partial shading 2

Simulation Modelling Practice and Theory 19 (2011) 1613–1626 Contents lists available at ScienceDirect Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat Modeling and simulation of photovoltaic (PV) system during partial shading based on a two-diode model Kashif Ishaque a, Zainal Salam a,⇑, Hamed Taheri a, Syafaruddin b a b Faculty of Electrical Engineering, Universiti Teknologi Malaysia, UTM 81310, Skudai, Johor Bahru, Malaysia Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan a r t i c l e i n f o Article history: Received 17 August 2010 Received in revised form 6 April 2011 Accepted 11 April 2011 Available online 7 May 2011 Keywords: PV module Partial shading Two-diode model Local maxima Global maxima a b s t r a c t This paper proposes accurate partial shading modeling of photovoltaic (PV) system. The main contribution of this work is the utilization of the two-diode model to represent the PV cell. This model requires only four parameters and known to have better accuracy at low irradiance level, allowing for more accurate prediction of PV system performance during partial shading condition. The proposed model supports a large array simulation that can be interfaced with MPPT algorithms and power electronic converters. The accurateness of the modeling technique is validated by real time simulator data and compared with the three other types of modeling, namely Neural Network, P&O and single-diode model. It is envisaged that the proposed work is very useful for PV professionals who require simple, fast and accurate PV model to design their systems. Ó 2011 Published by Elsevier B.V. 1. Introduction Photovoltaic (PV) power system is envisaged to become an important renewable energy source due to its pollution-free and inexhaustible nature. Large scale PV power systems have been commercialized in numerous countries due to their substantial long term benefits, generous fed-in tariff schemes and other initiatives provided by governments to promote sustainable green energy. However, due to the high investment cost on PV modules, optimal utilization of the available solar energy has to be ensured. This necessitates a precise and reliable simulation of the designed PV systems prior to installation. The most important component that affects the accuracy of the simulation is the PV cell model. Modeling of PV cell involves the estimation of the I–V and P–V characteristics curves to emulate the real cell under various environmental conditions. The most popular approach is to utilize the electrical equivalent circuit, which is primarily based on diode. Many models have been proposed by various researchers; the simplest is the basic single-diode model. It comprises of a linear independent current source in parallel to a diode [1–4]. The model only requires three parameters to completely characterize the I–V curve, namely short-circuit current (Isc), open circuit voltage (Voc) and diode ideality factor (a). An improvement of this model is done by the inclusion of one series resistance, Rs [5–10]. In literature, it is popularly known as the Rs-model. Due to its simplicity and computational efficiency, the Rs is by far the most widely used model in PV system simulation [6]. However it exhibits serious deficiencies when subjected to temperature variations; its accuracy is known to deteriorate at high temperature. Further extension of the Rs-model, called as the Rp-model, which includes an additional shunt resistance Rp was introduced [11–15]. Although some improvement is achieved, this model demands significant computing effort because the parameters have been increased to five. Furthermore its accuracy deteriorates at low irradiance, especially in the vicinity of the open circuit voltage, Voc. ⇑ Corresponding author. Tel: +60 7 5536187; fax: +60 7 5566272. E-mail addresses: kashif@fkegraduate.utm.my (K. Ishaque), zainals@fke.utm.my (Z. Salam). 