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Cheah-Mane, Marc, Sainz, Luis, Liang, Jun, Jenkins, Nick and Ugalde Loo, Carlos Ernesto 2017.

Criterion for the electrical resonance stability of offshore wind power plants connected through

HVDC links. IEEE Transactions on Power Systems 32 (6) , pp. 4579-4589.

10.1109/TPWRS.2017.2663111 file

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Transactions on Power Systems

IEEE TRANSACTIONS ON POWER SYSTEMS 1

Criterion for the Electrical Resonance Stability ofOffshore Wind Power Plants Connected through

HVDC LinksMarc Cheah-Mane,Student Member, IEEE,Luis Sainz, Jun Liang,Senior Member, IEEE,

Nick Jenkins,Fellow, IEEE,and Carlos E. Ugalde-Loo,Member, IEEE

Abstract—Electrical resonances may compromise the stabilityof HVDC-connected Offshore Wind Power Plants (OWPPs). Inparticular, an offshore HVDC converter can reduce the dampingof an OWPP at low frequency series resonances, leading to systeminstability. The interaction between offshore HVDC convertercontrol and electrical resonances of offshore grids is analyzedin this paper. An impedance-based representation of an OWPPis used to analyze the effect that offshore converters have onthe resonant frequency of the offshore grid and on systemstability. The positive-net-damping criterion, originally proposedfor subsynchronous analysis, has been adapted to determinethe stability of the HVDC-connected OWPP. The reformulatedcriterion enables the net-damping of the electrical seriesreso-nance to be evaluated and establishes a clear relationship be-tween electrical resonances of the HVDC-connected OWPPs andstability. The criterion is theoretically justified, with a nalyticalexpressions for low frequency series resonances being obtainedand stability conditions defined based on the total damping ofthe OWPP. Examples are used to show the influence that HVDCconverter control parameters and the OWPP configuration haveon stability. A root locus analysis and time-domain simulations inPSCAD/EMTDC are presented to verify the stability conditions.

Index Terms—electrical resonance, offshore wind power plant,HVDC converter, positive-net-damping stability criterion.

I. I NTRODUCTION

H ARMONIC instabilities have been reported in practi-cal installations such as BorWin1, which was the first

HVDC-connected Offshore Wind Power Plant (OWPP) [1],[2]. More recently, electrical interactions between offshoreHVDC converters and series resonances have been identifiedin DolWin1 and highlighted by CIGRE Working Groups aspotential causes of instability during the energization oftheoffshore ac grid [3], [4], [5]. Such interactions are knownas electrical resonance instabilities [6]. In HVDC-connectedOWPPs, the long export ac cables and the power transformerslocated on the offshore HVDC substation cause series reso-nances at low frequencies in the range of 100∼ 1000 Hz[3] - [5], [7]. Moreover, the offshore grid is a poorly dampedsystem directly connected without a rotating mass or resistiveloads [3], [2]. The control of the offshore HVDC converter canfurther reduce the total damping at the resonant frequenciesuntil the system becomes unstable.

This work was supported in part by the People Programme (Marie CurieActions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ (grant 317221 and project title MEDOW) and in part by the Ministeriode Economía y Competitividad (grant ENE2013-46205-C5-3-R).

M. Cheah-Mane, J. Liang, N. Jenkins and C.E. Ugalde-Loo are with theSchool of Engineering, Cardiff University, CF24 3AA Cardiff, UK. (e-mail:CheahM, LiangJ1, JenkinsN6, [email protected]).

L. Sainz is with Electric Engineering Department, Universitat Politecnicade Catalunya (UPC), ETS d’Enginyeria Industrial de Barcelona, 08028Barcelona, Spain (e-mail: [email protected]).

Electrical resonance instability has been studied with animpedance-based representation by several authors. In [8], [9]Voltage Source Converters (VSCs) are modeled as Thévenin orNorton equivalents with a frequency-dependant characteristic.Nyquist and Bode criteria are used to analyze electricalresonance stability [2], [10]. In [9], a clear relationshipbe-tween electrical resonances and the phase margin conditionisestablished, but the complexity of the loop transfer functionlimits further analysis of the elements that cause instability.Alternative approaches, the passivity conditions of the system[11], [12] and the positive-net-damping criterion [13], [14],have been used to define stability conditions.

A number of studies on electrical resonance instability inOWPPs have been reported in the literature. In [15] and [16],the impact of electrical resonances on Wind Turbine (WT)converters is investigated. However, few studies are focused onthe interactions between resonances and the offshore HVDCconverter. In [17], a modal analysis in a HVDC-connectedOWPP is used to characterize possible resonances and toassess the stability of the offshore converters. Also, in [9] and[18] the impact of resonances on offshore HVDC convertersis analyzed using an impedance-based representation.

In this paper, the impact that low frequency series reso-nances have on the voltage stability of HVDC-connected OW-PPs is analyzed and discussed. Preliminary work was reportedin [19], where the stability criterion presented in this paperwas assessed with examples. This paper furthers the initialcontributions of [19] by providing a formal framework for theanalysis of electrical resonance stability in HVDC-connectedOWPPs. An impedance-based representation is used to iden-tify resonances and to assess stability considering the effect ofthe offshore converters. The resonance stability of an OWPPis determined using an alternative approach to the positive-net-damping criterion [13]. This has been reformulated toevaluate the net-damping for electrical series resonancesandto provide a clear relationship between electrical resonances ofthe OWPP and stability. The main contributions of this paperare summarized as follows:

• The alternative approach to the positive-net-damping cri-terion is demonstrated using the phase margin condition.This criterion defines the relation between the dampingat electrical series resonances and system stability.

