An MPPT method using hybrid radial movement optimization with teaching-learning based optimization under fluctuating atmospheric conditions

Abstract

Due to its clean and abundant availability, solar energy is popular as a source to generate electricity. Solar photovoltaic (PV) technology converts sunlight incident on the solar PV panel or array directly into non-linear DC electricity. However, the non-linear nature of the solar panels’ power needs to be tracked for its efficient utilization. The problem of non-linearity becomes more prominent when the solar PV array is shaded, even leading to high power losses and concentrated heating in some areas (hotspot condition) of the PV array. Bypass diodes used to eliminate the shading effect cause multiple peaks of power on the power versus voltage (P-V) curve and make the tracking problem quite complex. Conventional algorithms to track the optimal power point cannot search the complete P-V curve and often become trapped in local optima. More recently, metaheuristic algorithms have been employed for maximum power point tracking. Being stochastic, these algorithms explore the complete search area, thereby eliminating any chance of becoming trapped stuck in local optima. This paper proposes a hybridized version of two metaheuristic algorithms, Radial Movement Optimization and teaching-learning based optimization (RMOTLBO). The algorithm has been discussed in detail and applied to multiple shading patterns in a solar PV generation system. It successfully tracks the maximum power point (MPP) in a lesser amount of time and lesser fluctuations.

Keywords

Maximum power point tracking metaheuristic algorithms partial shading photovoltaic

1 Introduction

Among all the renewable energy sources available, solar energy is one of the most promising. When photons from the sun strike the photovoltaic (PV) cell’s surface, movement of electrons and holes occurs, resulting in the production of DC current. A single solar PV cell is connected in series and parallel combinations to enhance the power generation capability. Although solar PV is a clean, renewable technology and quite reliable to use, it has lower efficiency than competitive renewable technologies. An efficiency of only 20-25% in field conditions is typically obtained using monocrystalline PV technology. But inefficient tracking of maximum power causes a loss in the generation, which becomes prominent due to varying insolation conditions.

The main effect which is responsible for the reduction in power is partial shading (PS) effect due to buildings, bird drops, tree branches, clouds, etc. which may impose their shadows on the PV array at particular times in a day. Due to this partial shading effect (even on a single cell), the performance of the complete solar array may drop. PS forces a cell to start working as a load absorbing excessive power and releasing it as heat. This condition is known as the hotspot condition. Modern PV arrays are equipped with bypass diodes that are able to eliminate this hotspot problem but the intervention results in multiple power peaks at the output of the array such that the problem of tracking the maximum power becomes highly non-linear. Hence, various optimization algorithms have been employed to track the highest peak in the multi-modal P-V curve.

Many conventional single-stage algorithms have been used for tracking maximum power [1] but conventional algorithms have the problem of settling down at local maxima, which is unwanted when tracking the maximum power. This local tracking issue was solved by employing soft computing techniques such as fuzzy logic control (FLC) and artificial neural network (ANN). These techniques provide desirable results for MPPT with varying insolation conditions.

However, because of the learning process in such approaches, a large amount of data is required that causes undesirable excessive burden on processors. Consequently, focus has moved towards maximum power point tracking (MPPT) using nature-inspired metaheuristic algorithms based on a random search criterion. Various algorithms of this category have already been used for tracking the MPP, such as particle swarm optimization (PSO), ant colony optimization (ACO), gravitational search algorithm (GSA), teaching learning-based optimization (TLBO), coyote optimization algorithm (COA), PSO hybrid with differential evolution (PSO-DE), adaptive algorithm, flower pollination (FP) and Jaya algorithm [2 –8].

The PV generation system tracking performance depends on various parameters such as the time taken for convergence to the MPP, the number of power fluctuations at the output, and efficiency for tracking the MPP. All these performance parameters are to be given careful consideration while developing an algorithm for MPPT. Hence, it is an open field for research with the need for newer algorithms with better performances to produce a more efficient PV system.

PSO has been used with various modifications by researchers since the conventional version of PSO has several drawbacks such as slower power tracking and getting stuck in local maxima. In [9], the Jaya algorithm, owing to its simplicity, was used for the MPPT, but the tracking time was much higher along with the large size power oscillations at the output, thereby resulting in huge power losses. The TLBO algorithm was suggested in [10] in 2012, which involves simulating the student and teacher teaching-learning behavior so that student performance improves—the learning of students was improved by two mechanisms: through lectures from teachers and from mutual discussion with other students.

