An Optimized PI Controller Design for Three Phase PFC Converters Based on Multi-Objective Chaotic Particle Swarm Optimization

Abstract

The compound active clamp zero voltage soft switching (CACZVS) three-phase power factor correction (PFC) converter has many advantages, such as high efficiency, high power factor, bi-directional energy flow, and soft switching of all the switches. Triple closed-loop PI controllers are used for the three-phase power factor correction converter. The control objectives of the converter include a fast transient response, high accuracy, and unity power factor. There are six parameters of the controllers that need to be tuned in order to obtain multi-objective optimization. However, six of the parameters are mutually dependent for the objectives. This is beyond the scope of the traditional experience based PI parameters tuning method. In this paper, an improved chaotic particle swarm optimization (CPSO) method has been proposed to optimize the controller parameters. In the proposed method, multi-dimensional chaotic sequences generated by spatiotemporal chaos map are used as initial particles to get a better initial distribution and to avoid local minimums. Pareto optimal solutions are also used to avoid the weight selection difficulty of the multi-objectives. Simulation and experiment results show the effectiveness and superiority of the proposed method.

I. INTRODUCTION

As a type of improved three phase Pulse-Width Modulation (PWM) converter, the compound active clamp zero voltage soft switching (CACZVS) three-phase power factor correction (PFC) converter [1] has wide application prospects, because it can realize zero voltage soft switching for all of the switches (including auxiliary switch) and suppress the diode reverse recovery current to reduce power losses.

Traditionally, a three phase PWM converter employs a Proportional Integral (PI) controller based on the synchronous rotating coordinate frame [2]. Using this control configuration, there are three PI controllers including the outer loop output DC voltage controller, the inner loop active current controller and the reactive current controller. Therefore, there are six control parameters that need to be determined to get good performance including a unity power factor, fast transient response and zero steady state error. These parameters are mutual influenced to obtain good performance. References [3] and [4] introduced some PI parameter calculation methods. Reference [5] also proposed an improved parameters tuning algorithm. However, the PI parameters obtained by these methods can only be used as a starting point for parameter tuning. It is necessary to further tune the PI parameters depending on designer experience to get satisfactory performance. In addition to the conventional PI control, many control methods have been proposed recently to improve the performance of three-phase PWM converters [6]-[10]. However, the difficulty of controller parameters tuning limits, to a certain extent, the performance of these control algorithms in practical applications. To avoid the above difficulty, fuzzy logic [11], genetic algorithm (GA) [12], and particle swarm optimization (PSO) theory [13], [14] have been applied to controller design and parameters optimization. However, the control parameter optimization of three phase PWM converters are a multi-dimensional (six-parameters) and multi-objective (including the unity power factor, fast transient response and zero steady state error) optimization problem, which is still a challenging task.

Generally speaking, there are two main problems restricting advanced optimization algorithms from getting good results. Firstly, to deal with a multi-objective condition, the traditional method gives different weights to different objectives. However, weights selection is very difficult. Secondly, the initial individual distribution has an important impact on the performance of optimization algorithms. In the case of the PSO method, a non-uniform distribution of the initial particles decreases the global convergence performance. To deal with the first problem, reference [14] proposed using the PSO method and the Pareto optimal solution theory to overcome the difficulty of weights selection. For the second problem, because the uniform distribution of the initial particles in the solution space accelerates the convergence speed and reduces the possibility of falling into a local optimum [15], references [16], [17] use chaotic maps to produce the initial particles, where the ergodic characteristic of the chaotic maps was used to get uniform distribution. However, the traditional chaotic map, such as the logistic map [18] and the self-logical map [19], can only achieve a one-dimensional uniform distribution of the particles. With an increasing number of parameters, the uniform distribution of the initial particles in the multi-dimensional solution space is required.

