IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 1, MARCH 2009

301

Power Engineering Letters Control of the Reactive Power Supplied by a Matrix Converter Roberto C´ardenas, Senior Member, IEEE, Rub´en Pe˜na, Member, IEEE, Jon Clare, Senior Member, IEEE, and Patrick Wheeler, Member, IEEE

Abstract—In this paper, the control of the reactive power supplied by a matrix converter to a grid is presented. A matrix converter is used to feed an induction generator using a space vector modulation algorithm. An input current observer, implemented using an estimation of the modulation matrix, is used in a nonlinear controller that regulates the reactive power supplied to the grid by a variable-speed wind energy conversion system (WECS). The results obtained are experimentally validated using a 2 kW experimental prototype implemented using a cage machine, a matrix converter and a wind turbine emulator. Index Terms—Induction generators, power generation control, wind energy.

Fig. 1.

Proposed variable speed WECS.

Fig. 2.

Symmetrical pattern used by the space vector modulation.

I. INTRODUCTION HEN A MATRIX converter is used in a grid-connected variable speed wind energy conversion system (WECS), it is a desirable aim to control the reactive power supplied to the grid [1]. In any practical application of matrix converters, voltage transducers at the grid side and current transducers at the load side are available [2]. To avoid additional transducers in a reactive power control system, a current observer is proposed in this paper. The input current estimated by the observer is used by a nonlinear control system that regulates the displacement angle φ at the matrix converter input. The WECS studied in this paper is shown in Fig. 1. A cage induction generator driven by a variable speed induction machine is utilized [3].

W

For each of the vectors used in the modulation algorithm, there is a 3 × 3 switching matrix relating the instantaneous output voltages to the instantaneous input voltages of the matrix converter, i.e. [2], [4]

va S11 vb = S21 vc S31

II. CURRENT OBSERVER When the space vector modulation proposed by Casadei et al. [4] is used to control a matrix converter, four non zero vectors are used in each sampling period. For a symmetrical pattern, the vector times are loaded into timer/counters using the arrangement shown in Fig. 2. T1 , T2 , T3 , and T4 are the times used by the vectors l1, l2 , l3 , and l4 in a given sampling time. T0 is the total time used by the zero vectors in the space vector algorithm [4]. The sum of all the vector times is equal to Ts, i.e., 2(T1 + T2 + T3 + T4 + 3T0 ) = Ts .

S12 S22 S32

vA S13 S23 vB S33 vC

where vA , vB , and vC are the input voltages and va , vb , and vc are the output voltages. Sij is zero when the corresponding switch is opened and Sij = 1 when the switch is closed [2]. Using the switching matrix of each vector, the low-frequency modulation matrix M (t) can be estimated in each sampling time as ˆ (t) = [Sl1 T1 + Sl2 T2 + Sl3 T3 + Sl4 T4 ] M

Manuscript received August 20, 2007. First published January 13, 2009; current version published February 19, 2009. This work was supported by Fondecyt, Chile, under Contract 1085289 and by the Industrial Electronics and Mechatronics Millennium Nucleus P04-048-F. Paper no. PESL-00103-2007. R. C´ardenas is with the Electrical Engineering Department, University of Magallanes, 113-D, Punta Arenas, Chile (e-mail: [email protected]). R. Pe˜na is with the Electrical Engineering Department, University of Concepci´on, 160-C, Concepci´on, Chile (e-mail: [email protected]). J. Clare and P. Wheeler are with the School of Electrical and Electronic Engineering, The University of Nottingham, Nottingham NG7 2RD, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2008.2003213

(1)

Ts 2

.

(2)

Note that the zero vectors are not used to estimate M (t). If the output current does not have zero sequence components, then the zero vectors do not influence the input current. Using (2), the input current is estimated as [2], [4]

0885-8969/$25.00 © 2009 IEEE

ˆ (t)T [iabc ] . ˆiA B C = M

(3)

302

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 1, MARCH 2009

Fig. 4. Fig. 3.

Steady-state performance of the current observer.

Proposed control system.

III. CONTROL OF THE REACTIVE POWER SUPPLIED TO THE GRID In a matrix converter, the relationship between the voltage ratio q and the input displacement factor is [4] q (4) cos(φ) = qm ax where q = vo /vi and qm ax is the maximum voltage ratio used in the modulation algorithm (usually qm ax < 0.866) [2]. The maximum value of the displacement angle is obtained from (4) as φm ax = cos−1 (q/qm ax ). The maximum reactive power supplied to the grid is Q1 = Pi tan(φm ax )

(5)

Fig. 5. Reactive power control considering a wind profile. (a) Active and reactive powers. (b) Maximum (φ m a x ) and real (φ) displacement angles.

