IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, …ir.nsfc.gov.cn/paperDownload/1000006901305.pdf · 2018. 7. 19. · IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, JULY

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, JULY 2013 1391

Converter System Nonlinear Modelling and Controlfor Transmission Applications—Part II: CSC Systems

Yonghe H. Liu, Senior Member, IEEE, Neville R. Watson, Senior Member, IEEE,Keliang L. Zhou, Senior Member, IEEE, and B. F. Yang

Abstract—The high-power self-commutated voltage-sourceconverter (VSC) and current-source converter (CSC) are the keycontrol devices in high-voltage direct current, flexible ac transmis-sion systems, and distribution flexible ac transmission systems. Toachieve the expected control objectives, suitable control strategiesmust be implemented based on the available devices, systemmodels, and control techniques. The self-commutated ac/dc con-verters control the electrical power by generating controllable acfundamental and dc average outputs. These controllable outputsare controlled by the conducting state combinations of the con-verter switching devices, driven by their gate signals. The gatesignals are specified by fundamental parameters of frequency,amplitude, and phase angle. The converter system model fordescribing the relation between the system-state variables and thegate signal parameters is essential for the converter system controlstrategies. The companion paper (Part I) derives the state variableequations for the transmission systems using voltage-source-typeconverters. Part II is for the transmission systems using cur-rent-source-type converters. The self-commutated convertersystems provide control flexibility of active and reactive powers,but their nonlinearity makes their control difficult. The linearizedstate equations using feedback linearization are presented toenable the controller design by using linear control theory.

Index Terms—Control, current-source converter (CSC), flexibleac transmission systems (FACTS), high-voltage direct current(HVDC), linearization, modeling, nonlinearity.

I. INTRODUCTION

T HEORETICALLY, either self-commutated voltage-source converters (VSCs) or self-commutated cur-

rent-source converters (CSCs) can be used for the control ofactive and reactive powers. However, most publications onself-commutated converters used for active and reactive power

Manuscript received March 04, 2012; revised July 15, 2012; accepted Feb-ruary 18, 2013. Date of publication April 09, 2013; date of current version June20, 2013. This work was supported in part by the State National Science Foun-dation (No. 51067006) and in part by the National Science Foundation of InnerMongolia (No. 2010ZD06). Paper no. TPWRD-00225-2012.Y. H. Liu is with the Inner Mongolia Electric Power Research Institute,

Hohhot 010020, China, and also with the the Department of Electrical andComputer Engineering, University of Canterbury, Christchurch 8140, NewZealand (e-mail: [email protected]).B. F. Yang is with the Inner Mongolia Electric Power Research Institute,

Hohhot 010020, China (e-mail: [email protected]).N. R. Watson and K. L. Zhou are with the Department of Electrical and Com-

puter Engineering, University of Canterbury, Christchurch 8140, New Zealand(e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRD.2013.2250529

control focus on VSC topology. The reasons for the preferenceof the VSC over the CSC are mainly as follows.1) VSCs control their ac voltages but never switch off their accurrent; therefore, an interfacing reactor is needed betweenthe VSC and ac source for suppressing current inrush;while PWM CSCs control their ac current by switchingON/OFF their ac current; therefore, an interfacing capacitoris needed for absorbing inductive energy stored in the acsystem, unless their ac currents are kept continuous (neverswitching them to zero).

2) The interfacing capacitors required for absorbing the en-ergy stored in the ac-side inductance could be bulky dueto high ac system inductance and, thus, not only adds tothe overall cost but also has the risk of causing harmonicresonance.

3) The switching devices required must be symmetrical involtage blocking capability. This rejects the use of theasymmetrical switches like insulated-gate bipolar transis-tors (IGBTs), or series combination with a diode and in-herent extra losses. The symmetrical switches are of lowswitching frequency but low ON-state voltage drop; the lowswitching frequency characteristics result in fundamentalswitching being preferred for CSC.

4) Comparing the dc-side inductor for dc current stiff in theCSCwith the capacitor used in VSC for dc voltage stiff, theinductor losses are higher than the capacitor. (The lossescan be reduced significantly by using the superconductiveinductor.)

