Direct Power Control for Three-Phase PowerConverters under Distorted Input Voltages

S. Vazquez, Student Member, IEEE, J.A. Sanchez, J.M. Carrasco, Member, IEEE, E. Galvan, Member, IEEE,J.I. Leon, Member, IEEE, E. Dominguez, Student Member, IEEE, and O.Martinez

Escuela Superior de Ingenieros. Dpto. de Ingenieria ElectronicaUniversity of Seville

Avd. Camino de los descubrimientos s/n CP: 41092Seville, Spain

Email: svazquez@gte.esi.us.es

Abstract—Conventional Direct Power Control (DPC) techniqueis a simple and efficient control strategy for three-phase powersynchronous rectifiers. However its performance is deterioratedwhen the power rectifier is working under distorted or unbal-anced input voltages. In this paper the distorted input voltagesoperation of the system is under concern, and a model-basedDPC for three-phase power converters is designed. The obtainedexpressions for the input control signal permit to design anadaptive control law which ensures the power reference trackingand the high performance operation of the power system. Besides,this adaptive control law minimizes the errors introduced bythe parameters uncertainties as the smoothing inductor value,the grid frequency and the input voltage harmonic content.Controller design process and stability of the system are shown.Besides, the instantaneous power reference definition to achievesinusoidal input currents are presented. Simulation results for asynchronous three-phase power rectifier prototype are shown tovalidate the proposed controller.

I. INTRODUCTION

P OWER synchronous rectifiers are well-known powersystems for industrial applications as DC-bus supply to

three-phase vector-controlled PWM inverters and DC-Loadsand for the integration of renewable energy applications. Thesynchronous converters can provide sinusoidal input currentsthat fulfil the new standards for the electric grid, IEEE-519for USA and IEC 61000-3-2 and 61000-3-4 for Europe [1]-[4]. To achieve this objective several control strategies havebeen developed [5]-[10], these control techniques deal theproblem of balanced input voltages with only fundamentalharmonic content. When these strategies are applied underdistorted input voltages, the performance of the system ishighly deteriorated, low order harmonic contents appear in theinput currents and DC-Link voltage ripple at twice of the linefrequency occurs. To achieve high performance under distortedinput voltages new control strategies have been developed.Different approaches has been used to solve the problem, in[11] four current control strategies are analyzed and compared,including controllers in the stationary and the synchronousreference frames. In [12] the behavior of the Direct PowerControl (DPC) strategy based on the Virtual Flux (VF) isstudied under distorted input voltages. In [13] it is pointed outthat not only the control strategy has influence on the converterperformance, but the currents reference has to be defined

Load

a

b

c

L

L

L

r

s

t

2C2

dcVI1

I2

I3

I4

I5

I6

ai

ci

bi

anv

cnv

convi i

capi

2C

o

2

dcVn

bnv

Load

Fig. 1. Three-phase two level power converter.

in order to obtain different control objectives as sinusoidalcurrents and constant DC voltage value simultaneously. In [14]is presented an analytical study of the behavior of the DC-Linkvoltage and input current under the unbalance and distortedinput voltages and in [15] a repetitive controller to avoid thelow order harmonic content in the input currents is proposed.In [16] a complete review of different current references toobtain full control of the active and reactive power injectedinto the grid under unbalance and distorted input voltages arepresented. To obtain this current references the use of a PLLis needed for the extraction of the positive a negative voltagesequences [17].

In this paper a SVM-DPC for a synchronous rectifier underdistorted input voltages is proposed. To achieve the controlobjectives the harmonic content of the input voltages is takeninto account in the controller design. The proposed DPCstrategy is based on [18], adaptive techniques are introducedto avoid system parameters uncertainties as the grid frequency,the smooth inductor values and the harmonic content of theinput voltages. Stability analysis of the proposed controlleris also shown. In addition to the proposed controller, ainstantaneous power references definition to achieve balancedsinusoidal input currents are proposed and an analysis of thepower converter performance is included. Finally simulationresults are shown to demonstrate the good behavior of theproposed DPC controller.