• The relationship between the total damping and resonantfrequencies with the poles of the system is demonstrated.This relationship shows that the pole analysis and thepositive-net-damping criterion provide the same informa-tion about resonance stability.





• Analytical expressions of the low frequency series reso-nances are proposed considering the effect of VSC con-trollers. These expressions are employed to calculate theresonant frequencies where the total damping is evaluatedto determine system stability.

• A stable area of an OWPP is defined as a function ofthe HVDC converter control parameters and the OWPPconfiguration. Such an area is obtained from the dampingof the OWPP and indicates conditions of stability.

The effect that the HVDC converter control parameters andthe OWPP configuration have on stability is shown usingexamples. For completeness, root locus analysis and time-domain simulations in PSCAD/EMTDC are used to validatethe stability conditions. The examples presented in this paperare complementary to those included in [19].

II. I MPEDANCE-BASED REPRESENTATION OF AN

HVDC-CONNECTEDOWPP

An impedance-based representation is suitable for the mod-eling of converters of an HVDC-connected OWPP wheneverdetailed design information is not available. Such a converterrepresentation offers advantages as it can easily be combinedwith the equivalent impedance of the offshore ac grid to char-acterize resonant frequencies. It is also possible to considerthe effect of the converter controllers. Moreover, the stabilityassessment methods for impedance-based representations aresimple and less computational intensive compared to othertraditional methods such as eigenvalue analysis [2], [10].

The configuration of an HVDC-connected OWPP is shownin Fig. 1. Type 4 WTs are connected to strings of thecollector system through step-up transformers from low tomedium voltage. Each WT grid side VSC has a couplingreactor and a high frequency filter represented as an equivalentcapacitor. The strings are connected to a collector substation,where transformers step-up from medium to high voltage. Thecollector transformer in Fig. 1 is an equivalent representationof 4 transformers that are connected in parallel [3]. Exportcables send the generated power to an offshore HVDC sub-station, where a VSC based Modular Multilevel Converter(MMC) operates as a rectifier and delivers the power to thedc transmission system. The dc transmission system and theonshore HVDC converters are not represented in this study.

Fig. 2 shows an impedance-based model of the HVDC-connected OWPP suitable for the analysis of electrical reso-nances and stability. The ac cables of the export and collectorsystem are modeled as singleπ sections with lumped param-eters and the transformers are modeled as RL equivalents.These models are accurate enough to characterize the lowfrequency resonances that are responsible for stability issues[3]. The VSCs are represented by equivalent circuits, whichinclude the frequency response of the controller. The offshoreVSC is represented by a Thévenin equivalent as it controlsthe ac voltage of the offshore grid [9], [20]; however, Nortonequivalents are used to represent the WT VSCs since theycontrol current [8], [20].

III. I MPEDANCE-BASED MODEL OFVSCS

The VSC models are represented in a synchronousdq frameand the Laplaces domain, where complex space vectors are

Fig. 1. General scheme of an HVDC-connected OWPP.

Fig. 2. Impedance-based model of an HVDC-connected OWPP forresonanceand stability analysis.

denoted with boldface letters for voltages and currents asv =vd + jvq and i = id + jiq.

A. Offshore VSC model

The offshore VSC controls the ac voltage of the offshoregrid. Fig. 3a describes the control structure of this converter.If the VSC uses a MMC topology, high frequency filters arenot required and only a voltage control loop is considered[18], [21]. Additionally, the internal MMC dynamics can beneglected if a circulating current control is implemented [18].A control action based on a PI controller is expressed as:

vhvsc = FPI,v(vr − vpoc) (1)

FPI,v = kp,v +ki,vs

(2)

wherevhvsc is the reference voltage for the offshore converter,

vr is the control reference voltage at the Point of Connection(POC),vpoc is the voltage measured at the POC andFPI,v isthe PI controller for the voltage control loop.

The dynamics across the equivalent coupling inductance ofthe offshore converter are expressed as:

vhvsc = vpoc + ic(R

hf + sLh

f + jω1Lhf) (3)

whereic is the current from the HVDC converter,Lhf is the

coupling inductance,Rhf is the equivalent resistance ofLh

f

andω1 = 2πf1 rad/s (f1 = 50 Hz). The coupling inductanceis equal toLh

f = Larm/2 + Lhtr, whereLarm is the arm





inductance of the MMC andLhtr is the equivalent inductance

of the offshore HVDC transformers.A Thévenin equivalent of the offshore VSC (see Fig. 2) is

obtained by combining (1) and (3):

vpoc = vr ·Ghc − ic · Zh

c (4)

Ghc =

FPI,v

1 + FPI,v; Zh

c =Rh

f + sLhf + jω1L

hf

1 + FPI,v

(5)

whereGhc is the voltage source transfer function andZh

c isthe input-impedance of the converter.