The first two problems can be solved by newer CSC topologyand control technology. The MLCR–CSC [11] uses the multi-level and reinjection concept as well as fundamental switchingto produce controllable and continuous ac output currentswithout the need for the capacitor to interface with the powergrids. The advantages provided by the CSC are its directcontrol of the dc current and fully controllable dc voltage fromnegative-rated dc voltage to positive-rated dc voltage, whichcould not be provided by the VSC. For some applications, thedc voltage must be fully controllable in the rated range.The nonlinear nature of self-commutated conversion under

the four-quadrant power control operation requirement presentsthe main difficulty to the control system design. For control ofactive and reactive powers of CSC in four quadrants, the activepower and reactive power .In the conventional line-commutated conversion (LCC) withoutreactive power control requirements, the two quadrant operationforces the firing delay angle to be within the limited operationregion . The active power, being proportional

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1392 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, JULY 2013

Fig. 1. Equivalent circuit of the CSC system.

to , can be linearized around without signifi-cant error for most of the operation region. This linearization hasbeen proven reliable in practical thyristor LCC system controlof the active power. For four quadrant control of active and reac-tive powers, the firing delay angle has to be within the region

. The triangle functions andcannot be linearized around a specific point, as their magnitudesand their derivative magnitudes as well as the signs of the func-tions and their derivatives change in the required region. Thelinearization around a point cannot be used for control of activeand reactive powers in four quadrant.VSC system modelling and control have been studied for

decades [2]–[10]. Relatively few publications address CSCsystem control and modeling. The CSC average models derivedin this paper are also based on the generalized state-spaceaveraging method and the principle of power balance. The ac-curacy of the CSC average model is ensured by the high-qualitywaveforms and the stiff dc current of the high-power CSCs,as the high-quality waveforms of the high-power CSCs areachieved by adopting PWM or multi-level techniques.

II. MODELING PWM CSCS

The circuit shown in Fig. 1 is a generalized connection ofthe PWM–CSC for different applications. The derivation in thissection obtains the average model of the CSC in the generalizedsystem which is suitable for different applications by setting theappropriate parameters of the respective components to eitherzero or infinitive.In Fig. 1, the CSC ac terminals are connected to the ac power

system, represented by a Thevenin equivalent; the inductanceand resistance between the ac source and CSC represent

the leakage reactance and losses of the interfacing transformer.Its dc terminal is connected to a smooth reactor, representedby and . The load of the CSC is a dc voltage source,representing another converter or dc source or load; the smoothreactor and the dc voltage source are connected by a T-network,which is used for modeling the transmission line or cable.The ac-side voltage balancing matrix equation of the three-

phase CSC is written in the form of

where is the three-phase current vector,is the three-phase ac-source voltage

vector, and is the three-phase voltagevector at the CSC ac terminal.

The current balance matrix equation at the CSC ac terminalsis

whereis the CSC ac output current vector. Its rms am-

plitude , the frequency , andthe phase angle are the CSC controllable variables. For con-necting to a constant frequency ac system, only and canbe used for control of the CSC operation.The CSC dc output current is directly injected to the

T-network, and the dc current from the T-network to the dcsource is

The dc-side voltage balance equations are

The dc- and ac-side power balancing yields

By substituting into, and rearranging all of the other equations,

the system three-phase equations are

(1)

(2)

(3)

(4)

(5)

where , ,, and , .

By performing the orthogonal transformation, the three-phasevector (1) to (3) can be transformed into the -frame. By ex-pressing the control variable vector of the PWM–CSCin the form of

LIU et al.: CONVERTER SYSTEM NONLINEAR MODELLING AND CONTROL FOR TRANSMISSION APPLICATIONS—PART II 1393

the state equations of the PWM–CSC for all of the applicationscan be written in the affine form of

For PWM–CSC HVDC terminal control, the state equationin affine form is

(6)

For the PWM–CSC STATCOM application, the state matrixequation in affine form is

(7)

For PWM–CSC back-to-back applications, the state matrixequation in the affine form is

(8)

