Shadow moving parallel to strings

When a shadow moves parallel to the strings of a PV generator array, every string of the array is in identical irradiance conditions with each other. Thus, every string has a common MPP voltage. Because of this, no reduction of mismatch losses is achieved by controlling every string individually. Thus, the mismatch losses of the MS configuration are equal to the mismatch losses of the SP configuration. Because every string is in identical irradiance conditions with each other, no current flows through the cross-connections of the TCT configuration and also the mismatch losses of the TCT configu-ration are equal to the ones of the SP configuconfigu-ration. Hence, every configuconfigu-ration of a PV generator has equal mismatch losses when a shadow is moving parallel to the strings of the PV generator array.

The relative mismatch losses of all three configurations with three different numbers of transition steps are presented as a function of the length of array sides in Figure 7.9. The relative mismatch losses of all the configurations increase rapidly at first and then turn down as the size of the array increases. At small array sizes, the

mis-match losses decrease with the increasing number of transition steps or with the de-creasing sharpness of the shadow. On the contrary, at large array sizes, the mismatch losses increase with the increasing number of transition steps. The maximum point of the relative mismatch losses is around the length of array sides which is twice the num-ber of transition steps, i.e., the maximum point goes to the higher lengths of array sides with the increasing number of transition steps. The causes of these phenomena are dis-cussed in the following.

Figure 7.9. The relative mismatch losses of the series-parallel, total-cross-tied and multi-string configurations as a function of the length of array sides when a shadow is

moving parallel to the strings of the PV generator array.

Because every string is in identical irradiance conditions with each other, no mismatch losses result from parallel connections either in the SP or in TCT configura-tion. All the mismatch losses result from series connections in strings. Thus, the causes of the mismatch losses of all the configurations can be explained by studying the mis-match losses of a single string.

When a shadow moves parallel to the strings of a PV generator array, a simula-tion period can be divided into three parts: the part when the edge of the shadow moves over the generator array, the part when the edge of the shadow moves from top of the array and the period between these two parts. Before the first part of the simulation pe-riod, all the modules of the array are exposed to the maximum irradiance. During the first part, the edge of a shadow moves over the array. When the edge of the shadow has moved over the array, one module of every string is fully shaded. During the second part of the simulation period, the edge of a shadow moves to the opposite side of the array. Before the third part of the simulation period, all the modules of a single string except for one module are shaded. During the third part, the edge of the shadow moves from top of the array. After the third part, all the modules are fully shaded and exposed to the irradiance of 10 % of the maximum. The mismatch losses of PV generators are studied separately during the three parts of the simulation period.

curves of the string in these situations are presented in Figure 7.10. The relative mis-match losses of the string are 22.38 % in the situation of the first part of the simulation period and 36.39 % in the situation of the third part. The power at MPP is 412.0 W in the case of the first part and 102.2 W in the case of the third part.

Figure 7.10. The P-I curves of the PV string in the single situation of the first part of the simulation period and in the corresponding situation of the third part of the

simula-tion period. The length of the string is four modules and the ambient temperature is 25 °C.

The total mismatch losses during the parts of the simulation period are the sum of the mismatch losses of all single shading situations during the parts of the simulation period. The number of single shading situations during the first and the third part of the simulation period is the number of different irradiance levels. For example, when the number of transition steps is four, there are five different irradiance levels and five sin-gle shading situations during the first and the third part of the simulation period. When the length of the string is four modules and the number of transition steps is four, the relative total mismatch losses are 15.71 % during the first part of the simulation period and 31.69 % during the third part. The relative total mismatch losses during the first part of the simulation period are smaller than during the third part with all lengths of the string larger than one module. Naturally, the mismatch losses during both parts of the simulation period are zero if the length of the string is one module. The relative total

mismatch losses of the PV generator during the first and the third part of the simulation period are presented as a function of the length of array sides in Figure 7.11. In addition, the combined mismatch losses of both parts of the simulation period are presented. In Figure 7.11, the number of transition steps is four.

Figure 7.11. The relative mismatch losses of the PV generator during the first and the third part of the simulation period as a function of the length of array sides when a shadow with four transition steps is moving parallel to the strings of the PV generator

array.

As can be seen from Figure 7.11, the relative mismatch losses during the first part of the simulation period are much smaller than during the third part. The relative combined mismatch losses of both parts of the simulation period are closer to the rela-tive mismatch losses of the first part than the ones of the third part because the produced power during the first part is substantially higher than during the third part. At the small lengths of array sides, the irradiance conditions of single strings become more uneven as the length of array sides increases. Thus, the relative mismatch losses increase at first as the length of array sides increases. As the length of array sides increases further, the portion of a single string that is in a transition region decreases and the irradiance condi-tions of the string become more even. Thus, the amount of the relative mismatch losses turns down at some point as the length of array sides increases. As can be seen from Figure 7.11, at the large lengths of array sides, the relative mismatch losses decrease as the length of array sides increases.

When the number of transition steps increases, the point where the portion of a single string that is in a transition region starts to decrease moves towards the higher lengths of array sides. Thus, the point where the relative mismatch losses turn down moves towards the higher lengths of array sides as the number of transition steps in-creases. When the number of transition steps increases, the differences between the ad-jacent transition irradiances decrease. Thus, at the small lengths of array sides, the rela-tive mismatch losses decrease as the number of transition steps increases. At the large

Figure 7.12. The relative mismatch losses of the PV string as a function of the length of the string when the centre of the edge of a static shadow is in the middle of the string.

