A Fast Reconfiguration Technique for Boost-Based DMPPT PV Systems Based on Deterministic Clustering Analysis

Abstract

Mismatching operating conditions affect the energetic performance of PhotoVoltaic (PV) systems because they decrease their efficiency and reliability. The two different approaches used to overcome this problem are Distributed Maximum Power Point Tracking (DMPPT) architecture and reconfigurable PV array architecture. These techniques can be considered not only as alternatives but can be combined to reach better performance. To this aim, the present paper presents a new algorithm, based on the joint action of the DMPPT and reconfiguration approaches, applied to a reconfigurable Series-Parallel-Series architecture, which is suitable for domestic PV application. The core of the algorithm is a deterministic cluster analysis based on the shape of the current vs. voltage characteristic of a single PV module combined with its DC/DC converter to perform the DMPPT function. Experimental results are provided to validate the effectiveness of the proposed algorithm and to demonstrate evidence of its major advantages: robustness, simplicity of implementation and time-saving.

1. Introduction

According to forecasts of the International Energy Agency (IEA), the world renewable capacity will increase by almost 2400 GW between 2022 and 2027; solar PV will represent over 60% of all renewable capacity expansion [1]. PV energy sources are the most attractive renewable sources, due to their availability and sustainability. Thanks to simpleness and modularity of installation, PV power plants of different sizes are now widespread all over the world. In a typical PV plant, modules are connected in series and create a string to provide the required output voltage; then, strings can be connected in parallel to form an array that can produce the required current. It is well known that, in the case of uniform temperature and irradiance conditions, the output current vs. voltage (I–V) characteristic of a PV array is non-linear and exhibits a unique Maximum Power Point (MPP) corresponding to the maximum power (P_MPP) that the array is able to provide [2,3]. To ensure the maximum utilization of the array, it is essential to work in the MPP. Unfortunately, a direct connection of the output PV generator to the load (i.e., battery, AC grid, etc.) is not an optimal solution because in this case the operating point does not match with the MPP. In addition, the position of the MPP changes during the day due to many effects, such as changes in temperature and variation in solar radiation. For this reason, a Centralized Architecture (CA) of the PV array is adopted, where a central inverter, as shown in Figure 1, is generally interposed between the output terminals of the generator and the load; it must be able to change its parameters (i.e., its duty cycle) to adapt the load to the source to operate at MPP [4]. In such conditions, the energy performance is optimized [3] and the PV system behaves like a synchronized complex network, in which the extracted power is the maximum allowable and is the sum of the maximum power of each individual PV module [5,6]. A great variety of methods are able to track the MPP in PV plants; these so-called Maximum Power Point Tracking Techniques (MPPT) have been proposed in the literature and are categorized and commented on in many books and review papers [4,7,8,9,10,11].

The classification includes the so-called “Hill Climbing”, “Curve Fitting”, “Dynamical” and “Machine Learning” MPPT techniques [10,12,13,14]. The above techniques are high-performing in the absence of mismatching operating conditions. Unfortunately, mismatching working conditions of the PV plant are very frequent. In such a case, the PV modules of an array operate in nonuniform conditions, due to shadowing of neighboring objects, dirtiness of modules, clouds, installation on irregular surfaces and different orientation angles of modules in the PV field, but also manufacturing tolerances or uneven aging, etc. In the case of mismatch, the PV modules are not synchronized and the maximum extractable power is lower than the sum of the maximum power over each individual PV module [15,16,17,18]. Moreover, the power vs. voltage (P–V) characteristic of the PV system may exhibit multiple peaks, one global maximum point (GMP) and many local peaks, due to the presence of bypass diodes used to protect PV modules from hot spots and thermal breakdown. Therefore, the efficiency of MPPT methods decreases dramatically [15,16,19]. For instance, the Perturb and Observe (P&O) technique, which is the most used method, may track local maxima, but not global maximum points [20] because they cannot differentiate the global maximum point with respect to local maxima. In conclusion, the impact and power loss due to mismatch depend on the PV modules’ operating points and on their electrical configuration. To maximize the efficiency of the module and track the global maximum point in case of mismatching conditions, a possible solution is offered by the dynamical reconfiguration of PV modules [21]. A basic scheme of a reconfigurable grid-connected PV system is shown in Figure 2. Like in complex networks, the topology of the network is dynamically changed using active switches to reach the synchronization of the modules and power maximization [22,23,24,25]. These techniques have proved to be efficient in mismatching conditions, although they are relatively expensive since they require the use of reliable devices able to switch high Direct Currents (DCs) [21,26,27]. An alternative solution to mitigate mismatching’s effects is the adoption of a Distributed Architecture (DA) in which, unlike the centralized one, each PV module, as shown in Figure 3, is equipped by an independent DC/DC converter and a dedicated Distributed Maximum Power Point Tracking (DMPPT) controller [28,29,30].

