three phase rectifier controllers design (step by step)
DESCRIPTION
AFE rectifier or Thyristor rectifier simulationTRANSCRIPT
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DEMO TITLES
Three Phase Rectifier Controllers Design Case I: Active Front End (AFE) Rectifier
Case II: Silicon-Controlled Rectifier (Thyristor)
Author: Tshibain Tshibungu Simsmart Technologies Inc.
Brossard, Quebec Canada
Software used: Simsmart Engineering Suite V6 (ES V6)
A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark
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1. OBJECTIVE AND DESCRIPTION
The following document will help the user:
in designing step by step controllers for AFE in abc and dq synchronous reference frame,
in designing step by step controller for a Thyristor rectifier.
Test cases are done in order to test and validate the theory using the power electronics components from the Engineering suite V6 Electrical library.
1.1. AFE RECTIFIER IN ABC REFERENCE FRAME
An AFE (Active Front End) rectifier is built with IGBT components to keep constant the DC
voltage and have fewer harmonics compares to the front rectifier (diode building a converter).
With an AFE rectifier, the load factor can be set as desired value and generally it set as unity.
DC voltage is kept constant by using a PI controller (block Gc see figure below).
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Voltage controller design
Formulation for general load
The input converter voltage is given by
(1)
(2)
(3)
The input power to the converter is given by
(4)
Equations (1) – (3) into (4)
(
) (
) (
) (5)
Assuming a balanced voltage source as follows:
√ ( )
√ (
⁄ )
√ (
⁄ )
Since the AFE rectifier has to output a current with fewer harmonics (generally less than 5%),
we assume that the input current is given as below:
√ ( )
√ (
⁄ )
√ (
⁄ )
Where
Phase delay (voltage versus current)
Current R.M.S
Voltage R.M.S
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NB: is time dependent
Equation (5) can be rewritten as follows:
(6)
The small signal analysis (around the operating point) gives
(7)
Where
The current at steady state (operating point)
The transfer function is
( )
(8)
Neglecting the converter power loss, the capacitor power is given as:
(9)
The small signal analysis (around the operating point) gives
(10)
The transfer function is
( )
(11)
The PI controller is given by
( )
(12)
The graph of the converter transfer function is given by using (8), (11) and (12)
+
+ -
+
-
+
+
+
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5
Then, the converter transfer function is given by
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( ) (13)
Taking only the first transfer function, we have
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( )
( ) (
)
( )
( )
To have a stable system, poles and zeros must be in the left half plane and the following
conditions should be taken:
1. To avoid a significant zero on the right half plane, we must have:
2. Applying the Routh-Hurwitz criterion, we must have:
Using the condition (1) and neglecting the resistance, the above transfer function becomes:
( )
( )
( )
The choice of PI parameters depends on the specification requirements. So, this transfer function is close to the well-known second order transfer function. So, the zero must be placed to almost at the infinity (compares to the dominant pole).
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Other formulation for Resistive load
For a define load as a resistor, the transfer function can be changed.
Since
, equation (9) can be written as follows:
For small signal analysis, we have
The transfer function is
( )
Hence, the converter transfer function is given by:
( ) ( )
( ) (
( )
)
(
)
Or
( ) ( ) ( )
( )
Where
( )
,
,
( )
Thus, the PI controller is calculated by using the IMC method design. So, we have:
( )
Where the time constant of the closed loop transfer function
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Since the system was linearized, the choice of is critical. Suggestion: Good start could be . One can try to select
The diagram block control of AFE in abc reference frame
1.2. AFE RECTIFIER IN DQ SYNCHRONOUS REFERENCE FRAME
Comparatively to previous method, this method will design two PI controllers respectively for
current and DC voltage.
Current controller design
Using the Park transformation in synchronous reference frame of equations (1)-(3), the matrix
form is given as follows:
[ ] [ ] [ ] [ ][ ][ ] [ ]
[ ] ([ ][ ] )
(14)
Where
(
( ⁄ ) ( ⁄ )
( ⁄ ) ( ⁄ )
)
The Park transformation with q axis leading d axis and the angle between the rotor d axis and the reference (stator d axis).
