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FUZZY CONTROLLER TUNING FOR UPFC BOOST RECTIFIER BY EXPERIMENTAL DESIGNS
FUZZY CONTROLLER TUNING FOR UPFC BOOST RECTIFIER BY EXPERIMENTAL DESIGNS Jérôme FAUCHER, Alexandre TONERO, Pascal MAUSSION, Laboratoire d'Electrotechnique et d'Electronique Industrielle Unité Mixte de Recherche INPT-ENSEEIHT/CNRS BP 7122 - 2 rue Camichel 31071 TOULOUSE Cedex 7 –France Tel and Fax : 00 33 5 61 62 44 83 [email protected], [email protected]
Abstract: This paper shows the validity of experimental designs as an efficient on-site tuning tool for fuzzy controller dedicated to electrical engineering applications with multi-objective criteria. Our purpose is to improve the input and output system characteristics as well as the quality of electrical power. The desirability notion combines here time dynamic and harmonic criteria for a boost rectifier with unity power factor correction and illustrates the trade-off that has to be made between the different properties. Copyright © 2006 IFAC Keywords: fuzzy control, designs, methodology, converters.
1. INTRODUCTION Our work deals with the tuning of fuzzy controller in order to control electric systems. Fuzzy-logic-based controllers (FLC) are widely used in various applications, mainly because of advantages such as the dynamic performances, the robustness or the possibility of taking into account an experimental knowledge of the process. Nevertheless, a main drawback have to be underlined, the huge number of parameters that have to be tuned, even for a very simple fuzzy structure. These numerous fuzzy controller parameters could of course be tuned trough trial-and-error procedure, but it could be quite long and rather delicate. On the other hand, some methods have already been proposed, for the on line tuning of fuzzy controllers, using adaptive algorithms, additional fuzzy rules, neural networks or genetic algorithms. Besides, a simple tuning methodology based upon experimental on-line designs for all the parameters of a PID-like-fuzzy-logic controller have already been proposed (Hissel, 1999) few years ago. This method, based on time criterion, for fuzzy controller tuning gave experimental and simple pre-established settings just like the well-known Ziegler-Nichols methods for classical PID controllers.
Our aim is now to show that experimental designs methodology could also be an efficient tool in order to tune fuzzy controllers for applications that require multi-objectives criteria. In this paper, the methodology will be applied to a single phase boost rectifier with unity power factor correction. This kind of converter is strongly nonlinear. It is well known that linear controllers are not truly efficient, especially when sudden and hard parameters variations are applied to the system. Then, a fuzzy controller should be an efficient solution for the control of such a converter. Moreover, two criteria have to be regarded: time-response-based criterion and the harmonic distortion of the electrical network signal. Fuzzy controllers have already been used for this system control (Yu, 1996), (Henry, 1999), (Pires, 1999) and (Mattavelli, 1995), but authors don’t give a simple, efficient and robust tuning method without any need of a model of the system for FLC parameters. LMI method can be used for fuzzy control of DC/DC converters (Lian, 2006). However, this paper will show how the experimental designs could be an easy to use and efficient tool for the on-site tuning of a nonlinear controller under those specifics constraints which is the main contribution of our work, without system model.
L boost
Irec
D
Iload K vmeas
V rec
Ln
V DC C out
T Vn
R
Cf
K vn Iref +
ABS
Reference VDC
+
Σ
-
-
Σ
Current control
PWM
Voltage Control Low pass filter
Fig. 1. System structure 2. SYSTEM DESCRIPTION
Fc =
1 2π L f .C f
(1)
A complementary study shows that this Cf value allows reducing the current distortion under no load.
2.1 System description The system is a single phase boost rectifier with unity power factor correction. The first step is to generate a DC voltage through a classical diode bridge. An input filter capacitor Cf reduces the voltage ripple. However, this capacitor also reduces the diode conduction angles and generates harmonic distortion on the electrical network described by Vn (network voltage) and Ln (network inductance). In order to solve theses problems, a boost converter is added to the system. The capacitor Cout is the output filter for the load that needs a constant voltage. The values of these different components are given by table 1.
Vn Ln Cf Lboost Cout
Table 1 System parameter values 325V Rload 100 Ω 0.1 mH R0 2000 Ω 25 µF Kvmeas 1/100 4 mH Kvn 1/325 500 µF Ref VDC 4V
Hard and sudden load variations are applied to the system in order to evaluate the performances of the control strategy. The benchmark test is the following: from steady-state operation under no load conditions (R = R0) to sudden maximum load connection (R = Rload), and sudden disconnection. In addition, it is important to notice that capacitor Cf prevents high frequency harmonics from going back to the network. A cut-off frequency Fc chosen above the higher frequency (the 100 Hz frequency of the rectified voltage Vrec) and around one decade below the switching frequency (20 kHz) is suitable. We fix Cf = 25 µF , that means Fc = 500 Hz (1).
