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Abstract—This paper presents a Fuzzy-PID controller for maximum power point tracking (MPPT) of photovoltaic (PV) systems. A DC/DC buck converter to ...
Fuzzy-PID controller for MPPT of PV system optimized by Big Bang-Big Crunch algorithm A.I. Dounis, S. Stavrinidis, P. Kofinas, D. Tseles Department of Automation Engineering Piraeus University of Applied Sciences (T.E.I. of Piraeus) Egaleo-Athens, Greece

Abstract—This paper presents a Fuzzy-PID controller for maximum power point tracking (MPPT) of photovoltaic (PV) systems. A DC/DC buck converter to regulate the output power of the photovoltaic system is considered. The Fuzzy-PID scheme operates on MPP and improves the performance of solar energy conversion efficiency. The offline optimization algorithm of Big Bang – Big Crunch (BB-BC) is applied on the parameters of the proposed controller and upgrades the overall performance of the controller. The prime contribution of this study is a simple and effective solution for MPPT. The optimized fuzzy-PID controller is compared with a fuzzy-PID controller set up by the trial and error method in order to highlight the contribution of the offline algorithm in the overall performance of the system. The tuned controller is also compared to the conventional perturbation and observation (P&O) method, with a Fuzzy-PID optimized by Particle Swarm Optimization algorithm (PSO). Keywords: Fuzzy-PID; Maximum Power Point Tracking; Photovoltaic system; Big Bang – Big Crunch

I.

INTRODUCTION

Nowadays, the rapid increase in the demand for electricity and the recent change in the environmental conditions led to a need for cheap and green energy. Therefore, solar energy which has no carbon emissions can be used for several applications [1]. Photovoltaic (PV) generators are the units that convert solar energy into electric energy. The maximization of the power extracted from the PV generator is usually carried out by means of mechanical and/or electronic systems. In the first case, sun tracking systems optimize the solar incidence angle while in the electronic approach a maximum power transfer at the PV array terminals is pursued by means of maximum power point tracking (MPPT) methods [2]. The PV generator delivers power which depends on solar irradiance, cell temperature and the electrical load. Also, the PV generators have a nonlinear characteristic, and the power has a Maximum Power Point (MPP) at a certain working point, with coordinates VMPP voltage and IMPP current for a certain pair of solar irradiance and cell temperature. So, as the solar irradiance and cell temperature change over time the MPP changes its coordinates over time too. For this reason, MPPT should be used to track its changes [3]. Furthermore, the efficiency of MPPT depends on both the MPPT control algorithm and the MPPT circuit. The MPPT

control algorithm is usually applied in the DC-DC converter, which is normally used as the MPPT circuit [4]. Regarding MPPT many techniques have been proposed in the literature. Some examples are the Perturb and Observe (P&O) methods [5,6], Incremental Conductance (IC) methods [7,8], Fuzzy Logic method [4,9,10], etc. MPPT fuzzy logic controllers (FLC) are relatively simple to design as they do not require the knowledge of the exact model. They also have the advantage of being fast robust and of having quite good performance (time response, stability, tracking speed, small oscillations) under varying atmospheric conditions. These controllers are more effective under sudden changes of atmospheric conditions compared to the traditional algorithms. However, the reliability depends on the knowledge of the control engineer who determines the fuzzy rule base and chooses the fuzzy parameters such as error metrics, membership functions and scaling factors with accuracy [11,12]. Consequently the FLC design is not optimum. In order to overcome the aforementioned limitations regarding the MPPT-FLCs, computational intelligence methods using fuzzy logic, neural networks, particle swarm optimizer (PSO) and genetic algorithms (GA) have been applied to optimize the FLC and to improve the tracking efficiency [13-19]. In this paper a metaheuristic algorithm is applied which is called BB-BC. This algorithm is very fast with high converge speed which relies on one of the theories of the evolution of the universe; namely, the Big Bang and Big Crunch Theory. In the Big Bang phase, energy dissipation producing disorder and randomness is the main feature of this phase; whereas, in the Big Crunch phase, randomly distributed particles are drawn into an order [20]. The conventional fuzzy control uses three types of controllers. The Fuzzy-Proportional-Integral (Fuzzy-PI) control, the Fuzzy-Proportional-Derivative (Fuzzy-PD) control and the Fuzzy-Proportional-Integral-Derivative (Fuzzy-PID) control. The Fuzzy-PI control has good performance in steady states as it minimizes the permanent error but lacks in performance at the transient states, while the Fuzzy-PD comes with good performance at the transient state but fails to eliminate the permanent error at steady states [21]. The Conventional Fuzzy-PID control has both the advantages of Fuzzy-PI and Fuzzy-PD control but has three inputs (error, change of error and acceleration of error) which increase the curse of dimensionality. The PID type fuzzy control (Fuzzy-

