Sensorless nonlinear control for a three-phase PWM AC-DC converter

Sensorless Nonlinear Control for a Three-Phase PWM AC-DC Converter Amira Marzouki, Mahmoud Hamouda, and Farhat Fnaiech, Senior, Member, IEEE SICISI Unit, ESSTT, 5 Av. Taha Hussein, 1008, Tunis, Tunisia [email protected], [email protected], [email protected]

Abstract-The main objective of this paper is to implement a nonlinear control for a PWM voltage source converter with a reduced number of sensors. For this purpose an input-output feedback linearization control strategy is firstly implemented in order to control both the line current and the DC output voltage. Next, an estimator of the AC mains voltage is implemented so as to reduce the number of the requested sensors, to avoid the undesired noise, and to minimize the converter’s cost. The efficiency of the proposed control law and estimation method is next validated through simulation results. Consequently, load independent unity input power factor operation and a perfect tracking of the DC bus voltage waveform are achieved even without using mains voltage sensors.

I.

reduce the sensors number by replacing them by equivalent software [12]-[14]. In this paper, the input-output linearization together with an AC mains voltage estimator were implemented in order to control the line currents and the DC output voltage, to avoid the sensor’s noise and to reduce the converter’s cost. The paper is organized as follows. In section II, the mathematical model expressed in a d-q synchronous reference frame of the Voltage Source Converter (VSC) model is presented. The input-output feedback linearization control strategy and AC mains voltage estimator were developed in section III. The performance evaluation and simulation results are presented in section IV. Finally, conclusions are drawn in section V.

INTRODUCTION

Three-phase PWM AC-DC converters are widely used in many power electronics fields. They feature several advantages such as providing constant DC bus voltage, low harmonic distortion of the utility currents, high power factor and bidirectional power flow. Many researchers focus on the control principle of three-phase PWM AC-DC converter of dealing with nonlinearity which is the origin of difficulties in the control of the above converters. For example, two classical linear PI controllers were used in [1][2] to regulate the DC bus voltage and the power factor separately. However, this method presents a long settling time and a difficulty to identify the parameters of PI controllers. Another method is to linearize the system around an operating point [3][4]. But, this method presents the problem that the controller design is dependent of the operating point. Other researchers have applied fuzzy logic and neural networks [5]. However, controller design is difficult because of the complexity of these approaches. In [6], a cascaded nonlinear PI controller was developed to synthesize the voltage and current control loop. This structure of the controller is easy to be implemented in practice and simulation results show its good performance, however it’s difficult to find the appropriate nonlinear function to cascade with PI controller. In [7]-[11], a feedback linearization control technique is employed to transform the nonlinear system into a linear decoupled one. Usually, complex nonlinear control process needs several sensors. This increases the cost of the implementation. Consequently, many researchers try to

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II.

THE MATHEMATICAL MODEL OF THE VSC

The power circuit of a three-phase PWM AC-DC voltage source converter without neutral connection is shown in Fig. 1 below. The state space model reported in a d-q rotating reference frame synchronized with the mains voltage is ⎡ ⎤ ⎡1 −R id + ωiq ⎡• ⎤ ⎢ ⎥ ⎢ i L ⎢ d ⎥ ⎢ ⎥ ⎢L • ⎢ ⎥ ⎢ −R ⎥ ⎢ iq − ωid ⎢ iq ⎥ = ⎢ ⎥+⎢0 L ⎢• ⎥ ⎢ ⎥ ⎢ Vdc ⎥ ⎢ 0 ⎢V dc ⎥ ⎢ 3 + − e i e i ( ) ⎣ ⎦ ⎢ d d q q Cdc Rdc ⎥⎦ ⎣⎢ ⎣ 2CdcVdc

Vd = Vdc d d Vq = Vdc d q

⎤ 0⎥ ⎥ 1 ⎥ ⎡ed −Vd ⎤ ⎢ ⎥ L ⎥ ⎣ eq −Vq ⎦ ⎥ 0⎥ ⎦⎥

(1)