1569-190X/$ - see front matter Ó 2011 Published by Elsevier B.V. doi:10.1016/j.simpat.2011.04.005 1614 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 With the availability of today’s vast computing power, more accurate (but complicated) PV models are proposed. One important example is the two-diode model, originally introduced by [16]. The inclusion of the additional diode increases the parameters from five (for Rp-model) to seven. The main challenge now is to estimate the values of all the model parameters while maintaining a reasonable simulation time. The key is to realize an efficient and fast computation method to calculate the values of these parameters. Several computational methods are proposed [17–20] but in all these techniques, new additional coefficients are introduced into the equations, increasing their computational burdens. Furthermore difficulty arises in determining the initial values of the parameters; in some cases heuristic solutions need to be sought. Another approach to describe the two-diode model is by investigating its physical characteristics such as the electron diffusion coefficient, minority carrier’s lifetime, intrinsic carrier density and other semiconductor parameters [21–24]. Whilst these models can be helpful in understanding the physical behavior of the cell, information about the semiconductor is not always available in commercial PV datasheets. Hence a useful simulator using such model is not feasible because in majority of the cases, PV system designers are not equipped with the detail knowledge of the semiconductor processes. Once the appropriate model and its computational model have been identified, a complete PV system simulation can be developed. A good PV simulation package should fulfill the following criteria: (1) it should be fast but can accurately predict the I–V and P–V characteristic curves; including special conditions such as partial shading (2) it should be a comprehensive tool to develop and validate the PV system design inclusive of the power converter and MPPT control. Although existing software packages like PSpice, PV–DesignPro, SolarPro, PVcad, and PVsyst are available in the market, they are expensive, unnecessarily complex and rarely support the interfacing of the PV arrays with power converters [25]. Over the years, several researchers have studied the characteristics of PV modules under partial shading conditions [26,27]. In [26], an experimental work was undertaken to characterize the I–V curve during partial shading but the scope was limited to module-level shading. In a real PV generation system, a large number of modules are interconnected to form arrays; thus module-level shading would not be effective to investigate the shading phenomena. The effect of shading on the output of the PV modules and the associated change in their I–V characteristics was investigated in [27]. However, the I–V and P–V characteristics do not visualize the occurrence of multiple peaks, which are usually present in the I–V and P–V characteristics when subjected to partial shading. In another work, a numerical algorithm was proposed in [28] to simulate the complex characteristics of a PV array by representing each element (each cell of the module, bypass diode, blocking diode, etc.) with mathematical expressions. The results were found to be attractive but at the cost of complicated numerical computation, thus limiting its application to a small PV systems. A MATLAB based modeling to study the effects of partial shading in a PV array was proposed in [25]. However, the work utilized the Rs-model. As stated earlier, the Rs-model exhibits serious deficiencies when subjected to high temperature variations. This can be very crucial when simulation of large PV array system is required. In [29], PSpice based modeling to study the effects of bypass diode configurations on PV modules was proposed. In this work, the authors used the conventional two-diode model with Bishop’s model [30]. However, the model requires additional parameters to characterize the I–V and P–V curves, which in turn increases the computation burden. In view on the importance