B. Wind Turbine VSC modelEach WT is equipped with a back-to-back converter, but

only the grid side VSC is represented in this study. Its controlis based on an ac current loop employing a PI controller asshown in Fig. 3b. The dc voltage outer loop is not representedin the WT VSC model since its dynamic response is slow;i.e.there is sufficient bandwidth separation with the inner currentloop [2], [22]. This ensures that there are no interactionsbetween harmonic resonances and the outer loops, which arenot of interest in this paper. A Norton equivalent of the WTconverter (see Fig. 2) is obtained as:

iwt = ir ·Gwc − vwt · Y w

c (6)

whereiwt is the current from the WT VSC,ir is the controlreference current,Gw

c is the current source transfer function,vwt is the voltage after the coupling filter andY w

c is the input-admittance of the VSC.Gw

c andY wc are expressed as [8]:

Gwc =

FPI,c

Rwf + sLw

f + FPI,c; Y w

c =1−Hv

Rwf + sLw

f + FPI,c(7)

FPI,c = kp,c +ki,cs

; Hv =αf

s+ αf

(8)

whereFPI,c is the PI controller of the current loop,Lwf the

coupling inductance,Rwf the equivalent resistance ofLw

f , Hv

the low pass filter of the voltage feed-forward term [8] andαf the bandwidth ofHv. The PI design is based on [8], [23],with proportional and integral gains given askp,c = αcL

wf and

ki,c = αcRwf , and the bandwidth of the current control byαc.

IV. STABILITY ANALYSIS OF HVDC-CONNECTEDOWPPS

The stability analysis considers the impedance-based circuitpresented in Fig. 4, where the offshore grid is modeled with anequivalent circuit (further explained in Section V). A similarrepresentation can be found in [9].

The impedances were expressed in the stationaryαβ frame[6], [11], which is denoted in boldface letters for voltagesandcurrents asvs = vα + jvβ and is = iα + jiβ. The current inthe stationaryαβ frame and the Laplaces-domain is given as:

isc = (vs

rGhc − i

srG

wc Z

wc )

Th

︷︸︸︷

1/Zg

1 + Zhc /Zg

(9)

whereZg = Zgrideq +1/Y w

c is the equivalent impedance of theOWPP from the offshore VSC andT h is the OWPP closedloop transfer function, which can be also expressed as:

T h(s) =M(s)

1 +M(s)N(s)=

M(s)

1 + L(s)(10)

(a)

(b)Fig. 3. Control structures: (a) Offshore HVDC converter and(b) WT gridside converter.

Equivalent offshore grid: Zg

vrGc

Zc vpoc ic iwt

Ycw

Gcirw

Zeqgrid

h

h

s

s s s

s

Fig. 4. Equivalent impedance-based circuit of an HVDC-connected OWPPwith representation of offshore grid circuit.

where M(s) = 1/Zg is the open loop transfer function,N(s) = Zh

c is the feedback transfer function andL(s) is theloop transfer function.

Assuming that the voltage and current sources in Fig. 4are stable when they are not connected to any load [10], thestability of the OWPP can be studied in the following ways:

• By analyzing the poles ofT h or the roots ofZg+Zhc = 0.

• By applying the Nyquist stability criterion ofZhc /Zg [10].

• By considering the passivity ofT h [6], [11].

In addition to the previous alternatives, a variation to thepositive-net-damping criterion given in [13], [14] is hereemployed instead to analyze system stability. The criterion hasbeen reformulated to evaluate electrical resonance stability asexplained in section IV-B.

A. Passivity

A linear and continuous-time systemF (s) is passive if [11]:

• F (s) is stable and,• ReF (jω) > 0 ∀ ω, which is expressed in terms of

the phase as−π2 < argF (jω) < π

2 . This conditioncorresponds to a non-negative equivalent resistance inelectrical circuits.

Passivity can be applied to determine the stability of closedloop systems [6], [11]. A system represented by the closedloop transfer function in (10) is stable ifM(s) andN(s) arepassive since−π < argL(jω) < π ∀ ω. This implies thatthe Nyquist stability criterion forL(s) is satisfied. Therefore,the OWPP is stable ifZg andZh

c are passive. When the HVDCconverter is connected to a passive offshore grid,Zg is passive





and the stability only depends on the passivity conditions ofthe converter input-impedance,Zh

c .In no-load operation (i.e. when only the passive elements

of the OWPP are energized), the passivity ofZg is ensuredas the WTs are assumed to be disconnected from the offshoregrid. However, the WTs represent active elements when theyare connected to the offshore grid (i.e.Zg can have a negativeresistance), which may compromise the OWPP stability.

B. Positive-net-damping stability criterionThe criterion states that a closed loop system is stable if

the total damping of the OWPP is positive at the followingfrequencies: (i) open loop resonant frequencies and (ii) lowfrequencies where the loop gain is greater than 1 [13]. How-ever, it does not provide a clear relation between electricalresonances of the OWPP and system stability. This increasesthe complexity of analyzing the impact that system parametershave on resonance stability.

