Study on a Simpliﬁed Structure of a Two-Stage Grid-Connected Photovoltaic System for Parameter Design Optimization

: Conventional parameter designs of two-stage grid-connected photovoltaic (PV) system relied on its mathematical model of the cascade structure (CS), but the procedure is excessively cumbersome to implement. Besides, for a two-stage converter system, the coupling interaction between the power converters can directly lead to a poor parameter design. To overcome this drawback, this paper uses a simpliﬁed structure (SS) of single-phase two-stage grid-connected PV system to better design the parameters of the front-stage dc-dc converter. After establishing the small-signal model for SS and CS in the PV system, the relative eigenvalue sensitivity is used as the criterion for judging the inﬂuence of some parameters on the stability of the two structures. The stable boundary of MPPT control parameters is compared and discussed in SS and CS, respectively. In addition, the relationship between the front-stage dc-dc converter and the rear-stage dc-ac inverter is analyzed by the modal participation factor calculated in CS. An experiment is also performed at the end of this paper to further verify the feasibility of using SS to design the parameters of the dc-dc converter in the PV system.


Introduction
Renewable energy sources, such as wind and solar energy, have begun to replace traditional energy sources as the main source of energy supply in many countries. With the decreasing cost of photovoltaic (PV) modules [1] and the growing utilization rate of the PV cells [2], the application of PV systems has greatly expanded. Compared with the centralized inverter for the high-power PV grid-connected system, the micro-inverter has some certain advantages, such as reducing the cost and ensuring working at the maximum power point, for low power distributed grid-connected PV systems. According to the arrangements and structural characteristics of the dc bus, the topology of the micro-inverter can be roughly categorized into three types, namely, a cascade converter with a dc bus, an inverter with a pseudo dc bus, and an inverter without a dc bus [3]. Furthermore, these structures involve various types of converters, including the Boost converter, Buck-Boost converter [4,5], Flyback converter [6], and resonant push-pull dc-dc converter [7]. Power electronic converter is an inherently strong nonlinear system and it presents various nonlinear phenomena [8][9][10]. Thus, in order to design the circuit parameters and control parameters of the PV system, it is necessary to establish the explicit model and analyze its stability.
At present, the small-signal method is usually employed to establish an effective model to analyze the PV power generation system. Wang et al. [11] introduced the small-signal model of the of the dc-dc Boost converter is considered to be constant voltage source in this paper. Based on the criterion that dc bus voltage ripple is kept within a certain control range, SS with MPPT controller is shown in Figure 2, in which Cin and Lb are the input filter capacitor and the energy storage inductance, respectively, and the equivalent series resistance (ESR) of Cin is represented by RCin, RLb is the winding series resistance of Lb.  To analyze the stability of SS, a suitable mathematical model must be established. Since the parameters of open circuit voltage UOC, short-circuit current ISC, maximum power point voltage UM and current IM are usually provided in the data sheet of PV array, the fitting model [27] to describe the nonlinear relationship between PV array output current ipv and voltage upv are employed in this research. The PV array model can be expressed as Suppose the system is always operating in continuous current mode (CCM). Using the state space averaging method, the averaged equations corresponding to Figure 2 can be derived as follows of the dc-dc Boost converter is considered to be constant voltage source in this paper. Based on the criterion that dc bus voltage ripple is kept within a certain control range, SS with MPPT controller is shown in Figure 2, in which Cin and Lb are the input filter capacitor and the energy storage inductance, respectively, and the equivalent series resistance (ESR) of Cin is represented by RCin, RLb is the winding series resistance of Lb.   To analyze the stability of SS, a suitable mathematical model must be established. Since the parameters of open circuit voltage UOC, short-circuit current ISC, maximum power point voltage UM and current IM are usually provided in the data sheet of PV array, the fitting model [27]  Suppose the system is always operating in continuous current mode (CCM). Using the state space averaging method, the averaged equations corresponding to Figure 2 can be derived as follows To analyze the stability of SS, a suitable mathematical model must be established. Since the parameters of open circuit voltage U OC , short-circuit current I SC , maximum power point voltage U M and current I M are usually provided in the data sheet of PV array, the fitting model [27] to describe the nonlinear relationship between PV array output current i pv and voltage u pv are employed in this research. The PV array model can be expressed as