of this issue, this paper proposes a practical modeling and simulation method, which can predict the I–V and P–V characteristics of large PV arrays. It can be used to study the effect of temperature and insolation variation, varying shading patterns, and the role of array configuration on the PV characteristics. The simulation is developed using the MATLAB environment. An important contribution of this work is the incorporation of the modified two-diode model as the main engine for the simulation. This model is known to have better accuracy, especially at low irradiance level. Despite its known advantages, previous researchers have avoided the use of the two-diode model, probably due to the significant increased in computational time. In this work, that problem is overcome by introducing an efficient computational method which requires only four parameters to characterize the I–V and P–V curves. In addition, the proposed work supports large array simulation that can be interfaced with MPPT algorithms and actual power electronic converters. The accurateness of the simulation model is compared with three modeling methods proposed by previous researchers, namely single-diode model [25], P&O [31], and Neural Network [32]. It is envisaged that the proposed work can be very useful for PV professionals who require simple, fast and accurate PV model to design their system. 2. PV model for partial shading 2.1. Two-diode model The single-diode models [5–15] were based on the assumption that the recombination loss in the depletion region is absent. In a real solar cell, the recombination represents a substantial loss which cannot be adequately modeled using a single diode. Consideration of this loss leads to a more precise model known as the two-diode model [16]. However, the inclusion of the additional diode increases the parameters to seven (new parameters: Io2, a2). The two-diode model is depicted in Fig. 1 [16]. Eq. (1) describes the output current of the cell: I ¼ IPV  ID1  ID2    V þ IRs Rp ð1Þ K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 1615 Fig. 1. Two-diode model. where     V þ IRS 1 ; ID1 ¼ Io1 exp a1 V T1     V þ IRS ID2 ¼ Io2 exp 1 a2 V T2 ð2Þ where IPV is the current generated by the incidence of light, Io1 and Io2 are the reverse saturation currents of diode 1 and diode 2, VT1 (=a1  NskT/q) and VT2 (=a2  NskT/q) are the thermal voltages having Ns cells connected in series, a1 and a2 represent the diode ideality constants, q is the electron charge (1.60217646  1019 C), k is the Boltzmann constant (1.3806503  1023 J/K), and T is the temperature of the p-n junction in K. In this work, current of the PV cell is used as the input of partial shading modeling. Therefore, Eq. (1) need to be expressed in terms of cell output voltage as:        V þ IRs IPV þ I01 þ I02  I V þ IRs V ¼ V T ln exp    1  IRs I02 V T2 Rp I02 ð3Þ where VT ¼ V T1 V T2 V T1  V T2 ð4Þ Although greater accuracy can be achieved using this model, it requires the computation of seven parameters, namely IPV, Io1, Io2, Rp, Rs, a1 and a2. To simplify computation effort, several researchers assumed a1 = 1 and a2 = 2. The latter is an approximation of the Schokley–Read–Hall recombination in the space charge layer in the photodiode [16]. Although this assumption is widely used but not always true [33]. As discussed in the introduction section, many attempts have been made to reduce the computational time of this model. However they appear to be unsatisfactory. 