The criterion presented in [13] has been reformulated toevaluate the net-damping for electrical series resonances. Theapproach proposed in this paper is developed from the phasemargin condition [9]. If stability is evaluated in terms ofthe phase margin,L(jω) = M(jω)N(jω) must satisfy thefollowing conditions at angular frequencyω:

|M(jω)N(jω)| = 1, (11)

−π < argM(jω)N(jω) < π ∀ ω. (12)M(jω) andN(jω) in (11) and (12) can be expressed in termsof equivalent impedances as:

1

M(jω)= Zg(jω) = Rg(ω) + jXg(ω) (13)

N(jω) = Zhc (jω) = Rh

c (ω) + jXhc (ω) (14)

Also, the equivalent impedance from the voltage sourcevsrG

hc in Fig. 4 is expressed as:

Zheq(jω) = Zh

c (jω) + Zg(jω) (15)

Phase margin condition (11) is equivalent to:

Rhc (ω)

2 +Xhc (ω)

2 = Rg(ω)2 +Xg(ω)

2 (16)The resistive components in ac grids and VSCs may be

usually neglected compared to the reactive components. There-fore, Rg ≪ Xg, Rh

c ≪ Xhc and (16) is simplified to:

Xhc (ω) = ±Xg(ω) (17)

The electrical series resonances observed from the voltagesourcevs

rGhc in Fig. 4 correspond to frequencies whereZh

eq in(15) has a dip or a local minimum. If the resistive componentsare neglected, the series resonance condition is reduced to:

ImZheq(jωres) ≈ 0 ⇒ Xh

c (ωres) ≈ −Xg(ωres) (18)

It can be observed that (18) is a particular case of (17);i.e. the series resonance condition ofZh

eq coincides with thestability condition|M(jω)N(jω)| = 1 given by (11).

Phase margin condition (12) can be expressed in terms ofthe imaginary part ofL(jω) as follows:

If d|L(jω)|dω

> 0 : 0 < argL(jω) < π ⇒⇒ Rg(ω)X

hc (ω)−Rh

c (ω)Xg(ω) > 0

If d|L(jω)|dω

< 0 : −π < argL(jω) < 0 ⇒⇒ Rg(ω)X

hc (ω)−Rh

c (ω)Xg(ω) < 0

(19)

If the resonance condition in (18) is combined with (19):

If d|L(jω)|dω

> 0 : Xhc (ωres)[Rg(ωres) +Rh

c (ωres)] > 0

If d|L(jω)|dω

< 0 : Xhc (ωres)[Rg(ωres) +Rh

c (ωres)] < 0(20)

It can be shown (see Appendix A) that if the offshore grid iscapacitive (i.e. Xg < 0) and the HVDC converter is inductive(i.e. Xh

c > 0), then d|L(jω)|dω

> 0. On the other hand, ifthe offshore grid is inductive (i.e. Xg > 0) and the HVDCconverter is capacitive (i.e. Xh

c < 0), then d|L(jω)|dω

< 0. Byconsidering the previous conditions, (20) is simplified to:

RT (ωres) = Rg(ωres) +Rhc (ωres) > 0 (21)

where resistanceRT represents the total damping of thesystem, resistanceRh

c the HVDC converter damping andresistanceRg the offshore grid damping.

It can be observed that (21) is equivalent to the positive-net-damping criterion in [13], but evaluated for the seriesresonances ofZh

eq. Therefore, the offshore HVDC VSC isasymptotically stable if the total damping of the system,RT , ispositive in the neighborhood of an electrical series resonance.The advantage of this criterion with respect to the passivityapproach is that the stability can be ensured even ifZg andZhc are not passive because it considers the contribution of

both terms in the closed loop system.It should be noted that if the resistive components of the

offshore grid and HVDC VSC are large compared to thereactive elements (e.g.Xg/Rg < 10 andXh

c /Rhc < 10), the

approximations in (17) and (18) are not valid and this criterioncannot be used.

C. Relation between total damping and poles of the system

The HVDC-connected OWPP is a high order system withseveral poles. However, the system response is governed by adominant poorly-damped pole pair. If this pole pair is relatedto the electrical series resonance, impedancesZh

c and Zg

around this resonance can be approximated as:

Zhc,res(s) ≈ Rh

c + sLhc ; Zg,res(s) ≈ Rg +

1

sCg

(22)

where Cg is the equivalent capacitor of the offshore gridimpedance when the frequency is close the resonance. Using(18), the series resonance reduces toωres = 1/

√

LhcCg.

The poles related to the series resonance are obtained from1 + Zh

c,res(s)/Zg,res(s) = 0, yielding:

s =−(Rh

c +Rg)Cg ±√

(Rhc +Rg)2C2

g − 4LhcCg

2LhcCg

(23)

Considering that(Rhc + Rg)

2C2g ≪ 4Lh

cCg, equation (23) isapproximated to:

s ≈ −Rhc +Rg

2Lhc

± j1

√

LhcCg

(24)

The imaginary part of the closed loop system poles cor-responds to the resonant frequency. Also, the real part of thepoles is correlated to the total damping,Rh

c+Rg, as mentionedin [14]. Therefore, there is a pair of poles that represent theseries resonance and can be used to identify instabilities.





V. RESONANCE CHARACTERIZATION

In this section, the low frequency series resonances of anOWPP are characterized. It is useful to identify resonantfrequencies in an OWPP since they can destabilize an offshoreHVDC converter. To this end, the frequency response ofZheq(jω) is here used to identify electrical resonances. Due

to the complexity of the VSC and offshore grid equations,simplifications are used to obtain analytical expressions of theresonant frequencies.

A. Simplifications of the OWPP impedance model

Fig. 5 shows that the frequency response of a VSCimpedance can be simplified to RL equivalents above 100Hz. The input-impedance of the VSCs was represented inan αβ frame (see Fig. 4). To achieve this, a reference frametransformation fromdq toαβ was performed using the rotations → s − jω1 [6], [15]. For frequencies higher thanω1, theoffshore VSC impedance,Zh

c (s− jω1), is approximated to:

Rhc =

Rhf

1 + kp,v; Lh

c =Lhf

1 + kp,v(25)

Similarly, the WT VSC impedance,Zwc (s−jω1) = 1/Y w

c (s−jω1), is approximated to:

Rwc = Rw

f + (αf + αc)Lwf ; Lw

c = Lwf (26)

The previous simplifications do not consider the VSCs as ac-tive elements sinceRh

c andRwc are positive for all frequencies.