PV Array
Here, expansion Equation (1) at maximum power point, it can be calculated that Suppose the system is always operating in continuous current mode (CCM). Using the state space averaging method, the averaged equations corresponding to Figure 2 can be derived as follows where d b is the duty cycle of the power switch in the dc-dc converter. Since the applicable scope of the transfer function is defined as a linear time-invariant system, the linearization of Equation (1) at the maximum power point is Marking the steady state values of u Cin and i Lb at maximum power point as U Cin and I Lb , respectively. Then, adding small signal disturbance to the steady-state value and ignoring the steady-state quantity, Equation (2) can be expressed as follows after performing the Laplace transform.
Here,û pv ,û Cin ,î Lb , andd b are all the small disturbance signal.
Thus, the open-loop transfer function from the control input to the PV array output voltage can be derived as where the denominator coefficients are In general, a classical Boost converter using small-signal model in CCM reveals that its G vd (s) contains two real zeros in the S-plane [28]. One is a left half plane zero (LPHZ) due to the parasitic capacitance resistance, another is a right-half-plane zero (RPHZ) being peculiar to Boost topology, which is related to filter inductance, load resistance, and duty cycle. As shown in Equation (5), when the dc bus capacitance can be replaced by a constant voltage source, a stable zero is retained in SS and the RPHZ is eliminated. Since dc-dc Boost converter is connected to a dc-ac inverter in the two-stage PV system, SS is used as the basis of subsequent analysis.

Parameter Design
The PV array parameters in SS are shown in Table 1. As usual, the duty cycle d b is set as 0.5. So, the dc bus rated voltage U dc can be inferred to be 36 V. To meet the requirement both of efficiency and noise interference, the switching frequency f s of the dc-dc converter is set at 20 kHz. The function of the input filter capacitor C in is to reduce the fluctuation of the voltage u pv in the PV array, which improves the output efficiency of the PV array. The value of the inductor L b in the Boost converter should be able to ensure that the inductor current ripple is limited to a reasonable range. The inductor current ripple rate is 20%, and the inductance value can be calculated by this paper takes L b as 2 mH in consideration of a certain margin. Furthermore, the ripple of u pv can be expressed as [1] In order to maintain the output power of the photovoltaic cell above 98% of the maximum power, the voltage ripple ∆u pv should be less than 8.5% of the maximum power point voltage [29]. Here, we set ∆u pv = 0.1% to make SS more accurate in stability analysis as voltage-source replace the bus capacitor, thus obtaining  The function of the input filter capacitor Cin is to reduce the fluctuation of the voltage upv in the PV array, which improves the output efficiency of the PV array.
The value of the inductor Lb in the Boost converter should be able to ensure that the inductor current ripple is limited to a reasonable range. The inductor current ripple rate is 20%, and the inductance value can be calculated by this paper takes Lb as 2 mH in consideration of a certain margin. Furthermore, the ripple of upv can be expressed as [1] Δupv= b pv In order to maintain the output power of the photovoltaic cell above 98% of the maximum power, the voltage ripple Δupv should be less than 8.5% of the maximum power point voltage [29]. Here, we set Δupv = 0.1% to make SS more accurate in stability analysis as voltage-source replace the bus capacitor, thus obtaining

