interconnection of two very weak ac systems by vsc-hvdc links using power-synchronization...

344 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 1, FEBRUARY 2011

Interconnection of Two Very Weak ACSystems by VSC-HVDC Links Using

Power-Synchronization ControlLidong Zhang, Member, IEEE, Lennart Harnefors, Senior Member, IEEE, and Hans-Peter Nee, Senior Member, IEEE

Abstract—In this paper, voltage-source converter (VSC) basedhigh-voltage dc (HVDC) transmission is investigated for intercon-nection of two very weak ac systems. By using the recently pro-posed power-synchronization control, the short-circuit capacitiesof the ac systems are no longer the limiting factors, but rather theload angles. For the analysis of the stability, the Jacobian transfermatrix concept has been introduced. The right-half plane (RHP)transmission zero of the ac Jacobian transfer matrix moves closerto the origin with larger load angles. The paper shows that, dueto the bandwidth limitation imposed by the RHP zero on the di-rect-voltage control of the VSC, high dc-capacitance values areneeded for such applications. In addition, the paper proposes acontrol structure particularly designed for weak-ac-system inter-connections. As an example, it is shown that the proposed con-trol structure enables a power transmission of 0.86 p.u. from asystem with the short-circuit ratio (SCR) of 1.2 to a system withan SCR of 1.0. This should be compared to previous results forVSC based HVDC using vector current control. In this case, only0.4 p.u. power transmission can be achieved for dc link where onlyone of the ac systems has an SCR of 1.0.

Index Terms—Control, converters, HVDC, power systems, sta-bility.

I. INTRODUCTION

L ARGE interconnected ac systems have many well-knownadvantages [1], [2]. However, larger interconnected ac

systems also increase the system complexity from the opera-tion point of view, and might adversely decrease the systemreliability. Large blackouts in America and Europe confirmedclearly that the close coupling of the neighboring systemsmight also include the risk of uncontrolled cascading effectsin large and heavily loaded systems [3], [4]. High-voltage dc(HVDC) transmission has been introduced since the 1950s [5].In addition to its unique role in submarine-cable transmission,asynchronous-system connection, etc., HVDC also has the“firewall” function in preventing cascaded ac-system outagesspreading from one system to another [6]. By utilizing HVDClinks as system interconnector, one can gain the benefits oflarger systems, yet keep the operation of each subsystem rela-tively independent. Therefore, HVDC transmission is generally

Manuscript received November 19, 2009; revised November 23, 2009. Firstpublished May 24, 2010; current version published January 21, 2011. This workwas supported by ELFORSK under the Electra program. Paper no. TPWRS-00897-2009.

L. Zhang and L. Harnefors are with ABB Power Systems, SE-771 80 Ludvika,Sweden (e-mail: [email protected]; [email protected]).

H.-P. Nee is with the School of Electrical Engineering, Royal Institute ofTechnology, SE-100 44 Stockholm, Sweden (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRS.2010.2047875

considered as a promising solution for future large ac-systeminterconnections [7].

However, there is an inherent weakness with the conventionalline-commutated HVDC system, i.e., the commutation of theconverter valve is dependent on the stiffness of the alternatingvoltage supplied by the ac system. The converter cannot workproperly if the connected ac system is weak. Substantial re-search has been performed in this field [8]–[12]. The most out-standing contribution on this subject is [8], which recommendsto use short-circuit ratio (SCR) as a description of the strengthof the ac system relative to the power rating of the HVDC link.Both [11] and [12] conclude that, for ac systems with an SCRlower than 1.5, synchronous condensers have to be installed toincrease the short-circuit capacity of the ac system. However,synchronous condensers can substantially increase the invest-ment and maintenance costs of an HVDC project.

The pulse-width modulation (PWM) based voltage-sourceconverter (VSC) is an emerging technology for HVDC trans-mission [13], [14]. Thanks to the gradually increased ratingsand reduced losses, the technology has reached maturity inrecent years. In contrast to the conventional thyristor-basedHVDC system, a VSC-HVDC system has the potential to beconnected to very weak ac systems, as well as the capability togenerate or consume reactive power depending on the operatingconditions. With the traditional vector current control, however,the potential of the VSC is not fully utilized [15]–[17], e.g.,[16] shows that the maximum power that a VSC-HVDC linkusing vector current control can transmit to the ac system with

is 0.4 p.u. On the other hand, the recently proposedpower-synchronization control for grid-connected VSCs hasbeen shown to be a superior solution for VSC-HVDC linksconnected to very weak ac systems [18], [19]. One of the majorfeatures of the power-synchronization control is that the VSCsynchronizes with the ac system through an active-power con-trol loop, similar to the operation of a synchronous machine.By using power-synchronization control, the VSC emulates asynchronous machine. Therefore, it basically has no require-ment on the short-circuit capacity of the ac system. Moreover,a VSC terminal can give the weak ac system strong voltagesupport, just like a normal synchronous machine does.

