Extension theory can be divided into extension mathematics and matter-element theory. Extension mathematics takes extension set and extension correlation function as the main indicators. Moreover, matter-element theory divides an object into characteristics and the range value of characteristic expansion through the matter-element model in the extension theory to represent its object information. By transforming the relationship of the objects, it is possible to know the proportion of each characteristic that affects the objects and clearly observe the importance of characteristics to the objects.

The matter-element model: the matter element is the basic element used to describe “objects,” determined by the name (N) of the objects, its characteristics (C), and the corresponding value (V). These three basic elements are composed to describe objects, and the matter element is represented by R, which is expressed in the mathematical model as follows: R = ( N , C , V ) , (1) where R is the basic element to describe objects, called matter element, and the three basic matter elements are the name N, characteristics C, and characteristic value V of objects.

This study combines the extension theory as the theoretical basis and proposes a smart maximum power tracking method (Chao et al., 2011; Chao et al., 2012). Analyze the P-V characteristic curve of the photovoltaic module array, take the slope error and the slope error change as the characteristics of the extension maximum power tracking method, and set up 12 categories and establish their matter-element models to calculate the correlation between them. Adjust the duty cycle (D) of the MPPT converter according to its category to achieve maximum power tracking (Chao and Li, 2010).

The characteristics of the extension theory maximum power tracking method are the slope error е, the slope error variation Δе, and the duty cycle variation ΔD. Figure 1 is a schematic diagram of the change of the slope error е and the slope error variation Δе of the P-V characteristic curve. Based on the P-V characteristic curve of the photovoltaic module array, two characteristics [slope error e (k) and the slope error change Δe(k)], defined in Eqs 5–7, have been divided into 12 categories. For example, for category #1, if slope error e is a positive and small value and the slope error change Δe is a native value, the operation point is on the left of the maximum power point (MPP). When it is close to the MPP, the increased voltage of the photovoltaic module arrays (PVMAs) can be controlled to shift the operation point to the right to track MPP. This will reduce the duty cycle of the boost converter, leading to a decrease in the output current of the module array. Accordingly, the output voltage of the PV module array will increase. As the operation point in category #1 is quite close to MPP, the duty cycle only needs to be slightly decreased to slightly increase the voltage of the PVMA. The operation point can then shift to the MPP in small steps, not likely causing great oscillation to and from the MPP or causing power generation loss. Therefore, if the operation point is in category #1, ΔD = −0.001. In other categories, the ΔD positive and negative values and sizes are determined by the same method. Take the operation point farther away from the left side of MPP as an example. In order to quickly track MPP, the voltage of the PVMA must be substantively increased. Therefore, the duty cycle D is substantively reduced, making the ΔD negative value enlarged, such as category #3 in Figure 1. At this time, ΔD = −0.05. On the contrary, for the operation point farther away from the right side of MPP, in order to quickly track MPP from the left side, the voltage of the photovoltaic module array must be substantively reduced. Therefore, the duty cycle D of the boost converter needs to be amplified, leading to a substantial increase in the output current of the PV module array. The output voltage of the PV module array can be quickly lowered, prompting the operation point to quickly shift toward the direction of MPP on the left side. By considering #12 in the category in Figure 1, for example, e is a negative value and Δe is a positive value. However, the operation point is on the right of the MPP and farthest away from MPP. Therefore, ΔD = +0.05 (positive maximum) is set. The slope error, slope error variation, and duty cycle variation of each category are shown in Table 1. According to the P-V characteristic curve of the photovoltaic module array obtained during solar irradiation 200 ∼1,000 W / m 2 , the matter-element model of 12 categories of classical domains can be established. In other words, using Eq. 1 extension matter-element model and Table 1 (12 categories classified using two characteristics e and Δe), the matter-element model of 12 categories of classical domains in Table 2 can be established. The results are shown in Table 2. The neighborhood domain established by the maximum and minimum values ​​of each characteristic classical domain is shown in Eq. 8: e ( k ) = P p v ( k ) − P p v ( k − 1 ) V p v ( k ) − V p v ( k − 1 ) , (5) e ( k + 1 ) = P p v ( k + 1 ) − P p v ( k ) V p v ( k + 1 ) − V p v ( k ) (6) Δ e ( k ) = e ( k + 1 ) − e ( k ) (7) R F = ( F , C , V q ) = [ F e < − 30,10 > Δ e < − 100,100 > ] (8)

Figure 2 is the architecture diagram of the maximum power tracking system based on the extension theory. A boost converter is designed between the photovoltaic power generation system and the load (Chao and Lee, 2011). The duty cycle D of the converter is changed to track the maximum power point of the photovoltaic power generation system.

