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Efficiency Optimization of Induction Motor Drive Based on
Dynamic Programming Approach
Presented by: Branko Blanu{a
University of Banja Luka, Faculty of Electrical Engineering
E-mail: [email protected]
Research director: Prof. Slobodan N. Vukosavi}, Ph.D
Niš
2007
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Main goal:
For a known operating conditions, define optimal
control so the drive operates with minimal energy
consumption
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FUNCTIONAL APPROXIMATION OF THE POWER
LOSSES IN THE INDUCTION MOTOR DRIVE.
Inverter losses:
PINV  RINV  is2  RINV  id2  iq2 
where id , iq are components of the stator current in d,q rotational system and
RINV is inverter loss coefficient.
Motor losses:
Main core losses:
PFe  c1we (i)y D2 (i)  c 2we2 (i)y D2 (i)
where yD is rotor flux, we supply frequency and c1 and c2 are hystresis and
eddy current loss coefficient.
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Copper losses:
p Cu  R s i s2  Rr i q2
where Rs is stator resistance, and Rr rotor resistance.
Stray losses:
The stray flux losses depend on the form of stator and rotor slots and are
frequency and load dependent. The total secondary losses (stray flux, skin
effect and shaft stray losses) must not exceed 5% of the overall lossesrequirments of the EU for 1.1-90kW motors
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Important conclusions
1. It is possible to minimize power losses by variation of
magnetizing flux in the machine.
2. For a given working point of the induction motor, only one
pair of the stator currents produce flux which gives minimum of
the power losses.
3. For a known operating conditions and for closed-cycle
operation, it is possible to define optimal control so the drive
operates with minimal energy consumption
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Optimal Control Computation Using Dynamic
Programming Aproach
In order to do that, it is necessary to define performance index, system
equations, constraints and boundary conditions for control and state
variables and present them in a form suitable for computer processing
1. Performance index:
N 1
J   x( N )   L( x(i), u(i))
(1)
i 1
where N=T/Ts, T is a period of close-cycled operation and Ts is sample
time. The L function is a scalar function of x-state variables and ucontrol variables, where x(i) , a sequence of n-vector, is determined by
u(i), a sequence of m-vector
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2. System equations
x(i  1)  f xi , ui , i  0..N  1
4. Boundary conditions
x(0) has to be knownn
3. Constrains
Equality contraints
C[u(i)]=0 (Control variable equality constrains)
C[x(i),u(i)]=0 (Equality constraints on function of control and state variables)
S[x(i)]=0 (Equality constraints on function of state variables)
Inequality contraints
C[u(i)]0 (Control variable inequality constrains)
C[x(i),u(i)]  0 (Inequality constraints on function of control and state variables)
S[x(i)] 0 (Inequality constraints on function of state variables)
i=0,1,..N-1
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Following the above mentioned procedure, performance index, system
equations, constraints and boundary conditions for a vector controlled
induction motor drive in the rotor flux oriented reference frame, can be
defined as follows
1. Performance index
J
 aid2 i   biq2 (i)  c1w e (i)y D2 (i)  c2w e2 (i)y D2 (i)
N 1
i 0
2. System equation (dynamic of the rotor flux)

y D i  1  y D i 1 

Ts
Tr
 Ts
 
Lm i d i 
 Tr
where Tr=Lr/Rr is a rotor time constant
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3. Constrains
2
3 p Lm
kid i i q i   Tem i , k 
,
2 2 Lr
For torque
i d 2 i   i q 2 i   I s2max  0,
Stator current
 w rn  w r  w rn ,
Rotor speed
y D i   y Dn  0,
y D min y D i   0.
Rotor flux
4. Boundary conditions
Basically, this is a boundary-value problem between two points the boundary
conditions of which are defined by starting and final value of state variables:
w r 0  w r N   0,
Tem 0  Tem N   0,
y Dn 0  y Dn N   free , considering constrains
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Following the dynamic programming theory , a system of differential
equations can be defined as follows:
 i    i  1


