Power Electronic Circuits Design: A Particle Swarm Optimization ...
Power Electronic Circuits Design: A Particle Swarm Optimization Approach* Jun Zhang, Yuan Shi, and Zhi-hui Zhan** Department of Computer Science, Sun Yat-sen University, China, 510275 [email protected]
Abstract. The development of power electronics results in a growing need for automatic design and optimization for power electronic circuits (PECs). This paper presents a particle swarm optimization (PSO) approach for the PECs design. The optimization problem is divided into two processes using a decoupled technique and PSO is employed to optimize the values of the circuit components in the power conversion stage (PCS) and the feedback network (FN), respectively. A simple mutation operator is also incorporated into PSO to enhance the population diversity. The algorithm is applied to the optimization of a buck regulator for meeting requirements under large-signal changes and at steady state. Compared with genetic algorithm (GA), PSO can yield more optimized values of circuit components with lower computational effort. Keywords: Particle swarm optimization (PSO), power electronic circuits (PECs), circuit optimization.
1 Introduction Power electronics is the technology to efficiently control the electric energy by supplying voltages or currents in a form that is suited for user loads. Power electronic circuits (PECs) have been widely used in various applications, such as mobile devices, computers, televisions and uninterruptible power supply. With the advances in semiconductor technology and electronic packaging as well as the increasing demand in industry, automatic design and optimization of PECs becomes a growing need [1]. The methods for automated synthesis and optimization of circuits are either deterministic or stochastic. Deterministic approaches, like the gradient method and the hillclimbing technique [1] [2], might easily be trapped into local optima, as high nonlinearity always involved in PECs optimization. Also, the performance of some deterministic methods is significantly dependent on the initial starting point of the search. On the other hand, stochastic methods tend to be more suitable for PECs optimization since they explore the search space more extensively. Recently, a class of stochastic methods, called evolutionary algorithms (EAs) has attracted many researchers’ attentions due to its high *
This work was supported by NSF of China Project No.60573066 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, P.R. China. ** Corresponding author. X. Li et al. (Eds.): SEAL 2008, LNCS 5361, pp. 605–614, 2008. © Springer-Verlag Berlin Heidelberg 2008
606
J. Zhang, Y. Shi, and Z.-h. Zhan
global search capability. EAs are population-based search techniques inspired by natural mechanisms or phenomenon. During the last several decades, many EAs, such as genetic algorithms (GAs) [3] and particle swarm optimization (PSO) [4] have been proposed and applied to plenty of real-world optimization problems. GAs perform search by simulating the natural evolutionary process. They adopt three evolutionary operations, namely selection, crossover and mutation. A variety of GAs-based design schemes have been used in analog circuits, like transconductance amplifier [5], voltage reference circuit [6], loaded wire antenna [7], etc. Recently, GAs have been successfully applied in the optimization of PECs [8]. Although the optimized results demonstrated the effectiveness of the GAs-based methods, more effective and efficient approaches are still required for the optimization of PECs. Different from GAs, PSO solves optimization problems by emulating the flocking behavior of birds. It employs a swarm of particles, which represent feasible solutions, to search in a cooperative manner. Each particle has its position and velocity, which are iteratively updated by tracing the personal best position found so far and the global best position found so far. PSO has also been used in analog circuit synthesis, such as RF circuit design [9], microwave filter design [10] and on-chip spiral inductor synthesis [11]. Compared with GA, PSO is easier to implement because of its simple concept. Moreover, PSO converges faster, which can considerably reduce the computational time. In this paper, a PSO approach for the optimization of PECs is presented. A PEC is decoupled into two parts and particles are employed to find optimal values for the components of each part, aiming to satisfy the specified static and dynamic requirements. To improve the population diversity, a mutation operator is incorporated into the PSO. The algorithm is illustrated with the design of a buck regulator. The optimized results are compared with those obtained by the GA approach [8], showing that PSO is a promising approach for design and optimization of PECs. The rest of this paper is organized as follows. Section 2 describes the decoupled structure of PECs. Section 3 gives a brief introduction of PSO. Section 4 shows the details of the algorithm for PECs optimization. The design of a buck regulator is illustrated in Section 5 and the conclusion is represented in Section 6.
