Evaluation of a Particle Swarm Optimization controller for DC-DC boost converters J. B. L. Fermeiro, J.A.N. Pombo, M. R. A. Calado, S.J.P.S. Mariano IT – Instituto de Telecomunicações Department of Electromechanical Engineering University of Beira interior Covilhã, Portugal
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[email protected] however the controller need time to train with a training function to adjust the population weights, and depending on the complexity of the controller it can be time-consuming. Optimal control is another approach [8, 9] that is very robust to changes in the system parameters, however in order to achieve the finest response from the controller, certain controller parameters need to be calculated, thus being timeconsuming. The sliding mode control also requires timeconsuming tuning to achieve satisfactory results [10-12].
Abstract—Particle swarm optimization has been used as a tuning algorithm for dc-dc converter controllers. In this work it is proposed a novel approach where the PSO is used as the controller itself for a dc-dc boost converter. The PSO controller uses a population of particles to achieve the necessary compensation to stabilize the output at reference voltage. The algorithm uses the gain formula to reposition the particles near the required position, to stabilize the output at reference voltage, according to the disturbances in the converter. Simulations have been carried on in order to observe the controller’s response to changes in reference voltage and in load variance.
In order to optimize certain controller approaches researchers can use hybrid techniques and particle swarm optimization (PSO). This last optimization algorithm is used to find the optimal parameters for the controllers [13, 14] and enhance its response. After some search we found that the PSO is only used to tune and optimize other controller approach and never as the controller itself.
Keywords—Dc-Dc boost converter; Pulsewidth modulation switching control; Particle Swarm Optimization; Space state model boost converter.
I.
INTRODUCTION
The optimization method PSO is a simple, robust method with great flexibility and rapid convergence. It allows the control of the response timing through a velocity equation where we can give different weights for the social and individual parameters. After the reference voltage is achieved from the controller the output doesn’t oscillate from that value, providing a more pure and clean voltage. With this in mind a PSO controller is proposed and tested. It was tested, comparing with a classic PI controller, in a simulation environment with reference voltage and load variations to observe its response.
Switching dc-dc converters are power electronic converters and therefore extremely important in most electronic devices, from consumer electronic to renewable energy systems [1, 2]. In order to achieve the high efficiency and quality required for the different applications, the control and modeling of the dc-dc converters have suffered extensive research and improvement in the last few decades. The most common control method for dc-dc converters is the pulsewidth modulation (PWM). Classical controllers that take advantage of conventional proportional-integralderivative (PID) approach are sensitive to the system parameter. They work best in systems with single input and output, thus to control systems that have multiple inputs/outputs several PID controllers are required. This strategy might work however is hard to implement because of the correlation between the variables[3]. Also the plant requiring the control might not behave in a linear fashion, meaning that the output for a given input might not exhibit a linear response. The plant behavior changes over time due to changes on plant loads and normal wear and tear, so the controller needs recalibration maintenance over time. Lastly the tuning of PID controllers might not even achieve optimal system performance [4].
This paper is organized as follows: in section II is presented a brief resume of the optimization algorithm PSO where the most common topologies are addressed; in section III is stated a study of the converter and the mathematic model in space state equations of the boost converter for the two stages of the converter operating in continuous mode; in section IV the proposed approach of the PSO controller for dc-dc converter is presented, with its characteristics and system operation; in section V the results of the simulations carried out are reported, with the response of the controller PSO, comparing with a classic PI controller, proposed to three different situations, considering changes in reference voltage and in load.
For all of these reasons researchers are searching for optimizations for dc-dc converters. Several approaches have been published like fuzzy control, optimal control or sliding mode control. Controllers that use neural networks and fuzzy are accurate and robust [5-7] to be used in dc-dc converters
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II.
PARTICLE SWARM OPTIMIZATION
The particle swarm optimization algorithm is inspired in cooperation and social behavior principles. This algorithm
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has particle populations, where each one represents a possible solution. For each particle there is a velocity related to it, which is adjusted with an update equation that consider the historic of individual and collective experiences [15]. The main idea is to change the position of each particle with certain velocities in order to find the best solution. In each iteration the particle and the population performances are evaluated with a fitness function and each particle velocity is incremented or decremented in the direction of their best solution (Pbest), as well as in the direction of the best performance of the global best solution (gbest), Fig. 1.