PID) has only two inputs (error and chaange of error). It composed by a Fuzzy-PI and a Fuzzy PD working w in parallel. The rule-base is two dimensional and it is juust like the FuzzyPI type control rule base. Due to the fact thaat the inputs of the Fuzzy-PID are limited to two, complexity is i decreased while the advantages of the Conventional Fuzzzy-PID control is equivalent with Fuzzy-PID with two inputs [22].

x c is the center of mass, l=10 is i the parameter’s upper limit, r is a normal random number, x is the new point and k is the iteration step. Step 4: The algorithm returns to Step 2 until stopping criteria s the flowchart of the BBhave been met [20]. Figure 1 shows BC algorithm.

a using The literature shows a wide variety of applications Fuzzy-PID controllers. Qiao [23] propooses a parameter adaptive fuzzy controller which has the capaability of tuning its parameters on line in order to further improvve the performance of the fuzzy controller, while Bouallegue proposes p a FuzzyPID controller tuned off-line using PSO in order o to control an electrical DC drive [24]. Kayacan proposes a grey predictionbased Fuzzy-PID controller that can oveercome the stated shortcomings in a non-linear liquid level sysstem [25] and Tsai proposes a Fuzzy-PID controller with paarameter adaptive methods for the control of a rotating active magnetic bearing [26]. ws: Section I gives Finally, this paper is organized as follow the introduction to the MPP techniques. Seection II gives the basic information about the theory of the BB B-BC optimization algorithm. The proposed control strategy (Fuuzzy-PID) and the off-line optimization of the proposed controoller are analyzed in Section III. Section IV includes the simuulation results and the comparison between the Fuzzy-PID optiimized by BB-BC, the conventional method of P&O andd the Fuzzy-PID optimized by PSO. Section V gives the concllusions. II.

BB-BC OPTIMIZATION MEETHOD

Big Bang-Big Crunch is a new optim mization technique developed by Erol and Eksin based on the t theory of the universe evolution which has high convergennce speed and low computational time. This technique has two phases, the Big Bang phase where new candidate solutioons are produced randomly around a “center of mass” that is later calculated in f values. Big the Big Crunch phase with respect to their fitness Bang-Big Crunch algorithm is comprised of four steps: mly generated and Step 1: The initial population is random spread all over the entire search space. Step 2: The fitness function values of all candidate solutions c by using are calculated and the center of mass xc is computed the following formula: xc

∑i 1 1i x i f ∑i 1 1i

(1)

f

where N is the population size, fi is a fitnesss function value of this point and xi is a point within an n-diimensional search space generated. Alternatively, the best fitness individual can be chosen as the center of mass. Step 3: New candidate solutions are geneerated around the center of mass by adding a random number according a to (2).

x

c

x

·

(2)

Fig. 1. Flowchart of the BB-BC algorithm

III. MPPT CONTRO OL SYSTEM DESIGN A. PV model and MPP descripption The equivalent circuit of the PV source is presented in figure 2. This PV model is the one diode model. The generated current from the current sourcee Ipv is described by: Ipv Isc - e

Vpv

(3)

where Isc Iscr

Gpv Gr

1 niscT Tpv -Tr Iscr e-bSTC Voc bSTC

1 nvocT Tpv -Tr

(4) (5) (6)

Imppr

bSTC =

l log(1) Iscr Vmppr -Vocr

(7)

t equivalent circuit, Gr is the and Vpv is the voltage across the reference solar irradiance at Standard S Test Conditions (STC) and equals to 1000w/m2, Tr is the reference Temperature at STC and equals to 25OC, Iscr iss the short circuit current of the PV source at STC, Gpv is the solar irradiance that incidents on the PV source, Tpv is the tempeerature of the PV source, Vocr is the voltage across the PV sourrce at STC, Vmppr is the voltage of the PV source at the MPP att STC, Imppr is the current of the PV source at the MPP at STC, S nvocT is the temperature coefficient of the open circuit voltage and niscT is temperature coefficient of the short circuitt current. The characteristics of the PV source are presented in Table I.