(2)

where, L R Cdc Vdc ω id, iq ed, eq dd, dq Vd, Vq

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inductance of the line; resistance of the line; capacitance of the DC bus; DC bus voltage; angular velocity of line voltage; line currents in d-q reference frame; line voltages in d-q reference frame; switching functions in d-q reference frame; input voltages of the rectifier in d-q reference frame;

⎡ x1 ⎤ x = ⎢⎢ x 2 ⎥⎥ ⎢⎣ x3 ⎥⎦ ⎡U d ⎤ U =⎢ ⎥= ⎣U q ⎦

⎡ id ⎤ ⎢ ⎥ = ⎢ iq ⎥ ⎢V ⎥ ⎣ dc ⎦ ⎡ ed − V d ⎤ ⎢e − V ⎥ q ⎦ ⎣ q

(7)

(8)

Since the mains voltage axis d was chosen as reference in the Park transformation, then ed = E and eq = 0 Thereafter, the control inputs are U d = E − Vd

(9)

U q = −V q

Fig. 1. Circuit diagram of a three-phase PWM AC-DC voltage source converter

The next step consists in differentiating each output many times until a control input U appears in the dynamic of the output. For the first output, it can be observed that the control input does not appear in the first derivative of y1 y1 = x 3

The system (1) is nonlinear regarding to Vdc. The number of state variables is three whereas there are only two control inputs. III.

A. Nonlinear control for three-phase PWM AC-DC converter Feedback linearization is recognized as a powerful approach to nonlinear control design. The central idea is to algebraically transform nonlinear systems dynamics into linear ones. Then, linear control techniques can be applied. Based on the input-output feedback linearization control approach already introduced in [9], the above model (1) can be rewritten in the following form

(3)

••

y1 =

⎡1 0 ⎤ ⎢L ⎢ g 2 ⎥⎥ = ⎢ 0 0 ⎦⎥ ⎢ 0 ⎢ ⎣

(11)

The above derivatives can be arranged in the following form ⎡ •• ⎤ u ⎢ y1 ⎥ = A ( x ) + E ( x ) ⎡ d ⎤ ⎢u ⎥ ⎢ • ⎥ ⎣ q⎦ ⎣ y2 ⎦

(13)

where, 1 ⎡ 3E f1 ( x) x1 f3 ( x) ⎤ ⎡ 3Eg1 − ( )− f3 ( x)⎥ ⎢ 2Cx RchC x32 = A( x) = ⎢⎢ 2C x3 ; E ( x ) 3 ⎥ ⎢ ⎢⎣ ⎥⎦ ⎢⎣ 0 f 2 ( x)

(4)

⎤ 0 ⎥ (14) ⎥ g2 ⎥⎦

Since E(x) is nonsingular, the control law is derived as follows

0 ⎤⎥ 1⎥ ⎥ L⎥ 0⎥ ⎦

(5)

⎡ y1 ⎤ ⎡Vdc ⎤ ⎡ x3 ⎤ h( x) = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎣ y 2 ⎦ ⎣ id ⎦ ⎣ x1 ⎦

(6)

⎡ g1 g ( x ) = ⎢⎢ 0 ⎣⎢ 0

x f ( x) 3 E f1 ( x ) + g 1 u d 1 − 1 32 ) − ( f3 ( x) 2C x3 x3 R ch C

For the second output the control input already exists in the first derivative of y2 so we don’t need to make a second derivative. y2 = x2 (12) • • y2 = x2 = f 2 ( x) + g 2 uq

where,

⎡ ⎤ −R x1 + ω x2 ⎢ ⎥ L ⎥ ⎡ f1 ( x) ⎤ ⎢ −R ⎢ ⎥ ⎢ ⎥ f ( x) = ⎢ f 2 ( x) ⎥ = ⎢ x2 − ω x1 ⎥ L ⎥ ⎢⎣ f 3 ( x) ⎥⎦ ⎢ ⎢ 3 (e x + e x ) − x3 ⎥ ⎢ 2C x d 1 q 2 R C ⎥ ch dc ⎦ ⎣ dc 3