Fig. 6 shows the equivalent model of the HVDC-connectedOWPP with the simplified VSC and cable models. The capac-itor Cec represents the export cable capacitance. The inductiveand resistive components of the export cable are small enoughto be combined with the RL equivalent of the transformers andthe HVDC converter. Also, the collector cables are removedbecause their equivalent inductance and capacitance are smalland only affect the response at high frequencies, which arenot considered in this study.

When the collector cables are removed, the aggregationof WTs is reduced to a combination of parallel circuitsindependent to the collector system topology. Fig. 7 showsthe OWPP model under this scenario, which is equivalent tothe model in Fig. 4. The parameters of the aggregated modelare defined as follows:

• Rcstr andLcs

tr are the RL values of the collector transform-ers.

• Rwtr,a andLw

tr,a are the RL values of the aggregated WTtransformers:

Rwtr,a = Rw

tr/N ;Lwtr,a = Lw

tr/N (27)

whereN is the number of WTs andRwtr andLw

tr are theRL values of one WT transformer.

• Rwc,a andLw

c,a are the RL values of the aggregated WTconverters:

Rwc,a = Rw

c /N ;Lwc,a = Lw

c /N (28)

• Cwf,a is the equivalent capacitance of the aggregated WT

low pass filters:Cw

f,a = Cwf ·N (29)

whereCwf is the capacitance of one WT low pass filter.

Frequency (Hz)100 102 104

|Zch| (

dB)

-50

0

50

100 Without simp.With simp.

(a) Offshore HVDC VSC

Frequency (Hz)100 102 104

|Zcw

| (dB

)

-50

0

50

100

150

(b) WT grid side VSC

Fig. 5. Frequency response with and without simplifications(parameters inAppendix B withkp,v= 1, ki,v= 500).

Fig. 6. Impedance-based model of an HVDC-connected OWPP with simpli-fied VSC and cable models (indicated in grey rectangles).

Fig. 7. Impedance-based model of an HVDC-connected OWPP with aggre-gation of collector system.

Fig. 8 shows the frequency response of the equivalentoffshore grid impedance,Zh

eq, with and without simplificationsto VSC and cable models. It can be observed that if simplifica-tions are made the 50 Hz resonance of the VSC control is notexhibited; however, the frequency response agrees well withthat of the un-simplifiedZh

eq over 200 Hz and up to 1 kHz.





Fig. 8. Frequency response of OWPP impedance without and with VSC andcable simplifications (parameters in Appendix B andkp,v= 1, ki,v= 500).

Additionally, the simplification of the collector cables slightlyshifts the series resonance from 459 Hz to 497 Hz. In light ofthese results, it can be concluded that the simplified frequencyresponse represents a good approximation for low frequencyresonances in the range of 200∼ 1000 Hz.

B. Analytical expression for the series resonant frequencyThe expression of the lowest series resonant frequency of

Zheq is obtained for no-load operation and when WTs are

connected. The resistances are neglected as they only havea damping effect on resonance (i.e. they barely modify theresonant frequency).

In no-load operation, the WTs are not connected and thecontribution of the collector system at low frequencies isnegligible. Therefore, the OWPP impedanceZh

eq in (15) isequivalent to an LC circuit with a resonant frequency:

fnloadres =

1

2π√

LhcCec

(30)

The lowest series resonant frequency when WTs are con-nected has been obtained following an algebraic calculationusing Fig. 7:

f loadres =

1

2π

√

b−√b2 − 4ad

2aa = CecL

wc,a(L

cstr + Lw

tr,a)LhcC

wf,a

b = CecLhc (L

wc,a + Lcs

tr + Lwtr,a)+

+Cwf,aL

wc,a(L

hc + Lcs

tr + Lwtr,a)

d = Lhc + Lw

c,a + Lcstr + Lw

tr,a

(31)

Expressions (30) and (31) are employed to calculate thefrequencies where the total system damping is evaluated todetermine stability.

VI. V OLTAGE STABILITY ANALYSIS

The modified positive-net-damping criterion was applied toanalyze the impact of electrical series resonances in the voltagestability of an HVDC-connected OWPP. The effects of theoffshore HVDC converter control and the OWPP configurationare considered in the study. For completeness, the root locus ofthe system and time-domain simulations in PSCAD/EMTDCare used to confirm the results.

The cable model simplifications considered in the resonancecharacterization are used in the stability analysis given that thelow frequency response is well-represented and the dampingcontribution from the cable resistances can be neglected.However, the VSC simplifications in (25) and (26) are notconsidered, because the converters are not represented as

active elements. The system is analyzed in no-load operationand when WTs are connected based on the OWPP describedin Appendix B.A. No-load operation

In no-load operation, the positive-net-damping stabilitycri-terion only includes the damping contribution of the offshoreconverter,Rh

c , because the export and collector cables arepassive elements with a small resistance and thus can beneglected (i.e.Rg ≈ 0). Therefore, condition (21) is reduced toRh

c (ωres) > 0, which is equivalent to analyzing the passivityof the HVDC converter control at a resonant frequency.

