① Equivalent mathematical model of photovoltaic array

A single photovoltaic cell module is composed of multiple photovoltaic cells in series and in parallel. If the characteristics of each photovoltaic cell in the cell module match, under the same temperature condition and the light is uniform, the engineering mathematical model of the photovoltaic cell module

It can be described by formula (1).

Among them, Iz is the output current of the photovoltaic cell module, V_{z} is the output voltage of the photovoltaic cell module, n_{s} is the number of battery cells in series, and n_{p }is the number of battery cells in parallel. In formula (1), let I_{ph}=n_{p}I_{pc}, ε=n_{p}I_{o} and ξ=q/n_{c}k_{c}Tn_{s}, then use formula (2)

To express I_{z}. Among them, I_{ph} is the equivalent photo-generated current of the battery module, and ε and ξ are constants at a certain temperature.

When I_{z}=0, the open circuit voltage V_{oc.z} of the photovoltaic cell module is shown in formula (3).

The output power P_{z} of the photovoltaic cell module can be obtained by formula (4).

Differentiating the V_{z} in equation 2) and equation (4) within the working voltage range of photovoltaic cell module can obtain equation (5) and equation (6).

Substituting formula (2) and formula (5) into formula (6), when dP_{z}/dV_{z}=0, that is, at this time, the output power P_{z} has an extreme point V_{z} equal to the extreme point voltage V_{mpp.z} so that the formula (7) describes The relationship is established. The extreme point

The range of voltage V_{mpp.z} is 0<Vmmp.z<Voc.zo

Further, the derivative of V_{z} in formula (6) within the working voltage range of photovoltaic cell module can be obtained by formula (8). Through the analysis of equations (6) to (8), it can be seen that the output power P_{mpp.z} corresponding to the voltage V_{z}=V_{mmp.z} point is the maximum output power of the battery assembly.

From equations (5) to (7), it can be seen that the output current of photovoltaic cell modules decreases monotonously with the increase of output voltage within its working voltage range, and the output power curve of battery modules has a single peak characteristic, and the peak point can be determined by equation ( 6) Obtained. According to the above analysis, the characteristics of the photovoltaic cell module and the photovoltaic cell unit are similar, and the P-V and I-V characteristic curves of the photovoltaic cell module are shown in Figure 1. Figure 1 shows that when the applied voltage is higher than its own open circuit voltage or withstand reverse voltage, the photovoltaic cell module is in a negative power state, that is, the load state. Therefore, in practice, avoid photovoltaic cells and photovoltaic cell components working at negative power. Status, otherwise the hot spot effect may occur, which may damage the photovoltaic cell materials.

②Analysis of series characteristics of photovoltaic cell modules

Two photovoltaic modules of the same type PV_{1} and PV_{2} form a series photovoltaic branch as shown in Figure 2. The output voltage and current of the two photovoltaic modules are V_{z1}, V_{z2}, I_{z1} and I_{z2}, respectively. PV_{1} and PV_{2} are generated when they are exposed to different light. The equivalent photocurrents are I_{ph1 }and I_{ph2}, respectively. Because of the series connection, there is formula (9).

Among them, V_{s} is the output voltage of the series photovoltaic branch, and I_{s} is the output current of the series photovoltaic branch. Let △I_{ph} =I_{ph1} -I_{ph2}≥0

Then from equation (2) and equation (9), equation (10) can be obtained.

Combine formula (9) and formula (10) to obtain formula (11).

The output current and output power of the Shenlian branch can be described by equation (12) and equation (13) respectively. Among them, P_{s} is the output power of the series photovoltaic branch.

Combining equations (8) and (12), we can see that the open circuit voltage V_{oc.z} of the series branch is equal to the open circuit voltage V_{oc.z1} of PV_{1}

Sum with the open circuit voltage V_{oc.z2} of PV_{2}.** V _{oc.z}=V_{oc.z1}+V_{oc.z2}** (14)

When the two battery modules receive uniform light, there is I

_{ph1}=I

_{ph2}, that is, △I

_{ph}=0. At this time, V

_{z1}=V

_{z2}can be obtained from equation (10). When the two battery modules receive uneven light, if there is I

_{ph1}>I

_{ph2}, then △I

_{ph}>0, at this time, according to formula (9), formula (10) and formula (11), we can get:

Therefore, when two battery modules receive uneven light, the weaker photovoltaic module may work in a negative voltage state, and the more uneven the light, the closer the working voltage of the weaker photovoltaic module is to zero, or even negative.