2 Decoupled Structure of PECs Fig. 1 shows the basic block diagram of a PEC, in which the circuit is decoupled into two parts, namely the power conversion stage (PCS) and the feedback network (FN). PCS consists of IP resistors, JP inductors and KP capacitors, which transfers the power from the input source to the output load. FN consists of IF resistors, JF inductors and KF capacitors, which keeps the output voltage vo to a reference voltage vref. The goal of the optimization is to find proper values for these circuit components, aiming to satisfy the static and dynamic responses. Although performing a wholecircuit optimization is feasible, it is computationally intensive since the number of variables is considerably large. In [8], a decoupled technique based on the circuit structure was proposed, which divided the optimization into two processes. One process optimized the component values in PCS and another process tackled FN. This decoupled technique is effective to reduce the computational effort and is adopted in this paper.
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Fig. 1. Block diagram of power electronics circuits
3 Particle Swarm Optimization PSO is an iterative approach that uses a swarm of M particles. Each particle has its own position and velocity that are vectors with N dimensions. The position of a particle represents a feasible solution in multidimensional search space, and its quality is evaluated by the fitness function. In addition, every particle memories two positions: the personal best position found by the particle so far (denoted as pbest) and the global best position found by the whole population so far (denoted as gbest). Thus, in PSO, the ith (i = 1, 2, …, M) particle is represented by three vectors: The current position: xi = ( xi1 , xi 2 ,..., xiN ) The current velocity: vi = (vi1 , vi 2 ,..., viN ) The personal best position: pbesti = ( pbesti1 , pbesti 2 ,..., pbestiN )
In every iteration, the velocity and position of each particle are updated following the equation below:
vij (t + 1) = wvij (t ) + c1r1 ( pbestij (t ) − xij (t )) + c2 r2 ( gbestij (t ) − xij (t ))
(1)
where t is the iteration index, i is the particle index, and j is the dimension index. r1 and r2 are two random values generated with uniform probability from [0,1]. Parameter w>0 is called the inertia weight, which determines the impact of the previous velocity: a large inertia weight favors exploration while a small value favors exploitation. Parameters c1 and c2 control the influence of the personal best position and the global best position, respectively. Typically, the velocity is further restricted by a maximum velocity Vmax. Mathematically, the restriction rule is expressed by: ⎧ vij (t + 1), ⎪ vij (t + 1) = ⎨Vmax, j , ⎪−V ⎩ max, j ,
− Vmax, j < vij (t + 1) < Vmax, j vij (t + 1) > Vmax, j vij (t + 1) < −Vmax, j
(2)
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where Vmax, j stands for the jth dimension of Vmax. Assume the search range of the jth variable is [lj, uj], Vmax, j can be set by: Vmax, j = α ⋅ (u j − l j )
(3)
where α is a positive parameter. After the velocity of the ith particle is updated, the particle moves to a new position according to the following equation: xij (t + 1) = xij (t ) + vij (t + 1)
(4)
Particles’ positions and velocities are updated repeatedly according to (1), (2) and (4) until meeting the terminal condition. Usually, the optimization process is terminated when the number of iterations reaches the maximum number MAX_ITER. PSO has been applied to a wide range of optimization problems due to its simple concept, easy implementation and fast convergence. However, PSO is sometimes found to be converged on local optima, the factor of which is the lack of the population diversity [12]. Therefore, incorporating an additional mutation operator can help prevent premature convergence, and thus enhance the algorithm’s global search characteristics [13].