II. MODEL OF THE SYSTEM

A three-phase two level power converter is depicted in Fig.1, where the neutral point is denoted by n. The system isconnected to the grid through smoothing inductors L, and it

TABLE ISYSTEM VARIABLES AND PARAMETERS

Variable Description

v = vavbvcT Phase to neutral input voltage vector

i = iaibicT Phase input current vector

δ = δaδbδcT Control input vector

ω Grid frequency

L Smoothing inductor

C Output capacitor

RL Load resistance

Vdc Output capacitor voltage

is assumed that a pure resistive load RL is connected at DC-Link capacitor C. The system parameters and variables aredescribed in Table I.

vαβ = Ldiαβ

dt+

Vdc

2δαβ (1)

Cd

dt(V 2

dc

2) =

Vdc

2δTαβiαβ − V 2

dc

RL(2)

A =√

23

[1 − 1

2 − 12

0√

32 −

√3

2

](3)

The equations that describe the input currents dynamicsand the output DC voltage dynamic can be derived from thesystem model. Following the technique presented in [19], thesystem model can be obtained in the stationary αβ frame[20]. Equations (1) and (2) describe three-phase power rectifierdynamics, the variables can be calculated as ·T

αβ = A·Tabc

where the matrix A is defined by (3)The control objectives for the three-phase power converter

are the following:(i) The instantaneous active power p and the instantaneousreactive power q, [21], should track their reference, p∗ and q∗

respectively. These references are calculated in such way thatthe capacitor output voltage is regulated towards its referenceand unity power factor is achieved.

p → p∗ (4)q → q∗ (5)

Thus, DPC controls indirectly the currents provided by thesource through the values of the instantaneous active andreactive power.(ii) The capacitor output voltage should be regulated towardsits reference.

Vdc → V ∗dc (6)

III. CONTROL DESIGN

The proposed controller for the system (1)-(2) is composedof an instantaneous power tracking (inner) loop and a voltageregulation (outer) loop. In what follows is described the designof the two control stages.

A. Instantaneous power tracking loop

The objective of the inner control loop is to guaranteetracking of p and q towards their references. The instantaneous

active power reference, p∗, and the instantaneous reactivepower reference, q∗, are the output of the outer control loop,and are calculated in order to achieve DC-Link voltage regula-tion, and unity power factor. The expressions of instantaneousactive and reactive power are

p = iTαβvαβ (7)

q = iTαβJvαβ (8)

J =[

0 −11 0

](9)

In what follows the controller design is described in detail.

Lp = |vαβ |2 + LiTαβ vαβ − Vdc

2vT

αβδαβ (10)

Lq = LiTαβJvαβ − Vdc

2vT

αβJδαβ (11)

The dynamics of the instantaneous active and reactive powercan be written as (10) and (11). These equations includea term with the first time derivate of vαβ over the time,when input voltages are balanced sinusoidal waveforms thisvalue can be calculated as Jwvαβ , however in presence oflow order harmonic this is no longer valid, leading to ainstantaneous power dynamics different from those shown in[18]. As a consequence if power dynamics from [18] are usedto derive the control signal the power tracking performance isdeteriorated.

δp→p∗

αβ =2

Vdc(|vαβ |2 + φ + kpp)

vαβ

|vαβ |2 + CpJvαβ (12)

δq→q∗

αβ =2

Vdc(ϕ + kq q)

Jvαβ

|vαβ |2 + Cqvαβ (13)

φ , LiTαβ vαβ − Lp∗ =∞∑

kεH

ρTk φk, H = 0, 2, 4, ..,∞ (14)

ϕ , LiTαβJvαβ − Lq∗ =∞∑

kεH

ρTk ϕk, H = 0, 2, 4, ..,∞ (15)

ρTk = cos (kωt) sin (kωt) (16)

The set of points in the stationary frame that makes p → p∗

and q → q∗ are (12) and (13) respectively, where the functionsφ and ϕ are defined in (14)-(16).

