A photovoltaic array simulator based on current feedback fuzzy PID control

Abstract

A photovoltaic array simulator is designed based on a Buck DC converter with a fuzzy PID control algorithm to account for current feedback. The conventional PID controller cannot self-tune the parameters, therefore the new algorithm uses a fuzzy controller to solve the problem. The difference between the reference current and the real-time feedback current of the Buck converter and its rate of change are regarded as two input variables of the fuzzy controller. The fuzzy controller outputs the adjustment quantities, which are then used to adjust the PID parameters. Next, the duty cycle of the electronic power switch is adjusted by a closed-loop fuzzy PID algorithm to make the output current of the simulator work at the anticipated point of the photovoltaic array on the I-V characteristic curve and realize the simulation of photovoltaic characteristics. Results of the simulation and experiment indicate that the proposed fuzzy PID control simulator can not only accurately simulate the static output characteristics of a photovoltaic cell, but can also rapidly realize the dynamic characteristics when the load or the external environment changes. The approximate error was less than 3.6% , the overshoot was less than 3.5% , the ripple coefficient was less than 3% , and the tracking time was approximately 0.3s. The fuzzy PID control simulator can work well as PV array experiment equipment for the research and development of photovoltaic systems.

1 Introduction

In general, a photovoltaic (PV) power generation system (PVPGS) is primarily composed of PV cells, controllers, power converters and batteries. Utilizing the PV cells to conduct PVPGS experiments is often restricted by seasons and weather, especially at night, as when the environment changes the PV cells cannot maintain a stable output test. Moreover, the PV cells greatly increase the cost of testing. Therefore, using PV cells to test the performance of PVPGS induces major inconvenience. As a consequence, the PV simulator [4, 11] is an ideal solution to simulate PV cell output I-V characteristics under different temperatures and illumination conditions, free from climate change influence, and provide favorable conditions for the performance test of PVPGS. Currently, depending on the different types of simulators, it may be an analog simulator, digital simulator or a mixed simulator. A digital simulator [7, 9] combines the power electronics technology and real-time control technology, presents high precision and controllable output characteristics, and simulates a large-capacity power system. Therefore, a digital simulator is well suited to the research and development of the PVPGS, and has been extensively studied [14]. Digital simulators use algorithms such as the secant method, numerical iteration method and point by point approximation (PPA) method. Among these, the secant method requires much computation, as it requires the solving of complex transcendental equations. The numerical iteration method and PPA are closely related to the step size; the system will produce greater oscillation if the step size used is too large. To achieve a stable output, the step size must be small, but will then induce a slow dynamic response.

Fuzzy logic [1, 16] plays an important role in accounting for uncertainty when making decisions in control systems. Until 2013, all studies on fuzzy derivative, fuzzy integral and fuzzy differential equations had been conducted based on type-1 fuzzy sets (T1FSs). Mehran Mazandareni define type-2 fuzzy fractional derivatives and present type-2 fuzzy fractional differential equations (T2FFDEs) [12, 13]. Housheng Zhang designed a simulator using a dual-loop control with a PI regulator [8], which can enhance the system performance. However, in case of a controlled system that is strongly time-varying and nonlinear, a pre-set PI parameter cannot meet its requirements. As a consequence, this paper presents a fuzzy PID control algorithm with a BUCK circuit used as a power converter. In the proposed method, the output current i and voltage u of the BUCK circuit are sampled in real-time. The voltage sampling value is substituted into the PV cell engineering mathematical model to generate a reference current i_ref. The diffrence e = iref-i and its rate of change de/dt are used as two input variables of the fuzzy controller. The outputs of the fuzzy controller are used to modify the PID parameters, and then control the BUCK power converter using a PWM method output corresponding current and voltage according to the I–V curve of PV cells. The operating point of the simulator will be made to match the output characteristics of the PV cells. The block diagram of the digital simulator is shown inFig. 1.

A fuzzy PID PV simulator consists of a BUCK converter and a controller based on TMS320F2812. The controller generates a PWM control signal to control the converter according to the PV cell I–V curve. Simulation and experiment results indicate that the simulator possesses a fast dynamic response and stable output characteristic under different conditions, with an overshoot less than 3.5% , a steady-state error less than 3.6% , a ripple factor less than 3% , and tracking time less than 0.3s.

