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OPTIMAL CONTROL FOR THREE-PHASE POWER CONVERTERS SVPWM BASED
ON LINEAR QUADRATIC REGULATOR
Article · May 2012
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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH
Vol. 4. No. 3. May, 2012
OPTIMAL CONTROL FOR THREE-PHASE POWER CONVERTERS
SVPWM BASED ON LINEAR QUADRATIC REGULATOR
1
2
3
3
Hari Sutiksno , Lie Jasa , Mochamad Ashari , Mauridhi Hery Purnomo
1
2
Sekolah Tinggi Teknik Surabaya (STTS) Surabaya, Udayana University Bali,
3
Institut Teknologi Sepuluh Nopember (ITS) (INDONESIA)
E-mails: hari@stts.edu, liejasa@unud.ac.id, ashari@ee.its.ac.id, hery@ee.its.ac.id
ABSTRACT
A three-phase power converter space vector PWM with current regulation using PI controller typically
produces low harmonics distortion and unity power factor with the load change. Nonetheless, its time response of
the output voltage is not optimal. This paper proposes an improvement to the conventional three-phase power
converter space vector PWM using an optimal control based on the linear quadratic regulator and integral action to
improve the output voltage responses. In this paper, the power converter is determined using a linearized statespace model and the parameters of plant model can be found online using the recursive least square algorithm.
The feedback and feed forward gains can be obtained based on the linear quadratic regulator to minimize the
performance index. In this setting, an integral action is required to eliminate the steady-state error. Using a
sinusoidal input, the simulation results demonstrate that, at steady state, the proposed method results in the
overshoot of time response of the output voltage of 1.67% with the settling time of 0.05 sec for load changes from
full load to half load.
Key words: optimal control, power converter, pulse width modulation, linear quadratic
1. INTRODUCTION
Space vector PWM technique has been used extensively in three phase power converters due to the low
harmonic distortion, high efficiency and bidirectional power flow [1-8]. In order for the line current and voltage to be
in-phase with certain amplitude, the current regulator must be required. Although the PI controller has typically
been used in three-phase power converters, obtaining the optimal time response of the voltage output with respect
to the load-change remains challenging due to the fact that the load will likely change the parameters of the plant
model. An optimization method has been proposed using the output regulation subspaces and Pulse Width
Modulation (PWM) technique taking the advantage of the direct power control strategy [9]. The outcomes show that
the overshoot of the dynamics is about 3% and the settling time is about 0.2 sec. The LQR with integral action has
also been applied in the design of three-phase three-wire shunt active power filters, giving the overshoot of the
output voltage on dynamic load of around 5% with a half cycle settling time [10]. Research dealing with constrained
optimal control of three-phase Voltage Source Converters (VSC) based on a mathematical model in the
synchronous reference frame has also been developed recently [11]. In this instance, the optimal control design to
improve the time response of dc voltage on load change based on linear quadratic regulator with integral action is
proposed whereby the model of the three phase power converter with current regulator must be linearized. As the
parameters of the model vary as the load changes, these parameters must be calculated online by using the
recursive least square algorithm. These parameters are used to calculate the performance index. The optimal time
response will then be obtained when the performance index reaches the minimum value. Based on the linear
quadratic regulator algorithm, both the feedback and feed forward gains can be obtained. The integral action can
be added to eliminate the error output voltage. The performance index used in this paper entails the output voltage
and input control of the line currents.
2. THREE-PHASE POWER CONVERTER SVPWM WITH CURRENT REGULATOR
A. Three-Phase Power Converter
Fig 1 shows the main circuit diagram of the three-phase power converter. The relationship between the
currents and the voltages of the ac side can be expressed using the following equations:
di s
dt
(1)
vs  va vb vc 
(2)
vs  v '  L
T