(10)

•

Differentiate once again y1, the control input ud appears as shown in equation (11) below

SENSORLESS NONLINEAR CONTROL

⎧⎪ • x = f ( x) + g ( x)u ⎨ y = h( x ) ⎪⎩

•

y1 = x 3 = f 3 ( x )

⎡ •• ⎤ ⎡u d ⎤ −1 ⎢ y1 ⎥ ⎢ u ⎥ = E ( x )[ A ( x ) + ⎢ • ⎥ ] ⎣ q⎦ ⎣ y2 ⎦

(15)

where, ⎡ 2 Cx3 ⎢ 3 Eg 1 E −1 ( x ) = ⎢ ⎢ ⎢ 0 ⎣

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⎤ 0 ⎥ ⎥ 1 ⎥ ⎥ g2 ⎦

(16)

The new linear control inputs are ⎡ •• ⎤ ⎡ v d ⎤ ⎢ y1 ⎥ ⎢v ⎥ = ⎢ • ⎥ ⎣ q⎦ ⎣ y2 ⎦

The VSC model in (a- b- c) reference frame is given by (24)

Once the system is linearised, we can apply linear control techniques by using PI controllers. With regard to the target error dynamics given by (18), the linear control laws vd and vq are calculated as (20) • ⎧ •• ⎪ e1 + k11 e1 + k12 e1 + k13 ∫ e1 dt = 0 ⎨ • ⎪ e 2 + k 21 e 2 + k 22 e 2 dt = 0 ∫ ⎩

(18)

where, e i = y i − y iref ; ( i = 1, 2) y1ref = x 3 = V dc = 600V y

ref 2

(19)

= x 2 = iq = 0

⎡ v d ⎤ ⎡ − k11 f 3 ( x ) − k 12 e1 − k 13 ∫ e1 dt ⎤ ⎥ ⎢ ⎥= ⎢ ⎥ − k 21 e 2 − k 22 ∫ e 2 dt ⎣ v q ⎦ ⎢⎣ ⎦

dia 1 3 ⎧ ⎪ ea = L dt + Ria + Vdc ( d1 − 3 ∑ d n ) n =1 ⎪ ⎪ dib 1 3 + Rib + Vdc ( d 2 − ∑ d n ) ⎨ eb = L dt 3 n =1 ⎪ ⎪ dic 1 3 + Ric + V dc ( d 3 − ∑ d n ) ⎪ ec = L dt 3 n =1 ⎩

(17)

Substituting (24) into (23) di di di ⎧ pˆ = L( a ia + b ib + c ic ) +Vdc (d1ia + d2ib + d3ic ) ⎪ dt dt dt ⎪ (25) ⎨ di di 1 ⎧ ⎪qˆ = ⎨3L( a i − c i ) −V [d (i − i ) + d (i − i ) + d (i − i )]⎫⎬ c a dc 1 b c a b 2 c 3 a ⎪⎩ dt 3 ⎩ dt ⎭

By using Concordia transformation it follows ⎧ p = eα iα + e β iβ ⎨ ⎩ q = e β iα − eα i β

(20)

1 ( v q − f 2 ( x )) uq = g2

(21)

⎡ˆ ⎢ eα ⎢ ⎢⎣ eˆ β

v qr e f = − u q

⎡ˆ ⎤ ⎢ ea ⎥ ⎢ ⎥ ⎢ eˆb ⎥ = ⎢ ⎥ ⎢ eˆc ⎥ ⎣ ⎦

(22)

The proposed nonlinear control block diagram of the PWM converter is shown in Fig. 2.

