Vector Current Controlled Grid Connected Voltage Source Converter— Influence of Non-Linearities on the Performance

Jan Svensson Michael Lindgren

Department of Electric Power EngineeringChalmers University of Technology

S-412 96 Göteborg, Sweden

Abstract— In the literature, the influence of non-linearitiesoccurring due to the effects of blanking time, the valves and thenon-ideal grid filter on the dynamic performance of currentcontrolling are usually not described. In this paper, small-signal analysis is used to study the influence of non-linearitiesfor a vector current controlled VSC connected to a grid. Toshow the influence of non-linearities, measured frequencyresponses are compared with responses from analyticalmodels. Measured responses are presented for operation inrectifier and inverter operation as well as static VAr operation.As displayed, the frequency dependent losses in the grid filteraffect the dynamic performance at high frequencies. A methodfor compensating for non-linearities due to blanking-time andnon-ideal valves is implemented. The compensation methodimproves the performance.

I. INTRODUCTION

In the evaluation of control systems for voltage sourceconverters (VSCs) connected to an AC grid, the dynamicperformance of the system should be studied. The discrete-timevector current controller with a sub-oscillating PWM [1] usesfeedforward and is sensitive to parameter variations and non-linearities. In VSCs, a blanking time is introduced to avoidshort-circuiting the dc-link voltage. This blanking time givesrise to a non-linearity in the PWM. In addition, a voltage dropoccurs across the valves when the valves are conducting.These effects introduce non-linearities into the system.

In steady-state and for slowly varying reference values, anintegration term in the controller allows some parametervariations and non-linearities. The performance of the currentcontrol system can be improved by using feedforwardcompensation for errors due to the blanking time and non-linearities of the valves. This has been investigated foradjustable-speed drives [2].

In addition to errors caused by the valves, the PWM-methodmay introduce non-linearities. The linearity of an analogcontrol system performing continuous natural sampling in thePWM has been investigated in [3]. Constant gain was obtainedexcept for frequencies in which harmonics due to the PWMoccur. In [4], both natural sampling and uniform samplingwere considered. As displayed, natural sampling is linear.With uniform sampling, the amplitude of the fundamentaloutput is a function of the frequency ratio. In [3] and [4],transfer functions in the fixed αβ-frame were considered. Influx-oriented control systems, errors may be introduced bycoordinate transformations. Since uniform sampling isperformed, a piecewise constant angle is used in thetransformation from the fixed frame to the rotating frame. Dueto the rotation of the dq-frame during the sample, a non-lineargain and coupling between d- and q-directions may occur iflow sampling frequencies are used. As displayed in [5], cross-coupling and non-linear gain may occur due to uniformsampling and non-ideal PWM.

The influence of valve compensation dynamically, i.e., in thefrequency domain, has not been focused on before for a grid-connected VSC. Moreover, the influence of the frequencydependent parameters of the grid filter on the current controllerand the effect of the operating point on the small-signalperformance is an important aspect which has not been studiedbefore.

In this paper, the performance of a discrete-time vector currentcontroller using a sub-oscillating PWM is investigated. Toreduce the influence of the valves both in steady state anddynamically, a compensation function is introduced. Thedynamic performance of the control system is studied in thefrequency domain. To evaluate the performance of P and PI-controllers and the influence of the compensation function forthe non-linearities, measured frequency responses arecompared with responses from analytical models. In the firstanalytical model, the grid filter has constant parameters. In thelatter model, measured frequency-dependent losses of the filterare taken into account. To display the influence of the losses inthe grid-filter inductors, two different inductors are used.Measured responses are also shown at four different operatingpoints.