Fig. 2. Equivalent circuit of PV sourrce

Table I: Parameters of PV soource PARAME-TERS Vocr(V) Iscr(A) Vmppr(V) Imppr(W) niscT nvocT

DESCRIPTION The open circuit voltage of the PV source at STC The short circuit current of the PV source at STC The voltage of the PV source at the Maximum Power Point at STC The current that the PV source produces at the maximum power point at STC Temperature coefficient of the short circuit current Temperature coefficient of the open circuit voltage

VALUES Fig. 4. Typical characteristicc P-V curve of photovoltaic source

20

The parameters of the buuck converter are presented in Table II. The power circuit is integrated i by a resistive load of about 0.5 ohms.

7.34 16

Table II: Parameters of buck converter

6.70 0.00067 -0.0035

A buck converter is connected in paraallel with the PV source. In order to force the source to operaate at the MPP, the converter is connected between the source and the load. The circuit of the buck converter is depicted in Fiigure 3.

PARAMETERS Cin Cou L

By adjusting the duty cycle “d” of the buck b converter via the switch, the operation point of the phottovoltaic source is changing [27]. The relation between the inpuut voltage (Vi) and the output voltage (Vo) of the converter is linear and depend on the value of the duty cycle. Vo dVi

(8)

The goal is to track and keep the operration point at its maximum value (MPP). In order to identify fy whether the PV source operates at its MPP, the current and the voltage of the PV source are measured and the derivative of power to voltage q of dP/dV is calculated (dP/dV). At the MPP the quantity equals to zero (Fig. 4).

VALUE 0.001F 0.001F 0.001H

B. Design strategy of Fuzzy--PID controller The block diagram of the whole w system is shown in Figure 5. The buck converter is inserrted between the PV source and the load. The switch of the converter is driven by the control system whose inputs are the phhotovoltaic voltage and current. The control system consists of the Fuzzy-PID, the signal conditioning and the Pulse Widdth Modulator (PWM) generator (Fig.6). The signal conditioning measures the photovoltaic current and voltage and calculates the error (e) and the derivative of error (de). The errror can be described as:

e k

Fig. 3. Circuit of buck converter

DESCRIPTION Input capacittance Output capaccitance Inductance

P

0- Vpv

k -Ppv k-1

pv k -Vpv k-1

dp

0- dv

(9)

where Ppv and Vpv is the pow wer and the voltage, which are generated by the photovoltaic source respectively. The zero is the desired value of the dP/dV (MPP) and the ‘e’ arises by this comparison. The inputs of the fuzzy-PID are the error ‘e’ and the derivative of error ‘de’=de//dt, multiplied by scaling factors Ge and Gde so as their values to t comply with the universe that the Membership Functions off the input have defined. The scaling factors are setup by the t trial and error method and equals to Ge=0.0125 and Gdde=0.00625. The output of the controller is the reference voltaage (Vref) which adjusts the duty cycle of the converter through the t PWM generator.

Fig. 5. Block diagram of the whole PV V system

Fig. 8. MF Fs of the inputs

Fig. 6. Block diagram of the proposed coontrol system

The block diagram of the Fuzzy-PIID developed in Matlab/Simulink is presented in Figure 7.

Fig. 9. Fuzzyy singleton outputs

Table III: Fuzzzy-PID rule base de Fig. 7. Block diagram of the Fuzzyy-PID

The fuzzy inference is carried out by using the Sugeno o the inputs of the method. The Membership Functions (MFs) of Fuzzy-PID controller are presented in Figuure 8. The MFs of the inputs are between the universe [-1 1]. The initial NB stands for Negative Big, the NM for Negaative Medium, the NS for Negative Small, the ZE for Zero, thhe PS for Positive Small, the PM for Positive Medium and thhe PB for Positive Big. The shape of the MFs is triangular. Thhe output MFs are presented in Figure 9 and their values are: NB=-1, N NM=-0.67, NS=-0.33, ZE=0, PS=0.33, PM=0.67 and PB B=1. The rule base of the controller (FLC) is shown in Table III. The factor ‘a’ a the factor ‘b’ which is the weight of the PD controller and which is the weight of PI controller are seet by the trial and error method. Their values are a=0.001 b=0.001