3. LINEAR CONTROL The control of this kind of system is usually done by linear controllers. Performances of such controllers will be the reference for a comparison with the fuzzy controller and its initial parameter values. There are two control loops (figure 1): one for the dc output voltage and the other for the rectified input current.
3.1 Current loop The objective of this loop is to get a sinusoidal current in phase with the electrical network. Thus, it reduces the harmonic rejection and maintains a unity power factor. A linear PI controller is used in combination with a PWM module. The high frequency harmonics are then reduced with this kind of control. The shape of the current reference is generated from the network voltage (via Kvn) and its amplitude from the DC voltage (via the voltage controller) as shown on figure 1. The transfer function of the current controller is:
H current ( p) = Gc .
(1 + p.Tic ) p.Tic
(2)
From different considerations like the bandwidth, the dynamic of the current loop and the reduction of the oscillations, the following coefficients are given through trial and error methodology:
Gc = 7 Tic = 0.0005
IAE = ∫ e(t ) .dt
(3)
Figure 2 shows the simulation results for a continuous controller. One can observe the regulation efficiency since the rectified current is closed to its reference and contains few high frequency harmonics.
(8)
This criterion applied to VDC will show the robustness and the dynamic performances of the controller at its output. In addition, the second criterion, the harmonic distortion rate (THD), represents the harmonic rejection quality at the input. 39
∑I THD% =
k =2
2 k
.100
39
∑I k =1
(9)
2 k
The CEI 61000-3-2 standard defines the electromagnetic compatibility and limits the harmonic current emissions for the 39 first harmonics. It gives the maximum allowable current for each of those harmonics. Figure 3 shows how and when the two different criteria are calculated during the benchmark test. The IAE criterion is taken into account throughout the test. The harmonic distortion is only computed during steady-state operation under rated load condition, i.e. rated current.
Fig. 2. Current control: Irec and Iref 3.2 Voltage loop
VDC
This loop must regulate the output voltage VDC with respect to load, input voltage and input current variations. The PI controller tuning is based on the average model of the system. This method relies on the equilibrium of the instantaneous powers between the output of the rectifier and the DC part (Yu, 1996). If the current loop is fast enough compared to the voltage loop, the following approximation could be done:
V DC ( p ) VDC ( p) ≈ I rec ( p ) I ref ( p)
(4)
Load
No load
t Harmonic Distortion for supply current
Then, the transfer function is:
V DC ( p ) Vrec = . I ref 4.V
Rload R .C 1 + p. load out 2
(5)
Fig. 3. Criterion measurement and benchmark test.
where V is the VDC average value. The transfer function of the voltage controller is:
H voltage ( p ) = Gv .
(1 + p.Tiv ) p.Tiv
4. FUZZY CONTROLLER (6)
From different considerations like the bandwidth, the dynamic of the current loop and the reduction of the oscillations, the following coefficient are calculated:
Gv = 7.5 Tiv = 0.062
IAE for output voltage
(7)
3.3 Criteria The control quality for the whole system will be evaluated trough two criteria. The first one is the IAE (integral of absolute error, e).
A fuzzy controller will be used in the rest of this paper in order to improve the dynamic performances. The controller is a classical PI-like fuzzy controller (FLC) (see figure 4). The inherent difficulty of such a kind of controller is the huge number of parameters. This structure was chosen because the error’s second derivative does not have to be calculated. Indeed, its value could be important as it may amplify noise. The fuzzy part consists in two inputs / one output Sugeno FLC (Hissel, 1999) with seven triangular membership functions on each input and seven singletons at the output. There is a normalisation factor for each input (em for the error signal and dem for the error derivative) and for the output (gm).
e
Control action
1 em
gm de
1
Spi Integral action
dem
Fuzzy controller Normalisation
Denormalisation
Fig. 4. Fuzzy controller structure A zero-symmetry is imposed for both membership functions and singletons in order to provide a similar response for positive and negative inputs. A classical antidiagonal rule table, with fixed parameters, is used. By Fixing em to the reference value, only 8 parameters have to be tuned among the initial 73 ones (7*7 rules, 3*7 membership functions and 3 gains): dem, gm, PSe and PVSe (membership functions on error), PSde and PVSde (membership functions on error derivative) and PSs and PVSs (output singletons). Anyway, the tuning problem remains effective as 8 control parameters are to be tuned according to two criteria. Fuzzy logic is only used for the PI controller on the voltage loop. Due to frequency limitation of our DSP, the current loop must be continuous (the sampling period is
Te = 1.10 −4 s ). Moreover, two fuzzy controllers for the same system would dramatically increase the number of parameters that have to be tuned.