v '  v a'
i s  i a
v b'
ib
v c'
ic 

T
(3)
T
(4)
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Vol. 4. No. 3. May, 2012
The currents of the dc side can be expressed as
irect  C
dvDC
 iL
dt
(5)
Meanwhile, the relationship between the voltages and currents of both the dc and ac sides can be obtained
respectively using the following equations:
M 
v '  MSv DC
(6)
i rect  S T i s
(7)
 2  1  1
1
 1 2  1
3
  1  1 2 
S  S a
Sb
(8)
Sc 
T
(9)
In equation (9), S indicates the switch position being 1 means ON and 0 means OFF. The space vector
PWM technique can then be applied to generate the three phase voltage of the ac side.
IL
Irect
Sa
L ia
Va
N
Vb
L
ib
Vc
L
ic
Sb
Sc
iC
+
C
S'a
S'b
_
RL
VDC
S'c
Fig. 1. Circuit Diagram
B. Current Regulator
In this case, the current regulator in Fig.2 uses an algorithm to produce the reference space vector voltage
such that the average line currents over the period Ts for next one period are in phase with the phase voltages of
the power source, with:
is (t  Ts )  uvs (t  Ts )
(10)
where u is the input signal. This input value is the ratio of the line currents to the phase voltages of the
power source. The voltage reference
'
vref
v' ref (t n )  (1 
will be generated as
2uL
uL
L
)vs (t n )  I (tn  Ts )  I (t )
Ts
Ts
Ts
IL
Vs
Vref
u
CURRENT
REGULATOR
(11)
Is
THREE-PHASE
POWER
CONVERTER
SVPWM
Fig. 2. The Plant Diagram
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VDC
INTERNATIONAL JOURNAL of ACADEMIC RESEARCH
Vol. 4. No. 3. May, 2012
3. OPTIMAL CONTROL BASED ON LINEAR QUADRATIC WITH INTEGRAL ACTION
There are a number of steps required in the design of the optimal control of three phase power converter
SVPWM based on linear quadratic regulator with integral action. The first step is to formulate the linearized state
space model of the plant whereby the parameters of the plant will be obtained by means of the recursive least
square (RLS) algorithm. The second step is to obtain the feedback and feed forward gains using Riccati equation in
order to minimize the performance index. The final step is to generate a control signal. Fig.3 shows the block
diagram of the proposed system.
Vabc
uref
u
IL
THREE-PHASE POWER
CONVERTER WITH
CURRENT REGULATION
u   K f v DC  K r u ref  K i  edt
vDC
ESTIMATOR
Kf Kr
A,B
LINEAR QUADRATURE
ALGORITHM
Fig. 3. Block Diagram of the proposed system
C. Linearized State Space Model
The mathematical model of three-phase power converter SVPWM with current regulation can be derived as
follows. The rectified current can be calculated as
irec (t ) 
3vmim
3v (u (t )vm ) 3vm2 u(t )
 m

2v DC (t )
2vDC (t )
2 vDC (t )
i L (t ) 
v DC (t )
RL
irec (t )  G1
G1 
(12)
(13)
u (t )
vDC (t )
(14)
3vm2
2
(15)
where the peak phase voltage vm and peak line current
im
are considered constant along the interval Ts .
The capacitor current can now be derived from the voltage as
iC (t )  C
dvDC (t )
dt
(16)
irec (t )  iC (t )  iL (t )
(17)
From equations (12) to (17), the state equation of the plant can be expressed as
dv DC (t )
G u (t )
1 v DC (t )
 F (u, v DC )  1

dt
C v DC (t ) C RL
(18)
Fig.4 shows the non-linear model of the three-phase power converter with current regulation.
Fig. 4. Non-linear Model of the Three-Phase Power Converter SVPWM with Current Regulator
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Vol. 4. No. 3. May, 2012
The linearized model of the state equation (18) can be therefore be formulated as follows
dvDC
F (u, vDC )
 F (uc ,VDC ) 
dt
vu
( uc ,VDC )
(u (t )  uC ) 
F (u, vDC )
vDC
(uc ,VDC )
(vDC (t )  VDC )
(19)
where
F (u, v DC )
u
F (u, v DC )
v DC
( uc ,VDC )
(u c ,VDC )

