Title : Feasible Performance Evaluations of Digitally-Controlled Auxiliary Resonant Commutation Snubber-Assisted Three Phase Vo



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Optimum Setting and Performance Evaluation of the Three Mode Controller Using Bio Inspired Soft Computing Algorithms

D. M. Mary Synthia Regis Prabha1 G. Glan Devadhas2 S. Pushpa Kumar3



1Associate Professor, EEE Department, Noorul Islam Centre for Higher Education, Kumaracoil, Tamilnadu, India, Pin 629180, email: regisprabha@gmail.com, mobile no: +91-9940877270 (Corresponding Author)

2Associate Professor, EIE Department, Noorul Islam Centre for Higher Education, Kumaracoil, India, glandeva@gmail.com

3Principal,Muthoot Institute of Technology and Science,Varikoli.P.O,Ernakulam, kerala,Puthencruz-682308, spushpakumar@gmail.com



Abstract: The best possible and desirable solution of the engineering problems can be found using optimization algorithms. The most widely used dc-dc converters are highly non-linear dynamical systems. Designing a control technique has been a most promising control area. One of the robust non-linear sliding mode controllers has been proved by researchers to be having large signal stability and providing good dynamic response. But the complicated design of the controller as well as the problems encountered during practical implementation gives rise to the necessity to design a controller which gives better performance than the former case and simple in structure and design. The traditional linear three mode PID (Proportional Integral Derivative) controllers which are the most common control algorithm in industries can be properly tuned so that they overcome the performance of the former case. The Bacterial Foraging Optimization Algorithm and Artificial Bee Colony Algorithm have been used for tuning purpose. The best optimization algorithm has been identified and the effectiveness of the PID controller tuned using the best optimization algorithm over the sliding mode controller has been proved.


Key words: PID controller, Bacterial Foraging Optimization Algorithm (BFOA), Artificial Bee Colony Algorithm (ABCA), DC-DC Buck Converter, PID Controller, Sliding Mode Controller



1 Introduction

The most commonly encountered mathematical problem in almost all the engineering disciplines is the optimization problem. The best possible and desirable solution of the engineering problems can be found using optimization algorithms. There are a wide range of optimization problems which are yet to be solved which makes this to be an active research area. Naturally, optimization algorithms are seen to be either deterministic or stochastic. Enormous computational efforts are to be put forward to solve optimization problems using the deterministic methods adopted. These methods tend to fail as the problem size increases. The nature inspired optimization algorithms are identified to be computationally efficient alternatives to deterministic methods. The sole inspiration of these algorithms is that derived from nature which reveals their real beauty. These algorithms require little or even no knowledge of the search space which starts from very simple initial conditions and rules for describing and resolving complex problems. Proper representation of the problem, choosing a correct fitness function for evaluating the quality of problem and creating new solution sets by suitably defining operators are the main steps involved in the design of a bio-inspired algorithm.

Switching mode dc-dc converters are widely used today in a variety of applications including power supplies for personal computers, mission critical space applications, laptop computers, dc motor drives, medical electronics as well as high power transmission [1]. The dc-dc converters have high power packing density and high efficiency. So, they are widely used in dc regulation problems. These converters are non-linear dynamical systems. The non-linearities arise primarily due to switching, power switching devices and passive components such as inductors and capacitors. A most promising control area is to design a control technique which is suitable for dc-dc converters to cope up with their intrinsic non-linearities and wide input voltage and load variations ensuring stability in any operating condition while providing fast transient response [2].

PID control which is one of the best known industrial processes is a traditional linear control. It can be easily implemented by engineers using current technologies which make it more popular than other controllers. Linear PID controllers for dc-dc converters are usually designed by classical frequency response techniques applied to the small signal models of converters [3]. Various approaches are available in the literature to determine the PID controller parameters which include Ziegler-Nichols method [4], Cohen-Coon method [5], Internal Model Control based method [6] and Gain-Phase Margin method [7]. In Gain-Phase Margin method the bode-plot of the converter is adjusted in the design to obtain the desired loop gain, cross-over frequency and phase margin.