In this paper, VSC-HVDC technology using the recently pro-posed power-synchronization control is investigated for inter-connection of two very weak ac systems. The major focusesof the paper are dynamic modeling and controller design. Thepaper is organized as follows. In Section II, the definition andcharacteristics of weak ac systems are discussed and followed

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ZHANG et al.: INTERCONNECTION OF TWO VERY WEAK AC SYSTEMS BY VSC-HVDC LINKS USING POWER-SYNCHRONIZATION CONTROL 345

Fig. 1. Main-circuit diagram of a VSC converter connected to ac systems.

by a brief review of the fundamental principle of power-syn-chronization control for grid-connected VSCs. In Section III,the design of direct-voltage controllers is described and the re-quirement on dc capacitance for weak-system connections isdiscussed. Finally, in Section IV, a control structure is proposedfor weak-ac-system interconnection. The proposed design andlinear model are verified by a VSC-HVDC model in the electro-magnetic transient simulation software PSCAD/EMTDC. Thesimulation software PSCAD/EMTDC is often considered as ad-equate for controller design in industry. Experiments on con-verters with power ratings of several MW are very costly andtime consuming. Such tests are likely to be performed in the fu-ture; however, they are beyond the scope of the present paper.

II. CONTROL OF VSC-HVDC LINKS

CONNECTED TO WEAK AC SYSTEMS

Fig. 1 shows the main-circuit diagram of a VSC-HVDC con-verter connected to an ac system. and are the inductanceand resistance of the phase reactor of the VSC, and andare the inductance and resistance of the ac system. is the accapacitor connected at the point-of-common-coupling (PCC).The bold letter symbols , , and represent the voltage vec-tors of the ac source, the PCC, and the VSC. , , and aretheir corresponding voltage magnitudes. The ac source, which isa stiff constant-frequency voltage source, is used as the voltagereference, and the phase angles of and are and ,respectively. and are the active power and reactive powerfrom the VSC to the ac system. The quantity is the currentvector of the phase reactor, and is the current vector to the acsource.

A. Characteristics of Weak AC Systems

A weak ac system is typically characterized by its highimpedance [8]. As the ac-system impedance increases, thevoltage magnitude of the ac system will become ever more sen-sitive to power variations of the HVDC system. This difficultyis usually measured by the short-circuit ratio (SCR), which isa ratio of the ac-system short-circuit capacity versus the ratedpower of the HVDC system. SCR is directly related to theac-system inductance . According to [8], SCR is defined as

(1)

where is the short-circuit capacity of the ac system at thefilter bus, while is the rated dc power of the HVDC link.

The short-circuit capacity of the ac system can be expressedas

(2)

where is the angular frequency of the ac system and isthe equivalent impedance of the ac system. To further simplifythe expression of SCR, the filter-bus voltage is assumed to beidentical to the base value, i.e., , and the ratedpower of the HVDC link is used as the base power of theac system, i.e., . If is expressed in per unit(p.u.), it follows from (1) and (2) that SCR can be expressed as

.Besides the voltage-control difficulty, SCR also imposes a

theoretical limitation on the maximum power that the HVDCsystem is able to transmit to or from the ac system. This can beshown by the following well-known power-angle equation:

(3)

where is defined as the load angle of the VSC-HVDC con-verter in this paper, and and are approximately equalto in normal operating conditions. Equation (3) showsthat the maximum load angle cannot be beyond 90 in steadystate, e.g., for VSC-HVDC links connected to a system with

, the maximum theoretical power transmission is.

B. Modeling of AC Systems

In a synchronous grid reference frame with the axischosen aligned with the ac source , the dynamic equations ofthe main circuit in Fig. 1 can be written as

(4)

(5)

(6)

and in -component form

(7)

For power-synchronization control, the active power and alter-nating voltage are controlled by the phase angle and voltagemagnitude of the VSC output. Thus, the real and imaginaryparts of the VSC vector and can be expressed as

(8)


The output variables are the active power and the voltagemagnitude at the PCC, which are expressed as

(9)

The state-space model can be obtained by linearizing (6), (8),and (9), which yields the following form:

(10)

where

(11)

The state-space representation (10) can also be written in input-output transfer matrix form

(12)

which yields

(13)

If the resistance and are neglected, the poles of , i.e.,the eigenvalues of the matrix, can be solved analytically withthe expressions

(14)

The poles of are independent of the operating points,but are usually very poorly damped due to the low resistancein transmission systems. contains a pair of transmissionzeros. If the resistances and are neglected, the zeros aregiven by

(15)

In contrast to the poles, the transmission zeros of are verymuch dependent on the operating points, mainly on the loadangle [18]. The higher the load angle, the closer the zerosare to the origin. In control theory, a zero on the right-half plane(RHP) of the process imposes a fundamental limitation on theachievable bandwidth of the closed-loop system [20], i.e., theclosed-loop system cannot achieve higher bandwidth than thelocation of the RHP zero. It is interesting to note that (15) isconsistent with the conclusion drawn from the steady-state re-lationship in (3) but with more dynamic insight. With a loadangle , the zeros of move to the origin. Fromfeedback-control point of view, such a process cannot be tightlycontrolled at low frequencies.

Due to the similarity with the Jacobian matrix in load-flowstudies in terms of inputs and outputs [21], is named Ja-cobian transfer matrix in this paper. The concept of Jacobian

Fig. 2. Power-synchronization control of grid-connected VSCs.

transfer matrix can be used to represent larger ac systems withmore input devices to study the interactions, especially if theelectromagnetic transients are considered.

