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

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 6, NOVEMBER 2017 4579 Criterion for the Electrical Resonance Stability of Offshore Wind Power Plants Connected Through HVDC Links Marc Cheah-Mane , Student Member, IEEE, Luis Sainz, Jun Liang, Senior Member, IEEE, Nick Jenkins, Fellow, IEEE, and Carlos Ernesto Ugalde-Loo , Member, IEEE Abstract—Electrical resonances may compromise the stability of HVDC-connected offshore wind power plants (OWPPs). In particular, an offshore HVDC converter can reduce the damping of an OWPP at low-frequency series resonances, leading to the system instability. The interaction between offshore HVDC converter control and electrical resonances of offshore grids is analyzed in this paper. An impedance-based representation of an OWPP is used to analyze the effect that offshore converters have on the resonant frequency of the offshore grid and on system stability. The positive-net-damping criterion, originally proposed for subsynchronous analysis, has been adapted to determine the stability of the HVDC-connected OWPP. The reformulated criterion enables the net damping of the electrical series resonance to be evaluated and establishes a clear relationship between electrical resonances of the HVDC-connected OWPPs and stability. The criterion is theoretically justified, with analytical expressions for low-frequency series resonances being obtained and stability conditions defined based on the total damping of the OWPP. Examples are used to show the influence that HVDC converter control parameters and the OWPP configuration have on stability. A root locus analysis and time-domain simulations in PSCAD/EMTDC are presented to verify the stability conditions. Index Terms—Electrical resonance, HVDC converter, offshore wind power plant, positive-net-damping stability criterion. I. INTRODUCTION HARMONIC instabilities have been reported in practical installations such as BorWin1, which was the first HVDC- connected Offshore Wind Power Plant (OWPP) [1], [2]. More recently, electrical interactions between offshore HVDC converters and series resonances have been identified in DolWin1 and highlighted by CIGRE Working Groups as potential causes Manuscript received July 8, 2016; revised December 21, 2016; accepted January 28, 2017. Date of publication February 20, 2017; date of current version October 18, 2017. This work was supported in part by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under Grant 317221 and Project title MEDOW and in part by the Ministerio de Econom´ ıa y Competitividad under Grant ENE2013-46205- C5-3-R. Paper no. TPWRS-01039-2016. M. Cheah-Mane, J. Liang, N. Jenkins, and C. E. Ugalde-Loo are with the School of Engineering, Cardiff University, Cardiff CF24 3AA, U.K. (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). L. Sainz is with Department of Electric Engineering, ETS d’Enginyeria Industrial de Barcelona, Universitat Politecnica de Catalunya, Barcelona 08028, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2017.2663111 of instability during the energization of the offshore ac grid [3]–[5]. Such interactions are known as electrical resonance instabilities [6]. In HVDC-connected OWPPs, the long export ac cables and the power transformers located on the offshore HVDC substation cause series resonances at low frequencies in the range of 100 ∼1000 Hz [3]–[5], [7]. Moreover, the offshore grid is a poorly damped system directly connected without a rotating mass or resistive loads [2], [3]. The control of the offshore HVDC converter can further reduce the total damping at the resonant frequencies until the system becomes unstable. Electrical resonance instability has been studied with an impedance-based representation by several authors. In [8], [9] Voltage Source Converters (VSCs) are modeled as Th´ evenin or Norton equivalents with a frequency-dependant characteristic. Nyquist and Bode criteria are used to analyze electrical resonance stability [2], [10]. In [9], a clear relationship between electrical resonances and the phase margin condition is established, but the complexity of the loop transfer function limits 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 in OWPPs 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 on the interactions