In this paper, an improved CPSO method is proposed for the optimization of the triple closed-loop PI controller parameters for CACZVS three-phase PFC converters. In the proposed method, a spatiotemporal chaos model, referred to as unilateral coupled map lattices, is used to produce multi-dimensional initial particles. The Pareto optimal solution theory is used to avoid the difficulty of weight selection and to achieve balance among the different control objectives such as a unity power factor, fast transient response and zero steady state error. When compared to the conventional random initialization method and the traditional logistic map initialization method, the proposed particles initialization method improves the global searching ability and reduces the time consumption. When compared to the weighted multi-objective optimization method, the proposed method avoids the difficulty of weight selection, and gets automatic balance among the different objectives. Simulation and experiment results verify the effectiveness and superiority of the proposed method.

This paper is organized as follows: Section II gives a brief analysis of the working principle of a CACZVS three-phase PFC converter and its mathematical model; Section III gives the PI controller design for the current loop and voltage loop in detail; Section IV gives the proposed multi-parameter multi-objectives CPSO algorithm; Section V gives a comparison of the simulation and prototype experimental results to show the effectiveness and superiority of the proposed method; Section VI gives some conclusions.

 

II. THREE PHASE PFC CONVERTER AND ITS MATHEMATICAL MODEL

The CACZVS three-phase PFC rectifier circuit topology is shown in Fig. 1, where Ua, Ub, and Uc denote the three-phase input voltages; ia, ib, and ic denote the three-phase input currents; the three-phase AC side filter inductor La=Lb=Lc=L; and R denotes the equivalent resistance of the filter inductor and the switches. The output DC voltage is Udc. The load resistance RL and the DC-link capacitor C are connected to the DC side of the converter, where iL=Udc/RL denotes the load current. The resonant inductor Lr is resonant with the switches parallel capacitors C1-C7 to create the condition of soft switching and to suppress the diode reverse recovery current. Cc denotes clamping capacitor, which forms the clamp branch together with switch S7 and resonant inductor Lr to reduce the voltage stress across the switches.

Fig. 1.CACZVS three-phase PFC rectifier.

The switching function of the bridge leg is defined as Si (i=a, b, c), where Sa=1 means that S1 is on, and S2 is off. Meanwhile, Sa=0 means that S1 is off, and S2 is on.

Based on the working principle of a CACZVS three phase PFC converter [1], it is known that during most of the operation time, the auxiliary switch S7 is conducting. S7 is turned off in a very short time to create the zero voltage switching condition for the main switches. This does not affect the main circuit during the rest time. Therefore, the state space equation of the CACZVS three phase PFC converter in the three phase stationary coordinate system can be written as [1]:

where UCc denotes the voltage crossing clamping capacitor Cc. By transforming Eq. (1) into the two-phase rotating coordinate system, the following is obtained:

where w=2πf denotes the angular frequency of the input sinusoidal voltage, Ud, Uq denote the active and reactive voltage components in the dq coordinate system, respectively, id, iq denote the active and reactive current components in the dq coordinate system, respectively, and Sd, Sq denote the switching functions in the dq coordinate system, respectively. Because Cc<

Based on Eq. (3), the PI controllers are designed in the next section.

 

III. TRIPLE CLOSE-LOOP PI CONTROLLERS DESIGN

A. Current Loop Controller Design

From Eq. (3), the current loop equation is given as:

where Urd=SdUdc and Urq=SqUdc denote the d-axis (active) and q-axis (reactive) current manipulating variables, respectively. The d-axis and q-axis current dynamics are nonlinear and strongly coupling. Therefore, traditional current feedback control based on the linear system theory cannot deal with such case. Here, it is possible to use the feed-forward decoupling PI control strategy for the current loop [3]. The controller equation is:

where idref is the d-axis current reference, and iqref is the q-axis current reference, respectively, and eid=id-idref, eiq= iq-iqref denote the corresponding current errors. kidp, kidi, kiqp, kiqi are PI parameters. By substituting (5) into (4), it is possible to obtain:

The stability analysis shows that, as long as kidp>0, kidi>0, kiqp>0, kiqi>0, the feed-forward decoupling PI control strategy can make id track idref, and iq track iqref. In order to achieve a unity power factor, the reactive current iq must be zero. Therefore, iqref=0. idref is determined by the voltage loop controller. The function of the current loop controller is to make the input current sinusoidal and synchronous with the input voltage. At the same time, it should also make the active power of the converter have a quick response to load variations.