The Sliding Mode Control (SMC) is a type of non-linear control. The SMC has been well known for its large signal stability, robustness and good dynamic response. Moreover the design of an SMC does not require an accurate model of the system. But Sliding Mode Controller even though proved to be more robust in the literatures [8-13] suffers from the drawback of its complicated design procedure. So, it becomes necessary to design a controller which is robust, stable as well as simple in design. So, necessity arises to upgrade the PID controller, which plays an eminent role in industries, to compete with that of the PID based Sliding Mode Controller. It is a well known fact that the PID controller’s performance is decided by the proper selection of the control parameters. So, in order to improve the performance of the controller, and making it superior than that of the sliding mode controller, it becomes necessary to tune the parameters of the PID controller using an algorithm which is efficient in solving hard and complex optimization problems with much less computational time.

The various steps involved for finding the optimal controller parameters is called tuning. A best optimization algorithm is one which takes care of the local minima, requires less number of computations and settles in minimum time. The social behaviour of a group of living organisms has attracted the interest of various researchers. In the recent decades, people have developed many population-based optimization algorithms which come under the bio-inspired algorithms for solving complicated engineering problems, imitating the collective foraging behaviour and life system of a group of social insects like ants, bees, termites and wasps or other animal societies. These algorithms are becoming more attractive owing to their immense parallelism, simple computation and its ability in finding near-optimal solutions to difficult optimization problems. Some of the popular population-based algorithms are the Bacterial Foraging Optimization Algorithm (BFOA) and the Artificial Bee Colony Algorithm (ABCA). The ABCA simulates the behaviour of real bees while finding food sources. BFOA is developed inspired by the social foraging behaviour of Escherichia coli. These optimization algorithms which are finding increasing applications in engineering field are chosen in this paper for optimizing the PID controller parameters because of their capability in handling complex optimization problems, thus rejecting the local minima and finding out the global minima in a wide search space.

The basic versions of BFOA and ABCA are utilized in this paper for optimizing the PID controller parameters of a dc-dc Buck converter.

In order to make the controllers produce better results, various optimization techniques have been proposed for properly tuning the controller parameters. In 2012, Rajinikanth and Latha [14] proposed an enhanced bacteria foraging optimization algorithm based PID controller tuning for a non-linear chemical process. In 2011, OzdenErcin et.al. [15] studied the performance of the Artificial Bee Colony and the Bees algorithms for proportional-integral-derivative (PID) controller tuning. In 2010, Altinozet. al. [16] applied PSO for achieving improved performance of PID controllers on a Buck converter. In 2010, H. Vahedi [17] compared the BFO algorithm with the Particle Swarm Optimization (PSO) technique for optimizing the parameters of a PID controller. In 2010, Jalilvand [18] illustrated the technique of optimally tuning a PID controlled dc-dc converter using BFOA. In 2010, Abachizadehet. al. [19] studied the performance of Artificial Bee Colony Algorithm optimized PID controller designed for benchmark plants of different orders and time delays and tuned it using Artificial Bee Colony Algorithm.

In the literature provided, it can be observed that optimization techniques are applied for tuning of a PID controller designed for various systems. In [18], a dc-dc converter system has been considered for analysis, but it does not provide much information regarding the robustness of the system. This paper gives a detailed procedure for tuning a dc-dc Buck converter using BFOA and ABCA also proves the robustness of the system during line and load disturbances.

In this paper, the problem formulation has been explained in section 2. A PID controller has been designed for a dc-dc Buck converter operating in Continuous Conduction Mode which is explained in section 3. Section 4 explains the procedure for optimizing the PID controller parameters using BFOA technique and a detailed performance analysis is made which is also presented in the same section. Section 5 explains the optimization of the controller parameters using ABCA and its performance analysis is also made. Section 6 gives the experimental results. Section 7 presents the results and discussions on the results obtained. It is revealed that the ABCA tuned Buck converter produces better results than that of the BFOA optimized Buck converter. The former case is also compared with Sliding Mode Controlled (SMC) Buck Converter which proves its superiority over the SMC. An experimental study is also made on the ABCA tuned PID controlled Buck Converter which proves its oneness with the simulation results.



  1. Problem Formulation

    1. Control Principle

The PID controller has several parameters that can be adjusted to make the control loop perform better. The controller’s performance deteriorates if the control parameters are not chosen properly. The procedure for finding the controller parameters is called tuning [20].