between resonances and the offshore HVDC converter. In [17], a modal analysis in an HVDC-connected OWPP is used to characterize possible resonances and to assess the stability of the offshore converters. Also, in [9] and [18] the impact of resonances on offshore HVDC converters is analyzed using an impedance-based representation. In this paper, the impact that low frequency series resonances have on the voltage stability of HVDC-connected OWPPs is analyzed and discussed. Preliminary work was reported in [19], where the stability criterion presented in this paper was assessed with examples. This paper furthers the initial contributions of [19] by providing a formal framework for the analysis of electrical resonance stability in HVDC-connected OWPPs. An impedance-based representation is used to identify resonances and to assess stability considering the effect of the offshore converters. The resonance stability of an OWPP is determined This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/ 4580 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 6, NOVEMBER 2017 using an alternative approach to the positive-net-damping criterion [13]. This has been reformulated to evaluate the netdamping for electrical series resonances and to provide a clear relationship between electrical resonances of the OWPP and stability. The main contributions of this paper are summarized as follows: 1) The alternative approach to the positive-net-damping criterion is demonstrated using the phase margin condition. This criterion defines the relation between the damping at electrical series resonances and system stability. 2) The relationship between the total damping and resonant frequencies with the poles of the system is demonstrated. This relationship shows that the pole analysis and the positive-net-damping criterion provide the same information about resonance stability. 3) Analytical expressions of the low frequency series resonances are proposed considering the effect of VSC controllers. These expressions are employed to calculate the resonant frequencies where the total damping is evaluated to determine system stability. 4) A stable area of an OWPP is defined as a function of the HVDC converter control parameters and the OWPP configuration. Such an area is obtained from the damping of the OWPP and indicates conditions of stability. The effect that the HVDC converter control parameters and the OWPP configuration have on stability is shown using examples. For completeness, root locus analysis and time-domain simulations in PSCAD/EMTDC are used to validate the stability conditions. The examples presented in this paper are complementary to those included in [19]. II. IMPEDANCE-BASED REPRESENTATION OF AN HVDC-CONNECTED OWPP An impedance-based representation is suitable for the modeling of converters of an HVDC-connected OWPP whenever detailed design information is not available. Such a converter representation offers advantages as it can easily be combined with the equivalent impedance of the offshore ac grid to characterize resonant frequencies. It is also possible to consider the effect of the converter controllers. Moreover, the stability assessment methods for impedance-based representations are simple and less computational intensive compared to other traditional methods such as eigenvalue analysis [2], [10]. The configuration of an HVDC-connected OWPP is shown in Fig. 1. Type 4 WTs are connected to strings of the collector system through step-up transformers from low to medium voltage. Each WT grid side VSC has a coupling reactor and a high frequency filter represented as an equivalent capacitor. The strings are connected to a collector substation, where transformers step-up from medium to high voltage. The collector transformer in Fig. 1 is an equivalent representation of 4 transformers that are connected in parallel [3]. Export cables send the generated power to an offshore HVDC substation, where a VSC based Modular Multilevel Converter (MMC) operates as a rectifier and delivers the power to the dc transmission system. Fig. 1. General scheme of an HVDC-connected OWPP. Fig. 2. Impedance-based model of an HVDC-connected OWPP for resonance and stability analysis. The dc transmission system and the onshore 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 resonances