2.2. Improved computational method 2.2.1. Simplification of saturation current equation The equation for PV current as a function of temperature and irradiance can be written as IPV ¼ ðIPV STC þ K i DTÞ G GSTC ð5Þ where IPV STC (in Ampere) is the light generated current at Standard Test Conditions (STC), DT ¼ T  T STC (in Kelvin, TSTC = 25 °C), G is the surface irradiance of the cell and GSTC (1000 W/m2) is the irradiance at STC. The constant Ki is the short-circuit current coefficient, normally provided by the manufacturer. The well known diode saturation current equation is given: I0 ¼ I0;STC  3    qEg T STC 1 1 exp  T ak T STC T ð6Þ where Eg is the band gap energy of the semiconductor and I0;STC is the nominal saturation current at STC. An improved equation to describe the saturation current which considers the temperature variation is given by [15]: I0 ¼ ðIsc STC þ K i DTÞ exp½ðV oc;STC þ K v DTÞ=aV T   1 ð7Þ The constant Kv is the open circuit voltage coefficient. This value is available from the datasheet. For the two-diode model, several researchers have calculated the values of Io1 and Io2 using iteration. The iteration approach greatly increases the computation time, primarily due to the non-suitable values of the initial conditions [34]. In general, Io2 is 3–7 orders of magnitude larger than Io1. Furthermore, most of the previous works consider the ideality factors a1 = 1 and a2 = 2. In this work, we propose a modification of Eq. (7) and apply it to the two-diode model. No attempt has been made to this equation to solve for the saturation currents. To maintain the equation in the same form as in Eq. (7), both reverse saturation currents Io1, Io2 are set to be equal in magnitude, i.e. 1616 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 I01 ¼ I02 ¼ ðIsc STC þ K I DTÞ exp½ðV oc;STC þ K V DTÞ=fða1 þ a2 Þ=pgV T   1 ð8Þ The equalization simplifies the computation as no iteration is required; the solution can be obtained analytically. Diode ideality factors a1 and a2 represent the diffusion and recombination current component, respectively. In accordance to Shockley’s diffusion theory, the diffusion current, a1 must be unity [16]. The value of a2, however, is flexible. Based on extensive simulation carried out, it is found that if a2 P 1.2, the best match between the proposed model and practical I–V curve is observed. Since (a1 + a2)/p = 1 and a1 = 1, it follows that variable p can be chosen to be P 2.2. The following expression for Io1, Io2 results: I01 ¼ I02 ¼ ðIsc STC þ K I DTÞ exp½ðV oc;STC þ K V DTÞ=V T   1 ð9Þ This generalization can eliminate the ambiguity in selecting the values of a1 and a2. Using Eqs. (5) and (8), five parameters of this model can be readily determined, i.e. IPV, Io1, Io2, a1 and a2. 2.2.2. Determination of Rp and Rs values The remaining two parameters in Eq. (1), i.e. Rp and Rs are obtained through iteration. Several researchers have evaluated these two parameters independently, but the results are unsatisfactory. In this work, Rp and Rs are calculated simultaneously, similar to the procedure proposed in [15]. This approach has not been applied for two-diode model. The idea is maximum power point (Pmp) matching; i.e. to match the calculated peak power (Pmp,C) and the experimental (from manufacturer’s datasheet) peak power (Pmp,E) by iteratively increasing the value of Rs while simultaneously calculating the Rp value. From Eq. (1) at maximum power point condition, the expression for Rp can be rearranged and rewritten as V þ Imp;STC Rs h   mp;STC i h   i o Rp ¼ n V mp;STC þImp;STC RS V þI Rs P max;E IPV  Io1 exp  1  Io2 exp mp;STCa2 Vmp;STC  1  a1 V T V T mp;STC ð10Þ The initial conditions for both resistances are given below: Rso ¼ 0; Rpo ¼ V mp;STC V oc;STC  V mp;STC  Isc;STC  Imp;STC Imp;STC ð11Þ The initial value of Rp is the slope of the line segment between short-circuit and the maximum power points. For every iteration, the value of Rp is calculated simultaneously using Eq. (10). With the availability of all the seven parameters, the output current of the cell can now be determined using the standard Newton–Raphson method. The flowchart that describes the Pmp matching algorithm is given in Fig. 2. Fig. 2. Matching algorithm. K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 1617 Fig. 3. SP, BL and TCT connections for 20  3 PV array. Fig. 4. (a) Module during normal conditions and (b) bypass operation during partial shading. 