This kind of approach allows the negative effect of mismatched PV modules to be counteracted. Different architectures, based on Boost, Buck, Buck–Boost or Cuk topology, have been proposed for the resulting Self-Controlled PV Module (SCPVM) [31]. The main problem of this promising solution is the reliability of the DMPPT DC/DC converter. Moreover, the efficiency of the PV system is also related to a non-optimal value of the string voltage [32,33] and to the finite ratings of devices used in the power stage [31]. In a recent paper [34], some of the authors have theoretically shown that dynamical reconfiguration and Distributed Architecture are not alternative techniques but can be combined (combined approach) to get the maximum efficiency of the PV system. In particular, the authors showed, as a “proof of concept”, how in different mismatching scenarios, the critical issues affecting the distributed approach can be overcome by acting on the configuration of a PV array. However, no experimental tests have been described in that paper. The authors have also recently proposed a flexible DMPPT emulator able to reproduce the output (I–V) characteristics of a Boost-based [35] DMPPT system, not only at different irradiance levels but also as a function of the maximum voltage that can be applied across the drain and source terminals (V_Dsmax) of the adopted silicon devices under turn-off conditions. However, in the paper the emulator was adopted only to explore the performance of the Boost-based DMPPT approach. To overcome the limitations of the previous papers, the aim of the present work is to introduce a new algorithm suitable for Series-Parallel-Series Reconfiguration Boost-based DMPPT architecture and to prove its effectiveness in experimental tests performed by using the flexible emulator. After the introduction, the paper is structured as follows: Section 2 explores the necessity of the joint adoption of DMPPT and reconfiguration approaches; Section 3 introduces a fast Boost-based DMPPT reconfiguration algorithm, based on clustering analysis; Section 4 and Section 5 present the conducted experimental activity; and lastly, the conclusion of this work is drawn in Section 6.

2. The Necessity of the Joint Adoption of Boost-Based DMPPT and Reconfiguration Approaches

The reason why the Boost-based DMPPT approach alone is not able to effectively counteract the mismatching operating conditions is linked to the fact that its energetic performance is affected by the adopted electrical configuration (electrical connection between SCPVMs). To better clarify the above aspect, which directly leads to the proposed combined approach (joint adoption of Boost-based DMPPT and reconfiguration approaches), a simple PV system, composed by four Boost-based SCPVMs, will be used as a “case study”. With the aim to obtain results with general validity, no reference to numerical values, as shown in Figure 4 and Figure 5, will be made. The curves shown in Figure 4 represent the I–V characteristics of the SCPVMs that are considered mismatched. The mismatching scenario is characterized by the fact that all SCPVMs show different short circuit currents (I_SC1, I_SC2, I_SC3 and I_SC4), suggesting different operating irradiance values. The guidelines allowing the curves that are reported in Figure 4 to be obtained have been discussed in detail in Appendix A. Once the I–V characteristics of the considered SCPVMs are known, it is possible to easily obtain the P–V characteristic of whatever configuration. In our case study, the following three different architectures are compared: (I) Series Architecture (SA), (II) Parallel Architecture (PA) and (III) Series-Parallel Architecture (SPA). In particular, SA (PA) refers to a PV system composed of the series (parallel) connection of all SCPVMs and SPA refers to a string in which the parallel connection between SCPVM 1 and SCPVM 4 is connected in series with the parallel connection between SCPVM 2 and SCPVM 3. From Figure 5 it is evident that the shape of the P–V curves, and hence the amplitude and the position of the Best Operating Region (region in which the power extractable assumes its maximum value), are significantly influenced by means of the electrical connection between SCPVMs. This result suggests that, by acting on the configuration architecture with a controllable switch matrix, it is possible to increase the energetic performance of the mismatched Boost-based DMPPT PV systems.