The boost converter in this study is designed in continuous conduction mode (CCM); that is, the operating conditions are I min ≥ 0 , so the inductance value of the boost converter is designed (Hart, 2016) as L ≥ D ( 1 − D ) 2 R 2 f . (9)

From the function D ( 1 − D ) 2 in Eq. 9, the inductance L reaches a maximum value when D = 1/3. Therefore, the design inductance value is used when D = 1/3.

The operating frequency of the maximum power tracking controller used in this paper is 20 kHz. Because the operating power of the subsequent test system is about 750 W, the rated power P of the converter is set to 900 W, and the maximum output operating voltage V _o is 400 V, which is determined by R = V _o ² /P and can calculate the resistance value of load R to be 178 Ω. Then, calculate the required L value of 329 μH according to Eq. 9, and then multiply this L value by a margin of 1.25 times to ensure that the inductor current operates in continuous conduction mode. Hence, the inductor value selected in this study is 0.5 mH.

The change in charge of the capacitor during the on-time of the switch is (Hart, 2016) c = D R f ( Δ V o / V o ) , (10)

where Δ V o / V o is the ripple ratio of the output voltage. In order to effectively reduce the ripple of the output voltage, the ripple ratio of the output voltage is set to Δ V o / V o = 0.01 , and this value is substituted into Eq. 10 to obtain the capacitance value of 9.3 μF. As for the withstand voltage of the capacitor, it must be greater than the output voltage V _o (400 V). By considering the convenience of part acquisition, a capacitor with a specification of 470 μF/450 V is selected.

Select the switch and the component part of the diode because the withstand voltage of both must be greater than the output voltage of 400 V when it is turned off. According to Hart (2016), calculate the switch component. The current rating needs to be greater than I _max (2.93A). Based on the convenience of laboratory acquisition, IRFP460 is selected as the switching element, and its specification is 500 V/20 A. The selected diode is FMPG5FS, and its specification is 1500 V/10 A.

This study proposes a single-phase full-bridge inverter circuit structure, as shown in Figure 3, for converting the DC power of the photovoltaic module to the power grid, which uses dual-loop control. The outer loop is a DC-link voltage loop, using the error between the reference voltage (here is set to 400 V) and the practical output voltage of the DC link to generate the current command of the inner loop. The inner loop is the inverter output voltage and current control loop, and the error is used to generate the PWM control signal to control the AC output voltage. PWM adopts the sine wave pulse width modulation switching method to generate the trigger signal of the switch, and L _f -C _f constitutes a low pass filter to attenuate the high-frequency switching of the inverter. The output end makes the output voltage a sine wave of the grid power frequency.

The inverter of the single-phase main system has only a single vector and cannot directly generate a dual-axis DC control system through Park’s transformation matrix (Krause et al., 2002). Therefore, if one wants to analyze the active power and reactive power of a single-phase system, one needs to use the imaginary quadrature-axis system (Azab, 2017). The single-phase inverter output voltage v g ( = 2 V r m s ⁡ sin ⁡ θ v ) is equal to the direct-axis (α axis) value v α of the stationary reference frame expressed as follows: v α = 2 V r m s × sin ⁡ θ v . (16)

Then, use the imaginary vector to generate the quadrature-axis (β-axis) component with a phase difference of 90^o from the direct axis as v β = v α ∠ 90 ° = 2 V r m s × cos ⁡ θ v (17)

Therefore, the output voltage of the single-phase inverter can be expressed as v g = v α + j v β (18)

In the same way, the current components of the direct axis (α-axis) and the quadrature axis (β-axis) of the stationary reference frame of the single-phase inverter output current i g can be expressed as i α = 2 I r m s × sin ⁡ θ i (19) i β = i α ∠ 90 ° = 2 I r m s × cos ⁡ θ i (20)

Therefore, the output current of the single-phase inverter can be expressed as i g = i α + j i β (21)

Figure 5 is a schematic diagram of the single-phase inverter system vector projection on a stationary reference frame, so the system’s active power and reactive power can be obtained from Eq. 22 and Eq. 23, respectively. The active output power of the single-phase inverter can be expressed as P = ℜ e ( 1 2 v g × i g ∗ ) = V r m s × I r m s × cos ( θ v − θ i )   = 1 2 ( 2 V r m s 2 I r m s ⁡ sin ⁡ θ v ⁡ sin ⁡ θ i + 2 V r m s 2 I r m s ⁡ cos ⁡ θ v ⁡ cos ⁡ θ ) = 1 2 ( v α i α + v β i β ) (22)

In the same way, the instantaneous reactive power of the single-phase inverter system can be obtained as Q = ℑ m ( 1 2 v g × i g ∗ ) = V r m s × I r m s × sin ( θ v − θ i )    = 1 2 ( 2 V r m s 2 I r m s ⁡ sin ⁡ θ v ⁡ cos ⁡ θ i − 2 V r m s 2 I r m s ⁡ cos ⁡ θ v ⁡ sin ⁡ θ i )    = 1 2 ( v β i α − v α i β ) (23)

From Eq. 22 and Eq. 23, the output active power and reactive power of the inverter can be calculated. When the direct-axis current component i α of the reference coordinate of the stationary frame is controlled, it will affect the phase angle difference θ ( = θ v − θ i ) between the voltage and current of the single-phase system at the same time and then affect the active and reactive power output of the inverter. In the same way, i β is the same when controlling the quadrature axis current component of the stationary reference frame.