A small signal model of (5) is obtained as ∆Q1 = ∆P1 tan(φm ax 0 ) + (Pi0 / cos2 (φm ax 0 ))∆φ

(6)

In a variable speed WECS, the power, Pi , is proportional to ωr3 [3], where ωr is the turbine rotational speed. Because of the high inertia of wind turbines, it is assumed that the active power changes slowly and ∆Pi could be considered ≈ 0. The proposed reactive power control system is shown in Fig. 3. The system is orientated along the input voltage angle. A proportional–integral (PI) controller regulates the displacement angle, φ, that is limited to a maximum value of φm ax . In Fig. 3, Qc is the reactive power supplied by the matrix converter inˆ (t) is calculated from (3) and λv is the put filter capacitors, M voltage vector angle used in the abc to d–q transformation. A variable gain, calculated using (6) and considering ∆Pi ≈ 0, is used in the PI controller. To compensate small nonlinear effects, for instance, commutation time [2], a compensating gain is obtained experimentally and applied to the estimated current.

IV. EXPERIMENTAL RESULTS The reactive power control system of Fig. 3 and the current observer of (2) and (3) have been experimentally tested using a 2 kW experimental prototype. A speed-controlled motor, driving the induction generator, is used to emulate a variable speed wind turbine. The space vector matrix modulation algorithm [3] is implemented in a digital signal processing (DSP) based system with a sampling frequency of 10 kHz. The induction gener-

ator is vector controlled with a constant magnetizing current of 3.12 A and an output power proportional to ωr3 [3]. Fig. 4 shows the estimated input current and the grid current measured by a transducer. Because of the filter capacitors, the grid current, ia grid , has a small phase angle with respect to the input voltage. The observed current, iai , is in phase with respect to the input voltage because a zero displacement angle was considered in the modulation algorithm. Fig. 5 shows the dynamic performance of the proposed control system. For this test a variable speed WECS is emulated using a typical wind profile. As shown in Fig. 5(a), for t < 20 s, a zero reactive power reference is used in the control system of Fig. 3. Between t > 20 s and t < 40 s, the matrix converter supplies ≈ 1 kVAR to the grid. For t > 40 s, the grid supplies 1 kVAR to the matrix converter. In all the cases, the control system operates with a good dynamic response and low settling time. Fig. 5(b) shows the real displacement angle, φ, and the maximum displacement angle φm ax , calculated from (4). In part of the wind profile (for instance, in t ≈ 50 s), the reactive power cannot be provided because the active power, Pi , is low [see (5)], and the displacement angle reaches the maximum value allowed by the variable limiter of Fig. 3. Fig. 6 shows the variation of the gain calculated from the small signal model of (6). For high values of φ, the gain variation is large and compensation of the PI gain (see Fig. 3) is indispensable.

´ CARDENAS et al.: CONTROL OF THE REACTIVE POWER SUPPLIED BY A MATRIX CONVERTER

303

tor induction generators fed by matrix converters. Experimental results show that excellent performance is achieved with the methodology. REFERENCES

Fig. 6.

Variable gain of (6) corresponding to the test of Fig. 5.

V. CONCLUSION A new control system for the regulation of the reactive power, supplied by the variable speed WECS, has been proposed. The new control system is also applicable to wound ro-

[1] G. Tapia, A. Tapia, and J. X. Ostolaza, “Proportional-integral regulatorbased approach to wind farm reactive power management for secondary voltage control,” IEEE Trans. Energy Convers., vol. 22, no. 2, pp. 488– 498, Jun. 2007. [2] P. W. Wheeler, J. C. Clare, L. Empringham, M. Bland, and K. G. Kerris, “Matrix converters,” IEEE Ind. Appl. Mag., vol. 10, no. 1, pp. 59–65, Jan. 2004. [3] R. C´ardenas and R. Pe˜na, “Sensorless vector control of induction machines for variable speed wind energy applications,” IEEE Trans. Energy Convers., vol. 19, no. 1, pp. 196–205, Mar. 2004. [4] D. Casadei, G. Serra, A. Tani, and L. Zarri, “Matrix converter modulation strategies: A new general approach based on space-vector representation of the switch state,” IEEE Trans. Ind. Electron., vol. 49, no. 2, pp. 370– 381, Apr. 2002.