TLBO was applied to the MPPT problem in [11]. An improvement of 23.8 % in tracking time using TLBO was reported in [30] against PSO for a combination of partial shading patterns. Although TLBO outperformed several state-of-the-art algorithms when it was proposed, subsequent evaluation at higher dimensions affected the convergence rate [12] so several improvements were incorporated into the algorithm to improve the performance. In [13], authors have suggested incorporating a two-step modification in TLBO by adding self-study and smart teaching in the conventional TLBO to improving performance.

A practical radial movement optimization (RMO) was proposed in [14] and applied to the MPPT problem in [15]. The particle movement is not over the entire solution search space; hence information about the current particle location does not need to be carried to the next iteration. RMO produces better results compared with InC, PSO, MPSO, and GWO on performance parameters such as simplicity, efficiency, and tracking speed.

However, the RMO has not been compared with the more recent robust JAYA algorithm. Consequently, this paper has developed a hybrid algorithm called radial movement optimization hybrid with a teaching-based optimization (RMOTLBO). The proposed hybrid algorithm is the hybrid of the RMO proposed in [14] and the TLBO in [10]. The RMOTLBO algorithm, due to its faster tracking time and lesser power fluctuations, has been found useful for MPPT. The algorithm was compared with a very recently proposed state-of-the-art Jaya algorithm [2]. The results showed that the proposed algorithm outperforms Jaya in terms of faster tracking time and fewer fluctuations in the output power. Fewer fluctuations and faster settling time provided a dual contribution towards increasing the overall efficiency of the system.

The results were taken for static and dynamic PS conditions; in the former case the insolation pattern remained constant throughout, while in the latter it was varied with time. The latter case was chosen to show the proposed algorithm’s performance under a practical scenario where insolation patterns change continuously due to fast-moving clouds and changes in the sun’s position. It was observed that, for all the conditions, the proposed algorithm produces stable and significantly better performance compared to Jaya, thereby becoming one of the more suitable options for the MPPT application.

The remainder of this paper is divided into five sections. Section 2 describes RMO. The application of TLBO in MPPT is discussed in Section 3. RMOTLBO is introduced in Section 4 and application to MPPT has been explained in Section 5. Section 6 provides the results and discussion, while Section 7 concludes the paper.

2 Radial movement optimization (RMO)

RMO [14] is an evolutionary optimization technique similar to other heuristic techniques such as PSO, ACO, and TLBO. However, RMO has a unique feature of particle movement, which is sprinkled from the center of a sphere along its radii. The different velocities of particles are updated in each iteration by evaluating each particle’s location in accordance with its fitness. This makes a suitable trade-off between the exploration and exploitation process of an optimization algorithm. The steps of RMO are described as follows:

Initialization: Randomly generate a matrix X with order n_p by n_d, where n_p and n_d are the numbers of particles and number of dimensions (variables), respectively. The elements of the matrix X may be evaluated using the following equation: $X_{i, j} = X_{\min, j} + rand (0, 1) * (X_{\max, j} - X_{\min, j}$ (1) where i=1, 2, 3...n_p and j=1, 2, 3,..., n_d, and X_min,j and X_max,j are the lower and upper limits of the jth variable. Here, rand (0, 1) is a randomly generated value between zero and one using the normal distribution.

Movements of the particles: After assigning the center point, the particles are sprinkled from the center along the radii in straight lines based on the velocity vector (V). The matrix has the same order as that of matrix X i.e., n_p by n_d. The elements of V may be evaluated using the following equation: $V_{i, j} = rand (0, 1) * V_{\max, j}$ (2) where i=1, 2, 3,..., n_p and j=1, 2, 3,..., n_d, and $V_{\max, j} = \frac{(X_{\max, j} - X_{\min, j})}{I}$ . The parameter i is an integer suitably selected by the operator. The convergence is controlled by an inertia weight (W), which decreases with increasing iterations. W may be evaluated using the following equation: $W_{k} = W_{\max} - (\frac{W_{\max} - W_{\min}}{{Generation}_{\max}}) * {Generation}_{k}$ (3)