From (Chen and Chien, 2014), we can use Park’s transformation to convert the stationary frame reference coordinate axis into a synchronous reference coordinate and fix the voltage vector v g in the synchronous rotation reference frame as shown in Figure 5. On the direct axis (d-axis), the relational formula can be obtained as v d = v g , v q = 0 . (24)

Eq. 23 can be simplified to P = 1 2 v d i d , Q = 1 2 v d i q (25)

It can be seen from Eq. 25 that the active power and reactive power have been decoupled. The direct axis current i d of the synchronously rotating reference frame is responsible for controlling the active power. In contrast, the quadrature-axis current i q is responsible for controlling reactive power.

Figure 6 is a diagram of the structure controlling the output voltage and current phase angle difference of the smart inverter. After an error occurs in the command value of the reactive power, the direct-axis current command I d ∗ in Eq. 22 and the quadrature-axis current command I q ∗ in Eq. 23 can be obtained through the proportional-integral controller. Then, Eq. 26 can be used to calculate the phase angle difference θ between the voltage and the current and obtain the θ value to generate a unit sine wave. Modify the unit sine wave to change the output active power and reactive power of the inverter: θ = tan − 1 ( I q I d ) ≜ θ v − θ i . (26)

When the smart inverter is parallel-connected to the grid system, it must comply with the grid-connected regulations. Because several renewable energy sources are connected to the grid, the grid power voltage will fluctuate greatly. Therefore, it is necessary to pay attention to the requirements for voltage fluctuations in the grid-connected regulations. When the smart inverter starts to perform voltage-power control due to the fluctuation of the grid voltage, if the grid voltage or frequency still cannot operate stably and exceeds the normal range, the smart photovoltaic inverter must trip by itself to avoid damage.

After calculating the error between the actual grid power terminal voltage and the reference voltage (220 V), the reactive power command Q   r e f can be obtained through the proportional-integral controller and the reactive power limiter. Then, substituting this command value into Figure 6 can make the smart inverter provide or absorb reactive power to achieve the purpose of stabilizing the grid power terminal voltage.

Figure 7 is the voltage-power control flowchart. First, detect the voltage at the grid-connected point of the inverter, and perform reactive power control according to the grid power voltage per unit (p.u.). When equal to 1, the PF is maintained at 1. At this time, no reactive power is absorbed or provided, and only the active power is provided. When the per unit value of the grid power voltage is greater than 1 p.u. and less than 1.03 p.u., the PF will be controlled to decrease to 0.9 gradually (lagging). When the per unit value of the grid power voltage is greater than 0.97 p.u. and less than 1 p.u., the PF will be controlled to decrease to 0.9 gradually (leading). The above control adjusts the reactive power to return the grid power voltage normal range. However, if the power factor has been adjusted to 0.9 leading or lagging, the grid power voltage cannot effectively restrain its rise or fall, and when the system voltage is greater than 1.03 p.u. or less than 0.97 p.u., the photovoltaic power generation system will be disconnected from the grid system.

Figure 8 shows the simulation waveforms of maximum power tracking using the traditional P&O and extension theory-based methods, respectively. It can be observed from Figure 8 that the convergence time of the P&O method is 0.741 s. However, the extension theory-based MPPT convergence time is 0.32 s. In particular, the perturbation factor in the P&O method is changed in the duty cycle of the boost converter, which is hereby set as △D = 0.02. As for the power ripple, when the P&O method is adopted, the value is 18.131 W. When the extension theory-based method is used, the value is only 4.127 W. The simulation results show that when the control system adopts the extension theory for MPPT, the output power of the photovoltaic module array can be quickly and stably controlled at the MPP P max = 744 W shown in Table 3, which proves the maximum power method of the extension theory does have a faster convergence rate and better steady-state performance than the traditional P&O method. In order to present the comparison results of the performance of extension theory-based and P&O MPPT methods, the simulation results under the standard test condition (STC) are listed in Table 5, showing that the extension theory-based MPPT method is superior to the P&O MPPT method, whether in tracking speed or steady-state ripple.