Stability
Let the MPPT controller variable xmppt be

Stability
Let the MPPT controller variable x mppt be Energies 2019, 12, 2193 6 of 16 and duty cycle d b can be expressed as where K p1 and T i1 are the gain and time constants of PI controller in the MPPT voltage loop, respectively. The state variable is denoted by x boost = [u Cin , i Lb , x mppt ] T . Furthermore, Equations (1), (2), (9), and (10) can be joint to obtain the state-space average model of SS To study the effect of MPPT controller parameters on the stability of SS, both of R Cin and R Lb are ignored for simplicity. Let the differential term be zero in Equation (11), and the equilibrium point x e can be obtained. Then, putting Taylor expansion of the state-space average model near x e and evaluating Equation (12) under first approximation according to Lyapunov method, small-signal model d∆x boost dt = A boost ∆x boost can be obtained, and is the Jacobian matrix. Where , A 31 = 1. So, the eigenvalue λ of the small signal model can be solved where I represent the unit matrix. It may be noted that, for every value of K p1 and T i1 , the system contains a real eigenvalue λ 1 and a pair of conjugate eigenvalues λ 2,3 , which means it involves one oscillating mode and one attenuation mode. For the eigenvalue of the conjugate complex pair σ±jω, the damping ratio of the system oscillation mode can be defined as The indicator ξ can be utilized to characterize the degree of stability of the oscillating mode. To better analyze the influence of the MPPT control parameters on the stability of SS, the eigenvalues and the damping ratio of the oscillating mode under five different examples are given in Table 2.
As the real part of λ 2,3 enter into the right half plane of the S-plane in Example-2, Example-4, and Example-5, ξ changes into a negative and in the meanwhile gain margin (GM) and phase margin (PM) are both less than zero. Therefore, the system works in an unstable state at that time and the unstable phenomenon is represented by the low-frequency oscillation of the voltage and current. Remaining K p1 is unchanged at Example-1 and Example-2, it can be observed that the decreasing of T i1 will lead λ 2,3 move to the right half plane, which reduces the stability of the system. Let T i1 = 0.01 at Example-1 and Example-3, the increasing of K p1 causes λ 2,3 to move the right half plane and λ 1 to move to the left half plane. Besides, the PM of Example-1 and Example-3 are 8.56 • and 14.8 • respectively, which means that the stability in Example-3 is actually stronger than the stability in Example-1.
Along with practical application, it is usually found that one or several parameters have a dominant influence on a particular mode of the system, while other parameters take little or no impact. Since multiple circuit parameters exist on the converter, relative eigenvalue sensitivity is used here to evaluate the trajectory alteration when the system parameters changed [22].
To obtain the eigenvalue sensitivity, it is possible to screen for parameters that have an important influence on system stability.
let Ψ = (ψ 1 , ψ 2 , . . . , ψ n ) be the right eigenmatrix of the state matrix A, and Λ be the diagonal matrix composed of the eigenvalues of the state matrix A. Hence, their relationship can be expressed as Thus, the eigenvalue sensitivity can be defined as where λ i is the ith eigenvalue of the feature matrix A, and α is a certain system parameter.
The first-order eigenvalue sensitivity of SS is given in Table 3 and it could be found that the sensitivity values of various parameters are significantly different due to the difference in parameter units. To address this problem, the concept of relative eigenvalue sensitivity is introduced to identity such differentiation between parameters. It should be noted that the real part of the eigenvalue can clearly reflect the change of the state. For this reason, the relative eigenvalue sensitivity is defined only for the real part of the eigenvalue to simplify the analysis, and then Equation (16) can be replaced by where RS Re(λi) reflects the impact of relative parameters changing. Table 4 gives the relative eigenvalue sensitivity of the four key parameters of SS. Furthermore, Figures 4 and 5 depict the eigenvalue locus as the parameters changing. As shown in Table 4 and Figure 4, when RS Re(λi) is positive, eigenvalue locus moves toward the imaginary axis as the parameters increasing. And when RS Re(λi) is negative, eigenvalue locus moves away from the imaginary axis as the parameters increasing. From Figure 5a,b either T i1 is increasing or K p1 is decreasing, and the real part of λ 1 is increasing. However, the system is still stable, because the real eigenvalue is always less than 0. Furthermore, the reduction of T i1 and the increasing of C in will make the real part of the oscillating mode λ 2,3 move from the left half plane into the right half plane shown in Figure 4a,c and Figure 5a,c. So, the system tends to be unstable along with the Hopf bifurcation. In addition, the effect of L b on the stability of the system is almost negligible, which is consistent with the analysis of the relative eigenvalue sensitivity of L b in Table 4. the real part of the oscillating mode λ2,3 move from the left half plane into the right half plane shown in Figure 4a,c and Figure 5a,c. So, the system tends to be unstable along with the Hopf bifurcation. In addition, the effect of Lb on the stability of the system is almost negligible, which is consistent with the analysis of the relative eigenvalue sensitivity of Lb in Table 4.