2 Output characteristic of PV cell

The I–V output characteristics [15] of a PV cell are nonlinear, and are affected by the illumination intensity and condition temperature, as shown in Fig. 2. The current-voltage curve of a PV cell at 25°C and 50°C are respectively shown in Fig. 2(a) and (b).

Simplified mathematical models to describe the output characteristics of PV cells generally use the short-circuit current I _sc , open circuit voltage V _oc , the maximum power point current I _m and the maximum power point voltage V _m under standard test conditions (S_ref = 1000 W/m², T_ref = 25°C). $I=I_{\mathrm{sc}}[1-C_1({\mathrm{e}}^{V/C_2{V}_{\mathrm{oc}}}-1\left)\right]$(1)$C_2=\left(\frac{V_{\mathrm{m}}}{V_{\mathrm{oc}}}-1\right){\left[\ln \left(1-\frac{I_{\mathrm{m}}}{I_{\mathit{sc}}}\right)\right]}^{-1}$(2)$C_1=(1-I_{\mathrm{m}}/I_{\mathrm{sc}}){\mathrm{e}}^{-V_{\mathrm{m}}/C_2{V}_{\mathrm{oc}}}$(3)

When the illumination intensity S and temperature T are non-standard test conditions, I_sc, V_oc, I_m, V_m and the coefficients C₁, and C₂ will change. Setting ΔS, ΔT as the variation from the standard test conditions of illumination and temperature respectively, four basic parameters can be calculated by utilizing the following formula, known as the PV cell engineering mathematical model: ${\mathit{Isquo}}_{\mathrm{sc}}=I_{\mathrm{sc}}(1+a\Delta T)\frac{S}{S_{\mathrm{ref}}}$(4)$I_m^{\prime }=I_{\mathrm{m}}(1+a\Delta T)\frac{S}{S_{\mathrm{ref}}}$(5)${\mathit{Vsquo}}_{\mathrm{oc}}=V_{\mathrm{oc}}(1-c\Delta T)(1+b\Delta S)$(6)$V_{\mathrm{m}}^{\prime }=V_{\mathrm{m}}(1-c\Delta T)(1+b\Delta S)$(7) where a = 0.0025/°C; b = 0.5; c = 0.00288/°C

Replacing I_sc, V_oc, I_m and V_m in Formulas (1) through (3) with $I_{\mathrm{sc}}^{\prime }$, $V_{\mathrm{oc}}^{\prime }$, $I_{\mathrm{m}}^{\prime }$ and $I_{\mathrm{m}}^{\prime }$, PV cell output V - I characteristics can be obtained.

3 Fuzzy PID controller

3.1 Control objective of the simulator

When the illumination intensity and temperature remain constant, the PV cell demonstrates a unique I-Vcharacteristic curve. Corresponding to a determined load R, I-V curves of load must meet the output I-V curve of the PV cells at a point; for example, if the load changes from R₂ to R₁, the intersection must change from point E to point C (R₂ is less than R₁). The control target of the simulator is to allow the output characteristic of the converter run on the point C from point E by controlling the duty cycle d of the DC-DC circuit, as shown in Fig. 3.

3.2 Design of fuzzy PID controller

Due to its nonlinearity, it is very difficult to describe the characteristics of PV cells using an accurate mathematical model. As the PV generation system has strong uncertainty, a fuzzy PID control is used to resolve these problems and realize the tracking of the output current. The structure diagram of a two-dimensional fuzzy PID controller is shown in Fig. 4. In Fig. 4(a), V_n and I_n are the output voltage and current of the simulator in sampling time n, respectively. V_n is substituted into the above engineering mathematical model, which is composed of the Formulas (1) through (7), to calculate the theoretical current I_ref of PC cell. Set e (n) is the error of I_ref and I_n, and ec(n) is the rate of change of e (n). E (n) and EC (n) are the current error and current error variances respectively quantified by the quantized factor K_e and K_ec, and serve as two input variables of the fuzzy controller. The variable quantities (ΔK_p,  ΔK_i,  ΔK_d) corresponding to the proportional, differential and integral coefficient in the PID controller can be obtained by means of fuzzy inference, defuzzification and range conversion (Ka, Kb, Kc). The duty cycle adjustment value ΔD of the DC / DC converter is calculated by the PID controller, and a new duty cycle D(n) = D(n-1)+ΔD can be obtained. The converter operating state is adjusted according to the duty cycle D to make the output operating point of the simulator run on the I-V characteristic curve of the PV cell at point C, as shown in Fig. 3.