Tr  T S
 2 c1w e i   c 2 w e2 i y D i 
Tr
2biq i    i kid i   0
2aid i    i kiq i    i  1
(1)
TS
Lm  0
Tr
Lm i q i 
kid i i q i   Tem i , w e i   w r i  
Tr y D i 
i  0, 1, 2,.., N  1,
where  and  are Lagrange multipliers.
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By solving the system of equations (1) and including boundary
conditions, we come to the following system:
TS 3
2b 2
i 
i d (i )  2 Tem
Tr
k
Ts
Tr
y D i  
y D i  1 
Lm i d (i )
Tr  T s
Tr  T s
2aid4 (i )   (i  1)
(2)
Tem i 
Lm i q i 
i q (i ) 
, w e i   w r i  
,
kid i 
Tr y D i 


 i   2 c1w e i   c 2 w e2 i y D i    i  1
Tr  T s
Tr
i  0,1,2,.., N  1.
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Every sample time values of wr(i) and Tem(i) defined by operation
conditions is used to compute the optimal control (id(i), iq(i), i=0,..,N1) through the iterative procedure and applying the backpropagation
rule, from stage i =N-1 down to stage i =0. Value of YD and  have
to be known. In this case, YD(N)=YDmin and
(N ) 
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
y D N 
 0.
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Simulation Results
Operation conditions (speed reference and load torque ) are given
in Fig. 1. and Fig. 2.
Graph of power loss for given operation condition are presented in
Fig. 3.
Graph of power loss and speed response during transient process
and for different methods are presented in Fig. 4. and 5.
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Fig.2 Speed reference.
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Fig 3. Load torque reference.
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Fig. 3. Graph of total power loss.
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Fig. 4. Graph of total power loss during transient process.
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Fig. 5. Speed response to step change of load torque.
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Expermental results
Experimental tests have been performed in the laboratory station for
digital control of induction motor drives which consists of:
induction motor (3 MOT, D380V/Y220V, 3.7/2.12A,
cosf=0.71, 1400o/min, 50Hz)
incremental encoder connected with the motor shaft,
three-phase drive converter (DC/AC converter and DC
link),
PC and dSPACE1102 controller board with TMS320C31
floating point processor and peripherals,
interface between controller board and drive converter.
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1.0
0.9
0.8
0.7
0.6
0.5
time (5s/div)
a)
time (5s/div)
b)
Fig. 6 Graph of magnetization flux for LMC method a), dynamic
programming approach b)
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mechanical speed p.u.
0.5
0.4
0.3
0.2
0.1
0
time (5s/div)
a)
time (5s/div)
b)
Fig. 7 Graph of mechanical speed for LMC method a), dynamic
programming approach b)
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power loss (W)
140
120
100
80
60
40
20
time (5s/div)
a)
time (5s/div)
b)
Fig. 8 Graph of power loss for dynamic programming a), nominal flux b)
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Conclusions
1. If load torque has a nominal value or higher in steady state,
magnetization flux is also nominal regardless of whether an algorithm for
efficiency optimization is applied or not.
2. At low loads in steady state, power loss for the LMC method and
method based on dynamic programming is practically the same but
significantly less than when the drive runs with nominal flux.
3. The method based on dynamic programming works in a way that
magnetization flux starts to rise before the increase of load torque and
keeps a higher value of magnetization flux during the transient
processes than other methods for efficiency optimization. As a result,
transient loss is lower and speed response is better.
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4. The procedure of on-line parameter identification has been carried out in
the background. In case the parameters change, a new optimal control value
is computed for the next cycle of the drive operation. This increases the
robustness of the algorithm in response to parameter variations.
5. Few simplifications in the computation of optimal control for the dynamic
programming method have been made. Therefore, the computation time is
significantly reduced. Some theoretical and experimental results show that
some effects like nonlinearity of magnetic circuit for YDYDn has negligible
influence in the calculation of optimal control.
6. One disadvantage of this algorithm is its off-line control computation.Yet, it
is not complicated in terms of software.
7. This algorithm is applicable to different close-cycled processes of electrical
drives, like transport systems, packaging systems, robots, etc.
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Thank you
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