3. Fast Series-Parallel-Series Reconfiguration Algorithm

In the following, for the sake of simplicity, but without any loss of generality, a Boost-based DMPPT PV field made of N = 8 self-controlled PV modules will be considered. In Italy, most domestic PV plants are characterized by such architecture, with a power capability less than or equal to 3 kW. In addition, the monocrystalline solar panel capability of between 360 and 380 W represents, currently, an Italian market standard. As shown in Figure 6, the output of a proposed reconfiguration process, carried out with a proper controlled switch matrix, can be a Series-Parallel-Series (SPS) architecture in which several parallel clusters of SCPVMs are connected in series.

In the considered network topology, in order to identify the configuration that represents the best compromise between efficiency and reliability, it will also be considered a possibility to exclude one or several SCPVMs; in particular, each SCPVM may belong to one of the identified series or parallel clusters or be disconnected. The only constraint, called string voltage constraint, is to ensure that the Optimal String Voltage Range (OSVR), which depends not only on the considered atmospheric conditions, in terms of irradiance and temperature values, but also on the topology [35], must exhibit an intersection (no-zero intersection) with the Optimal Working Range of the Inverter (OWRI). This means that the following condition must be satisfied:

$$OSVR\cap OWRI=\varnothing$$

(1)

where, OWRI is the voltage interval [V_min, V_max] between the minimum input voltage V_min at which the inverter starts pulling power from the PV system and the maximum input voltage V_max, that must never be exceeded to prevent damage to the inverter. Typical commercial values of, respectively, V_min and V_max are reported in

Table 1. Input voltage range of commercial PV inverters.

Manufacturer	Vmin	Vmax
Growatt (Shenzhen, China)	150 V	550 V
Huawei (Shenzhen, China)	90 V	560 V
SMA (Niestetal, Germany)	80 V	600 V
SolarEdge (Herzliya, Israel)	380 V	480 V
ABB (Zurich, Switzerland)	140 V	580 V
Sungrow (Hefei, China)	40 V	560 V

.

Based on the previous considerations, the proposed deterministic reconfiguration algorithm is made from the following three operating steps: (1) measurement; (2) identification of the optimal series configuration; (3) identification of the optimal SeriesPparallel-Series architecture.

Step 1 (Measurement step): The measurement step is voted to measure the short circuit current I_SC,i of each SCPVM of the PV system. In fact, as shown in Appendix A, the knowledge of such current, together with the data obtained from the adopted PV module and Boost-based DC/DC converter’s datasheets, allows the determination of the approximate static I–V characteristic of the i-th SCPVM and, hence, the corresponding approximate Best Current (Voltage) Operating Range. In particular, once all the short circuit currents I_SC,i (i = 1, 2, …, N) are known, the approximate i-th Best Operating Ranges are defined as follows:

$$BCO{R}_i=\beta ·I_{SC,i}·\left[\frac{V_{MPP\_C}}{V_{DSmax}},1\right]$$

(2)

$$BVO{R}_i=\left[V_{MPP\_C},V_{DSmax}\right]$$

(3)

where BCOR_i, and BVOR_i represent the i-th Best Current Operating Range and the i-th Best Voltage Operating Range, respectively. The meanings of β and V_{MPP_C} are clarified in the Appendix section (Appendix A). Since the position and amplitude of such ranges are time-varying, depending on the atmospheric operating conditions, it is necessary to perform the measurement step periodically at fixed time intervals T^∗. It is obvious that with higher f^∗ = 1/T^∗, there will also be a higher speed of variation in mismatching conditions which can be efficiently addressed by the proposed algorithm. On the other hand, the higher f^∗ makes for a higher occurrence of short-circuit operating conditions in which the SCPVMs are not able to deliver power (higher waste of energy). In the following, for the sake of simplicity, the speed of tracking in the proposed algorithm will not be a subject of discussion. Our attention will be completely focused on the capability of the algorithm to identify the configuration that represents the best compromise between efficiency and reliability. In other words, an ideal lossless Boost DC/DC converter will be considered. Consequently, the efficiency of the Boost DC/DC converter will be assumed to be equal to one and the settling time will be argued as negligible (instantaneous system response). Step 1 ends with the creation of a list in which all SCPVMs are sorted in descending order, namely that the first SPCVM (SCPVM_,1) has the highest irradiance value and, hence, the highest corresponding short-circuit current, while the N-th SCPVM (SCPVM_,N) has the lowest short circuit current.

