are shown in Appendix D. To ensure a reliable numericalolution, the EMT-type simulation package discretises the powerystem and the control system with a sample time of 7.406 �s and4.06 �s, respectively, whereas for the developed RMS-type model,n integration step of 1 ms is used. All the results shown in p.u.alues are based on the HVDC station’s rating.

Initially the rectifier station is set to control the active powerransmission at Psch = 200 MW (1 p.u.), the inverter is responsibleor controlling the DC voltage at EDCInom = 200 kV (1 p.u.). The rec-ifier and inverter stations are set to comply with a fixed reactiveower command of 0 p.u. and −0.1 p.u., respectively. In order toeach the steady-state equilibrium point in Simulink®, the simula-ion is run up to t = 1 s. At this point, the active power transmissions reduced from 200 MW to 100 MW, that is, a −50% step is appliedo the reference scheduled DC power. Furthermore, at t = 3 s, a stephange of −5% is applied to the reference DC voltage of the inverter,.e., the DC voltage is decreased from 1 p.u to 0.95 p.u.

The DC voltages at the converters’ DC terminals are shown inig. 7 corresponding to cases where step changes in the referenceC power and DC voltage are applied. As expected, some differ-nces can be seen from the results obtained using both solutionechniques. The dynamic performance of the DC voltages of theMS-type model follows well the dynamic pattern obtained by

he switching-based HVDC model simulated in Simulink®. A veryonsiderable difference exists between the two approaches at thetart of the simulation (0.5 s of the simulation), a fact that cane explained by the very different manner in which both power

Time [s]

Fig. 7. DC voltage performance for the proposed and Simulink VSC-HVDC model.

system simulations are initialised; our proposed VSC-HVDC systemuses an accurate starting condition furnished by a power flow solu-tion whereas the Simulink® model starts from its customary zeroinitial condition, i.e., the currents and voltages of the inductors andcapacitors, respectively, are set to zero at t = 0 s.

Similar conclusions can be drawn when analysing the dynamicresponse of the DC power following the application of the stepchanges in DC power and DC voltage, as shown in Fig. 8. As for thechange in the DC power reference, it can be seen that the power sta-bilises in no more than 0.5 s; this shows the rather quick responseand robustness afforded by the dynamic controls of the VSC-HVDClink even in the event of a drastic change in the transmitted DCpower. On the other hand, the step change in the DC voltage ref-erence causes momentary power flow oscillations in the DC linkwhich are also damped out quite rapidly.

The dynamic behaviour of the HVDC’s modulation indices aredepicted in Fig. 9. The negative step change in the DC power refer-ence yields a very noticeable variation in the modulation indices;the dynamic performance of the modulation indices as calculatedby both Simulink® and the proposed HVDC model, follow the sametrend although an exact match was not expected. After the firstdisturbance, a steady-state error of 0.87% and 1.67% is obtainedfor the modulation indices of the rectifier and inverter, respec-tively. Similarly, once the oscillations due to the step change inthe reference DC voltage have been damped, the differences in themodulation indices stand at 0.07% and 1.16%, respectively. These

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time [s]

Fig. 8. DC power performance for the proposed and Simulink VSC-HVDC model.

L.M. Castro et al. / Electric Power Systems Research 126 (2015) 111–120 117

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.8

0.9

1

1.1

Time [s]

Mod

ulat

ion

inde

x

maR

maI

maR

maI

Proposed model

Step change in DC power

Step change in DC voltageSimulink model

Fig. 9. Modulation indices performance for the proposed and Simulink VSC-HVDCmodel.

tff

aetVwEjHstv

4

Ft4a

TC

Fig. 10. Relevant area of the New England test system.

he DC voltage behaviour as this has a strong impact on the per-ormance of the modulation indices. More importantly, the resultsurnished by the two software simulations follow the same trend.

For the sake of comparison, Table 1 shows the VSC-HVDC resultss obtained by the new model and the Simulink® model at differ-nt points in time of simulation. Table 1 also shows the computingimes required to simulate the test system using both the RMS-typeSC-HVDC model and the EMT-type simulation tool Simulink®,ith the new model being approximately nine times faster than the

MT simulation. The significant computational time saving withouteopardising the accuracy of the results makes the developed VSC-VDC link model a suitable option for large-scale power system

imulations, specifically in studies that require longer simulationimes such as those involving synchronous generators’ frequencyariations and long-term voltage stability issues.