These affine nonlinear equations do not satisfy the state-inputlinearizable condition, therefore the exact linearized state equa-tions cannot be given. By ignoring some terms in the dq-framestate equations, the linearized equations could be obtained butaccuracy is reduced. The approximations for linearization pro-posed by many researchers are focused on linearizing the lastrow equation in these state equations. The last equation is de-rived based on the ac and dc active power balancing; the ac-sideactive power of the CSC at its ac terminals can be expressed by

or by:

The term corresponds to ac-side resistivelosses, and it is a small percentage of the power con-trolled by the CSC, therefore it is negligible. As the terms

and corre-spond to the changing rate of the energy stored in inductance

and capacitance respectively, their energy are directlyrelated to the rms levels of the inductance current and capac-itance voltage ( and ). Ifand only if the current rms-levels change slowly or and

are relatively small, ignoring them will not cause great dif-ference in use of these model in practice. That is to say ignoringthe two terms will not be appropriate for system analysis andcontroller design with the fast changing ac voltage and current.After removing the three terms, the linearized state matrix

equation for STATCOM and back-to-back applications are

(9)

and


(10)

where and arethe new control input vectors, is the new state variable in-stead of .

III. MODELING GROUPED CSCS WITHFUNDAMENTAL SWITCHING

Pulsewidth modulation (PWM) provides the control freedomfor regulating the CSC ac current amplitude by turning theswitching devices ON/OFF at higher frequencies. For high-powerapplications, the switching losses can be minimized by usingfundamental switching, but without the control freedom overthe ac current amplitude. For high-power applications, it iscommon that more than one set of CSCs is needed. By com-bining them in an appropriate way the grouped fundamentalswitching CSCs can provide the ac current amplitude controlfreedom. This is achieved by parallel connection on the ac sideand series connection on the dc side of the CSCs. The parallelconnection on ac side provides the combination freedom ofadding ac currents with the different phase shifts to form theamplitude controllable ac current output; the series connectionon dc-side provides the combination freedom of adding thedc voltages with the different levels to form higher dc voltageoutput.The MLCR–CSC topology (refer to [11]) is preferred to be

used in the grouped CSC configuration as it uses multi-level andreinjection concept as well as fundamental switching to producephase controllable and continuous ac output currents without theneed of capacitors to interface with power grids.The grouped configuration of the two MLCR–CSCs (or any

type of CSC without need of interfacing capacitance) is shownin Fig. 2, their ac-side terminals are parallelled and the dc-sideterminals are series connected. The voltages ( , ) acrossthe two CSC ac terminals are for representing their equivalentac output voltages.In the diagram (Fig. 3) the ac current phasors of two CSCs are

denoted as and , respectively, the combined ac current

Fig. 2. Equivalent circuit of the grouped two CSC system.

Fig. 3. Phasor diagram of the two CSCs system.

phasor are vectorial sum of these two currents,. Since the amplitude of and is exactly

the same and determined by the converter dc-side current, thefollowing relation can be written:The combined ac current amplitude and its phase angleare

(11)

It is clear that the amplitude of can be controlledby , and its phase angle is determined by

. Thus, the combined CSC group consisted oftwo CSCs and provides phase and amplitude controllabilityindependently, similar to how PWM converters control theac current amplitude through the modulation index and phaseangle through the phase shift.In practice, the two CSCs are identical. Therefore, to derive

the state equations of the grouped CSCs, the interfacing trans-formers are modelled by the same impedance (series connec-tion of the resistance and inductance and ). The ac-sidevoltage balance equations are

where is the three-phasecurrent vector, is the three-phaseac source voltage vector, and and

are the three-phase current vectorsof the two CSCs ac current outputs, respectively. Under the


fundamental switching control, the ac output currents of thetwo CSCs have a strict relation to the common dc current

where is a constant related to the CSC topology.By adding and subtracting these two equations, respec-

tively, and substituting , ,, into the equations,

the ac-side expressions are obtained, i.e.,

where , ,, . The dc-side equations are

(12)

(13)

(14)

The ac- and dc-side power balance gives

Substituting the expression into (13) andrewriting the three-phase equations yields

(15)

(16)

(17)

(18)

(19)

where , and

The state equations in the -frame can be obtained by per-forming the orthogonal transformation (15)–(17). They are