4 PSO for PECs Optimization Based on the decoupled technique, PSO is employed, respectively, for the optimization of PCS and FN. The position of each particle is coded as a vector containing the component values. 4.1 Fitness Function
The fitness function is used to evaluate the quality of a particle’s position. Here, we adopt the fitness function in [8] which is based on a time-domain simulation of PECs. For PCS, the fitness function is defined considering the following objectives with each one expressed by an objective function (OFx). 1) steady-state error of the output voltage – OF1 2) operation constraints of the circuit components – OF2 3) steady-state ripple voltage on the output – OF3 4) intrinsic factors concerning with the components – OF4 Hence, the fitness function for PCS is defined as: Φ P ( xPCS ) =
RL ,max
∑δ
Vin ,max
∑
RL = RL ,min , Rl vin =Vin ,min ,δ vin
[OF1 ( RL , vin , xPCS ) + OF2 ( RL , vin , xPCS )
(5)
+ OF3 ( RL , vin , xPCS ) + OF4 ( RL , vin , xPCS )]
where xPCS represents the position vector for PCS. For FN, the fitness function is defined based on the following considerations: 1) steady state error of output – OF5 2) maximum overshoot and undershoot, damping ratio, and the settling time – OF6 3) steady-state ripple voltage on output – OF7
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4) dynamic behaviors during the large-signal change – OF8 Thus, the fitness function for FN is defined as: Φ F ( xFN ) =
RL ,max
Vin ,max
RL = RL ,min ,δ Rl
vin =Vin ,min ,δ vin
∑
∑
[OF5 ( RL , vin , xFN ) +OF6 ( RL , vin , xFN )
(6)
+ OF7 ( RL , vin , xFN ) + OF8 ( RL , vin , xFN )]
where xFN represents the position vector for FN. 4.2 Steps of Optimization
The optimization processes for PCS and FN are similar except for the definition of the fitness function. The steps of optimization are illustrated as follows and the flowchart of the algorithm is provided in Fig. 2.
Fig. 2. Flowchart of PSO for PECs optimization
Step 1) Initialize a number of M particles. For the jth circuit component, randomly generate a feasible value within the user-defined search range of [lj, uj]. The jth velocity is assigned by a random value in the range of [–Vmax, j, Vmax, j]. Step 2) Calculate the fitness value for each particle. Use the fitness function (5) for PCS and (6) for FN, respectively. Step 3) Update the personal best position (pbest) for each particle according to its fitness value. Update the global best position (gbest) for the whole population. Step 4) Update the position and velocity of every particle. Step 5) Apply the mutation operator to enhance the population diversity. The operator is performed as follows. For every dimension of each particle, a random value
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r∈[0,1] is generated and is compared with the predetermined mutation probability Pm. If r is smaller than Pm, the corresponding component value is randomly reinitialized. Step 6) If the number of iterations exceeds a maximum value MAX_ITER, the optimization process will be ended, otherwise go to Step 2).
5 Design Example In this section, the above algorithm is illustrated with the design of a buck regulator with overcurrent protection [14] as shown in Fig. 3. The required specifications are listed in Table 1. For the optimization of PCS, L and C are the designed components and RL, rC, rE are assumed to be known. For FN, all component values are required to be optimized.
Fig. 3. Schematic of a buck regulator with over current protection Table 1. Required specifications
Attribute Input voltage range Output load range Nominal output voltage Switching frequency Maximum setting time
Value 40V±20V 5Ω~10Ω 5V±1% 20kHz 20ms
The parameter setting for PSO is given in Table 2, in which w is decreased linearly from 1.2 to 0.6 though the optimization process. As suggested in [15], this strategy well balances the exploration and exploitation of the search.
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Table 2. Parameter setting for PSO
Parameter POPSIZE MAX_ITER w c1 c2 α Pm
Value 30 500 1.2→0.6 2.0 2.0 0.5 0.02
The fitness value versus the number of iterations in the optimization of FN is shown in Fig. 4. Compared with the related figure given in [8] which was optimized by a GA (the optimizing curve of GA is not presented here, refer to Fig. 7(b) in [8]), PSO converges much faster and achieves significantly higher fitness value. Apparently, using PSO for PECs optimization can considerably lower the computational effort.
Fig. 4. Fitness values versus the number of iterations in the optimization of FN Table 3. Optimized results after 500 iterations
Part PCS
FN
Parameter L C RC3 C2 C3 R2 C4 R4 R1
Value 200μH 100μF 470Ω 0.2μF 5.56μF 3MΩ 0.18μF 7.644kΩ 60Ω
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(a) vo and vcon
(b) iL Fig. 5. Simulated startup transient responses when vin is 20V and RL is 5Ω
(a) vo and vcon
(b) iL Fig. 6. Simulated transient responses when vin is changed from 20V to 40V and RL is 5Ω
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(a) vo and vcon
(b) iL Fig. 7. Simulated transient responses when RL is changed from 5Ω to 10Ω and vin is 40V
Table 3 gives the optimized values of PCS and FN obtained by the proposed PSO. Fig. 5 shows the simulated startup transients, when the input voltage is 20V and the output load is 5Ω. The setting time of the circuit optimized by PSO is less than 7ms, which is shorter than that in [8]. Fig. 6 illustrates the simulated transient responses when the input voltage is changed from 20V to 40V. Fig. 7 shows the theoretical transients when the output load is changed from 5Ω to 10Ω. From the figures, it can be seen that, the circuit optimized by PSO has much smaller disturbances and shorter response time than the ones in [8], confirming the advantage of the PSO approach.