Generally, W_max and W_min are taken as 1 and 0, respectively. The impact of velocity vectors on the movement of particles is determined by these two limiting values of W. Thus, incorporating the impact of W, Equation (2) may now be rewritten as: $V_{i, j} = W_{k} * rand (0, 1) * V_{\max, j}$ (4)

After sprinkling all the particles, the fitness of each particle is evaluated using the objective function. The best fitness value is stored in R_best (radial best), and the location of the corresponding particle is saved. There exists G_best (global best), which stores the maximum value and the corresponding location among all R_bes_t values of previous iterations. Using G_best and R_best, the center is updated through the Update vector, using the following equations: ${Centre}_{new} = {Centre}_{old} - Update$ (5)

$\begin{matrix} Update = C_{1} * (G_{best} - {Centre}_{old}) \\ + C_{2} * (R_{best} - {Centre}_{old}) \end{matrix}$ (6)

The coefficients C₁ and C₂ are set before running the algorithm. Once the Center is updated, the above procedure is repeated using the new Center unless the termination criterion (such as the maximum number of generations, the predefined minimum value of G_best) is met.

3 TLBO-based MPPT

Developed by authors in [10], TLBO simulates the teaching-learning mechanism in the classroom. The population of the student with the subject assigned to them represents the population. Marks obtained by the students represent the fitness function while the optimized solution is the teacher in each phase. The entire algorithm involves two stages, namely students and teacher stage. While in the teacher stage, the mean marks of the students in the class is enhanced by direct instruction or knowledge sharing by the teacher. In the student stage, the interaction within students improves the marks. Further, the best learner becomes the teacher in the next step. When applied to the maximum power point tracking problem, the DC converter’s duty ratio to DC Converter can be selected as the students while the output PV power is considered the fitness function. The teacher stage involves updating the duty ratio of the converter by Equation 1 $D (i + 1) = D (i) + r * (D_{i, best} - T_{F} D_{i, mean})$ where: $\begin{matrix} D (i + 1), D (i) are the duty ratio in (i + 1) th \\ iteration an current duty ratio \\ D_{i, best} is the value of the teacher \\ D_{i, mean} is the mean value of the students \\ T_{F} is the teaching factor \end{matrix}$

In the student interaction stage, random students D_x and D_y interact to generate a new value of Duty ratio. $\begin{matrix} D_{new} = D_{old} + (D_{y} - D_{x}) if [P_{PV}]_{at D_{y} >} [P_{PV}]_{at D_{x}} \\ D_{new} = D_{old} + (D_{x} - D_{y}) if [P_{PV}]_{at D_{x} >} [P_{PV}]_{at D_{y}} \end{matrix}$

4 Radial Movement Optimization hybrid with the Teaching Learning Based Optimization (RMOTLBO)

In this modification, the particle position is further updated using a switching probability between bringing the solution closer to the best value and the learner phase of TLBO proposed in [29]. The idea of using the switching probability is obtained from [27]. However, incorporating the switching probability with the TLBO is the authors’ own idea. This further modification was useful in ensuring convergence to a better solution. The pseudo-code for the RMOTLBO can be given as: $if rand < p$ $s_{i}^{w} (t + 1) = s_{i}^{w} (t) + v_{i}^{w} (t + 1)$ $\begin{matrix} else \\ iff (s_{j}^{w} (t)) > f (s_{k}^{w} (t)) \\ s_{i}^{w} (t + 1) = s_{i}^{w} (t) + rand (s_{j}^{w} (t) - s_{k}^{w} (t)) \\ else \\ s_{i}^{w} (t + 1) = s_{i}^{w} (t) + rand (s_{k}^{w} (t) - s_{j}^{w} (t)) \end{matrix}$

where rand is a random number between 0 and 1, p is the probability switch kept equal to 0.8 in this study, and j and k are two randomly chosen particles other than the current particle.

5 RMOTLBO Applied to MPPT

For MPPT, a DC-DC boost converter was used as a medium in between the load and the PV array. The boost converter’s duty ratio for a particular insolation condition was optimized such that it sent the voltage and current at the output corresponding to the maximum power value. The duty ratio here was similar to the particle in RMOTLBO. Real-time simulation results were taken using the Typhoon HIL device. RMOTLBO was compiled in the Typhoon’s advanced C function block, which is similar to a microcontroller. Power corresponding to four initial duty ratio values (0.1, 0.3, 0.5, 0.9) was first found in each iteration, then all the four duty ratio values were updated using the equations of RMOTLBO. The process continued until the simulation runtime was over. A flowchart showing the working of RMOTLBO is in Fig. 2.