.2. New England test system with embedded VSC-HVDC link

The New England test system [17] is modified, as shown in

ig. 10, to incorporate the model of a VSC-HVDC with the parame-ers shown in Appendix D. The transmission line connecting nodes

and 14 is replaced by a VSC-HVDC link. The DC cable resistance isssumed to be 0.24% on the VSCs’ base: Snom = 300 MVA, resulting

able 1omparison of VSC-HVDC variables for the proposed model and the Simulink® model.

Time (s) Proposed model

EDCR EDCI maR maI PDCI

t = 1− 1.0105 1.0000 0.8553 0.8296 0.9895t = 3− 1.0053 1.0000 0.8389 0.8172 0.4974t = 5 0.9556 0.9500 0.9329 0.8711 0.4971

Computing time: 19.76 s

Fig. 11. Voltage performance at different nodes of the network.

in the same resistance value as that of the replaced transmissionline for the system’s base: 0.08%. The rectifier and inverter stations,VSCR and VSCI, exert voltage control at their respective AC terminalsat VvR = 1.01 p.u and VvI = 1.03 p.u, respectively. For the steady-stateconditions, the high-voltage side of the LTC transformers, whichcorrespond to nodes 4 and 14, are held fixed at the same voltagelevel as those for the converters’ terminals, VvR and VvI; under theseconditions, the LTC’s taps are computed through the steady-statepower flow algorithm; their values are kept constant during thedynamic solution. In addition to providing reactive power control,the HVDC link performs active power regulation at the rectifierstation’s DC bus at Psch = 100 MW, which implies that the activepower is drawn from node 4 and injected to node 14, as depictedin Fig. 10. The active and reactive powers presented in the analysisfor the steady-state and dynamic operating regimes are given atthe high-voltage side of each LTC transformer. All results shown inp.u. values are based on the HVDC station’s ratings.

During steady state, the rectifier station is delivering 153.359MVAr to the network so as to uphold its target voltage magnitudewith a modulation ratio of 0.8282, whereas the inverter stationoperates with a modulation ratio of 0.8423, injecting 27.306 MVArto the grid. In the case of the active power flowing through the HVDCsystem, the power entering the rectifier station stands at 101.159MW and the power leaving the inverter station takes a value of99.641 MW. It is clear that the difference between these two pow-ers is the total power loss incurred by the HVDC system includingthat produced by the DC link cable. Taking as a reference the nomi-nal apparent power for each converter Snom, the total power lossesstand at 0.504% of which 0.386% corresponds to the rectifier stationand 0.112% to the inverter station whilst the power loss producedby Joule’s effect in the DC cable stands at 0.006%, recalling that itsmagnitude is dependent on the length of the DC transmission line.Table 2 shows the main VSC-HVDC results as given by the steady-state power flow solution which serves the purpose of initialisingthe dynamic simulation.

Hence, using values from the steady-state power flow solution,it is easy to proceed with the calculation of the initial values for thecontrol variables taking part in the dynamics of the VSC-HVDC link;these are employed to initialise the dynamic simulation of the test

network, when subjected to the disconnection of the transmissionlines linking nodes 25-2, 2-3 and 3-4, at t = 0.1 s.

Simulink® model

EDCR EDCI maR maI PDCI

1.0007 0.9968 0.8499 0.8301 0.9897 1.0044 0.9996 0.8476 0.8005 0.4977 0.9554 0.9494 0.9336 0.8827 0.4954

Computing time: 180.39 s

118 L.M. Castro et al. / Electric Power Systems Research 126 (2015) 111–120

Table 2Computed VSC-HVDC variables by the power flow solution.

Qgen EDC ma � Beq LTC’s tap Ploss

VSCR 153.359 MVAr 1.0002 p.u. 0.8282 −2.6655◦ 0.5182 p.u. 1.0252 1.1595 MWVSCI 27.306 MVAr 1.0000 p.u. 0.8423 6.8416◦ 0.0918 p.u. 1.0044 0.3383 MW

Fig. 12. Dynamic behaviour of the converters’ modulation indices.

F

atAiobta

epttalDi

Fig. 15. DC voltage behaviour for the rectifier and inverter.

ig. 13. Reactive power generated by both converters making up the HVDC link.