The number of transition steps is four and the ambient temperature is 25 °C.

As can be seen from Figure 7.12, the behaviour of the relative mismatch losses as the length of the string increases is similar than in the case of the first and the third part of the simulation period. At the small lengths of the string, irradiance conditions of the string become more uneven with the increasing length of the string. Thus, the rela-tive mismatch losses increase at first as the length of the string increases. The portion of modules that are in a transition region decreases as the length of the string increases further. Thus, the amount of the relative mismatch losses turns down and decreases as the length of the string increases further.

If half of the modules of a string are exposed to the maximum irradiance 1000 W/m² and the other half are exposed to the irradiance of 10 % from that, the rela-tive mismatch losses of the string are about 15.94 %. The relarela-tive mismatch losses of a string, when the centre of the edge of a shadow is in the middle of the string, approach that value as the length of the string approaches infinity. This is illustrated in Figure 7.13, where the relative mismatch losses of a string in both of these cases are presented as a function of the length of the string on the interval from nine to 1600 modules.

Figure 7.13. The relative mismatch losses of the PV string when the centre of the edge of a shadow with four transition steps is in the middle of the string (blue curve) and when half of the modules are exposed to the maximum irradiance 1000 W/m² and the other half are exposed to the relative irradiance of 10 % (green dashed line) as a

func-tion of the length of the string. The ambient temperature is 25 °C.

The relative total mismatch losses of the PV generator during the second part of the simulation period are presented as a function of the length of array sides in Figure 7.14. The number of transition steps is four and the length of array sides is increased from seven to thirty modules. When the number of transition steps is four, there is no period between the first and the third part of the simulation period at lengths of array sides less than seven.

Figure 7.14. The relative mismatch losses of the PV generator during the second part of the simulation period as a function of the length of array sides when a shadow with four

transition steps is moving parallel to the strings of the PV generator array.

the number of transition steps increases.

As can be noticed by comparing Figures 7.14 and 7.11, the relative mismatch losses during the second part of the simulation period are significantly higher than the ones during the first part of the simulation period or than the relative combined mis-match losses of the first and the third part. However, the relative mismis-match losses during the third part of the simulation period are higher than the ones during the second part.

The relative mismatch losses during the parts of the simulation period and the whole simulation period are presented as a function of the length of array sides in Figure 7.15 when the number of transition steps is four.

Figure 7.15. The relative mismatch losses of the PV generator during the parts of the simulation period and the whole simulation period as a function of the length of array sides. A shadow with four transition steps is moving parallel to the strings of the PV

generator array.

At the small lengths of array sides, the first and the third part of the simulation period overlap partly and there is no period between them. The overlapping of the first and the third part of the simulation period means that the third part of the simulation period starts before the first part has ended. Because of the overlapping, at the small lengths of array sides, the total mismatch losses during the whole simulation period are not equal to the sum of the mismatch losses of the first and the third part of the simula-tion period. When the number of transisimula-tion steps is four, the combined mismatch losses

of the first and the third part of the simulation period are higher than the total mismatch losses at lengths of array sides less than six modules as can be seen from Figure 7.15.

At the large lengths of array sides when there are all three parts of the simulation period, the total mismatch losses during the whole simulation period are the sum of the mismatch losses of the parts of the simulation period. As aforementioned, the maximum of the relative mismatch losses during every part of the simulation period moves to-wards the higher lengths of array sides as the number of transition steps increases. Thus, also the maximum of the relative mismatch losses during the whole simulation period moves towards the higher lengths of array sides as the number of transition steps in-creases. The reason for this is that the minimum length of array sides from which all possible irradiance levels occur simultaneously increases while the number of transition steps increases. This phenomenon can be seen from Figure 7.9.

The absolute mismatch power losses and the absolute mismatch losses during the second part of the simulation period, during the whole simulation period and during the first and the third part of the simulation period are presented as a function of the length of array sides in Figures 7.16 and 7.17, respectively. The number of transition steps is four.

Figure 7.16. The absolute mismatch power losses of the PV generator during the parts of the simulation period and the whole simulation period as a function of the length of array sides. A shadow with four transition steps is moving parallel to the strings of the

PV generator array.

Figure 7.17. The absolute mismatch losses of the PV generator during the parts of the simulation period and the whole simulation period as a function of the length of array sides. A shadow with four transition steps is moving parallel to the strings of the PV

generator array.

As can be seen from Figure 7.16, the absolute mismatch power losses during the second part of the simulation period are higher than the ones during the whole simula-tion period or than the combined mismatch power losses of the first and the third part of the simulation period. The absolute mismatch power losses and the absolute mismatch losses during the whole simulation period and during the second part of the simulation period increase intensely as the length of array sides increases. The increase of the com-bined absolute mismatch power losses and the comcom-bined absolute mismatch losses of the first and the third part of the simulation period is much lower.

The absolute mismatch power losses during the whole simulation period with three different numbers of transition steps are presented as a function of the length of array sides in Figure 7.18. The absolute mismatch power losses of the generator in-crease as the size of the array inin-creases. Likewise the relative mismatch losses in Figure 7.9, naturally also the absolute mismatch losses decrease as the number of the transition steps increases at the small sizes of the array, and increase as the number of the transi-tion steps increases at the large sizes of the array.

Figure 7.18. The absolute mismatch power losses as a function of the length of array sides when a shadow is moving parallel to the strings of the PV generator array.