The PID algorithm can be described as





Where u is the control variable and is the control error. The control variable is thus a sum of three terms: the P term (which is proportional to error), the I-term (which is proportional to the integral of the error), and the D-term (which is proportional to the derivative of the error). While the integral action takes care of the steady state error under steady state condition making it zero, the derivative action improves the closed-loop stability. Proper selection of the controller parameters and makes the controller perform optimally.

This paper is aimed at optimizing the PID controller parameters kp, ki and kd. Some of the figures of merit which are used for evaluating the performance of the control system are Integral of Square Error (ISE), Integral of Absolute Error (IAE) and Integral of Time-Weighted Absolute Error (ITAE). The Integral of the Square of the Error with respect to time (ISE) is often used to evaluate the response of the control system which is defined as,

where is the ‘reference voltage’ and is the ‘actual output voltage’.

For a perfectly designed system, with zero offset, this ISE is a single real value. Proper selection of the parameter to produce a minimum ISE value is the main objective in this optimization design problem. Thus the objective of this optimization function can be expressed as

The ISE value will be more for responses having large errors which persist for a long time. It provides a compromise between the reduction of rise time, thus limiting the effect of a large initial error, reduction of the peak overshoot and settling time.

The other criteria used in the control process are as follows.

Integral of Absolute Value of Error



Integral of Time-Weighted Absolute Error



ISE strongly suppresses large errors and IAE suppresses small errors more effectively. ITAE works more effectively on errors that persist for a long time. So ISE is used for penalizing the response that has large errors, which is usually the case, in the dynamic portion of the response and ITAE is used for penalizing a response which occurs during the steady state condition. The figure of merit, namely the ISE is often used in optimal control theory because it can be used more easily in mathematical operations than the errors which use the absolute value of error.



    1. Specification Of Buck Converter

The range of input voltage is selected to be from 13V to 30V. The desired output voltage is 12V. The range of load current is selected to be from 0.5A to 4A. The variation of load resistance is from 3Ω to 24Ω. The acceptable output voltage ripple is chosen to be 50mV. The peak-to-peak ripple current of the inductor is normally 25% of the maximum load current which is calculated to be 1A. The time at which the change in the inductor current occurs is given by. The peak-to-peak ripple voltage of the capacitor is The ESR value of an electrolytic capacitor used is chosen as 0.03Ω as specified by the manufacturer and the ESR value of the inductor is chosen as 0.12Ω. Hence, the capacitance value is calculated to be 125μF. The capacitor value is selected as 150μF which is normally chosen to be greater than the calculated value. The critical value for inductance is calculated using the formula

(7)

3. Design of a PID Controller in Continuous Conduction Mode

The open loop transfer function G(s) H(s) is given as,



The prototype Buck converter is made to operate with the given nominal operating point. Input Voltage, Vin =24V, Desired Output Voltage, Vod = 12 V, Duty ratio D = 0.5. The switch is made to operate at a frequency of 200 kHz. The converter is designed for a variation in load resistance ranging from 3Ω to 24Ω. For a load resistance of Ro=6 Ω, Filter Inductance, L =35 μH, which is greater than, the value required to make it to operate in CCM, Filter Capacitance, C =150 μF, ESR of Inductor, RL=0.12 Ω and ESR of Capacitor, RC=0.03 Ω, the transfer function becomes,



(9)

This model has poles at 135.355x103rad/sec (21.5424 kHz) and 1.7417x103rad/sec (277.2Hz) and a zero at 22.222x103rad/sec (3.5367 kHz). The open loop response of the Buck converter is shown in Figure 1.



Fig. 1 Open loop response of a Buck converter

The system response has a peak overshoot of 20.682V and settles at 1.756x10-3s.

A PID compensator is designed such that system operates with a desired phase margin at a desired cross-over frequency with a steady state error for unit ramp input of 0.035%. The switching frequency of the Buck converter is chosen to be 200 kHz. The cross-over frequency of the PID compensator should be chosen such that . Here and. Substituting the values, x103 << 35.3678x103 < 100x103. So is chosen as 25 kHz and 157.08x103rad/s. A PID controller whose transfer function is is introduced to get the desired phase margin at the desired frequency. For the controller to meet the requirements,





So

Also

Representing eq.(3.11) in triangle form,



So, (15)

And


(16)

is decided from the steady state requirements and is found that . Substituting this value of in eq. (12) and (16), the values of and are found to be 5.80125, and 0.0869x10-4 respectively.