(20)

(21)

(22)

(23)

(24)

where

The last vector (24) shows that there is a stable self-governedsubsystem inside the system, as the state variables andhave no interaction with the other equations. They are only re-lated to the two CSCs control variables and . Thefirst five equations are sufficient for the study of the dynamicand steady-state operation of the system related to the ac- anddc-side variables .The vector is related to the system-state variables and

control inputs, and by using the relations

it can be derived

(25)

where


By substituting it into the -frame equations, the resultingequations in affine form are

(26)

where

The equations in (26) can be directly used for the control of anHVDC terminal or dc/ac conversion. They can also be modifiedfor a particular application. For STATCOM application,, , , and , the stateequation in affine form is

(27)

For a back-to-back (BTB) link , ,, , and the last two rows in (26) are removed.

The dc voltage-source block in Fig. 2 is an equivalent mirrorsystem as it is the same as that of the ac system and CSC onthe left-hand side. The ac-side equations for the two ac systems(left side is denoted A, right side is denoted B) are

(28)

(29)

As and are the CSC powers from the ac todc sides, they are

and these two dc powers on dc side are related to equation:

This results in

(30)

where

Eliminating , , and in theright-hand sides of (28)–(30), gives the Link state equa-tion in the affine form

(31)

where


These equations can also be used for a UPFC system by makingsmall changes of .These equations present the affine nonlinearity and they could

be linearized around a specific operation state and then a con-troller designed, however it is then only being valid in a smallregion around this state. In practical CSC operation, its dc-sidecurrent may be controlled within a small margin or in a largeregion changing slowly, but its ac-side current need to becontrolled to vary in the region of zero to rated level graduallyor sharply. The challenge is to find a design method which isvalid globally and independently on the operation states.It is possible to linearize an affine nonlinear system by using

the feedback linearization [1]. A nonlinear controller is designedfor an affine nonlinear system based on the nonlinear trans-

formation to convert the affine nonlinear system into a linearsystem. Therefore the transformed state variables can be con-trolled by the transformed input variables, to vary in desiredtraces by using the well developed linear control techniques.The reversible transformation from the transformed input vari-ables to the original input variables enables the CSC systems tobe controlled through the practical control variables, achievingthe desired performance.The main difficulty for transforming the affine nonlinear

equations into linear form is in synthesizing the nonlineartransformations, but for the grouped-CSCs systems this canbe achieved based on the electrical technology. The followingderivations are presented to give the transformations and theprocedures.To transform the grouped-CSCs STATCOM (27) into linear

form the energy stored in the ac- and dc-side inductors and theirderivation, as well as the dc current are chosen as the trans-formed state variables

The transformed state equation for grouped-CSCsSTATCOM application is

(32)

where

To transform the grouped-CSCs state (26) for HVDC ter-minal control, the energy stored in the ac- and dc-side induc-tors and their derivation as well as the dc current, the T-networkcapacitor voltage, and inductor current are chosen as the trans-formed state variables. Hence


The transformed state equation for HVDC terminal controlapplication is thus

(33)

and

if exists,

is the transformation to obtain the original

control inputs for HVDC and STATCOM applications.To transform the grouped-CSCs state (31) for a back-to-back

(BTB) system, the energy stored in the dc- and both ac-sideinductor currents and their derivation as well as the right-side accurrents , and the left-side ac current are chosen.The transformations are

The transformed state matrix equation for a back-to-backsystem is

(34)

and

exists,

is the transformation to obtain the original control inputs forBTB link application.

IV. NONLINEAR CONTROL OF CSC CONVERTER SYSTEMS

In power transmission and distribution the active and reac-tive power need to be controlled to vary in full rated range andbidirectionally. The nonlinear nature and wide range operationrequirements of the CSC systems are the main difficulties incontrol system designs. For a linearizable affine system, a non-linear controller can be designed based on the well-developedlinear control system theory. The linearized CSC systems canbe described by two sets of state equations of multi-inputs andmulti-outputs. They are:

The nonlinear control system is expected to generate the re-quired control inputs which keeps the system stable and forcesthe converter to operate at the referenced level as well as toprovide the required steady-state and dynamic performance. If