3. Partial shading modeling A PV array is arrangement of several PV modules, connected in various interconnected topologies. Three types of interconnections structure are typically used namely, series–parallel (SP), bridge link (BL), and total cross tied (TCT). Fig. 3 depicts 1618 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 Fig. 5. Flow chart of partial shading modeling. the aforementioned configurations for a 20  3 PV array. For simplicity, four types of shading patterns, labeled A–D are shown. For the series connected modules in SP configuration, due to the partial shading condition, the optimum operating point (Mpp) is being forced to move from the non-shaded to the shaded module [35]. However, for the BL and TCT interconnections, due to the additional wires in the modules connections, new current paths are created and the PV output power can be increased under the non-uniform insolation conditions. This kind of connections can be useful under certain shading patterns [36]. The characteristics of PV modules under shading conditions with bypass diodes connected at module terminal are explained as follows. In normal condition, i.e. when modules are not shaded, the bypass diodes are reversed biased. The current flows through each module, as shown in Fig. 4a. Under partially shaded conditions, the shaded cells behave as a load instead of generator and create the hot spot problem. The hot spot effect can be avoided by driving the current away from the nonshaded cells through a bypass diode as shown in Fig. 4b. In the shaded area, the bypass diode is in forward biased; therefore it conducts the current produced by the non-shaded part. Since the shaded modules are bypassed, multiple peaks in the I–V and P–V characteristics curves are created. For a large PV array, the ability of the simulation tool to resolve partial shading problem is very crucial. This is due to the fact that in large array configuration, the likelihood for partial shading to occur is large. The flow chart in Fig. 5 shows the procedure to compute the I–V and P–V curves for any array size during partial shading. For simplicity, only SP configuration is modeled. Once the shading pattern and temperature of the modules are generated, a software routine will be executed which will carry out the following tasks: (1) determination of shading patterns and temperature for a particular shading group, (2) calculation of voltage and current for each group based on the two-diode model, subjected to a known shading pattern and (3) performing linear interpolation with extrapolation techniques to form the continuous I–V and P–V curves. 4. Results and discussion 4.1. Verification of two-diode model The two-diode model described in this paper is validated by measured parameters of selected PV modules. The specifications of these modules are summarized in Table 1. The computational results are compared with the Rs [6] and Rp [11] models. Table 2 shows the parameters for the proposed two-diode model. Although the model has more variables, the actual number of parameters computed are only four because Io1 = Io2 while a1 and a2 can be chosen arbitrarily from (8). For brevity only MSX-60 [37] and KC200GT [38] will be used in the model verification. The SM55 [39] will be included in the validation of partial shading modeling. Fig. 6 shows the I–V curves for a single KC200GT module for different levels of irradiation (per unit quantity: Sun = 1 equivalent to 1000 W/m2). The calculated values from the proposed two-diode and K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 1619 Table 1 STC specifications for the three modules used in the experiments. Parameter BP Solar MSX-60 Kyocera KG200GT Siemens SM55 Isc Voc Imp Vmp Kv Ki Ns 3.8 A 21.1 V 3.5 A 17.1 V 80 mV/°C 3 mA/°C 36 8.21 A 32.9 V 7.61 A 26.3 V 123 mV/°C 3.18 mA/°C 54 3.45 A 21.7 V 3.15 A 17.4 V 77 mV/°C 1.2 mA/°C 36 Table 2 Parameters for the proposed two-diode model. Parameter BP Solar MSX-60 Kyocera KG200GT Siemens SM55 Isc Voc Imp Vmp Io1 = Io2 IPV Rp Rs 3.8 A 21.1 V 3.5 A 17.1 V 4.704  1010 A 3.80 A 176.4 X 0.35 X 8.21 A 32.9 V 7.61 A 26.3 V 4.218  1010 A 8.21 A 160.5 X 0.32 X 3.45 A 21.7 V 3.15 A 17.4 V 