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Thus, we have:
(15)
(16)
Since the system is coupled, let’s decoupled it by using the following new variables:
(17)
(18)
Where
Thus, the PI controllers that control both current axes (d and q) are calculated using the IMC
(Internal Model Control) and are given as follows:
( )
Where
Switching frequency
Proportional gain
Integral gain
Voltage controller design
Neglecting the converter power loss, we have:
(9)
Where
(
)
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The small signal analysis (around the operating point) where , gives:
(
)
Hence, the converter transfer function is given by:
( ) ( ) ( )[
( )
]
(
)
( )
Where
( )
( )
The converter transfer function becomes:
( ) ( ) ( )
( )( )
Where
( )
,
,
( )
Thus, the PID controller is calculated by using the IMC method design. So, we have:
( ),
( )
( ),
Where the time constant of the closed loop transfer function Since the system was linearized, the choice of is critical. Suggestion: Good start could be . One can try to select
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The diagram block control of AFE in dq synchronous reference frame
1.3. SELECTION OF SOURCE IMPEDANCE FOR AN AFE RECTIFIER
Starting with
Since for unity power factor and using a PLL that will locked , we have in steady
state:
Or
Neglecting the resistance voltage drop, we have:
The magnitude of the voltage is
( )
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Where
The input converter peak voltage
The source peak voltage
Since,
, where is the modulation index, We have:
(
)
Thus,
√( )
Having selected the inductance, the resistance can be selected as follows:
1.4. THYRISTOR RECTIFIER FEEDING GENERAL LOAD
General equation
The AC side parameters are function of DC voltage side of a rectifier as follows:
(
) (1)
Where
Source resistance
Source Inductance
Source line to line voltage
On the DC side and assuming a general load RL and E (EMF), we have:
(2)
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Where
Load resistance
Load Inductance
Load internal voltage source
Rectifier transfer function
In order to linearized the control, the control input delay angle is modified to be
(
) (3)
Where
The control voltage
Maximum value of the control voltage
Then (3) into (1) yields:
(
) (4)
However, there is always a time delay between the command and the rectifier response which
is one-twelfth of a period of AC source. So, the Laplace function of DC voltage is given as
follows:
( )
(
) ( ) (5)
Where
Transfer function gain
Time delay
Load transfer function
The general load RL and E (emf) transfer function is given by
( ) ( ) ( ) ( ) (6)
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Where
Internal voltage of the load
Resistor and inductance of the load
PI transfer function
The PI transfer function is given by
( )
(7)
Global transfer function
Combining (5) and (6), we have:
( )( )
( )
( ) ( ) (8)
Where
(
)
Using equation (8), the graph of the converter transfer function is given below
Where
( )
( ) Rectifier transfer function
( )
( ) Load transfer function
( ) The perturbation
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Controller PI design Classical method of PI design
Then, the converter transfer function is given by
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
Assuming that the internal voltage is constant and taking only the first transfer function, we
have
( ) ( ) ( ) ( )
( ) ( ) ( )
The open loop transfer function is given as:
( ) ( ) ( ) ( )
( )( )
To reduce the characteristic equation from order 3 to order 2, the following assumption can be
made
So,
( ) ( ) ( ) ( )
( )
Using the pole-zero cancellation, we have:
( ) ( ) ( )
Then, the closed loop transfer function is given as:
( )
For a desired raising time , we have:
Proportional gain
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Proportional gain
Internal Model Control (IMC) of PI design The following study shows how IMC is implemented. For more information about IMC theory, see appropriate books. The process open loop transfer function is
( ) ( )
( )( )
Since , we can approximate
( ) ( )
( )
Where
So, the PI which represents the IMC is given by
( )
( )( )
( )
(
)
Where
Proportional gain
Proportional gain
NB: Internal voltage source is supposed to be constant. If the internal voltage source is time
dependent, an estimated value can be compensated in feedforward with a time delay of the
converter.
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Where
( ) ( ) Filter function for internal voltage source feedforward compensation.
2. PROCESSES DATA
Active Front End Rectifier
Two simulations will be performed for each case by assuming a pre-charged and not pre-charged capacitor. For the AFE in dq reference frame the configuration is slightly different when capacitor is pre-charged or not. Example 1
A three phase voltage source supplies an AFE three phase rectifier that feeds a resistive load. The DC voltage is set to 500 Volts, and the load resistance jumps at 0.4 s from 50 ohms to 20 Ohms. The IGBTs are modeled by ideal switches in parallel with diodes. The capacitor . The AFE should be designed in abc and dq references frame. Here below the characteristic for each design: AFE in abc reference frame The IGBTs are triggered by a hysteresis control which is set to .
AFE in dq synchronous reference frame The IGBTs are triggered by PWM modulation with a carrier of
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Example 2 A three phase voltage source supplies an AFE three phase rectifier that feeds an equivalent 2.5 kW resistive load. The DC voltage is set to 700 Volts. The IGBTs are modeled by ideal switches in parallel with diodes. The capacitor . The AFE should be designed in dq references frame. AFE in dq synchronous reference frame The IGBTs are triggered by PWM modulation with a carrier of Example 3 A three phase voltage source supplies an AFE three phase rectifier that feeds an equivalent 11 kW resistive load. The DC voltage is set to 450 Volts. The IGBTs are modeled by ideal switches in parallel with diodes. The capacitor . The AFE should be designed in dq references frame. AFE in dq synchronous reference frame The IGBTs are triggered by PWM modulation with a carrier of
Thyristor Rectifier
A three phase voltage source supplies a
three phase rectifier that feeds a load . The load reference
current is set to 15 A. At 0.1 s the reference current jumps to 30 A and at 0.2 s and the internal
voltage jumps to 50 V. Simulate the model without and with feedforward compensation. The
maximum DC voltage that can be applied to the load is 220 V.