Step 2 (Identification of the optimal series configuration): The starting point of Step 2 is the construction of an intersection matrix (I_m) that identifies the SCPVMs that can be connected in series to form efficient and reliable strings. The matrix I_m is an [N × N] matrix, whose generic element I_m(i,j) can take the value 0 or 1. The element I_m(i,j) = 1 when i-th SCPVM and j-th SCPVM can be connected in series, that is, when the Best Current Operating Range of both SCPVMs have common values and I_m(i,j) = 0 when the intersection between the Best Current Operating Range of both SCPVMs is an empty interval. The intersection matrix can, therefore, be defined as follows:

$$I_{\mathrm{m}}\left(i,j\right)=\left\{\begin{array}{cc}1& \mathrm{if}\left(BCO{R}_i\cap BCO{R}_j\ne \varphi \right)\\ 0& \mathrm{if}\left(BCO{R}_i\cap BCO{R}_j=\varphi \right)\end{array}\right.$$

(4)

where i = 1, 2, …, N; and j = 1, 2, …, N.

Remembering that, after Step 1, all SCPVMs have been sorted in descending order, with decreasing irradiance values (I_SC,i > I_SC,j if (i > j)), the resulting matrix I_m is an upper triangular sparse matrix, whose rows represent the set of all possible optimized series configuration clusters (C_S,i with i = 1, 2, …, N). In particular, the generic i-th row of the intersection matrix identifies a reconfiguration architecture in which the maximum extractable power is equal to the sum of the maximum power that all the involved SCPVMs can provide in the considered atmospheric conditions (efficiency point of view). Of course, the identified series clusters are not exclusive because a single SCPVM may belong to one or more clusters. Then the generic i-th row identifies the number of SCPVMs that must be connected in series but also those who must be necessarily disconnected, to prevent one or more SCPVMs, belonging to i-th cluster (C_S,i), from operating in the saturation state (reliability point of view). By denoting with j_max,i the maximum column index in which I_m(i,j_max,i) = 1, the number of included (N_in,i) and excluded (N_ex,i) SCPVMs for each C_S,i are, respectively, as follows:

$$N_{in, i}=(j_{ max,i}-i)+1$$

(5)

$$N_{ex, i}=N-N_{in, i}$$

(6)

Once the N series configuration clusters have been identified, and the Maximum Series Cluster Extractable Power (MSCEP_i) has been calculated for each of them, it is also necessary to evaluate the Optimal Series Cluster Voltage Range (OSCVR_i) and the Optimal Series Cluster Current Range (OSCIR_i), which are the optimal voltage (and current) range at which they have to operate in order to ensure optimal energetic performances. They can be calculated as [35] follows:

$$MSCE{P}_i\cong \beta ·V_{MPP C}·{{\displaystyle \sum }}_{k=i}^N{I}_{\mathrm{m}}\left(\mathrm{i},\mathrm{k}\right)·I_{SC,k}$$

(7)

$$OSCV{R}_i=\left[\frac{MSCE{P}_i}{\beta ·I_{\mathrm{m}}\left(\mathrm{i},j_{ max, i}\right)·I_{SC,j_{ max, i}}},\frac{MSCE{P}_i}{\beta ·V_{MPP\_C}·I_{\mathrm{m}}\left(\mathrm{i},\mathrm{i}\right)·I_{SC,i}}\right]$$

(8)

$$OSCI{R}_i=\left[\frac{\beta ·V_{MPP\_C}·I_{\mathrm{m}}\left(\mathrm{i},\mathrm{i}\right)·I_{SC,i}}{V_{DSmax} },\beta ·I_{\mathrm{m}}\left(\mathrm{i},j_{ max, i}\right)·I_{SC,j_{ max, i}}\right]$$

(9)

where i = 1, 2, …, N; and j = 1, 2, …, N.

In conclusion, the aim of Step 2 is to assemble a set of SCPVMs so that SCPVMs belonging to the same group, called a series configuration cluster, satisfy the first condition of Equation (4). In this way, N series configuration clusters can be identified, and the resulting matching strings are both efficient (the relative maximum extractable power is coincident with the relative maximum available power) and reliable (no SCPVM works in the saturation state). Another advantage of the above identification process is the low computational complexity since it is based on the definition of a triangular sparse matrix.