and stability. The ac cables of the export and collector system are modeled as single πsections with lumped parameters and the transformers are modeled as RL equivalents. These models are accurate enough to characterize the low frequency resonances that are responsible for stability issues [3]. The VSCs are represented by equivalent circuits, which include the frequency response of the controller. The offshore VSC is represented by a Th´ evenin equivalent as it controls the ac voltage of the offshore grid [9], [20]; however, Norton equivalents are used to represent the WT VSCs since they control current [8], [20]. III. IMPEDANCE-BASED MODEL OF VSCS The VSC models are represented in a synchronous dq frame and the Laplace sdomain, where complex space vectors are denoted with boldface letters for voltages and currents as v= vd+jvqand i=id+jiq. CHEAH-MANE et al.: CRITERION FOR THE ELECTRICAL RESONANCE STABILITY OF OFFSHORE WIND POWER PLANTS CONNECTED 4581 Fig. 3. Control structures: (a) Offshore HVDC converter and (b) WT grid side converter. A. Offshore VSC Model The offshore VSC controls the ac voltage of the offshore grid. Fig. 3(a) describes the control structure of this converter. If the VSC uses a MMC topology, high frequency filters are not required and only a voltage control loop is considered [18], [21]. Additionally, the internal MMC dynamics can be neglected if a circulating current control is implemented [18]. A control action based on a PI controller is expressed as: vh vsc =FPI,v(vr−vpoc)(1) FPI,v =kp,v +ki,v s(2) where vh vsc is the reference voltage for the offshore converter, vris the control reference voltage at the Point of Connection (POC), vpoc is the voltage measured at the POC and FPI,v is the PI controller for the voltage control loop. The dynamics across the equivalent coupling inductance of the offshore converter are expressed as: vh vsc =vpoc +ic(Rh f+sLh f+jω1Lh f)(3) where icis the current from the HVDC converter, Lh fis the coupling inductance, Rh fis the equivalent resistance of Lh fand ω1=2πf1rad/s (f1=50 Hz). The coupling inductance is equal to Lh f=Larm/2+Lh tr, where Larm is the arm inductance of the MMC and Lh tr is the equivalent inductance of the offshore HVDC transformers. ATh ´ evenin equivalent of the offshore VSC (see Fig. 2) is obtained by combining (1) and (3): vpoc =vr·Gh c−ic·Zh c(4) Gh c=FPI,v 1+FPI,v ;Zh c=Rh f+sLh f+jω1Lh f 1+FPI,v (5) Fig. 4. Equivalent impedance-based circuit of an HVDC-connected OWPP with representation of offshore grid circuit. where Gh cis the voltage source transfer function and Zh cis the input-impedance of the converter. B. Wind Turbine VSC Model Each WT is equipped with a back-to-back converter, but only the grid side VSC is represented in this study. Its control is based on an ac current loop employing a PI controller as shown in Fig. 3(b). The dc voltage outer loop is not represented in the WT VSC model since its dynamic response is slow; i.e. there is sufficient bandwidth separation with the inner current loop [2], [22]. This ensures that there are no interactions between harmonic resonances and the outer loops, which are not of interest in this paper. A Norton equivalent of the WT converter (see Fig. 2) is obtained as: iwt =ir·Gw c−vwt ·Yw c(6) where iwt is the current from the WT VSC, iris the control reference current, Gw cis the current source transfer function, vwt is the voltage after the coupling filter and Yw cis the inputadmittance of the VSC. Gw cand Yw care expressed as [8]: Gw c=FPI,c Rw f+sLw f+FPI,c ;Yw c=1−Hv Rw f+sLw f+FPI,c (7) FPI,c =kp,c +ki,c s;Hv=αf s+αf (8) where FPI,c is the PI controller of the current loop, Lw fthe coupling inductance, Rw fthe equivalent resistance of Lw f,Hv the low pass filter of the voltage feed-forward term [8] and αfthe bandwidth of Hv. The PI design is based on [8], [23], with proportional and integral gains given as kp,c =αcLw fand ki,c =αcRw f, and the bandwidth of the current control by αc. IV. STABILITY ANALYSIS OF HVDC-CONNECTED OWPPS The stability analysis considers the impedance-based circuit presented in Fig. 4, where the offshore grid is modeled with an equivalent circuit (further explained in Section V). A similar representation can be found in [9]. The impedances were expressed in the stationary αβ frame [6], [11], which