.
III.
(3)
STATE SPACE MODEL BOOST CONVERTER
The Boost Converter is a non-isolated converter to rise the voltage, and it is capable of providing output voltage higher or equal to the input voltage. In Fig. 2 it is represented the electric circuit and the control of the converter.
pbest ,i xik +1
xik xik -1
gbest
Fig. 1. Graphical representation of the particle evolution.
The most important aspect for the algorithm performance is its topology, the way that the particles communicate between each other. There are many topologies in the literature where the more common are: • Star topology - where every particle communicates with each other. • Ring topology – each particle only communicates with a certain k number of adjacent neighbors. • Cluster topology – certain particles perform the communication between the clusters and inside each cluster the particles communicate with each other. • Von Neumann topology – the particles are connected in a way where the particles in one extremity communicate with particles in the opposite extremity.
Fig. 2. Electric circuit and control of the Boost Converter.
Looking closely at Fig.2, it can be stated two stages in the converter operating, in continuous mode: •
the On the first stage where 0 mosfet is in electric conduction state and the diode is in reverse polarization state, Fig. 3.a.
•
On the second stage where 0 the mosfet is in cut off voltage state and the diode is in direct polarization state, Fig 3.b.
The velocity for each particle can be defined by the equation (1), (1) where , ~
,
(2)
Fig. 3. Boost converter electric conduction stages operating in continuous mode.
The positive constants and are the acceleration and the inertial weight, they should respect the equation (3) so that the particle velocities and the particle positions do not diverge.
The differential equations that characterize the first stage of conduction considering ideal semiconductors (mosfet and diode) are:
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0 0
(4)
(9)
If equations (4) are rewritten in state-space form, considering the state vector , it is obtained:
The mathematical model of the converter can be obtained doing an arithmetic average weighted by the modulation parameter of the converter from the first and the second stages, so:
(5)
(10) Resulting in,
Writing the system (5) in the matrix form it is obtained: …
(6)
…
, the On the second stage, where differential equations that characterize the conduction stage considering that the semiconductors are ideal (mosfet and diode) are:
(11)
The component in permanent regime from the previous system is given from: 0
(12)
0 From the previous equations we can obtain the equation of the realistic converter gain, equation (13). Considering and equal to zero, the gain expression is given by equation (14)
0
1
(13) (7) (14)
0
If this is written in the equation state form where the state vector is it is obtained:
The gain variation of the converter as function of the duty cycle for the system parameters used can be visualized in Fig. 4. It is non-inverting, i. e, there is not polarity inversion, the graphic shows a curve with higher slopes for duty cycle above 0.5 and a more aligned behavior below that mark.
(8)
Another relevant aspect to be accounted for is that the converter should not be designed with a nominal duty cycle close to the maximum gain, because if there is the need to increase the output voltage, by increasing the duty cycle, the controller will be operating in the descending part of the curve, lowering the output voltage instead of increasing it, and this will lead to a converter failure.
Writing the previous system in the matrix form it is obtained:
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Fig. 6. Graphical representation of the particle positioning and repositioning – A when submitted to a reference voltage variation and B when submitted to load variation.
After the particles positioning it evaluate the performance of each particle and determines the best Particle and global performances. In the third stage it updates the position of each particle with the equation (1), Fig. 7. Fig. 4. Gain variation as function of the duty cycle with Rload=1000Ω, C=200e-6F, RC=0.05 Ω, L=1.6e-3 H, RL=0.5Ω.
IV.
PROPOSED APPROACH
The controller proposed provides the output voltage or feedback voltage through mathematical formulation of the determined duty cycle for each k moment, for a more realistic response of the controller, Fig. 5. The algorithm has 3 particles in a star topology, which means that all the particles communicate with each other. The algorithm in a first stage positions the particles (converter duty cycle) as shown in Fig.6.
Fig. 5. The general schematic of the controller system.