Fig. 1

Typical P-V characteristics for varying Insolation.

Fig. 2

TLBO Algorithm.

6 Comparison and hardware-in-loop implementation of the proposed RMOTLBO algorithm

In this section, a comparison of RMOTLBO is made with the very recently proposed Jaya algorithm [27]. The comparison is made based on the time taken for convergence to the MPP and the number of power oscillations at the output on the Typhoon HIL 402 platform.

The specification of the panels taken for HIL implementation is given in Table 1. The results are divided into two sections. In the first section, the results were taken for weak PS conditions (fewer or no modules were partially shaded). In the next section, different cases of strong PS (3 out of 4 panels partially shaded) were considered to show the proposed algorithm’s tracking capability under high shading conditions.

Table 1
Specification of the PV panels used in HIL implementation

Specification Solar PV panel At STC of 1000W/m² and Temperature of 20°C

Number of PV Panels in series 4

V_oc 5.425 volts

I_sc 5.34 amperes

V _mp 4.35 volts

I _mp 5.02 amperes

P _mp 21.837 watts

Temperature coefficient of V_oc in% /deg/C –0.37501

Temperature coefficient of I_scin %/deg/C 0.075

Specification	Solar PV panel At STC of 1000W/m² and Temperature of 20°C
Number of PV Panels in series	4
V_oc	5.425 volts
I_sc	5.34 amperes
V _mp	4.35 volts
I _mp	5.02 amperes
P _mp	21.837 watts
Temperature coefficient of V_oc in% /deg/C	–0.37501
Temperature coefficient of I_scin %/deg/C	0.075

6.1 Static PS conditions

In this condition, the PS scenarios were kept constant throughout the simulation runtime. Under two different conditions, the results were taken: the light PS condition and the heavy PS condition described in the upcoming subsections.

6.1.1 Light PS conditions

Three different cases were chosen, comprising of full and partially shaded conditions. Under the full insolation condition, all four panels received the full insolation. The two other cases were chosen such that a total of two and three modules were receiving full insolation, respectively. A summary of all the patterns for these conditions is given in Table 2.

Table 2
Pattern summary under light PS conditions

Shading patterns Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Full insolation 1000 1000 1000 1000

1 panel Shaded 1000 1000 1000 700

2 panels Shaded 1000 1000 900 750

Shading patterns	Insolation across modules (W/m²)
Full insolation	1000	1000	1000	1000
1 panel Shaded	1000	1000	1000	700
2 panels Shaded	1000	1000	900	750

Figure 3(a) shows the comparison results under full insolation conditions. The proposed algorithm tracked the MPP of 87.17W in 0.7 seconds. The tracking time of Jaya was 1.2 seconds at a power of 86.97W. Hence, it is clear that the proposed algorithm tracked the MPP at a faster rate. Moreover, large fluctuations at the output were also much fewer in the proposed algorithm than the Jaya algorithm that produced a larger number of higher fluctuations, resulting in high power losses. The reduced tracking time also contributed to power reductions, thereby increasing the system’s overall efficiency significantly compared to the case when Jaya was employed.

Fig. 3

Comparison graphs for RMOTLBO and Jaya under (a) full insolation, (b) 1 panel shaded and (c) 2 panel shaded conditions.

Figure 3(b) showed the comparison results under PS conditions when only one module was shaded. The proposed algorithm tracked the MPP of 68.39W in 1.7 seconds. The tracking time of the Jaya was 1.8 seconds at 68.4W. Hence, the proposed algorithm tracked the MPP at a faster rate. Also, it is clear from the graphs that the large size fluctuations were much fewer in number in the case of RMOTLBO, which increases the overall efficiency.

Figure 3(c) showed the comparison results under PS condition when two modules were shaded. The proposed algorithm tracked the MPP of 71.27W in 1.2 seconds. The tracking time of Jaya was 1.3 seconds. Again, the proposed algorithm tracked the MPP at a faster rate. The number of output fluctuations was also much lesser in the case of RMOTLBO, which increases the overall efficiency. Overall, a summary of settling to MPP is provided in Table 3.