6 Conclusions This paper applies a PSO approach for the design and optimization of PECs. No complicated mathematical analysis of the whole circuit is required in the presented method. A design example on a buck regulator with overcurrent protection demonstrates the three advantages of the PSO approach: easy implementation, fast convergence and high global search capability, which can significantly improve the efficiency and effectiveness of PECs optimization and give highly-optimized values of circuit components.
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References 1. Verghese, G.C., Bruzos, C.A., Mahabir, K.N.: Averaged and sampled-data model for current-mode control: A reexamination. In: Proceedings PESC 1989, pp. 484–491 (1989) 2. Massara, R.E.: Optimization Methods in Electronic Circuit Design. Longman, New York (2000) 3. Holland, J.H.: Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor (1975) 4. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings IEEE Int. Conf. Neural Networks, vol. 4, pp. 1942–1948 (1995) 5. Dhanwada, N.N., Nunez-Aldana, A., Vemuri, R.: A genetic approach to simultaneous parameter space exploration and constraint transformation in analog synthesis. In: Proceedings IEEE Int. Sym. Circuits Systs., pp. 362–365 (1999) 6. Nam, D., Seo, Y., Park, L., Park, C., Kim, B.: Parameter optimization of a voltage reference circuit using EP. In: Proceedings IEEE Int. Conf. Evolutionary Computation, pp. 301–305 (1998) 7. Lee, K.C.: Genetic algorithms based analyses of nonlinearly loaded antenna arrays including mutual coupling effects. IEEE Trans. on Antennas and Propagation 5, 776–781 (2003) 8. Zhang, J., Chung, H., Lo, W., Hui, S., Wu, A.: Implementation of a decoupled optimization technique for design of switching regulators using genetic algorithms. IEEE Trans. Power Electron 16, 752–763 (2001) 9. Jinho, P., Kiyong, C., David, J.A.: Parasitic-Aware RF Circuit Design and Optimization. IEEE Trans. on Circuits and Systems-I 51, 1953–1966 (2004) 10. Wen, W., Yilong, L., Jeffrey, S.F., Yong, Z.: Particle Swarm Optimization and FiniteElement Based Approach for Microwave Filter Design. IEEE Trans. on Magnetics 41, 1800–1803 (2005) 11. Sushanta, K., Mandal, S.S., Amit, P.: ANN- and PSO-Based Synthesis of On-Chip Spiral Inductors for RF ICs. IEEE Trans. on Computer-aided Design of Integrated Circuits and Systems 27, 188–192 (2008) 12. Ratnaweera, A., Halgamuge, S.K., Watson, H.C.: Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans. on Evolutionary Computation 8(3), 240–255 (2004) 13. Paul, S.A.: An Investigation into Mutation Operators for Particle Swarm Optimization. In: Proceedings IEEE Congress on Evolutionary Computation, pp. 1044–1051 (2006) 14. Bedrosian, D., Vlach, J.: Time-domain analysis of networks with internally controlled switches. IEEE Trans. Circuits Systs. I. 39, 199–212 (1992) 15. Shi, Y.H., Eberhart, R.C.: Parameter selection in particle swarm optimization. In: Proceedings of the 7th Ann. Conf. on Evolutionary Programming, San Diego, CA, pp. 591–600 (1998)
Abstract. The development of power electronics results in a growing need for automatic design and optimization for power electronic circuits (PECs). This paper presents a particle swarm optimization (PSO) approach for the PECs design. The optimization problem is divided into two processes using a decoupled technique and PSO is employed to optimize the values of the circuit components in the power conversion stage (PCS) and the feedback network (FN), respectively. A simple mutation operator is also incorporated into PSO to enhance the population diversity. The algorithm is applied to the optimization of a buck regulator for meeting requirements under large-signal changes and at steady state. Compared with genetic algorithm (GA), PSO can yield more optimized values of circuit components with lower computational effort. Keywords: Particle swarm optimization (PSO), power electronic circuits (PECs), circuit optimization.