In case of the reference voltage change the controller repositions the particles in the following manner, Fig 6. A, if Vref(k) is different from Vref(k-1), the central particle is positioned according equation (14) and the other two particles are positioned in a configurable predetermined range, originating a small convergence time and little overshoot. In case of the load change the controller repositions the particles in the following manner, Fig 6. B, if the difference between Vrefk) and Vfbk(k) is higher than a configurable predetermined value, and the particles velocities are lower than another configurable predetermined value, the particles are repositioned. This last is required because the particles converge to the reference voltage and their velocities tend to zero, at this point the controller will not respond to this kind of perturbations without repositioning the particle positions. In this situation the central particle is positioned again according to equation (14) and the other two particles are positioned in a range determined with the difference between the reference voltage and the feedback voltage.
Fig. 7. Fluxogram of the controller algorithm.
In Fig. 8 we can observe with great detail the particle behavior when the controller is subjected to reference voltage variations. In the first iteration the particles are positioned as mentioned before, this happens within four sampling times (with Ts = 1/25kHz), one for each particle and the last one for the velocities calculation and new position determination, with equation (1).
Fig. 8. Representation of the algorithm particle positioning.
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V. TEST RESULTS The proposed controller was compared to a classic PI controller. The PI parameters used were Kp= 0.6 and Ki= 2.5. Comparing both controllers the PSO controller revealed a reduced convergence time and lower overshoot. However the biggest advantage of this approach is that after the system converge it stabilizes at reference voltage with no oscillations. At first the controller was subjected to respond to a single reference voltage given of 100 V, Fig. 9 and 10. In this case the convergence time for both controllers was very little but the PSO controller revealed lower overshoot and no oscillation after reaching the reference voltage.
Fig. 11. Response, in output voltage, of the PSO controller and classic PI controller to two changes in reference voltage.
Fig. 9. Response, in output voltage, of the PSO controller and classic PI controller to a single reference voltage.
Fig. 12. Response, in duty cycle, of the PSO controller and classic PI controller to two changes in reference voltage.
In the last trial the controller was submitted to changes in the reference voltage, starting at 100 V from 0 s to 0.015 s and finishing with 120 V from 0.015 s to 0.03 s, and a load increase by a factor of 5 at 0.0075 s and then decreased back to the initial load at 0.0225 s, Fig. 13 and 14. To ensure the correct convergence of the particles at this moment, the parameters used for the range of the particles positioning might not be ideal and with the improvement of the algorithm the controller will obtain better results. This influenced the performance of the PSO controller when comparing with the classic PI controller at the moment when the load is changed, showing a little more overshoot and convergence time. However and as said above this behavior can be improved with the PSO parameters optimization, also it is important to mention that the classic PI controller responded worst when the reference voltage was changed with the new load, with higher convergence time and increased overshoot for the same PI parameters.
Fig. 10. Response, in duty cycle, of the PSO controller and classic PI controller to a single reference voltage.
In a second test the controller was submitted to changes in reference voltage starting with 100 V from 0 s to 0.01 s, then lowering to 60 V from 0.01 s to 0.02 s and finishing with 120 V from 0.02 s to 0.03 s, Fig. 11 and 12. Again the PSO controller exhibited a better response than the classic PI controller, with less overshoot, oscillation and settling time.
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this use. In a future work we expect to optimize the PSO controller and test it in a real circuit to obtain the response and performance of this controller in a real environment. REFERENCES [1]
[2]
[3] [4] [5]
Fig. 13. Response, in output voltage, of the PSO controller and classic PI controller to a changes in reference voltage and in load.
As we can see in Fig. 14 the controllers adjusted the duty cycle to compensate the current consumption, increasing it when load increases and lowering when load decreases, in order to provide the required output voltage.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Fig. 14. Response, in duty cycle, of the PSO controller and classic PI controller to changes in reference voltage and in load.
VI.
CONCLUSION [13]
The particle swarm optimization is a great algorithm to rapidly and accurately find the best parameters for common controllers of dc-dc converters, and it is been used only with that purpose. However it can be, as shown in this work, used itself as a controller for dc-dc boost converter. Simulations have been carried with component parameters in order to obtain the most reliable response from the controller. The proposed approach is a PSO controller with a population of particles in a star topology, where they communicate with each other. These are positioned and repositioned when the controller detects reference voltage and load disturbances. Overall the results of this approach are very promising, when compared to a classic PI controller, the controller response is accurate, with rapid convergence time, little overshoot and no oscillation after convergence, proving its robustness for
[14] [15] .
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