Stable Boundary
In this section, the model of PV grid-connected CS introduced in [20] is used, and then the stable boundary of SS and CS are compared. The parameters of the PV array and the converter are shown in Tables 1 and Table 5, respectively. Simulating the circuit of CS circuit, in which MPPT control parameters are assigned according to Example-2 and Example-4 shown in Table 2, respectively. The time domain waveform diagram of the PV voltage upv can be seen in Figure 6, and the corresponding fast fourier transform (FFT) analysis diagram is presented in Figure 7. A large peak amplitude appears at the low-frequency oscillation 422.9 Hz and 325.7 Hz of Example-2 and Example-4 respectively in Figure 7, which is roughly the same as the calculated oscillation frequencies of 443.38 Hz and 332.51 Hz shown in Table 2. Therefore, consistent with the stability analysis of SS, low-frequency oscillation occurs in CS when the parameter in Example-2 and Example-4 are adopted. It indicates that the system is in an unstable state.

Stable Boundary
In this section, the model of PV grid-connected CS introduced in [20] is used, and then the stable boundary of SS and CS are compared. The parameters of the PV array and the converter are shown in Tables 1 and 5, respectively. Simulating the circuit of CS circuit, in which MPPT control parameters are assigned according to Example-2 and Example-4 shown in Table 2, respectively. The time domain waveform diagram of the PV voltage u pv can be seen in Figure 6, and the corresponding fast fourier transform (FFT) analysis diagram is presented in Figure 7. A large peak amplitude appears at the low-frequency oscillation 422.9 Hz and 325.7 Hz of Example-2 and Example-4 respectively in Figure 7, which is roughly the same as the calculated oscillation frequencies of 443.38 Hz and 332.51 Hz shown in Table 2. Therefore, consistent with the stability analysis of SS, low-frequency oscillation occurs in CS when the parameter in Example-2 and Example-4 are adopted. It indicates that the system is in an unstable state. Next, to further investigate the similarity of the stability of the two structures, Figure 8 depicts the stable boundary of SS and CS with respect to the parameter regions of Kp1 and Ti1. It reveals that the almost same results of the stability analysis described in Section 3 are obtained. Besides, two parameter stability domains of SS are almost identical with CS, and just slight differences exist in the vicinity of the boundary. Thus, it can be concluded that the stability analysis of the dc-dc converter in CS can be carried out in SS.

Stable Zone
Unstable Zone Next, to further investigate the similarity of the stability of the two structures, Figure 8 depicts the stable boundary of SS and CS with respect to the parameter regions of Kp1 and Ti1. It reveals that the almost same results of the stability analysis described in Section 3 are obtained. Besides, two parameter stability domains of SS are almost identical with CS, and just slight differences exist in the vicinity of the boundary. Thus, it can be concluded that the stability analysis of the dc-dc converter in CS can be carried out in SS. Next, to further investigate the similarity of the stability of the two structures, Figure 8 depicts the stable boundary of SS and CS with respect to the parameter regions of K p1 and T i1 . It reveals that the almost same results of the stability analysis described in Section 3 are obtained. Besides, two parameter stability domains of SS are almost identical with CS, and just slight differences exist in the vicinity of the boundary. Thus, it can be concluded that the stability analysis of the dc-dc converter in CS can be carried out in SS. Next, to further investigate the similarity of the stability of the two structures, Figure 8 depicts the stable boundary of SS and CS with respect to the parameter regions of Kp1 and Ti1. It reveals that the almost same results of the stability analysis described in Section 3 are obtained. Besides, two parameter stability domains of SS are almost identical with CS, and just slight differences exist in the vicinity of the boundary. Thus, it can be concluded that the stability analysis of the dc-dc converter in CS can be carried out in SS.