⎤ ⎡ iα 1 ⎥ = 2 2 ⎢ ⎥ iα + i β ⎣ i β ⎥⎦

⎡ ⎢ 1 ⎢ 2 ⎢ 1 − 3 ⎢ 2 ⎢ ⎢ 1 ⎢⎣ − 2

AC mains voltage estimation The proposed control strategy requires a considerable number of sensors. To reduce this number, we suggest a sensorless nonlinear control. This technique consists in estimating power-source voltages from line currents as well as the DC voltage and by using active and reactive power as intermediate variables [12]. The active and reactive powers are expressed as (23)

(23)

(27)

⎤ ⎥ ⎥⎡ 3 ⎥ ⎢ eˆα 2 ⎥ ⎢ eˆ ⎥ ⎢⎣ β 3⎥ − 2 ⎥⎦ 0

⎤ ⎥ ⎥ ⎥⎦

(28)

Then, by considering the relationships (27) and (28), the expressions of estimated AC mains voltage are derived as follows

B

p = ea ia + eb ib + ec ic ⎧ ⎪ 1 ⎨ ⎪ q = 3 [( eb − ec )ia + ( ec − ea ) ib + ( ea − eb )ic ] ⎩

− i β ⎤ ⎡ pˆ ⎤ ⎢ ⎥ iα ⎥⎦ ⎢ ⎥ ⎣ qˆ ⎦

By applying the inverse Concordia transformation we obtain

According to (9), the resulting voltage references to be modulated by the PWM converter are derived as follows v dr e f = E − u d

(26)

Then,

and kij are the gains allowing to impose the desired dynamics. Substituting (18) and (19) into (15) yields x 1 2C 2 [ ud = x3 v d + f 3 ( x )( x3 + 1 ) − f1 ( x )] 3 R ch E g1 3 E x3

(24)

2 ⎡ eˆa ⎤ 3 ⎢ eˆ ⎥ = ⎢ b ⎥ i2 + i2 ⎢⎣ eˆc ⎥⎦ α β

⎡ ⎢ 1 ⎢ ⎢− 1 ⎢ 2 ⎢ ⎢ 1 ⎢⎣ − 2

⎤ 0 ⎥ ⎥ 3 ⎥ ⎡ iα ⎢ 2 ⎥ ⎣ iβ ⎥ 3⎥ − 2 ⎥⎦

− iβ ⎤ ⎡ pˆ ⎤ iα ⎥⎦ ⎢⎣ qˆ ⎥⎦

(29)

Once three line voltages are estimated, we can eliminate their corresponding sensors in the control loops.

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Fig. 2. The proposed nonlinear control block diagram of the PWM converter

200

SIMULATION RESULTS

ea (V) and ia (A)

IV. A

Simulation without estimator In order to show the effectiveness of the proposed sensorless control technique, numerical simulations are carried out using Matlab/ Simulink software wherein the controller shown in Fig. 2 was implemented. The simulation parameters are listed in Table 1 below. The steady state waveforms of the mains voltage (ea), line current (ia), the reactive current (iq), and the output DC bus voltage (Vdc) are depicted in Figs. 3, 4, 5, and 6 respectively. It can be seen that current and voltage are in phase. Moreover, the reactive current iq is zero which means that the converter operates at unity input power factor. On the other hand, the output DC voltage is stable and varies around the target reference. The stability of the voltage regulation loop is also verified during in transient operation by applying a step increase of the target output voltage from 600V to 300V as shown in Fig. 6. The settling time is about 0.15s.

ea ia

100 0 -100 -200 0.405

0.415

0.435

0.425

Time (s)

Fig. 3. Line current ia and line voltage ea 20

iq (A)

10 0 -10 -20 0.2

0.3

0.5

0.4

Time (s)

Fig. 4. Reactive current iq

602 601

TABLE I

800

PARAMETERS FOR SIMULATION

600

110 V (rms) 3.3 mH 0.5 Ω 660 µF 60 Ω 60 Hz

599

700

Vdc (V)

Source voltage Inductance of the line reactor Resistance of the line reactor Capacitance of the DC bus DC link resistance Network frequency

0.4

0.4

0.42

0.43

600

500

400

0.3

0.4

0.5

Time (s)

Fig. 5. DC bus voltage for V*dc = 600V

1055

301

500

Vdc (V)

Vdc (V)