Fig. 4. Control system structure of the VSC or CSC system.

the required state variable vector can be expressed by thereferenced output (i.e., ), the required trans-formed state reference can be obtained by the transformation

. The control system can be constructed based onlinear state feedback. Fig. 4 shows the structure.In Fig. 4, the closed-loop system from to can be treated

as a full-state feedback controlled linear system, is the statefeedback constantmatrix, which can be uniquely found based onthe expected poles of . The part representedby a dashed line is used to design the state feedback matrix. In real time, it will not be involved. In real-time control,

the required control for the linear system is continuouslyconverted to the original input by , and sendsto the converter gate firing logic to force the converter to thedesired operation state.For transmission and distribution applications, the power

converters need to be controlled effectively in normal andfault grid conditions on their dc and ac sides. In the proposedmodel, the ac and dc grids are represented by the equivalentsource voltages and impedances. These grid parameters mayvary enormously in practice. The proposed control systemmust be capable of providing proper control actions whenthese parameters change. The grid fault handling capabilityand grid parameter sensitivity of the control system must beacceptable to ensure the system performances under practicalgrid conditions.The CSC direct current control nature can be used to regulate

the dc current during the fault on dc side, and can control thesystem states quickly to the previous state by the state controlafter fault being cleared; with the dc current limitation providedby the dc source the CSC system state control can control theconverter ac and dc currents during fault on ac side, and return tothe previous state smoothly and quickly after fault being cleared.The robust control performance against large variations of the

ac grid impedance is very important for transmission and distri-bution applications. The proposed control system is based onthe feedback linearization. In the linearized system the ac gridequivalent parameters and all of other system parameters areconverted into equivalent inputs, and the linearized system isa system of its state variables being directly driven by theseequivalent inputs and without coupling interaction. The pro-posed controller for this linear system can provide perfect ro-bust performance, the parameter sensitivity of the transforma-tion from the equivalent inputs to the original converter controlinputs in the proposed controller is investigated by simulations.The simulation results show that the proposed control system isquite robust against the grid impedance variations.

Fig. 5. Simulated waveforms.

V. SIMULATION VERIFICATION

A grouped CSC for integrating 1 MW-PV system into a gridis simulated to test the proposed model and controller, using thefollowing parameters: , 60 mH, 10kV, for representing the ac grid systems; ,16 mH, for the interfacing transformers; MLCR CSCs with adc reactor of 190 mH and ; 5mH, and for collecting cable. Thevariables shown in the results are in per unit with reference tothe following normalizing bases: 0.1 kA for the ac current, 1MW for real and 1 MVA for reactive powers, and 1.4 kA for thedc current.In Fig. 5(a)–(h), the PV–CSC is controlled to follow the ac-

tive and reactive power orders in different forms. The sinusoidalactive power order and the zero reactive power order are for sim-ulating the active power variation and unity power-factor oper-ation; the step active power order, and the quickly changing re-active power order are for testing the active and reactive powercontrollability of the PV–CSC system.In Figs. 5(a)–(c), the CSC active power , the dc cur-

rent , and CSC reactive power follow their referenceorder , , and , respectively. As the unitypower-factor operation is achieved.


Fig. 6. Robust performances of the proposed control system.

In Fig. 5(d)–(f), also , , and follow ,and , as and in changing the active and re-active power independent control is achieved. Fig. 5(g) and (h)shows the ac current waveforms in reactive power dynamics,(g) for reactive power from capacitive to inductive and (h) forinductive to capacitive, respectively.In Fig. 5(i)–(l), the control strategy for the CSC–PV system

is constant dc current and constant voltage (CCCV) at its acterminal for the different source voltage conditions. The CCCVstrategy is applied from 3 s, and the previous strategy is constantdc current and unity power factor (CCUPF). The source voltageconditions are set to: 1 p.u. (2 to 4 s), 0.85 p.u. (4 to 5 s), 1 p.u.(5 to 6 s), 0.5 p.u. (6 to 7 s), 1 p.u. (7 to 8 s), 0 p.u. (8 to 9 s),1 p.u. (9 to 10 s), to test the terminal voltage controllability andlow-voltage ridethrough capability.The PV–CSC active power and reactive power are