301

Power Engineering Letters Control of the Reactive Power Supplied by a Matrix Converter Roberto C´ardenas, Senior Member, IEEE, Rub´en Pe˜na, Member, IEEE, Jon Clare, Senior Member, IEEE, and Patrick Wheeler, Member, IEEE

Abstract—In this paper, the control of the reactive power supplied by a matrix converter to a grid is presented. A matrix converter is used to feed an induction generator using a space vector modulation algorithm. An input current observer, implemented using an estimation of the modulation matrix, is used in a nonlinear controller that regulates the reactive power supplied to the grid by a variable-speed wind energy conversion system (WECS). The results obtained are experimentally validated using a 2 kW experimental prototype implemented using a cage machine, a matrix converter and a wind turbine emulator. Index Terms—Induction generators, power generation control, wind energy.

Fig. 1.

Proposed variable speed WECS.

Fig. 2.

Symmetrical pattern used by the space vector modulation.

I. INTRODUCTION HEN A MATRIX converter is used in a grid-connected variable speed wind energy conversion system (WECS), it is a desirable aim to control the reactive power supplied to the grid [1]. In any practical application of matrix converters, voltage transducers at the grid side and current transducers at the load side are available [2]. To avoid additional transducers in a reactive power control system, a current observer is proposed in this paper. The input current estimated by the observer is used by a nonlinear control system that regulates the displacement angle φ at the matrix converter input. The WECS studied in this paper is shown in Fig. 1. A cage induction generator driven by a variable speed induction machine is utilized [3].

W

For each of the vectors used in the modulation algorithm, there is a 3 × 3 switching matrix relating the instantaneous output voltages to the instantaneous input voltages of the matrix converter, i.e. [2], [4]

va S11 vb = S21 vc S31

II. CURRENT OBSERVER When the space vector modulation proposed by Casadei et al. [4] is used to control a matrix converter, four non zero vectors are used in each sampling period. For a symmetrical pattern, the vector times are loaded into timer/counters using the arrangement shown in Fig. 2. T1 , T2 , T3 , and T4 are the times used by the vectors l1, l2 , l3 , and l4 in a given sampling time. T0 is the total time used by the zero vectors in the space vector algorithm [4]. The sum of all the vector times is equal to Ts, i.e., 2(T1 + T2 + T3 + T4 + 3T0 ) = Ts .

S12 S22 S32

vA S13 S23 vB S33 vC

where vA , vB , and vC are the input voltages and va , vb , and vc are the output voltages. Sij is zero when the corresponding switch is opened and Sij = 1 when the switch is closed [2]. Using the switching matrix of each vector, the low-frequency modulation matrix M (t) can be estimated in each sampling time as ˆ (t) = [Sl1 T1 + Sl2 T2 + Sl3 T3 + Sl4 T4 ] M

Manuscript received August 20, 2007. First published January 13, 2009; current version published February 19, 2009. This work was supported by Fondecyt, Chile, under Contract 1085289 and by the Industrial Electronics and Mechatronics Millennium Nucleus P04-048-F. Paper no. PESL-00103-2007. R. C´ardenas is with the Electrical Engineering Department, University of Magallanes, 113-D, Punta Arenas, Chile (e-mail: [email protected]). R. Pe˜na is with the Electrical Engineering Department, University of Concepci´on, 160-C, Concepci´on, Chile (e-mail: [email protected]). J. Clare and P. Wheeler are with the School of Electrical and Electronic Engineering, The University of Nottingham, Nottingham NG7 2RD, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2008.2003213

(1)

Ts 2

.

(2)

Note that the zero vectors are not used to estimate M (t). If the output current does not have zero sequence components, then the zero vectors do not influence the input current. Using (2), the input current is estimated as [2], [4]

0885-8969/$25.00 © 2009 IEEE

ˆ (t)T [iabc ] . ˆiA B C = M

(3)

302

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 1, MARCH 2009

Fig. 4. Fig. 3.

Steady-state performance of the current observer.

Proposed control system.

III. CONTROL OF THE REACTIVE POWER SUPPLIED TO THE GRID In a matrix converter, the relationship between the voltage ratio q and the input displacement factor is [4] q (4) cos(φ) = qm ax where q = vo /vi and qm ax is the maximum voltage ratio used in the modulation algorithm (usually qm ax < 0.866) [2]. The maximum value of the displacement angle is obtained from (4) as φm ax = cos−1 (q/qm ax ). The maximum reactive power supplied to the grid is Q1 = Pi tan(φm ax )

(5)

Fig. 5. Reactive power control considering a wind profile. (a) Active and reactive powers. (b) Maximum (φ m a x ) and real (φ) displacement angles.