and each column is indexed to a factor, an output singleton position or a gain for example. For each experiment, the selected criterion value is calculated through simulation results or measured during experiments. Experiment number 1 2 3 4
Factor 1
Factor 2
+ +
+ +
Interaction 12 + +
Effects
E1
E2
E12
Criterion y1 y2 y3 y4
Figure 5 : Example of an experimental table − y1 + y 2 − y 3 + y 4 E1 = 4
E12 =
(10)
y1 − y 2 − y 3 + y 4
(11) 4 According to the experimental design methodology (Dey, 1999), the effect of a factor is obtained through expression (10). For example, E1 = 0.12 means that factor 1 at high level has an effect of +0.12 on the criterion. Moreover, the effect of interactions between factors 1 and 2 can also be investigated with this methodology. Expression (11) leads to the effect of interaction between factor 1 and 2, on the desired criterion. Furthermore, the interaction E12 between factors 1 and 2 could also be used to study a third factor. From these effects, an optimal tuning could be done. It requires a trial in order to confirm the design. If the obtained results are irrelevant, then the hypotheses must be reconsidered.
5. EXPERIMENTAL DESIGNS PRINCIPLES The History of experimental designs began in the 30’s in England with M. Fisher, but it had an increasing development since Taguchi published predefined tables. This methodology principle is to realize a schedule of the experiments in order to obtain the most accurate information for a specific problem with a minimum number of experiments (dey, 1999). The idea is to modify the level of each factor for each experiment according to a specific procedure. It allows a drastic reduction of the number of required experiments, the possibility of taking into account much more parameters, the detection of interactions between factors and gives an optimized solution. Considering for example only two levels for each of the 8 factors described above, the classical experimental tuning method that consists in varying one of the parameters when all the others are maintained constant leads to 28=256 required experiments. With experimental designs methodology, only 16 experiments are necessary to find the suitable combination of factor levels in order to minimize the selected criterion We use centred reduced variables, i.e. -1 for the low level and +1 for the high level of each factor. Then, an experimental table, as the one shown on figure 5, could be used. Each line represents an experiment
6. TUNING AND RESULTS
6.1 Parameter values In this controller, 8 parameters have to be tuned. The initial levels of parameters are difficult to choose. An accurate expertise on the system is required. The values of the continuous PI controller parameters will be used as initial values. Regarding SPI as the fuzzy controller output, on figure 4, equation 12 can be defined. Kp is the proportional gain and kd is the derivative one of the first part of the fuzzy controller.
S PI = ∫ (k p (e, de).e + k d (e, de).de ).dt 1 e(t ).k1 (e, de).dt em 1 de(t ) + gm.∫ . .k 2 (e, de).dt dem dt gm = . e(t ).k1 (e, de).dt em ∫ gm. de(t ) + . .k 2 (e, de).dt dem ∫ dt
S PI = gm.∫
S PI
(12)
Then, the last expression in equation 12 reveals two different actions: integral and proportional of the complete PI-like-fuzzy controller which can be used for initial tuning. The values are sampled at the sampling period Te:
e(t ) ⇒ e( k ) e(k ) − e( k − 1) de(t ) ⇒ Te
(13)
Then, from the transfer function Hvoltage, it comes:
gm Gv em = T iv gm.Te = G v dem
(14)
The levels of the membership functions are chosen on both sides of the values of the equidistributed membership function positions. Similar choices are made for gm and dem coefficients.
Fig.6 Elementary desirability Accounting for the 39 first harmonics, the composite desirability Dh, is then calculated and defined by: 1
39 39 Dh = ∏ dhi i =1
(18)
The final criterion Y is therefore:
Y = IAE * Dh
6.2 Desirability The desirability notion was introduced by E.C. Harrington (Harrington, 1965). It combines several different properties Yi with different scales and units (Derringer, 1994). Each of them is transformed in an elementary desirability function di. A desirability function is ranged between zero and one. A zero level corresponds to an unacceptable value for the criterion while a desirability of one represents the maximum desired performance. Many different transformations could be chosen. The most classical one was adopted due to its simplicity. It is described below. The value of Yi,p is the minimum acceptable value for Yi and Yi,c is the value above whom an amelioration of Yi is not very interesting. 0 ⇔ Yi ≤ Yi , p
Y − Y ri i i, p di = ⇔ Yi , p < Yi < Yi ,c (15) Y − Y i ,c i , p 1 ⇔ Yi ≥ Yi,c
The parameters ri allow to balance the importance of the increase of the property on the elementary desirability (see figure 6). Then, all the elementary desirabilities are combined into a composite desirability such as D:
]
]
As there are 8 parameters, a 28-4IV experimental table is used, that means only 16 experiments. Two successive designs are carried out. The first one is a “rough” design which gives significant levels for parameters. The second one allows improving the tuning. The experimental design, described in table 3 gives the following optimal tuning (table 2).