Step 3 (Identification of the optimal Series-Parallel-Series architecture): Step 3 is devoted to exploring, for all the N identified series configuration clusters, the possibility of also including the discarded N_ex,i (with i = 1, 2,…, N) SCPVMs, while avoiding their operation in saturation state. Since the possibility of connecting each discarded panel in series with the selected cluster has already been evaluated and excluded from Step 2, a possible strategy involves the identification of the available N_ex,i SCPVMs that can be connected in parallel with each other and then connected in series with the previous identified i-th series cluster. The obtained group of parallel panels is named parallel configuration cluster. To achieve this goal, Step 3 starts with the calculation of the total number N_CP,i of parallel configuration clusters that can be associated to the i-th series configuration cluster:

$$N_{CP,i}=1+N_{ex,i}!·{{\displaystyle \sum }}_{k=2}^{N_{ex,i}-1}\frac{1}{k!·\left(N_{ex,i}-k\right)!}$$

(10)

Soon after, a cluster matrix (CL_m,i) can be constructed, which identifies all the possible parallel clusters that can be associated with each series cluster. It is a [N_{CP, i} × N] matrix defined as follows:

$$C{L}_{\mathrm{m},\mathrm{i}}\left(\mathrm{k},\mathrm{t}\right)=\left\{\begin{array}{cc}1& \mathrm{if}\left(SCPV{M}_t\in k\text{-}th cluster\right)\\ 0& \mathrm{if}\left(SCPV{M}_t\notin k\text{-}th cluster\right)\end{array}\right.$$

(11)

where k = 1, 2, …., N_Cl,i; and t = 1, 2, …, N.

The generic element is CL_m,i(k,t) = 1 if the t-th SCPVM belongs to the k-th cluster, otherwise it is equal to zero. The cluster matrix is a sparse matrix since the number of non-zero elements for each row is between 2 and N_ex,i, as the possibility of considering clusters with only one SCPVM has been already excluded in the previous steps (Steps 1 and 2) of the proposed algorithm. Moreover, for each row the minimum number of zero elements is N_in,i because they are related to the SCPVMs that belong to the i-th identified series configuration cluster that, obviously, cannot be included in the k-th parallel cluster. For the i-th series parallel cluster, it is definitely possible to determine N_cl,i parallel configuration clusters whose properties, in terms of Optimal Parallel Cluster Voltage Range (OPCVR_i,k), Optimal Parallel Cluster Current Range (OPCIR_i,k) and Maximum Parallel Cluster Extractable Power (MPCEP_i,k), are obtained using the following equations:

$$MPCE{P}_{i,k}=\beta ·V_{MPP C}·{{\displaystyle \sum }}_{t=1}^{N}C{L}_{\mathrm{m},\mathrm{i}}\left(k,\mathrm{t}\right)·I_{SC,t}$$

(12)

$$OPCV{R}_{i,k}=\left[V_{MPP C},V_{DSmax} \right]$$

(13)

$$OPCI{R}_{i,k}=\left[\frac{MPCE{P}_{i,k}}{V_{DSmax}}, \frac{MPCE{P}_{i,k}}{V_{MPP C}} \right]$$

(14)

At this point, the proposed algorithm is trained to restart Steps 1 and 2 by treating each identified parallel cluster as an equivalent SCPVM whose electrical characteristics are those obtained using Equations (12)–(14). Step 3 ends with the identification of the SPS configuration that exhibits the maximum extractable power included in the input inverter voltage range. In terms of computational complexity, the same considerations drawn for Step 2 also hold for Step 3. Both steps are based on the storage and manipulation of sparse matrices. Moreover, the ability to terminate the finding process in advance once the SPS architecture, in which all SCPVMs are electrically connected, has been identified, allows consideration of the proposed iterative algorithm, based on the binomial expansion, as a time-saving reconfiguration algorithm. The above statement is clearer if we compare the performance of the proposed algorithm with the commonly used stochastic reconfiguration algorithms, that, generally, employ a much higher number of trials to reach convergence.