A small signal model of (5) is obtained as ∆Q1 = ∆P1 tan(φm ax 0 ) + (Pi0 / cos2 (φm ax 0 ))∆φ

(6)

In a variable speed WECS, the power, Pi , is proportional to ωr3 [3], where ωr is the turbine rotational speed. Because of the high inertia of wind turbines, it is assumed that the active power changes slowly and ∆Pi could be considered ≈ 0. The proposed reactive power control system is shown in Fig. 3. The system is orientated along the input voltage angle. A proportional–integral (PI) controller regulates the displacement angle, φ, that is limited to a maximum value of φm ax . In Fig. 3, Qc is the reactive power supplied by the matrix converter inˆ (t) is calculated from (3) and λv is the put filter capacitors, M voltage vector angle used in the abc to d–q transformation. A variable gain, calculated using (6) and considering ∆Pi ≈ 0, is used in the PI controller. To compensate small nonlinear effects, for instance, commutation time [2], a compensating gain is obtained experimentally and applied to the estimated current.

IV. EXPERIMENTAL RESULTS The reactive power control system of Fig. 3 and the current observer of (2) and (3) have been experimentally tested using a 2 kW experimental prototype. A speed-controlled motor, driving the induction generator, is used to emulate a variable speed wind turbine. The space vector matrix modulation algorithm [3] is implemented in a digital signal processing (DSP) based system with a sampling frequency of 10 kHz. The induction gener-

ator is vector controlled with a constant magnetizing current of 3.12 A and an output power proportional to ωr3 [3]. Fig. 4 shows the estimated input current and the grid current measured by a transducer. Because of the filter capacitors, the grid current, ia grid , has a small phase angle with respect to the input voltage. The observed current, iai , is in phase with respect to the input voltage because a zero displacement angle was considered in the modulation algorithm. Fig. 5 shows the dynamic performance of the proposed control system. For this test a variable speed WECS is emulated using a typical wind profile. As shown in Fig. 5(a), for t < 20 s, a zero reactive power reference is used in the control system of Fig. 3. Between t > 20 s and t < 40 s, the matrix converter supplies ≈ 1 kVAR to the grid. For t > 40 s, the grid supplies 1 kVAR to the matrix converter. In all the cases, the control system operates with a good dynamic response and low settling time. Fig. 5(b) shows the real displacement angle, φ, and the maximum displacement angle φm ax , calculated from (4). In part of the wind profile (for instance, in t ≈ 50 s), the reactive power cannot be provided because the active power, Pi , is low [see (5)], and the displacement angle reaches the maximum value allowed by the variable limiter of Fig. 3. Fig. 6 shows the variation of the gain calculated from the small signal model of (6). For high values of φ, the gain variation is large and compensation of the PI gain (see Fig. 3) is indispensable.

´ CARDENAS et al.: CONTROL OF THE REACTIVE POWER SUPPLIED BY A MATRIX CONVERTER

303

tor induction generators fed by matrix converters. Experimental results show that excellent performance is achieved with the methodology. REFERENCES

Fig. 6.

Variable gain of (6) corresponding to the test of Fig. 5.

V. CONCLUSION A new control system for the regulation of the reactive power, supplied by the variable speed WECS, has been proposed. The new control system is also applicable to wound ro-

[1] G. Tapia, A. Tapia, and J. X. Ostolaza, “Proportional-integral regulatorbased approach to wind farm reactive power management for secondary voltage control,” IEEE Trans. Energy Convers., vol. 22, no. 2, pp. 488– 498, Jun. 2007. [2] P. W. Wheeler, J. C. Clare, L. Empringham, M. Bland, and K. G. Kerris, “Matrix converters,” IEEE Ind. Appl. Mag., vol. 10, no. 1, pp. 59–65, Jan. 2004. [3] R. C´ardenas and R. Pe˜na, “Sensorless vector control of induction machines for variable speed wind energy applications,” IEEE Trans. Energy Convers., vol. 19, no. 1, pp. 196–205, Mar. 2004. [4] D. Casadei, G. Serra, A. Tani, and L. Zarri, “Matrix converter modulation strategies: A new general approach based on space-vector representation of the switch state,” IEEE Trans. Ind. Electron., vol. 49, no. 2, pp. 370– 381, Apr. 2002.