PSe PVSE PSde PVSde
(16)
Table 2 Fuzzy parameter values 0.4 NSs 0.1 NVSs 0.6 Gm 0.1 dem
-0.8 -0.5 570 45e-4
6.3 Simulation results Figure 7 and 2 show the simulation results for the fuzzy controller and the analogical PI. The dynamic performances with the fuzzy controller are drastically improved and the harmonic distortion is worth but remains low. 520 500 480
Fuzzy PI
460
1
∑ wi
So, each harmonic value of rank i is transformed into elementary desirabilities dhi ( i ∈ [1,39] ) with Yhi,c = 0 as the objective is to reject harmonic distortion and Yhi,p the CEI 61000-3-2standard limit value. We fix rhi << 1 in order to react only near the standard values. ∀i ∈ 1,39 , rhi = rh = 0.05 << 1 (17)
[
6.3 Tuning
Voltage (V)
[
D = ∏ di wi
(19)
440 420 400 380 360 340 Analogical PI
320 300 0
0.5
1
1.5
2 2.5 Time (s)
3
3.5
Fig.7 Output voltage responses in simulations
4
4.5
Table 3 28-4IV experimental table F1
F2
F3
F4
PSe
PVSe
PSde
PVSde
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
exp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
F6 = 134 PVSs + + + + + + + +
F5 = 234 PSs + + + + + + + +
F7 = 123 gm + + + + + + + +
F8 =124 dem + + + + + + + +
Experimental results
500
Linear PI
440
Voltage (V)
Table 4 Experimental results Fuzzy Linear controller PI IAE voltage (10-2V.s) 10.9 53.1 TDH (%) 3.8 4.1 Load connection: Tr 5% (ms) 67 193 VDC min (V) 367 335 Load disconnection: 169 1236 Tr 5% (ms) 439 485 VDC max (V) 384 344 VDC min (V)
7. CONCLUSION The experimental design methodology could be an efficient tool for on-line tuning of fuzzy controller according to multi-objective criteria. The controller was first tuned through simulations and some differences with experimentations appear due to some modeling errors. The next step will consist in an experimental tuning, using the experimental response surface methodology.
480 460
modified with a modification of the desirability on the harmonic criterion.
420 400 380 360
Fuzzy PI
340
REFERENCES
320 300 0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
Fig.8 Experimental output voltage responses 10
Current (A)
5
0
-5
Established Settings for PID-like Fuzzy Logic Controllers, EPE'99, 8th European Conference on Power Electronics and Applications, Lausanne.
Irec Inet
-10 2.265
2.27
2.275
2.28
2.285
2.29
Derringer, G., Suich, R. (1994). A Balancing Act : Optimizing a Product’s Properties, Quality Progress, Pages 51-58. Dey A., Mukerjee R. (1999). Fractional Factorial Plans, Wiley, NewYork. Harrington, E., JR (1965). The Desirability Function, Industrial Quality Control, pp 494-498. Henry S.H. Chung, Eugene P.W. Tam, S.Y.R. Hui (1999). Development of a Fuzzy Logic Controller for boost Rectifier with Active Power Factor Correction, PESC 1999, Vol. 1, Pages 149 – 154. Hissel D., Maussion P., Faucher J. (1999). Robust Pre-
2.295
Fig.9 Experimental input currents
6.4 Experimental results Figure 8 and 9 show experimental results for the fuzzy controller and the numerical linear PI. It appears that fuzzy controller gives better dynamic performances with an equivalent harmonic distortion rate. The values of the numerical criteria are given by table 4. These results illustrate the compromise that has to be reach between harmonic quality and dynamic performances. The balance could be
Lian, KY, Liou JJ, Huang CH. (2006). LMI Based Integral Fuzzy Control of DC-DC Converters, IEEE transactions on fuzzy systems, vol. 14, no. 1, February 2006. Mattavelli, P., Buso, S., Spiazzi, G., Tenti, P. (1995). Fuzzy control of power factor preregulators, Industry Applications Conference, 1995, Vol. 3, Pages 2678 – 2685. Pires, V.F., Amaral, T.G., Silva, J.F., Crisostomo, M. (1999). Fuzzy logic control of a single phase AC/DC buck-boost converter, EPE 1999, Lausanne. Yu Qin, Shanshan Du (1996). Comparison of fuzzy logic and digital PI control of single phase power factor pre-regulator for an on-line UPS, IECON 1996, Vol. 3, Pages 1796 – 1801.