C. Power-Synchronization Control of Grid-Connected VSCs

The fundamental idea of power-synchronization control ofgrid-connected VSCs is that VSCs synchronize with the acsystems through the active-power control instead of using aphase-locked loop (PLL), similar to the operation of a syn-chronous machine [18]. Fig. 2 shows the control overviewof power-synchronization control. The active power and thealternating voltage are controlled by the phase angle and themagnitude of the VSC through a multivariable controller [19]

(16)

In Fig. 2, the “Current control” block is a standard proportional-type vector current controller with the expression

(17)

where is the desired closed-loop bandwidth of the inner-cur-rent controller, is the converter current reference, andis the voltage reference of the VSC. The superscript denotesthe converter frame, which leads the grid frame with thephase angle . With power-synchronization control, instead ofgiving a constant current reference to (17), the value of in(17) is given by

(18)where is a nominal value, e.g., Equation (18)corresponds to the “Current reference control” in Fig. 2. Thecurrent reference in (18) is designed in such a way that the con-trol law in (17) becomes

(19)

in normal operation. This can be easily verified by substituting(18) into (17). However, the current reference in (18) givesan indication of the actual converter current. During ac systemfaults, current limitation is automatically achieved by limiting


the modulus of to the maximum current limit . A de-tailed analysis of this has been given in [18]. The output vari-ables , , and are the three-phase reference voltagesof the VSC for pulsewidth modulation. An important part of thepower-synchronization control structure is the high-pass currentcontroller in (19). As mentioned in Section II-B, theresonant (complex) poles of are usually poorly damped,which requires the control system of the VSC to provide ad-ditional damping. This is especially important for VSC-HVDClinks connected to weak ac systems, where the resonance fre-quency tends to be lower. Therefore, a high-pass current controlwas proposed in [18] to add “active damping” to the resonantpoles. The high-pass current controller has the transfer function

(20)

where should be chosen to cover the frequency rangeof all the possible resonances in ac systems, typically

to also cover the subsynchronousresonance in ac systems. The gain determines the level ofdamping as shown below. With the control functionapplied, if the switching-time delay of the converter is neglectedand it is assumed that does not exceed the maximumvoltage modulus, then . The dynamic equation in(4) should be expressed as

(21)

To fit (21) in state-space form, a new state variable is intro-duced. Consequently, (21) is expressed as

(22)

Replacing the phase-reactor dynamic equations in (4) by (22)and following the same procedure as in Section II-B, the transfermatrix including can be obtained. By varying

, Fig. 3 shows the effect of which shifts the reso-nant poles of towards the left-half plane (LHP) withoutaffecting the locations of the two transmission zeros.

Power-synchronization control is fundamentally ofmulti-input multi-output nature, i.e., both the active powerand the alternating voltage are controlled by the phase angleand voltage magnitude of the VSC. In this paper, the multi-variable internal model controller (IMC) designed in [19] isapplied.

VSCs using power-synchronization control basically emu-late the operation of a synchronous machine. Therefore, it con-tributes short-circuit capacity to the ac system at the PCC. Inorder to obtain a simple estimate of the impact of the VSC onthe short-circuit capacity of ac system, the effects of the al-ternating-voltage control and the ac filter can be disregarded.

Fig. 3. Damping effect of � �� on the resonant poles of the ac system rep-resented by � �� (“�”: pole, “�”: zero). Main circuit parameters: � � �� , � � �� , � � � �� , � � �� , � � �� . Initial conditions: � � �� , � �� , � � .� ��: � � �� , from 0.0 to 0.6.

Doing so, the short-circuit capacity of the ac system includingthe VSC at the PCC can be expressed as

(23)

where the resistances and have also been neglected. How-ever, VSCs do not necessarily increase short-circuit currents tothe ac system during ac system faults thanks to the current lim-itation function [18].

III. DIRECT-VOLTAGE CONTROL

For a VSC-HVDC link, at least one of the converter stationshas to control the direct voltage, while the other converter stationcontrols the active power. The active power is thus automaticallybalanced between the two converter stations. In this section, thevarious aspects of the direct-voltage control are discussed.

A. Modeling of DC Systems

Fig. 4(a) shows a dc-link circuit that is simplified as one dccapacitor. Such a representation is applicable for dc cable trans-mission or back-to-back HVDC links. The capacitor bank at thedc link is an energy storage. The time derivative of the stored en-ergy must equal the sum of the instantaneous power infeed fromthe two converters (neglecting the losses). The direct-voltagedynamics can thus be written as

(24)

where is dc-link capacitance and is the direct voltage.and are the instantaneous power from Converter 1 and

Converter 2. If the direct-voltage controller were to operate di-rectly on the error , the closed-loop dynamics wouldbe dependent on the operating point . This inconvenienceis avoided by selecting the direct-voltage controller operating


Fig. 4. DC-link representation of the VSC-HVDC system. (a) dc capacitor(b)�-link.

instead on the error as suggested in [22]. Conse-quently, the dc-link dynamics can be written in the linearizedform

(25)

If the dc transmission line is a long overhead line, then the resis-tance and inductance of the line have to be taken into account.Fig. 4(b) shows a -link model of the dc circuit. Based on Kirch-hoff’s voltage and current laws, the dynamic equations of the dccircuit can be written as

(26)

where and are the lumped capacitances representingthe capacitance and the dc capacitors at the two converter sta-tions, and and represent the direct voltages at two con-verter stations. and are the resistance and inductance ofthe dc line. If the linearized deviation variables andof and are chosen as the inputs, and and of

and are chosen as the outputs, a state-space model canbe obtained by linearization of (26)

(27)

where

(28)

Fig. 5. Root-loci of � �� regarding operating-point variations (“�”: pole,“�”: zero). DC-link parameters: � � �� , � � �� , � �� , � � �� . Initial conditions: � � �� , variationof � from �� to �� .