δαβ =2(|vαβ |2 + φ + kpp)

Vdc

vαβ

|vαβ |2 +2(ϕ + kq q)

Vdc

Jvαβ

|vαβ |2 (17)

L ˙p + kpp = 0 (18)L ˙q + kq q = 0 (19)

Thus if the control signal is defined as the intersection pointof these lines, (17), the dynamic of the instantaneous activeand reactive power errors are (18)-(19), so these errors tendtoward to zero exponentially.

δαβ , 2(|vαβ |2 + φ + kpp)Vdc

vαβ

|vαβ |2 +2(ϕ + kq q)

Vdc

Jvαβ

|vαβ |2 (20)

φ ,∞∑

kεH

ρTk φk, H = 0, 2, 4, ..,∞ (21)

ϕ ,∞∑

kεH

ρTk ϕk, H = 0, 2, 4, ..,∞ (22)

The problem with this control law is that it depends onthe system parameter, L, and on the first time derivate ofthe input voltage, the first time derivate of the instantaneousactive power reference and the first time derivate of the instan-taneous reactive power reference. These values are unknownor difficult to calculate, however it can be assumed that thefrequency components of φ and ϕ are constants or slowlyvariants. For this reason is proposed the control law (20),where two estimator functions (21)-(22) have been includedcontaining all the unknown values.

L ˙p = −kpp−∞∑

kεH

ρTk φk (23)

L ˙q = −kq q −∞∑

kεH

ρTk ϕk (24)

φk , φk − φk (25)ϕk , ϕk − ϕk (26)

When this control law is used the dynamics of the powererrors are (23) and (24). Where the kth value of the estimatorerrors φk and ϕk are defined in (25) and (26) respectively

H =Lp2

2+

Lq2

2+

∞∑

kεH

(φT

k φk

)

2γk+

∞∑

kεH

(ϕT

k ϕk

)

2ξk(27)

To derive an adaptive law to reconstruct the values φk andϕk a Lyapunov approach is followed. For this purpose thepositive-definite function (27) is proposed, in this equation theparameters γk and ξk are positive constants that represents theadaptation gains for each harmonic.

H =−kpp2−kq q

2+∞∑

kεH

(( ˙φT

k

γk−pρT

k

)φk +

(˙ϕTk

ξk−qρT

k

)ϕk

)(28)

˙φk = γk p ρT

k (29)˙ϕk = ξk q ρT

k (30)˙φk , γk p ρT

k (31)˙ϕk , ξk q ρT

k (32)

H = −kpp2−kq q

2 (33)

The time derivate of (27) along the trajectories of (23) and(24) is equation (28) which is made negative semidefinite byproposing expressions (29) and (30) to ˙

φk and ˙ϕk respectively.Now taken into account that φk and ϕk are constants or slowlyvariant values, φk and ϕk can be reconstructed using (31) and(32). Finally the time derivate of (27) is given by (33).

Following Lasalle´s theorem arguments it can be stated thatp → 0 and q → 0 as t → ∞ asymptotically. Moreover, from(29) and (30) p ≡ 0 and q ≡ 0 imply that φk and ϕk areconstants. According to (23) and (24) these constants shouldbe zero. This guarantees convergence of the estimated valuestowards its actual value.

ψrk = ρT

k φk (34)ψi

k = ρTk Jφk (35)

φ =∞∑

kεH

γks

s2 + k2ω2p (36)

ϕ =∞∑

kεH

ξks

s2 + k2ω2q (37)

The estimator functions can be implemented more easily inthe Laplace domain. Using the transformations (34) and (35)for φk conduct to (36). The same manipulation can be donewith ϕk and finally (37) is obtained.