Abstract: This paper shows the validity of experimental designs as an efficient on-site tuning tool for fuzzy controller dedicated to electrical engineering applications with multi-objective criteria. Our purpose is to improve the input and output system characteristics as well as the quality of electrical power. The desirability notion combines here time dynamic and harmonic criteria for a boost rectifier with unity power factor correction and illustrates the trade-off that has to be made between the different properties. Copyright © 2006 IFAC Keywords: fuzzy control, designs, methodology, converters.
1. INTRODUCTION Our work deals with the tuning of fuzzy controller in order to control electric systems. Fuzzy-logic-based controllers (FLC) are widely used in various applications, mainly because of advantages such as the dynamic performances, the robustness or the possibility of taking into account an experimental knowledge of the process. Nevertheless, a main drawback have to be underlined, the huge number of parameters that have to be tuned, even for a very simple fuzzy structure. These numerous fuzzy controller parameters could of course be tuned trough trial-and-error procedure, but it could be quite long and rather delicate. On the other hand, some methods have already been proposed, for the on line tuning of fuzzy controllers, using adaptive algorithms, additional fuzzy rules, neural networks or genetic algorithms. Besides, a simple tuning methodology based upon experimental on-line designs for all the parameters of a PID-like-fuzzy-logic controller have already been proposed (Hissel, 1999) few years ago. This method, based on time criterion, for fuzzy controller tuning gave experimental and simple pre-established settings just like the well-known Ziegler-Nichols methods for classical PID controllers.
Our aim is now to show that experimental designs methodology could also be an efficient tool in order to tune fuzzy controllers for applications that require multi-objectives criteria. In this paper, the methodology will be applied to a single phase boost rectifier with unity power factor correction. This kind of converter is strongly nonlinear. It is well known that linear controllers are not truly efficient, especially when sudden and hard parameters variations are applied to the system. Then, a fuzzy controller should be an efficient solution for the control of such a converter. Moreover, two criteria have to be regarded: time-response-based criterion and the harmonic distortion of the electrical network signal. Fuzzy controllers have already been used for this system control (Yu, 1996), (Henry, 1999), (Pires, 1999) and (Mattavelli, 1995), but authors don’t give a simple, efficient and robust tuning method without any need of a model of the system for FLC parameters. LMI method can be used for fuzzy control of DC/DC converters (Lian, 2006). However, this paper will show how the experimental designs could be an easy to use and efficient tool for the on-site tuning of a nonlinear controller under those specifics constraints which is the main contribution of our work, without system model.
L boost
Irec
D
Iload K vmeas
V rec
Ln
V DC C out
T Vn
R
Cf
K vn Iref +
ABS
Reference VDC
+
Σ
-
-
Σ
Current control
PWM
Voltage Control Low pass filter
Fig. 1. System structure 2. SYSTEM DESCRIPTION
Fc =
1 2π L f .C f
(1)
A complementary study shows that this Cf value allows reducing the current distortion under no load.
2.1 System description The system is a single phase boost rectifier with unity power factor correction. The first step is to generate a DC voltage through a classical diode bridge. An input filter capacitor Cf reduces the voltage ripple. However, this capacitor also reduces the diode conduction angles and generates harmonic distortion on the electrical network described by Vn (network voltage) and Ln (network inductance). In order to solve theses problems, a boost converter is added to the system. The capacitor Cout is the output filter for the load that needs a constant voltage. The values of these different components are given by table 1.
Vn Ln Cf Lboost Cout
Table 1 System parameter values 325V Rload 100 Ω 0.1 mH R0 2000 Ω 25 µF Kvmeas 1/100 4 mH Kvn 1/325 500 µF Ref VDC 4V
Hard and sudden load variations are applied to the system in order to evaluate the performances of the control strategy. The benchmark test is the following: from steady-state operation under no load conditions (R = R0) to sudden maximum load connection (R = Rload), and sudden disconnection. In addition, it is important to notice that capacitor Cf prevents high frequency harmonics from going back to the network. A cut-off frequency Fc chosen above the higher frequency (the 100 Hz frequency of the rectified voltage Vrec) and around one decade below the switching frequency (20 kHz) is suitable. We fix Cf = 25 µF , that means Fc = 500 Hz (1).