The subscript 0 in (28) denotes the operating-point value.The state-space representation (27) can also be written ininput-output transfer matrix form by applying (12), yielding

(29)

The poles of are the eigenvalues of the A matrix, whichcan be solved analytically for the operating point where

with the expression

(30)

As shown in Fig. 5, the poles of are dependent on the op-erating point. With the increase of the loading, the frequenciesof the two complex poles are reduced, and the real poleat the origin moves into the right-half plane. In control theory,the RHP pole of the process imposes a fundamental lower limiton the bandwidth of the controller, i.e., the closed-loop systemof the direct-voltage control has to achieve a bandwidth that ishigher than the location of the RHP pole of to stabilizethe process. Recalling the upper limit imposed by the RHP zeroof on the bandwidth of the power-synchronization con-troller in Section II-B, it is generally more complicated to op-erate VSC-HVDC links at high load angles.

The instability of has to do with the resistance of the dclink. With dc powers as the inputs, as the way how VSCs workto a dc link, the dc resistance gives a destabilizing effect. Ofcourse, such destabilizing effects only become apparent if the dcresistance is large enough, e.g., long-distance HVDC transmis-sions. The analytical solutions of the poles of with otheroperating point than are difficult to obtain.However, Appendix B gives a rigorous mathematical proof ofthe instability of for other operating points than


or . The mathematical proof also shows the role of thedc resistance to the instability of .

The expression for the transmission zero of is surpris-ingly simple, i.e., . The location of the zero is inde-pendent of the operating points. This implies that the dc systemis always a minimum-phase system. Due to the similarity withthe ac Jacobian transfer matrix in Section II-B, the transfermatrix may be conveniently named dc Jacobian transfermatrix. Similar to the concept of ac Jacobian transfer matrix forac systems, the concept of dc Jacobian transfer matrix can alsobe used to represent larger dc systems with several or many VSCterminals.

B. Direct-Voltage Controller Design

The connection between the ac Jacobian transfer matrixand dc Jacobian transfer matrix is the equivalence be-tween the instantaneous power at the ac side and that at thedc side of the VSC

(31)

i.e., the output of the ac Jacobian transfer matrix becomes theinput of the dc Jacobian transfer matrix, and the minus sign in(31) is due to the definition of the power direction. However, adirect connection of the two matrices may introduce an error,since the active power derived in (9) is the power from the PCCto the ac system, which is somewhat different from the activepower flowing through the VSC due to the energy stored inthe phase reactor. Therefore, the ac power should be ob-tained from the linearization of . InSection IV, the active power from the VSC is named todistinguish from the active power from the PCC into the grid.

Another possible error is that (31) does not consider the lossesin the VSC. Experiment tests show that this error might havean impact if dc resonance is a concern. The losses of the VSCare of nonlinear nature and shall not be simply represented by aconstant resistor. This issue certainly needs to be addressed infuture research.

For the direct-voltage controller, a proportional-integral (PI)controller is proposed with the control law

(32)

If the power-synchronization controller is assumed to be suffi-ciently fast, i.e., , and the dc link is assumed tobe a pure capacitor, i.e., , then the closed-loopsystem is expressed as

(33)

The control parameters are selected as

(34)

which place two real poles of the closed-loop system at. For long overhead lines, the dc-line inductance and the

Fig. 6. Notch filter to reduce the dc resonance peak (dashed: without notchfilter, solid: with notch filter). DC-link parameters: the same as Fig. 5. Initialconditions: � � �� , � � �� . Direct-voltage controller: � �

�� . Notch filter: � � �� , � � ��, � � �� .

Fig. 7. Control block diagram of direct-voltage controller.

dc capacitance may create resonance peak in the low-frequencyrange. If the resonance peak appears in the bandwidth where thedirect-voltage controller is active, an effective way to mitigatethe dc resonance is to apply a notch filter in the direct-voltagecontrol loop. A notch filter commonly has the expression

(35)

where the three adjustable parameters are , , and . Theratio of sets the depth of the notch, and is the resonancefrequency. Fig. 6 shows the effect of the notch filter by time stepresponse with change of the direct-voltage reference. Itshould be noted that the notch filter may adversely affect thephase margin of direct-voltage control.

The block diagram of the direct-voltage controller is shownin Fig. 7. A prefilter is added on the reference to removethe overshooting with reference tracking [20]. The multivariablepower-synchronization controller is used as an inner loop.