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3. CONTROLLERS DESIGN
Active Front End Rectifier
Example 1
Current calculations The power consumption by the load for load resistance 50 Ohms is
On the converter,
Since we want the unity power factor = 1, then converter current should be
√
Since DC link is constant, the power consumption by the load for load resistance 20 Ohms is
Then the converter current should be
√
Since DC link is constant, the power consumption by the load for load resistance 20 Ohms is
Then the converter current should be
√
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Impedance selection For the worst case scenario when the load resistance is set to 20 Ohms, we have: , , ,
√( )
, , ,
√( )
Thus, we select
Having selected the inductance, the resistance can be selected as follows:
PLL PI controller
AFE Rectifier in abc reference frame is rounded to
( )
( )
( )
One can select
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So, the PI controller is:
( ) ,
AFE Rectifier in dq synchronous reference frame Current controller design
( )
Where
Switching frequency
Proportional gain
Integral gain
( )
( )
( )
One can select
Thus, the PID controller is calculated by using the IMC method design. So, we have:
( ) ,
( )
( ) ,
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Example 2 Current calculations Since we want the unity power factor = 1, then converter current should be
√
Impedance selection , , ,
√( )
Thus, we select
Having selected the inductance, the resistance can be selected as follows:
AFE Rectifier in dq synchronous reference frame
Current controller design
( )
Where
Switching frequency
Proportional gain
Integral gain
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( )
( )
( )
One can select
Thus, the PID controller is calculated by using the IMC method design. So, we have:
( ) ,
( )
( ) ,
PLL PI controller
,
Example 3
Current calculations Since we want the unity power factor = 1, then converter current should be
√
Impedance selection , , ,
√( )
Thus, we select:
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Having selected the inductance, the resistance can be selected as follows:
Current controller design
( )
Where
Switching frequency
Proportional gain
Integral gain
( )
( )
One can select
Thus, the PID controller is calculated by using the IMC method design. So, we have:
( ) ,
( )
( ) ,
PLL PI controller
,
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Thyristor Rectifier
Let first define the . Then, the following parameters are calculated:
(
)
Since the maximum DC voltage is 220 V, the PI must be anti-windup with a limit of
Using the formulas above where (
), we have:
, we have
4. SIMULATION PARAMETERS
The simulation was run in time domain with sample time of
5. PROCESSES REPRESENTATION IN ES V6
See the end of the document
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6. ENGINEERING SUITE V6 RESULTS
Example 1: AFE Rectifier in abc reference frame
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Example 1: AFE Rectifier in dq synchronous reference frame
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Example 2: AFE Rectifier in dq synchronous reference frame
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Example 3: AFE Rectifier in dq synchronous reference frame
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Thyristor Rectifier
Green without feedforward compensation load internal voltage Blue with feedforward compensation load internal voltage
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Conclusion
The following conclusions can be made:
for AFE rectifier, both methods show how to design controllers. The choice of a closed- loop parameter is sensitive since the system was linearized,
for AFE rectifier, the current and source voltage are in phase as desired,
for AFE the inrush current can be eliminated by using pre-charged capacitor as displayed above,
for AFE rectifier, a LC filter can be also added to reduce harmonics,
for thyristor rectifier the load internal voltage compensation has an impact on the response.
for thyristor rectifier the method can be extended to the control of DC motor.
7. REFERENCE BOOKS
1. Electric motor drives: modeling, analysis, and control.
R. Krishnan.
2. Internal model control: a comprehensive view. Daniel E. Rivera
3. Internal model control PID control design and AC Drives. Daniel E.Rivera, Manfred Morari and Skogestad
4. Power electronics handbook. 2nd edition. Muhammad R. Rashid
V ΦVC
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A/D
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ACTIVE FRONT END RECTIFIER IN ABC REFERENCE FRAME
Vdc reference
Vdc measured
PI Voltage Controller
Ia template
Ib template
Ic template
Hysteresis control
Hysteresis control
Hysteresis control
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Ib measured
Ic measured
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Δ∑
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ACTIVE FRONT RECTIFIER IN DQ REFERENCE FRAME
(CAPACITOR NOT PRE-CHARGED)
Vdc reference
Vdc measured
PI Voltage Controller
∫
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c
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PLLPhi angle from PLL
Iabc to Idq
Vabc to Vdq
Iabc to dq tranformation block
Triangular source 5.2 kHz
Vdr*
Vqr*
-wsLs
wsLs
PI d axis
Current Controller
PI q axis
Current Controller
V ΦVC
T
V
Δ∑
K
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ACTIVE FRONT RECTIFIER IN DQ REFERENCE FRAME
(CAPACITOR PRE-CHARGED)
Vdc reference
Vdc measured
PI Voltage Controller
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3
3
3
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b
c
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ΦI bI c
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Iabc to dq tranformation block
Triangular source 5.2 kHz
Vdr*
Vqr*
-wsLs
wsLs
PI d axis
Current Controller
PI q axis
Current Controller
Thyristor Rectifier with feedforward compensation load internal voltage
Thyristor Rectifier without feedforward compensation load internal voltage