3. LINEAR CONTROL The control of this kind of system is usually done by linear controllers. Performances of such controllers will be the reference for a comparison with the fuzzy controller and its initial parameter values. There are two control loops (figure 1): one for the dc output voltage and the other for the rectified input current.
3.1 Current loop The objective of this loop is to get a sinusoidal current in phase with the electrical network. Thus, it reduces the harmonic rejection and maintains a unity power factor. A linear PI controller is used in combination with a PWM module. The high frequency harmonics are then reduced with this kind of control. The shape of the current reference is generated from the network voltage (via Kvn) and its amplitude from the DC voltage (via the voltage controller) as shown on figure 1. The transfer function of the current controller is:
H current ( p) = Gc .
(1 + p.Tic ) p.Tic
(2)
From different considerations like the bandwidth, the dynamic of the current loop and the reduction of the oscillations, the following coefficients are given through trial and error methodology:
Gc = 7 Tic = 0.0005
IAE = ∫ e(t ) .dt
(3)
Figure 2 shows the simulation results for a continuous controller. One can observe the regulation efficiency since the rectified current is closed to its reference and contains few high frequency harmonics.
(8)
This criterion applied to VDC will show the robustness and the dynamic performances of the controller at its output. In addition, the second criterion, the harmonic distortion rate (THD), represents the harmonic rejection quality at the input. 39
∑I THD% =
k =2
2 k
.100
39
∑I k =1
(9)
2 k
The CEI 61000-3-2 standard defines the electromagnetic compatibility and limits the harmonic current emissions for the 39 first harmonics. It gives the maximum allowable current for each of those harmonics. Figure 3 shows how and when the two different criteria are calculated during the benchmark test. The IAE criterion is taken into account throughout the test. The harmonic distortion is only computed during steady-state operation under rated load condition, i.e. rated current.
Fig. 2. Current control: Irec and Iref 3.2 Voltage loop
VDC
This loop must regulate the output voltage VDC with respect to load, input voltage and input current variations. The PI controller tuning is based on the average model of the system. This method relies on the equilibrium of the instantaneous powers between the output of the rectifier and the DC part (Yu, 1996). If the current loop is fast enough compared to the voltage loop, the following approximation could be done:
V DC ( p ) VDC ( p) ≈ I rec ( p ) I ref ( p)
(4)
Load
No load
t Harmonic Distortion for supply current
Then, the transfer function is:
V DC ( p ) Vrec = . I ref 4.V
Rload R .C 1 + p. load out 2
(5)
Fig. 3. Criterion measurement and benchmark test.
where V is the VDC average value. The transfer function of the voltage controller is:
H voltage ( p ) = Gv .
(1 + p.Tiv ) p.Tiv
4. FUZZY CONTROLLER (6)
From different considerations like the bandwidth, the dynamic of the current loop and the reduction of the oscillations, the following coefficient are calculated:
Gv = 7.5 Tiv = 0.062
IAE for output voltage
(7)
3.3 Criteria The control quality for the whole system will be evaluated trough two criteria. The first one is the IAE (integral of absolute error, e).
A fuzzy controller will be used in the rest of this paper in order to improve the dynamic performances. The controller is a classical PI-like fuzzy controller (FLC) (see figure 4). The inherent difficulty of such a kind of controller is the huge number of parameters. This structure was chosen because the error’s second derivative does not have to be calculated. Indeed, its value could be important as it may amplify noise. The fuzzy part consists in two inputs / one output Sugeno FLC (Hissel, 1999) with seven triangular membership functions on each input and seven singletons at the output. There is a normalisation factor for each input (em for the error signal and dem for the error derivative) and for the output (gm).
e
Control action
1 em
gm de
1
Spi Integral action
dem
Fuzzy controller Normalisation
Denormalisation
Fig. 4. Fuzzy controller structure A zero-symmetry is imposed for both membership functions and singletons in order to provide a similar response for positive and negative inputs. A classical antidiagonal rule table, with fixed parameters, is used. By Fixing em to the reference value, only 8 parameters have to be tuned among the initial 73 ones (7*7 rules, 3*7 membership functions and 3 gains): dem, gm, PSe and PVSe (membership functions on error), PSde and PVSde (membership functions on error derivative) and PSs and PVSs (output singletons). Anyway, the tuning problem remains effective as 8 control parameters are to be tuned according to two criteria. Fuzzy logic is only used for the PI controller on the voltage loop. Due to frequency limitation of our DSP, the current loop must be continuous (the sampling period is
Te = 1.10 −4 s ). Moreover, two fuzzy controllers for the same system would dramatically increase the number of parameters that have to be tuned.