1 Introduction Power electronics is the technology to efficiently control the electric energy by supplying voltages or currents in a form that is suited for user loads. Power electronic circuits (PECs) have been widely used in various applications, such as mobile devices, computers, televisions and uninterruptible power supply. With the advances in semiconductor technology and electronic packaging as well as the increasing demand in industry, automatic design and optimization of PECs becomes a growing need [1]. The methods for automated synthesis and optimization of circuits are either deterministic or stochastic. Deterministic approaches, like the gradient method and the hillclimbing technique [1] [2], might easily be trapped into local optima, as high nonlinearity always involved in PECs optimization. Also, the performance of some deterministic methods is significantly dependent on the initial starting point of the search. On the other hand, stochastic methods tend to be more suitable for PECs optimization since they explore the search space more extensively. Recently, a class of stochastic methods, called evolutionary algorithms (EAs) has attracted many researchers’ attentions due to its high *
This work was supported by NSF of China Project No.60573066 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, P.R. China. ** Corresponding author. X. Li et al. (Eds.): SEAL 2008, LNCS 5361, pp. 605–614, 2008. © Springer-Verlag Berlin Heidelberg 2008
606
J. Zhang, Y. Shi, and Z.-h. Zhan
global search capability. EAs are population-based search techniques inspired by natural mechanisms or phenomenon. During the last several decades, many EAs, such as genetic algorithms (GAs) [3] and particle swarm optimization (PSO) [4] have been proposed and applied to plenty of real-world optimization problems. GAs perform search by simulating the natural evolutionary process. They adopt three evolutionary operations, namely selection, crossover and mutation. A variety of GAs-based design schemes have been used in analog circuits, like transconductance amplifier [5], voltage reference circuit [6], loaded wire antenna [7], etc. Recently, GAs have been successfully applied in the optimization of PECs [8]. Although the optimized results demonstrated the effectiveness of the GAs-based methods, more effective and efficient approaches are still required for the optimization of PECs. Different from GAs, PSO solves optimization problems by emulating the flocking behavior of birds. It employs a swarm of particles, which represent feasible solutions, to search in a cooperative manner. Each particle has its position and velocity, which are iteratively updated by tracing the personal best position found so far and the global best position found so far. PSO has also been used in analog circuit synthesis, such as RF circuit design [9], microwave filter design [10] and on-chip spiral inductor synthesis [11]. Compared with GA, PSO is easier to implement because of its simple concept. Moreover, PSO converges faster, which can considerably reduce the computational time. In this paper, a PSO approach for the optimization of PECs is presented. A PEC is decoupled into two parts and particles are employed to find optimal values for the components of each part, aiming to satisfy the specified static and dynamic requirements. To improve the population diversity, a mutation operator is incorporated into the PSO. The algorithm is illustrated with the design of a buck regulator. The optimized results are compared with those obtained by the GA approach [8], showing that PSO is a promising approach for design and optimization of PECs. The rest of this paper is organized as follows. Section 2 describes the decoupled structure of PECs. Section 3 gives a brief introduction of PSO. Section 4 shows the details of the algorithm for PECs optimization. The design of a buck regulator is illustrated in Section 5 and the conclusion is represented in Section 6.
2 Decoupled Structure of PECs Fig. 1 shows the basic block diagram of a PEC, in which the circuit is decoupled into two parts, namely the power conversion stage (PCS) and the feedback network (FN). PCS consists of IP resistors, JP inductors and KP capacitors, which transfers the power from the input source to the output load. FN consists of IF resistors, JF inductors and KF capacitors, which keeps the output voltage vo to a reference voltage vref. The goal of the optimization is to find proper values for these circuit components, aiming to satisfy the static and dynamic responses. Although performing a wholecircuit optimization is feasible, it is computationally intensive since the number of variables is considerably large. In [8], a decoupled technique based on the circuit structure was proposed, which divided the optimization into two processes. One process optimized the component values in PCS and another process tackled FN. This decoupled technique is effective to reduce the computational effort and is adopted in this paper.