γk , 4 ωkrp

π, ∀kεH (38)

ξk , 4 ωkrp

π, ∀kεH (39)

φ =2 ωkrp

π

1sp + krp coth

( πs

2 ω

)p (40)

ϕ =2 ωkrp

π

1sq + krp coth

( πs

2 ω

)q (41)

These expressions include a bank of resonant filter for theeven harmonics. If adaptation gains γk and ξk are defined as(38) and (39), then the bank of resonant filter can be calculatedwith a repetitive scheme [22]. The final expressions for theestimator functions are (40) and (41)

B. Voltage regulation loop

The outer control loop is designed to regulate the outputcapacitor voltage towards its reference value . This problemcan be solved using a PI controller as in [20].

pav =aKp

s + bz +

Ki

sz (42)

z , z∗ − z (43)

z , V 2dc

2(44)

z∗ =(V ∗

dc)2

2(45)

The proposed controller for the outer control loop is (42),where Kp, Ki, a and b are design parameters which arepositive constants. This controller includes a low pass filterin the proportional term to reduce the high frequency noise.The output of this controller is the average active power, pav ,which is needed to regulate the DC-Link voltage.

p∗

q∗

,

pav

qav =0

(46)

The instantaneous active and reactive power references toachieve DC-Link voltage regulation with unity power factorin the input currents, can be established as (46).

IV. POWER TRACKING PERFORMANCE

To evaluate the power tracking performance of the proposedDPC controller under distorted input voltages, it is made acomparison between the results obtained using the controllerpresented in [18], designed for balanced input voltage situa-tion, and the results achieved with the proposed DPC strategy.Both controllers are simulated under input voltage with 10%

TABLE IIELECTRICAL AND CONTROL PARAMETERS FOR THE SYSTEM

Parameter Value

Phase to neutral Voltage 230 V

Smoothing Inductance 1 mH

DC-Link Capacitor 2200 µF

Resistive Load 56.25 Ω

DC-Link Reference Voltage 750 V

Switching Frequency 10 kHz

Sampling Frequency 10 kHz

of 5th harmonic, and using (46) to define the instantaneouspower references. The simulation results have been carriedout using the electrical parameters shown in Table II.

Fig. 2 and 3 show respectively, the instantaneous activepower reference with the actual value drawn by the the powerconverter and the instantaneous reactive power reference withthe actual value flowing between the power converter and thegrid, when the input voltages have 10% of 5th order harmonic.

Fig. 2a and 3a show the performance achieved in theinstantaneous power tracking with the controller designedassuming balanced sinusoidal input voltages and Fig. 2b and3b the performance obtained with the proposed DPC strategy.It can be noticed that DPC strategy [18] presents an errorwhen it tries to track the instantaneous power references, incomparison the proposed DPC controller achieve the powertracking although the input voltages are distorted.

V. INPUT CURRENTS PERFORMANCE

The converter input currents waveforms depend on boththe power tracking controller performance as well as theinstantaneous power reference definition.

A. Conventional power references

The conventional instantaneous active and reactive powerreferences used to achieve DC-Link voltage regulation withunity power factor in the input currents are (46). In thatcase, the input voltages harmonic content has influence in theinput currents harmonic content as shown in the simulationspresented in Fig. 4 and Fig. 5.

Fig. 4 shows the currents waveforms achieved with theproposed controller when the input voltages are balanced andsinusoidal, it can be observed that input currents show asinusoidal shape with a Total harmonic distortion (THD) of1.0%. The simulations results presented in Fig. 5 have beencarried out when input voltages have a 10% of the 5th orderharmonic,under this conditions the input currents lose qualityand present a THD of 8.7%, with a of 9.31% of 7th harmonic.

B. Power references with sinusoidal input currents

To avoid the effects of the distorted input voltages overthe input currents, the instantaneous active and reactive powerreferences should be calculated taking into account that activeand reactive power should be provided only by the fundamen-tal harmonic of the input voltages and currents.

0.5 0.52 0.54 0.56 0.58 0.65

10

15

0.5 0.52 0.54 0.56 0.58 0.65

10

15

Time (s)

Time (s)

p, p (kW)

p, p (kW)

*

*

(a)

(b)

Fig. 2. Instantaneous active power and instantaneous active power reference:a) DPC controller designed for balanced input voltages. b) Proposed DPCcontroller.