B. Voltage Loop Controller Design

From Eq. (3), the voltage loop equation is given as:

The voltage loop controller is:

where Udcref is the voltage reference, and kvp, kvi are the proportional and integral gains of the voltage PI controller, respectively.

To sum up, a block diagram of the triple closed loop PI controllers is shown in Fig. 2. Six control parameters, including kidp, kidi, kiqp, kiqi, kvp, kvi need to be tuned to obtain the desired performance. Moreover, the six parameters are mutually influenced. Therefore, they should be tuned coordinately to get better performance.

Fig. 2.Schematic of triple closed loop PI controller.

The traditional control parameter tuning method [3] simplified the synchronous reference-frame current control plant into a first-order time lag block. Then the linear control theory was used to design the control parameters. A block diagram of the traditional current control loop design is shown in Fig. 3, where Udis denotes the voltage disturbance, and the PWM rectifier is treated as a first-order subsystem given by KVSR/(0.5Tss+1), where KVSR denotes the equivalent gain of the rectifier, and Ts denotes the sampling period. The sampling process is modeled as a first-order subsystem given by 1/(Tss+1). The AC block is modeled as a integral subsystem given by 1/Ls , where the equivalent resistance R is omitted.

Fig. 3.Block diagram of the traditional PI current controller design [3].

The traditional parameter tuning method, based on a lot of hypothesis and simplification, the voltage loop control parameters and the current loop control parameters are designed separately. Then the PI parameters need to be adjusted in a large range according to the designer experience. PI parameters obtained by the traditional method, generally speaking, make it impossible to achieve optimal performance.

 

IV. TRIPLE CLOSE LOOP PI CONTROLLER PARAMETER OPTIMIZATION BASED ON CPSO

A. Basic PSO

PSO is an optimization tool based on the principle of bird food searching, where each bird is treated as a particle, and each particle is a potential N-dimensional solution of the problem under consideration. There are M particles forming a population, the particles in the population coexist and cooperate. Each particle, with its velocity determined by the experience of itself and the "best experience" of the adjacent particles, flies to a "better" position in the solution space. In this way, the optimal solution is eventually found. The main parameters used in the PSO are shown in Table I.

TABLE IPSO PARAMETERS DESCRIPTION

The searching algorithm of particle m is given as follows:

where Xmn(k), Vmn(k) denote the nth coordinate position value and the nth coordinate speed value of particle m at the kth iteration, respectively. In the algorithm, Vmn is limited to within ±Vmax, which is set as 20% of the searching range of the particle. In this paper, a particle presents the 6-dimensional PIs parameters of the triple closed loop PI controllers.

B. Initial Population Generation Using the Unilateral Coupled Map Lattice

Based on the result in [21], the parameters w, c1 and c2 affect the convergence of the classical PSO algorithm. However, c1 and c2 are constant parameters, while the linearly or exponentially variant w can not ensure the ergodicity of the solution. In fact, the initial distribution of the particles has an important effect on the performance of the PSO. Non-uniform distribution of the initial particles might cause instability of the algorithm. In order to obtain a better initial distribution of the particles, chaotic logistics and a self-logical map have been proposed to generate the initial particles [16]-[19]. However, the most commonly used chaotic maps can only generate a one-dimensional uniform distribution of particles. For the application of a multi-dimensional optimization problem, such as the multiple control parameter optimization problem in this paper, the ideal situation is that the initial population (N-dimensional M-particles) is uniformly distributed in the entire solution space. However, the existing methods can hardly fulfill this requirement.

In this paper, a unilateral coupled lattices spatiotemporal map is proposed to generate the initial particles. The unilateral coupled map lattices model is given as:

where f [Ln(k)] is the Logistic map, which is given by:

where Ln(m) denotes the state variables, n denotes the space position (corresponding to the dimension of the particle), m denotes the discrete-time (corresponding to the population of the particle), μ is a constant, and εn denotes the coupling strength. Here, Xmn=Ln(m). As a result, it is possible to use Eqs. (11) and (12) to generate the initial population of the proposed PSO.