Stability is ensured if the electrical series resonance islocated in a frequency region with positive resistance. Thisregion is determined using the zero-crossing frequencies ofRh

c (i.e.Rhc (ω) = ReZh

c (ω) = 0) in (5). The two followingsolutions are obtained:

ωcut1 = ω1 = 2π(50)

ωcut2 =ω1

1−ki,vL

hf

Rhf (1 + kp,v)

(32)

Whenωcut2 < 0, the only zero-crossing frequency consideredis 50 Hz andRh

c is negative forω > 2π(50). Therefore, theconverter is always unstable for resonant frequencies above50 Hz. If ωcut2 > 0, thenRh

c is negative for2π(50) < ω <ωcut2 and positive forω > ωcut2. In this case, the converteris stable for frequencies higher thanωcut2 since the resonanceis located in a positive-resistance region. Thus, the offshoreHVDC converter is stable whenRh

c has two zero-crossingfrequencies (ωcut2 > 0 and ωres > ωcut2). The followinginequalities are obtained by combining (30) and (32):

ωcut2 > 0 ⇒ Rhf (1 + kp,v)− ki,vL

hf > 0

ωres > ωcut2 ⇒ Rh2f (1 + kp,v)

2 − 2RhfL

hf (1 + kp,v)ki,v−

−ω21R

h2f Lh

fCec(1 + kp,v) + k2i,vLh2f > 0

(33)

Fig. 9 shows the stability area (Rhc (ωres) > 0) defined by

(33) as a function of the control parameters of the offshoreHVDC converter,kp,v and ki,v, and the export cable length,lcb. It is observed that when the cable length increases thestable area is reduced.

Fig. 10 shows the root locus of the low frequency resonantpoles for parametric variations ofkp,v, ki,v and lcb. It shouldbe emphasized that these poles are not complex conjugate dueto the transformation of the VSC input impedance from asynchronousdq to a stationaryαβ reference frame, whichintroduces complex components. The increase of cable lengthmoves the resonance to lower frequencies sinceCec increases.As kp,v increases, the resonance shifts to higher frequenciesgiven thatLh

c in (25) decreases. Changes inki,v do not affectthe resonant frequency. The system becomes unstable whenone of the resonant poles moves to the positive side of thereal axis; this is equivalent to have a negative damping. It canbe observed that the stability conditions of the resonant polesagree with the stable areas shown in Fig. 9.

Figs. 11 and 12 show examples of stable and unstablecases whenki,v is modified. The intersection betweenZh

c





Fig. 9. Stable area of offshore HVDC converter in no-load operation asfunction of kp,v, ki,v and lcb (the stable and unstable examples of Fig. 11and 12 are marked with circles).

(a) Cable length variation from 1 to100 km (kp,v = 0.1, ki,v = 3).

(b) Variation of ki,v from 1 to 5(kp,v = 0.1, lcb = 10 km).

(c) Variation of kp,v from 0 to 1(ki,v = 5, lcb = 10 km)

Fig. 10. Root locus of OWPP in no-load operation for variations of exportcable length and ac voltage control parameters.

andZg (i.e.1/|M(jω)| = |N(jω)|) approximately determinesthe series resonant frequency, as defined in (18). When thesystem is stable the resonant frequency is located in a positive-resistance region ofZh

c , as shown in Fig. 11a. Also, followingthe Nyquist criterion, the Nyquist curve encircles(−1, 0) inanti-clockwise direction and the open loop system does nothave unstable poles. Therefore, the system is stable as it doesnot have zeros with positive real part. Although the ac voltagecontrol can be designed to ensure stability, all the poles havea low damping. This slows down the dynamic response, asshown in Fig. 11c, which is not acceptable for the operationof the offshore converter.

When the system is unstable the resonant frequency islocated in the negative-resistance region ofZh

c , as shown inFig. 12a. Following the Nyquist criterion, the Nyquist curveencircles(−1, 0) in clockwise direction and the open loopsystem does not have unstable poles. Therefore, the system isunstable because the total number of zeros with positive realpart is 1. In Fig. 12c, the voltage at POC shows oscillations at309 Hz due to the resonance instability identified in Fig. 12a.B. Connection of Wind Turbines

When the WTs are connected to the offshore ac grid, theWT converters modify the low frequency resonance locationand the total damping. The stability conditions are discussed,

(a) Frequency response:Rhc , Zh

eq , Zhc , Zg .

(b) Nyquist curve ofZhc /Zg (positive freq.).

(c) Instantaneous and RMS voltages at POC. Step change is applied at 1 s

Fig. 11. Stable example in no-load operation withkp,v = 0.1, ki,v = 3 andlcb = 10 km.

(a) Frequency response:Rhc , Zh

eq , Zhc , Zg .

(b) Nyquist plot ofZhc /Zg (pos freq.).

(c) Instantaneous and RMS voltages at POC. Step change is applied at 1 s

Fig. 12. Unstable example in no-load operation withkp,v = 0.1, ki,v = 5

and lcb = 10 km.





but the expressions for the zero-crossing frequencies ofRT arenot obtained analytically due to the complexity of the system.

Fig. 13 shows the stable area defined byRT (ωres) > 0.There is a significant increase of the stable region whenthe WTs are connected. Therefore, the ac control parameterscan be modified for a larger range of values to improve thedynamic response without compromising stability.