Fig. 11 shows the voltage magnitudes at various nodes following change in the network’s topology. During the transient period, thearget voltage set point is achieved very quickly by the action of theC-bus voltage controllers that regulate the converters’ modulation

ndices maR and maI, as shown in Fig. 12. The rather prompt actionf both controllers leads to very rapid reactive power injection atoth converters AC terminals, as can be seen in Fig. 13, resulting inhe very effective damping of the voltage oscillations and enabling

smooth voltage recovery throughout the grid.As soon as the disturbance takes place in the AC system, the

nergy balance in the DC link is broken; the voltage sags that takelace at both converters AC terminals reduce the active power beingransferred through the DC link; the current -IDCR that flows fromhe rectifier station VSCR towards the inverter station VSCI dropsbruptly from 0.166 p.u to 0.146 p.u, as illustrated by the blue

ine in Fig. 14. A momentary mismatch between both convertersC currents is then produced because the DC current cannot be

nstantly re-established due to the time constants involved in the

Fig. 14. DC current behaviour for the rectifier and inverter.

Fig. 16. VSC-HVDC’s AC active power and DC-power transfer behaviour.

current controller of the inverter station; as a result, DC voltagedeviations take place, as shown in Fig. 15, reaching a minimumvalue of 0.978 p.u. during the transient event. Nevertheless, oncethis controller starts responding to the DC voltage variations, thecurrent IDCI, depicted by the green line in Fig. 14, starts tracingthe DC current of the rectifier IDCR to compensate for the voltagedrop, enabling a speedy recovery of the DC link voltage. It shouldbe remarked that in view of the fact that the cable resistance isrelatively small, so is the voltage drop along the DC transmissionline, resulting in quite similar magnitudes and, of course, dynamicbehaviours of the voltages at both DC buses, EDCR and ECDI.

The simulation results for the active power and DC-power trans-fer following the disconnection of the transmission lines close tothe HVDC link are illustrated in Fig. 16. The blue line represents theactive power entering the high-voltage side of the load-tap changertransformer coupled to the rectifier station whereas the greenline characterizes the active power performance at the inverter’sLTC’s high-voltage side. The power difference represents the powerlosses incurred by the VSC-HVDC link, including those produced bythe DC cable. The DC-power transfer Psch consisting of the productof voltage EDCR and current −IDCR, is also shown in the same graph.Since the voltage and current controls have been shown to operateefficiently, as illustrated in Fig. 14 and Fig. 15, then a fast powerrecovery is achieved in spite of the severe disturbance occurredin the network. Given that the power flowing from the rectifiertowards the inverter station has been brought back to its initialtarget power transfer of 0.333 p.u, the deviation of the power angle�R suffers a mere marginal increase, as can be seen in Fig. 17, only

to agree with the new reached steady-state conditions where dif-ferent currents and, therefore, active power losses are produced.Fig. 17 also shows the behaviour of the phase-shifting angles of

L.M. Castro et al. / Electric Power System

Fi

tavn

5

sapidHAD

sesadwmct

hwascab

liowaawa

A

tV

u

ig. 17. Dynamic performance of various angles involved in the VSC-HVDC dynam-cs.

he rectifier and inverter converters, �R and �I. As expected, thesengles follow the same pattern as those obtained by the network’soltage angles �, at the nodes where the VSC-HVDC system is con-ected.

. Conclusions

A new VSC-HVDC model for RMS dynamic simulations of large-cale power systems has been introduced in this paper. This isn all-encompassing model that facilitates the back-to-back andoint-to-point representation of the VSC-HVDC by simply mod-

fying the DC cable resistance value. The model possesses fouregrees of freedom, a characteristic that conforms to actual VSC-VDC links, i.e., it exerts simultaneous voltage control on its twoC terminals and at its DC bus and transmitted power through theC link.

The model solution is carried out using the NR method whicholves simultaneously the algebraic and differential equations atach time step. The point-to-point VSC-HVDC model comprises twoeries-connected VSC stations and a DC cable. Each VSC model usesn ideal phase-shifting transformer as its core element. The con-uction losses and the switching losses of the HVDC converters areell captured in the model. Furthermore, the VSC-HVDC dynamicodel is fitted with independent controllers for the AC and DC cir-

uits to represent the quite distinct dynamic performances of thewo control circuits.

The point-to-point VSC-HVDC model introduced in this paperas been validated using the EMT simulation tool Simulink®,here the results obtained from the two fundamentally different

pproaches agreed quite well with each other. Nevertheless, it washown that the greater level of detail needed in an EMT solutionomes with an onerous price-tag to pay in terms of a very consider-ble computational time compared to the computing time incurredy the new RMS-type VSC-HVDC model.

