[IEEE 34th Annual Meeting of the IEEE Industry Applications - Phoenix, AZ, USA (3-7 Oct. 1999)]...

Vector Modeling and Control of Unbalanced Electrical Systems

C.B. Jacobinal, M.B.R. Correa2, T.M.Oliveiral, A.M.N. Lima', E.R.C da Silva' lLaborat6rio de EletrSnica Industrial e Acionamento de Mkuinas

Departamento de Engenharia ElBtrica, Universidade Federal da Paraiba Caixa Postal 10105,58109-970 Campina Grande, PB, Brazil

Fax: ++55(83)310-1015 Email: [email protected] 2Departamento de Eletricidade, Escola TBcnica Federal de Alagoas, Cefet, AL, Brazil

Abstmct: In this paper a vector modeling a p proach is employed to represent the transient and steady-state behavior of unbalanced three-phase (three wires), unbalanced two-phase and single- phase systems containing PWM converters. In the case of the three-phase system, it is decomposed on dq components, by using an appropriated coordinate transformation, and then the vector model is obtained. While in the case of the single-phase system, it is introduced a fictitious second single-phase system to obtain the vector model. From the vector model two vector controllers on direct and on indirect synchronous reference frame are defined. The digital version of this controller in the stationary reference frame is also presented. Simulations and experimental results are presented for a V S I converter system.

I. INTRODUCTION

Unbalanced three-phase systems are quite common in power systems [l]. Unbalanced two-phase systems like a single-phase motor [2] are also quite common. Moreover, single-phase systems are very common in low-power low- cost applications [3].

It is well known, in agreement with the symmetrical- component theory, that an unbalanced steady-state operation of three-phase systems creates positive and negative sequences. To compensate for this unbalancing, some authors have proposed some specific compensation techniques like, for instance, Hsu and Behnke [4] and Kim et al. [5]. In [5] the compensation of the unbalancing is achieved by adding a negative sequence controller. In [4] the unbalancing is modeled as disturbance term which must be compensated by the controller.

In the present paper a vector modelling approach is employed to represent the transient and steady-state behavior of unbalanced three-phase (three wires), unbalanced two-phase and single-phase systems containing PWM converters. In the case of the unbalanced three-phase system, it is transformed in an equivalent unbalanced dq vector system by selecting a suitable coordinate transformation. In the case of the single-phase system, the equivalent unbalanced dq vector system is obtained by introducing a

fictitious second single-phase system. The proposed models are mathematically elegant and

very useful to understand the operation of an unbalanced systems as well as to derive a suitable control scheme. The control scheme for unbalanced operation is implemented using two controllers in the synchronous reference frame. The paper also presents a discrete-time implementation of the proposed scheme in the stationary reference frame. Simulations and experimental results with a V S I converter system are provided to illustrate the operation of the proposed methodology.

11. VECTOR MODELS

A . Unbalanced three-phase and two-phase systems

Fig.1 shows the circuit representation of a typical three- phase (eye connected - 3 wire) system. As an example, if the voltages sources U ; , U; and U$ represent a three-phase voltage source inverter, the voltages e;, e; and e; and the RL part represent the equivalent grid voltages and load, the system illustrated in Fig.1 may be considered as an equivalent representation of a power system that includes an active power filter. The following equation can be written for this system:

The superscript s indicates the stationary reference frame has been selected. Note that only u12 (= U; - U ; ) and U&

(= U; - U ; ) are used to represent the system behavior. If uf2 and are given, i.e., at the output of a controller, the phase voltages U!, U; and U$ can be determined by using a restricting equation like U! +U; +U$ = 0 or another similar condition.

The three-phase components in (1) may be transformed into an equivalent orthogonal dq components as given in (2)-(7), which are obtained from the odq-123 conservative

0-7803-5589-X/99/$10.00 0 1999 IEEE 1011

S -U]-

S

N

Fig. 1. Three-phase unbalanced system.

transformation [SI.

The dq components can be transformed into two complex vectors x& and %& in the stator reference frame

where the variable x indicates generically the voltage or the current of the system.