2.232  1010 A 3.45 A 144.3 X 0.47 X KC200GT Multi-Crystalline PV Module 9 Proposed Two-diode Model Rp-Model Experimental Data 8 Sun=1 7 6 I (A) Sun=0.8 5 Sun=0.6 4 3 Sun=0.4 2 1 0 Sun=0.2 0 5 10 15 20 25 30 V (V) Fig. 6. I–V curves of Rp-model and proposed two-diode model of the KC200GT PV module for several Irradiation levels. Rp-models are evaluated against measured data from the manufacturer’s datasheet. Comparison to the Rs-model is not included to avoid overcrowding of plot. However, the results for the Rs-model will be analyzed later in the performance evaluation between the three models. The proposed two-diode model and the Rp-model exhibit similar results at STC. This is expected because both models use the similar maximum power matching algorithm to evaluate the model parameters at STC. However, as irradiance goes lower, more accurate results are obtained from the two-diode model, especially in the vicinity of the open circuit voltage. At Voc, the Rp-model shows departure from the experimental data, suggesting that Rp-model is inadequate when dealing with low irradiance level. This is envisaged to have significant implication during partial shading. The performance of the models when subjected to temperature variation is considered next. All measurements are conducted at STC irradiance of 1000 W/m2. The proposed model is compared to the Rs-model. The comparison specifically is chosen to highlight the significant problems with the Rs-model when subjected to temperature variations. The Rp-model is not shown for simplicity, but will be included later in the analysis that compares all the three models together. Two modules are 1620 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 KC200GT Multi-crystalline PV Module 9 8 7 I (A) 6 Proposed 2-D Model Model Rp Experimental Data 5 25oC 50oC 4 75oC 3 2 1 0 0 5 10 15 20 25 30 35 V (V) Fig. 7. I–V curves of Rs and proposed two-diode model of the KC200GT PV module for several temperature levels. MSX-60 Multi-crystalline PV Module 4 I (A) 3 2 25oC Proposed Two-Diode Model Rp-Model Experimental Data 1 0 0 5 10 50oC 75oC 15 20 V (V) Fig. 8. I–V curves of Rs and proposed two-diode model of the MSX 60 PV module for several temperature levels @ 1 kW/m2. tested, namely the KC200GT and MSX-60. As can be seen in Figs. 7 and 8, respectively, the curves I–V computed by the twodiode model fit accurately to the experimental data for all temperature conditions. In contrast, at higher temperature, results from the Rs-model deviates from the measured values quite significantly. Fig. 9a shows the absolute error of the all the three modeling methods with respect to the experimental data. The proposed model is compared to the single-diode model with Rs [6] and Rp [15], respectively. The absolute errors are evaluated at irradiance of 200 W/m2. For clarity, the errors are plotted on the same graphs. As can be seen, for the whole range of experimental data, the proposed model is superior, especially at the vicinities of the open circuit voltage. Fig. 9b analyzes the three modeling methods with respect the temperature changes. It can be observed that Rs-model yields inaccurate results particularly at the neighborhood of the open circuit voltage. This is to be expected as this model does not account for the open circuit voltage coefficient, KV [6]. The proposed and Rp-models offers an almost equivalent performance for the whole range of temperature variations. 1621 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 0.8 Proposed Two-Diode Model Rp Model Rs Model Absolute Error (A) 0.6 0.4 0.2 0 0 10 20 30 V (V) Proposed Two-Diode Model 1.2 Rp Model Absolute Error (A) Rs Model 0.8 0.4 0 0 10 20 30 V (V) Fig. 9. Absolute errors for proposed two-diode model, Rp- and Rs-models. (a) For KC200GT PV module and (b) for MSX-60 PV module. Table 3 Shading pattern of the array used in illustration. Curves Shading pattern A (kW/m2) Shading pattern B (kW/m2) Shading pattern C (kW/m2) Shading pattern D (kW/m2) a b c d 1 0.9 0.75 0.8 0.75 0.75 0.5 0.6 0.5 0.5 0.25 0.4 0.25 0.1 0.1 0.2 The extensive experimental verification above has proven that the two-diode model is superior that the single-diode model with Rs and Rp. This justifies its usage in a more complicated situation such as partial shading. 