4. Reconfigurable Series-Parallel-Series Emulator

The proposed algorithm has been verified experimentally by the laboratory setup whose schematic and physical implementation are shown in Figure 7. The test bed consists of two fundamental blocks, i.e., the Emulation Block (EB) and the Acquisition and Generation Block (AGB). The emulation block is carried out by three commercial controlled power supplies (Kepco BOP). The first two (PS1 and PS2) being, respectively, the power stage of the Series Configuration Cluster (SCC) emulator and the Parallel Configuration Cluster (PCC) emulator. The third one is used as an “Electronic Load” to scan the I–V characteristics of the proposed Reconfigurable Series-Parallel-Series Emulator (RSPSE), featured by the series connection of the above two emulators (SCC emulator and PCC emulator). The three power supplies can work in all four quadrants of the current–voltage plane. They are linear power supplies with two bipolar control channels (voltage or current mode), selectable and individually controllable by either front panel controls or remote signals. The electrical characteristics of the overall system are reported in

Table 2. Electrical characteristics of the proposed RSPSE.

Maximum allowed current	IPV MAX = 4 A
Maximum allowed voltage	VPV MAX = 100 V

. It is evident that the proposed experimental setup is suitable for the emulation of reconfigurable domestic PV systems, characterized by severe mismatching scenarios.

Moreover, the AGB consists of a modular National Instruments (Austin, TX, USA) USB system (NICompactDAQ 9174 with NI 9215 modules characterized by 16-bit resolution and a maximum sampling frequency of 100 kS/s) that allows backup and management of the experimental data in a MATLAB (R2022b) environment by MathWorks, Inc (Massachusetts, USA). Each input channel has its own analog to digital converter enabling the simultaneous recording of the PV voltage V_PV(t) and the PV current I_PV(t).

5. Experimental Results

In the following, the results obtained using the proposed algorithm will be considered with reference to two different mismatching scenarios: Case I and Case II. In both cases, the emulated PV system is characterized by the presence of N = 8 SCPVMs, whose electrical characteristics are listed in

Table 3. Electrical characteristics of the SCPVMs.

Open circuit voltage (standard test conditions)	VOC_STC = 13.9 V
Short circuit current (standard test conditions)	ISC_STC = 3.1 A
Maximum power point voltage (standard test conditions)	VMPP_STC = 10.7 V
Maximum power point current (standard test conditions)	IMPP_STC = 2.8 A
Maximum allowed voltage (standard test conditions)	VDS MAX = 15 V

. As far as the electronic load is concerned, it will be assumed that it is characterized by an operating range whose values are shown in

Table 4. Input voltage range of the emulated inverter.

Minimum inverter voltage	40 V
Maximum inverter voltage	100 V

.

5.1. CASE I

Case I refers to the following ordered short circuit current vector, which can be obtained by applying Step 1 of the proposed algorithm: I_{SC_V} = [2.04 1.13 0.92 0.82 0.7 0.7 0.7 0.7 0.5]A. The corresponding values of BCORs are listed in

Table 5. Best operating current ranges for Case I.

	BCOR		Reference
	Min [A]	Max [A]
SCPVM1	1.3268	1.8600	Equation (2)
SCPVM2	0.7496	1.0509
SCPVM3	0.5971	0.8370
SCPVM4	0.5307	0.7440
SCPVM5	0.4644	0.6510
SCPVM6	0.4644	0.6510
SCPVM7	0.4644	0.6510
SCPVM8	0.3317	0.4650

.

With reference to Case 1, the intersection matrix, obtained by applying the conditions expressed in (4), appears as follows:

$${\mathit{I}}_{\mathrm{m}}=\left(\begin{array}{c}10000000\\ 01100000\\ 00111110\\ 00011110\\ 0 0 0 0 1 1 1 1\\ 0 0 0 0 0 1 1 1\\ 0 0 0 0 0 0 1 1\\ 0 0 0 0 0 0 0 1\\ \end{array}\right)$$

(15)

As shown before, each row of the identified intersection matrix represents a series configuration vector whose characteristics are reported in

Table 6. Electrical characteristics of the identified CSs for Case I.