0.5 0.52 0.54 0.56 0.58 0.6−4

−2

0

2

4

0.5 0.52 0.54 0.56 0.58 0.6−4

−2

0

2

4

Time (s)

Time (s)

q, q (kVAr)

q, q (kVAr)

*

*

(a)

(b)

Fig. 3. Instantaneous reactive power and instantaneous reactive powerreference: a) DPC controller designed for balanced input voltages. b) ProposedDPC controller.

0.55 0.56 0.57 0.58 0.59 0.6−40

−20

0

20

40

0 5000 10000 150000

2

4

6

8

10

Frequency (Hz)

Time (s)

ia (A)

ia harmonic spectrum (%)

Fig. 4. Input currents of the power converter using the proposed DPC controland the coventional instantaneous active and reactive power references wheninput voltages are ideal.

0.55 0.56 0.57 0.58 0.59 0.6−40

−20

0

20

40

0 5000 10000 150000

2

4

6

8

10

Frequency (Hz)

Time (s)

ia (A)

ia harmonic spectrum (%)

7 harmonicth

Fig. 5. Input currents of the power converter using the proposed DPC controland the coventional instantaneous active and reactive power references wheninput voltages have 10% of 5th order harmonic.

i∗αβ , 1|vαβ,1|2

vα,1 −vβ,1

vβ,1 vα,1

pav

qav

(47)

In that case, the input currents to be provided by the grid canbe calculated as (47), where vα,1 and vβ,1 are the fundamental

+

+

αβδ+ +

+

+

+ +

2 rpk

s

ω

π

+

+

+

2

vαβ

-+

*p

p

s

eπω

−

pk

rpk

2

v

v

αβ

αβ

+ +

+

+

+ +

+

+

-

+

*q

q

s

eπω

−

qk

2 rqk

s

ω

π

rqk

2

Jv

v

αβ

αβ

2

dcV

*,1 ,1

2*,1 ,1

,1

1 av

av

v v v v pp

v v v v qq v

∆α β α β

β α β ααβ

− = −

+

-

( )2

2*

dcV

2

2

dcV avp

s

K

bs

aKip +

+

Resonant

filtervαβ

,1vαβ

avq

Fig. 6. Diagram block of the proposed adaptive DPC controller for the overall system.

components of the input voltages in the stationary frame,which can be evaluated using a resonant filter tuned at thegrid frequency.

p∗

q∗

, 1|vαβ,1|2

vα vβ

−vβ vα

vα,1 −vβ,1

vβ,1 vα,1

pav

qav =0

(48)

Now, following the instantaneous active and reactive powerdefinitions (7) and (8), the references p∗ and q∗ can becalculated as (48)

Fig. 6 shows the block diagram of the proposed DPCcontroller including the adaptive law implemented throughthe repetitive scheme. In this diagram, the power referencesare calculated through the proposed expression (48), using aresonant filter tuned at the grid frequency.

Fig. 7 shows the input currents of the power converter whenthe input voltages have 10% of 5th order harmonic and theproposed instantaneous active and reactive power referencesare used. It can be noticed that the 7th harmonic almostdisappear (0.67%) and the quality in the input currents hasbeen restored presenting a THD of 1.7%.

VI. DC-LINK VOLTAGE PERFORMANCE

The DC-Link voltage performance can be evaluated analyz-ing the influence of the input voltages harmonic content in theoutput DC voltage. For this purpose the DC-Link voltage is

0.55 0.56 0.57 0.58 0.59 0.6−40

−20

0

20

40

0 5000 10000 150000

2

4

6

8

10

Frequency (Hz)

Time (s)

ia (A)

ia harmonic spectrum (%)

Fig. 7. Input currents of the power converter using the proposed DPC controland the proposed instantaneous active and reactive power references wheninput voltages have 10% of 5th order harmonic.

shown in Fig. 8 when the the proposed DPC control is usedunder different input voltages conditions.