II. SYSTEM CONFIGURATION

The system configuration of the VSC connected to the grid byan L-type grid filter is shown in Fig. 1. The controller tracksthe reference active current i q

∗ (t ) and the reference reactivecurrent id

∗ (t) . The grid currents and the grid voltages aresampled and transformed into the two-axis αβ-coordinatesystem and then into the rotating dq-coordinate system. The d-axis of the dq-frame is synchronized with the grid flux vector.The reference voltage vector from the controller is transformedback into the three-phase system. To take full advantage of thedc-link voltage udc (t) , triplen harmonics are added to thereference voltage values in the block OPT.

i 1(t )VSC

+

–udc (t)

u1 (t)

R s L s

u3( t)

u2 (t)

3/23/2

i 2 (t )

i3 (t)

6

PWM

OPT

current controller

estimatorθ

transformationangle for αβ / dq

θ(k )

αβ/dq

αβ/dq 2/3dq/αβ

sample and holdsample and hold

i (αβ )(k)e(αβ )(k)

θ(k )

θ(k )

e(dq )(k) i(dq )(k)

udc(k)

id∗ (k) iq

∗ (k)

grid

e1(t)

e3( t)

e2 (t)

+– ~

~

~

+–

+–

θ +32

ωN TS(k+1)

ud∗ (k+1) uq

∗ (k+1)

u∗( )(k+1)123

uopt∗( )(k+1)123

Fig. 1. Overview of the system, which contains the VSC, the grid filterand the controller.

The reference voltages for the VSC are transformed into apulsewidth pattern in the block PWM. The resistance and theinductance of the grid filter are denoted by R S and L S ,respectively. The phase voltages and currents of the grid aredenoted by e1(t) , e2 (t) and e3(t) and i1(t) , i2 (t) and i3(t),respectively. The phase voltages of the VSC are denoted byu1(t), u2 (t) and u3(t). The experimental system consists of aVSC with IGBT valves (Toshiba MG400Q1US41 1200V,400A) and a TMS320C30 control computer. The characteristicparameters of the system are displayed in Table 1. The samplefrequency, fS , is equal to the switching frequency, fSW . Thesample time is denoted by TS . The frequency of the gridvoltages is 50 Hz, which gives the angular frequency ωg inTable 1. The base impedance is denoted by Z base . In themeasurements of the currents, LEM LA50-S/SP1 modules areapplied. In the inputs to the DSP-system, first-order, low-passfilters with a crossover frequency of 75 kHz are used. In theoptical fibers between the control system and the drive circuitsof the valves, a time delay of 0.5 µs is introduced. Finally, atime-delay of 3 µs occurs in the gate circuits of the valves.

TABLE ICHARACTERISTIC PARAMETERS OF THE SYSTEM.

L S =0.071 pu E =1.0 pu ωg=100π rad/s fsw =6 kHz

R S=0.012 pu udc=1.5 pu TS =166.7 µs Z base=6.28 Ω

III. VSC AND GRID MODELS

To evaluate the performance of the vector-control system,measured small-signal responses are compared with responsesfrom analytical models. By forming state-space models in therotating dq-frame, time-invariant models of the closed-loopsystem are obtained. The valves in the converter are modelledas ideal switches by the use of the average switch model(ASM) technique [6]. Due to the harmonic content in theconverter output voltages, analytical models based onaveraging predict the states of the system accurately only at thesampling instants .

A. Grid Filter and Converter ModelsThe continuous-time state equation for the grid filter in the dq-coordinate system is

d

dt

id (t)iq (t)

= Aid (t)iq (t)

+ Bu

ud (t)uq (t)

+ Be

ed (t)eq (t)

(1)

where the matrices are given by

A =–

R s

L s

ωg

–ωg –R s

L s

Bu =

1Ls

0

01Ls

Be =–

1Ls

0

0 –1Ls

The vector current controller operates in discrete time. Tomodel the closed-loop system, the grid model is transformedinto a zero-order held sampled system with the sample time TS .The state equation is

id (k + 1)iq (k + 1)

= Fid (k )iq (k )

+ Gu

ud (k )uq (k )

+ Ge

ed (k )eq (k )

(2)

where the matrices are

F = eATS

Gu = –Ge = eAτ

0

TS

∫ dτ Bu

(3)

The VSC is modelled by using the ASM method and isassumed to provide the requested voltage vector in the dq-

frame. Two different inductor models are used. In the firstmodel, the inductors have a constant inductance and resistance.In the second one, the influence of the losses on the equivalentseries resistance and inductance is taken into account.