4.2. Validation of modeling for partial shading In order to verify the accurateness of the proposed model for partial shading conditions, the result is compared with the work carried out in [32]. The latter is based on Artificial Neural Network. In [32], a real-time PV emulator was designed to emulate P–V characteristics curves with a special focus on partial shading conditions. For the PV modules model, the P–V 1622 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 I-V characteristics of 20x3 SP array a b 12 10 10 8 25oC 6 75oC 2 0 100 6 4 50oC 4 0 I (A) I (A) 8 2 200 300 0 400 0 100 V (V) 300 400 300 400 V (V) c d 8 8 6 6 I (A) I (A) 200 4 2 0 4 2 0 100 200 300 400 0 0 100 V (V) 200 V (V) Fig. 10. I–V characteristics curves for Table 3. characteristic was generated based on Sandia’s PV model. The main component of the developed emulator consists of two personal computers (PCs) with Analog–Digital (AD) and Digital–Analog (DA) hardware under a real-time dSPACE platform. Several inhomogeneous irradiance distributions were used to investigate the behavior of the proposed system. For simplicity, only 20  3 SP configuration is discussed here. However, it can be increase for any number M  N array, where M and N represent the number of series and parallel modules, respectively. Four shading patterns are considered; they are depicted in Table 3. For every curve, the evaluation is done at three different temperatures, i.e. 25 °C, 50 °C, and 75 °C. Figs. 10 and 11 show the I–V and P–V characteristics curves for the shading patterns described above. It expected, the curves exhibit multiple number of peaks that are equal to the number of irradiance levels imposed on the array. However, more precisely, it depends on the temperature of the modules, the insolation level, the shading pattern, and the array configuration [25]. In order to verify the accurateness of the proposed modeling approach comparison is made with 10 shading patterns. The results are shown in Tables 4–6 for temperatures at 25°, 50° and 75°, respectively. It can be seen that values obtained for Vmp,G and Pmp,G (global peak values) are in close agreement with the results [32]. Fig. 12 shows the relative error of Pmp,G for the proposed and single-diode model. It can be observed that, in general, the proposed method gives comparable Pmp,G errors to the single-diode method (less than 5%, except for a few cases). 4.3. Simulation with converter and controller Another important aspect of the PV simulation model is the ability to interface with the power electronic converters. Fig. 13 depicts a simulation example of a PV system, in which a boost-type dc–dc converter (with MPPT controller) is included. The SM55 PV modules are used in the simulation for a 20  3 array configuration. The MPPT controller utilizes the conventional Perturbation and Observe (P&O) algorithm to track the MPP. The performance of P&O is tested for shading pattern of curve (a) of Fig. 11 at 25oC. The simulation results are shown in Fig. 14. Initially the array receives a uniform irradiation of (G = 1 kW/m2). It is observed that, prior to the shading occurrence, i.e. at t = 0.5 s, the array’s voltage and current are retained at 350 V and 9.44 A, respectively. This corresponds to the maximum power point, i.e. 3220 W. Due to the shading conditions (at t = 0.5 s), operating point is shifted to a new minima at 820 W. This clearly highlights the limitation of the P&O scheme when multiple peaks are present. It cannot distinguish between the global and local maxima. 