The distribution chart of particles generated by the unilateral coupled map lattices is shown in Fig. 4.

Fig. 4.Initial population generated by unilateral coupled map lattices.

Fig. 5 is a distribution comparison between the traditional logistic map and the proposed unilateral coupled map lattices spatiotemporal map. Figures 5 (a) and (b) are the box-plot curves of the proposed method and the logistic map, respectively, where m is chosen as different values to see the degree of the uniform distribution, and the box length indicates the uniform distribution degree, i.e., the longer the box is, the better the uniform distribution is. From Figs. 5 (a) and (b), it can be seen that, for every value of m, the length of the corresponding box in Fig. 5(a) is larger than that in Fig. 5(b). This indicates the better performance of the proposed map when compared to the traditional logistic map. Figure 5(c) shows a comparison of the N-dimension elements between the proposed chaotic map and the traditional logistic map when m=30. From Fig. 5 (c), it can be seen that when compared to the traditional logistic map, the proposed method has tighter boundary conditions in the sense of no boundary point appearing in the “red star points” in the left subplot of Fig. 5 (c) (by using the logistic map).

Fig. 5.Distribution comparison between the traditional logistic map and the proposed unilateral coupled lattices spatiotemporal map.

In summary, by using the unilateral coupled lattices spatiotemporal chaotic map, the proposed CPSO algorithm in this paper can achieve a better distribution for the initial particles, which helps accelerate the convergence speed and avoid falling into a local optimum.

C. Objective Functions and Pareto Optimum Based Best Solution Selection

The objectives of the optimization problem in this paper are multi-fold, because the ideal performance of triple closed loop PI controllers needs to fulfill the following requirements: (1) a small raising time and a small overshoot; (2) zero steady state error; and (3) a unity power factor. The objectives of the voltage loop controller include two parts: firstly, the output DC voltage should approach the reference as soon as possible, and the corresponding overshoot should be as small as possible, i.e., the shadow area in Fig. 6 should be as small as possible. This objective is denoted as Jv1 given by Eq. (13). Secondly, the steady-state error of the output voltage should be 0. This objective is denoted as Jv2 given by Eq. (14). The objective of the current loop is to achieve a unity power factor. This is denoted as Ji given by Eq. (15).

Fig. 6.Calculation of objective functions.

where ei=Udcref(iT)-Udc(iT) and ej=PF(jT)-1 denote the voltage error and the power factor error, respectively, PF(jT) denotes the power factor at the jth sampling time, T is the sampling interval, and ε denotes the boundary of the steady-state error, which is used to distinguish the transient and steady state. L1 represents the data length considered for the output DC voltage in the transient state. L2 represents the data length considered for the steady state of the output DC voltage, and L3 is the data length considered for the power factor PF.

Clearly, for the optimization problem in this paper, there are three objectives. These objectives might have conflicts, and the increasing of one objective function might cause a decreasing of the other objectives (such as Jv1 and Jv2). Balancing these objectives is a challenge task. The simplest way to solve a multi-objective optimization problem is giving different weights to different objectives and summing them together to form a single objective. In this way, it is possible to use the traditional single objective optimization algorithm to find the optimal solution. The drawback of this approach is the difficulty in choosing optimal weights for the multi-objectives. In fact, there exists a proper weight set to get the optimum solution only for convex optimization problems [20]. In order to solve these problems, an objective function based on the Pareto optimal solution theory is proposed.

To begin with, a definition about the domination (for the maximization problem) is given as follows:

Domination: one solution X1 dominates another solution X2 for a optimizing problem, denoted as X1> X2, if and only if

A non-dominated solution is a solution that is not dominated by any other solutions. A non-dominated solution is the best solution to a problem in the sense of no other solution being better than it. All non-dominated solutions form the Pareto non-dominated solutions set.

Unlike the classical PSO, non-dominated solutions are used as the individual best Ym, and the global best Yg. In other words, the global best and individual best are selected from the Pareto non-dominated solutions set for the proposed PSO.