is denoted in boldface letters for voltages and currents as vs=vα+jvβand is=iα+jiβ. The current in the stationary αβ frame and the Laplace s-domain is given as: is c=(vs rGh c−is rGw cZw c) Th   1/Zg 1+Zh c/Zg (9) 4582 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 6, NOVEMBER 2017 where Zg=Zgrid eq +1/Y w cis the equivalent impedance of the OWPP from the offshore VSC and This the OWPP closed loop transfer function, which can be also expressed as: Th(s)= M(s) 1+M(s)N(s)=M(s) 1+L(s)(10) where M(s)=1/Zgis the open loop transfer function, N(s)= Zh cis the feedback transfer function and L(s)is the loop transfer function. Assuming that the voltage and current sources in Fig. 4 are stable when they are not connected to any load [10], the stability of the OWPP can be studied in the following ways: 1) By analyzing the poles of Thor the roots of Zg+Zh c=0. 2) By applying the Nyquist stability criterion of Zh c/Zg[10]. 3) By considering the passivity of Th[6], [11]. In addition to the previous alternatives, a variation to the positive-net-damping criterion given in [13], [14] is here employed instead to analyze system stability. The criterion has been reformulated to evaluate electrical resonance stability as explained in Section IV-B. A. Passivity A linear and continuous-time system F(s)is passive if [11]: 1) F(s)is stable and, 2) Re{F(jω)}>0∀ω, which is expressed in terms of the phase as −π 2<arg{F(jω)}<π 2. This condition corresponds to a non-negative equivalent resistance in electrical circuits. Passivity can be applied to determine the stability of closed loop systems [6], [11]. A system represented by the closed loop transfer function in (10) is stable if M(s)and N(s)are passive since −π<arg{L(jω)}<π∀ω. This implies that the Nyquist stability criterion for L(s)is satisfied. Therefore, the OWPP is stable if Zgand Zh care passive. When the HVDC converter is connected to a passive offshore grid, Zgis passive and the stability only depends on the passivity conditions of the converter input-impedance, Zh c. In no-load operation (i.e. when only the passive elements of the OWPP are energized), the passivity of Zgis ensured as the WTs are assumed to be disconnected from the offshore grid. However, the WTs represent active elements when they are connected to the offshore grid (i.e. Zgcan have a negative resistance), which may compromise the OWPP stability. B. Positive-Net-Damping Stability Criterion The criterion states that a closed loop system is stable if the total damping of the OWPP is positive at the following frequencies: (i) open loop resonant frequencies and (ii) low frequencies where the loop gain is greater than 1 [13]. However, it does not provide a clear relation between electrical resonances of the OWPP and system stability. This increases the complexity of analyzing the impact that system parameters have on resonance stability. The criterion presented in [13] has been reformulated to evaluate the net-damping for electrical series resonances. The approach proposed in this paper is developed from the phase margin condition [9]. If stability is evaluated in terms of the phase margin, L(jω)=M(jω)N(jω)must satisfy the following conditions at angular frequency ω: |M(jω)N(jω)|=1,(11) −π<arg{M(jω)N(jω)}<π∀ω. (12) M(jω)and N(jω)in (11) and (12) can be expressed in terms of equivalent impedances as: 1 M(jω)=Zg(jω)=Rg(ω)+jXg(ω)(13) N(jω)=Zh c(jω)=Rh c(ω)+jXh c(ω)(14) Also, the equivalent impedance from the voltage source vs rGh c in Fig. 4 is expressed as: Zh eq(jω)=Zh c(jω)+Zg(jω)(15) Phase margin condition (11) is equivalent to: Rh c(ω)2+Xh c(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. Therefore, RgXg,Rh cXh cand (16) is simplified to: Xh c(ω)=±Xg(ω)(17) The electrical series resonances observed from the voltage source vs rGh cin Fig. 4 correspond to frequencies where Zh eq in (15) has a dip or a local minimum. If the resistive components are neglected, the series resonance condition is reduced to: Im{Zh eq(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 of Zh eq coincides with the stability condition |M(jω)N(jω)|=1given by (11). Phase margin condition (12) can be expressed in terms of the imaginary part of L(jω)as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ If d|L(jω)| dω >0: 0<arg{L(jω)}<π⇒ ⇒Rg(ω)Xh c(ω)−Rh c(ω)Xg(ω)>0 If