3.3 Fuzzy algorithm design

The universal ranges of input and output variables are restricted to {3, 2, 1, 0, 1, 2, 3}; fuzzy subsets {NB, NM, NS, ZO, PS, PM, PB} correspond to negative big, negative middle, negative small, zero, positive small, positive middle, and positive big, respectively. The membership functions of input E, EC and output ΔK_p/ΔK_i/ΔK_d adopt the isosceles triangle function, as shown in Fig. 5.

In addition, in order to ensure the input linguistic variance E and EC in its universal range, the quantitative factors K_e and K_ec are used respectively before the E and EC are inputted to the fuzzy controller. The process of determining the fuzzy quantization factors is detailed below.

As shown in Fig. 4, the following equation can be obtained Δe (n) = I_ref (n) - I (n) ,  ec (n) = e (n) - e (n - 1) ,  e (n) _max = Isctake K_e = 3/I_sc. Then: $\begin{array}{ccc}\hfill \mathrm{E}& \hfill =\hfill & e\left(n\right)K_{\mathrm{e}}=3e\left(n\right)/I_{\mathrm{sc}}\hfill \\ \hfill & \hfill =\hfill & 3e\left(n\right)/e{\left(n\right)}_{\mathit{max}}\in [-3,3]\hfill \end{array}$(8)$e\left(n\right)=I_{\mathit{sc}}-C_1{I}_{\mathit{sc}}(e^{V_B/C_2{V}_{\mathit{oc}}}-1)-I_B$(9)$e\left(n-1\right)=I_{\mathit{sc}}-C_1{I}_{\mathit{sc}}(e^{V_A/C_2{V}_{\mathit{oc}}}-1)-I_A$(10)$\begin{array}{ccc}\hfill \mathit{ec}\left(n\right)& \hfill =\hfill & e\left(n\right)-e(n-1)\hfill \\ \hfill & \hfill =\hfill & C_1{I}_{\mathit{sc}}(e^{V_A/C_2{V}_{\mathit{oc}}}-e^{V_B/C_2{V}_{\mathit{oc}}})\hfill \\ \hfill & \hfill \hfill & +V_A-V_B/R_1\hfill \end{array}$(11)

Which can simplify to:

$\begin{array}{ccc}\hfill \mathit{ec}\left(n\right)& \hfill =\hfill & e\left(n\right)-e(n-1)\hfill \\ \hfill & \hfill =\hfill & C_1{I}_{\mathit{sc}}{e}^{V_B/C_2{V}_{\mathit{oc}}}(e^{V_A-V_B/C_2{V}_{\mathit{oc}}}-1)\hfill \\ \hfill & \hfill \hfill & -V_B-V_A/R_1\hfill \end{array}$(12)

Since V_B >  V_A, a minimum value 1v of each voltage variation is set in the program. Thus, (V_B >  V_A) min = 1, the maximum value of V_B is approximately equal to the open circuit voltage V_OC, and the following equation can be obtained from Formula (12): $\mathit{ec}{\left(n\right)}_{\mathit{max}}=C_1{I}_{\mathit{sc}}{e}^{1/C_2}(e^{-1/C_2{V}_{\mathit{oc}}}-1)-1/R_1$(13)

Take K_ec = 3/ ec(n)_max: $\mathit{EC}=\mathit{ec}\left(n\right)K_{\mathit{ec}}=3\mathit{ec}\left(n\right)/\mathit{ec}{\left(n\right)}_{\mathit{max}}\in [-3,3]$(14)

According to the output characteristics of the PV cell and the control objective of the simulator, self-tuning rules of K_p, K_i and K_d follow three basic principles:

When the error E is large, in order to accelerate the response speed of system and prevent system loss of control due to the large deviation, system should adopt the larger K_p and smaller K_d. In order to protect the system from a large overshoot, the value of the K_i also should be smaller.

When the error E and error change rate EC is medium, in order to reduce overshoot of the system and ensure an ideal dynamic response speed, K_p and K_d should be smaller; the value of the K_i is appropriate.