D. Steps of the Proposed Algorithm

The detailed steps of the proposed multi-objective CPSO method are given as follows:

In [22], the global best is selected by calculating the average value of all of the particles in the global Pareto optimal solution set. Reference [23] divided the searching space into different hypercubes, where each hypercube had a chance of being selected (by the Roulette method). Within one hypercube the global best would be randomly selected from the intersection of this hypercube and the global Pareto optimal solution set. This method needs a greater computation cost. Reference [24] calculated the Euclidean distances, in every iteration, from particle m to each row in the global Pareto optimal solution set. The row in the Pareto optimal solution set which had the smallest distance is selected as the global guide for particle m. This method is very complicated in terms of computation. Compared with the existing methods used to select the global best in [22] [23] [24], the method proposed in this paper has a faster search speed and a larger possibility to obtain the true global best.

 

V. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulation Results

A simulation model of a CACZVS three phase PFC converter with triple closed loop PI controllers is built by integrating PSIM and MATLAB as done in reference [25]. This simulation configuration has the following benefits: (1) Compared to MATLAB, PSIM uses an ideal device to build the model and it uses a simpler trapezoidal method to solve the system equation. Thus, the simulation speed is faster. In this paper, the optimization algorithm requires many iterations. Therefore, using this configuration can effectively save the simulation time. (2) When compared to the PSIM simulation environment, MATLAB provides greater flexibility in terms of the controller design, and it provides a facility for running the proposed CPSO algorithm.

The converter parameters used in the simulation are given as follows: the three-phase input voltage value Uin=220V; the three-phase input filter inductor L=20mH; the inductor equivalent resistance R=1Ω; the switch parallel capacitor C1=C2=…=C7=2nF; the resonant inductor Lr=100μH; the clamp capacitor CC=40μF; the output filter capacitor C=1500μF; the load resistor RL=300Ω; the reference of the output voltage Udcref=600V; and the switching frequency f=10kHz.

The parameters used in the proposed multi-objective CPSO are given as follows: the population size M=100, the maximum number of iterations kmax=10, the initial value of the particles are generated in [0, 1] by the unilateral coupled lattices spatiotemporal map, then they are transformed into the parameter range [0, PKmax] as the initial position, PKmax =5 in the following simulation, the learning factors c1=2.5 and c2=1.5, the inertia weight w=0.9, and the speed limit Vmax=1. The coupling strength εn of the unilateral coupled map lattices is 0.85, and the initial values of the lattices are randomly chosen in [0, 1].

In this paper, the proposed chaotic initialization method has the advantage of global convergence. In order to prove this, based on the same multi-objective PSO condition above but with different particles initialization methods, the comparison results using the random number in [14], the logistic map in [16], and the proposed chaotic map are shown in Table II. The results shown in Table II are the average values for running the optimization algorithm 50 times, where kr denotes the average iteration times used for the different algorithms to reach stable optimized solutions. As can be seen from Table II, the traditional multi-objective PSO method in [14] (using the random initial particles) makes it easy to find a local optima because of the non-uniform distribution of the particles. Obviously, the proposed method has the highest average objective function and the fewest average iterations.

TABLE IIPERFORMANCE COMPARISON OF DIFFERENT METHODS

In order to verify the effectiveness of the proposed multi-objective optimization method, the PI parameters derived by the conventional method in [3] and fine tuning by the weighted multi-objective CPSO (the weighted function is J=2Jv1+ 3Jv2+ Ji), and by the proposed multi-objective CPSO, are shown in Table III. It can be seen from Table III that the objective function values of the proposed method are the best.