On the other hand, the complex vector xzq and Z;q may be transformed to an arbitrary reference frame by

x;q = e- j6ax;q (10) Z;q = (11)

The superscript a indicates that the reference frame is associated with the angular position 6, (d6aldt = wa). Now, equation (1) can be rewritten in terms of the complex vectors xzq and Z:q. It should noted that uiq is the complex conjugate of ii8 and then for simplicity reasons only the model for ut, is given below dq

where

- Rdq = (Rd+Rq)/2, &q = (Rd-R, + j 2 R d q ) / 2 , Rd = (4R1 + R2 + R3)/6, Rq = ( R E + & ) / 2 , Rdq = (R3 -

Ld = (4Li L2 + L3)/6, Lq = (L2 + & ) / a and Ldq = R 2 ) / ( 2 f i ) , z d q = (Ld+Lq)/27 Ldq = (Ld-Lq +j2Ldq)/2,

(L3 - L 2 ) / ( 2 4 ) . In steady-state, even if the input vectors v ; ~ and etq

have constant amplitude and rotate at constant speed W e - wa (where W e is the frequency of system), the complex vector Ezq presents constant amplitude, but rotates at -we - War i.e., corresponds to the inverse sequence. The Ezq term represents the unbalancing of the RL part of the load.

It is important to remark that the model given by (12)- (13) represents a generic unbalanced two-phase system, then it can be used to study also such a system.

B. Single-phase system

law for this circuit gives the following equation: Fig. 2a shows a single-phase circuit. Kirchoff’s voltage

It is not possible to define an equivalent dq vector model for the single-phase system based only on equation (14). However, as proposed in this paper, if a fictitious Q circuit is introduced, the definition of the equivalent dq vector model now becomes possible. Lets consider the single- phase circuit of Fig. 2b. As i t has been done for the actual circuit, its behavior can be described by:

di$ U; = R Q i $ + L Q - + e $ dt ( 1 5 )

where RQ, LQ and e6 are the parameter of the fictitious model.

In this paper it is considered that the control scheme discussed is implemented in a microcomputer-based en- vironment, which is equipped with an adequate interface cards to interact with electric power system. In this case, the model given by (15) is written in discrete-time form and simulated in the microcomputer software to provide

1012

' I

1'

I CO>

Fig. 2. Single-phase circuits for the vector model. (a) Actual single- phase circuit. (b) Fictitious single-phase circuit.

the Q component required to define the vector model to- gether with the D component measured from the actual circuit.

NOW, introducing the complex vectors xgQ and ZgQ by using (8) to ( l l ) , the equations of the vector model can be derived from (14) and (15) as given below

- * a - di$Q VgQ = RDQ~DQ + LDQ - + jwazDQigQ + dt

egQ + EgQ (16) where

- RDQ = (RD+RQ)/2, EDQ - - (RD-%)/2, ~ D Q =

The E& term is a disturbance that must be compensated by the controller. The disturbance term decreases when RQ, LQ are close to RD, LD, respectively. Sequence negative disturbance also appears if le; I and Le; - n/2 are different of Iebl and Le&, respectively. However, as it will be shown, the controller is very robust and is able to compensate the disturbance term even with large dif- ferences between RQ, LQ, le;/ and Le; - r / 2 and RD, Lo, leg1 and Leg, respectively.

(LD-LQ)/B and z D Q = (LD-LQ)/2.

111. CONTROL SCHEMES Lets consider the system of Fig. 1 in which the three-

phase RLE load is supplied through a voltage source inverter represented by the voltage sources U!, U; and U;. If the load is balanced and a proportional-integral controller is employed to control the load current, the use of the synchronous reference frame has proved to be the best choice just because the disturbance terms (direct sequence) are transformed to dc quantities that are easily compensated by the controller itself. However, if the load is unbalanced the use the synchronous reference frame (wa = w e ) only solves the disturbance rejection for the direct sequence term that rotates at the frequency we - wa = 0 and after coordinate transformation becomes a dc component. The inverse sequence term, after coordinate transformation becomes a component that rotates at -2we and con- sequently cannot be compensated by a single controller.

The controller structure proposed in this paper employs two different synchronous controllers. The positive sequence synchronous controller rotates at +we and is designed to compensate the direct sequence term. The negative sequence synchronous controller rotates at -we and is designed to compensate the inverse sequence term. These two controllers operate simultaneously and its outputs are added.

Assuming that PI controllers are used, the continuous- time state space control law can be described by

where t:,, = iE - i& is the load current error and v z is the controller output (which is synthesized through the inverter and indicated by the superscript *). The parameters kp and ki are the gains of the controller. This continuous-time controller can be emulated at the stationary reference frame as proposed by Rowan and Kerk- man [7]. In this paper the discretetime version of the controller at the synchronous and at the stationary reference frame will be discussed.

The exact discretetime version of the above control law, in terms of the d and q components, is given by

where h is sampling time. From (18) and (19) and by using the coordinate trans-

formers ejae and e-jae, the positive sequence stationary continuous-time version can be obtained, that is

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. s* b !