Fig. 13. Stable area of offshore HVDC converter as a functionof kp,v andki,v and the number of connected WTs (the stable and unstable examples ofFig. 15 and 16 are marked with a circle).

(a) Variation of kp,v from 0 to 80(ki,v = 3, N = 80)

(b) Variation ofki,v from 1 to 1200(kp,v = 0.1, N = 80)

(c) Variation of WTs from 1 to 80(kp,v = 1, ki,v = 500)

Fig. 14. Root locus of OWPP for variations of ac voltage control parametersand number of WTs (N = 80).

Fig. 14 shows the root locus of the low frequency resonantpoles for different ac voltage control parameters and numberof WTs. The connection of WTs improves the resonancestability because the associated poles move to the left handside of the real axis and increase the damping of those lowfrequency modes. This damping contribution of the WTs isalso mentioned in [2]. The stability conditions of the resonantpoles agree with the stable area shown in Fig. 13. Also, theresonance moves to higher frequencies whenkp,v and thenumber of WTs increases, as shown in Fig. 14.

Figs. 15-17 describe two situations where the ac voltagecontrol is designed to have a fast dynamic response (e.g.kp,v = 1 andki,v = 500) and the number of WTs decreasesfrom 40 to 20. When all the WTs are connected, the offshoreconverter is stable because the resonance is located in apositive-resistance region, as shown in Fig. 15a. The converterintroduces a negative resistance at the resonant frequency, but

(a) Frequency response ofRhc + Rg , Zh

eq , Zhc andZg

(b) Frequency response ofRhc andRg

(c) Nyquist plot ofZhc /Zg (positive freq.)

Fig. 15. Stable example when 40 WTs are connected,kp,v = 1 andki,v =

500.

(a) Frequency response ofRhc + Rg , Zh

eq , Zhc andZg .

(b) Frequency response ofRhc andRg

(c) Nyquist plot ofZhc /Zg (positive freq.)

Fig. 16. Unstable example when 20 WTs are connected,kp,v = 1 andki,v = 500.





the total damping is compensated byRg, as shown in Fig. 15b.When the number of WTs reduces to 20 the offshore converterbecomes unstable since the resonance lies in the negative-resistance region, as shown in Fig. 16a. In this case,Rg cannotcompensateRh

c , as shown in Fig. 16b. Also, the Nyquistcurve agrees with the positive-net-damping criterion in bothsituations (Figs. 15c and 16c). In Fig. 17, the instantaneousvoltages at POC show oscillations at 444 Hz when the numberof WTs is reduced at 1 s; this is due to the resonance instabilityidentified in Fig. 16a.

The variation of connected WTs can be caused by switchingconfigurations during commissioning phases or during outagesdue to maintenance or contingencies [3]. As shown by theprevious examples, a sudden reduction in the number ofWTs should be carried out with care as this can lead toinstability. Active damping can be implemented as a virtualresistor in the offshore HVDC converter to compensate thenegative resistance introduced by the ac voltage control for alloperational states. This will allow a design of the ac voltagecontrol to have a fast dynamic response without compromisingthe stability.

Fig. 17. Instantaneous and RMS voltages at POC when the number of WTs isreduced from 40 to 20 at 1 s. The ac voltage control parametersarekp,v = 1

andki,v = 500

VII. C ONCLUSION

Instabilities in BorWin1 have increased interest in electricalresonance interactions in HVDC-connected OWPPs. The CI-GRE Working Groups suggest that series resonances can befound in the range of a few hundred Hz. These resonances caninteract with the offshore HVDC converter control leading tosystem instability.

This paper has reformulated the positive-net-damping cri-terion to define the conditions of stability of an HVDC-connected OWPP as a function of the ac voltage controlparameters of the HVDC converter and the configuration ofthe OWPP. The modified criterion is evaluated for electricalseries resonances based on the phase margin condition. Thisreduces the complexity of the stability analysis. In addition,expressions for the low frequency resonance are obtained fromsimplified VSC and cable models.

Risk of detrimental resonance interaction increases in no-load operation and when a limited number of WTs are con-nected. This is due to the poor damping exhibited by the seriesresonance of the offshore grid and the resonance location atthe lowest frequencies. The HVDC converter reduces the totaldamping at the resonant frequency if the control is designedto have a fast dynamic response. Resistive elements or activedamping are necessary to compensate the negative resistance

of the converter control for all possible operational states andto allow a fast dynamic response.

APPENDIX AIf Rh

c (ω) ≪ Xhc (ω) andRg(ω) ≪ Xg(ω), the loop transfer func-

tion is approximated asL(jω) ≈ Xhc (ω)/Xg(ω) and its derivative

as a function ofω is:

d|L(jω)|

dω≈

1

|Xg(ω)|3d|Xh

c (ω)|

dω−

|Xhc (ω)|

|Xg(ω)|2d|Xg(ω)|

dω(34)

Specific conditions ford|L(jω)|dω

can be defined depending on theoffshore grid and HVDC converter impedances:

• If the offshore grid impedance is inductive,Xg > 0, and theHVDC converter impedance is capacitive,Xh

c < 0:

d|Xg(ω)|

dω> 0 &

d|Xhc (ω)|

dω< 0 ⇒

d|L(jω)|

dω< 0 (35)

• If the offshore grid impedance is capacitive,Xg < 0, and theHVDC converter impedance is inductive,Xh

c > 0:

d|Xg(ω)|

dω< 0 &

d|Xhc (ω)|

dω> 0 ⇒

d|L(jω)|

dω> 0 (36)

APPENDIX BOWPPDESCRIPTION

A 480 MW OWPP is considered in this study. A total number of80 WTs is distributed in 16 strings of 5 units.