600

400

300

299 0.405

300 0.2

0.3

0.4

0.415

0.425

0.435

Time (s)

0.5

Time (s)

Fig. 6. Transient response of DC bus voltage

Figs. 7 and 9 show the capability of leading and lagging reactive power operation of the converter. This operation is achieved by imposing a reactive line current reference equal to +3A and -3A respectively. In both cases the line current remains sinusoidal and the DC voltage waveform is stable and close to the target reference as shown in Figs. 8 and 10. These results confirm that with this control the converter is able to operate as a static power compensator on the electric network.

ea (V) and ia (A)

200

ea 10 ia

100

0

-100

-200 0.4

0.42

0.41

0.43

Time (s)

Fig. 7. Line current ia and line voltage ea for V*dc = 300V and iqref = +3A

Fig. 10. DC bus voltage for V*dc = 300V and iqref = -3A

B

Simulation with estimator In this part we are going to test our estimator before elimination of sensors. After that, the estimated mains voltage will be used in the control algorithm so as to replace the ones delivered by voltage sensors. Fig. 11 and Fig. 12 show that the AC mains voltage is successfully estimated. Indeed, the error between ea and its estimate is about 1.3%. After insertion of the estimator in the control loop, it can be seen according to Fig. 13 that voltages waveforms present some noises, but they have the wished fundamental. Figs. 14 and 15 illustrate the responses of the DC bus voltage and dq axis current with the sensorless control. It can be observed that Vdc is well regulated and remains close to the target reference value equal to 300V. Besides, the reactive current iq is kept at zero in an average sense, i.e. no reactive power is exchanged between the rectifier and the source. Fig. 16 illustrates how the converter responds to a step change in the DC bus voltage reference with the sensorless control. It can be seen that Vdc reaches the reference value rapidly. êa êb êc

200

êa, êb and êc (V)

Vdc (V)

301

300

299

0.405

0.415

0.425

0.435

100 0 -100

Time (s) -200

0.404

0.4

0.42

Fig. 11. The estimated voltages before sensors’ elimination

ea 10 ia

2

100

1

0

era (V)

ea (V) and ia (A)

0.416

Time (s)

Fig. 8. DC bus voltage for V*dc = 300V and iqref = +3A

200

0.412

0.408

0

-1

-100 -2

0.405

0.415

0.425

0.435

Time (s)

-200 0.4

0.42

0.41

0.43

Time (s)

Fig. 12. Error between the line voltage ea et its estimate

Fig. 9. Line current ia and line voltage ea for V*dc = 300V and iqref = -3A

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400

V.

êa êb êc

êa, êb, and êc (V)

200

0

-200

-400

0.4

0.404

0.408

Time (s)

Fig. 13. The estimated voltages after sensors’ elimination

In this paper a sensorless nonlinear control strategy for Voltage Source Converters is developed. The proposed controller combines a feedback linearization law with an estimator of the AC mains voltage. Simulation results show its good steady state and dynamic performances. Indeed, this control law guarantees a decoupled control of the DC term and the reactive power. In addition, sensorless control can improve the dynamic response in the control of three-phase PWM AC-DC converter, while sinusoidal AC current with unity input power factor operation are maintained. As further work, this technique will be applied on a back-to-back PWM converter used in variable speed wind turbine applications.

302

REFERENCES

300

380

[1]

298

0.37

0.41

0.45

Vdc (V)

340

[2]

300 260

[3]

220 0.2

0.3

0.4

0.5

Time (s)

[4]

Fig. 14. DC bus voltage with sensorless control [5]

12

[6]

id iq

id, and iq (A)

8

[7] [8]

4

[9] 0

0.3

0.34

0.38

0.42

0.46

0.5

[10]

Time (s)

Fig. 15. Reponses of the d and q axis current with sensorless control [11] 700 600

Vdc (V)

[12] 500

400

[13]

300 0.2

0.25

0.3

0.35

CONCLUSION

0.4

0.45

0.5

Time (s)

[14]

Fig. 16. Transient response of DC bus voltage with sensorless control

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