Connection and Distinction of SS and CS
According to the method mentioned in Section 3, the relative eigenvalue sensitivity of CS is obtained in Table 6. From Table 6, its influence on different oscillation modes is different for one parameter. For example, the most influential parameter of oscillation mode λ 6,7 is C in , K p1 , and T i1 have the greatest influence on the attenuation mode λ 8. Compared with Table 4, it can be found that λ 8 and λ 6,7 in CS are analogous to λ 1 and λ 2,3 in SS, respectively. So, it means the performance of C in , K p1 , and T i1 are almost identical in the two structures.
In the small disturbance analysis method, the eigenvalue analysis can judge the stability of the system. However, the eigenvalue is based on the analysis performed at steady-state. To analyze the correlation between the state variables of SS and CS under the transient condition after the disturbance completed, the modal participation factor of the state variable is introduced.
According to the nature of the eigenvalues, each eigenvalue corresponds to a modality of the system. The real eigenvalue corresponds to the attenuation mode of the system, and the conjugate complex eigenvalue corresponds to the oscillating mode. Among the basic modal listed in Table 6, there are three attenuation modes λ 1 , λ 8 , and λ 9 and four oscillation modes λ 2,3 , λ 4,5 , λ 6,7 , and λ 10,11 , which play a decisive role in the dynamic behavior of the system. In addition, λ 10,11 is independent of the stability of the system due to the introduction of virtual state variables, and it is also independent of the inherent characteristics of the system.
The modal participation factor is a measure that combines the left and right eigenvectors as the degree of interaction between the state variables and the modalities. The correlation between the kth state variable to the ith mode can be represented by a modal participation factor [30] as follows where φ ik represents the kth element of the row vector φ i ; ψ ki represents the kth element of the column vector ψ i . P ki describes the scale of the effect of the ith mode and the kth state variable in the case of the kth state variable is under unit perturbation. Equation (18) indicates that the modal participation factor is only related to the structural parameters of the system. And it has nothing to do with the disturbance, which is similar to the property of the sensitivity of the eigenvalue. The modal participation factors of the system are given in Table 7. It can be seen clearly that the attenuation mode λ 1 is mainly related to the voltage deviation signal u e of the voltage outer loop, the oscillation modes λ 2,3 and λ 4,5 are mainly related to i o and u c2 , the oscillation mode λ 6,7 is mainly related to i Lb and u c1 ,the attenuation mode λ 8 is mainly related to u Cin and u c1 , the attenuation mode λ 9 is mainly related to u dc , and the undamped oscillation mode λ 10,11 is only related to the constructed virtual state variables g 1 and g 2 . According to the modal participation factor of the system, the basic mode closely related to a certain state variable of the system can be known, thereby a certain state of the system can be affected by regulating the basic mode. Additionally, the nonlinear interaction between the front-stage dc-dc converter and the rear-stage dc-ac inverter can be found through modal participation factor. In Table 7, the corresponding three state variables of SS are u Cin , i Lb , and u c1 . Based on the above analysis, u Cin , i Lb , and u c1 are the main state variables affecting λ 6,7 and λ 8 . However, they also play almost no effect on the other modes, which indicates that parameters of the front-stage converter designed by SS will not adversely affect the rear-stage inverter in CS.
The bold sections represent the main key data.