B. Controllers and Closed-loop ModelsThe controller presented here has partially been adopted from[7]. The integration part of the PI-controller is introduced tocompensate for static errors caused by non-linearities, noisymeasurements and non-ideal components. Eq. (1) can berewritten as

ud (t) = ed (t) + R rid (t) – ωgL riq (t) + L r

d

dtid (t) (4)

uq (t) = eq (t) + R riq (t) + ωgL rid (t) + L r

d

dtiq (t) (5)

where the inductance and the resistance of the series coil,which is used in the controller is denoted by subscript r toemphasize that the parameters of the controller can differ fromthe real values. The mean voltages over the sample period k tok+1 are derived by integrating (4) and (5) from kTS to(k + 1)TS and dividing by TS . Dead-beat control is used.During one sample period, the current variations are assumedto be linear and the grid voltage components constant. Theoutput reference voltages are delayed one sample due to thecomputer calculation time. A time delay compensation term isincluded in the dead-beat controller to avoid currentoscillations. When using dead-beat control, it takes twosamples before the grid currents are equal to the referencevalues of the current. The equations of the controller can bewritten as

ud∗ (k + 1) = ed (k ) + R rid (k ) – ωgL riq (k )

+k P id∗ (k ) – id (k )( ) + ∆uId (k ) – ∆ud (k )

(6)

uq∗ (k + 1) = eq (k ) + R riq (k ) + ωgL rid (k )

+k P iq∗ (k ) – iq (k )( ) + ∆uIq (k ) – ∆uq (k )

(7)

where the gain for the dead-beat control is

k P = L r

Ts

+ R r

2(8)

The compensation terms for one-sample time delay [8] are

∆ud (k ) = k p id∗ (k − 1) – id (k − 1)( ) – ∆ud (k − 1) (9)

∆uq (k ) = k p iq∗ (k − 1) – iq (k − 1)( ) – ∆uq (k − 1) (10)

The integration term of the PI-controller is a sum of the currenterrors for all the old samples and can be written as

∆uId (k + 1)∆uIq (k + 1)

= ∆uId (k )∆uIq (k )

+ k I

id∗ (k – 2)

iq∗ (k – 2)

– k I

id (k )iq (k )

(11)

where the integration constant is defined as k I = TSk PR r / L r.The closed-loop state equation with the PI-controller becomes

x(k + 1) = FCLPIx(k ) + GCLPI id∗ (k ) iq

∗ (k ) ed (k ) eq (k )[ ]T(12)

where the state vector x(k ) has 12 states.

IV. SYSTEM LINEARIZATION AND INFLUENCEOF NON-LINEARITIES

Since most systems are non-linear, a small-signal analysis isgenerally performed by analysing the linearized system aroundan operating point denoted by the subscript 0. Perturbationsaround the operating point are denoted as ∆ . In this paper, alinear model of the system is used. However, due to non-

linearities in the PWM and non-linear grid-filter parameters, theresult is assumed to be valid only at the operating point.

A. Parameters of the Grid FilterThe grid filter consists of an inductor in each phase. The 50 Hzvalues of the series resistance and inductance of the inductorare used in the current controller. To display the influence ofthe frequency dependent losses in the grid filter on the dynamicperformance, two different inductors are used. The iron core ofinductor 1, is made of 0.5 mm non-oriented laminations, andthe iron core of inductor 2 is made of 0.35 mm orientedlaminations. Both cores have air-gaps. When the frequencyincreases, the I2R losses increase due to the skin effect, butthe iron losses will dominate due to increased eddy-current andhysteresis losses. If the inductors are modelled by equivalentseries resistance and inductance in the frequency domain, theequivalent series parameters vary, according to measurements,as shown in Fig. 2. Both inductors have similar total losses.By using thinner and oriented iron in the core, the hysteresislosses and eddy current losses in the iron are reduced and amore constant series inductance is obtained. The I2R lossesare similar with both filters.