The performance of the new VSC-HVDC model was tested in aarger network which is widely used in academic circles, compris-ng 39 nodes. The VSC-HVDC link model performed well in termsf its modelling flexibility and in attaining the set control targets. Itas shown that the DC current controller and the rectifier’s angular

perture controller operate efficiently to stabilize both the DC volt-ge and the DC power, respectively. Likewise, the AC voltage controlas quickly achieved due to the proper action of the controllers

cting upon the modulation indices of the converters.

ppendix A.

EDC: DC voltage. IDC: DC current. CDC: DC capacitance. RDC: DCransmission line resistance. Inom: VSC’s nominal current. Snom:SC’s nominal power. Psch: Scheduled active power. ma: VSC’s mod-

lation index. ϕ: VSC’s phase-shifting angle. k2 =√

3⁄8: Constant

s Research 126 (2015) 111–120 119

for a two-level, three-phase VSC. G0: Shunt resistor which accountsfor the switching losses of the VSC. Beq: Equivalent variable shuntsusceptance. Y1: VSC’s series admittance associated with conduc-tion losses and interface magnetics. V: Complex nodal voltage.I: Complex current injection. S: Complex nodal power injection.Kpedc, Kiedc: Proportional and integral gains for the DC voltage con-trol. Kppdc, Kipdc: Proportional and integral gains for the DC powercontrol. Kma, Tma: Proportional gain and time constant for the mod-ulation index control. Subscripts R and I stand for rectifier andinverter, respectively.

Appendix B.

In connection with Fig. 2, the voltage and current relationshipsin the ideal phase-shifting transformer are:

V1

EDCI= k2maI∠�I

1and

k2maI∠ − �I

1= I2

I1(A.1)

The current through the impedance connected between vI and1 is:

I′1 = Y1 (VvI − V1) = Y1VvI − k2maI∠�IY1EDCI = IvI (A.2)

where Y1 = (R1I + jX1I)−1.At node 0vI, the following relationship holds,

I0vI = −I2 + GswIEDCI = −k2maI∠ − �IY1VvI + k22m2

aIY1EDCI

+ jBeqIk22m2

aIEDCI + GswIEDCI (A.3)

Rearranging Eqs. (A.2) and (A.3) yields:[IvI

I0vI

]=[

Y1 −k2maI∠�IY1

−k2maI∠ − �IY1 k22m2

aI

(Y1 + jBeqI

)+ GswI

] [VvI

EDCI

]

(A.4)

Therefore the power injections would be,(SvI

S0vI

)=(

VvI

EDCI

)

×{(

Y∗1 −k2maI∠ − ϕIY∗

1

−k2maI∠ϕIY∗1 k2

2m2aI

(Y∗

1 − jBeqI

)+ GswI

) (V∗

vI

EDCI

)}

(A.5)

Appendix C.

As a first step, the implicit trapezoidal method calls for alge-braizing any differential equation y by means of expressing itsstep-by-step solution as an integral form,

y(t) − F(X(t), Y(t)) = 0 (B.1)

Y(t) − Y(t − �t) −t∫

t−�t

F(X(t), Y(t))dt = 0 (B.2)

Assuming that all functions F(·) vary linearly over the time inter-val [t − �t, t], the area under the integral can be approximated by atrapezium; the differential algebraic equation given in the form of

a mismatch equation is then,

FY = Yt−�t + �t

2Yt−�t −

(Yt − �t

2Yt

)= 0 (B.3)

1 System

A

sBdpXKp

tsdHpHK

L

Q

tv

R

[

[

[

[

[

[

20 L.M. Castro et al. / Electric Power

ppendix D.

Parameters used in Section 4.1: (i) The parameters of the powerystem can be found in the section of ‘demos’ in Simulink® as: VSC-ased HVDC Transmission System (Detailed Model). (ii) VSC-HVDCata based on the HVDC rating Snom = 200 MVA: RDC = 0.042704.u.; EDCInom = 1.0 p.u.; G0I = G0R = 2e-3 p.u.; R1I = R1R = 2e-3 p.u.;1I = X1R = 1e-3 p.u.; Hc = 0.014; Kpedc = 0.6; Kiedc = 35; Kppdc = 0;ipdc = 5; KmaI = KmaR = 25; TmaI = TmaR = 0.02; ZLTCtransf = 0.005 + j0.15.u.