Experiment Verification
To experimentally evaluate the consistency of the stability analysis of the PV system in SS, an experimental setup is implemented as shown in Figure 9. The parameters are consistent with Tables 1 and 5. where  ik represents the kth element of the row vector  i ;  ki represents the kth element of the column vector  i . Pki describes the scale of the effect of the ith mode and the kth state variable in the case of the kth state variable is under unit perturbation.
Equation (18) indicates that the modal participation factor is only related to the structural parameters of the system. And it has nothing to do with the disturbance, which is similar to the property of the sensitivity of the eigenvalue. The modal participation factors of the system are given in Table 7. It can be seen clearly that the attenuation mode λ1 is mainly related to the voltage deviation signal ue of the voltage outer loop, the oscillation modes λ2,3 and λ4,5 are mainly related to io and uc2, the oscillation mode λ6,7 is mainly related to iLb and uc1,the attenuation mode λ8 is mainly related to uCin and uc1, the attenuation mode λ9 is mainly related to udc, and the undamped oscillation mode λ10,11 is only related to the constructed virtual state variables g1 and g2. According to the modal participation factor of the system, the basic mode closely related to a certain state variable of the system can be known, thereby a certain state of the system can be affected by regulating the basic mode. Additionally, the nonlinear interaction between the front-stage dc-dc converter and the rear-stage dcac inverter can be found through modal participation factor. In Table 7, the corresponding three state variables of SS are uCin, iLb, and uc1. Based on the above analysis, uCin, iLb, and uc1 are the main state variables affecting λ6,7 and λ8. However, they also play almost no effect on the other modes, which indicates that parameters of the front-stage converter designed by SS will not adversely affect the rear-stage inverter in CS.

Experiment Verification
To experimentally evaluate the consistency of the stability analysis of the PV system in SS, an experimental setup is implemented as shown in Figure 9. The parameters are consistent with Table 1 and Table 5.
The PV analog power supply uses the Chroma programmable dc current supply 62150H-1000S. The input capacitor Cin is Panasonic's 25SEPF330M. The diode VD is General Semiconductor's FES8JT. The drive power supply uses the IR2101 chip. In the control circuit of the dc-dc converter, the amplifier used a four-channel LM324AD, and the comparator uses a four-channel LM2901. In the dual-loop control of single-phase full-bridge inverter, it is achieved by using the TMS320F28027 micro-processor. The output current is sampled by a WCS2705 Hall sensor. Furthermore, the grid voltage and dc bus voltage sampling circuit are sampled by differential amplifier circuit, and the amplifier adopts TLV2374.  Figure 9. Experimental setup. Figure 9. Experimental setup.