G1 1
C VDC

G1 uc
1

2
C VDC RL C
(20)
(21)
At steady state, the dc output voltage will move towards a constant value, so the function F (u c , V DC ) must
be equal to zero. From equations (19) to (21)
dvDC (t )
  AvDC  Bu(t )
dt
(22)
G1 uc
1

2
C VDC RLC
(23)
G1 1
C VDC
(24)
where
A
B
Finally, it can be seen from equations (22) and (23), the parameters of power converter model are
influenced by the load resistance. The linearized model of the power converter (Fig.5) demonstrates that the
parameters of the three-phase power converter model vary with the load changes.
Fig. 5. Linearized Model of the Three-Phase Power Converter SVPWM
Recursive Least Square Algorithm
For the simplicity reason, the three-phase power converter SVPWM with current regulator model may be
expressed as a linear model. A state space realization of the ARX model is given in discrete equivalent by:
v DC (k  1)  Av DC (k )  Bu(k )
(25)
The parameters of the plant (A and B) can be estimated by means of the Recursive Least Square (RLS)
algorithm, where
v DC (k )
and
u (k )
are the regressors.
  [ A B]
T
 (k )  [ y (k  1) u (k  1)]T
(26)
(27)
The load change will affect the output voltage. Both control signal u and dc voltage y are used to estimate
the parameters of the three-phase power converter with current regulator. Fig.6 shows the block diagram of the
RLS estimator to obtain A and B.
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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH
Vol. 4. No. 3. May, 2012
Fig. 6. RLS Estimator
The following equations are used within the RLS algorithm.
K (k )  P(k  1) (k )[ I   (k )T P(k  1) (k )]1
(28)
 ( k )  v DC ( k )   ( k ) T  ( k  1)
(29)
 (k )   (k  1)  K (k ) (k )
(30)
P(k )  [ I  K (k ) (k ) T ]P(k  1)
(31)
The initial value of the parameters
 (0)  0 0
T
(32)
Fig. 7 shows the RLS algorithm to estimate the power converter model.
Fig. 7. Diagram of Recursive Least Square Method
Linear Quadratic Regulator with integral action
For the system model v DC (k
 1)  Av DC ( k )  Bu (k ) , where v DC is the output dc voltage and u is the
input signal (control), and the scalar performance index is defined below
J
1 
((v DC ( k )  u ref ) T Q ( y ( k )  u ref )  u ( k ) Ru (k ))