1623 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 P-V characteristics of an array 1500 a b 1500 o 50oC 1000 1000 P (W) P (W) 25 C 75oC 500 0 500 0 100 200 300 0 400 0 100 V (V) c 300 400 300 400 V (V) d 1000 1200 1000 P (W) 800 P (W) 200 600 800 600 400 400 200 0 200 0 100 200 300 0 400 0 100 V (V) 200 V (V) Fig. 11. P–V characteristics curves for Table 3. Table 4 The Vdc and Pdc outputs of 20  3 PV array under partially shaded conditions at 25 °C. Time 1 2 3 4 5 6 7 8 9 10 Shading pattern 1 = (1000 W/m2) P&O [31] Single-diode model [25] Proposed modeling A B C D Vdc Pdc Vdc Pdc Vdc Pdc Vdc Pdc 1 0.75 1 0.8 0.9 0.6 0.75 1 1 1 0.75 0.25 0.5 0.6 0.6 0.5 0.5 0.6 1 0.5 0.5 0.25 0.3 0.4 0.3 0.4 0.2 0.3 0.5 0.5 0.25 0.1 0.1 0.2 0.1 0.3 0.1 0.15 0.25 0.2 384.72 373.67 382.51 380.3 382.51 369.25 375.88 382.51 386.93 382.51 971.44 377.78 387.11 770.38 387.7 1117.4 381.06 580.18 977.16 774.62 276.38 263.12 174.67 274.17 172.46 369.25 172.46 174.67 165.83 267.54 1383.1 646.13 866.76 1100 1020.6 1117.4 847.77 1030.4 1566.3 1301.6 271 247 178 265 176 345 172 177 173 259 1350.8 595.76 872.02 1058.9 1030.7 1046.4 844.17 1044.5 1621.4 1247.1 275 263 180 272 178 360 177 179 171 266 1359.6 620.9 883.4 1077.9 1047.7 1076.8 867.6 1055.2 1632.4 1282.4 ANN [32] Table 5 The Vdc and Pdc outputs of 20  3 PV array under partially shaded conditions at 50 °C. Time 1 2 3 4 5 6 7 8 9 10 Shading pattern 1 = (1000 W/m2) P&O [31] A B C D Vdc Pdc Vdc 1 0.75 1 0.8 0.9 0.6 0.75 1 1 1 0.75 0.25 0.5 0.6 0.6 0.5 0.5 0.6 1 0.5 0.5 0.25 0.3 0.4 0.3 0.4 0.2 0.3 0.5 0.5 0.25 0.1 0.1 0.2 0.1 0.3 0.1 0.15 0.25 0.2 338.29 325.03 336.08 336.08 336.08 322.81 329.45 336.08 340.5 336.08 856.99 329.46 339.24 677.21 339.93 979.39 333.01 509.23 863.12 681.71 241.01 229.95 152.56 241.01 150.35 322.81 150.35 150.35 143.72 232.16 ANN [32] Single-diode model [25] Proposed modeling Pdc Vdc Pdc Vdc Pdc 1213.4 560.97 755.81 964.26 885.98 979.39 735.27 895.62 1351.1 1136.5 240 217 158 236 156 307 153 159 155 229 1206 520.76 775.9 940.89 914.9 925.6 746.31 929.17 1435.9 1101.9 244 231 161 243 158 323 158 160 152 235 1220.2 553.01 790.67 965.92 935.20 964.16 774.04 943.68 1446.9 1143.5 1624 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 Table 6 The Vdc and Pdc outputs of 20  3 PV array under partially shaded conditions at 75 °C. 1 2 3 4 5 6 7 8 9 10 Shading pattern 1 = (1000 W/m2) Rs-model [25] Proposed modeling A B C Vdc Vdc Pdc Vdc Pdc Vdc Pdc Vdc Pdc 1 0.75 1 0.8 0.9 0.6 0.75 1 1 1 0.75 0.25 0.5 0.6 0.6 0.5 0.5 0.6 1 0.5 0.5 0.25 0.3 0.4 0.3 0.4 0.2 0.3 0.5 0.5 0.25 0.1 0.1 0.2 0.1 0.3 0.1 0.15 0.25 0.2 338.29 325.03 336.08 336.08 336.08 322.81 329.45 336.08 340.5 336.08 856.99 329.46 339.24 677.21 339.93 979.39 333.01 509.23 863.12 681.71 241.01 229.95 152.56 241.01 150.35 322.81 150.35 150.35 143.72 232.16 1213.4 560.97 755.81 964.26 885.98 979.39 735.27 895.62 1351.1 1136.5 240 217 158 236 156 307 153 159 155 229 1206 520.76 775.9 940.89 914.9 925.6 746.31 929.17 1435.9 1101.9 244 231 161 243 158 323 158 160 152 235 1220.2 553.01 790.67 965.92 935.20 964.16 774.04 943.68 1446.9 1143.5 P&O [31] ANN [32] Proposed Two-Diode Model Rs-Model 8 6 o 25 C 4 2 0 Relative Error @ Pmp (%) Time 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 10 o 50 C 5 0 1 10 o 75 C 5 0 1 Time (s) Fig. 12. Relative error of Pmp for proposed and Rs models for Tables 4–6. Fig. 13. PV system description utilizing proposed model. 1625 K. Ishaque et al. / Simulation Modelling Practice and Theory 19 (2011) 1613–1626 Shading ocuurs here a Array Voltage (V) 600 400 200 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 1 10 Array Current (A) b 5 0 Power (W) c 4000 2000 Local Maxima Pboost Parray 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (s) Fig. 14. (a–c) Output voltage, current, and output power from the PV array. 5. Conclusion In this paper, a partial shading modeling based on an improved two-diode model is proposed. The proposed modeling supports large array simulation that can be interfaced with MPPT algorithms and actual power electronic converters. It is observed that the two-diode model is superior to the Rp- and Rs-models. The accurateness of the partial shading modeling is compared with the three types of modeling methods. 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