0.8

0.85

0.9

0.95

1

0

0.2

0.4

0.6

0.8

0 200 400 600 800 1000

Nor

mal

ized

ser

ies

indu

ctan

ce

Seri

es r

esis

tanc

e [Ω

]

Frequency [Hz]

inductor 2: Ls

inductor 1: Ls

inductor 1: Rs

inductor 2: Rs

Fig. 2. The measured grid filter inductance and resistance as a function ofthe frequency for two different inductors.

B. Non-linear Valves and Valve CompensationFunctionThe non-linearities of the valves and PWM occur due to twodifferent phenomena: blanking time and on-state voltage drop.To avoid short-circuiting the dc-link, each commutation isdivided into two steps. First, the conducting valve is turnedoff. After the blanking time, the other valve in the phase leg isturned on. If the transistor is conducting in the valve that isturned off, the diode in the valve that is to be turned on starts toconduct as soon as the transistor is turned off. Thus, theswitching time instant is not influenced by the blanking time. Ifthe sign of the current is not changed, the next commutation isdelayed by the blanking time since a commutation from a diodeto a transistor occurs. Consequently, the average phase voltagedeviates from that requested by the PWM modulator.

The influence of the blanking time and the on-state voltagedrop can be reduced by adding compensation voltages ∆u∗ tothe reference phase voltages u∗, i.e., by introducing a valvecompensation function. The equations for the compensation ofphase 1 are

∆u1,Tb∗ (k ) = sign i1(k )[ ]Tb / TS (13)

u1,comp∗ (k ) = ∆u1,Tb

∗ (k ) + D1uonT1 + 1 – D1( )uonD2 + u1∗(k ) (14)

u1,comp∗ (k ) = ∆u1,Tb

∗ (k ) – 1 – D1( )uonT 2 – D1uonD1 + u1∗(k ) (15)

Equation (13) compensates only for the blanking time.Equations (14) and (15) also compensate for the voltage dropacross the valves in the on-state for positive and negativecurrent, respectively. The blanking time and the sampling timeare denoted as Tb and TS , respectively. The phase 1 duty ratiois D1 = 2u1

∗ / udc . The transistor and diode voltage drops aredenoted as uonT and uonD . These voltage drops have beenmeasured and are displayed in Table 2. The voltage dropsconsist of a constant term u0 and a current-dependent termR oni . As discussed above, the compensation is only validwhen the current is positive or negative during a whole sampleperiod. When the amplitude of the phase current sample issmaller than the amplitude of the current ripple, the valvecompensation is not included.

TABLE IIMEASURED PARAMETERS FOR THE VALVE COMPENSATION FUNCTION.

u0T =1.08 V u0D =1.05 V TS =166.7 µs

R onT =12 mΩ R onD =10 mΩ Tb=1.4 µs

V. ANALYSIS OF FREQUENCY RESPONSES

To examine the influence of the non-linearities on the dynamicperformance of the system, small-signal frequency responseshave been measured at different operating points. The influenceof the non-linear valves is described by displaying responseswith and without the valve compensation function. Insection A, the influence of the losses in the grid-filterinductors is described by comparing measured responses withdifferent inductors at a selected operating point. The frequencyresponses in section A were both measured and calculated bymeans of analytical models. The objective in section B is toverify that the linear model of the system is valid both in therectifier operation and the inverter operation, as well as staticVAr compensation. In section B, responses measured at fourdifferent operating points are presented.