When the error E is small, in order to provide a good steady-state performance, the values of K_p and K_ishould increase. Simultaneously, in order to avoid oscillation of the output response and consider the anti-jamming performance of the system, K_d should be appropriate. When the EC is small, K_d takes on a larger value; when EC is large, K_d should be smaller. The fuzzy rules of the controller are concluded as shown in Tables 1 through 3.

Finally, using the gravity method of defuzzication, the precise controlled quantity is obtained.

4 Main parameters of the simulator test circuit

The basic circuit diagram and experimental platform are shown in Fig. 6. The main circuit of the system is composed of a step-down rectifier circuit, detection circuit, power supply, control circuit and PWM driving circuit. The output voltage of the rectifier circuit is 62∼67 V. The controller uses TMS320F2812 as its core. The parameters of the PV cell at 25°C and an irradiation of 1000 W/m² are listed in Table 4. The parameters of the fuzzy PID controller are as follows: K_a = 0.1, K_b = 1, K_c = 1, K_p = 0.1; K_i = 50; K_d = 100; K_e and K_ec are detailed in Section 3.3.

5 Analysis and discussion of simulation and experiment results

5.1 Analysis of simulation results

Simulation experiments are conducted using the proposed fuzzy PID algorithm. Figure 7 depicts the tracking waveforms during the system startup process and the load mutation. Figures 8 and 9 depict the tracking performance during illumination intensity change and temperature altering, respectively. In each figure, the load current is shown by a dotted line and the theory current is shown by a solid line.

5.1.1 Tracking performances during the system startup process and load mutation

To study the tracking performance of the simulator during the startup process and the load mutation, a simulation under standard test conditions was initically conducted. Figure 7(a) shows the waveforms during the system startup from the initial operating point, at which the load R = 10Ω and then transforms to 16Ω at 0.2s. Figure 7(b) shows the waveforms during the system startup from the initial operating point, at whichwhich the load R = 16Ω and then transforms to 10Ω at 0.2s.

Figure 3 depicts the I-V characteristic curve of the PV cell. If the load R of the simulator is small, the working point is located at a point with higher voltage and lower current. On the contrary, if the load R of the simulator is large, the working point will be a point with higher current and lower voltage. According to the working principle, the dynamic performance of the simulator is related to the difference between the load current and the target point current; a greater difference requires a longer settling time.

Figure 7(a) shows the current tracking waveform for the startup with a small load; the settling time of the startup process is 0.07s, and its tracking time is 0.04s when the load changes from 10Ω to 16Ω at 0.2s.Figure 7(b) shows the current tracking waveform for the startup with a large load; its settling time of the startup process is 0.05s, and its tracking time is 0.1s when the load changes from 16Ω to 10Ω at 0.2s. The entire simulation approximate error is 0.8% , and the ripple coefficient is 0.6% .

5.1.2 Simulation under varying illumination intensity

The ambient temperature T = 30°C, load R = 10Ω, and the illumination intensity S mutates at 0.2s from 1000 W/m² to 800 W/m². The simulation waveforms are shown in Fig. 8. Figure 8 demonstrates that the tracking time of the simulation is approximately 0.12s, and the steady-state error and ripple factor are approximately 0.8% and 0.6% , respectively. Results indicate that when the weather varies, the fuzzy PID algorithm possesses good dynamic response speed and steady-state accuracy.

5.1.3 Simulation under varying temperature

The ambient temperature S = 1000 W/m², load R = 8Ω, and the temperature T changes at 0.2s from 70°C to 10°C. The simulation waveforms are shown in Fig. 9. Figure 9 indicates that the tracking time of the simulation is approximately 0.08s,with a steady-state error and ripple factor of approximately 1% and 0.6% , respectively. The simulation results indicate that when the weather varies, the fuzzy PID algorithm possesses a preferable dynamic response speed and excellent tracking performance.

5.2 Analysis of experiment results

The experiments were conducted on the experimental platform shown in Fig. 6. In this section, three kinds of experiments are proposed and analyzed: (1) System startup process; (2) Dynamic performance with load mutation; (3) Dynamic performance with illumination mutation.