d|L(jω)| dω <0: −π<arg{L(jω)}<0⇒ ⇒Rg(ω)Xh c(ω)−Rh c(ω)Xg(ω)<0 (19) If the resonance condition in (18) is combined with (19): ⎧ ⎨ ⎩ If d|L(jω)| dω >0:Xh c(ωres)[Rg(ωres)+Rh c(ωres)] >0 If d|L(jω)| dω <0:Xh c(ωres)[Rg(ωres)+Rh c(ωres)] <0 (20) It can be shown (see Appendix A) that if the offshore grid is capacitive (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, if the offshore grid is inductive (i.e. Xg>0) and the HVDC converter is capacitive (i.e. Xh c<0), then d|L(jω)| dω <0. By considering the previous conditions, (20) is simplified to: RT(ωres)=Rg(ωres)+Rh c(ωres)>0(21) CHEAH-MANE et al.: CRITERION FOR THE ELECTRICAL RESONANCE STABILITY OF OFFSHORE WIND POWER PLANTS CONNECTED 4583 where resistance RTrepresents the total damping of the system, resistance Rh cthe HVDC converter damping and resistance Rg the offshore grid damping. It can be observed that (21) is equivalent to the positive-net- damping criterion in [13], but evaluated for the series resonances of Zh eq. Therefore, the offshore HVDC VSC is asymptotically stable if the total damping of the system, RT, is positive in the neighborhood of an electrical series resonance. The advantage of this criterion with respect to the passivity approach is that the stability can be ensured even if Zgand Zh care 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 the reactive elements (e.g. Xg/Rg<10 and Xh c/Rh c<10), the approximations in (17) and (18) are not valid and this criterion cannot be used. C. Relation between Total Damping and Poles of the System The HVDC-connected OWPP is a high order system with several poles. However, the system response is governed by a dominant poorly-damped pole pair. If this pole pair is related to the electrical series resonance, impedances Zh cand Zgaround this resonance can be approximated as: Zh c,res(s)≈Rh c+sLh c;Zg,res(s)≈Rg+1 sCg (22) where Cgis the equivalent capacitor of the offshore grid impedance when the frequency is close the resonance. Using (18), the series resonance reduces to ωres =1/Lh cCg. The poles related to the series resonance are obtained from 1+Zh c,res(s)/Zg,res(s)=0, yielding: s=−(Rh c+Rg)Cg±(Rh c+Rg)2C2 g−4Lh cCg 2Lh cCg (23) Considering that (Rh c+Rg)2C2 g4Lh cCg, equation (23) is approximated to: s≈−Rh c+Rg 2Lh c±j1 Lh cCg (24) The imaginary part of the closed loop system poles corresponds to the resonant frequency. Also, the real part of the poles is correlated to the total damping, Rh c+Rg, as mentioned in [14]. Therefore, there is a pair of poles that represent the series resonance and can be used to identify instabilities. V. RESONANCE CHARACTERIZATION In this section, the low frequency series resonances of an OWPP are characterized. It is useful to identify resonant frequencies in an OWPP since they can destabilize an offshore HVDC converter. To this end, the frequency response of Zh eq(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 the resonant frequencies. Fig. 5. Frequency response with and without simplifications (parameters in Appendix B with kp,v=1, ki,v =500). (a) Offshore HVDC VSC. (b) WT grid side VSC. A. Simplifications of the OWPP Impedance Model Fig. 5 shows that the frequency response of a VSC impedance can be simplified to RL equivalents above 100 Hz. The inputimpedance of the VSCs was represented in an αβ frame (see Fig. 4). To achieve this, a reference frame transformation from dq to αβ was performed using the rotation s→s−jω1 [6], [15]. For frequencies higher than ω1, the offshore VSC impedance, Zh c(s−jω1), is approximated to: Rh c=Rh f 1+kp,v ;Lh c=Lh f 1+kp,v (25) Similarly, the WT VSC impedance, Zw c(s−jω1)=1/Y w c(s− jω1), is approximated to: Rw c=Rw f+(αf+αc)Lw f;Lw c=Lw f(26) The previous simplifications do not consider the VSCs as active elements since Rh cand Rw care positive for all frequencies. Fig. 6 shows the equivalent model of the HVDC-connected OWPP with the simplified VSC and cable models. The capacitor Cec represents the export cable capacitance. The inductive and resistive components of the export cable are small enough to be combined with the RL equivalent of the transformers and the HVDC converter. Also, the collector cables are removed because their equivalent inductance and capacitance are small and only affect the response at high frequencies, which are not considered in this study. When the collector cables are removed, the aggregation of WTs is reduced to a combination of parallel circuits independent to the collector system topology. Fig. 7 shows the OWPP model under this scenario, which is equivalent to the model in Fig. 4. The parameters of the aggregated model are defined as follows: 4584 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 6, NOVEMBER 2017 Fig. 6. Impedance-based model of an HVDC-connected OWPP with simplified VSC and cable models (indicated in grey rectangles). Fig. 7. Impedance-based model of an HVDC-connected OWPP with aggregation of collector system. 