$ .I .s

3 - PHASE is ii i PWM-VSI-LOAD

I I

3 - PHASE PWM - VSI - LOAD

U;;*

Eq. 9

U;;

@)

Fig. 3. Block diagram of the two synchronous current controller for three-phase system. (a) synchronous frame. (b) stationary frame.

In this case the discretetime version of the controller cannot be exact, since dxiqi /dt depends of x : , ~ . The discrete-time version has to be determined by using

O0 Ak-lhk )B (24)

O0 Akhk F = I + and H = (E - k! k=1 k=l

where A and B are the continuous-time matrices and F and H are the corresponding discrete-time matrices.

To use (24) t o obtain the discretetime version of (22)- (23) we first define A = j w , and B = ki, and then decide up to what order the power series should be truncated. If F and H are calculated with k = 1 we obtain the first- order discrete-time controller which, in terms of the d and q components, is given by

xzgi(t) = ~ : , i ( t - h) + hk&,(t - h) + jheX&i (t - h) (25)

v z (t> = x:qi ( t ) + kPGq ( t ) (26) Note that there is an additional term j h w , ~ : , ~ ( t - h ) in the controller. Even by this additional term, this controller is simpler than the synchronous controller since it does not require any coordinate transformation. It is possible to improve the discretetime representation by truncating F and H at k = 2 to obtain the second-order discretetime controller which, in terms of the d and q components, is given by

x;,i(t) 1 (1 - h2w:/2)x:,i(t - h) + hk&:,(t - h) + jh,x:,i(t - h) + jh2weki[z,(t - h) /2 (27)

As in the previous case, the expression to compute v:;(t) is also given by (26). The negative sequence stationary controller is obtained replacing we by -we in the previous equations.

Fig. 3a present a corresponding block diagram for the two synchronous current controller for a three-phase system. Blocks ,jag and e-jae perform the coordinate transformation from the stationary to the positive and negative synchronous reference frame, respectively. The superscript e+ and e- indicate the positive sequence and negative sequence reference frame, respectively. Block dq/123 represents the transformation from vector to the three-phase components. Block PWM + VSI+ Load represents the pulse width modulator, inverter and load, and blocks R:+ and R:- represent two P I controllers. The emulating version of these synchronous controllers in the stationary frame is presented in Fig. 3b. Blocks Rf represent the stationary controllers where x,$gj is calculated by (25) and (26) or by (27) and (26).

The controller structure for a single-phase case is presented in Fig. 4 only for the stationary version. It must

1 - PHASE PWM - VSI - LOAD .

t Fig. 4. Block diagram of the two synchronous current controller for single-phase system at the stationary frame.

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0 0.1 0.2 0.3 0.4 0.5

4t 1

0 0.1 0.2 0.3 0.4 0.5 t i m e (s)

Fig. 5. d-axis and q-axis currents errors for the two synchronous controller.

be noted that in this case the block diagram includes a simulation block which represent the calculation of the fictitious discrete-time single-phase model.

IV. SIMULATION RESULTS

Figures 5 and 6 show the current error as obtained with two controllers (Fig. 4), a single synchronous controller and a scalar standard stationary controller (the &axis component in (22)-(23) with we = 0). In this case the load is a single-phase R L E load (RD = 2.1R, LD = 40mH, e$ = E,cos(w,t), E, = 200V) being supplied by a ideal voltage source and h = loops. The reference currents are given by i:& = I,cos(w~~), 0 < t < tm,/2, and i:b = (1,/2)~0s(w,t), tmw/2 5 t 5 tm,, where I, = 10A, we = 1207rrad/s, and tm, = 0.5s. The RLE parameters of the fictitious Q model are RD = 4.20, LD = 120mH, e& = 0 and ir;d = i:;. These waveforms show that the error with the single synchronous controller and with the standard stationary controller are no longer zero, however the error with the proposed controller tends to zero.

V. EXPERIMENTAL RESULTS

Fig. 7 shows the reference (ir and ii*) and actual phase (i: and ill) for an unbalanced three-phase load being supplied by a VSI. The unbalanced three-phase load has been implemented by using a three-phase in- duction machine. To create the unbalancing one resis- tor of 260 has been added in series with one of the machine windings and one inductor of 23mH has been added in series with another windings. The parameters of the motor is R, = 26.80, R: = 26.8R2, L,, = 23mH, Lb, = 23mH and L, = 0.498H. The control system is based on the diagram sketched in Fig. 3 and is implemented by software in the microcomputer with h = loops.