5.2.1 Tracking features during the system startup process

Figure 10 shows the waveforms during the startup process of simulations under the following conditions: ambient illumination S = 1100 W/m², load R = 10Ω, temperature T = 30°. Figure 10(a) shows the waveforms of the proposed fuzzy PID controller, and Fig. 10(b) represents the traditional point-by-point approximation (PPA) method.

According to Fig. 10(a), the tracking time of the proposed fuzzy PID is approximately 200 ms, the steady-state error is approximately 2% and the ripple coefficient is 2.8% . The tracking time of the traditional method is approximately 260 ms, the steady-state error is approximately 3% and the ripple coefficient is 3% , as shown in Fig. 10(b). As a consequence, the fuzzy PID algorithm demonstrates better rapidity and steady-state performance.

5.2.2 Dynamic tracking performance with load change

In this section, the dynamic response performance is analyzed when the load changes under the following conditions: ambient illumination S = 1100 W/m², load R = 10Ω, temperature T = 30°. Figures 11 and 12 represent the waveforms corresponding to the proposed fuzzy PID algorithm and the traditional PPA method, respectively. Figures 11(a) and 12(a) show the waveforms when the load changes from 8Ω to 16Ω. Figures 11(b) and 12(b) show the waveforms when the load changes from 16Ω to 8Ω.

Based on Fig. 11(a), the response time of the fuzzy PID is nearly 180 ms when the load changes from 8Ω to 16Ω; Fig. 11(b) shows that the response time is approximately 320 ms when the load changes from 16Ω to 8Ω, and the steady-state errors under the two conditions are approximately 2.2% . According to Fig. 12(a), the response time of the PPA method is approximately 200 ms when the load changes from 8Ω to 16Ω,; Fig. 12(b) shows that the response time is approximately 320 ms when the load changes from 16Ω to 8Ω, and the steady-state errors under the two conditions are approximately 2.6% .

5.2.3 Dynamic tracking performance with illumination change

This experiment verifies the dynamic response performance when the illumination intensity changes from 1200 W/m² to 800 W/m² under the following conditions: ambient temperature T = 30°, load R = 11Ω. According to Fig. 13(a), the tracking time of the fuzzy PID is approximately 110 ms when the illumination intensity changes, and the steady-state error is approximately 2.2% . According to Fig. 13(b), the tracking time of the traditional PPA algorithm is approximately 200 ms, and the steady-state error is approximately 3% .

5.2.4 Dynamic performance with environment temperature change

This experiment verifies the dynamic tracking performance when the environment temperature changes from 30°C to 70°C under the following conditions: ambient illumination S = 1100 W/m², load R = 6Ω. According to Fig. 14(a), the tracking time of the fuzzy PID is approximately 120 ms when the environment temperature changes, and the steady-state error is approximately 2.2% . According to Fig. 14(b), the tracking time of the traditional PPA algorithm is approximately 300 ms, and the steady-state error is approximately 3.2% .

5.3 Static characteristics of simulator during experiments under different conditions

To test the ability of the simulator to simulate the PV cell I-V characteristic curve, three experiments are conducted under the folloing weather conditions: (1) S = 1100 W/m², T = 30°C; (2) S = 800 W/m², T = 30°C; (3) S = 900 W/m².

During the experiment, the load resistance changed gradually from 0Ω–200Ω, and the controller using the proposed control algorithm output current and voltage according to the I-V characteristic curve of PV cells. Experimental results were measured in addition to load resistance changes. Figure 15 shows the PV cell theoretical I-V curve and experimental data points under different weather conditions. Experimental results indicate that the simulator output agrees well with the PV cell theoretical I-V curve.

6 Conclusion

A fuzzy PID control PV array simulator based on current feedback is designed and investigated. The fuzzy controller outputs the adjusted quantities, which are used to adjust the PID parameters. Then, the duty cycle of the electronic power switch is adjusted by a closed-loop fuzzy PID algorithm. Simulation and experimental results indicate that the approximate error is 3.6% , the overshoot is less than 3.5% , the ripple coefficient is less than 3% , and the tracking time approximately 0.3s. The proposed algorithm can not only accurately simulate the static output characteristics of PV cells, but can also rapidly realize the dynamic characteristics when the load or the external environment changes. The fuzzy PID control simulator can work well as PV array experimental equipment for the research and development of photovoltaic systems.