NB

NM

NS

ZE

PS

PM

PB

NB NB NB NM NS NS ZE

NB NM NM NM NS ZE PS

NB NM NS NS ZE PS PM

NB NM NS ZE PS PM PB

NM NS ZE PS PS PM PB

NS ZE PS PM PM PM PB

ZE PS PS PM PB PB PB

e NB NM NS ZE PS PM PB

C. BB-BC optimization of Fuuzzy PID (structure, MFs) The Fuzzy-PID can be further f tuned by adjusting the parameters ‘a’, ‘b’ and the output MFs by the BB-BC optimization algorithm. So thhe optimization variable x is a vector which it is given by x a b NB NM NS PS PM PB . The cost function that is used in i order to find the best solution is:

f

En

1- En

max

2

ITAE ITAEmax

(10)

where En is the total energy prroduced by the system, Enmax is the theoretical maximum outpuut energy of the system for solar irradiance 1000W/m2 and tem mperature 25OC, ITAE is the Integral Time Absolute Error and ITAEmax is the maximum theoretical Integral Time Absoolute Error for a specific amount

of time and for different environmental conditions. The changes of the environmental conditions wiith respect to time are presented in Figure 10. The ITAE is evaluated as: (11)

ITAE= t|e| dt

i number is The initial population N is set to 15 and the iteration set to 30.

D. PSO optimization of Fuzzy zy PID (structure, MFs) The Fuzzy-PID optimized by b BB-BC is compared with the Fuzzy-PID optimized by PSO. The candidate solutions produced by the PSO are calcculated according the following equations [28]: vi

k 1

wvi

k

k

c1 y1 Pbi -xi xi

k 1

xi

k

k

k

c2 y2 Pg i -xi vi

k

k 1

(12) (13)

where is the componentt of the i particle velocity at iteration k, w=0.48 is the inertiia weight, y1 and y2 are random factors in the interval [0,1], c1 =2 = and c2 =2 are constant weight factors, is the componennt of the i particle position at iteration k, Pbi is the best posittion achieved so far by particle i and Pgi is the best known possition of the entire swarm. The PSO starts with the initializatioon of a population on N particles numbered i = 1,…,N. Figure 122 presents the values of the cost function for each iteration off the PSO. At the twenty third t minimum value. The values iteration the cost function has the of the parameters that arise frrom the optimization algorithm are presented in Table V. Fig. 10. Environmental conitions with resspect to time

Figure 11 presents the values of the cost funnction with respect to iterations. At the twenty-one iteration thee cost function has the minimum value. The values of the parrameters that arise from the optimization algorithm as well as the values of the parameters before the optimization, are show wn in Table IV

Fig. 12. Cost function with w respect to iterations (PSO)

Table V: Parametters optimized by PSO PARAMETERS a b NB NM NS PS PM PB

Fig. 11. Cost function with respect to iterattions (BB-BC)

BC Table IV: Parameters optimized by BB-B PARAMETERS a b NB NM NS PS PM PB

VALUES WITHOUT OPTIMIZATION 0.01 0.001 -1 -0.67 -0.33 0.33 0.67 1

V VALUES WITH OP PTIMIZATION 0.1243 0.0144 -0.7295 -0.2632 -0.1329 0.1767 0.3005 0.7054

IV.

VALUES WITH OPTIMIZATION 0.0042 0.0041 -0.6194 -0.2000 -0.3523 0.2442 0.5607 0.7475

SIMULATION RESSULTS AND COMPARISONS

The proposed controller is tested under various environmental conditions. Thhe temperature and the solar irradiance are changing throuugh time and are presented in Figure 10. All the simulationn results arise by applying this scenario. Figure 13.a presents the voltage, the current and the power generated by the PV source s under the control of the Fuzzy-PID (optimized by BB-BC). Figure 13.b presents the control signal produced by the controller.

Fig. 13. a) Current (green), voltage (red) and power (bllue) of the PV source. b) Control Signal (Vref)

In Figure 14.a the power produced by the phhotovoltaic system under the control of the Fuzzy-PID set up by trial and error D can be seen. In method and under the optimized Fuzzy-PID Figure 14.b the control signals produceed by these two controllers are presented. The control signall of the optimized Fuzzy-PID has more oscillations than the simple s Fuzzy-PID but the overall performance of the opttimized controller regarding the power production is better. Botth controllers have good performance at steady states wheere there are no oscillations around the MPP.

Fig. 14. a) Power produced under the Fuzzy-PIID (red) and power produced under the optimized Fuzzy-PID (blue). b) Coontrol Signal (Vref) of Fuzzy-PID (red) and optimized Fuzzy-PID D (blue).