Four different controllers are used. Measured responses aredenoted as:• M(p) P-controller• M(pi) PI-controller• M(p,c) P-controller with valve compensation• M(pi,c) PI-controller with valve compensation

The corresponding responses from the analytical models, allwithout compensation functions, are denoted as:• A(p) P-controller with constant parameters• A(pi) PI-controller with constant parameters• B(p) P-controller with frequency-dependent grid filter

parameters• B(pi) PI-controller with frequency-dependent grid filter

parameters

In the measurements, a sinusoidal small-signal current ∆i∗ of0.1 pu was added to one of the reference currents at theoperating point. Measurements were performed at 27 differentfrequencies between 10 Hz and 1200 Hz. The referencecurrents as well as the d- and q-currents were sampled at 10kHz. Anti-alias filters with a crossover frequency of 3 kHzwere used. In the analysis of the measured results, the FFT-algorithm was used to evaluate the amplitude gain and phaseshift between the sinusoidal reference current ∆id

∗ or ∆iq∗ and

the measured currents ∆id and ∆iq .

A. Responses with Different Grid Filter InductorsIn this section, the influence of the losses in the grid-filterinductors is described. The measured transfer functions areobtained at the operating point id0

∗ =0.22 pu and iq0∗ =

–0.67 pu. Responses for inductors with high iron-losses aredenoted as M1(·) and responses for inductors with low ironlosses are denoted as M2(·). The Bode-diagrams below arescaled in dB. The gains 1 dB and 2 dB correspond to 12 %and 26 %, respectively.

Direct Coupling in q-directionThe gain characteristics of the direct coupling in the q-directionare very similar for all controller types. As shown in Fig. 3,the gain of the response M1(·) is close to 0 dB at lowfrequencies. However, when the frequency exceeds about400 Hz, the gain is increased for all controllers M1(·).However, at 1 kHz the gain is 2 dB, which corresponds to anerror of 26 %. Thus, the gain deviates from the gain predictedby the linear analytical models A(·). As displayed, the increasein the gain can be predicted by using models B(·). In theresponses M2(·), the gain is much closer to the ideal responseat all frequencies. The analytical models A(·) give a slightlylarger phase lag than the measured one, but the analyticalmodels B(·) track the measured phase correctly.

Cross Coupling from q-direction to d-directionIn Fig. 4, the gains of the cross coupling from the q-directionto the d-direction are shown. The gains for the responsesobtained from measurements with compensation for the non-linear valves M1(p,c) and M1(pi,c) are lower than themeasured gains without compensation, M1(p) and M1(pi).Consequently, the valve compensation decreases the cross-coupling. The theoretical gains with models B(·) are slightlyhigher than those of models A(·) but lower than the measuredgains. The cross-coupling is significantly affected by the lossesin the inductor only at high frequencies, where lower couplingis displayed in the response M2(pi,c). There is also a deviationat low frequencies but very low coupling is obtained with bothinductors.

Direct Coupling in d-directionDirect coupling in the d-direction is displayed in Fig. 5. At lowfrequencies, the responses with the compensation functionfollow the analytical responses most accurately. At highfrequencies, the measured responses do not follow theresponses from any of the analytical models. This is due to anon-modelled error in the converter system at high frequencies.In the responses M1(·), the frequency dependent losses in theinductor cause an error that partly cancels the error caused bythe converter system. In the measured response M2(pi,c), thegain is reduced at high frequencies. The phase-shifts of themeasured frequency responses follow the theoretical phase-shifts B(p), B(pi), which have a smaller phase lag than themodels A(p) and A(pi).

Cross Coupling from d-direction to q-directionFigure 6 shows that cross coupling from the d-direction to theq-direction is reduced by using the valve compensationfunction in responses M1(·). However, at low frequencies,low coupling is obtained for both inductors. The coupling athigh frequencies is lower in M2(pi,c) in comparison withM1(·).

-1

-0.5

0

0.5

1

1.5

2

101

102

103

Gai

n [d

B]

Frequency [Hz]

M1(p)

M1(p,c)

M1(pi,c) M1(pi)

A(p)A(pi)

B(p)B(pi)

M2(pi,c)

-180

-120

-60

0

101

102

103

Phas

e [d

eg]

Frequency [Hz]

M1(p) M1(p,c)M1(pi,c)M1(pi)

A(p)A(pi)

B(p) B(pi)M2(pi,c)

Fig. 3. The Bode diagram from the reference q-current ∆iq∗ to the q-current

∆iq .