C. DC-Capacitance Requirements

In Section III-B, it has been shown that, by explicitly in-cluding the dc capacitance in the direct-voltage control param-eters, one can freely place the poles of the closed-loop system.This implies that, as long as its size is known, the dc-capaci-tance does not affect the linear (small signal) stability of the di-rect-voltage control. However, for disturbance reduction, the dccapacitance has an important role. In this section, the dc-capac-itance requirement for weak-system connections is discussed.To simplify the analysis, the dc-link representation shown inFig. 4(a) is assumed, i.e., [cf. (25)]. If thepower-synchronization loop has the bandwidth which is con-siderable higher than that of the direct-voltage controller, the


transfer function from the load disturbance to the error signalas shown in Fig. 7 becomes

(36)

If the worst scenario is considered, i.e., the active power outputto the other converter station is changed stepwise from the max-imum active power to 0, e.g., the converter is suddenlyblocked. The step response of the error signal in the timedomain becomes

(37)

The time derivative of is

(38)

The time derivative of becomes zero at , whichcorresponds to the peak of the voltage error. By substitutingthis into (37), the maximum error is found to be

. By considering the maximum direct-voltage allowed, then , the requireddc capacitance is

(39)

A common expression for dc capacitance is using its time con-stant , which is defined as

(40)

where is the direct-voltage base value (kV) and isthe rated (base) power (MW) of the VSC, and is the dccapacitance . Substituting (40) into (39) yields

(41)

It should be noted that, for the dc-capacitance requirement, onlyper unit values of the direct voltage and the dc power are appli-cable for (41), while either per unit values or real values can beapplied in (39). As mentioned in Section II-A, for weak-systemconnections, the VSC-HVDC link needs to operate with largeload angles, where the RHP transmission zero of the ac Ja-cobian transfer matrix moves closer to the origin. TheRHP zero limits the bandwidth of the power-synchronizationcontroller, and eventually limits the bandwidth of the direct-voltage controller. A rule of thumb is that the bandwidth ofthe power-synchronization controller should be lower than halfof the location of the RHP zero [20]. If it is further assumedthat the direct-voltage controller is four times slower than the

Fig. 8. Direct-voltage control bandwidth reduced by the inner loop.

Fig. 9. DC-capacitance requirement in weak-ac-system connection with� � �� (solid: � � �� , dashed: � � �� ,dotted: � � �� ).

power-synchronization loop, then can be solved by (15) and(3). Accordingly

(42)

Another issue that needs to be taken into account is that, if thebandwidth of the direct-voltage controller is chosen four timesslower than the power-synchronization loop, the inner loop willaffect the maximum voltage variation. Fig. 8 shows the ratio ofbandwidth reduction by the inner loop, which is approximatedby a first-order filter with bandwidth . Considering this effect,(41) is adjusted as

(43)

Given the maximum loading and maximum allowed di-rect voltage , the relationship between dc-capacitancerequirement and SCR of the ac system can be established by(42) and (43). If the worst scenario is considered, the maximumloading should be chosen as . However, with veryweak-ac-system connection, it is recommended that the loadangle shall not be above 60 to maintain a reasonable stabilitymargin. For instance, if the SCR of the ac system is 1.0, thenthe maximum loading is . Fig. 9shows the plots of dc-capacitance requirement for ac systemshaving with different allowed .

IV. INTERCONNECTION OF TWO VERY WEAK AC SYSTEMS

This section shows a design example. A 350-MWVSC-HVDC link is applied to interconnect two very weak acsystems, where the inverter station (station 1, power receivingend) connected to an ac system having , while therectifier station (station 2, power sending end) connected to anac system having . Since the VSC-HVDC systemshave mostly been applied with cable transmissions, the dc link


Fig. 10. Control block diagram of a VSC-HVDC link interconnecting twoweak ac systems.

is supposed to be represented by the single dc-capacitor modelshown in Fig. 4(a). The detailed parameters of the converterstations are listed in Appendix A.

From Fig. 7, it can be easily observed that, if one converterstation controls the direct voltage, while the other converter sta-tion controls the active power, the two converter stations are, infact, linearly independent. This implies that the stability of oneconverter station does not affect the stability of the other con-verter station. One might consequently conclude that there isnothing more special for a VSC-HVDC link connected to twoweak systems than it is only connected to one weak system. Thismight be true if only linear effects are considered. However,the real system is nonlinear. For VSC-HVDC operation, the di-rect voltage has to be carefully maintained around its nominalvalue. For instance, a big direct-voltage drop might temporarilylimit the capability of the alternating-voltage controller and neg-atively affect the linear stability of the control systems.

The proposed control structure for weak-system interconnec-tion is shown in Fig. 10. The basic idea of the design is that bothof the two converter stations have direct-voltage controllers,while the active-power controlling station controls the activepower by adding an additional contribution to the reference ofthe direct-voltage controller, and its output shall be limited. Thismay seem unnecessarily complicated since two cascaded con-trol layers have been applied. However, the proposed schemehas shown to be superior to other solutions since it gives thehighest priority to direct-voltage control, which is very impor-tant for weak ac system interconnection. Once the direct voltagecomes out of bounds, the nonlinear phenomena will create sub-stantial difficulties for the control system. With the proposedcontrol structure, the linear independence between the two con-verter stations is lost. However, the bandwidth of the direct-voltage controllers is much higher than the bandwidth of the acsystem dynamics. This implies that the “firewall” effect of theHVDC link is still in force.

A. Power-Synchronization Inner Loop

Due to the very low SCR of the ac system connected to theconverter station 1, the maximum allowed power is

, which corresponds to approximatelyload angle. The ac system connected to the converter station

Fig. 11. Step response of the direct-voltage controllers. (a) Direct-voltage stepat station 1. (b) Direct-voltage step at station 2 (solid: nonlinear simulation,dashed: linear model).