Parameters used in Section 4.2: (i) Synchronous genera-ors are equipped with exciter, automatic voltage regulator,peed governor and hydro turbine. Generators and networkata are available in [17]. (ii) VSC-HVDC data based on theVDC rating Snom = 300 MVA: RDC = 0.0024 p.u.; EDCInom = 1.0.u.; G0I = G0R = 2e-3 p.u.; R1I = R1R = 2e-3 p.u.; X1I = X1R = 0.01 p.u.;c = 0.007; Kpedc = 0.05; Kiedc = 1.0; Kppdc = 0.002; Kipdc = 0.075;maI = KmaR = 25.0; TmaI = TmaR = 0.02; XLTCtransf = 0.05 p.u. (iii)

oad models: PL = PL0

(0.2 + 0.4

(V/V0

)+ 0.4

(V/V0

)2)

and

L = QL0

(0.2 + 0.4

(V/V0

)+ 0.4

(V/V0

)2)

, where, PL0 and QL0 are

he nominal active and reactive powers drawn by the load at ratedoltage V0.

eferences

[1] S. Dodds, B. Railing, K. Akman, B. Jacobson, T. Worzyk, B. Nilsson, HVDC VSC

(HVDC light) transmission–operating experiences, Cigré (2010) 1–9 (B4-2032010).

[2] T. Larsson, A. Edris, D. Kidd, F. Aboytes, Eagle pass back-to-back tie: a dualpurpose application of voltage source converter technology, IEEE Power Eng.Soc. 3 (2001) 1686–1691.

[

[

s Research 126 (2015) 111–120

[3] D.V. Hertem, M. Ghandhari, Multi-terminal VSC-HVDC for the Euro-pean supergrid: obstacles, Renew. Sust. Energy Rev. 14 (2010)3156–3163.

[4] H.F. Latorre, M. Ghandhari, L. Söder, Active and reactive power control of aVSC-HVDC, Electr. Power Syst. Res. 78 (2008) 1756–1763.

[5] E. Acha, C.R. Fuerte-Esquivel, H. Ambriz-Perez, C. Angeles-Camacho, FACTSModeling and Simulation in Power Networks, John Wiley & Sons,Chichester, England, 2005.

[6] S. Ruihua, Z. Chao, L. Ruomei, Z. Xiaoxin, VSCs based HVDC and its controlstrategy, IEEE/PES Trans. Dist. Conf. Exhibit. (2005) 1–6.

[7] X.P. Zhang, Multiterminal voltage-sourced converter-based HVDCmodels for power flow analysis, IEEE Trans. Power Syst. 19 (2004)1877–1884.

[8] S.P. Teeuwsen, Simplified dynamic model of a voltage-sourced converter withmodular multilevel converter design, IEEE/PES Power Syst. Conf. Expo. PSCE(2009) 1–6.

[9] S. Cole and R. Belmans, Modelling of VSC HVDC using coupled current injectors,IEEE PES General Meeting—Conversion and Delivery of Electrical Energy in the21st Century (2008) 1–8.

10] S. Cole, R. Belmans, A proposal for standard VSC HVDC dynamic mod-els in power system stability studies, Electr. Power Syst. Res. 81 (2011)967–973.

11] S. Chiniforoosh, J. Jatskevich, A. Yazdani, V. Sood, V. Dinavahi, J.A. Mar-tinez, A. Ramirez, Definitions and applications of dynamic average modelsfor analysis of power systems, IEEE Trans. Power Deliv. 25 (2010)2655–2669.

12] M.M.Z. Moustafa, S. Filizadeh, A VSC-HVDC model with reduced computationalintensity, IEEE PES Gen. Meet. (2012) 1–6.

13] M. Rafian, M.J.H. Sterling, M.R. Irving, Real-time power system simulation, IEEProc. Gen. Trans. Dist. 134 (1987) 206–223.

14] H.W. Dommel, N. Sato, Fast transient stability solutions, IEEE Trans. PowerAppar. Syst. PAS-91 (1972) 1643–1650.

15] E. Acha, B. Kazemtabrizi, L.M. Castro, A new VSC-HVDC model for powerflows using the Netwon–Raphson method, IEEE Trans. Power Syst. 28 (2013)2602–2612.

16] M.M. de Oliveira, Power Electronics for Mitigation of Voltage Sags and ImprovedControl of AC Power Systems, Royal Institute of Technology (KTH), Stockholm,2000 (Doctoral Dissertation).

17] M.A. Pai, Energy Function Analysis for Power System Stability, Kluwer AcademicPublishers, New York, USA, 1989.