-60

-40

-20

101

102

103

Gai

n [d

B]

Frequency [Hz]

M1(p)

M1(p,c)M1(pi,c)

M1(pi)

A(p)A(pi)

B(p)

B(pi)

M2(pi,c)

A(p)A(pi)

-240

-120

0

120

240

101

102

103

Phas

e [d

eg]

Frequency [Hz]

M1(p)

M1(p,c)M1(pi,c)

M1(pi)

A(p) A(pi)

B(p)B(pi)

M2(pi,c)

Fig. 4. The Bode diagram from the reference q-current ∆iq∗ to the d-current

∆id .

-2.5

-1.25

0

1.25

2.5

101

102

103

Gai

n [d

B]

Frequency [Hz]

M1(p)

M1(p,c)

M1(pi,c)M1(pi)

A(p)A(pi)B(p)

B(pi)

M2(pi,c)

-180

-120

-60

0

101

102

103

Phas

e [d

eg]

Frequency [Hz]

M1(p) M1(p,c)M1(pi,c)M1(pi)

A(p)A(pi)

B(p)B(pi)

M2(pi,c)

Fig. 5. The Bode diagram from the reference d-current ∆id∗ to the d-current

∆id .

-50

-45

-40

-35

-30

-25

-20

-15

-10

101

102

103

Gai

n [d

B]

Frequency [Hz]

M1(p)

M1(p,c)

M1(pi,c)

M1(pi)

A(p)

A(pi)

B(p)B(pi)

M2(pi,c)

-360

-240

-120

0

101

102

103

Phas

e [d

eg]

Frequency [Hz]

M1(p)M1(p,c)M1(pi,c)

M1(pi)

A(p)A(pi)

B(p)B(pi)

M2(pi,c)

Fig. 6. The Bode diagram from the reference d-current ∆id∗ to the q-current

∆iq .

B. Responses at Different Operating PointsThe objective in this section is to examine the influence of theoperating point on the dynamic performance. The inductor withlow iron-losses is used. To illustrate the influence of the non-linearities caused by the valves, measurements with andwithout the valve compensation function are displayed. Asshown in the responses in the previous section, the phase isusually predicted well by a linear analytical model. The phase-shifts are, thus, not displayed. At each operating point,

responses to sinusoidal perturbations in the d- and q-directionsare presented. Responses for rectifier and inverter operationsare displayed in Figs. 7 to 10. Moreover, responses forreactive power compensation are shown in Figs. 11 to 14. Inall cases, the Bode-diagrams are scaled in dB. The gains of1 dB and 2 dB correspond to 12 % and 26 %, respectively.

Responses in Rectifier and Inverter OperationsThe operating points corresponding to rectifier and inverteroperations are: id0

∗ =0.22 pu, iq0∗ = –0.67 pu, denoted as –P;

and id0∗ = –0.22 pu, iq0

∗ = 0.67 pu, denoted as P.

In Fig. 7, responses from the q- to q-direction are displayed.The best responses are obtained in the rectifier operation. Atlow frequencies, similar performances are obtained. At highfrequencies, the best results are obtained without thecompensation function.

According to Fig. 8, similar couplings from the q- to d-direction are obtained at both operating points. At lowfrequencies, without compensation, the highest couplingoccurs in the inverter operation. At high frequencies, the samecoupling is obtained at both operating points. The valvecompensation function reduces the coupling at frequenciesabove 100 Hz.

As illustrated in Fig. 9, almost exactly the same responsesfrom the d- to d-direction are obtained for both operatingpoints. The valve compensation function improves theperformance at low frequencies. The best responses areobtained without compensation at frequencies from 600 Hz to900 Hz.

The coupling from the d-to q-direction is displayed in Fig. 10.In the inverter operation, a significant difference betweenresponses with and without the valve compensation function isobtained at low frequencies. However, a very low coupling isobtained in the inverter operation. At 1 kHz, the lowestcoupling is obtained without the valve compensation functionat both operating points.