TABLE IIIOPTIMIZATION RESULTS OF DIFFERENT METHODS

The advantage of the Pareto optimal solution is automatically achieving balance among the different objectives in order to achieve better performance. The CACZVS three phase PFC converter output voltage waveforms are shown in Fig. 7 using the different PI parameters obtained by different methods in Table III. The input A-phase voltage and the corresponding current waveforms are shown in Fig. 8. From Figs. 7 and 8, the following conclusions can be obtained: the traditional PI parameter tuning method is very complicated and time consuming since it depends heavily on the designer’s experience. The weighted multi-objective optimization method is simple and easy to operate. However, it is difficult to obtain the ideal weights to get the best results. The proposed method has the best performance. The output DC voltage has a smaller overshoot and no static state error. The input current of the proposed method settles down very quickly and achieves a unity power factor. Considering the performance of the voltage loop and current loop comprehensively, there is no doubt that the PI control parameters obtained by the proposed CPSO method are the best.

Fig. 7.Output DC voltage waveforms.

Fig. 8.The input A-phase voltage and the corresponding current waveforms.

A load RL variation from 300Ω to 450Ω is simulated, and the output DC voltage waveforms using the different PI parameters in Table III are shown in Fig. 9. From Fig. 9, it can be seen that the output perturbation is large and recovery time is long using the traditional PI tuning parameters. The performance of the weighted CPSO method is better than the traditional PI tuning method. The performance of the parameters obtained by the proposed method is the best.

Fig. 9.Output dc voltage simulation waves when load changes.

B. Experiment Results

A 1.2kW hardware prototype is built as shown in Fig. 10. The control algorithms are programmed on a DSP28335 digital controller. The circuit parameters of the prototype are the same as those used in the simulation.

Fig. 10.Experimental platform photo.

The CACZVS three phase PFC converter experiment results using the PI parameters obtained by the proposed multi-objective CPSO method are shown in Fig. 11. Fig. 11(a) shows the input A-phase voltage and the corresponding current waveforms, where the input current is synchronized with the input voltage. This indicates a unity power factor. Fig. 11(b) shows the display of a HIOKI3197 three-phase power quality analyzer, where the average three phase power factor is 0.991 and the three-phase total harmonic distortion THD <5%. This shows that the PI parameters obtained by the proposed optimization method work very well.

Fig. 11.Experiment result: (a) A-phase input voltage and current waveforms (b) result displayed by HIOKI 3197 power quality analyzer.

The experimental results corresponding to the simulation results in Fig. 9 are shown in Fig. 12. Fig. 12(a) shows the output DC voltage (channel 1) and the A-phase input current (channel 2) waveforms using the traditional tuning PI parameters. The output DC voltage perturbation is 24V and the voltage recovery time is about 0.36s. Fig. 12(b) shows the output DC voltage (channel 1) and the A-phase input current (channel 2) waveforms using the control parameters obtained by the weighted CPSO method. The output DC voltage perturbation is 16.5V and the voltage recovery time is about 0.3s. Figure. 12 (c) shows the output DC voltage (channel 1) and the A-phase input current (channel 2) waveforms using the control parameters obtained by the proposed method. The output DC voltage perturbation is 15V and the voltage recovery time is about 0.27s. Comparing three subplots in Fig. 12, it can be seen that the proposed method has the smallest output voltage perturbation and the shortest recovery time. The experiments results are consistent with the simulation results, which verifies the effectiveness and superiority of the proposed method.

Fig. 12.Output DC voltage experimental waves when load changes. (a) Traditional PI tuning method. (b) Weighted CPSO method. (c) The proposed method.

 

VI. CONCLUSION

A CACZVS three phase PFC converter is used as an example for triple close loop PI controller parameter optimization. A multi-dimension multi-objective CPSO method is proposed to optimize the control parameters of the triple close loop PI controllers. For multi-dimensional particles, the unilateral coupled map lattices is proposed to produce the uniformly distributed initial particles in an N-dimension solution space. At the same time, for multi-objective functions, the Pareto optimal solution theory is applied to achieve balance among multiple objectives. The proposed chaos particle initialization method can realize a better initial distribution of particles in a multi-dimension solution space when compared to the traditional random number or one dimensional chaotic map. The proposed method avoids the difficulty of weight selection of the traditional weighted multi-objective optimization method. The propose method can be applied to optimize the control parameters. It can also be applied to the topology and parameter optimization of the circuits of the whole converter.