Offshore HVDC VSC: MMC-VSC; rated power, 560 MVA; ratedvoltage, 320 kV; arm inductance,Larm = 183.7 mH.

Offshore HVDC transformer: 2 units in parallel; rated power, 280MVA; rated voltages, 350 kV/220 kV; equivalent inductance andresistance at 220 kV,Lh

tr = 99.03 mH, Rhtr = 0.86 Ω.

Export cables: 2 cables in parallel; length,lec = 10 km; equivalentlumped parameters per cable,Lec = 4 mH, Rec = 0.32 Ω, Cec =1.7 µF.

Collector transformers: 4 units in parallel; rated power of eachunit, 140 MVA; rated voltages, 220 kV/33 kV; equivalent inductanceand resistance of the transformers at 220 kV,Lcs

tr = 20.63 mH,Rcs

tr = 0.22 Ω.WT transformers: rated power, 6.5 MVA; rated voltages, 33 kV/0.9

kV; equivalent inductance and resistance at 33 kV,Lwtr = 31 mH,

Rwtr = 1.46 Ω.WT grid side VSC: 2-level VSC; rated power, 6.5 MVA; rated

voltage, 0.9 kV; coupling inductance,Lwf = 50 µH; coupling

resistance,Rwf = 0.02 mΩ; equivalent capacitance of high frequency

filter, Cwf = 1 mF; low pass filter bandwidth,αf = 50; current

control bandwidth,αc = 1000.

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[3] Working Group B3.36, “Special Considerations for AC Collector Sys-tems and Substations Associated with HVDC - Connected Wind PowerPlants,” CIGRE, Tech. Rep. March, 2015.

[4] Working Group B4.55, “HVDC Connection of Offshore Wind PowerPlants,” CIGRE, Tech. Rep. May, 2015.

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[7] M. Bradt, B. Badrzadeh, E. Camm, D. Mueller, J. Schoene, T. Siebert,T. Smith, M. Starke, and R. Walling, “Harmonics and resonance issuesin wind power plants,” in2011 IEEE Power and Energy Society GeneralMeeting. IEEE, Jul. 2011, pp. 1–8.





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Marc Cheah-Mane (S’14) received the degree inindustrial engineering from the School of IndustrialEngineering of Barcelona (ETSEIB), Technical Uni-versity of Catalonia (UPC), Barcelona, Spain, in2013. He was a researcher in Centre d’InnovacioTecnologica en Convertidors Estatics i Acciona-ments (CITCEA) from 2010 to 2013. Since Octoberof 2013, he is pursuing a Ph.D at Cardiff University,Cardiff, Wales. His research interests include renew-able energies, power converters, high-voltage directcurrent systems, electrical machines and microgrids.

Luis Sainz was born in Barcelona, Spain, in 1965.He received his BS degree in Industrial Engineeringand his PhD degree in Industrial Engineering fromUPC-Barcelona, Spain, in 1990 and 1995 respec-tively. Since 1991, he has been a Professor at theDepartment of Electrical Engineering of the UPC.His research interest lies in the areas of powerquality and grid-connected VSCs.

Jun Liang (M’02-SM’12) received the B.Sc. degreein Electrical Power Engineering from HuazhongUniversity of Science and Technology, Wuhan,China, in 1992 and the M.Sc. and Ph.D. degreesfrom China Electric Power Research Institute, Bei-jing, China, in 1995 and 1998, respectively. From1998 to 2001, he was a Senior Engineer with ChinaElectric Power Research Institute. From 2001 to2005, he was a Research Associate at ImperialCollege, London, U.K.. From 2005 to 2007, he wasa Senior Lecturer at the University of Glamorgan,

Wales, U.K.. Currently, he is a Reader at the School of Engineering, CardiffUniversity, Wales, U.K.. His research interests include FACTS devices/HVDC,power system stability and control, power electronics, andrenewable powergeneration.

Nick Jenkins (M’81-SM’97-F’05) receivedthe B.Sc. degree from Southampton University,Southampton, U.K., in 1974, the M.Sc. degree fromReading University, Reading, U.K., in 1975, andthe Ph.D. degree from Imperial College London,London, U.K., in 1986. He is currently a Professorand Director, Institute of Energy, Cardiff University,Cardiff, U.K. Before moving to academia, hiscareer included 14 years of industrial experience,of which five years were in developing countries.While at university, he has developed teaching and

research activities in both electrical power engineering and renewable energy.

Carlos E. Ugalde-Loo (M’02) was born in MexicoCity. He received the B.Sc. degree in Electron-ics and Communications Engineering from ITESM,MÃl’xico (2002); the M.Sc. degree in ElectricalEngineering from IPN, MÃl’xico (2005); and thePh.D. degree in Electronics and Electrical Engineer-ing from the University of Glasgow, Scotland, U.K.(2009). In 2010 he joined the School of Engineeringin Cardiff University, Wales, U.K., and is currentlya Senior Lecturer in Electrical Power Systems. Hisacademic expertise includes power system stability

and control, grid integration and control of renewables, HVDC transmission,modeling of dynamic systems, and multivariable control.

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