Responses in Reactive Power Compensation ModesReactive power compensation corresponds to the points: id0

∗ =–0.67 pu and iq0

∗ = 0.22 pu, denoted as –Q; id0∗ = 0.67 pu, iq0

∗ =–0.22 pu denoted as Q.

As displayed in Fig. 11, at low frequencies, the response fromthe q- to q-direction is significantly improved by using valvecompensation. The best responses are obtained without thecompensation function in the frequency range from 600 Hz to1100 Hz.

In Fig. 12, the coupling from the q- to d-direction is displayed.The influence of the compensation function varies with thefrequency. At low frequencies and with a positive reactivecurrent, the lowest coupling occurs without the valvecompensation function. At the other operating point, the mosteffective compensation is obtained at low frequencies. Theopposite effect is displayed at frequencies from 600 Hz to1 kHz.

In the responses plotted in Fig. 13, approximately the sameresponses from the d- to d-direction are obtained in bothoperating points. The valve compensation function improvesthe performance below 600 Hz.

As illustrated in Fig. 14, the lowest coupling from the d-direction to the q-direction occurs with the negative reactivecurrent. At this operating point, the compensation functionyields very low coupling at low frequencies and improves

performance below 600 Hz. At the operating point with thepositive reactive current, the coupling is reduced below300 Hz. At higher frequencies, a similar coupling is obtainedwith and without compensation.

-1.5

-1

-0.5

0

0.5

1

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi) (-P)M2(pi,c)(-P)

M2(pi)(P)

A(pi)

M2(pi,c)(P)

Fig. 7. The frequency response gain from the reference q-current ∆iq∗ to the

q-current ∆iq .

-50

-40

-30

-20

-10

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi)(-P)

M2(pi,c)(-P) M2(pi)(P)

A(pi)

M2(pi,c)(P)

Fig. 8. The frequency response gain from the reference q-current ∆iq∗ to the

d-current ∆id .

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi) (-P)

M2(pi,c)(-P)

M2(pi)(P)

A(pi)M2(pi,c)(P)

Fig. 9. The frequency response gain from the reference d-current ∆id∗ to the

d-current ∆id .

-50

-40

-30

-20

-10

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi)(-P)M2(pi,c)(-P)

M2(pi)(P)

A(pi)

M2(pi,c)(P)

Fig. 10. The frequency response gain from the reference d-current ∆id∗ to

the q-current ∆iq .

-2.5

-2

-1.5

-1

-0.5

0

0.5

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi) (-Q)

M2(pi,c)(-Q)

M2(pi)(Q)

A(pi)

M2(pi,c)(Q)

Fig. 11. The frequency response gain from the reference q-current ∆iq∗ to

the q-current ∆iq .

-50

-40

-30

-20

-10

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi)(-Q)

M2(pi,c)(-Q)

M2(pi)(Q)

A(pi)

M2(pi,c)(Q)

Fig. 12. The frequency response gain from the reference q-current ∆iq∗ to

the d-current ∆id .

-2.5

-2

-1.5

-1

-0.5

0

0.5

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi) (-Q) M2(pi,c)(-Q)

M2(pi)(Q)

A(pi)

M2(pi,c)(Q)

Fig. 13. The frequency response gain from the reference d-current ∆id∗ to

the d-current ∆id .

-50

-40

-30

-20

-10

101

102

103

Gai

n [d

B]

Frequency [Hz]

M2(pi)(-Q)

M2(pi,c)(-Q)

M2(pi)(Q)

A(pi)

M2(pi,c)(Q)

Fig. 14. The frequency response gain from the reference d-current ∆id∗ to

the q-current ∆iq .