and each column is indexed to a factor, an output singleton position or a gain for example. For each experiment, the selected criterion value is calculated through simulation results or measured during experiments. Experiment number 1 2 3 4
Factor 1
Factor 2
+ +
+ +
Interaction 12 + +
Effects
E1
E2
E12
Criterion y1 y2 y3 y4
Figure 5 : Example of an experimental table − y1 + y 2 − y 3 + y 4 E1 = 4
E12 =
(10)
y1 − y 2 − y 3 + y 4
(11) 4 According to the experimental design methodology (Dey, 1999), the effect of a factor is obtained through expression (10). For example, E1 = 0.12 means that factor 1 at high level has an effect of +0.12 on the criterion. Moreover, the effect of interactions between factors 1 and 2 can also be investigated with this methodology. Expression (11) leads to the effect of interaction between factor 1 and 2, on the desired criterion. Furthermore, the interaction E12 between factors 1 and 2 could also be used to study a third factor. From these effects, an optimal tuning could be done. It requires a trial in order to confirm the design. If the obtained results are irrelevant, then the hypotheses must be reconsidered.
5. EXPERIMENTAL DESIGNS PRINCIPLES The History of experimental designs began in the 30’s in England with M. Fisher, but it had an increasing development since Taguchi published predefined tables. This methodology principle is to realize a schedule of the experiments in order to obtain the most accurate information for a specific problem with a minimum number of experiments (dey, 1999). The idea is to modify the level of each factor for each experiment according to a specific procedure. It allows a drastic reduction of the number of required experiments, the possibility of taking into account much more parameters, the detection of interactions between factors and gives an optimized solution. Considering for example only two levels for each of the 8 factors described above, the classical experimental tuning method that consists in varying one of the parameters when all the others are maintained constant leads to 28=256 required experiments. With experimental designs methodology, only 16 experiments are necessary to find the suitable combination of factor levels in order to minimize the selected criterion We use centred reduced variables, i.e. -1 for the low level and +1 for the high level of each factor. Then, an experimental table, as the one shown on figure 5, could be used. Each line represents an experiment
6. TUNING AND RESULTS
6.1 Parameter values In this controller, 8 parameters have to be tuned. The initial levels of parameters are difficult to choose. An accurate expertise on the system is required. The values of the continuous PI controller parameters will be used as initial values. Regarding SPI as the fuzzy controller output, on figure 4, equation 12 can be defined. Kp is the proportional gain and kd is the derivative one of the first part of the fuzzy controller.
S PI = ∫ (k p (e, de).e + k d (e, de).de ).dt 1 e(t ).k1 (e, de).dt em 1 de(t ) + gm.∫ . .k 2 (e, de).dt dem dt gm = . e(t ).k1 (e, de).dt em ∫ gm. de(t ) + . .k 2 (e, de).dt dem ∫ dt
S PI = gm.∫
S PI
(12)
Then, the last expression in equation 12 reveals two different actions: integral and proportional of the complete PI-like-fuzzy controller which can be used for initial tuning. The values are sampled at the sampling period Te:
e(t ) ⇒ e( k ) e(k ) − e( k − 1) de(t ) ⇒ Te
(13)
Then, from the transfer function Hvoltage, it comes:
gm Gv em = T iv gm.Te = G v dem
(14)
The levels of the membership functions are chosen on both sides of the values of the equidistributed membership function positions. Similar choices are made for gm and dem coefficients.
Fig.6 Elementary desirability Accounting for the 39 first harmonics, the composite desirability Dh, is then calculated and defined by: 1
39 39 Dh = ∏ dhi i =1
(18)
The final criterion Y is therefore:
Y = IAE * Dh
6.2 Desirability The desirability notion was introduced by E.C. Harrington (Harrington, 1965). It combines several different properties Yi with different scales and units (Derringer, 1994). Each of them is transformed in an elementary desirability function di. A desirability function is ranged between zero and one. A zero level corresponds to an unacceptable value for the criterion while a desirability of one represents the maximum desired performance. Many different transformations could be chosen. The most classical one was adopted due to its simplicity. It is described below. The value of Yi,p is the minimum acceptable value for Yi and Yi,c is the value above whom an amelioration of Yi is not very interesting. 0 ⇔ Yi ≤ Yi , p
Y − Y ri i i, p di = ⇔ Yi , p < Yi < Yi ,c (15) Y − Y i ,c i , p 1 ⇔ Yi ≥ Yi,c
The parameters ri allow to balance the importance of the increase of the property on the elementary desirability (see figure 6). Then, all the elementary desirabilities are combined into a composite desirability such as D:
]
]
As there are 8 parameters, a 28-4IV experimental table is used, that means only 16 experiments. Two successive designs are carried out. The first one is a “rough” design which gives significant levels for parameters. The second one allows improving the tuning. The experimental design, described in table 3 gives the following optimal tuning (table 2).