PV Arrays
The PV analog power supply uses the Chroma programmable dc current supply 62150H-1000S. The input capacitor C in is Panasonic's 25SEPF330M. The diode VD is General Semiconductor's FES8JT. The drive power supply uses the IR2101 chip. In the control circuit of the dc-dc converter, the amplifier used a four-channel LM324AD, and the comparator uses a four-channel LM2901. In the dual-loop control of single-phase full-bridge inverter, it is achieved by using the TMS320F28027 micro-processor. The output current is sampled by a WCS2705 Hall sensor. Furthermore, the grid voltage and dc bus voltage sampling circuit are sampled by differential amplifier circuit, and the amplifier adopts TLV2374.
We first investigated what happens when the corresponding circuit of CS is applied. Figure 10 gives a description of the experimental waveform and FFT analysis results of u dc in CS under the two sets of MPPT parameters. When K p1 = 0.11 and T i1 = 0.01 shown in Figure 10a, the u dc waveform approximates a complete sine wave. And the peak voltage appears only at the fundamental frequency multipliers of 100 Hz and 200 Hz in the FFT spectrum, therefore, the system is in a stable state. When K p1 = 0.11 and T i1 = 0.00068, u dc has a certain degree of distortion, and its FFT spectrum shows a peak at 450 Hz in Figure 10b. The system shows an unstable oscillation phenomenon. control parameters. When Kp1 = 0.11 and Ti1 = 0.01, upv is basically going to keep the voltage at the maximum power point as shown in Figure 11a. When Kp1 = 0.11 and Ti1 = 0.001, upv exhibits a small oscillation. It can be seen from the FFT diagram of Figure 11b that there is an intermediate frequency oscillation about at 530 Hz. When Kp1 = 0.11 and Ti1 = 0.00068, as shown in Figure 11c, the voltage waveform appears more severely distorted for the oscillation amplitude is increased, and a mediumfrequency oscillation nearly appears at 420 Hz. Thus, the system becomes unstable.
Comparing Figure 10 with Figure 11, we obtained the same stability analysis results when the two structures operate in the same MPPT parameters. It is in accordance with the description of stable boundary from Section 4. Additionally, the oscillation frequencies of Figure 11b,c are substantially the same as those of the theoretical values obtained in Table 2   Next, we try to find a similar performance when the Boost converter is worked alone. Figure 11 illustrates the experimental waveform and FFT analysis of u pv in SS from the three sets of MPPT control parameters. When K p1 = 0.11 and T i1 = 0.01, u pv is basically going to keep the voltage at the maximum power point as shown in Figure 11a. When K p1 = 0.11 and T i1 = 0.001, u pv exhibits a small oscillation. It can be seen from the FFT diagram of Figure 11b that there is an intermediate frequency oscillation about at 530 Hz. When K p1 = 0.11 and T i1 = 0.00068, as shown in Figure 11c, the voltage waveform appears more severely distorted for the oscillation amplitude is increased, and a medium-frequency oscillation nearly appears at 420 Hz. Thus, the system becomes unstable.
Comparing Figure 10 with Figure 11, we obtained the same stability analysis results when the two structures operate in the same MPPT parameters. It is in accordance with the description of stable boundary from Section 4. Additionally, the oscillation frequencies of Figure 11b,c are substantially the same as those of the theoretical values obtained in Table 2

Conclusions
For two-stage PV system, this research uses SS instead of CS to simplify the parameter design of the front-stage dc-dc converter, which can avoid the influence of the parameter coupling in CS. To linearize the nonlinear characteristics of the output current and output voltage in PV array, the average model of SS is established by small-signal analysis method, and, further, the eigenvalue matrix is obtained. Relative eigenvalue sensitivity measure indicates that Kp1, Ti1, Cin, and Lb of the dc-Kp1=0. 11

Conclusions
For two-stage PV system, this research uses SS instead of CS to simplify the parameter design of the front-stage dc-dc converter, which can avoid the influence of the parameter coupling in CS. To linearize the nonlinear characteristics of the output current and output voltage in PV array, the average model of SS is established by small-signal analysis method, and, further, the eigenvalue matrix is obtained. Relative eigenvalue sensitivity measure indicates that K p1 , T i1 , C in , and L b of the dc-dc converter can exert the equally major effect on one oscillating mode and one attenuating mode in SS and CS. It is also verified through damping ratio and root-locus analysis. By curving the stability boundaries of MPPT controller parameters K p1 and T i1 in the aforementioned two structures, the comparison of the two broken-line shows that they have approximate stability domains, and five sets of MPPT examples analyzed in the two structures have the similar stability, which further validates the feasibility of SS instead of CS to design the front-stage dc-dc converter. Furthermore, the modal participation factor is used to describe the interaction between SS and CS. It shows the mode related to the parameters in dc-dc converter is hardly affected by other parts of two-stage PV system. Finally, two sets of MPPT examples tested in CS and three sets of MPPT examples tested in SS perform show almost identical stability, and it is also in accordance with the analysis above in the oscillation frequency. All of which point toward the conclusion that, in two-stage PV system with dc bus, the simplified structure can be used to replace the cascade structure for parameter design optimization of the dc-dc converter.