2 has slightly higher SCR, but considering the losses of theconverter station and of the dc cable, it has maximum loading

, which corresponds approximately to loadangle of .

The power-synchronization controller is used as the innerloop for the direct-voltage controller as shown in Fig. 7. In thisapplication, the multivariable internal model control (IMC) in-vestigated in [19] is applied to parameterize the power-synchro-nization controller. According to the discussion in Section II-B,the bandwidth of the power-synchronization controller shouldbe lower than half of the RHP-zero location of . By using(15), the bandwidth of the active-power controller of power-syn-chronization control at the two converter stations can be calcu-lated as and . The band-widths of the alternating-voltage controllers, however, do notneed to be very high. Thus, they have been chosen as

.

B. Direct-Voltage Controller

As mentioned in Section III-C, the bandwidth of the direct-voltage controllers should be chosen at least four times slowerthan the active-power controller of the power-synchronizationcontrol. Alternatively, the bandwidth of the direct-voltage con-trollers can be calculated directly by substituting the SCR andmaximum loading into (42), which yields and

. However, the linear analysis shows that thedirect-voltage controller at the rectifier side (station 2) tends tohave less phase margin than the inverter side (station 1). There-fore, the bandwidth of both of the direct-voltage controllers havebeen chosen as . Fig. 11 shows the stepresponse of the direct-voltage controllers at the two converterstations with the comparison between the linear model and thenonlinear model in PSCAD/EMTDC.

Fig. 12 shows the plots of disturbance reduction of the di-rect-voltage controller with the comparison between the linearmodel and the nonlinear model in PSCAD/EMTDC. The max-imum direct-voltage peak in Fig. 12(b) is below the specification


Fig. 12. Disturbance reductions of the direct-voltage controllers with max-imum load changes. (a) Converter block at station 2. (b) Converter block at sta-tion 1 (solid: nonlinear simulation, dashed: linear model).

. The discrepancies between the linear resultand nonlinear simulation result are due to that the linear result isobtained at one operating point, while the disturbance reductionin PSCAD/EMTDC involves operating point changes. The testsin PSCAD/EMTDC are performed by blocking the converter atthe other station with maximum loading.

C. Active-Power Controller

In principle, either of the VSC-HVDC converter stations canbe the power-controlling station. In this example, the converterstation 1 controls the active power by a PI controller with thecontrol law

(44)

The parameters of the active-power controller are tuned bythe root-locus technique. The proposed PI controller has twoparameters. Thus, the first step is to determine the value ofthe . A higher integral gain of a PI controller is neces-sary for eliminating error in steady state, as well as reducingthe power-recovery time after ac-system faults. However, alarge integral gain may also reduce the phase margin of theclosed-loop system. Thus, the initial integral gain is chosenas . It should be noted that might need to bere-adjusted based on the time simulation result.

Fig. 13 shows the root-loci of the closed-loop system byvarying from 0.0 to 1.0; three dominant pole pairs areaffected. The pole pair are shifted from the origin towardsthe left-half plane, which can be viewed as the stabilizingeffect of the feedback control. However, both the pole pairs

and ( , )are shifted towards the right-half plane. Frequency domainanalysis show that are related to the gain margin, while

are related to the phase margin of the active-power con-trol. is chosen to get a balance of stability and

Fig. 13. Root-loci of the dominant poles of the closed-loop system by applyingPI-type active-power control. � � ��, � from 0 to 1.0.

Fig. 14. Step response of the active-power controller at low and high powerlevels at the converter station 1. (a) Step from � � �� to � � ��

(b) Step from � � �� to � � �� . (solid: nonlinear simulation,dashed: linear model). Active-power controller: � � ��, � � ��.

response time, which places the three dominant pole pairs at, , and

.Fig. 14 shows the step response of the active-power control at

low and high power levels, respectively. The step response at thehigh power level corresponds to the operating point applied forthe root-locus tuning. The output of the active-power controllershould be limited to avoid too large direct-voltage variations.In this example, the limitation of the output of the active-powercontroller is chosen as , which corresponds to approxi-mately and direct-voltage variations.

D. AC-System Faults

The fault ride-through capability of the VSC-HVDC link istested in PSCAD/EMTDC by applying three-phase ac-systemfaults at both of the converter stations. The VSC-HVDC link ini-tially operates with maximum loading, i.e., . TheVSC-HVDC link is supposed to ride through ac-system faults


Fig. 15. Fault ride-through capability of the VSC-HVDC link with a three-phase ac-system fault at the inverter station (station 1).

without relying on telecommunications between the two con-verter stations.

In Fig. 15, a three-phase ac fault with 0.2 s duration isapplied at the converter station 1, i.e., the power-controllingstation in this example, at 0.1 s. One consequence of theac fault is that the direct voltage increases to approximately1.3 p.u. due to the loss of power output. The converter station2 brings down the direct voltage to its nominal value afteran initial overshooting. Another consequence of the ac faultis the increase of the modulus of the valve current .A current limitation controller proposed in [18] has beenapplied during the ac-system fault. After detecting the fault,the current limiter reduces the valve current to half of themaximum load current except a very short current spikeat the fault inception. For VSC applications, regardless of thecontrol principle, the converter always tries to protect itselffrom excessive over currents. This fast protection is oftenimplemented as a low-level hardware system, and its objectiveis to protect the converter in cases where the higher levels ofcontrol fail. Since the current spike in Fig. 15 is so short( in magnitude and ) in duration, it neitherdoes any harm to the converter valve, nor contributes much tothe short-circuit current to the ac system.