3.1Simulation and Performance Analysis

The PID controlled Buck Converter is simulated in the MatLab environment with and. The output response of this Buck converter with an input voltage of 24V is shown in Figure 2. It is clear that the converter produces a large peak overshoot which is 54.375% of its final value and settles at 4.635x10-4s.


Fig. 2 Output response of PID controlled Buck Converter



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Fig.3 Dynamic response for a step input change from 24V to 18V at 0.5ms

The converter is subjected to a dynamic line disturbance from 24 V to 18V which is shown in Figure 3 and from 18V to 24V which is shown in Figure 4. When the input voltage is changed from 24V to 18V at 0.5ms, the system takes 0.04ms to settle at the steady state value.

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Fig. 4 Dynamic response for a step input change from 18V to 24V at 0.5ms



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Fig. 5 Dynamic response for a load step change from 6Ω to 25Ω

The converter is then subjected to load step variation from 6Ω to 25Ω and the dynamic response is shown in Figure 5. It can be noticed that the response undergoes a small overshoot and settles within 0.6x10-4s.

Figure 6 shows the dynamic response of the converter for a load step change from 25Ω to 6Ω. It can be noticed that the converter settles within 5.8x10-5s during that disturbance.

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Fig. 6 Dynamic response for a load step change from 25Ω to 6Ω



4. Bacterial Foraging Optimization Algorithm (BFOA)

The BFOA [21-23] is based on search and optimal foraging decision making capabilities of the E.Coli bacteria. A set of trial solutions is performed making it to converge towards the optimal solution following the foraging group dynamics of the bacterial population which involves four basic processes namely swim, tumble, reproduction and elimination-dispersion [24]. The computational chemotactic process is the main driving force of the bacterial foraging phenomenon. This chemotactic movement is continued until a bacterium goes in the direction of a positive nutrient gradient (i.e. increasing fitness). After a certain number of complete swims, the best half of the population undergoes reproduction, eliminating the rest of the population. In order to escape the local optima, an elimination-dispersion event is carried out, where some bacteria are liquidated at random with a very small probability and new replacements are initialized at random locations of the search space.

The BFO algorithm basically has three steps, the first being the execution of the inner chemotactic loop. The second step is the execution of the loop outside the chemotactic loop, which is the reproduction loop. The last step is the execution of the outermost loop which is the elimination-dispersion loop. These three steps are described below in detail.

Step 1: Chemotactic Loop

In this process, the movement of an E.Coli bacterium is simulated. It moves in two different ways: tumbling and swimming [23, 24]. It can move in the same direction for a period of time or it can randomly change its direction of movement by tumbling. It alternates between these two modes throughout it life time. Each time it performs an operation, the objective function value is calculated. The bacterium changes its position only if the modified objective function value is less than the previous one. On completion of chemotaxis, the bacterium will be swarming around a point in search space with the least objective function value. The movement of the bacterium may be represented in computational chemotaxis [24] by

Where is a random vector whose elements lie within the range [-1, 1], represents the th bacterium at th chemotactic, th reproductive and th elimination-dispersal step, is the initial random location of each bacterium and is the step size taken during tumbling taken in the random direction.

Step 2: Reproduction Loop

In this reproduction process, the objective function value of each and every bacterium is calculated and sorted out in ascending order. The worst half of the bacterial population with the highest value dies out and the other half of the population with the lowest value splits out and takes the same position of their parents [23, 24]. For this new generation of bacterium, the inner chemotactic loop is started and this process continues for a specific number of reproduction steps.

Step 3: Elimination-Dispersion Loop

This loop simulates the gradual or sudden unexpected changes in the environment which either kills the bacterial population or makes them dispersed to a new location. Hence, in this process, some bacterium with very small probability is eliminated and dispersed to a random location on the optimization domain. This process is executed in order to keep the number of bacterium constant [23, 24].

The optimization process of BFOA algorithm is shown in figure 7.


Fig.7 Optimization Process of BFO algorithm




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