Table 3

Power and settling time summary under light PS conditions

Shading patterns	RMOTLBO		Jaya
	Power (W)	Time (s)	Power (W)	Time (s)
Full insolation	87.17	0.7	86.97	1.2
1 panel Shaded	68.39	1.7	68.4	1.8
2 panels Shaded	71.27	1.2	71.62	1.3

6.1.2 Heavy PS conditions

Figure 4(a) shows the comparison results under heavy PS condition1. The proposed algorithm tracked the MPP in 0.91 seconds at a power of 58.21W. The settling time of the Jaya was 1.55 seconds at a power of 58.21W.

Fig. 4

Comparison graphs for RMOTLBO and Jaya under (a) PS1 and (b) PS2.

Therefore, under PS 1, the proposed algorithm tracked the MPP at a much faster rate and with very fewer large size fluctuations compared to the Jaya. Thus, the overall efficiency is increased due to the dual contribution of lesser fluctuations and faster tracking as usual.

Figure 4(b) shows the comparison results under heavy PS2. The proposed algorithm tracked the MPP in 1.7 seconds at a power of 36.7W. The time taken by the Jaya in settling to the MPP was 1.3 seconds at a power of 36.9W. This is the only case for which the Jaya algorithm settled to the MPP at a lesser time. However, the large size fluctuations were still a problem with the Jaya, which reduces efficiency. Hence, this case’s comparison could be considered equivalent because of a single contribution from both sides. The pattern summary under heavy PS condition along with setting time is shown in Tables 4 and 5.

Table 4

Pattern summary under heavy PS conditions

Shading patterns	Insolation across modules
	(W/m²)
	Module 1	Module 2	Module 3	Module 4
PS1	1000	850	700	600
PS2	1000	800	500	300

Table 5

Power and settling time summary under heavy PS conditions

Shading patterns	RMSOTLBO		Jaya
	Power (W)	Time (S)	Power (W)	Time (S)
PS 1	58.21	0.91	58.21	1.55
PS2	36.7	1.7	36.9	1.3

Nevertheless, overall, the proposed strategy’s performance was much better compared to the Jaya algorithm in terms of faster settling time and much fewer large size fluctuations, which contributed to the overall increased efficiency.

6.2 Dynamic PS conditions

The comparative analysis in this section considers a real-world scenario. In the actual world, the shading patterns on a PV array keeps on varying due to fluctuating atmospheric conditions such as moving clouds, changing sun insolation and position, etc. Hence, for a more comprehensive analysis, an algorithm is to be tested considering these conditions. Two different dynamic shading conditions were chosen. In the first case, the shading was initially kept lower and then increased, and vice versa, in the second case.

6.2.1 Dynamic Case 1

Table 6 summarizes PS patterns at different instants of this case. Figure 5(a) shows the results for this case. Each instant of PS was of 3 seconds. It is seen from the results that the proposed algorithm tracked the MPP of 36.09W at 0.95 seconds and 66.96W at 0.55 seconds, respectively at instants 1 and 2. Jaya tracked the MPP of 35.96W at 1.2seconds and 66.96W at 2.15 seconds, respectively at instants 1 and 2. Therefore, it is clear that the proposed algorithm works much better under real-time situations than Jaya in terms of faster tracking and producing much fewer large-size power fluctuations. Table 7 summarizes tracking time and power for both the algorithms. Figure 5(c) is shows the exact value for RMOTLBO for dynamic case 1.

Table 6
Shading patterns under dynamic insolation 1

Shading instants Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Instant 1 1000 600 500 200

Instant 2 1000 900 800 700

Shading instants	Insolation across modules (W/m²)
Instant 1	1000	600	500	200
Instant 2	1000	900	800	700

Fig. 5

Comparison under dynamic case 1 of (a) RMOTLBO and (b) Jaya.