Fig. 16 shows a three-phase ac fault with 0.2 s duration ap-plied at the converter station 2, i.e., the direct-voltage controllingstation in this example, at 0.1 s. During the ac-system fault, thepower-controlling station controls the direct voltage to a lowervoltage level which is 13% less than the nominal value. The 13%is a result of the limitation of the power controller. Since the acfault is applied at the converter station 2, there is no over-cur-rent problem at the converter station 1. A specific amount ofactive power flows in the reverse direction during the fault pe-riod. However, the current limitation controller at the rectifier(station 2) limits the fault current to half of the maximum loadcurrent, or any other desired value.

A benchmark with vector current control is performed in [18].It was found that vector current control could not operate in theconditions corresponding to Figs. 15 and 16.

Fig. 16. Fault ride-through capability of the VSC-HVDC link with a three-phase ac-system fault at the rectifier station (station 2).

V. CONCLUSIONS

In this paper, VSC-HVDC system using power-synchro-nization control is investigated for interconnection of two veryweak ac systems. By using power-synchronization control, theVSC-HVDC link is possible to be applied in more challengingconditions. In addition, the VSC-HVDC link contributesshort-circuit capacity to the ac system at the PCC withoutincreasing the short-circuit current thanks to its current limitingcapability during ac-system faults. The major considerationfor VSC-HVDC links connected to weak ac systems is thatthe RHP transmission zero of the ac Jacobian transfer matrix,which moves closer to the origin with larger load angles. TheRHP zero imposes a fundamental limitation on the achiev-able bandwidth of the VSC, which implies the following forVSC-HVDC links interconnecting two very weak ac systems.

• At either of the converter stations, the VSC-HVDC con-verter shall not operate with too large load angles in thesteady state to maintain a reasonable stability margin.

• A higher dc capacitance is necessary to keep the variationof the direct voltage within the allowed range.

• The active-power controller at the power-controlling sta-tion should use the direct-voltage controller as the innerloop and limit its output in order not to affect the direct-voltage level too much from its nominal value.

APPENDIX A

AC system base value: power , line-to-linevoltage , nominal frequency .

DC system base value: power , pole-to-ground voltage .

VSC-HVDC link: rated power , ratedalternating voltage , maximum valve cur-rent , maximum allowed direct voltage

, , , dc capaci-tance per pole (15 ms in total), switching frequency

.


APPENDIX B

In the following, a mathematical proof of the instability ofthe dc Jacobian transfer matrix for operating points that

and is given. is a linear model of thedc -link in Fig. 4.

The poles of are the eigenvalues of the matrix in(28) which has the characteristic equation

(45)

with the coefficients

(46)

To prove the instability of , it is sufficient that any oneof the three coefficients in (46) is negative. The following is aproof of

(47)

It is obvious that the denominator of in (46) is positive. Thus,(47) is identical to

(48)

From the main circuit of the -link model in Fig. 4, the fol-lowing equality is established:

(49)

For VSC-HVDC applications, it is apparent that andhave the same polarity. Consequently, it follows from (49) that

(50)

or

(51)

Based on the inequality in (51), by dividing at bothsides, (48) can be rewritten as

(52)

Substituting

(53)

into (52) yields

(54)

which apparently holds. In other words, for other operatingpoints when and , is unstable as longas the dc-transmission line is not lossless.

REFERENCES

[1] B. Weedy and B. Cory, Electric Power Systems, 4th ed. West Sussex,U.K.: Wiley, 1998.

[2] W. Breuer, M. Lemes, and D. Retzmann, “Perspectives of HVDC andFACTS for system interconnection and grid enhancement,” in Proc.Cigre Workshop: Brazil-China-India Summit Meeting on HVDC andHybrid Systems, Rio de Janeiro, Brazil, 2006.

[3] G. Andersson, P. Donalek, R. Farmer, and N. Hatziargyriou, “Causesof the 2003 major grid blackouts in North America and Europe, andrecommended means to improve system dynamic performance,” IEEETrans. Power Syst., vol. 20, no. 4, pp. 1922–1928, Nov. 2005.

[4] B. Yang, G. Vittal, and G. Heydt, “Slow-coherency-based controlledislanding-a demonstration of the approach on the August 14, 2003blackout scenario,” IEEE Trans. Power Syst., vol. 21, no. 4, pp.1840–1847, Nov. 2006.

[5] J. Arrillaga, High Voltage Direct Current Transmission, 2nded. London, U.K.: Inst. Elect. Eng., 1998.

[6] M. Bahrman and B. Johnson, “The ABCs of HVDC transmission tech-nologies,” IEEE Power Energy Mag., vol. 5, no. 2, pp. 32–44, Mar./Apr.2007.

[7] V. Ramaswami, D. Retzmann, and K. Uecker, “Prospects of bulk powerEHV and UHV transmission,” in Proc. GRIDTECH 2007 Int. Conf.,New Delhi, India, 2007.

[8] IEEE Guide for Planning DC Links Terminating at AC LocationsHaving Low Short-Circuit Capacities, 1997, IEEE Std. 1204–1997,Tech. Rep.