1) Rcs tr and Lcs tr are the RL values of the collector transformers. 2) Rw tr,a and Lw tr,a are the RL values of the aggregated WT transformers: Rw tr,a =Rw tr/N ;Lw tr,a =Lw tr/N (27) where Nis the number of WTs and Rw tr and Lw tr are the RL values of one WT transformer. 3) Rw c,a and Lw c,a are the RL values of the aggregated WT converters: Rw c,a =Rw c/N ;Lw c,a =Lw c/N (28) 4) Cw f,a is the equivalent capacitance of the aggregated WT low pass filters: Cw f,a =Cw f·N(29) where Cw fis the capacitance of one WT low pass filter. Fig. 8 shows the frequency response of the equivalent offshore grid impedance, Zh eq, with and without simplifications to VSC and cable models. It can be observed that if simplifications are made the 50 Hz resonance of the VSC control is not exhibited; however, the frequency response agrees well with that of the un-simplified Zh eq over 200 Hz and up to 1 kHz. Additionally, the simplification of the collector cables slightly shifts the series resonance from 459 Hz to 497 Hz. In light of these results, it can be concluded that the simplified frequency response represents a good approximation for low frequency resonances in the range of 200 ∼1000 Hz. Fig. 8. Frequency response of OWPP impedance without and with VSC and cable simplifications (parameters in Appendix B and kp,v=1, ki,v =500). B. Analytical Expression for the Series Resonant Frequency The expression of the lowest series resonant frequency of Zh eq is obtained for no-load operation and when WTs are connected. The resistances are neglected as they only have a damping effect on resonance (i.e. they barely modify the resonant frequency). In no-load operation, the WTs are not connected and the contribution of the collector system at low frequencies is negligible. Therefore, the OWPP impedance Zh eq in (15) is equivalent to an LC circuit with a resonant frequency: fnload res =1 2πLh cCec (30) The lowest series resonant frequency when WTs are connected has been obtained following an algebraic calculation using Fig. 7: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ fload res =1 2πb−√b2−4ad 2a a=CecLw c,a(Lcs tr +Lw tr,a)Lh cCw f,a b=CecLh c(Lw c,a +Lcs tr +Lw tr,a) +Cw f,aLw c,a(Lh c+Lcs tr +Lw tr,a) d=Lh c+Lw c,a +Lcs tr +Lw tr,a (31) Expressions (30) and (31) are employed to calculate the frequencies where the total system damping is evaluated to determine stability. VI. VOLTAGE STABILITY ANALYSIS The modified positive-net-damping criterion was applied to analyze the impact of electrical series resonances in the voltage stability of an HVDC-connected OWPP. The effects of the offshore HVDC converter control and the OWPP configuration are considered in the study. For completeness, the root locus of the system and time-domain simulations in PSCAD/EMTDC are used to confirm the results. The cable model simplifications considered in the resonance characterization are used in the stability analysis given that the low frequency response is well-represented and the damping contribution from the cable resistances can be neglected. However, the VSC simplifications in (25) and (26) are not considered, because the converters are not represented as active elements. CHEAH-MANE et al.: CRITERION FOR THE ELECTRICAL RESONANCE STABILITY OF OFFSHORE WIND POWER PLANTS CONNECTED 4585 Fig. 9. Stable area of offshore HVDC converter in no-load operation as function of kp,v,ki,v and lcb (the stable and unstable examples of Figs. 11 and 12 are marked with circles). The system is analyzed in no-load operation and when WTs are connected based on the OWPP described in Appendix B. A. No-Load Operation In no-load operation, the positive-net-damping stability criterion only includes the damping