C. Summary of the Frequency Response AnalysisTo determine the influence of non-linearities on dynamicperformance, small signal frequency responses are determined.In section A, the influence of the losses in the inductors are infocus. As shown, the measured responses in the q-directionfor inductors with high losses deviate from the responsesobtained from the linear analytical model at high frequencies.This deviation can be predicted by including the frequencydependent losses of the grid filter in the analytical model. In thed-direction, the deviation does not appear. According to theanalytical model with the losses included, a deviation shouldalso occur in the d-direction. This indicates that the convertersystem introduces an error in the d-direction. The responsesobtained with inductors with low losses show that the responsein the q-direction is considerably improved. However, thedeviation is increased in the d-direction. Thus, an error isdisplayed in the d-direction. The conclusion is that the dynamicperformance is improved by using inductors with low losses.This is predicted by analytical models and also verified in theq-direction. In the d-direction, the responses are distorted dueto a non-modelled error due to the time delay between thePWM circuit and the switching of the valves in the convertersystem.

In section B, the influences of the compensation for the non-linearities are displayed at different operating points. Theobjective is to show that the performance is improved by usingthe compensation method proposed. The responses areimproved at all operating points at low frequencies. At highfrequencies, the best results are obtained however, without thecompensation function. This should be the result of the non-modelled error mentioned above. The distortion could also bethe result of non-ideal sampling. However, the errors occur inthe d-direction. In the d-direction, the sampling can actually beshifted several µs since the current is not changed in the d-direction when the voltage vectors 000 and 111 are active.Thus, the error should be due to the time delay between thePWM circuit and the switching of the valves.

VI. CONCLUSION

The obtained frequency responses correspond well to theanalytical model. However, the focus has been on the smalldeviations in the gains and in the phases from their presumedvalues. The influence of non-linearities caused by non-idealvalves and grid filter inductors on the frequency response of avector controlled grid connected VSC are investigated. Non-linearities caused by losses in the grid filter, by blanking timeand by non-ideal IGBT-valves are described and their influenceon the frequency responses are verified by measurements. Byusing a compensation function for non-linearities caused by theblanking time and the valves, the frequency responses areimproved at frequencies up to about 600 Hz. At higherfrequencies, the best results are often obtained without thecompensation function. This may be a result of the time delaybetween the PWM-circuit and the switching of the valves. Thenon-linearities result in different responses at differentoperating points.

VII. REFERENCES

[1] J. Holtz, "Pulsewidth Modulation for Electronic PowerConversion," Proc. of IEEE, Vol. 82, No. 8, August1994, pp. 1194-1214.

[2] J. K. Pedersen, F. Blaabjerg, J. W. Jensen, P.Thogersen, "An Ideal PWM-VSI Inverter withFeedforward and Feedback Compensation," 5thEuropean Conference on Power Electronics andApplications (EPE'93), Brighton, England, 13-16September 1993, pp. 501-507.

[3] J. Ollila, "Analysis of PWM-converters Using SpaceVector Theory: Application to a Voltage SourceRectifier," Doctoral Thesis, Tampere University ofTechnology, Tampere, Finland, 1993.

[4] S. R. Bowes, "New Sinusoidal Pulsewidth-modulatorInverter," Proc. IEE, Vol. 122, No. 11, November1975.

[5] S. Hiti, D. Boroyevich, "Small-Signal Modeling ofThree-Phase PWM Modulators," Power ElectronicsSpecialists Conference (PESC'96), Baveno Italy, 23-27June 1996, pp. 550-555.

[6] R. Kagalwala, S. S. Venkata, P. O. Lauritzen, "ATransient Behavioral Model (TBM) for PowerConverters," 5th Workshop on Computers in PowerElectronics, IEEE Power Electronics Society, PortlandUSA, 11-14 August 1996, pp. 18-24.

[7] J. Svensson, "Inclusion of Dead-Time and ParameterVariations in VSC Modelling for Predicting Responsesof Grid Voltage Harmonics." 7th European Conferenceon Power Electronics and Applications (EPE'97),Trondheim, Norway, 8-10 September 1997,Proceedings, Vol. 3, pp. 216-221.

[8] M. Lindgren, "Feed forward – Time Efficient Control ofa Voltage Source Converter Connected to the Grid byLowpass Filters," Power Electronics SpecialistsConference (PESC'95), Atlanta, 18-22 June 1995, Vol.2, pp. 1028-1032.