Fig. 8a shows the output voltage when the input voltagesare composed only with the fundamental harmonic and theinstantaneous active and reactive power references are derivedwith the conventional definitions (46). As is expected from theinput currents (Fig. 4), the DC-Link voltage does not presentsripple as p∗ is set as a constant.

When the input voltages have low order harmonic contentand the instantaneous active and reactive power references areestablished using the conventional definitions, the DC-Linkvoltage is again ripple free as it is shown in Fig. 8b, thereason is the same as for the ideal input voltages conditions,i.e. p∗ is again constant. As a consequence the input currentsare distorted (Fig. 5) due to it is necessary to compensatethe effect of the input voltages harmonic content to achieveinstantaneous active power constant [14]-[16].

Finally, when the proposed instantaneous active and reactivepower references (48) are used under distorted input voltages,p∗ is not constant anymore, thus the output voltage presentsthe associated ripple, as it is shown in Fig. 8c. It should benotice that the ripple of the output voltage in Fig. 8c appears inthe 6th harmonic due to the 5th harmonic of the input voltages

0.55 0.56 0.57 0.58 0.59 0.6748

750

752

0.55 0.56 0.57 0.58 0.59 0.6748

750

752

0.55 0.56 0.57 0.58 0.59 0.6748

750

752

Vdc (V)

(a)

Time (s)Vdc (V)

(b)

Time (s)Vdc (V)

(c)

Time (s)

Fig. 8. DC-Link voltage using the proposed DPC control strategy. Fromtop to bottom: a) Using coventional instantaneous active and reactive powerreferences when input voltages are ideal. b) Using coventional instantaneousactive and reactive power references when input voltages have 10% of 5thorder harmonic. c) Using proposed instantaneous active and reactive powerreferences when input voltages have 10% of 5th order harmonic.

[14]-[16]. Although the ripple amplitude in the presentedresults is small, due to the harmonic content of the inputvoltages is not too large, this amplitude depends on the valueof the demanded active power, increasing as the load does. Theadvantage of the proposed instantaneous power references isthat they provide sinusoidal and balanced input currents (Fig.7), thus this control strategy causes less distortion in the pointof common coupling (PCC) and consequently disturbing lessto other loads connected in the PCC.

The presented analysis shows that both instantaneous powerreferences have good features. The conventional power refer-ences (46) present no ripple in the output DC-Link voltagebut have low harmonic content in the input currents underdistorted input voltages conditions, being suitable for systemsthat require a constant DC voltage. The proposed powerreferences (48) present balanced sinusoidal input currents buthave ripple in the output DC-Link voltage under distorted inputvoltages conditions, being suitable for system where sensitiveloads are connected at the PCC . Therefore there is a trade-off between the input currents and output DC-Link voltageperformance.

VII. CONCLUSION

DPC strategies are simple and efficient control strategiesthat can be applied to power systems as power rectifiersallowing the use of low cost microprocessors for their imple-mentation. However, the system performance is deterioratedwhen the input voltages present low order harmonics content.The proposed DPC strategy based on the system model permitsto enhance the performance of the system under distorted inputvoltages. To achieve this purpose, adaptive techniques havebeen applied in the controller design to overcome problemsassociate with system parameters uncertainties, which leadsthe proposed controller to achieve unity power factor, pro-viding a better performance of the overall system. Besidesa new instantaneous power references are defined using thefirst harmonic of the input voltages, that are obtained us-ing a resonant filter tuned at the grid frequency. This newinstantaneous power references allow to achieve balancedand sinusoidal input currents under distorted input voltagescondition. Simulations with a three-phase two-level powerconverter have been carried out to illustrate the good behaviorof the proposed control technique, and an analysis of theperformance of the system has been added to evaluate theconvenience of using the conventional or the new proposedinstantaneous power references.

ACKNOWLEDGMENT

This work has been developed thanks to the financial sup-port of Andalusian Government within the researching projectTIC 1172.

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