contribution of the offshore converter, Rh c, because the export and collector cables are passive elements with a small resistance and thus can be neglected (i.e. Rg≈0). Therefore, condition (21) is reduced to Rh c(ωres)>0, which is equivalent to analyzing the passivity of the HVDC converter control at a resonant frequency. Stability is ensured if the electrical series resonance is located in a frequency region with positive resistance. This region is determined using the zero-crossing frequencies of Rh c(i.e. Rh c(ω)=Re{Zh c(ω)}=0) in (5). The two following solutions are obtained: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ωcut1=ω1=2π(50) ωcut2=ω1 1−ki,vLh f Rh f(1 + kp,v) (32) When ωcut2<0, the only zero-crossing frequency considered is 50 Hz and Rh cis negative for ω>2π(50). Therefore, the converter is always unstable for resonant frequencies above 50 Hz. If ωcut2>0, then Rh cis negative for 2π(50) <ω<ω cut2 and positive for ω>ω cut2. In this case, the converter is stable for frequencies higher than ωcut2since the resonance is located in a positive-resistance region. Thus, the offshore HVDC converter is stable when Rh chas two zero-crossing frequencies (ωcut2>0 and ωres >ω cut2). The following inequalities are obtained by combining (30) and (32): ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ωcut2>0⇒Rh f(1 + kp,v)−ki,vLh f>0 ωres >ω cut2⇒Rh2 f(1 + kp,v)2−2Rh fLh f(1 + kp,v)ki,v −ω2 1Rh2 fLh fCec(1 + kp,v)+k2 i,vLh2 f>0 (33) Fig. 9 shows the stability area (Rh c(ωres)>0) defined by (33) as a function of the control parameters of the offshore HVDC converter, kp,v and ki,v, and the export cable length, lcb.Itis observed that when the cable length increases the stable area is reduced. Fig. 10. Root locus of OWPP in no-load operation for variations of export cable length and ac voltage control parameters. (a) Cable length variation from 1 to 100 km (kp,v =0.1,ki,v =3). (b) Variation of ki,v from 1 to 5 (kp,v =0.1, lcb =10km). (c) Variation of kp,v from 0 to 1 (ki,v =5,lcb =10km). Fig. 10 shows the root locus of the low frequency resonant poles for parametric variations of kp,v,ki,v and lcb. It should be emphasized that these poles are not complex conjugate due to the transformation of the VSC input impedance from a synchronous dq to a stationary αβ reference frame, which introduces complex components. The increase of cable length moves the resonance to lower frequencies since Cec increases. As kp,v increases, the resonance shifts to higher frequencies given that Lh cin (25) decreases. Changes in ki,v do not affect the resonant frequency. The system becomes unstable when one of the resonant poles moves to the positive side of the real axis; this is equivalent to have a negative damping. It can be observed that the stability conditions of the resonant poles agree with the stable areas shown in Fig. 9. Figs. 11 and 12 show examples of stable and unstable cases when ki,v is modified. The intersection between Zh cand Zg (i.e. 1/|M(jω)|=|N(jω)|) approximately determines the series resonant frequency, as defined in (18). When the system is stable the resonant frequency is located in a positive-resistance region of Zh c, as shown in Fig. 11(a). Also, following the Nyquist criterion, the Nyquist curve encircles (−1,0) in anti-clockwise direction and the open loop system does not have unstable poles. Therefore, the system is stable as it does not have zeros with positive real part. Although the ac voltage control can be designed to ensure stability, all the poles have a low damping. This slows down the dynamic response, as shown in Fig. 11(c), which is not acceptable for the operation of the offshore converter. When the system is unstable the resonant frequency is located in the negative-resistance region of Zh c, as shown in Fig. 12(a). Following the Nyquist criterion, the Nyquist curve encircles (−1,0) in clockwise direction and the open loop system does not have unstable poles. Therefore, the system is unstable because the total number of zeros with positive real part is 1. In Fig. 12(c), the voltage at POC shows oscillations at 309 Hz due to the resonance instability identified in Fig. 12(a). 