2 k 0
(33)
where J is a scalar performance index
Q is a positive semi-definite matrix
R is a definite matrix
uref is the command signal (assumed to be constant)
vDC is the output signal
The optimal control sequence that minimizes J is
The feedback gain
Kf
u ( k )   K f v DC ( k )  K r u ref
(34)
in equation (34) is given by
K f  ( B T SB  R) 1 B T SA
(35)
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where S is the solution of the algebraic Riccati equation
S  AT [ S  SB( B T SB  R ) 1 B T S ] A  Q
and the feed-forward gain
Kr
(36)
is given by
K r  ( B T SB  R) 1 B T [ I  ( A  BK f ) T ]1 C T Q
(37)
The problem is that equation (34) cannot eliminate the steady state error. A simple way to overcome this is
by adding a term proportional to the accumulated error to the controller output. It can be expressed as
u ( k )   K f v DC ( k )  K r u ref  K i z ( k )
where
Ki
(38)
is a constant chosen by the designer and signal z(k) is given by the following recursive formula
z ( k )  z ( k  1)  (u ref  v DC ( k ))
(39)
Fig.8 shows the optimal algorithm to calculate the feedback and feed-forward gains. In a conventional PI
controller, the proportional gain and integral gain will remain constant on the load change and therefore achieving
the optimal time response using conventional PI controller is unfeasible.
Fig. 8. Linear Quadrature Algorithm and Control
4. RESULTS AND DISCUSSION
In this simulation experiment, the performance of the optimal control based on linear quadrature with
integral action will be compared to that of the PI controller for three-phase power converter SVPWM with current
regulation. The parameters of the three-phase power converter SVPWM with current regulation are shown in Table
1. The dynamic performance measures include the time response of the dc voltage generated on load change and
the phase difference between the input line current and the phase voltage for both controllers. The load resistance
will be varied from the initial load RL=60 Ω to 120 Ω at time 0.5 sec.
Table 1. Parameters of three-phase power converter
Parameter
Phase voltage
Frequency
DC bus voltage
Inductance of Reactor
Internal Resistance of Reactor
DC bus capacitance
Load resistance on full load
Sampling Frequency
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Value
100 V peak
50 Hz
300V
300mH
0.2 ohm
1000uF
60 ohm
10kHz
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Vol. 4. No. 3. May, 2012
Fig. 9 shows the Matlab Simulink block diagram of the power converter using the PI controller. The
proportional and the integral gain settings are 0.006 and 0.1 respectively. These gains have been chosen in order
to obtain the best time response of the output voltage. Fig.10 shows the simulation results of the time response of
the dc voltage on load change using the PI controller, showing the overshoot of approximately 7.5 volt (2.5%) at
t=0.5 sec and the settling time of 0.3 sec. In Fig.11, the line current (phase A) is shown between 0.46 sec and 0.56
sec. The peak current changes from 10A to 5A. The wave form of the line current is sinusoidal with 3% total
harmonic distortion (THD) and unity power factor.
Fig. 9. Matlab Simulink Diagram of Power Converter with PI Controller
Fig. 10. Time response of output voltage using PI Controller
Fig. 11. The input line current using PI Controller
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Fig.12 shows the Simulink block diagram of the power converter with optimal control using LQ regulator and
integral action. From Fig.13 and Fig.14, it can be seen that the parameters of the power converter model (A and B)
and gains (feedback gain Kf and feed forward gain Kr) move towards constant values at less than 0.05 sec at the
initial condition and on load change. This means the time required by the algorithm to estimate the parameters and
to obtain the gains is short.
Fig. 12. Matlab Simulink of the Power Converter with Optimal Control using LQ regulator and Integral Action
Fig. 13. The coefficients A and B of converter model
Fig. 14. The feedback and feed-forward gain
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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH
Vol. 4. No. 3. May, 2012
Fig.15 shows the simulation result of the time response of dc voltage on load change using LQR optimal
control (Q=1 and R=0.1). The time response of the output voltage oscillates (underdamped) with overshoot of
approximately 4 volt (1.33%) and settling time of 0.05 sec. Fig.16 shows the input line current (phase A) changes
from 10A peak value to 5A with a sinusoidal waveform with small distortion on steady state. During the transition,
however, the waveform of line current is not sinusoidal which occurs in half a cycle.
Fig. 15. Time response of the output voltage on load change with LQR optimal control
Fig. 16. The input line current (phase A) on load change with LQR Optimal Control
5. CONCLUSION
The overshoot of the optimal control of the three-phase power converter SVPWM with current regulation
based on linear quadratic with integral action on load change from full load to half load is about 1.33% which is
lower than that of the PI controller (2.5%). The settling time of the optimal control is also lower (about 0.05 sec
compared to 0.3 sec). Both the input line currents are in phase to the phase voltage (unity power factor) with a
relatively low THD of less than 5%. Using the optimal control method, the parameters of the plant model can be
obtained online and this allows the controller to be used for various plants. Oscillation occurs at the starting time
and every load changes of the optimal control method; however, the interval of the oscillation is less than half a
cycle.
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