When there are multiple battery modules connected in series to form a series photovoltaic branch, as shown in Figure 3, m (m≥2) battery modules are connected in series, and the output voltages of the battery modules are V_{z1}, V_{z2}, …. Vzm, and the output current respectively. They are I_{z1}, I_{z2}, …, Izm. Because of the series connection, there are:

Assuming that the equivalent photocurrent generated by the photovoltaic cell module under different light conditions is I_{ph1}, I_{ph2}, …, I_{phm}, and I_{ph1}≥ I_{ph2}≥…≥ Iphm, then there is now: △ I_{ph1}= I_{ph1}– I_{ph2}≥0 , △ I_{ph2}= I_{ph2}– I_{ph3}≥0,…,△ I_{ph(m﹣1})= I_{ph(m﹣1)}﹣ I_{phm}≥0, by formula (2), formula (16) we can get formula (17) and Formula (18).

It can be seen that when the series-type photovoltaic cell module receives uneven light, only the cell module with the weakest light may work in a negative voltage state, and the more uneven the light is, the greater the possibility that it will consume power as a load.

From the above analysis, it can be seen that some battery components in the series-type photovoltaic branch circuit may output negative voltage and become a load under uneven lighting conditions, thereby reducing the power generation efficiency of the overall photovoltaic power generation system, and may even damage the battery components. Improve the negative voltage characteristics of photovoltaic modules to overcome the negative power characteristics in the negative voltage state. Generally, a diode is used in anti-parallel connection with the battery assembly to short-circuit the battery assembly in the negative voltage characteristic so as to prevent it from consuming power as a load, as shown in Figure 4. When the battery assembly is working in a negative voltage state, the anti-parallel diode is turned on and clamps the battery assembly terminal voltage, so that the negative voltage working range of the battery assembly is limited to the diode’s conduction voltage drop range, so the series connection condition The maximum working current of the lower photovoltaic branch is less than or equal to the short-circuit current of the battery module with the strongest light condition.

Figure 5 shows that two photovoltaic modules using anti-parallel diodes are connected in series to form a photovoltaic branch. Suppose PV_{1} receives stronger light than PV_{2}, that is, △ I_{ph}= I_{ph1}﹣ I_{ph2}>0. According to the above analysis, combining equations (2) and (14) shows that when the photovoltaic branch output voltage Vs satisfies equation (19), the diode VD_{1} is reversely blocked and the diode VD_{2} is turned on, which is equivalent to bypassing PV_{2}. The output voltage V_{s }and the output current I_{S} of the photovoltaic branch are respectively equal to the output voltage V_{z1 }of the PV_{1 }and the output current I_{z1 }of the PV_{1}.

At this time, if the maximum power point voltage V_{mpp.z1} of PV_{1} satisfies:

According to formula (5) ~ formula (8), it can be seen that the output power of the photovoltaic branch increases monotonically within the working voltage range represented by formula (19). The I area in Figure 5 (a) shows the photovoltaic characteristic curve under this condition , The maximum power point appears at the boundary of zone I. If the maximum power point voltage V_{mpp.z1} of PV_{1} satisfies the condition expressed by equation (21), then the output power of the photovoltaic branch has a peak point within the working voltage range expressed by equation (19), and the peak power is equal to the maximum output power of PV_{1} , The peak power point voltage is equal to the maximum power point voltage V_{mpp.z1} of PV_{1}. The I area in Figure 5 (b) shows the characteristic curve under this condition, and there is a maximum power point in the I area.

If the photovoltaic branch output voltage Vs satisfies the formula (22), both VD_{1} and VD_{2} are reversely cut off, the branch output voltage V_{s} is equal to the sum of the PV1 and PV_{2} output voltages V_{z1} and V_{z2}, and the branch output current Is is equal to the PV_{1} output current I_{z1 }or PV_{2} outputs current I_{z2}. At this time, the output characteristics of the photovoltaic branch are the same as the case of not connecting the anti-parallel diode. The photovoltaic branch always has another peak power point within the working voltage range represented by equation (22), and the voltage corresponding to the peak power point is V_{mpp.s} , As shown in Figure 5 (a), (b) shows the peak power point existing in zone II.

In the same way, the above method can be used to analyze the series branches of multiple battery modules, and the following conclusions can be drawn: the anti-parallel diodes introduced to protect the battery modules from negative voltage will cause the output of the photovoltaic branch under uneven lighting conditions. The characteristics are discontinuous, resulting in the possibility of multiple peak power points in the output power curve of the photovoltaic branch.