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Fig. 1. Block diagram of power electronics circuits
3 Particle Swarm Optimization PSO is an iterative approach that uses a swarm of M particles. Each particle has its own position and velocity that are vectors with N dimensions. The position of a particle represents a feasible solution in multidimensional search space, and its quality is evaluated by the fitness function. In addition, every particle memories two positions: the personal best position found by the particle so far (denoted as pbest) and the global best position found by the whole population so far (denoted as gbest). Thus, in PSO, the ith (i = 1, 2, …, M) particle is represented by three vectors: The current position: xi = ( xi1 , xi 2 ,..., xiN ) The current velocity: vi = (vi1 , vi 2 ,..., viN ) The personal best position: pbesti = ( pbesti1 , pbesti 2 ,..., pbestiN )
In every iteration, the velocity and position of each particle are updated following the equation below:
vij (t + 1) = wvij (t ) + c1r1 ( pbestij (t ) − xij (t )) + c2 r2 ( gbestij (t ) − xij (t ))
(1)
where t is the iteration index, i is the particle index, and j is the dimension index. r1 and r2 are two random values generated with uniform probability from [0,1]. Parameter w>0 is called the inertia weight, which determines the impact of the previous velocity: a large inertia weight favors exploration while a small value favors exploitation. Parameters c1 and c2 control the influence of the personal best position and the global best position, respectively. Typically, the velocity is further restricted by a maximum velocity Vmax. Mathematically, the restriction rule is expressed by: ⎧ vij (t + 1), ⎪ vij (t + 1) = ⎨Vmax, j , ⎪−V ⎩ max, j ,
− Vmax, j < vij (t + 1) < Vmax, j vij (t + 1) > Vmax, j vij (t + 1) < −Vmax, j
(2)
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where Vmax, j stands for the jth dimension of Vmax. Assume the search range of the jth variable is [lj, uj], Vmax, j can be set by: Vmax, j = α ⋅ (u j − l j )
(3)
where α is a positive parameter. After the velocity of the ith particle is updated, the particle moves to a new position according to the following equation: xij (t + 1) = xij (t ) + vij (t + 1)
(4)
Particles’ positions and velocities are updated repeatedly according to (1), (2) and (4) until meeting the terminal condition. Usually, the optimization process is terminated when the number of iterations reaches the maximum number MAX_ITER. PSO has been applied to a wide range of optimization problems due to its simple concept, easy implementation and fast convergence. However, PSO is sometimes found to be converged on local optima, the factor of which is the lack of the population diversity [12]. Therefore, incorporating an additional mutation operator can help prevent premature convergence, and thus enhance the algorithm’s global search characteristics [13].
4 PSO for PECs Optimization Based on the decoupled technique, PSO is employed, respectively, for the optimization of PCS and FN. The position of each particle is coded as a vector containing the component values. 4.1 Fitness Function
The fitness function is used to evaluate the quality of a particle’s position. Here, we adopt the fitness function in [8] which is based on a time-domain simulation of PECs. For PCS, the fitness function is defined considering the following objectives with each one expressed by an objective function (OFx). 1) steady-state error of the output voltage – OF1 2) operation constraints of the circuit components – OF2 3) steady-state ripple voltage on the output – OF3 4) intrinsic factors concerning with the components – OF4 Hence, the fitness function for PCS is defined as: Φ P ( xPCS ) =
RL ,max
∑δ
Vin ,max
∑
RL = RL ,min , Rl vin =Vin ,min ,δ vin
[OF1 ( RL , vin , xPCS ) + OF2 ( RL , vin , xPCS )
(5)
+ OF3 ( RL , vin , xPCS ) + OF4 ( RL , vin , xPCS )]
where xPCS represents the position vector for PCS. For FN, the fitness function is defined based on the following considerations: 1) steady state error of output – OF5 2) maximum overshoot and undershoot, damping ratio, and the settling time – OF6 3) steady-state ripple voltage on output – OF7
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4) dynamic behaviors during the large-signal change – OF8 Thus, the fitness function for FN is defined as: Φ F ( xFN ) =
RL ,max
Vin ,max
RL = RL ,min ,δ Rl
vin =Vin ,min ,δ vin
∑
∑
[OF5 ( RL , vin , xFN ) +OF6 ( RL , vin , xFN )
(6)
+ OF7 ( RL , vin , xFN ) + OF8 ( RL , vin , xFN )]
where xFN represents the position vector for FN. 4.2 Steps of Optimization
The optimization processes for PCS and FN are similar except for the definition of the fitness function. The steps of optimization are illustrated as follows and the flowchart of the algorithm is provided in Fig. 2.
