5.2. Light concentration effect on PV performance and efficiency

Let us find out how the concentration of light affects the I-V characteristics of a solar cell. We remember from Lesson 4 that the generation current of a solar cell (I_L) is a function of number of photons (N) hitting the photovoltaic surface:

I_{L} = q N A

(5.1)

where q is the electron charge, and A is the surface area of the cell. When light is concentrated, the number of photons increases according to the optical concentration ratio, so does the cell current. So, for the short circuit current of a solar cell (I_sc), we can write:

$I_{s c} (concentrated light) = C_{o p t} I_{s c} (incident light, 1 Sun)$

(5.2)

where C_opt is the optical concentration ratio (its definition was covered in Lesson 3). For convenience, we can denote cell performance parameters at concentrated light with an asterisk:

$I_{s c}^{*} = C_{o p t} I_{s c}$

(5.3)

This equation essentially shows how much the cell short circuit current will change when the available light is concentrated C_opt times. Then, we can substitute this equation to the I-V characteristic equation, which describes the cell performance over ranges of voltage and current:

V_{o c}^{*} = \frac{k T}{q} \ln (\frac{C_{o p t} I_{s c}}{I_{o}} + 1)

(5.4)

where V_oc* is the open circuit voltage (at concentrated light), k is the Boltzmann constant, T is the absolute temperature, and I_o is the dark saturation current. Now, we are going to modify this equation because we want to find how the open circuit voltage at concentrated light would be related to the open circuit voltage at ambient light. We know that the short circuit current is the highest current a solar cell can show, while the dark current is a very low number, so the quotient in the parenthesis should be much greater than 1, and therefore, a simplified form of Equation (5.4) should be true:

V_{o c}^{*} \approx \frac{k T}{q} \ln (\frac{C_{o p t} I_{s c}}{I_{o}})

(5.5)

Next, this equation can be modified by extending the natural log as follows:

V_{o c}^{*} = \frac{k T}{q} {lnC}_{o p t} + \frac{k T}{q} \ln (\frac{I_{s c}}{I_{o}})

(5.6)

The second term here is equal to V_oc - the open circuit voltage without concentration, so we can write finally:

V_{o c}^{*} = V_{o c} + \frac{k T}{q} {lnC}_{o p t}

(5.7)

From Equation (5.7), it is obvious that there is logarithmic dependence between the cell open circuit voltage and the light concentration ratio. For example, if $C_{o p t} = 10$ , the $(k T / q) \ln C_{o p t}$ term would be equal to 60 milivolts at 25^oC - this is by how much the cell voltage will increase with tenfold light concentration. In case of higher concentration, for example, C_opt = 1000, the voltage increase would be expected to be closer to 178 mV at 25 °C, which is relatively modest compared to current increase.

To estimate the concentration effect on maximum power output, we will use the equation (which was introduced in Lesson 4):

P_{m a x}^{*} = V_{o c}^{*} \times I_{s c}^{*} \times F F

(5.8)

Substituting here Equations (5.3) and (5.7) and re-arranging, we obtain:

\[P_{\max }^* = {P_{\max }}{C_{opt}}\left( {1 + \frac{{kT}}{q}\frac{{\ln {C_{opt}}}}{{{V_{oc}}}}} \right)\]

(5.9)

Self-Check Question 1

A solar cell generates maximum power of 2.3 W at regular light conditions at 25 °C. The open circuit voltage is measured at 0.55 V. Can you apply Equation (5.9) to estimate the maximum power of the solar cell if the light is concentrated 10 times (C_opt = 10)?

As you can see, the cell power can raise dramatically because of light concentration, mainly because the cell current is significantly increased.

From the maximum power equation, we can further derive the effect of concentration on cell efficiency:

η = \frac{P_{\max}}{G}

(5.10)

In this equation efficiency, (η) is expressed as the ratio of maximum cell power output to the irradiance on the cell surface. So, for concentrated light, the irradiance will be amplified to G* (which is proportional to C_opt). The maximum power output at the concentrated light, P_max, can be expressed as V_oc*I_sc*FF according to equation (4.9) in Lesson 4. Therefore, the expression for efficiency at the concentrated light can be modified as follows:

η^{*} = \frac{V_{o c}^{*} I_{s c}^{*} F F}{G^{*}} = η (1 + \frac{k T}{q} \frac{\ln C_{o p t}}{V_{o c}})

(5.11)

The algebraic transformation above is done by substituting Equations (5.3) and (5.7) into the equation (you can check). As a result, we see how "concentrated" efficiency (η*) is related to "non-concentrated" efficiency (η) through the optical concentration ratio. Try to apply this equation to find out what happens with the efficiency if you concentrate light ten times:

Self-Check Question 2

A solar cell has efficiency of conversion 15% at 25 ^oC (298 K). Open circuit voltage of the cell is 0.55 V. What efficiency ideally can we expect from it, when light is concentrated ten times (C_opt = 10)? Use equation (5.11) and type your number (in percent) below:

As you can see, the efficiency of the solar cell increases slightly in concentrated light, but this increase is not as apparent as for absolute output parameters (e.g. power). This is because in efficiency we always consider a ratio of the output to input energy. Both output and input energies increase due to concentration, so based on Eq. (5.10) the efficiency does not change much. Moreover, the efficiency of real solar cells cannot increase indefinitely because of power losses to heat. The amount of those losses is determined by the cell series resistance (R_s). The higher the series resistance, the bigger the power losses:

P_{l o s s} = I^{2} \times R_{S}

(5.12)

Because the current flowing through the cell is proportional to the light concentration ratio, the power wasted can be presented as:

P_{l o s s} ≅ C_{o p t}^{2} \times I s c^{2} \times R_{S}

(5.13)

The power loss will grow very rapidly as the concentration ratio increases because of the exponent factor. So, there is no sense to increase concentration infinitely because those efforts may not pay off in terms of useful power increase. According to some studies (Luque, 1989), there is an optimum concentration ratio for each type of cells. It is pretty much dictated by the cell series resistance and can be expressed as follows:

C_{o p t} ≅ (\frac{k T}{q}) \frac{1}{I_{s c} R_{s}}

(5.14)

Example

We are going to use Equation 5.14 to estimate the optimal concentration ratio for a solar cell of internal series resistance of 0.01 Ohm and producing short circuit current of 150 mA (at regular light).

The factor (kT/q) at 25 °C will be equal to 0.026 V, so for the optimal concentration, we can write:

$C_{o p t} = 0.026 / (0.01 O h m × 0.150 A) = 17.33$
That means that concentrating light at much greater than x17 ratios becomes unfeasible because of excessive losses.

Many solar cells designed for concentrated light in fact have special features to reduce the series resistance, but the limits of design may still be dependent on the cell material. For silicon, for example, it is hard to create cells that would be efficient at concentration ratios higher than 200 (Markvart, 2000).

5.2. Light concentration effect on PV performance and efficiency