Series Configuration Vector	MSCEPi [W]	OSCVRi [V]		Nin,i	Nex,i	NCP,i
		Min	Max
C1	19.9020	10.700	15.000	1	7	120
C2	20.2025	24.134	26.946	2	6	57
C3	37.8138	58.085	63.333	5	3	4
C4	28.8579	44.328	54.375	4	4	11
C5	25.8726	55.640	55.714	4	4	11
C6	18.9069	40.660	40.714	3	5	26
C7	11.9412	25.680	25.714	2	6	57
C8	4.9755	10.700	15.000	1	7	120

. As an example, cluster 6, represented at row 6 of the intersection matrix I_m, is composed of panels SPVCM₆, SPVCM₇ and SPVCM₈. The maximum extractable power from this cluster is MSCEP₆ = 18.9069 W, the optimal series cluster voltage range is OSCVR₆ = [40, 40-66, 714] V and the N_ex,6 excluded panels are 5 and can be connected in N_CP,6 = 26 possible parallel combinations. Figure 8 shows the obtained clusters and the corresponding optimal series voltage range.

Once the series configuration clusters are known, the proposed algorithm starts the identification of the corresponding parallel configuration clusters through a binomial expansion approach. The number of parallel configuration clusters that can be associated with the i-th series configuration is reported in the last column of Table 6. In the considered case, the investigation phase ends by analyzing the performances of SPS architectures built starting from the series configuration cluster C₁, since the possibility of including all SCPVMs, as shown in Figure 9, belongs to the above group.

In conclusion, to further highlight the results obtained using the proposed algorithm, Figure 10 compares the P–V characteristics of the string just identified (grey line) with that obtained by using the “classical approach”, that is, connecting in series all the eight SCPVMs (black line).

From Figure 10 it is evident that there is a sensible increase in the extractable power in the operating range of the inverter, estimated as ~41%; a percentage that may exceed 60% in comparison with the performance of the CA architecture (dot curve of Figure 10), in which all PV modules are connected in series.

The increased extractable power is not the only advantage of using the proposed approach. Referring to the identified SPS architecture (Figure 11a), the saturation and reverse bias working conditions are completely avoided, since in the Optimal String Current Range (OSCR), that in this case collapses at one point, all SCPVMs can provide the maximum available power. The same considerations do not hold for the standard series architecture, since the OSCR (red curve of Figure 11b) does not intersect the branches of the hyperbole of all SCPVMs. The SCPVM₁, SCPVM₂ and SCPVM₈ are excluded. In particular, the first two work in the saturation state, while SCPVM₈ works in a reverse bias condition since the OSCR is higher than its short circuit current. In conclusion, in the identified SPS architecture, the negative effects of mismatching conditions are completely mitigated.

5.2. CASE II

Case II refers to the following short circuit current vector: I_{SC_V} = [1.13 1.13 1.13 1.13 1.13 0.37 0.37 0.37]A. Repeating the same steps previously illustrated led to the SPS architecture shown in Figure 12, and whose energetic performances are reported in Figure 13 and Figure 14. Even in this second considered case, the operating performance of the identified SPS architecture is higher than that obtained with the classical approaches, both in terms of efficiency (20% increase in extracted power) and reliability (no SCPVM works in the reverse and/or in the saturation state). On this last point, it is evident that, in the OSCR of the classical architecture (Figure 14b) the SCPVM₆, SCPVM₇ and SCPVM₈ work in reverse bias conditions with consequent reduction of the reliability of the whole system. Even in this considered case, the proposed reconfiguration algorithm is able to face the efficiency and reliability reduction when mismatching conditions occur.

6. Conclusions

The aim of this work is to prove that the combined action of DMPPT and reconfiguration approaches represents a promising solution to maximize the energetic performances of PV systems in any mismatching operating conditions. For this purpose, a new algorithm, based on the complementary action of the DMPPT and reconfiguration approaches, has been proposed. Such an algorithm is based on the periodic measurement of the short circuit currents of all PV modules. The considered topology is the reconfigurable Series-Parallel-Series that is suitable for domestic PV application. The experimental results fully confirm the effectiveness of the proposed technique and provide evidence of its major advantages: robustness, simplicity of implementation and time-saving. The main disadvantage of the proposed algorithm is the waste of energy during the measurement step in which the PV system works in the short circuit condition. Further work is in progress to extend the above approach, not only to more complex topologies, but also to more severe mismatching operating conditions in which the measurement time step negatively affects the performance of the whole PV system.