Fig. 2. Flowchart of PSO for PECs optimization
Step 1) Initialize a number of M particles. For the jth circuit component, randomly generate a feasible value within the user-defined search range of [lj, uj]. The jth velocity is assigned by a random value in the range of [–Vmax, j, Vmax, j]. Step 2) Calculate the fitness value for each particle. Use the fitness function (5) for PCS and (6) for FN, respectively. Step 3) Update the personal best position (pbest) for each particle according to its fitness value. Update the global best position (gbest) for the whole population. Step 4) Update the position and velocity of every particle. Step 5) Apply the mutation operator to enhance the population diversity. The operator is performed as follows. For every dimension of each particle, a random value
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r∈[0,1] is generated and is compared with the predetermined mutation probability Pm. If r is smaller than Pm, the corresponding component value is randomly reinitialized. Step 6) If the number of iterations exceeds a maximum value MAX_ITER, the optimization process will be ended, otherwise go to Step 2).
5 Design Example In this section, the above algorithm is illustrated with the design of a buck regulator with overcurrent protection [14] as shown in Fig. 3. The required specifications are listed in Table 1. For the optimization of PCS, L and C are the designed components and RL, rC, rE are assumed to be known. For FN, all component values are required to be optimized.
Fig. 3. Schematic of a buck regulator with over current protection Table 1. Required specifications
Attribute Input voltage range Output load range Nominal output voltage Switching frequency Maximum setting time
Value 40V±20V 5Ω~10Ω 5V±1% 20kHz 20ms
The parameter setting for PSO is given in Table 2, in which w is decreased linearly from 1.2 to 0.6 though the optimization process. As suggested in [15], this strategy well balances the exploration and exploitation of the search.
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Table 2. Parameter setting for PSO
Parameter POPSIZE MAX_ITER w c1 c2 α Pm
Value 30 500 1.2→0.6 2.0 2.0 0.5 0.02
The fitness value versus the number of iterations in the optimization of FN is shown in Fig. 4. Compared with the related figure given in [8] which was optimized by a GA (the optimizing curve of GA is not presented here, refer to Fig. 7(b) in [8]), PSO converges much faster and achieves significantly higher fitness value. Apparently, using PSO for PECs optimization can considerably lower the computational effort.
Fig. 4. Fitness values versus the number of iterations in the optimization of FN Table 3. Optimized results after 500 iterations
Part PCS
FN
Parameter L C RC3 C2 C3 R2 C4 R4 R1
Value 200μH 100μF 470Ω 0.2μF 5.56μF 3MΩ 0.18μF 7.644kΩ 60Ω
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(a) vo and vcon
(b) iL Fig. 5. Simulated startup transient responses when vin is 20V and RL is 5Ω
(a) vo and vcon
(b) iL Fig. 6. Simulated transient responses when vin is changed from 20V to 40V and RL is 5Ω
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(a) vo and vcon
(b) iL Fig. 7. Simulated transient responses when RL is changed from 5Ω to 10Ω and vin is 40V
Table 3 gives the optimized values of PCS and FN obtained by the proposed PSO. Fig. 5 shows the simulated startup transients, when the input voltage is 20V and the output load is 5Ω. The setting time of the circuit optimized by PSO is less than 7ms, which is shorter than that in [8]. Fig. 6 illustrates the simulated transient responses when the input voltage is changed from 20V to 40V. Fig. 7 shows the theoretical transients when the output load is changed from 5Ω to 10Ω. From the figures, it can be seen that, the circuit optimized by PSO has much smaller disturbances and shorter response time than the ones in [8], confirming the advantage of the PSO approach.
6 Conclusions This paper applies a PSO approach for the design and optimization of PECs. No complicated mathematical analysis of the whole circuit is required in the presented method. A design example on a buck regulator with overcurrent protection demonstrates the three advantages of the PSO approach: easy implementation, fast convergence and high global search capability, which can significantly improve the efficiency and effectiveness of PECs optimization and give highly-optimized values of circuit components.
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J. Zhang, Y. Shi, and Z.-h. Zhan
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