PSe PVSE PSde PVSde
(16)
Table 2 Fuzzy parameter values 0.4 NSs 0.1 NVSs 0.6 Gm 0.1 dem
-0.8 -0.5 570 45e-4
6.3 Simulation results Figure 7 and 2 show the simulation results for the fuzzy controller and the analogical PI. The dynamic performances with the fuzzy controller are drastically improved and the harmonic distortion is worth but remains low. 520 500 480
Fuzzy PI
460
1
∑ wi
So, each harmonic value of rank i is transformed into elementary desirabilities dhi ( i ∈ [1,39] ) with Yhi,c = 0 as the objective is to reject harmonic distortion and Yhi,p the CEI 61000-3-2standard limit value. We fix rhi << 1 in order to react only near the standard values. ∀i ∈ 1,39 , rhi = rh = 0.05 << 1 (17)
[
6.3 Tuning
Voltage (V)
[
D = ∏ di wi
(19)
440 420 400 380 360 340 Analogical PI
320 300 0
0.5
1
1.5
2 2.5 Time (s)
3
3.5
Fig.7 Output voltage responses in simulations
4
4.5
Table 3 28-4IV experimental table F1
F2
F3
F4
PSe
PVSe
PSde
PVSde
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + + + +
exp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
F6 = 134 PVSs + + + + + + + +
F5 = 234 PSs + + + + + + + +
F7 = 123 gm + + + + + + + +
F8 =124 dem + + + + + + + +
Experimental results
500
Linear PI
440
Voltage (V)
Table 4 Experimental results Fuzzy Linear controller PI IAE voltage (10-2V.s) 10.9 53.1 TDH (%) 3.8 4.1 Load connection: Tr 5% (ms) 67 193 VDC min (V) 367 335 Load disconnection: 169 1236 Tr 5% (ms) 439 485 VDC max (V) 384 344 VDC min (V)
7. CONCLUSION The experimental design methodology could be an efficient tool for on-line tuning of fuzzy controller according to multi-objective criteria. The controller was first tuned through simulations and some differences with experimentations appear due to some modeling errors. The next step will consist in an experimental tuning, using the experimental response surface methodology.
480 460
modified with a modification of the desirability on the harmonic criterion.
420 400 380 360
Fuzzy PI
340
REFERENCES
320 300 0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
Fig.8 Experimental output voltage responses 10
Current (A)
5
0
-5
Established Settings for PID-like Fuzzy Logic Controllers, EPE'99, 8th European Conference on Power Electronics and Applications, Lausanne.
Irec Inet
-10 2.265
2.27
2.275
2.28
2.285
2.29
Derringer, G., Suich, R. (1994). A Balancing Act : Optimizing a Product’s Properties, Quality Progress, Pages 51-58. Dey A., Mukerjee R. (1999). Fractional Factorial Plans, Wiley, NewYork. Harrington, E., JR (1965). The Desirability Function, Industrial Quality Control, pp 494-498. Henry S.H. Chung, Eugene P.W. Tam, S.Y.R. Hui (1999). Development of a Fuzzy Logic Controller for boost Rectifier with Active Power Factor Correction, PESC 1999, Vol. 1, Pages 149 – 154. Hissel D., Maussion P., Faucher J. (1999). Robust Pre-
2.295
Fig.9 Experimental input currents
6.4 Experimental results Figure 8 and 9 show experimental results for the fuzzy controller and the numerical linear PI. It appears that fuzzy controller gives better dynamic performances with an equivalent harmonic distortion rate. The values of the numerical criteria are given by table 4. These results illustrate the compromise that has to be reach between harmonic quality and dynamic performances. The balance could be
Lian, KY, Liou JJ, Huang CH. (2006). LMI Based Integral Fuzzy Control of DC-DC Converters, IEEE transactions on fuzzy systems, vol. 14, no. 1, February 2006. Mattavelli, P., Buso, S., Spiazzi, G., Tenti, P. (1995). Fuzzy control of power factor preregulators, Industry Applications Conference, 1995, Vol. 3, Pages 2678 – 2685. Pires, V.F., Amaral, T.G., Silva, J.F., Crisostomo, M. (1999). Fuzzy logic control of a single phase AC/DC buck-boost converter, EPE 1999, Lausanne. Yu Qin, Shanshan Du (1996). Comparison of fuzzy logic and digital PI control of single phase power factor pre-regulator for an on-line UPS, IECON 1996, Vol. 3, Pages 1796 – 1801.