shown in Fig. 5(i), where the PV–CSC ac terminal voltage andcurrent rms levels in (j), are all in per unit. With 0.6-p.u.activepower injected to the 1-p.u. source voltage grid, the CCUPFrises the terminal rms voltage to 1.06 p.u. (2 to 3 s); for the sameactive power injected, the CCCV controls the terminal voltageto 1.0 p.u. (3 s to 4 s). For 0.85-p.u. source voltage (4 to 5 s), theCCCV controls the reactive power at the PV–CSC terminal tomaintain the terminal voltage to 1.0 p.u. (4 to 5 s). For 0.5 p.u.and 0-p.u. source voltages (6 to 7 s and 8 to 9 s), the PV–CSCis controlled by CCCV to inject the maximum reactive currentinto the grid, rising the terminal voltage to 0.66 and 0.18 p.u.,respectively. Fig. 5(k) and (l) shows the ac current waveformsin source voltage changing dynamics, and the PV–CSC currentsare well controlled due to the CSC characteristics and the con-troller performance.Fig. 6 shows the proposed control system robust performance

against the ac grid impedances variations. The control system isdesigned to control the dc current and the converter ac ter-minal voltage rms level for verifying the control systemperformances. In the simulations the control system is designedwith grid inductance 50 mH for Fig. 6(a) and (b);

100 mH for Fig. 6(c) and (d), but the source induc-tance is set to different values in the different time intervals,respectively. In these time intervals, the dc current and acterminal voltage rms level are observed and compared. Theirresponses caused by the grid inductance changes and the dc cur-rent order variations are shown in Fig. 6.

In Fig. 6(a) and (b), the control system of 50 mHis performed for the entire simulation period of 7 s, the gridinductance is set to 50 mH in the time interval (0 to 2s), 10 mH in the time interval (2 to 4.5 s) and 250mH in the time interval (4.5 to 7 s).In these three time intervals after the source inductance

has changed, the variations of the converter ac terminal voltagerms level are observable, but limited in a small marginand lasted shortly by the control action. Even for the largechange (from 10 to 250 mH), the maximum variation isless than 0.05 p.u.; the source inductance change causes dccurrent change too, but is very small (all less than 0.01 p.u.).In these three time intervals, the dc current order is increased

0.05 p.u. for 0.5 s to compare the dynamic differences fordifferent grid inductances. The control system is robust forac grid inductance variations, nearly the same dynamic isobserved.In Fig. 6(c) and (d), the control system of 100 mH

is performed for the whole simulation period of 7 s, the gridinductance is set to 100 mH in the time interval (0 to2 s), 10 mH in the time interval (2 to 4.5 s) and 250mH in the time interval (4.5 to 7 s).The three time intervals of Fig. 6(c) and (d) demon-

strated the same responses of and as those shownin Fig. 6(a) and (b), which are caused by grid inductancevariations and dc current order changes. These imply that theproposed control system is robust and suitable for grid-con-nected converter applications.

VI. CONCLUSION

This paper describes a generalized theory for the modellingand state control of nonlinear three-phase self-commutatingCSC systems at the converter-state control level. The CSCsystem uses a general equivalent suitable for the derivationof the state average model. The derivations are based on thegeneralized state-space averaging method and the principle ofac and dc power balance. This general model can be adaptedfor the specific applications by setting the parameters of theappropriate components to zero or infinite.The converter system-linearized models are derived by using

feedback linearization and, as a result, the nonlinear convertercontrol system can be designed based on linear control theory.PV power integration simulations are performed to verify theproposed model and control system. The simulation resultsshow that the proposed controller can control the active andreactive power independently and can achieve the desired per-formance for normal and abnormal grid conditions. The controlsystem is robust against the ac grid parameter variations; thisis required by the transmission and distribution applications inpractice. Further study is needed to ensure the input transfor-mation existence (from linear control to original control )in global state space.

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Yonghe H. Liu (SM’10), photograph and biography not available at the time ofpublication.

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, …ir.nsfc.gov.cn/paperDownload/1000006901305.pdf · 2018. 7. 19. · IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 3, JULY