0.1 0.2 0.3 0.4 0.5 U -4;

e o 2 -2 :I I 4

0. I 0.2 0.3 0.4 0.5

0. I 0.2 0.3 0.4 0.5 U -4; t i m e (s)

Fig. 6. d-axis and q-axis currents errors (top) for the single synchronous controller and d-axis current err0 (bottom) for the standard stationary controller.

I I 0 0.02 0.04 0.06 0.08

t i m e (s)

I

0 0.02 0.04 0.06 0.08 t i m e (s)

Fig. 7. Experimental reference and load currents for a three-phase system (two synchronous controller).

The reference currents are balanced and for the phase 1 it is given by ii* = Imcos(wet), 0 < t < tm,/2 and

1015

I I

I I 0 0.02 0.04 0.06 0.08

t i m e ( s ) 0 0.02 0. 04 0.06 0.08

t i m e (5)

I 0 0.02 0.04 0.06 c . 2

t i m e Is)

Fig. 8. Experimental reference and load currents for a three-phase system (single synchronous controller).

ir = (I,/2) cos(wet), tmax/2 5 t < tmax, where I, = lA, we = 100nrad/s and tmax = 0.08s. These waveforms show that the actual current tracks quite well the reference current. Fig. 8 show equivalent tests using a single synchronous controller. It can be noted, in this case, that the obtained result was not so good, the maximum steady-state error is about 6%.

Figs. 9 shows the reference currents, ig = ir and i:, and the actual phase current, ib = if and ib, su- perimposed for a single-phase load. The load is supplied by a VSI. The control system is based on the diagram sketched in Fig. 4 and is implemented by software in the microcomputer with h = loops. The reference current is given by if' = Imc0s(wet), 0 < t < tmax/2, and i;* = (I,/2) COS(W~~) , tm,/2 < t 5 tmax, where I, = lA, we = 100nrad/s and tmax = 0.08s. These waveforms show that both the actual and the fictitious current tracks quite well the reference current. Fig. 10 show equivalent tests using a single synchronous controller. In this case it can be noted also that the obtained result was not so good (the maximum steady-state error is about 6%).

This results demonstrates the correctness of the

I 0 0.02 0.04 0.06 6. "8

t i m e (5)

Fig. 9. Experimental reference and load currents for a single-phase system (two synchronous controller).

methodology and warrant the feasibility of the proposed technique.

VI. CONCLUSION

This paper presented a vector modeling approach to represent the transient and steady-state behavior of unbalanced three-phase (eye connected - 3 wires), unbalanced two-phase and single-phase systems containing PWM converters. In the case of the three-phase system, it was decomposed on dq components, by an appropriated transformation, and then the vector model is obtained. While in the case of the single-phase system, it is introduced a fictitious second single-phase system to obtain the vector model. From the vector model two vector controllers on direct and on indirect synchronous reference frame are defined. The digital version of this controller in the stationary reference frame is also presented. The simulations and experimental results demonstrate the correctness of the methodology and warrant the feasibility of the proposed techniques.

1016

Energy Conversion. John Wiley and Sons, 1959. A new syn-

chronous current regulator and an analysis of a current-regulated pwm inverter. IEEE Transactions on Industry Applications, 22(4):678-690, July/Aug. 1986.

[7] T. M. Rowan and R. J. Kerkman.

0 0.02 0.04 0.06 9.08 t i m e (s)

0 0.02 0. 04 0.06 0.08 t i m e Is)

Fig. 10. Experimental reference and load currents for a single-phase . system (single synchronous controller).

REFERENCES

J.W. Dixon, J.J. Garcia, and L.Morb. Control system for three-phase active power filter which simultaneously compensates power factor and unbalanced load. IEEE Transactions on Industrial Electronics, 42(6):636-641, December 1995. M. F. Rahman and L. Zhong. A current-forced re- versible rectifier fed single-phase variable speed induc- tion motor drive. In Conf. Rec. PESC, pages 114-119, 1990. P. N. Enjeti and Ashek Rahman. A new single-phase to three-phase converter with active input current shaping for low cost ac motor drives. In Conf. Rec. IAS, pages 935-939,1990. P. Hsu and M. Behnke. A three-phase synchronous frame controller for unbalanced load. In Conf. Rec. PESC, pages 1369-1374,1998. H.S. Kim, H.S. Mok, G.H. Choe, D.S. Hyun, and S.Y. Choe. Design of current controller for 3-phase pwm converter with unbalanced input voltage. In Conf. Rec. PESC, pages 503-509, 1998. D. C. White and H. H. Woodson. Electromechanical

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[IEEE 34th Annual Meeting of the IEEE Industry Applications - Phoenix, AZ, USA (3-7 Oct. 1999)]...

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