The optimized Fuzzy-PID is com mpared with the conventional MPPT method of P&O. In Figuure 15.a the power produced by the photovoltaic system under the control of the D is illustrated. In P&O and under the optimized Fuzzy-PID Figure 15.b the control signal produced by these two methodologies are presented. It is obvious that under abrupt changes of solar irradiance and under suudden changes of temperature, the Fuzzy-PID controller perforrms better than the conventional method of P&O; the MPPT is achieved faster and the amplitude of the oscillations is extreemely small in the steady state.

Fig. 15. a) Power produced undeer the P&O (red) and power produced under the optimized Fuzzy-PID (blue). b) Control Signal (Vref) of P&O (red)

In Figure 16.a the powerr produced by the photovoltaic system under the control of the optimized Fuzzy-PID with PSO and the optimized Fuzzy-PID with BB-BC can be seen. In Figure 16.b the control signals produced by these two controllers are presented. The Fuzzy–PID optimized by BBBC is better regarding the amoount of the produced energy but the control signal has more amplitude oscillation than the optimized Fuzzy PID by PSO.

Fig. 16. a) Power produced under the Fuzzy-PID optimized by PSO (red) and power produced under the Fuzzyy-PID optimized by BB-BC (blue). b) Control Signal (Vref) of Fuzzy-PID optimized o by PSO (red) and Fuzzy-PID optimized by BB-BC (blue).

In Table VI the energy prooduced by the PV source under the control of the Fuzzy-PID D, the Fuzzy-PID optimized by BB-BC, the Fuzzy-PID optim mized by PSO and the P&O are presented. The amount of enerrgy produced by the Fuzzy-PID optimized by BB-BC is greatter than the amount of energy produced by the Fuzzy-PID opptimized by the PSO and much greater than the other two MP PPT controllers. This Table also presents the Integral Absolutee Error (IAE) and the ITAE of these controllers. The IAE and ITAE of the controller optimized by PSO are lower thhan the optimized Fuzzy-PID by BB-BC and much less than thee non optimized Fuzzy-PID and the P&O. The IAE can be evaluuated as follows: IAE= |(e)| dt

(14)

Table VI: Energy, IAE and ITAE for simpple Fuzzy-PID, Fuzzy-PID (BB-BC), Fuzzy PID (PSO) andd P&O

ENERGY (W Sec) IAE ITAE

Proposed Fuzzy-PID (BB-BC) 7.559 0.006736 0.0001369

7.541

N Non optim mized Fuzzyy-PID 7.3 358

7.103

0.006486 0.0001237

0.0 0098 0022 0.00

0.0246 0.00077

Fuzzy-PID (PSO)

P&O

The proposed controller is also testedd under different environmental conditions. The changes of the t environmental conditions with respect to time are presennted in Figure 17. Figure 18.a presents the voltage, the curreent and the power generated by the PV source under the conttrol of the FuzzyPID (optimized by BB-BC). Figure 12.b prresents the control signal produced by the controller. The optimized controller successfully track the MPP in smooth envirronmental changes and the amplitude of the oscillations of thee control signal is very low.

mplitude of oscillations in the significantly short and the am steady state, is consideraably diminished. The total performance of the optimized Fuzzy-PID F by BB-BC are better than the conventional P&O, thhe non optimized fuzzy-PID and the Fuzzy-PID optimized by PS SO. Unique characteristics of thhe BB-BC algorithm over PSO method are its simplified numerical structure and its dependence on a relatively small of number parameters. Outside of common parameterrs, such as population size and convergence criteria the perforrmance of the BB-BC algorithm is controlled by the upper limit l on the search space. In contrast, a single PSO requuires at least three values of parameters. In the future research, thiss work will be further extended by adding into the optimizationn procedure both the input MFs and the scaling factors of the inputs of the Fuzzy-PID controller for MPPT of PV syystem. Another future study can be concerned with a detailed comparison c of the metaheuristic algorithms for optimization of Fuzzy-PID. F

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Fig. 17. Environmental conitions with resspect to time

Fig. 18. a) Current (green), voltage (red) and power (bllue) of the PV source. b) Control Signal (Vref)

V.

CONCLUSIONS

f MPPT issues The development of the Fuzzy-PID for successfully deals with the problem. In thiss study an off-line mechanism is successfully applied to thee controller. The parameters of the Fuzzy-PID optimized by BB-BC algorithm can lead to even better performance. Thhis architecture is simple, easy for hardware implementtation with low computational power. From the simulation results, it can be concluded that the Fuzzy-PID is effective inn achieving a highquality MPP operation under differennt environmental conditions. The optimized controller caan achieve high performance as the response time in the transient state is

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