Table 7

Tracking time and MPP under dynamic case 1

RMOTLBO		Jaya
Shading instants	Power (W)	Time (s)	Power (W)	Time (s)
Instant 1	36.09	0.95	35.96	1.2
Instant 2	66.95	0.55	66.96	2.15

6.2.2 Dynamic case 2

A summary for PS patterns at different instants for this case is shown in Table 8 and Fig. 6 shows the results. It can be seen from Fig. 6 that the proposed algorithm tracked the MPP of 65.91W at 1.45 seconds and 31.11W at 1.45 seconds, respectively, at instants 1 and 2. Jaya tracked the MPP of 65.9W at 1.8 seconds for instant 1; however, for instant 2 Jaya was unable to track the MPP, which clearly shows the unreliability of the Jaya algorithm under the real-time insolation fluctuation conditions. Unlike Jaya, RMOTLBO could track the MPP successfully. Therefore, it is clear that the proposed algorithm works much better under real-time situations than the Jaya algorithm in terms of faster tracking and producing a smaller number of large-size power fluctuations. The summary of settling time and MPP for different instants are shown in Table 9.

Table 8
Shading patterns under dynamic case 2

Shading instants Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Instant 1 1000 800 750 700

Instant 2 1000 700 500 250

Shading instants	Insolation across modules (W/m²)
Instant 1	1000	800	750	700
Instant 2	1000	700	500	250

Fig. 6

Comparison under dynamic case 1 of (a) RMOTLBO and (b) Jaya.

Table 9

Tracking time and MPP under dynamic insolation 2

Shading instants	RMOTLBO		Jaya
	Power (W)	Time (s)	Power (W)	Time (s)
Instant 1	65.91	1.45	65.9	1.8
Instant 2	31.11	1.45	untracked

7 Conclusion

The paper reports on a successful implementation of RMOTLBO for MPPT of a PV array. Performance validation was completed both for static conditions and more realistic dynamically varying insolation for various PS patterns. The proposed algorithm’s superior performance was proved over the very recently proposed Jaya technique in faster tracking time and a much smaller number of large-size power fluctuations at the load. Faster settling time and fewer fluctuations at the load side are necessary as the fewer the fluctuations and faster the tracking, the lower will be the power losses that contribute significantly to the system’s overall efficiency, especially under fast-changing dynamic insolation conditions. Consequently, the proposed algorithm can be a better choice when designing an MPPT controller for residential, commercial, and industrial uses.

Footnotes

Acknowledgements

The authors would like to acknowledge the financial support by the collaborative research grant scheme (CRGS) project, Hardware-In-the-Loop (HIL) Lab, Department of Electrical Engineering, Aligarh Muslim University, India, with project numbers CRGS/MOHD TARIQ/01 and CRGS/MOHD TARIQ/02.

The authors also acknowledge the technical support provided by the Hardware-In-the-Loop (HIL) Lab, Department of Electrical Engineering, Aligarh Muslim University, India.

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An MPPT method using hybrid radial movement optimization with teaching-learning based optimization under fluctuating atmospheric conditions

Abstract

Keywords

1 Introduction

2 Radial movement optimization (RMO)

4 Radial Movement Optimization hybrid with the Teaching Learning Based Optimization (RMOTLBO)

5 RMOTLBO Applied to MPPT

6.1.1 Light PS conditions

Table 2 Pattern summary under light PS conditions Shading patterns Insolation across modules (W/m2) Module 1 Module 2 Module 3 Module 4 Full insolation 1000 1000 1000 1000 1 panel Shaded 1000 1000 1000 700 2 panels Shaded 1000 1000 900 750

6.2.1 Dynamic Case 1

Table 6 Shading patterns under dynamic insolation 1 Shading instants Insolation across modules (W/m2) Module 1 Module 2 Module 3 Module 4 Instant 1 1000 600 500 200 Instant 2 1000 900 800 700

Table 8 Shading patterns under dynamic case 2 Shading instants Insolation across modules (W/m2) Module 1 Module 2 Module 3 Module 4 Instant 1 1000 800 750 700 Instant 2 1000 700 500 250

Footnotes

Acknowledgements

References

Table 2
Pattern summary under light PS conditions

Shading patterns Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Full insolation 1000 1000 1000 1000

1 panel Shaded 1000 1000 1000 700

2 panels Shaded 1000 1000 900 750

Table 6
Shading patterns under dynamic insolation 1

Shading instants Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Instant 1 1000 600 500 200

Instant 2 1000 900 800 700

Table 8
Shading patterns under dynamic case 2

Shading instants Insolation across modules (W/m²)

Module 1 Module 2 Module 3 Module 4

Instant 1 1000 800 750 700

Instant 2 1000 700 500 250