4586 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 6, NOVEMBER 2017 Fig. 11. Stable example in no-load operation with kp,v =0.1,ki,v =3and lcb =10km. (a) Frequency response: Rh c,Zh eq,Zh c,Zg. (b) Nyquist curve of Zh c/Zg(positive freq.). (c) Instantaneous and RMS voltages at POC. Step change is applied at 1 s. B. Connection of Wind Turbines When the WTs are connected to the offshore ac grid, the WT converters modify the low frequency resonance location and the total damping. The stability conditions are discussed, but the expressions for the zero-crossing frequencies of RTare not obtained analytically due to the complexity of the system. Fig. 13 shows the stable area defined by RT(ωres)>0. There is a significant increase of the stable region when the WTs are connected. Therefore, the ac control parameters can be modified for a larger range of values to improve the dynamic response without compromising stability. Fig. 14 shows the root locus of the low frequency resonant poles for different ac voltage control parameters and number of WTs. The connection of WTs improves the resonance stability because the associated poles move to the left hand side of the real axis and increase the damping of those low frequency modes. This damping contribution of the WTs is also mentioned in [2]. The stability conditions of the resonant poles agree with the stable area shown in Fig. 13. Also, the resonance moves to higher frequencies when kp,v and the number of WTs increases, as shown in Fig. 14. Figs. 15–17 describe two situations where the ac voltage control is designed to have a fast dynamic response (e.g. kp,v = 1and ki,v = 500) and the number of WTs decreases from Fig. 12. Unstable example in no-load operation with kp,v =0.1,ki,v =5 and lcb =10km. (a) Frequency response: Rh c,Zh eq,Zh c,Zg. (b) Nyquist plot of Zh c/Zg(pos freq.). (c) Instantaneous and RMS voltages at POC. Step change is applied at 1 s. Fig. 13. Stable area of offshore HVDC converter as a function of kp,v and ki,v and the number of connected WTs (the stable and unstable examples of Figs. 15 and 16 are marked with a circle). 40 to 20. When all the WTs are connected, the offshore converter is stable because the resonance is located in a positive-resistance region, as shown in Fig. 15(a). The converter introduces a negative resistance at the resonant frequency, but the total damping is compensated by Rg, as shown in Fig. 15(b). When the number of WTs reduces to 20 the offshore converter becomes unstable since the resonance lies in the negative-resistance region, as shown in Fig. 16(a). In this case, Rgcannot compensate Rh c, as shown in Fig. 16(b). Also, the Nyquist curve agrees with the positive-net-damping criterion in both situations [Figs. 15(a) CHEAH-MANE et al.: CRITERION FOR THE ELECTRICAL RESONANCE STABILITY OF OFFSHORE WIND POWER PLANTS CONNECTED 4587 Fig. 14. Root locus of OWPP for variations of ac voltage control parameters and number of WTs (N=80). (a) Variation of kp,v from 0 to 80 (ki,v = 3,N=80). (b) Variation of ki,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. 15. Stable example when 40 WTs are connected, kp,v =1and ki,v = 500. (a) Frequency response of Rh c+Rg,Zh eq,Zh cand Zg. (b) Frequency response of Rh cand Rg. (c) Nyquist plot of Zh c/Zg(positive freq.). Fig. 16. Unstable example when 20 WTs are connected, kp,v =1and ki,v = 500. (a) Frequency response of Rh c+Rg,Zh eq,Zh cand Zg. (b) Frequency response of Rh cand Rg. (c) Nyquist plot of Zh c/Zg(positive freq.). Fig. 17. Instantaneous and RMS voltages at POC when the number of WTs is reduced from 40 to 20 at 1 s. The ac voltage control parameters are kp,v =1 and ki,v = 500. and 16(c)]. In Fig. 17, the instantaneous voltages at POC show oscillations at 444 Hz when the number of WTs is reduced at 1 s; this is due to the resonance instability identified in Fig. 16(a). The variation of connected WTs can be caused by switching configurations during commissioning phases or during outages due to maintenance or contingencies [3]. As shown by the previous examples, a sudden reduction in the number of WTs should be carried out with care as this can lead to instability. Active damping can be implemented as a virtual resistor in the offshore HVDC converter to compensate the negative resistance introduced by the ac voltage control for all operational states. This will allow a design of the ac voltage control to have a fast dynamic response without compromising the stability.