[9] B. Franken and G. Andersson, “Analysis of HVDC converters con-nected to weak systems,” IEEE Trans. Power Syst., vol. 5, no. 1, pp.235–242, Feb. 1990.

[10] A. Gavrilovic, “AC/DC system strength as indicated by short circuitratios,” in Proc. AC/DC Power Transmission Int. Conf., London, U.K.,1991.

[11] O. B. Nayak, A. M. Gole, D. G. Chapman, and J. B. Davies, “Dynamicperformance of static and synchronous compensators at an HVDC in-verter bus in a very weak ac system,” IEEE Trans. Power Syst., vol. 9,no. 3, pp. 1350–1358, Aug. 1994.

[12] D. Jovcic, N. Pahalawaththa, and M. Zavahir, “Investigation of the useof inverter control strategy instead of synchronous condensers at in-verter terminal of an HVDC system,” IEEE Trans. Power Del., vol. 15,no. 2, pp. 704–709, Apr. 2000.

[13] G. Asplund, K. Eriksson, and H. Jiang, “DC transmission based onvoltage source converters,” in Proc. Cigre Conf. 14–302, Paris, France,1998.

[14] N. Flourentzou, V. G. Agelidis, and G. D. Demetriades, “VSC-basedHVDC power transmission systems: An overview,” IEEE Trans. PowerElectron., vol. 24, no. 3, pp. 592–602, Mar. 2009.

[15] D. Jovcic, L. Lamont, and L. Xu, “VSC transmission model for analyt-ical studies,” in Proc. IEEE Power Eng. Soc. General Meeting, Toronto,ON, Canada, 2003.

[16] M. Durrant, H. Werner, and K. Abbott, “Model of a VSC HVDC ter-minal attached to a weak ac system,” in Proc. IEEE Conf. Control Ap-plications, Istanbul, Turkey, 2003.

[17] P. Fischer, “Modelling and control of a line-commutated HVDC trans-mission system interacting with a VSC STATCOM,” Ph.D. disserta-tion, Royal Inst. Technol., Stockholm, Sweden, 2007.

[18] L. Zhang, L. Harnefors, and H.-P. Nee, “Power-synchronization con-trol of grid-connected voltage-source converters,” IEEE Trans. PowerSyst., vol. 25, no. 2, pp. 809–820, May 2010.

[19] L. Zhang and H.-P. Nee, “Multivariable feedback design ofVSC-HVDC connected to weak AC systems,” in Proc. PowerTech2009, Bucharest, Romania, 2009.

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Lidong Zhang (M’07) received the B.Sc. degreefrom the North China Electric Power University,Baoding, China, in 1991 and the Tech.Lic degreefrom Chalmers University of Technology, Gothen-burg, Sweden, in 1999. Since 2007, he has beenpursuing the Ph.D. degree in industrial engineeringpart time at the Royal Institute of Technology,Stockholm, Sweden.

From 1991 to 1996, he worked as an engineer withthe Leda Electric Co., Beijing, China. Since 1999,he has been with ABB Power Systems, Ludvika,

Sweden. His research interests are HVDC, power system stability and control,and power quality.

Lennart Harnefors (S’93–M’97–SM’07) was bornin 1968 in Eskilstuna, Sweden. He received theM.Sc., Licentiate, and Ph.D. degrees in electricalengineering from the Royal Institute of Technology,Stockholm, Sweden, and the Docent (D.Sc.) degreein industrial automation from Lund University,Lund, Sweden, in 1993, 1995, 1997, and 2000,respectively.

From 1994 to 2005, he was with Mälardalen Uni-versity, Västerås, Sweden, where he, in 2001, was ap-pointed as a Professor of electrical engineering. He is

currently with ABB Power Systems, Ludvika, Sweden. From 2001 to 2006, hewas also a part-time Visiting Professor of electrical drives at Chalmers Univer-sity of Technology, Gothenborg, Sweden. His research interests include appliedsignal processing and control, in particular, control of power electronic systemsand ac drives.

Prof. Harnefors was the recipient of the 2000 ABB Gunnar Engström En-ergy Award and the 2002 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Best Paper Award. He is an Associate Editor of the IEEE TRANSACTIONS ON

INDUSTRIAL ELECTRONICS.

Hans-Peter Nee (S’91–M’96–SM’04) was born in1963 in Västerås, Sweden. He received the M.Sc., Li-centiate, and Ph.D degrees in electrical engineeringfrom the Royal Institute of Technology, Stockholm,Sweden, in 1987, 1992, and 1996, respectively.

In 1999, he was appointed Professor of PowerElectronics in the Department of Electrical Engi-neering at the Royal Institute of Technology. Hisinterests are power electronic converters, semicon-ductor components and control aspects of utilityapplications, like FACTS and HVDC, and vari-

able-speed drives.Prof. Nee was awarded the Energy Prize by the Swedish State Power Board in

1991, the ICEM’94 (Paris) Verbal Prize in 1994, the Torsten Lindström ElectricPower Scholarship in 1996, and the Elforsk Scholarship in 1997. He has servedin the board of the IEEE Sweden Section for many years and was the chairmanof the board during 2002 and 2003. He is also a member of EPE and serves inthe Executive Council and in the International Steering Committee. Addition-ally, he is active in IEC and the corresponding Swedish organization SEK in thecommittees TC 25 and TK 25, respectively.

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